Atomic and Molecular Physics

Comprehensive Study of Atomic and Molecular

Physics

Elisha Franks

Comprehensive Study of Atomic and Molecular Physics

This page is Intentionally Left Blank

Edited by Elisha Franks

Comprehensive Study of Atomic and Molecular Physics Edited by Elisha Franks

The publisherrsquos policy is to use permanent paper from mills that operate a sustainable forestry policy Furthermore the publisher ensures that the text paper and cover boards used have met acceptable environmental accreditation standards

Trademark Notice Registered trademark of products or corporate names are used only for explanation and identification without intent to infringe

This book contains information obtained from authentic and highly regarded sources Copyright for all individual chapters remain with the respective authors as indicated A wide variety of references are listed Permission and sources are indicat-ed for detailed attributions please refer to the permissions page Reasonable efforts have been made to publish reliable data and information but the authors editors and publisher cannot assume any responsibility for the validity of all materials or the consequences of their use

Published by University Publications 5 Penn Plaza19th FloorNew York NY 10001 USA

copy 2017 University Publications

International Standard Book Number 978-1-9789-2193-1

Copyright of this ebook is with University Publications rights acquired from the original print publisher NY Research Press

Contents

Preface VII

Chapter 1 Analysis of Water and Hydrogen Bond Dynamics at the Surface of an Antifreeze Protein 1 Yao Xu Ramachandran Gnanasekaran and David M Leitner

Chapter 2 Temperature and HD Isotopic Effects in the IR Spectra of the Hydrogen Bond in Solid-State 2-Furanacetic Acid and 2-Furanacrylic Acid 7 Henryk T Flakus and Anna Jarczyk-Jedryka

Chapter 3 An Analytic Analysis of the Diffusive-Heat-Flow Equation for Different Magnetic Field Profiles for a Single Magnetic Nanoparticle 24 Brenda Dana and Israel Gannot

Chapter 4 Helmholtz Bright Spatial Solitons and Surface Waves at Power-Law Optical Interfaces 46 J M Christian E A McCoy G S McDonald J Saacutenchez-Curto and P Chamorro-Posada

Chapter 5 The Effect of Nonnative Interactions on the Energy Landscapes of Frustrated Model Proteins 67 Mark T Oakley David J Wales and Roy L Johnston

Chapter 6 Proton Transfer Equilibria and Critical Behavior of H-Bonding 76 L Sobczyk B Czarnik-Matusewicz M Rospenk and M Obrzud

Chapter 7 Polymorphism Hydrogen Bond Properties and Vibrational Structure of 1H-Pyrrolo[32-h]Quinoline Dimers 86 Alexandr Gorski Sylwester Gawinkowski Roman Luboradzki Marek Tkacz Randolph P Thummel and Jacek Waluk

Chapter 8 Effective Potential for Ultracold Atoms at the Zero Crossing of a Feshbach Resonance 97 N T Zinner

Chapter 9 Transition Parameters for Doubly Ionized Lanthanum 106 Betuumll Karaccediloban and Leyla Oumlzdemir

Chapter 10 Relativistic Time-Dependent Density Functional Theory and Excited States Calculations for the Zinc Dimer 121 Ossama Kullie

__________________________ WORLD TECHNOLOGIES __________________________

Chapter 11 Millimeter-Wave Rotational Spectra of trans-Acrolein (Propenal) (CH2CHCOH) A DC Discharge Product of Allyl Alcohol (CH2CHCH2OH) Vapor and DFT Calculation 137 A I Jaman and Rangana Bhattacharya

Chapter 12 The Effect of Nanoparticle Size on Cellular Binding Probability 146 Vital Peretz Menachem Motiei Chaim N Sukenik and Rachela Popovtzer

Chapter 13 Electron-Pair Densities with Time-Dependent Quantum Monte Carlo 153 Ivan P Christov

Chapter 14 Multispark Discharge in Water as a Method of Environmental Sustainability Problems Solution 158 E M Barkhudarov I A Kossyi Yu N Kozlov S M Temchin M I Taktakishvili and Nick Christofi

Chapter 15 The Advantages of Not Entangling Macroscopic Diamonds at Room Temperature 170 Mark E Brezinski

Chapter 16 Energies Fine Structures and Hyperfine Structures of the 1s22snp 3P (n = 2ndash4) States for the Beryllium Atom 179 Chao Chen

Chapter 17 Statistical Complexity of Low- and High-Dimensional Systems 185 Vladimir Ryabov and Dmitry Nerukh

Chapter 18 A First-Principles-Based Potential for the Description of Alkaline Earth Metals 191 Johannes M Dieterich Sebastian Gerke and Ricardo AMata

Chapter 19 Uniformly Immobilizing Gold Nanorods on a Glass Substrate 199 Hadas Weinrib Amihai Meiri Hamootal Duadi and Dror Fixler

Permissions

List of Contributors

VI Contents

Preface

The field of physics is a vast and detailed one that has many sub-divisions that branch off in diverse directions There has always been a part of the physical world that escapes our eyes and cannot really be explained through simple terms The components of matter are minuscule and need especially devoted arenas of research and study to be understood Atomic and molecular study is an essential part of this world of physics Atomic physics is an area of study that focuses on the study of atoms in the manner of an isolated system of nucleus and electrons Primarily focused on the arrangement of electrons around the nucleus it also studies how these arrangements change Molecular Physics on the other hand is a field of study that is focused on the physical properties of molecules It looks at molecules as well as molecular dynamics and bonds Both fields could be said to be closely related but they also overlap with physical chemistry chemical physics and theoretical chemistry Both atomic and molecular physics are essentially concerned with the electronic structure of atoms and molecules and the dynamic processes through which these structures arrange themselves The rapidly advancing research techniques bode well for the future of atomic and molecular physics

This book is an attempt to compile and collate all current and proposed research and data in the field of atomic and molecular physics I am thankful to all those whorsquos hard work and effort went into these studies I wish to personally thank all the contributing authors who shared their knowledge in this book and with me throughout the editing process It was an honour working with you all I also wish to thank my family who have always been my support system

Editor

Analysis of Water and Hydrogen Bond Dynamics at the Surface ofan Antifreeze Protein

Yao Xu Ramachandran Gnanasekaran and David M Leitner

Department of Chemistry and Chemical Physics Program University of Nevada Reno NV 89557 USA

Correspondence should be addressed to David M Leitner dmlunredu

Academic Editor Keli Han

We examine dynamics of water molecules and hydrogen bonds at the water-protein interface of the wild-type antifreeze proteinfrom spruce budworm Choristoneura fumiferana and a mutant that is not antifreeze active by all-atom molecular dynamicssimulations Water dynamics in the hydration layer around the protein is analyzed by calculation of velocity autocorrelationfunctions and their power spectra and hydrogen bond time correlation functions are calculated for hydrogen bonds between watermolecules and the protein Both water and hydrogen bond dynamics from subpicosecond to hundred picosecond time scales aresensitive to location on the protein surface and appear correlated with protein function In particular hydrogen bond lifetimes arelongest for water molecules hydrogen bonded to the ice-binding plane of the wild type whereas hydrogen bond lifetimes betweenwater and protein atoms on all three planes are similar for the mutant

1 Introduction

While the complex dynamics of large biological moleculesand the connection to function have fascinated physical sci-entists for some time in more recent years researchers haveturned their attention to the interface of biomolecules withwater Coupling of protein and water dynamics for examplehas been examined by molecular simulations [1ndash10] and agrowing number of experimental probes [11ndash14] and a widevariety of dynamical time scales have been found [15 16]due to the heterogeneity of protein-water interactions Oneclass of proteins for which protein-water interactions arecritical to function is antifreeze proteins (AFPs) AFPs arewidely distributed in certain plants vertebrates fungi andbacteria to provide cells protection in cold environments[17ndash20] but the mechanism for antifreeze activity is stillnot well understood In this paper we analyze by all-atommolecular dynamics (MD) simulations the dynamics ofwater molecules and hydrogen bonds at the protein-waterinterface of the AFP from the spruce budworm Choristoneurafumiferana and a mutant that has little antifreeze activityWe calculate velocity autocorrelation functions and theirpower spectra for water molecules around the protein and

we compute hydrogen bond time correlation functions forbonds between the protein and water We obtain distinctspectra for the water around different regions of the proteinwhich are affected by mutation Moreover we observe longerhydrogen bonding between water molecules and the ice-binding plane of this AFP compared to other parts of theprotein a difference that nearly disappears with mutationindicating a correlation between hydrogen bond lifetimesand activity of this AFP

AFPs were first discovered in several Antarctic fish species[21] AFPs that have since been classified as Type I Thegenerally accepted mechanism for the Type I AFP is theadsorption-inhibition mechanism [22ndash24] which proposesthat AFPs adsorb onto the preferred growth sites of anice surface thereby preventing new ice growth [25] Itwas initially thought that ice and AFP interacted throughhydrogen bonding [22] However when parts of the proteinthat were thought to facilitate this hydrogen bonding weremutated the hypothesized decrease in antifreeze activity wasnot observed and hydrophobic interactions were suggestedinstead [26] MD simulations have been carried out tosort out the possible mechanisms [6] but there is still noconsensus on which sites of the protein interact with ice

THR-51 THR-38THR-21

Figure 1 The structure of wild-type AFP from spruce budwormChoristoneura fumiferana indicating the location of the fourthreonine residues on the ice-binding plane (Plane 1) which in ourmutation studies we replace with leucine Plane 2 is in the front andPlane 3 is in the back

or whether the protein inhibits growth of ice locally at theprotein-water interface or over a larger number of waterlayers near the protein Recent THz studies [27] indicate thatat least for AFP in winter flounder the effect appears to bedelocalized

The antifreeze activity of the AFP from the sprucebudworm Choristoneura fumiferana [28] shown in Figure 1can apparently be attributed in part to specific residueslocated on part of the surface of the protein This proteinis not a member of the Type I family The protein structurecontains three planes and mutation studies demonstrate thatthreonine-rich Plane 1 is the ice-binding plane Mutationof just a few of the threonines to leucines (Figure 1) dra-matically diminishes antifreeze activity [28] Nutt and Smith[29] recently carried out MD simulations to examine thewater dynamics in the hydration layer around the proteinand found distinct dynamics around each of the threeplanes and noticeably slower dynamics around Plane 1 Inthis study we observe like Nutt and Smith quite distinctdifferences for the water dynamics around each of the threeplanes of the protein and in the hydrogen bond lifetimesfor hydrogen bonds between the water molecules and theprotein Moreover we examine a mutant that is antifreezeinactive and find that the mutation affects the hydrogen bonddynamics that is hydrogen bond lifetimes around the threeplanes are much closer to each other than in the wild type

In the following section we provide details of the com-putational methods and analysis We then report results ofour calculation of power spectra for water molecules near thethree distinct planes of the protein and of our investigation ofhydrogen bond lifetimes for bonds between water moleculesand the protein Concluding remarks are given in the finalsection

2 Computational Methods

The initial coordinates of the antifreeze protein from thespruce budworm Choristoneura fumiferana were taken from

the Protein Data Bank file 1L0S Missing residues andhydrogen atoms were built into the structure and the iodatedtyrosine Y26 required for the structure determination wasreverted to a standard tyrosine using Swiss PDB Viewer [30]For the mutant four threonine residues on Plane 1 weremutated to leucines (Figure 1) a mutation that significantlyreduces the antifreeze properties of the protein [17] toexplore the effect of this mutation on the water dynamics andhydrogen bond lifetimes

Both the wild-type and mutated structure were firstminimized for 1000 steps with the steepest descent algorithmusing the AMBER03 force field [31] after its solvation ina 70 A cubic water box of TIP5P water model Then thesystems each of which contained 10539 water moleculeswere equilibrated for 400 ps For the first 100 ps the positionsof the proteins were restrained and in the latter 300 psthey were released Constraints were applied to all bonds tohydrogen with the SHAKE algorithm and periodic boundaryconditions were applied All the classical MD simulationswere performed on the systems in canonical (NVT) ensemblewith the GROMACS software package [32] Followingequilibration trajectories of 2 ns were obtained at 300 K witha Nose-Hoover thermostat [22 23] Nonbonded interactionswere gradually brought to zero by a shift function for theelectrostatics as well as a switch function for van der Waalsinteractions between 10 and 12 A [24 25] All the simulationswere performed by integrating Newtonrsquos equations of motionwith the Verlet algorithm [26] using 1 fs time steps Thesystem coordinates and velocities were stored every 5 fs andthe velocity autocorrelation function (VACF) was averagedover 15 ps time segments of the trajectory for the oxygenatoms that survive in the first hydration shell of thickness5 A as well as for those that hydrogen bond to the proteins(Criteria for hydrogen bonds are specified below) The VACFis defined as

CV (t) = 〈vi(t) middot vi(0)〉〈vi(0) middot vi(0)〉 (1)

where vi(t) is the velocity vector of the oxygen atom attime t The angular brackets denote averaging over all atomsof the particular type present in the hydration shell andover different reference initial times Power spectra wereobtained by Fourier transform of CV (t) The power spectracorrespond to the vibrational density of the water Thevibrational density of protein molecules has been discussedelsewhere [11 33ndash36]

Hydrogen bond time correlation functions CHB(t) werealso computed for bonds between water molecules and theprotein at 300 K CHB(t) is defined as the probability that if ahydrogen bond between donor D and acceptor A exists att = 0 then it still exists at time t even if the bond broke atsome intermediate time [37] We adopt a standard criterionfor hydrogen bonds that is a DA distance of 35 A and a D-H-A angle greater than 150 [1 38 39]

3 Results and Discussion

31 VACF Power Spectra A protein molecule perturbs theregular water-water hydrogen bond network in bulk water

2 Comprehensive Study of Atomic and Molecular Physics

0 2 4 6 8 10 12 14 16

Frequency (THz)

0 2 4 6 8 10 12 14 16

Frequency (THz)

0 2 4 6 8 10 12 14 16

Frequency (THz)

Figure 2 Power spectra of the velocity autocorrelation function of water in hydration layers around Plane 1 (red) Plane 2 (green) and Plane3 (blue) of the protein as well as for bulk water (black) at 300 K (a) Power spectrum of water in the hydration layer taken to be 5 A fromthe surface of the wild-type AFP (b) Power spectrum of water hydrogen bonding to wild type AFP (c) Same as (b) but for mutant AFP

with the formation of protein-water hydrogen bonds andinfluences the water dynamics in the hydration layer aroundthe protein surface We have calculated at 300 K the velocityautocorrelation function and its power spectra for the watermolecules in the hydration layer around the protein whichcan provide insights into THz spectra of solvated proteins[40] We have carried out this calculation both for the watermolecules that form hydrogen bonds with the amino acidresidues of the three planes of the protein and for the largernumber of water molecules within a layer of thickness 5 Afrom the protein [41] Power spectra are plotted in Figure 2for the wild-type and mutant at 300 K The results of aseparate MD simulation of pure TIP5P water under the sameconditions are also included for comparison

We consider first the power spectra for bulk water whichappears in each of the panels in Figure 2 We observetwo bands in the power spectra of water at about 2 and8 THz The lower frequency band has been interpreted [4243] as corresponding to the Omiddot middot middotOmiddot middot middotO bending modefrom triplets of hydrogen-bonded water molecules and the

higher frequency band as Omiddot middot middotO stretching mode betweenpairs of hydrogen-bonded water molecules Turning to thehydration water the results plotted in Figure 2(a) reveal aclear blue shift in SO (ω) for the band corresponding to theOmiddot middot middotOmiddot middot middotO bending for water The shift is very similar forthe water molecules in the 5 A hydration layer around eachof the three planes A blue shift in the same spectral regionhas been observed for water molecules in the hydration layeraround helices of the villin headpiece subdomain HP-36[44] Figure 2(b) gives the result for the hydration layeraround the wild-type AFP and we observe similar results forthe mutant (not shown) Overall we find that for the watermolecules in the 5 A hydration layer around the protein thereis little difference among the spectra obtained for the waternear Planes 1 2 or 3

For the water molecules hydrogen bonded to the proteinwe observe distinct differences in the power spectra of thevelocity autocorrelation function for each of the planes Thepower spectra for the water hydrogen bonded to the proteinexhibit again peaks near 2 and 8 THz but the intensity of

3Analysis of Water and Hydrogen Bond Dynamics at the Surface of an Antifreeze Protein

Time (ps)

50 100 150 200

Figure 3 The hydrogen-bond time correlation function plotted forwater hydrogen bonded to Plane 1 (red) Plane 2 (blue) and Plane 3(green) of wild-type (solid) and mutant (dotted) AFP at 300 K Theresult for hydrogen bonds between water molecules in the bulk isplotted (black) for comparison

the 2 THz peak is smaller than for the hydration waterand the peak corresponding to the Omiddot middot middotOmiddot middot middotO bendingappears even further broadened and blue shifted comparedto bulk water than the peak for the hydration water in the5 A layer around the protein This could be related to themore restricted dynamics of the water molecules hydrogenbonded to the protein Indeed we have computed the powerspectrum for bulk water at 250 K and for water molecules inthe hydration shell and found the first peak for bulk waterat this lower temperature to have a smaller intensity andsimilar to that for the hydration water [45] For the wild typewe observe that Plane 1 exhibits a greater intensity on theblue edge of the lower frequency band compared to the otherplanes whereas for the mutant the intensity is also greaterbut on the red side of the band Because the power spectrafor the wild type and the mutant are distinct we expectthat differences in the THz spectra of the wild type and themutant can be detected

32 Hydrogen Bond Correlation Function We plot in Figure 3results for the hydrogen bond correlation function CHB(t)defined in Section 2 to times of 200 ps for hydrogen bondsbetween water molecules and protein atoms on Planes1 2 and 3 of the wild type and mutant as well asbetween water molecules in the bulk for comparison All thesimulations were carried out at 300 K Overall the observedslow rearrangement times for hydrogen bonds between watermolecules and the protein compared to hydrogen bondsbetween water molecules in the bulk are consistent withexpectations for water molecules in the hydration layeraround a protein [46 47] Nutt and Smith [29] computedthe hydrogen bond correlation function for bonds between

water and the three planes of the wild type and we focushere mainly on comparison of the wild-type results withthe results for the mutant The hydrogen bond lifetime forbonds between water and the protein survive longer thanhydrogen bonds between water molecules in the bulk as seenin numerous previous simulation studies [1 9 38 48 49]However we also observe differences for hydrogen bondsbetween water and atoms on different planes of the proteinConsidering first the wild type we find as did Nutt andSmith [29] that CHB(t) for hydrogen bonds between watermolecules and atoms of Plane 1 decays significantly slowerthan CHB(t) for bonds between water molecules and theother two planes

Interestingly we find the hydrogen bond correlationfunctions for hydrogen bonds between water and the threeprotein planes to be noticeably closer to each other for themutant than for the wild type out to the 100 picosecondtime scale The antifreeze activity of the protein decreasesdramatically when replacing four of the threonines on Plane1 indicated in Figure 1 with leucines [28] and we observein our MD simulations that the hydrogen bond lifetimes forbonds between water molecules and atoms of each planebecome similar to one another with this mutation Only fourpoint mutations have a sizable effect on the hydrogen bonddynamics indicating the effect may not simply be local butmay influence the orientation of many water molecules Sucha nonlocal effect on the orientation of hydration waters bypoint mutation has been illustrated recently for a simpleprotein-sized model system [50]

4 Concluding Remarks

In this work we examined the power spectrum of the velocityautocorrelation function for water molecules near the surfaceof the antifreeze protein (AFP) from the spruce budwormChoristoneura fumiferana and analyzed the hydrogen bondlifetimes for bonds between water molecules and the proteinWe explored effects of the heterogeneity of the proteinsurface in particular the distinctive properties of the waterand protein-water interactions on the three planes of theprotein one of which is vital to the function of this AFP andhow the dynamics is affected by mutation

For the power spectra of the water in the hydration layerof the AFP and the subset of that water that hydrogen bondsto the protein we find a blue shift of the roughly 2 THzband compared to the same band in bulk water with a morepronounced shift for the water molecules that are hydrogenbonded to the protein residues Although the power spectrafor the water molecules within 5 A of each of the planesof the protein appear quite similar power spectra for thewater molecules hydrogen bonded to different planes of theprotein exhibit distinct spectra in the range 1ndash4 THz Thedifferences among the power spectra for the water moleculeshydrogen bonding to each of the three planes are influencedby mutation We expect that THz measurements which arehighly sensitive to the hydration water [48 51ndash55] will revealdifferences between the wild type and mutant Recent THzexperiments [49] on a λ-repressor fragment indicate that

only a few point mutations can give rise to very different THzspectra

The hydrogen bond time correlation function was com-puted for hydrogen bonds between water molecules andeach of the planes of the protein For wild type AFP weobserve differences in the hydrogen bond lifetimes for bondsbetween water and the three planes The longest lifetimesare found for hydrogen bonds between water molecules andPlane 1 the ice-binding plane of the protein consistentwith results of previous simulations [29] We observe thatby introducing only four mutations to Plane 1 mutationsthat have been observed to substantially diminish the AFPactivity of the protein [28] the hydrogen bond correlationfunction for bonds between water molecules and each of thethree planes are similar to one another Overall mutationis seen to modify hydrogen bonding over a wide range oftime scales observable both in the power spectra and analysisof hydrogen bond lifetimes These measures of hydrogenbonding at the protein-water interface aid in quantifying thecomplexity and heterogeneity of the interactions betweenwater and the antifreeze protein and reveal regions of theprotein-water interface important for antifreeze activity

Acknowledgments

Support from the National Science Foundation (NSF CHE-0910669) and from the Volkswagen Foundation (VWStiftung Az I84 302) is gratefully acknowledged

References

[1] D J Tobias N Sengupta and M Tarek ldquoMolecular dynamicssimulation studies of coupled protein and water dynamicsrdquo inProteins Energy Heat and Signal Flow D M Leitner and J EStraub Eds pp 361ndash386 Taylor amp Francis Boca Raton FlaUSA 2009

[2] M E Johnson C Malardier-Jugroot R K Murarka andT Head-Gordon ldquoHydration water dynamics near biologicalinterfacesrdquo Journal of Physical Chemistry B vol 113 no 13pp 4082ndash4092 2009

[3] A R Bizzarri and S Cannistraro ldquoMolecular dynamics ofwater at the protein-solvent interfacerdquo Journal of PhysicalChemistry B vol 106 no 26 pp 6617ndash6633 2002

[4] P J Steinbach and B R Brooks ldquoProtein hydration eluci-dated by molecular dynamics simulationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 90 no 19 pp 9135ndash9139 1993

[5] D N LeBard and D V Matyushov ldquoFerroelectric hydrationshells around proteins electrostatics of the protein-waterinterfacerdquo Journal of Physical Chemistry B vol 114 no 28 pp9246ndash9258 2010

[6] X Yu J Park and D M Leitner ldquoThermodynamics of proteinhydration computed by molecular dynamics and normalmodesrdquo Journal of Physical Chemistry B vol 107 no 46 pp12820ndash12828 2003

[7] F Despa A Fernandez and R S Berry ldquoPublisherrsquos notemdashdielectric modulation of biological waterrdquo Physical ReviewLetters vol 93 no 26 Article ID 228104 1 pages 2004

[8] R Gnanasekaran J K Agbo and D M Leitner ldquoCommu-nication maps computed for homodimeric hemoglobincomputational study of water-mediated energy transport in

proteinsrdquo Journal of Chemical Physics vol 135 no 6 ArticleID 065103 10 pages 2011

[9] R Gnanasekaran Y Xu and D M Leitner ldquoDynamics ofwater clusters confined in proteins a molecular dynamics sim-ulation study of interfacial waters in a dimeric hemoglobinrdquoJournal of Physical Chemistry B vol 114 no 50 pp 16989ndash16996 2010

[10] A Lervik F Bresme S Kjelstrup D Bedeaux and J M RubildquoHeat transfer in protein-water interfacesrdquo Physical ChemistryChemical Physics vol 12 no 7 pp 1610ndash1617 2010

[11] D M Leitner M Havenith and M Gruebele ldquoBiomoleculelarge-amplitude motion and solvation dynamics modellingand probes from THz to X-raysrdquo International Reviews inPhysical Chemistry vol 25 no 4 pp 553ndash582 2006

[12] L Mitra N Smolin R Ravindra C Royer and R WinterldquoPressure perturbation calorimetric studies of the solvationproperties and the thermal unfolding of proteins in solutionmdashexperiments and theoretical interpretationrdquo Physical Chem-istry Chemical Physics vol 8 no 11 pp 1249ndash1265 2006

[13] S K Pal J Peon and A H Zewail ldquoBiological water atthe protein surface dynamical solvation probed directly withfemtosecond resolutionrdquo Proceedings of the National Academyof Sciences of the United States of America vol 99 no 4 pp1763ndash1768 2002

[14] W Doster and M Settles ldquoThe dynamical transition inproteins the role of hydrogen bondsrdquo in Hydration Processesin Biology Experimental and Theoretical Approaches M-CBellissent-Funel Ed pp 177ndash195 IOS Press Amsterdam TheNetherlands 1999

[15] E Persson and B Halle ldquoCell water dynamics on multiple timescalesrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 105 no 17 pp 6266ndash6271 2008

[16] H Frauenfelder P W Fenimore G Chen and B HMcMahon ldquoProtein folding is slaved to solvent motionsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 103 no 42 pp 15469ndash15472 2006

[17] S P Graether Biochemistry and Function of Antifreeze ProteinsNova Science New York NY USA 2011

[18] J G Duman K R Walters T Sformo et al ldquoAntifreeze andice-nucleator proteinsrdquo in Low Temperature Biology of InsectsD L Delinger and R E Lee Eds pp 59ndash90 CambridgeUniversity Press New York NY USA 2010

[19] B Moffatt V Ewart and A Eastman ldquoCold comfort plantantifreeze proteinsrdquo Physiologia Plantarum vol 126 no 1 pp5ndash16 2006

[20] L Pham R Dahiya and B Rubinsky ldquoAn in vivo study ofantifreeze protein adjuvant cryosurgeryrdquo Cryobiology vol 38no 2 pp 169ndash175 1999

[21] A L DeVries and D E Wohlschlag ldquoFreezing resistance insome antarctic fishesrdquo Science vol 163 no 3871 pp 1073ndash1075 1969

[22] J A Raymond and A L DeVries ldquoAdsorption inhibition as amechanism of freezing resistance in polar fishesrdquo Proceedingsof the National Academy of Sciences of the United States ofAmerica vol 74 no 6 pp 2589ndash2593 1977

[23] J A Raymond P W Wilson and A L DeVries ldquoInhibitionof growth of nonbasal planes in ice by fish antifreezesrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 86 no 3 pp 881ndash885 1989

[24] C A Knight C C Cheng and A L DeVries ldquoAdsorptionof α-helical antifreeze peptides on specific ice crystal surfaceplanesrdquo Biophysical Journal vol 59 no 2 pp 409ndash418 1991

[25] J Duman and A L DeVries ldquoIsolation characterization andphysical properties of protein antifreezes from the winterflounder pseudopleuronectes americanusrdquo Comparative Bio-chemistry and Physiology vol 54 no 3 pp 375ndash380 1976

[26] A D Haymet L G Ward M M Harding and C A KnightldquoValine substituted winter flounder ldquoantifreezerdquo preservationof ice growth hysteresisrdquo FEBS Letters vol 430 no 3 pp 301ndash306 1998

[27] S Ebbinghaus K Meister B Born A L Devries MGruebele and M Havenith ldquoAntifreeze glycoprotein activitycorrelates with long-range protein-water dynamicsrdquo Journalof the American Chemical Society vol 132 no 35 pp 12210ndash12211 2010

[28] S P Graether M J Kuiper and S M Gagne ldquoBeta-helixstructure and ice-binding properties of a hyperactive an-tifreeze protein from an insectrdquo Nature pp 325ndash328 2000

[29] D R Nutt and J C Smith ldquoDual function of the hydrationlayer around an antifreeze protein revealed by atomisticmolecular dynamics simulationsrdquo Journal of the AmericanChemical Society vol 130 no 39 pp 13066ndash13073 2008

[30] N Guex and M C Peitsch ldquoSWISS-MODEL and the Swiss-PdbViewer an environment for comparative protein model-ingrdquo Electrophoresis vol 18 no 15 pp 2714ndash2723 1997

[31] Y Duan C Wu S Chowdhury et al ldquoA point-charge forcefield for molecular mechanics simulations of proteins based oncondensed-phase quantum mechanical calculationsrdquo Journalof Computational Chemistry vol 24 no 16 pp 1999ndash20122003

[32] H J C Berendsen D Spoel and R V Drunen ldquoGROMACSa message-passing parallel molecular dynamics implementa-tionrdquo Computer Physics Communications vol 91 no 1ndash3 pp43ndash56 1995

[33] X Yu and D M Leitner ldquoVibrational energy transfer and heatconduction in a proteinrdquo Journal of Physical Chemistry B vol107 no 7 pp 1698ndash1707 2003

[34] X Yu and D M Leitner ldquoAnomalous diffusion of vibrationalenergy in proteinsrdquo Journal of Chemical Physics vol 119 no23 pp 12673ndash12679 2003

[35] X Yu and D M Leitner ldquoHeat flow in proteins computationof thermal transport coefficientsrdquo Journal of Chemical Physicsvol 122 no 5 Article ID 054902 11 pages 2005

[36] D M Leitner ldquoVibrational energy transfer and heat conduc-tion in a one-dimensional glassrdquo Physical Review B vol 64 no9 Article ID 094201 9 pages 2001

[37] B Bagchi ldquoWater dynamics in the hydration layer aroundproteins and micellesrdquo Chemical Reviews vol 105 no 9 pp3197ndash3219 2005

[38] M Tarek and D J Tobias ldquoRole of protein-water hydrogenbond dynamics in the protein dynamical transitionrdquo PhysicalReview Letters vol 88 no 13 Article ID 138101 4 pages 2002

[39] A Luzar and D Chandler ldquoHydrogen-bond kinetics in liquidwaterrdquo Nature vol 379 no 6560 pp 55ndash57 1996

[40] M Heyden and M Havenith ldquoCombining THz spectroscopyand MD simulations to study protein-hydration couplingrdquoMethods vol 52 no 1 pp 74ndash83 2010

[41] S Bandyopadhyay S Chakraborty and B Bagchi ldquoSecondarystructure sensitivity of hydrogen bond lifetime dynamics inthe protein hydration layerrdquo Journal of the American ChemicalSociety vol 127 no 47 pp 16660ndash16667 2005

[42] G E Walrafen and Y C Chu ldquoLinearity between structuralcorrelation length and correlated-proton Raman intensityfrom amorphous ice and supercooled water up to densesupercritical steamrdquo Journal of Physical Chemistry vol 99 no28 pp 11225ndash11229 1995

[43] G E Walrafen Y C Chu and G J Piermarini ldquoLow-fre-quency Raman scattering from water at high pressures andhigh temperaturesrdquo Journal of Physical Chemistry vol 100 no24 pp 10363ndash10372 1996

[44] S Chakraborty S K Sinha and S Bandyopadhyay ldquoLow-frequency vibrational spectrum of water in the hydration layerof a protein a molecular dynamics simulation studyrdquo Journalof Physical Chemistry B vol 111 no 48 pp 13626ndash136312007

[45] Y Xu R Gnanasekaran and D M Leitner (published results)[46] N Nandi and B Bagchi ldquoDielectric relaxation of biological

waterrdquo Journal of Physical Chemistry B vol 101 no 50 pp10954ndash10961 1997

[47] S K Pal J Peon B Bagchi and A H Zewail ldquoBiologicalwater femtosecond dynamics of macromolecular hydrationrdquoJournal of Physical Chemistry B vol 106 no 48 pp 12376ndash12395 2002

[48] S Ebbinghaus S J Kim M Heyden et al ldquoAn extendeddynamical hydration shell around proteinsrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 104 no 52 pp 20749ndash20752 2007

[49] S Ebbinghaus S J Kim M Heyden et al ldquoProtein se-quence- and pH-dependent hydration probed by terahertzspectroscopyrdquo Journal of the American Chemical Society vol130 no 8 pp 2374ndash2375 2008

[50] A D Friesen and D V Matyushov ldquoNon-Gaussian statisticsof electrostatic fluctuations of hydration shellsrdquo Journal ofChemical Physics vol 135 no 10 Article ID 104501 7 pages2011

[51] B Born S J Kim S Ebbinghaus M Gruebele and MHavenith ldquoThe terahertz dance of water with the proteins theeffect of protein flexibility on the dynamical hydration shell ofubiquitinrdquo Faraday Discussions vol 141 pp 161ndash173 2008

[52] U Heugen G Schwaab E Brundermann et al ldquoSolute-induced retardation of water dynamics probed directly byterahertz spectroscopyrdquo Proceedings of the National Academyof Sciences of the United States of America vol 103 no 33 pp12301ndash12306 2006

[53] M Heyden E Brundermann U Heugen G Niehues D MLeitner and M Havenith ldquoLong-range influence of carbo-hydrates on the solvation dynamics of watermdashanswers fromterahertz absorption measurements and molecular modelingsimulationsrdquo Journal of the American Chemical Society vol130 no 17 pp 5773ndash5779 2008

[54] J Knab J Y Chen and A G Markelz ldquoHydration dependenceof conformational dielectric relaxation of lysozymerdquo Biophys-ical Journal vol 90 no 7 pp 2576ndash2581 2006

[55] N Q Vinh S J Allen and K W Plaxco ldquoDielectric spec-troscopy of proteins as a quantitative experimental testof computational models of their low-frequency harmonicmotionsrdquo Journal of the American Chemical Society vol 133no 23 pp 8942ndash8947 2011

Temperature and HD Isotopic Effects in the IR Spectra ofthe Hydrogen Bond in Solid-State 2-Furanacetic Acid and2-Furanacrylic Acid

Henryk T Flakus and Anna Jarczyk-Jedryka

Institute of Chemistry University of Silesia 9 Szkolna Street 40-006 Katowice Poland

Correspondence should be addressed to Henryk T Flakus flakusichusedupl

Academic Editor Joanna Sadlej

Polarized IR spectra of 2-furanacetic acid and of 2-furanacrylic acid crystals were measured at 293 K and 77 K in the vOminusH andvOminusH band frequency ranges The corresponding spectra of the two individual systems strongly differ one from the other by thecorresponding band shapes as well as by the temperature effect characterizing the bands The crystal spectral properties remain ina close relation with the electronic structure of the two different molecular systems We show that a vibronic coupling mechanisminvolving the hydrogen bond protons and the electrons on the π-electronic systems in the molecules determines the way inwhich the vibrational exciton coupling between the hydrogen bonds in the carboxylic acid dimers occurs A strong couplingin 2-furanacrylic acid dimers prefers a ldquotail-to-head-rdquo type Davydov coupling widespread by the π-electrons A weak through-space coupling in 2-furanacetic acid dimers is responsible for a ldquoside-to-side-rdquo type coupling The relative contribution of eachexciton coupling mechanism in the dimer spectra generation is temperature and the molecular electronic structure dependentThis explains the observed difference in the temperature-induced evolution of the compared spectra

1 Introduction

Infrared spectroscopy still constitutes a basic tool in theresearch of the hydrogen bond dynamics The νXminusH bandsmeasured in the highest frequency range of the mid-infraredattributed to the proton stretching vibrations in XndashHmiddot middot middotYhydrogen bonds are the source of wealth data system in thismatter Complex fine structure patterns of these bands areconsidered as the result of anharmonical coupling mecha-nisms involving the proton stretching vibrations and othernormal vibrations occurring in associated molecular sys-tems mainly the low-frequency Xmiddot middot middotY hydrogen bridgestretching vibrational motions [1ndash5] The band contourshapes are extremely susceptible on the influences exertedby diverse physical factors such as changes of temperaturechanges in the matter state of condensation pressure andsolvents [1ndash5]

Among the contemporary theories of the IR spectra ofthe hydrogen bond formed in molecular systems quanti-tative theoretical models elaborated for the description of

the νXminusH band generation mechanisms are of the particularimportance There are two most advanced quantitativetheoretical models namely the ldquostrong-couplingrdquo theory [6ndash8] (the elder theory) and the ldquorelaxationrdquo (linear response)theory the novel model [9 10] Both models are of a purelyvibrational nature Over the last four decades by using ofthese theories IR spectra of diverse hydrogen bond systemshave been reproduced satisfactorily The model calculationsconcerned quantitative interpretation of spectra of singleisolated hydrogen bonds [7 11] spectra of cyclic dimerichydrogen bond systems [7 12ndash14] and the IR spectra ofhydrogen-bonded molecular crystals [15] Simultaneouslythe HD isotopic effects observed in the spectra of thedeuterium-bonded corresponding systems have been inter-preted [7ndash15]

Nevertheless despite the doubtless successes achieved inthis area when interpreting the hydrogen bond system spec-tra it seems that a number of basic theoretical problems stillremain unsolved It also seems that the main source in theunderstanding of many spectral phenomena characterizing

systems consisting with a number of mutually coupledhydrogen bonds in terms of the two different quantitativeapproaches is in the early history of these studies In practiceup to the beginning of the 90s of the 20th century thesestudies were restricted to the interpretation of spectra ofa number of very simple hydrogen bond systems mainlyto the spectra of cyclic acetic acid dimers formed in thegaseous phase [7 12ndash14] The extension of this research overother more diversified and complex hydrogen bond aggre-gates allowed us to recognize numerous puzzling spectraleffects attributed to these systems Interpretation of theseeffects seemed to be beyond the contemporary quantitativetheoretical models of the hydrogen bond IR spectra withoutassuming that some not revealed yet mechanisms codecide inthe spectra generation

For the last decade spectroscopy in polarized lightof hydrogen-bonded molecular crystals has provided keyexperimental data in this area By measuring of polarized IRspectra of spatially oriented molecular crystals characterizedby a rich diversity of hydrogen bond arrangements met intheir lattices the most complete information has been beobtained about the coupling mechanisms involving hydro-gen bonds in these systems It appeared that the investigationof spectra of even so simple mutually interacting hydrogenbond aggregates like cyclic dimers (eg carboxylic aciddimers) allowed to reveal new HD isotopic effects namelythe HD isotopic self-organization effects They depend ona nonrandom distribution of protons and deuterons in thecrystal lattices of isotopically diluted hydrogen bond systemsThese spectral effects may be considered as the manifes-tation of a new kind of cooperative interactions involvinghydrogen bonds that is the so-called dynamical cooperativeinteractions [16ndash18] This revealing has emphasized the roleof the vibronic coupling between the electronic and theproton vibrational motions taking place in hydrogen bondaggregates in the generation of the very nature of thehydrogen bond as the natural phenomenon and in theinterhydrogen bond interaction mechanisms [17 18]

In the lattices of carboxylic acid crystals centrosymmet-ric hydrogen bond dimers present in the (COOH)2 cyclesare frequently met [19 20] These dimers are the bearersof the main crystal spectral properties in the frequencyranges of the νOminusH bands attributed to the proton stretchingvibrations One might expect that regardless of the molecularstructure of carboxylic acids in their fragments placedoutside the carboxyl groups the νOminusH band contour shapesshould be fairly similar one to the other This presumption isbased on the considerations of the classic vibrational analysiswhich predicted that the proton stretching vibrations in thesemolecules practically do not mix with vibrations of otheratomic groups [21] The experiment learns however thatspectra of diverse carboxylic acid crystals considerably differone from the other with regard to their νOminusH band contourshapes as well as with regard to the temperature effectsmeasured in the spectra Qualitatively similar conclusion isvalid for the νOminusD bands in the spectra of the deuterium-bonded species [22ndash27] Our hitherto estimations resultingfrom the comparison of the IR crystalline spectra of diversecarboxylic acid molecular systems ascribe the differences

between the compared spectra in relation to the differencesin the electronic structure of carboxylic acid moleculesFor instance π-electronic systems of aromatic rings orother larger conjugated π-electronic systems linked directlyto carboxyl groups strongly change the basic spectralproperties of carboxylic acid dimers in comparison withthe analogous properties of aliphatic carboxylic acids [22ndash27] The generation mechanism of these effects still remainsunknown

This paper deals with IR spectra of the hydrogen bondin crystals of two different carboxylic acids namely of 2-furanacetic acid and 2-furanacrylic acid In these crystallinesystems associated molecules form hydrogen-bonded cycliccentrosymmetric dimers (Complete crystallographic data for2-furanacetic acid and (excluding structure factors) havebeen deposited at the Cambridge Crystallographic DataCentre under the number CCDC-885823 Copies can beobtained free of charge from CCDC 12 Union RoadCambridge CB2 1EZ UK (Fax Int+1223-336-033 e-maildepositccdccamacuk)) The crystallographic data for 2-furanacrylic acid can be found in [28 29] Molecules of thesetwo individual molecular systems differ one from the otherby their electronic structures In the latter case the carboxylgroups are directly linked to the large π-electronic systems Inthe 2-furanacetic acid crystal case methylene groups separatethe hydrogen bonds formed by the associated carboxylgroups from the π-electronic system of furan rings

The aim of the study reported in this paper was toprovide new arguments of experimental nature about therole of the electronic structures of carboxylic acid moleculesin the generation of IR spectra of cyclic hydrogen bonddimers The investigation results presented constitute a partof results obtained in the frames of a wider project whichalso assumed measuring of crystalline spectra of othercarboxylic acids mainly of furan and thiophene derivativesOur choice of these model molecular systems was stronglysupported by advantageous well-developed νOminusH and νOminusD

band contour shapes in the IR spectra of these systems Weexpected that the quantitative analysis of the polarized IRspectra of 2-furanacetic acid and 2-furanacrylic acid crystalsand also of the spectra of relative carboxylic acid crystalsshould provide new arguments for the formulation of a newtheoretical approach for the description of the hydrogenbond dimer spectra The understanding of the temperatureeffects and the generation mechanism of the intensitydistribution patterns in the νOminusH and νOminusD bands in thespectra of diverse carboxylic acid crystals are of the particularinterest and importance in this project

2 X-Ray Structures of 2-Furanacetic Acidand 2-Furanacrylic Acid

Crystals of 2-furanacetic acid are monoclinic and the space-symmetry group is P21c Z = 4 The lattice constants at100 K a = 130525(4) A b = 485360(10) A c = 94107(3) Aβ = 103832(3) In a unit cell four translationally nonequiv-alent molecules form two plain centrosymmetric cyclichydrogen-bonded dimers (Complete crystallographic datafor 2-furanacetic acid (excluding structure factors) have been

Figure 1 The X-ray structure of 2-furanacetic acid crystal Projec-tion of the lattice onto the ldquoacrdquo plane

Figure 2 The X-ray structure of 2-furanacrylic acid crystal Projec-tion onto the ldquoacrdquo plane

deposited at the Cambridge Crystallographic Data Centreunder the number CCDC-885823) The molecules of 2-furanacetic acid in the lattice are linked together by the OndashHmiddot middot middotO hydrogen bonds forming centrosymmetric dimersA view of the crystal lattice of 2-furanacetic acid is shown inFigure 1

Crystals of 2-furanacrylic acid are also monoclinic thespace-symmetry group is C2c and Z = 8 The unit cellparameters are a= 18975 A b = 3843 A c = 20132 A β =1139 The molecules of 2-furanacrylic acid in the lattice arelinked together by the OndashHmiddot middot middotO hydrogen bonds formingcyclic approximately centrosymmetric dimers [28 29] TheX-ray structure of 2-furanacrylic acid crystals is shown inFigure 2

3 Experimental

2-Furanacetic acid (C4H3OndashCH2ndashCOOH) and 2-furanacrylicacid (C4H3OndashCH=CHndashCOOH) used for our studies werethe commercial substance (Sigma-Aldrich) 2-furanace-tic acid was employed without further purification while 2-furanacetic acid was purified by crystallization from itsacetone solution The d1 deuterium derivatives of the

compounds (C4H3OndashCH2ndashCOOD and C4H3OndashCH=CHndashCOOD) were obtained by evaporation of D2O solution ofeach compound at room temperature and under reducedpressure It was found that the deuterium exchange rate forthe COOH groups varied from 60 to 90 and from 70 to90 for different samples respectively

Crystals suitable for further spectral studies wereobtained by melting solid samples between two closely com-pressed spaces CaF2 windows followed by a very slow coolingof the liquid film By that means reasonably thin crystalscould be received characterized by their maximum absorb-ance at the νOminusH band frequency range near to 05 at roomtemperature From the crystalline mosaic adequate mono-crystalline fragments having dimensions of at least 2times2 mmwere selected and then spatially oriented with the help of apolarization microscope It was found that in each systemcase the crystals most frequently developed the ldquoacrdquo crys-talline face These crystals were selected to the experimentby use of a thin tin plate diaphragm with a 15 mm diameterhole and then IR spectra of these crystalline fragments weremeasured by a transmission method Spectral experimentswere accomplished at room temperature and also at thetemperature of liquid nitrogen using polarized IR radiationIn each measurement two different mutually perpendicularorientations of the incident beam electric field vector ldquoErdquowere applied with respect to the developed face of the crystallattice The solid-state polarized spectra were measured witha resolution of 2 cmminus1 for the normal incidence of the IRradiation beam with respect to the crystalline face The IRspectra were measured with the Nicolet Magna 560 FT-IRspectrometer Measurements of the spectra were repeated forca 8 crystals of each isotopomer of an individual compoundSpectra were recorded in a similar manner for the deuteriumderivatives

The Raman spectra of polycrystalline samples of 2-furanacetic acid and 2-furanacrylic acid were measured atroom temperature with the use of the Bio-Rad FTS-175C FT-IR spectrometer at the 1 cmminus1 resolution

4 Results

The preliminary experimental studies of spectral proper-ties of 2-furanacetic acid and 2-furanacrylic acid based onthe measurements in CCl4 solution in the frequency rangeof the νOminusH proton stretching vibration bands The resultsare shown in Figure 3

In Figure 4 are shown the νOminusH bands from the IR spectraof the polycrystalline acid samples in KBr pellets measuredat 298 K and 77 K and in Figure 5 the νOminusD bands spectraof the deuterium derivatives samples in the same conditionsThe comparatively wealth spectrum of νOminusH and νOminusD bandsfor 2-furanacrylic acid molecules may be predictable basedon earlier results for cinnamic acid crystals [24] while theνOminusH and νOminusD bands for 2-furanacetic acid crystals arerelatively poorer similarly as in the phenylacetic acid crystalcase [25]

Polarized IR spectra of the two crystalline systems mea-sured at the room temperature in the νOminusH band frequency

9Temperature and HD Isotopic Effects in the IR Spectra of the Hydrogen Bond in Solid-State 2-Furanacetic Acid and 2-Furanacrylic Acid

3000 2400

Wavenumbers (cmminus1)

Figure 3 The νOminusH band in the IR spectra of (a) 2-furanacetic acidand (b) 2-furanacrylic acid in CCl4 solution

3000 2500

RamanRaman

Figure 4 The νOminusH bands in the IR spectra of polycrystalline sam-ples of (a) 2-furanacetic acid and (b) 2-furanacrylic acid dispersedin KBr pellets Temperature effect in the spectra The Raman spectrameasured for polycrystalline samples of the compounds at roomtemperature are also shown

range are presented in Figure 6 whereas the correspondinglow-temperature spectra are shown in Figure 7

The corresponding spectra of isotopically diluted crystalsrecorded in the νOminusD band range are shown in Figures 8 and9

The temperature effect in the crystalline spectra in themost intense polarized components of the νOminusH bands isshown in Figure 10 and in the νOminusD bands is given inFigure 11

3000 2000

Figure 5 The νOminusD bands in the IR spectra of polycrystallinesamples of (a) d1-2-furanacetic acid (ca 45 H and 55 D) and(b) d1-2-furanacrylic acid (ca 20 H and 80 D) dispersed in KBrpellets Temperature effect in the spectra

5 Isotopic Dilution Effects inthe Crystalline IR Spectra

On comparing the spectra in Figures 3 and 6ndash9 it can benoticed that the replacement of the major part of the hydro-gen bond protons by deuterons changed the dichroic prop-erties in the ldquoresidualrdquo νOminusH band substantially The bandshapes no longer depended on the crystal orientation inves-tigated and resembled the spectrum measured for the CCl4solution of the compounds Regardless of the increase in therates of deuterium substitution in the samples the ldquoresidualrdquoνOminusH band still retained its ldquodimericrdquo character This is dueto the fact that the hydrogen-bonded dimeric spectrummeasured in the ldquoresidualrdquo νOminusH band range is still underthe influence of the interhydrogen bond vibrational excitoninteractions occurring within each individual carboxylic aciddimer [22ndash27]

The unusual properties of the ldquoresidualrdquo νOminusH bandshave proved that the distribution of protons and deuteronsbetween the hydrogen bonds of the isotopically diluted crys-talline samples is nonrandom and in an individual dimerthe coexistence of two identical hydrogen isotope atomsproton or deuterons is preferred As a result the interhy-drogen bond exciton interactions still occur in each dimericsystem and consequently the ldquoresidualrdquo νOminusH bands retaintheir ldquodimericrdquo properties These spectral effects that isthe so-called HD isotopic ldquoself-organizationrdquo effects are

3000 2500

Figure 6 Polarized IR spectra of 2-furanacetic acid and 2-furana-crylic acid crystals measured at room temperature in the νOminusH bandfrequency range for the IR radiation of the normal incidence withrespect to the ldquoacrdquo crystal faces (a) 2-furanacetic acid crystal (I) Theelectric field vector E of the incident beam of IR radiation parallelto the a-axis (II) The E vector parallel to the clowast-axis (the clowast-symboldenotes the vector in the reciprocal lattice) (b) 2-furanacrylic acidcrystal (I) The electric field vector E parallel to the c-axis (II) TheE vector parallel to the alowast-axis

the attribute of the ldquodynamical cooperative interactionsrdquoinvolving hydrogen bonds in the dimers [16ndash18]

In the case of high excess of protons in the crystals quali-tatively similar spectral effects can be identified in the ldquoresid-ualrdquo νOminusD bands located in the range of 1900ndash2300 cmminus1 asthose observed in the ldquoresidualrdquo νOminusH bands In the low con-centration of deuterons the ldquoresidualrdquo νOminusD bands still retainthe characteristic linear dichroic effects accompanying them(see Figures 4ndash9) For the two compared ldquoresidualrdquo bandsνOminusH and νOminusD not only the linear dichroic but also thetemperature effects appear to be similar to the correspondingeffects measured in the spectra of isotopically neat crystals

This property results from the ldquodynamical cooperativeinteractionsrdquo in the hydrogen-bonded systems which leadto the appearance of the so-called HD isotopic self-organization effects in the hydrogen bond IR spectra [17 18]The source of these nonconventional interactions in thehydrogen bond dimers is a vibronic coupling mechanisminvolving the totally symmetric proton stretching vibrationsand the electronic motions in the systems [17 18] Accordingto the theory of the ldquodynamical cooperative interactionsrdquo the

3000 2500

Figure 7 Polarized IR spectra of (a) 2-furanacetic acid and (b)2-furanacrylic acid crystals measured at 77 K in the νOminusH bandfrequency range (a) 2-furanacetic acid crystal (I) The electric fieldvector E parallel to the a-axis (II) The E vector parallel to the clowast-axis (b) 2-furanacrylic acid crystal (I) The electric field vector Eparallel to the c-axis (II) The E vector parallel to the alowast-axis

symmetric hydrogen bond dimers of the HH or DD-typewith identical hydrogen isotope atoms are thermodynami-cally more stable than the non-symmetric dimers of the HDtype The distribution of the HH- or DD-type dimers inthe lattice sites is random The energy difference betweenthe two forms of dimers the HH and the HD types wasestimated as approximately equal to 15 kcalmole of thedimers Therefore the relative concentration of the HD-typedimers is negligibly low and practically nondetectable withthe use of the IR spectroscopic methods [16ndash18]

From the experimental studies presented in Figures 3ndash11it also results that hydrogen-bonded cyclic centrosymmetricdimers are the bearers of the crystal spectral properties sincethe inter-dimer vibrational exciton interactions are negligiblysmall

6 Model

61 Carboxylic Acid Dimers the Basic Idea The problemof the quantitative theoretical treatment of the spectralproperties of systems composed with mutually interactinghydrogen bonds still constitute a real challenge in the areaof the hydrogen bond research There are still many problemsto solve in this matter since even the most advanced theories

3000 2000

Figure 8 Polarized IR spectra of (a) d1-2-furanacetic acid and (b)d1-2-furanacrylic acidcrystals measured at room temperature in theνOminusD band frequency range (a) 2-furanacetic acid crystal (ca 10 Hand 90 D) (I) The electric field vector E parallel to the a-axis (II)The E vector parallel to the clowast-axis (b) 2-furanacrylic acid crystal(ca 65 H and 35 D) (I) The electric field vector E parallel tothe c-axis (II) The E vector parallel to the alowast-axis

elaborated for the description of the IR spectra of hydrogenbond systems are unable to reliably explain a number ofeffects observed in the dimeric spectra Despite of spectacularachievements in the quantitative description of the intensitydistribution in the νXminusH bands which are the attribute of theproton stretching vibrations in the XndashHmiddot middot middotY bridges and inthe description of the HD isotopic effects the understandingof temperature effects in the spectra seems to be totallyincomplete

Cyclic hydrogen bond dimers formed by associatedcarboxyl groups of diverse carboxylic acid molecules are themost frequently studied model systems investigated in thisresearch area They exhibit some unusual spectral propertiesin IR connected with the highly abnormal thermal evolutionof the νOminusH and νOminusD band contour shapes One could expectthat the hydrogen bond spectra of diverse carboxylic aciddimers measured in the νOminusH and νOminusD band frequencyranges should be fairly similar one to another due to theidentical structural units of the molecular dimers namelythe (COOH)2 rings in which two hydrogen bonds existforming hydrogen bond dimers However on comparison ofthe crystalline spectra of diverse carboxylic acids a consider-able variation degree of the analyzed band contour shapescan be found This fact undoubtedly remains in a close

3000 2500

3000 2000

Figure 9 Polarized IR spectra of (a) d1-2-furanacetic acid and (b)d1-2-furanacrylic acid crystals measured at 77 K in the νOndashD bandfrequency range (a) 2-Furanacetic acid crystal (ca 10 H and 90D) (I) The electric field vector E parallel to the a-axis (II) The Evector parallel to the clowast-axis (b) 2-Furanacrylic acid crystal (ca 65H and 35 D) (I) The electric field vector E parallel to the c-axis(II) The E vector parallel to the alowast-axis

connection with differences in the electronic structuresof diverse carboxylic acid molecules Simultaneously thesespectra strongly differ one from the other by temperatureeffects characterizing them Also these effects undoubtedlyremain in a close relation with the electronic structuresof the associating molecules The basic experimental factssupporting the hypothesis given above are presented in thefollowing

62 Electronic Structure of Carboxylic Acid Molecules versusthe Temperature Effects in Their Crystalline IR Spectra Basedon our previous studies at this point let us summarize thebasic properties of the νOminusH bands in the IR spectra of thehydrogen bond cyclic dimers formed by diverse carboxylicacid molecules in relation to their electronic structures

(a) In the case of carboxylic acid molecules in which thealiphatic fragments are connected directly with car-boxyl groups (eg aliphatic monocarboxylic acids[11ndash13 30 31] and dicarboxylic acids [22]) theνOminusH bands are characterized by different inten-sity distribution patterns when compared with thecorresponding band properties in the IR spectraof arylcarboxylic acids [23 26] In the first case

3000 2500

3000 25000

Figure 10 The νOminusH bands in the IR spectra of monocrystallinesamples of (a) 2-furanacetic acid and (b) 2-furanacrylic acid Tem-perature effect in the spectra

the higher-frequency branch of the νOminusH band ismore intense in relation to the intensity of the lower-frequency band branch

(b) In the case of hydrogen-bonded molecular systemsin which carboxyl groups are directly linked to π-electronic systems (eg arylcarboxylic [23 26] andarylacrylic acids [24]) the νOminusH band contours are aldquomirror reflectionrdquo of the band shapes of systems fromthe point ldquoardquo In this case the lower-frequency branchof the band is the most intense one Similar propertycharacterizes spectra of carboxylic acids with otherlarge π-electronic systems in their molecules forexample cinnamic acid [24] 2-naphthoic acid [26]and 1-naphthylacrylic acid [32]

(c) For other carboxylic acids in which aromatic radicalsare separated from carboxyl groups by fragmentsof aliphatic hydrocarbon chains (eg arylacetic acid[25 27] and styrylacetic acid [33]) the νOminusH bandcontour shapes are fairly similar to the correspondingband characteristics from the point ldquoardquo that is tothe corresponding spectra of aliphatic monocarboxylicacids [30 31] and dicarboxylic acids [22])

The νOminusH bands in the spectra of the hydrogen bondof carboxylic acid crystals from the ldquoardquo and ldquocrdquo groupsmeasured at room temperature are characterized by rela-tively low intensity of the lower-frequency branch of he bandin comparison with the higher-frequency band branch

3000 2000

Figure 11 The νOminusD bands in the IR spectra of monocrystallinesamples of (a) d1-2-furanacetic acid (ca 10 H and 90 D) and (b)d1-2-furanacrylic acid (ca 65 H and 35 D) Temperature effectin the spectra

intensity On the decrease of temperature to 77 K onlya relatively small growth of the relative intensity of thelower-frequency branch of each band can be observed Thisband branch still remains of the lower intensity in the low-temperature spectra

In the case ldquobrdquo even at room temperature spectra theνOminusH bands exhibit relatively high intensity of their lower-frequency branch in relation to the higher-frequency branchOn the temperature decrease up to 77 K a considerablegrowth of the relative intensity of the lower-frequency branchof each analyzed band can be observed As the result ofthe band contour thermal evolution in the low-temperaturespectra of carboxylic acid crystals of this group the lower-frequency branch is of the dominant intensity in the bands

According to the ldquostate-of-artrdquo in our contemporaryknowledge about the quantitative description of the IRspectra of the hydrogen bond in carboxylic acid dimers thefollowing interpretation of the νOminusH band generation mech-anisms seemed to be valid the lower-frequency branch of theνOminusH band is generated by the transition occurring to theAg-symmetry excited state of the totally symmetric protonstretching vibrations in the dimers This transition forbid-den by the symmetry rules becomes allowed via a vibronicmechanism which is a kind of reverse of the familiarHerzberg-Teller mechanism originally responsible for thepromotion of forbidden electronic transitions in UV spectra

of aromatic hydrocarbons [34] Within this approach ofthe reverse Herzberg-Teller vibronic coupling mechanismelectronic properties of single hydrogen bonds themselvesas well as electronic properties of the whole associatedmolecules and the proton vibration anharmonicity areresponsible for the magnitude of the forbidden transitionpromotion effects in the dimeric spectra [35] The promo-tion mechanism determines a unique property of centrosym-metric hydrogen bond dimeric system This effect foundno counterpart in the vibrational spectroscopy of singlecentrosymmetric molecules

On the other hand the higher-frequency spectral branchof the band corresponds with the symmetry-allowed tran-sition to the Au-state of the nontotally symmetric protonvibrations in the centrosymmetric hydrogen bond dimersOne should expect that the higher-frequency branch of theνOminusH band attributed to the allowed transition should bemore intense than the other band branch related with theforbidden transition Therefore based on these intuitivepredictions the spectral properties of the carboxylic aciddimers from the ldquobrdquo group seem to be highly surprisingcontradicting the interpretation of the spectra of systemsbelonging to the ldquoardquo and ldquocrdquo groups The particular electronicproperties of the carboxylic acid molecules from the ldquocrdquogroup can anyway explain the extremely high integral inten-sity of the forbidden lower-frequency branch of the band andits strong temperature dependence

In order to propose a reliable explanation of this paradoxin our analysis one should also recall the hydrogen bond IRspectra of other hydrogen bond dimeric systems includingspectra of hydrogen-bonded heterocycles On comparisonof the IR spectra of diverse crystalline systems containingcyclic hydrogen bond dimers as the structural units of theirlattices the following general conclusions can be made mostof centrosymmetric hydrogen bond dimers exhibit regularenough spectral properties characterizing their hydrogenbond spectra Usually the νXminusH bands have the lower-frequency (ie the ldquoforbiddenrdquo) branch of a lower intensityeven in their low-temperature spectra However in some rarecases for example 3-hydroxy-4-methyl-2(3H)-thiazolethione[36] 2-tiopyridone [37] and 2-pyridone [38] the νOminusH andνNminusH bands are characterized by an abnormal that is bya ldquoreverserdquo intensity distribution patterns in their contoursIn the latest cases the lower-frequency branch of each bandis more intense when compared with the higher-frequencyband intensities It fairly resembles the properties of thespectral properties at 77 K of carboxylic acid crystals ofthe ldquobrdquo group In the case of the dimeric spectra of thereverse intensity distribution patterns in the bands forexample 3-hydroxy-4-methyl-2(3H)-thiazolethione [36] and2-tiopirydone [37] this effect was ascribed previously tothe influence of the extreme lengths of the OndashHmiddot middot middot S andNndashHmiddot middot middot S hydrogen bonds in the dimeric systems

The recent considerations aiming to explain these phe-nomena were performed in terms of the dipole-dipole modelof the vibrational exciton interactions involving the hydrogenbonds in the dimers In the case of the interpretationof the spectra of 3-hydroxy-4-methyl-2(3H)-thiazolethione[36] and 2-tiopirydone [37] the hydrogen bond geometry

EAuEAg

EAu gt EAg

minus minus

Figure 12 The ldquoside-to-siderdquo (SS) exciton coupling involving theproton stretching vibrations in a cyclic centrosymmetric hydrogenbond dimer

was considered to be responsible for the unusual spectralproperty of these dimers However this approach fails in theinterpretation of the spectra of 2-pyridone cyclic dimers [38]in which the NndashHmiddot middot middotO hydrogen bonds are considerablyshorter when compared with the NndashHmiddot middot middot S bond lengthsin 2-thiopyridone cyclic dimers [37] and their spectraqualitatively fairly resemble the corresponding spectra of 2-pyridone [38] On the other hand even among the hydrogenbond dimers of diverse molecular systems with the NndashHmiddot middot middot S hydrogen bonds for which the extreme spectralproperties were found a substantial diversification in theanalyzed spectral properties has been found despite theextremely long hydrogen bonds in these cases The IR spectraof 2-mercaptobenzothiazole cyclic dimers [39] exhibit regularproperties of the intensity distribution pattern in their νNminusH

band contours similarly as the carboxylic acid dimers in thecrystals of the groups ldquoardquo an ldquobrdquo regardless of the extremeNndashHmiddot middot middot S bond lengths like these found in 2-thiopyridonedimers [37]

63 Spectra of Cyclic Dimers versus Spectra of Chain HydrogenBond Systems It is surprising that spectra of cyclic hydrogenbond dimers in 3-hydroxy-4-methyl-2(3H)-thiazolethione[36] 2-thiopyridone [37] and 2-pyridone [38] crystals fairlyresemble by their intensity distribution patterns of the νNminusH

bands the spectra of chain hydrogen bond systems in aparticular group of molecular crystals In the hydrogen bondspectra of pyrazole [40] and 4-thiopyridone [41] crystals withhydrogen-bonded molecules forming infinite chains in theirlattices strong linear dichroic effects can be observed whichprove a considerable influence of the exciton interactionsinvolving the adjacent hydrogen bonds in each chain Figures12 and 13 explain the source of the differences in thehydrogen bond dimers the cyclic and the chain ones

The analysis of this inter-hydrogen-bond coupling incase of cyclic centrosymmetric dimers and in linear dimersrequires taking into consideration two situations of thevibrational transition moment directions for hydrogenbonds in the dimers For cyclic dimers the parallel mutualorientation of the dipole transition moments the excitoninteraction energy EAu in the limits of the dipole-dipole modelis of the positive sign The vibrational transition correspond-ing to such arrangement of the vibration dipole moments is

Eminus

Eminus gt E+

minusminus

minus minus

Figure 13 The ldquohead-to-tailrdquo (TH) exciton coupling involvingthe proton stretching vibrations in an infinite chain of associatedhydrogen bonds

responsible for generation of the intense symmetry-allowedshorter-wave branch of the dimeric spectra In contrastwhen the dipole transition moments are of the antiparallelarrangement (see Figure 12) the energy exciton interactionenergy value EAg is negative so the band generated bythis situation is placed at the lower frequency and itcorresponds to the symmetry-forbidden excitation of thetotally symmetric proton vibrations Such sequence of thespectral branches in the hydrogen bond stretching bands istypical for cyclic centrosymmetric hydrogen bond dimers

When the vibrating transition moment dipoles in a lineardimer in the case of the totally symmetric proton vibrationsare oriented axially as ldquotail-to-headrdquo (Figure 5) the signof the exciton interaction energy value E+ is negative sothe intense branch corresponding to the symmetry-allowedtransition is placed at the lower-frequency range On thecontrary the forbidden by the symmetry rules spectralbranch situated at the higher frequency is generated bythe antiparallel orientation of the vibrating dipoles (seeFigure 13) In this case the exciton coupling energy Eminus isof the positive sign The sequence and the properties of thebranches in the proton stretching vibration bands in thediscussed case are reverse to those observed in the IR spectraof hydrogen bond cyclic dimeric systems

Therefore the following problem demands explanationwhy do some individual cyclic hydrogen bond dimericsystems exhibit similar spectral properties to the correspond-ing properties of a particular group of crystals with chainstructures of hydrogen-bonded associates (formic acid [31]pyrazole [40] and 4-thiopyridone [41] crystals) Undoubt-edly this property remains in a close connection with theπ-electronic properties of the associating molecules In theassociated molecular systems vibrational exciton couplingsare of the ldquotail-to-headrdquo (TH) type They involve the adjacenthydrogen bonds within each individual chain in the lattice

The electronic structure of molecules of this group is mostprobably the key factor governing these interhydrogen bondinteractions

Nevertheless the majority of crystals with hydrogen-bonded molecular chains in their lattices surprisingly exhibitthe spectral properties similar to the analogous properties ofcyclic hydrogen bond dimer spectra from the ldquoardquo and ldquocrdquogroups (eg acetic acid [30] N-methylthioacetamide [42]or acetanilide [18] crystals) In the latest case the excitoninteractions of the ldquoside-to-siderdquo (SS) type involve the closelyspaced hydrogen bonds where each moiety belongs to adifferent chain In molecules of this group large π-electronicsystems are absent Only carbonyl or thiocarbonyl groupseach with a small π-electronic system are present in thesemolecules

From the above-presented data it results that the way ofrealization of the vibrational exciton interactions in varioushydrogen bond aggregates (cyclic dimers infinite chains)affecting the νXminusH and νXminusH band fine structures does notdirectly depend on the hydrogen bond system geometryIt is rather determined by the electronic structure of theassociating molecules

7 Theoretical Approach Proposed

The dipole-dipole interaction model widely used for asimplified description of the exciton interactions betweenhydrogen bonds seems to be nonadequate in the explanationof the wide diversity of the spectra of cyclic hydrogen bonddimers There is some experimental data indicating that thesecouplings do not always occur as ldquothrough-spacerdquo and theyare also widespread by the hydrogen bond electrons as well asby electrons of the molecular skeletons Therefore in termsof the theory of molecular vibrational excitons [43 44]the exciton interaction integrals in some cases may alsoconsiderably strongly depend on the electronic coordinatesIn advantageous circumstances resulting from a properelectronic structure of the associating molecules the protonstretching vibrations can induce electric current oscillatingaround a cyclic hydrogen bond dimer or in the other caseoscillating along a hydrogen bond chain However only thetotally symmetric proton vibrations are able to effectivelyinduce the electric current in the ring or in the chainwhile the nontotally symmetric vibrations are inactive inthis mechanism since currents induced in each individualhydrogen bond are annihilated in a dimer The formalismof the model of the electric current generated by oscillatingprotons in cyclic hydrogen bond dimers was proposed byNafie three decades ago [45]

In the scope of the considerations given above it seemsjustified to treat formally a cyclic hydrogen bond dimerby the following two ways taking into account the excitoninteractions in the system

(1) As a closed chain in which the adjacent hydrogenbonds are strongly exciton-coupled similarly as inthe chain associates in pyrazole [40] and 4-thio-pyridone [41] crystals This is the coupling of theTH type occurring around the molecular cycle This

way the coupling occurs via the easy-polarizableelectrons on the π-orbitals Therefore the cyclicdimer spectrum is fairly similar to the spectrum ofa chain system with a low intensity of the higher-frequency band branch

(2) As a pair of partially independent hydrogen bondswhich remains only ldquothrough-spacerdquo exciton coupledIt can be considered as a coupling of the SS typewithout the generation of the ring electric current inthe dimer This behavior characterizes the associatedmolecular systems with no large π-electronic systemsin their structures where only small π-electronicsystems are present in carbonyl and thiocarbonylgroups In these circumstances the dimeric spectraare of the standard form with a low intensity of thelower-frequency νXminusH band branch For the quantita-tive description of the exciton interactions involvinghydrogen bonds influencing the dimer spectra thedipole-dipole model is sufficiently adequate

The νXminusH band shapes in the two types of the dimerspectra are related one with the other by the approximatemirror reflection symmetry In the case 1 the lower-intensityspectral branch appears in the higher-frequency range and isgenerated by the quasiforbidden vibrational transition in adimer occurring to the excited state of the totally symmetricproton stretching vibrations In case 2 the lower intensityspectral branch appears in the lower-frequency range Itcorresponds with the quasi-forbidden vibrational transitionin a dimer The above-presented spectral properties ofdiverse hydrogen bond cyclic dimers may allow explainingthe thermal evolution effects in the hydrogen bond IR spectraof carboxylic acid crystals

It seems that in order to explain the temperature effects inthe IR spectra of cyclic hydrogen bond dimers the followinghypothesis concerning the mechanisms of the spectra gener-ation should be accepted let us assume that two competingmechanisms of vibrational exciton interactions involvinghydrogen bonds in cyclic dimers are simultaneously respon-sible for the formation of the νXminusH band contour shapesThe contribution of each individual mechanism dependson the electronic structure of the associating molecules onthe electronic properties of the heavy atoms forming thehydrogen bridges as well as on temperature

(A) The first mechanism depends on the ldquoside-to-siderdquo(SS)-type vibrational exciton coupling between thehydrogen bonds in cyclic dimers In this case thedimer hydrogen bonds interact one with the other asthrough-space via the van der Waals forces

(B) The other mechanism assumes a ldquotail-to-headrdquo (TH)-type exciton coupling involving the hydrogen bondsin the dimers These interactions occur around thecycles via electrons

The ldquoBrdquo mechanism seems to be privileged in the caseof the particular kind of associated molecules in whichhydrogen bonds couple with large π-electronic systems forexample for aromatic carboxylic acid molecules The ldquoArdquo

mechanism seems to dominate in the case of molecularsystems with small π-electronic systems for example foraliphatic carboxylic acid molecules

It seems obvious that for an individual hydrogen-bondeddimeric system the contribution of each mechanism is tem-perature dependent For molecules with large π-electronicsystems directly coupled with the hydrogen bonds the ldquoBrdquomechanism should be privileged at very low temperaturesTemperature growth influencing the increase of atomicvibration amplitudes should annihilate the electric currentinduced by the totally symmetric proton vibrations in thecycles In these circumstances the role of the ldquoArdquo mechanismincreases namely of the ldquothrough-spacerdquo vibrational excitoncoupling between the hydrogen bonds in a dimer Thisshould therefore result in a particularly strong temperature-induced evolution of the νXminusH bands especially in the case ofthe spectra of 2-thiopyridone [37] and 2-pyridone [38] typedimers Even when the lower-frequency branch of the bandis less intense when compared with the higher-frequencyone the temperature decrease till 77 K causes its considerableintensity growth and in these circumstances the lower-frequency branch becomes more intense than the higher-frequency band branch

In the spectra of cyclic dimers with only small π-electronic systems in the associating molecules the tempera-ture decrease usually does not cause a considerable intensitygrowth of the lower-frequency band branch It still remainsless intense when compared with the higher-frequencybranch of the band It means that due to the molecularelectronic properties of this group of dimers the ldquoBrdquo mecha-nism cannot be activated effectively enough even at very lowtemperatures

8 Spectral Consequences of the Model forCarboxylic Acids

From the above assumptions it results the choice of theproper way of the model calculations of the νXminusH and νXminusD

band contours in IR spectra of hydrogen bond dimers In thelimits of the proposed approach a theoretical spectrum ofthe model system can be derived formally treated as a super-position of two component spectra where each individualspectrum corresponds with a different mechanism of theexciton interactions SS (A) and TH (B) involving the dimerhydrogen bonds In terms of the ldquostrong-couplingrdquo theory[6ndash8] in each exciton interaction mechanism case the νXminusH

band in the dimeric spectrum is a superposition of twocomponent bands ldquoPlusrdquo and ldquoMinusrdquo each of a differentorigin

The ldquoPlusrdquo band is generated by the dipole allowed tran-sition to the excited state of the nontotally symmetric protonstretching vibrations in a centrosymmetric dimer belongingto the Au representation On the other hand the ldquoMinusrdquoband is connected with the symmetry forbidden transitionto the Ag-symmetry state of the totally symmetric protonvibrations in the dimers activated by a vibronic mechanism[35] In the case when the mechanism ldquoArdquo exclusively decidesabout the dimer spectra generation mechanism the ldquoMinusrdquoband appears in the lower ldquoBrdquo mechanism frequency range in

relation to the ldquoPlusrdquo band location In the other case whenthe ldquoBrdquo mechanism governs the dimer spectra generationthe two component bands appear in the reverse sequencethan in the case ldquoArdquo It means that in the case of the ldquoBrdquomechanism governing the spectra generation the ldquoMinusrdquoband representing the forbidden transition appears in thehigher-frequency range than the ldquoPlusrdquo band connected withthe allowed transition

9 Model Calculations of the Band Contours

In the two cases A and B model calculations aiming atreconstituting of the ldquoresidualrdquo νOminusH and νOminusD band shapeswere performed within the limits of the ldquostrong-couplingrdquotheory for a model centrosymmetric OndashHmiddot middot middotO hydrogenbond dimeric system [6ndash8 46] We assumed that themain νOminusH and νOminusD band shaping mechanism involvedstrongly anharmonically coupled the high-frequency proton(or deuteron) stretching vibrations and the low-frequencyOmiddot middot middotO hydrogen bridge stretching vibrational motionsCalculation of the hydrogen bond system IR spectra interms of the ldquostrong-couplingrdquo model allows to obtain resultsfairly comparable with the results of the spectra calculationperformed using the ldquorelaxationrdquo theory [9 10 47ndash49]

According to the formalism of the ldquostrong-couplingrdquotheory [6ndash8 46] the νOminusH band shape of a dimer depends onthe following system of dimensionless coupling parameters(i) on the distortion parameter ldquobHrdquo and (ii) on theresonance interaction parameters ldquoCOrdquo and ldquoC1rdquo The ldquobHrdquoparameter describes the change in the equilibrium geometryfor the low-energy hydrogen bond stretching vibrationsaccompanying the excitation of the high-frequency protonstretching vibrations νOminusH The ldquoCOrdquo and ldquoC1rdquo parametersare responsible for the exciton interactions between thehydrogen bonds in a dimer They denote the subsequentexpansion coefficients in the series on developing theresonance interaction integral ldquoCrdquo with respect to the normalcoordinates of the νOmiddotmiddotmiddotO low-frequency stretching vibrationsof the hydrogen bond This is in accordance with the formula

C = CO + C1Q1 (1)

whereQ1 represents the totally symmetric normal coordinatefor the low-frequency hydrogen bridge stretching vibrationsin the dimer This parameter system is closely related to theintensity distribution in the dimeric νNminusH band The ldquobHrdquoand ldquoC1rdquo parameters are directly related to the dimeric νNminusH

component bandwidth The ldquoCOrdquo parameter defines thesplitting of the component bands of the dimeric spectrumcorresponding to the excitation of the proton vibrationalmotions of different symmetries Ag and Au In its simplestoriginal version the ldquostrong-couplingrdquo model predicts reduc-tion of the distortion parameter value for the deuteriumbond systems according to the relation

bH = radic2bD (2)

For the ldquo COrdquo and ldquoC1rdquo resonance interaction parametersthe theory predicts the isotopic effect expressed by the 10

toradic

2-fold reduction of the parameter values for D-bondeddimeric systems

As the consequence of the ldquostrong-couplingrdquo model theνOminusH and νOminusD band contour fine structures were treated asa superposition of two component bands They correspondto the excitation of the two kinds of proton stretchingvibrations each exhibiting a different symmetry In the caseof the A exciton coupling mechanism and for the Ci pointsymmetry group of the model dimer the excitation of theAg vibrations in the dimer generates the lower-frequencytransition branch of the νOminusH band when the Au vibrationsare responsible for the higher-frequency band branch In thecase of the B mechanism the component subbands appear inreverse sequence

Here we consider an identical anharmonic couplingparameter system for the two individual mechanism casesA and B although diversification of the coupling parametervalue systems seems to be better justified We assume the con-tribution of each mechanism as governed by a Boltzmann-type relation In addition for the statistical weight param-eters of each individual mechanism PA(T) and PB(T) onemust distinguish which state is dominant that is when theSS (A) state is of the lower energy and the TH (B) state isof a higher energy value and vice versa In order to repro-duce the temperature dependence of experimental spectraparticularly for its width and the position of its first momentwe used for the PAB

A (T) exponential temperature dependenceaccording to

PABA (T) = 1minus exp

(minus α

where is αAB the activation energy parameter when the SSstate is dominant and kB is the constant of Boltzmann Insuch circumstance PAB

B (T) takes the following expression

PABB (T) = exp

(minus α

It is interesting to note that in the case of A for verylow temperatures the statistical weight PAB

A (T) parameteris close to 10 and PAB

B (T) is almost equal 00 In thesecircumstances the SS-type interaction is the basic type ofthe exciton coupling involving the dimer hydrogen bondsFor high temperatures the PAB

B (T) parameter values aredifferent from 00 and they are intermediate between 00and 10 (rather closer to 05) and PAB

A (T) approaches 05When the temperature increases PAB

B (T) also increases Itmeans that the TH coupling occurring via the electriccurrent in the ring is activated in higher temperatures ina magnitude depending of the energy gap between thesetwo states of the vibrationally excited dimer From ourexperimental estimations the energy gap for some dimericsystem cases is relatively large and in another cases it may berelatively low

In the case B where the TH state is of a lowerenergy value we assume the same formula but the energybarrier αBA height is relatively low In such a circumstance

the statistical weight parameters PA(T) and PB(T) may bewritten as follows

PBAA (T) = exp

(minus α

PBAB (T) = 1minus exp

(minus α

As we can see for very low temperatures PABA (T) may be

practically equal to 10 For this kind of dimeric systems theTH-type exciton coupling is the basic natural way in whichthe inter-hydrogen bond interactions occur The growth intemperature annihilates this way of the coupling due to thevanishing of the electronic current induced in the cyclesaccompanied by large-amplitude thermal motions of atomsin the dimers For high temperatures PAB

A (T) decreases andbecomes of an intermediate value between 00 and 10 (rathercloser to 05) while the statistical weight PAB

A (T) growsdeclining from 00 up to 05 The energy gap between thetwo states in some molecular cases is usually relatively largeand in other cases it may be relatively small It dependsof the electronic properties of the associating moleculesforming the dimers From our experimental data it can beconcluded that the cases A and B represent the extremecases of the interhydrogen bond coupling in cyclic hydrogenbond dimers There are also many systems exhibiting anintermediate behavior For a relatively small magnitude ofthe absolute values of the energy barrier height the two casesA and B are practically nondistinguishable

The theoretical spectra reconstituting the νOminusH bandcontours measured at the two different temperatures 293 Kand 77 K were calculated in terms of the two differentindividual coupling mechanisms SS and TH which generatethe two component bands ldquoplusrdquo and ldquominusrdquo in a differentsequence The following coupling parameter values identicalin both molecular system cases were used

For the 2-furanacetic acid crystal spectra bH = 16 C0 =15 C1 = minus02 F+ = 10 Fminus = 02 ΩOmiddotmiddotmiddotO = 100 cmminus1 andwe used the same parameter system for calculation of the 2-furanacrylic acid crystal spectra bH = 16C0 = 15C1 =minus02F+ = 10 Fminus = 02 ΩOmiddotmiddotmiddotO = 100 cmminus1

The F+ and Fminus symbols denote the statistical weightparameters for the ldquoplusrdquo and ldquominusrdquo theoretically derivedsubspectra contributing at the band formation

The coupling parameter values used for calculation of theνOminusD band contour shapes were as follows

For 2-furanacetic acid crystal spectrum bD = 07 C0 =07 C1 = minus01 F+ = 10 Fminus = 02 ΩOO= 100 cmminus1 and for2-furanacrylic acid crystal spectrum bH = 07 C0 = 07 C1 =minus01 F+ = 10 Fminus = 02 ΩOmiddotmiddotmiddotO = 100 cmminus1

For the 2-furanacetic acid crystal spectra the statisticalweight parameter ratio PA(T) PB(T) for the SS and THmechanisms was estimated as equal to 10 00 in the case ofthe room temperature spectrum reconstitution For the low-temperature spectrum case this parameter ratio value is verysimilar and equal to 10 00 Among various parameter ratiovalues for the SS and TH mechanisms contributing in theband generation this parameter ratio value allowed for the

3 minus70

minus11 minus3 minus5

ωOmiddotmiddotmiddotO

IIIIII

ty3 minus7

Figure 14 The theoretically derived νOminusH band contours calculatedin terms of the ldquostrong-couplingrdquo theory in the limits of the twodifferent vibrational exciton coupling mechanisms involving thecyclic dimer hydrogen bonds that is ldquoside-to-siderdquo (SS) and ldquotail-to-headrdquo (TH) (a) The SS coupling mechanism (b) The TH couplingmechanism (I) The ldquominusrdquo band (II) The ldquoplusrdquo band (III)Superposition of the I and II spectra each taken with its appro-priate individual statistical weight parameter Fminus and F+ In bothmechanism cases the same coupling parameter value system wasused for calculations bH = 14 C0 = 15 C1 = minus02 F+ = 10Fminus = 02 ΩOmiddotmiddotmiddotO = 100 cmminus1 The transition frequencies are in theωOmiddotmiddotmiddotO vibrational quantum units and the transition frequenciesare expressed with respect to the gravity center of the hypotheticalspectrum of a monomeric hydrogen bond in the cyclic hydrogenbond dimer Transition intensities are in arbitrary units

most adequate reproduction of the temperature effect in thecrystal spectra

For the 2-furanacrylic acid crystal spectra the statisticalweight parameter ratio PA(T) PB(T) for the SS and THmechanisms were estimated as equal to 035 065 in the caseof the room temperature spectrum reconstitution For thelow-temperature spectrum case this parameter ratio valueis equal to 055 045

In Figures 14 and 15 we present the theoretical νOminusH andνOminusD band contours calculated in terms of the two individualmechanisms of the vibrational exciton interactions involvingthe dimer hydrogen bonds SS and TH

In Figures 16 and 17 the evolution of the νOminusH andνOminusD band contour shapes accompanying the variation in therelative contribution of the SS and TH coupling mechanismsin generation of a dimeric spectra is shown Similar band

2 minus30

01 minus1 minus2

2 minus30

01 minus1 minus2

Figure 15 The theoretically derived νOminusD band contours calculatedin terms of the ldquostrong-couplingrdquo theory in the limits of the twodifferent vibrational exciton coupling mechanisms involving thecyclic dimer hydrogen bonds that is ldquoside-to-siderdquo (SS) and ldquotail-to-headrdquo (TH) (a) The SS coupling mechanism (b) The TH couplingmechanism (I) The ldquominusrdquo band (II) The ldquoplusrdquo band (III)Superposition of the spectra I and II each taken with its appro-priate individual statistical weight parameter Fminus and F+ In bothmechanism cases the same coupling parameter value system wasused for calculations bH = 07 C0 = 07 C1 = minus02 F+ = 10Fminus = 02 ΩOmiddotmiddotmiddotO = 100 cmminus1 The transition frequencies are in theωOmiddotmiddotmiddotO vibrational quantum units and the transition frequenciesare expressed with respect to the gravity center of the hypotheticalspectrum of a monomeric hydrogen bond in the cyclic deuteriumbond dimer Transition intensities are in arbitrary units

shape evolution accompanies temperature changes duringthe spectral experiments

From the comparison of the corresponding calculatedand experimental spectra it results that the intensity distri-bution patterns and the temperature effects in the spectraof the two different crystalline systems have been at leastsemiquantitatively reproduced via the model calculations

10 Spectra of 2-Furanacetic and 2-FuranacrylicAcid Crystals

On comparing the IR spectra of the hydrogen bond for thetwo crystalline systems essential differences analyzed crys-talline spectra othe νOminusH and νOminusD bands In the case of 2-furanacetic acid spectra the fine structure pattern of eachband νOminusH and νOminusD is relatively simple Each band consistsof a low number of well-separated spectral lines In the

3 minus70

3000 2500

ty3 minus7

3000 2500

Figure 16 Temperature-induced evolution of the νOminusH bandcontour shapes accompanying the variation in the contribution rateof the two different exciton coupling mechanisms that is ldquoside-to-siderdquo (SS) and ldquotail-to-headrdquo (TH) Numerical reproduction ofthe temperature effect in the spectra of hydrogen-bonded (a) 2-furanacetic acid crystal (b) 2-furanacrylic acid crystal The relativecontribution ratio of the SS and TH mechanisms in the νOminusH bandgeneration is for 2-furanacetic acid crystal 095 005 at 293 K and095 005 at 77 K and for 2-furanacrylic acid crystal 065 035 at293 K and 040 060 at 77 K The experimental spectra are shownin inset

spectra of 2-furanacrylic acid each considered band iscomposed of a noticeably larger number of lines (ca 2 timeslarger) It seems to prove a more complex mechanism ofthe spectra generation in the case of 2-furanacrylic acid inrelation to the mechanism governing the spectra generationof 2-furanacetic acid

The analyzed crystalline spectra of 2-furanacetic acidseem to fully belong to the case A On the other hand thecrystalline spectra of 2-furanacrylic acid seem to satisfy thedemands of the case B The analyzed difference in the spectralproperties of arylacetic acid dimers and the arylacrylic aciddimers most probably results from the influences exerted onto the hydrogen bond dimers present in the (COOH)2 cyclesby the aromatic rings The direct contact between the furanrings with carboxyl groups (arylcarboxylic furanacrylic andthiopheneacrylic acids) most likely influences the electriccharge density in the (COOH)2 cycles This in turn strength-ens the vibronic mechanism of the electronic current gen-eration in the hydrogen bond cycles [45] Separation ofthe carboxyl groups from aromatic rings by methylene

2 minus30

01 minus1 minus2

2400 22002600

2 minus30

01 minus1 minus2

2200 20002400

Figure 17 Temperature-induced evolution of the νOminusD band-contour shapes accompanying the variation in the contribution rateof the two different exciton coupling mechanisms that is ldquoside-to-siderdquo (SS) and ldquotail-to-headrdquo (TH) Numerical reproduction ofthe temperature effect in the spectra of deuterium-bonded (a) 2-furanacetic acid crystal (b) 2-furanacrylic acid crystalThe relativecontribution ratio of the SS and TH mechanism in the νOminusD bandgeneration is for 2-furanacetic acid crystal 095 005 at 293 K and095 005 at 77 K and for 2-furanacrylic acid crystal 065 035 at293 K and 040 060 at 77 K The experimental spectra are shownin inset

groups (arylacetic acids furanacetic acids and thiopheneaceticacids) effectively weakens the vibronic coupling mechanismTherefore these latter systems belong to the A case

The analyzed spectral properties of the two differentcrystalline systems 2-furanacetic acid and 2-furanacrylic acidare in a good agreement with the described above vibrationalexciton interaction mechanisms of the spectra generationfor cyclic hydrogen bond dimer This remains in a closerelation to the electronic properties of the two carboxylicacid molecules For 2-furanacetic acid dimers the excitoninteractions involving the dimer hydrogen bonds of a SS-type is only weakly temperature dependent In the case of 2-furanacrylic acid dimers due to their electronic structure theinterhydrogen bond exciton coupling mechanism changes itscharacter along with the changes in temperature At very lowtemperatures the TH-type interactions transferred in the(COOH)2 cycles via electrons are dominating When tem-perature increases this mechanism becomes less privilegedas being annihilated by the hydrogen-bond atom thermalvibrational motions It is replaced by the other mechanismdepending of the SS-type interactions Each individual

mechanism generates its own spectrum characterized byits unique intensity distribution pattern Therefore theνOminusH and νOminusD bands in the spectra of 2-furanacrylic acidcrystals exhibit more complex fine structure patterns sincethey are superposition of two different spectra where eachcomponent spectrum is of a different origin Each com-ponent spectrum contributing to the νOminusH and νOminusD bandformation with its statistical weight parameter depended oftemperature corresponds with another exciton interactionmechanism in the cyclic hydrogen bond dimers in the lattice

Spectra of 2-thiopheneacrylic acid crystals [50] exhibitqualitatively fairly similar properties as the spectra of 2-furanacrylic acid crystals Their νOminusH and νOminusD bands alsodemonstrate complex and dense fine structure patterns Theyalso show very similar temperature effects when comparedwith the corresponding spectra of 2-furanacrylic acid crystals

In turn the spectra of 2-thiopheneacetic acid crystals [50]exhibit qualitatively very similar properties as the spectraof 2-furanacetic acid crystals Their νOminusH and νOminusD bandsalso exhibit relatively simple fine structure patterns Theyalso demonstrate fairly similar temperature effects whencompared with the corresponding spectra of 2-furanaceticacid crystals

From the comparison of the spectra of the two differentgroups of carboxylic acid crystals it results that the electronicstructure of the associating molecules is the main factordetermining the crystal spectral properties in IR differenti-ating the spectral properties of the two groups of hydrogen-bonded systems Namely the temperature effects registeredin IR spectra of the hydrogen bond in carboxylic acid crystalsremain in a close connection with the electronic spectra ofthe associating molecules forming cyclic hydrogen-bondeddimers in the lattices

11 The Problem of the Vibrational SelectionBreaking in IR Spectra of CentrosymmetricHydrogen Bond Dimers

The mechanism proposed in this paper for understandingthe sources of temperature effects in the IR spectra of cycliccentrosymmetric hydrogen bond dimers explains the gener-ation of the lower-frequency νOminusH and νOminusD band branchesof extremely high intensities in IR spectra of carboxylicacid crystals However at this stage the relation with theformerly published vibronic mechanism of the vibrationalrule selection breaking in the IR spectra of centrosymmetrichydrogen bond dimers [35] ought to be discussed since bothmechanisms can generate and also explain qualitatively fairlysimilar spectral effects

The vibronic mechanism was originally elaborated in thepast for the understanding of the fine structure patterns ofthe published earlier IR spectra of the cyclic centrosym-metric NndashHmiddot middot middot S bond dimers formed by 2-thiopyridoneand 2-mercaptobenzothiazole molecules as well as extremelynonregular HD isotopic effects in the spectra [37 3951] The isotopic effects were expressed by the unusuallynarrow νNminusD bands in correspondence to the very wideνNminusH bands characterized by complex fine structure pat-terns In terms of the vibronic model these effects were

explained by the disappearance of the intensity of thelower-frequency branch of the νNminusD bands attributed to theNndashD bond totally symmetric stretching vibrations in thedimers due to the weakening of the forbidden transitionpromotion mechanism [35] In the case of the νNminusH bandsthe promotion mechanism was effective enough generatingthe forbidden transition spectral branch of noticeably highintensity Nevertheless this branch appeared to be lessintense when compared with the allowed transition higher-frequency branch of the νNminusH band The vibronic modelascribed these effects to the difference in the proton anddeuteron vibration anharmonicity and to the extremelyhigh polarizability of the NndashHmiddot middot middot S hydrogen bonds in 2-thiopyridone and 2-mercaptobenzothiazole dimers Thesefactors were considered as responsible for the magnitude ofthe vibrational selection rule breaking effects in the dimericIR spectra [35]

The IR spectra of carboxylic acid crystals with cyclicdimers in their lattices considerably differ by the analogousHD isotopic effects from the spectra of the NndashHmiddot middot middot Sbonded dimers [22ndash27 37 39] In the case of carboxylicacid crystals practically no impact of the isotopic substitutiononto the relative intensity of the lower-frequency bandbranch intensities of the νOminusH and νOminusD bands in relation tothe corresponding higher-frequency band branch intensitiescan be noticed Also the incidentally observed very highintensities of the forbidden transition bands distinguish theseIR spectra of carboxylic acid crystals This proves that thespectra generation mechanism for the carboxylic acid dimersin the crystals essentially differs from the vibronic selectionrule breaking mechanism [35]

The following question arises in the scope of our latestestimations should the vibronic mechanism be definitivelyrejected as inadequate in the description of the IR spectralproperties of centrosymmetric hydrogen bond dimers espe-cially carboxylic acid dimers in the solid state

From our hitherto studies of IR spectra of hydrogen-bonded molecular crystals it results that the two differentmechanisms forming the band structures act parallel eachwith its individual statistical weight depending of theelectronic properties of the molecular systems forming thedimers In the case of cyclic dimeric NndashHmiddot middot middot S bondedmolecular systems the vibronic mechanism appeared to berelatively very sufficient leading to the appearance of intenseforbidden transition νNminusH band branches On the otherhand the νNminusD bands are extremely narrow as practicallydevoid of the forbidden band branch [37 39] The vibronicmechanism is also effective in the generation of IR spectraof crystals with infinite open chains of hydrogen bondedmolecules for example N-methylthioacetamide [42] or N-phenylacrylamide [52] crystals Also the HD isotopic effectsin their spectra are fairly similar to the analogous isotopiceffects in the corresponding spectra of the NndashHmiddot middot middot S bondedcyclic dimers In these chain structures centrosymmetrichydrogen bond dimeric systems are composed of hydrogenbonds where each moiety belongs to another chain of asso-ciated molecules penetrating a unit cell Most probably thechain structure of the molecular associates which excludesthe possibility of the induction to circulating electric currents

in such dimers as well as the polarization properties ofthese hydrogen bonds is responsible for the existence of thevibronic mechanism [35] in the pure form influencing theband contour formation

For the carboxylic acid dimer spectra the mechanismproposed in this work is dominant regardless of the elec-tronic structure of the substituent atomic groups linked tothe carboxyl groups in the molecules On the basis of theldquostate-of-artrdquo in the spectral studies of the hydrogen bondsystems in molecular crystals the HD isotopic effects in thespectra seem to be the main criterion for distinguishing thesetwo individual mechanisms However this problem demandsfurther intensive studies in the future

12 Conclusions

In this paper we report experimental and theoretical studyof IR spectra of 2-furanacetic acid and of 2-furanacrylic acidcrystals measured at 293 K and 77 K in the νOminusH and νOminusD

band frequency ranges The corresponding spectra of thetwo individual systems strongly differ Indeed in the caseof 2-furanacetic acid spectra the fine structure pattern ofeach band νOminusH and νOminusD is relatively simple Each bandconsists of a low number of well-separated spectral lines Inthe spectra of 2-furanacrylic acid each considered band iscomposed of a noticeably larger number of lines In additionthe temperature effect characterizing the bands is not thesame for the two compounds The results presented in thispaper for 2-furanacetic acid and 2-furanacrylic acid allow forthe following observations and conclusions

(1) The crystal IR spectral properties remain in a closerelation with the electronic structure of the two dif-ferent molecular systems The vibronic couplingmechanism involving the hydrogen bond protonsand the electrons on the π-electronic systems in themolecules determines the way in which the vibra-tional exciton coupling between the hydrogen bondsin the carboxylic acid dimers occurs

(2) The analyzed spectral properties of the two dif-ferent crystalline systems 2-furanacetic acid and 2-furanacrylic acid are in a good agreement with thevibrational exciton interaction mechanisms of thespectra generation for cyclic hydrogen bond dimer

(3) For 2-furanacetic acid dimers the exciton interac-tions involving the dimer hydrogen bonds of theSS type are only weakly temperature dependent Aweak ldquothrough-spacerdquo coupling in 2-furanacetic aciddimers of a van der Waals type is responsible for theSS-type coupling

(4) In the case of 2-furanacrylic acid dimers due to theirelectronic molecular structure the interhydrogenbond exciton coupling mechanism strongly changesits character along with the changes in temperatureStrong coupling in 2-furanacrylic acid dimers prefersa TH-type Davydov coupling widespread by the π-electrons At very low temperatures the TH-typeinteractions transferred in the (COOH)2 cycles via

electrons are dominating This mechanism becomesless privileged at higher temperature as annihilatedby the hydrogen-bond atom thermal vibrationalmotions

(5) Each individual mechanism that is the TH andSS generates its own spectrum characterized by itsunique individual intensity distribution pattern Aswe can see the νOminusH and νOminusD bands in the spectra of2-furanacrylic acid crystals exhibit more complex finestructure patterns since they are superposition of twodifferent spectra where each component spectrumis of a different origin Each component spectrumcontributing to the νOminusH and νOminusD bands formationwith its temperature-dependent statistical weightcorresponds with the different exciton interactionmechanism TH or SS acting in the cyclic hydrogenbond dimers in the lattice This explains the observeddifference in the temperature-induced evolution ofthe compared spectra

References

[1] C Pimentel and A L McClellan The Hydrogen Bond W HFreeman and Co San Francisco Calif USA 1960

[2] P Schuster G Zundel and C Sandorfy The Hydrogen Bondvol 1ndash3 North-Holland Amsterdam The Netherlands 1976

[3] G L Hofacker Y Marechal and M A Ratner ldquoThe dynamicalaspects of hydrogen bondsrdquo in In The Hydrogen Bond RecentDevelopments in Theory and Experiment W P Schuster GZundel and C Sandorfy Eds vol 1 p 295 North-HollandAmsterdam The Netherlands 1976

[4] P Schuster and W Mikenda Hydrogen Bond Research Monat-shefte fur Chemie Chemical Monthly vol 130 Springer NewYork NY USA 8th edition 1999

[5] D Hadzi Ed Theoretical Treatments of Hydrogen BondingWiley New York NY USA 1997

[6] A Witkowski ldquoInfrared spectra of the hydrogen-bonded car-boxylic acidsrdquo The Journal of Chemical Physics vol 47 no 9pp 3679ndash3680 1967

[7] Y Marechal and A Witkowski ldquoInfrared spectra of H-bondedsystemsrdquo The Journal of Chemical Physics vol 48 no 8 pp3697ndash3705 1968

[8] S F Fischer G L Hofacker and M A Ratner ldquoSpectralbehavior of hydrogen-bonded systems quasiparticle modelrdquoThe Journal of Chemical Physics vol 52 no 4 pp 1934ndash19471970

[9] O Henri-Rousseau and P Blaise ldquoThe infrared spectral den-sity of weak hydrogen bonds within the linear response the-oryrdquo Advances in Chemical Physics vol 103 pp 1ndash137 1998

[10] O Henri-Rousseau and P Blaise ldquoThe VXminusH line shapesof centrosymmetric cyclic dimers involving weak hydrogenbondsrdquo Advances in Chemical Physics vol 139 pp 245ndash4962008

[11] M J Wojcik ldquoTheoretical interpretation of infrared spectra ofthe ClndashH stretching vibration in the gaseous (Ch3)2Omiddot middot middotHClcomplexrdquo International Journal of Quantum Chemistry vol29 no 4 pp 855ndash865 1986

[12] J L Leviel and Y Marechal ldquoInfrared spectra of H-bondedsystems anharmonicity of the H-bond vibrations in cyclicdimersrdquo The Journal of Chemical Physics vol 54 no 3 pp1104ndash1107 1971

[13] J Bournay and Y Marechal ldquoDynamics of protons in hydro-gen-bonded systems propynoic and acrylic acid dimersrdquo TheJournal of Chemical Physics vol 55 no 3 pp 1230ndash12351971

[14] P Excoffon and Y Marechal ldquoInfrared spectra of H-bondedsystems saturated carboxylic acid dimersrdquo SpectrochimicaActa A vol 28 no 2 pp 269ndash283 1972

[15] M J Wojcik ldquoTheory of the infrared spectra of the hydrogenbond in molecular crystalsrdquo International Journal of QuantumChemistry vol 10 no 4 pp 747ndash760 1976

[16] HT Flakus and A Banczyk ldquoAbnormal distribution of pro-tons and deuterons between the hydrogen bonds in cyclic cen-trosymmetric dimers in partially deuterated samplesrdquo Journalof Molecular Structure vol 476 no 1ndash3 pp 57ndash68 1999

[17] H T Flakus ldquoVibronic model for HD isotopic self-organi-zation effects in centrosymmetric dimers of hydrogen bondsrdquoJournal of Molecular Structure vol 646 no 1ndash3 pp 15ndash232003

[18] H T Flakus and A Michta ldquoInvestigations of interhydrogenbond dynamical coupling effects in the polarized IR spectra ofacetanilide crystalsrdquo Journal of Physical Chemistry A vol 114no 4 pp 1688ndash1698 2010

[19] R W G Wyckoff Crystal Structures vol 5 Wiley New YorkNY USA 1972

[20] Z Berkovitch-Yellin and L Leiserowitz ldquoAtom-atom potentialanalysis of the packing characteristics of carboxylic acids Astudy based on experimental electron density distributionsrdquoJournal of the American Chemical Society vol 104 no 15 pp4052ndash4064 1982

[21] E B Wilson J C Decius and P C Cross Molecular Vibra-tions The Theory of Infrared and Raman Vibrational SpectraMcGraw- Hill New York NY USA 1955

[22] H T Flakus and A Miros ldquoInfrared spectra of the hydrogenbonded glutaric acid crystals polarization and temperatureeffectsrdquo Journal of Molecular Structure vol 484 no 1ndash3 pp103ndash115 1999

[23] H T Flakus and M Chelmecki ldquoInfrared spectra of thehydrogen bond in benzoic acid crystals temperature andpolarization effectsrdquo Spectrochimica Acta A vol 58 no 1 pp179ndash196 2002

[24] H T Flakus and M Jabłonska ldquoStudy of hydrogen bondpolarized IR spectra of cinnamic acid crystalsrdquo Journal ofMolecular Structure vol 707 no 1ndash3 pp 97ndash108 2004

[25] H T Flakus and M Chełmecki ldquoPolarization IR spectra ofthe hydrogen bond in phenylacetic acid crystals HD isotopiceffects-temperature and polarization effectsrdquo SpectrochimicaActa Part A vol 58 no 9 pp 1867ndash1880 2002

[26] H T Flakus and M Chełmecki ldquoPolarization IR spectraof hydrogen bonded 1-naphthoic acid and 2-naphthoic acidcrystals electronic effects in the spectrardquo Journal of MolecularStructure vol 659 no 1ndash3 pp 103ndash117 2003

[27] H T Flakus and M Chełmecki ldquoPolarization IR spectra of thehydrogen bond in 1-naphthylacetic and 2-naphthylacetic acidcrystals HD isotopic effects Temperature and polarizationeffectsrdquo Journal of Molecular Structure vol 705 no 1ndash3 pp81ndash89 2004

[28] S E Filippakis and G M J Schmidt ldquoTopochemistry PartXVI The crystal structure of trans-β-2-furylacrylic acidrdquo Jour-nal of the Chemical Society B pp 229ndash232 1967

[29] M Danish S Ali M Mazhar A Badshah and E R T Tiek-ing ldquoCrystal structure of 3-(2-Furyl)acrylic Acid C7H6O3rdquoZeitschrift fur Kristallographie vol 210 no 9 p 703 1995

[30] H T Flakus and A Tyl ldquoPolarized IR spectra of the hydrogenbond in acetic acid crystalsrdquo Chemical Physics vol 336 no 1pp 36ndash50 2007

[31] H T Flakus and B Stachowska ldquoA systematic study of polar-ized IR spectra of the hydrogen bond in formic acid crystalsrdquoChemical Physics vol 330 no 1-2 pp 231ndash244 2006

[32] A Tyl E Chełmecka M Jabłonska et al ldquoX-ray analysis at150 K synthesis and theoretical calculations of 1-naphthalene-acrylic acidrdquo Structural Chemistry vol 23 no 2 pp 325ndash3232012

[33] H T Flakus M Jabłonska and PG Jones ldquoStudy of polarizedIR spectra of the hydrogen bond system in crystals of styry-lacetic acidrdquo Spectrochimica Acta A vol 65 no 2 pp 481ndash4892006

[34] G Fisher Vibronic Coupling Acadamic Press London UK1984

[35] H T Flakus ldquoOn the vibrational transition selection rulesfor the centrosymmetric hydrogen-bonded dimeric systemsrdquoJournal of Molecular Structure C vol 187 pp 35ndash53 1989

[36] H T Flakus A Pyzik A Michta and J Kusz ldquolsquoReversalrsquo exci-ton coupling effect in the IR spectra of the hydrogen bondcyclic dimers polarized IR spectra of 3-hydroxy-4-methyl-2(3H)-thiazolethione crystalsrdquo Vibrational Spectroscopy vol44 no 1 pp 108ndash120 2007

[37] H T Flakus and A Tyl ldquoStrong vibrational exciton couplingeffects in polarized IR spectra of the hydrogen bond in 2-thiopyridone crystalsrdquo Vibrational Spectroscopy vol 47 no 2pp 129ndash138 2008

[38] H T Flakus A Tyl and A Maslankiewicz ldquoElectron-inducedphase transition in hydrogen-bonded solid-state 2-pyridonerdquoJournal of Physical Chemistry A vol 115 no 6 pp 1027ndash10392011

[39] H T Flakus A Miros and P G Jones ldquoInfluence of molecularelectronic properties on the IR spectra of dimeric hydrogenbond systems polarized spectra of 2-hydroxybenzothiazoleand 2-mercaptobenzothiazole crystalsrdquo Journal of MolecularStructure vol 604 no 1 pp 29ndash44 2002

[40] H T Flakus and A Machelska ldquoPolarization IR spectra ofhydrogen bonded pyrazole crystals self-organization effectsin proton and deuteron mixture systems Long-range HDisotopic effectsrdquo Spectrochimica Acta Part A vol 58 no 314pp 553ndash566 2002

[41] H T Flakus A Tyl and P G Jones ldquolsquoSelf-organizationrsquo pro-cesses in proton and deuteron mixtures in open-chain hydro-gen bond systems Polarization IR spectra of 4-mercaptopyri-dine crystalsrdquo Spectrochimica Acta A vol 58 no 2 pp 299ndash310 2002

[42] H T Flakus W Smiszek-Lindert and K Stadnicka ldquoStrongvibronic coupling effects in polarized IR spectra of the hydro-gen bond in N-methylthioacetamide crystalsrdquo Chemical Phys-ics vol 335 no 2-3 pp 221ndash232 2007

[43] C A Davydov Teorya Molekularnykh Ekscitonov Nauka Mos-cow Russia 1968

[44] R L Hochstrasser Molecular Aspects of Symmetry W ABenjamin Inc New York NY USA 1966

[45] L A Nafie ldquoAdiabatic molecular properties beyond the Born-Oppenheimer approximation Complete adiabatic wave func-tions and vibrationally induced electronic current densityrdquoThe Journal of Chemical Physics vol 79 no 10 pp 4950ndash49571983

[46] HT Flakus ldquoThe effect of strong coupling between vibrationsin hydrogen bonds on the polarized spectra of the mer-captobenzothiazole crystal an ldquoanomalousrdquo isotopic effectrdquoChemical Physics vol 62 no 1-2 pp 103ndash114 1981

[47] P Blaise M J Wojcik and O Henri-Rousseau ldquoTheoreticalInterpretation of the Lineshape of the Gaseous Acetic AcidDimerrdquo Journal of Chemical Physics vol 122 Article ID064306 2005

[48] N Rekik H Ghalla H T Flakus M Jablonska P Blaise andB Oujia ldquoPolarized infrared spectra of the H(D) bond in 2-thiophenic acid crystals a spectroscopic and computationalstudyrdquo ChemPhysChem vol 10 no 17 pp 3021ndash3033 2009

[49] R Najeh G Houcine H T Flakus M Jablonska and OBrahim ldquoExperimental and theoretical study of the polarizedinfrared spectra of the hydrogen bond in 3-thiophenic acidcrystalrdquo Journal of Computational Chemistry vol 31 no 3 pp463ndash475 2010

[50] H T Flakus N Rekik and A Jarczyk ldquoPolarized IR spectra ofthe hydrogen bond in 2-thiopheneacetic acid and 2-thiophe-neacrylic acid crystals HD isotopic and temperature effectsrdquoThe Journal of Physical Chemistry A vol 116 no 9 pp 2117ndash2130 2012

[51] L J Bellamy and P E Rogasch ldquoProton transfer in hydrogenbonded systemsrdquo Proceedings of the Royal Society A vol 257pp 98ndash108 1960

[52] H T Flakus A Michta M Nowak and J Kusz ldquoEffects ofdynamical couplings in IR spectra of the hydrogen bond inN-phenylacrylamide crystalsrdquo Journal of Physical Chemistry Avol 115 no 17 pp 4202ndash4213 2011

An Analytic Analysis of the Diffusive-Heat-FlowEquation for Different Magnetic Field Profiles for a SingleMagnetic Nanoparticle

Brenda Dana1 and Israel Gannot2

1 Department of Electrical Engineering Faculty of Engineering Tel-Aviv University 69978 Tel-Aviv Israel2 Department of Biomedical Engineering Faculty of Engineering Tel-Aviv University 69978 Tel-Aviv Israel

Correspondence should be addressed to Israel Gannot gannotengtauacil

Academic Editor Yuval Garini

This study analytically analyzes the changes in the temperature profile of a homogenous and isotropic medium having the samethermal parameters as a muscular tissue due to the heat released by a single magnetic nanoparticle (MNP) to its surroundingswhen subject to different magnetic field profiles Exploring the temperature behavior of a heated MNP can be very useful predictingthe temperature increment of it immediate surroundings Therefore selecting the most effective magnetic field profile (MFP) inorder to reach the necessary temperature for cancer therapy is crucial in hyperthermia treatments In order to find the temperatureprofile caused by the heated MNP immobilized inside a homogenous medium the 3D diffusive-heat-flow equation (DHFE) wassolved for three different types of boundary conditions (BCs) The change in the BC is caused by the different MF profiles (MFP)which are analyzed in this article The analytic expressions are suitable for describing the transient temperature response of themedium for each case The analysis showed that the maximum temperature increment surrounding the MNP can be achieved byradiating periodic magnetic pulses (PMPs) on it making this MFP more effective than the conventional cosine profile

1 Introduction

Magnetic Hyperthermia (MH) is one of many approachescurrently being tested for cancer therapy [1ndash3] The goal ofthis approach is to specifically heat the regions containing thecancerous cells by means of the magnetic losses caused by thephysical properties of the magnetic nanoparticles (MNPs)when being exposed to an external magnetic field (MF)

The MNPs that are often used in MH are usually madeof ferromagnetic or ferrimagnetic materials which stronglyreact to the externally applied MF [4] This magnetic reactionis converted by the two dominant relaxation mechanismsthe Neel mechanism and Brownian mechanism into powerdissipation or heat [1 4]

The eddy currents losses contribution may be neglecteddue to the low electrical conductivity that characterizes ferro-or ferrimagnetic materials and due to the small particleradius [5ndash9]

Fannin et al [10] pointed out that for small enoughparticles the anisotropy energy barrier Ea may becomeso small that thermal energy fluctuations can overcomeit and spontaneous reverse the magnetization of a particlefrom one easy direction to the other even in the absenceof an applied field The time it takes for a spontaneouslyfluctuation to reverse the magnetization after overcoming theenergy barrier is characterized by a time constant referred toas the Neel relaxation time or τN The probability of such atransition is proportional to exp(σ) where σ is the ratio ofanisotropy energy to thermal energy or (EakBT) [11]

The other distinct mechanism by which the magneti-zation of MNPs may relax after an applied field has beenremoved is the physical rotational Brownian motion of theparticle immobilized inside a medium When a magneticfield is applied to MNPs they rotate and progressively alignwith the magnetic field due to the torque generated by theinteraction of the magnetic field with the magnetization [12]

The time associated for an MNP to align with a small externalmagnetic field is given by the Brownian relaxation time τB[13]

Because these relaxation mechanisms happen simultane-ously they both contribute to the total magnetization andthe heat losses and their total influences can be express byan effective relaxation time τe which is a combination of τNand τB [14ndash16]

The two relevant mechanisms to change the magnetiza-tion of magnetic particles in an external field are given inFigure 1

Moreover our interests in MNPs as heat sources derivefrom the fact that they are vastly used as MRI agents [17]and their controllable size ranging from few nanometers totens of nanometers [18] This means that the MNP size issmaller than or as same as that of a protein (5ndash50 nm) a virus(20ndash450 nm) or a gene (2 nm wide and 10ndash100 nm long)[11] which enables them to penetrate through the leakypathological vasculature into the tumor interstitial easilyreaching any cell of interest in the body including cancerouscells [19]

In addition the MNPs can be attached to a specifictype of cancerous cells causing a controllable elevation oftemperature in them with almost no effects on healthycells [20 21] By selectively heating the cancerous cells toa temperature ranging from 42 to 46C one can damagethe tumors without causing vast harm to the healthysurrounding tissue [17 19 22]

Furthermore in order to ensure that the treatment isbiologically noninvasive and thermally tolerated for extendedperiod of time an experimentally measured tolerable limitof induced heating by an alternative MF was conducteddefining a limit to the product of the MF strength (H) andthe frequency ( f ) of the MF (eg H middot f le 485 middot 108 Ammiddots[23] or a less rigid criterion H middot f le 5 middot 109 Ammiddots [24])

Due to the MNPs submicron length size the conven-tional approach to heat conduction problems using macro-scopic empirical laws such as Fourierrsquos law or Joulersquos law ofheat generation requires justification and even breakdownwhen the length scale of the system is comparable to thecarrier mean free path or when the time scale of the physicalprocess is smaller than the relaxation time of the heat carriers[25] In this case transport of heat carriers must be treatedusing the Boltzmann transport equation as Chen et alpointed out [26]

Chen [27] suggested that heat is transported by carrierscomprising of electrons phonons and photons In dielectricmaterials the heat conduction is dominated by phononsin pure metals predominantly by electrons and in impuremetals or alloys by both phonons and electrons [27 28]Therefore the mean free path of the heat carriers for anMNP with a Fe core is approximately 08 nm [29 30] andfor a biological tissue 05 nm [26 31 32] allowing theconventional approach to be used for particles having aradius bigger than 10 nm

Consequently the temperature gradient caused by therelease of the magnetic energy which an MNP absorbs toits immediate surroundings can be found analytically whenapplying Fourier transforms (FTs) to the DHFE [33] as

Figure 1 (a) Neel rotation of magnetization inside a fixed magneticparticle due to the spontaneously reversing the magnetization fromone easy direction to the other (the particle does not rotate)(b) Brownian rotation of an MNP due to the rotation of thetorque generated by the interaction of the magnetic field with themagnetization (the particle rotates as a whole) [17]

Shih et al [34] and Yuan et al [35] suggested Using thistechnique Liu and Xu [36] analyzed the influences that asinusoidal heat flux source placed on the skin surface haveon the temperature inside it and Tjahjono [37] analyticallyanalyzed the heating temperature of a slab embedded withgold NPs due to a constant magnetic flux

By analytically solving the DHFE for different boundaryconditions one can easily describe the dependence of thesolution on each parameter composing it such as the radiusof the MNP the frequency of the MF and the material prop-erties [38] This allows us to optimize the solution for betterperformances reaching the highest temperature elevationunder specific constrains for example the radius of the MNPMoreover when exploring the solution analytically otherparameters and their influence may be observed more clearlywhich are usually neglected or not explored (eg the MFPand its effects on the temperature gradient)

Until recently the MFP was poorly analyzed in context ofhyperthermia treatments and how it influences the tempera-ture distribution concerning biological materials and tissuessurrounding MNPs Previous work focused on exploring theinfluences of different magnetic profiles on biological tissuesThese studies were mostly experimental and did not focuson the MNPs contribution to the temperature change whenexposed to different types of magnetic field profiles [39ndash43]

Recently a numerical simulation model based on theLandau-Lifshitz-Gilbert equation was created for simulatingMNPs ensembles when exposed to an incident square wave[44] as opposed to the usual sine wave This work showedan increase in the normalized heat released by MNPs byat least 50 as well as a more constant heating efficiencyover the spectrum of particle anisotropies due to the infinitenumber of harmonics contained in an ideal square wavewith the possibility of much greater improvement dependingon the magnetic anisotropies volumes and angles to theincident radiation However Morgan and Victora [44] didnot elaborate on the temperature dependencies on spaceand time near the MNP surface but mostly focused on

25An Analytic Analysis of the Diffusive-Heat-Flow Equation for Different Magnetic Field Profiles for a Single Magnetic Nanoparticle

the dependencies between the angle of the incident wave rel-ative to the anisotropy axis of the MNPs and the magnitudeof the normalized output power released from them

Therefore the primary aim of this paper is to explorethe transient analysis of the changes in temperature (fromthe steady state temperature Tb = 31015K) as a functionof the external MFP applied to a single MNP By doing soone can select the most efficient MFP that may improve theefficiency of MH treatments allowing the MF strength andthe frequency reductions in order to meet the requirementH middot f le 485 middot 108 Ammiddots [23]

The aims of this paper are as follows

(i) To construct a theoretical model of the magneticlosses for the three different MF profiles studied inthis article as follows

(1) Case 1mdasha cosine profile [18 37](2) Case 2mdasha PMP [45 46](3) Case 3mdasha discontinuous cosine profile

(ii) To explore the maximal temperature elevation andthe rate of the temperature change near the MNPrsquossurface and into the tissue-like surrounding it foreach of the above cases

(iii) To investigate who the core radius influences themaximal rate of the temperature change and themaximal temperature value in order to find theoptimal core radius that should be used for each ofthe above cases [5 47]

(iv) To study the effective confining heat depth (ECHD)symbolized as δ (see Figure 2) where the tempera-ture elevation has a significant influence for each ofthe above cases

2 Methods

In this study we analytically model the transient temperaturefield (TTF) produced by a single MNP inside a homogenousand uniform medium having the same thermal parametersas a cancerous muscle cell The analysis for each of the threeMFPs mentioned in Section 1 is presented after solving theDHFE inside the medium surrounding the MNP with theproper BC corresponding to its specific MFP

In order to simplify the solution of DHFE that gave us theTTF and the temperature rate change due to the magneticlosses some assumptions were made

(a) The properties of the surrounding medium areconstant and homogeneous having the same thermalproperties as the macroscopic-scale muscular tissue[48]

(b) The temperature on the surface of the MNP isuniform

(c) There is a negligible emission and evaporation

(d) There are no ldquothermally significantrdquo blood vesselsnear the zone of interest therefore the perfusion isnegligible

(e) The metabolic-heat generation is neglected

Medium

Control volume

Penetrationdepthregion

ρmd cmd

Figure 2 The control volume where the conductive analysis ispreformed qprimeprimes is the constant heat flux released from the MNPafter absorbing the magnetic energy and δ is the thickness of thepenetration region [37]

21 The Thermodynamic Analysis The TTF originating fromthe surface of a single MNP can be found after solving the 3DDHFE in the homogenous medium surrounding it [39] Thegeneral DHFE can be written in spherical coordinates (dueto the problemrsquos symmetry) as follows

kmnabla2Tm(r t) = ρmcmpartTm(r t)

partt (1)

where ρm (kg mminus3) is the mass density cm (J kgminus1Cminus1) thespecific heat and km (Wmminus1Cminus1) the thermal conductivityof the phantom tissue

This equation was also used by Keblinski et al [38] andGovorov et al [49] for solving nanoscale heat problems

The general BC for this heat problem is given as follows

minuskm middot nablaTm (r t)|r=a = qprimeprimes (t) (2)

where a is the radius of the MNP in meters and qprimeprimes (t)(Wmminus2) is the heat flux

The DHFE is valid if the mean free path of the heat carrierphonon or electron is smaller than the characteristic featuresize as mentioned in Section 1 For amorphous solids andliquids due to lack of crystalline the mean free path is veryshort and of the order of atomic distances Consequently theheat flow in cells can be well described by the diffusive heatequation even when nanoscopic length scales are involved[31]

Based on the above considerations we evaluate thetemperature field arising from continuous heating of a singleparticle by solving the heat equation (1) in the region outsidethe solid sphere surrounding the MNP where there are noheat sources using a constant heat-flux-boundary conditionat the particle surface caused by the magnetic losses insidethe MNP The constant heat flux from the MNPrsquos surfacebecomes heat input to the medium which is then storedwithin the volumetric penetration depth region as shown inFigure 2

After solving (1) and (2) (see the detailed formulations inAppendix A (A1)ndash(A12)) the temperature elevation inside

the medium surrounding the MNP can be expressed using(A12) as follows

ΔTm(r t) = θ(R t)kmr

radicαm = φ(R t)lowastqprimeprimes (t)

radicαm (3)

where lowast symbolizes the convolution between two functionsR = radic1αmr and αm = kmρmcm

In order to analytically calculate (3) the general expres-sion of qprimeprimes (t) must be found for each case which depends onH(t) (Amminus1) the magnetic field and on M(t) (Amminus1) themagnetization

When a linear and isotropy material is assumed therelation between M(t) and H(t) in the frequency domain(using FTs) may be described by the magnetic susceptibility[5]

M(ω) =intdtprimeχ(tprime)eminusiωt

primeintdtH(t)eminusiωt = χ(ω)H(ω) (4)

The magnetic susceptibility χ(ω) in the frequency domaincan be expressed as [4]

χ(ω) = χ0

1 + iωτe= φμ0Ms

2vp3kBT

11 + iωτe

where χ0 is the static susceptibility τe = τNeel||τBrown isthe effective relaxation time given by Fannin [14] φ is thevolume fraction solid [4] vp the particle volume μ0 thevacuum permeability kB is the Boltzmann constant and Ms

is the magnetic saturationMoreover in order to calculate the total heat generated

by a single MNP caused by the conversion of the absorbedmagnetic energy to heat inside a linear ferromagneticmedium we must introduce Poyntingrsquos theorem for smallelectric fields and neglecting ohmic losses [5 6] as follows

nabla middot Sprimeprimeout(t)

= minusintdωintdωprimeH(ω) middotH(ω)ωμ0 Im

(μ(ω)

)ei(ωminusω

prime)t

minus partU(t)partt

= minusPLoss(t)minus partU(t)partt

where Sprimeprimeout(t) represents the energy flowing out through theboundary surfaces of the volume per unit time H(ω) is theconjugate of H(ω) μ0 = 4π10minus7 (VsAm) is the vacuumpermeability μ(ω) = μr(ω) minus iμim(ω) = μ0(1 + χ(ω))is the complex magnetic permeability [5 6] Im( ) is theimaginary part of μ(ω) partU(t)partt is the time rate change ofthe effective electromagnetic energy density given by (7) andPLoss(t) represents the conversion of the magnetic energy intoheat not counting conductive losses It is worth mentioningthat only the imaginary part of the complex permeability is

causing heat losses and partU(t)partt can be found using [5 6]as follows

H middot partBpartt=intdωintdωprimeH(ω) middotH(ω)ωμ0 Imμ(ω)ei(ωminusω

prime)t

+μ0part

2partt

intdωintdωprimeH(ω) middotH(ω)

[ωμ(ω)

times ei(ωminusωprime)t

= PLoss(t) +partU(t)partt

where μ(ω) is the conjugate of μ(ω)Next the explicit analytic expressions for the temper-

ature gradient profile from the equilibrium tempertaureΔTm(r t) and partTm(r t)partt are deduced for three different BCderived from the three MFPs mentioned earlier in Section 1

22 The Analytic Expressions of the TTF for

Three Different MFPs

Case 1 (a cosine profile) The magnetic field has a cosineprofile so

H(t) = A cos(ω0t) (8)

Taking the inverse FT of (A19) deduced in Appendix A using(A13)ndash(A19) one can find that θ(R t) can be written in thiscase as

θ(R t) = aμ0A2ω0

6ω0χ0τ

1 + (ω0τ)2

middot(

aR0ei2ω0t

2(R0radici2ω0 + 1

) exp(minusradici2ω0R

+aR0eminusi2ω0t

2(R0radicminus2ω0i + 1

) exp(minusradicminusi2ω0R

where θ(R t) is a function in the complex domain thereforethe temperature profile has a magnitude and phase as oftenoccurs in many problems of physics or engineering such astheory of heat conduction particularly when nonsteady heatconduction is concerned [50 51] Moreover (9) is related tothe TTF by (3)

Sometimes the derivative of the temperature profile orthe rate of the change in the temperature surrounding theMNP is taken in consideration in order to verify that thetreatment is safe for inducing controlled MH [47 52] ForCase 1 this equals

partθ(R t)partt

= iω0aμ0A2ω0

6ω0χ0τ

1 + (ω0τ)2

middot(

aR0ei2ω0t(R0radici2ω0 + 1

minus aR0eminusi2ω0t(R0radicminus2ω0i + 1

Case 2 (a PMP) The magnetic field has a rectangular pulseshape profile with a period of Ts = 2πω0 and a pulse widthof Δ so

H(t) = 2A 0 le t le Δ H(t) = 0 Δ le t le Ts(11)

The amplitude of the pulse wave was chosen to be twice theamplitude of the cosine single in order to maintain the samepeak-to-peak value for this case and the previous one Forthis case the temperature elevation as a function of timecan be expressed using the inverse FT of (A25) found inAppendix A that was deduced using (A20)ndash(A25) to receivethe following

θ(R t)

4μ0A2ω0

sumsum sin(mπΔTs)m

sin(nπΔ

)nω0χ0τ

1+(nω0)2τ2

middot(

aR0ei(n+m)ω0t

R0radici(n +m)ω0 + 1

exp(minusradici(n +m)ω0R

+aR0eminusi(n+m)ω0t

R0radicminusi(n +m)ω0 + 1

exp(minusradicminusi(n +m)ω0R

+aR0ei(mminusn)ω0t

R0radici(mminus n)ω0 + 1

exp(minusradici(mminus n)ω0R

+aR0eminusi(mminusn)ω0t

R0radicminusi(mminus n)ω0 + 1

exp(minusradicminusi(mminus n)ω0R

34ω0μ0A

2 middot Δ

sumsin(nπΔ

)nω0χ0τ

1 + (nω0)2τ2

times(

aR0einω0t

R0radicinω0 + 1

exp(minusradicinω0R

+aR0eminusinω0t

R0radicminusinω0 + 1

exp(minusradicminusinω0R

Again (12) is related to the TTF by (3)

As for Case 1 we can calculate the rate of the changein the temperature surrounding the MNP and receive thefollowing

partθ(R t)partt

= iω20a

34μ0A2

times sin(nπΔ

)nω0χ0τ

1 + (nω0)2τ2

middot(

(n +m)aR0ei(n+m)ω0t

times exp(minusradici(n +m)ω0R

minus (n +m)aR0eminusi(n+m)ω0t

times exp(minusradicminusi(n +m)ω0R

+ (mminus n)aR0ei(mminusn)ω0t

times exp(minusradici(mminus n)ω0R

minus (mminus n)aR0eminusi(mminusn)ω0t

times exp(minusradicminusi(mminus n)ω0R

+ iω20a

34μ0A

sumsin(nπΔ

middot n2ω0χ0τ

1 + (nω0)2τ2

times(

aR0einω0t

R0radicinω0 + 1

minus aR0eminusinω0t

Case 3 (a discontinuous cosine profile) The magnetic fieldhas a periodic discontinuous cosine profile with a timeconstant of Ts = 2πω1 and a pulse width of Δ so

H(t) = A cos(ω0t) 0 le t le Δ

H(t) = 0 Δ le t le Ts ω0 =ω1(14)

For this third case the temperature elevation as a function oftime can be expressed using the inverse FT of (A35) found inAppendix A that was deduced using (A30)ndash(A35) to receive

θ(R t)

timessumsum[( sin((Δ2)(ω0 +mω1))

ω0 +mω1

+sin((Δ2)(ω0minusmω1))

ω0minusmω1

times(

sin((Δ2)(ω0 +nω1))ω0 +nω1

+sin((Δ2)(ω0minusnω1))

ω0minusnω1

middot (nω1)2χ0τ

1 + (nω1)2τ2

times(

ei(n+m)ω1t

+eminusi(n+m)ω1t

+ei(mminusn)ω1t

+eminusi(mminusn)ω1t

As for Cases 1 and 2 we can calculate the rate of thechange in the temperature surrounding the MNP to receivethe following

partθ(R t)partt

= iω1a

ω0 +mω1

+sin((Δ2)(ω0 minusmω1))

ω0 minusmω1

times(

sin((Δ2)(ω0 + nω1))ω0 + nω1

+sin((Δ2)(ω0 minus nω1))

ω0 minus nω1

middot (nω1)2χ0τ

1 + (nω1)2τ2

times(

For Cases 2 and 3 there is a limitation regarding the MFSand the frequency in order for the MH treatment to be safe(see (A30) and (A39)) Moreover for frequencies lower than10 MHz there is essentially no attenuation of the MFS withinmuscle-equivalent materials limiting the maximal harmonicfrequency to 10 MHz [16]

In conclusion (9)ndash(16) can be used to predict the TTPand the special temperature profile for a single-MNP subjectto three different magnetic field profiles and using the sameequations we can also explore the influence that the coreradius has on the temperature profile estimating the ECHDfor each case

23 The Simulations Parameters The mathematical expres-sions of the TTP were simulated using MATLAB andCOMSOL (COMSOL results can be seen in Appendix B) fora single MNP immobilized inside a uniform and isotropicphantom medium having the same biological thermal prop-erties as a muscular tissue [48] and are summarized inTable 1 These assumptions were made in order to simplifythe theoretical calculations

The thermal parameters are considered to be constantwith temperature and space as will be latter proven More-over the magnetic parameters of the MNP were measured atTb = 31015K based on the findings of Fannin [14] and aresummed up in Table 2

The external magnetic field strength (MFS) for all threecases was chosen as 88 kAmminus1 and the MF frequency as f0 =400 KHz These values are based on previous works made byKettering et al [52] Hergt et al [53] and Hilger et al [54]

For all three profiles mentioned in this section thesimulations were plotted for 0 le r minus a le 10 nm and 0 let le 5μs where r is the distance from the center of the MNPand a its radius The upper value for distance simulation waschosen accordantly to the thickness of the cell membranethatis about 5ndash10 nm [55ndash57] and damaging it can cause thedestruction of the cell [58] The upper time value was chosenso several cycles of the magnetic field could be simulated andplotted

For all the simulations the volume fraction solid wasdefined as φ = 0032 This value is been justified in Section 4

In Section 3 as already mentioned in Section 1 themaximal temperature elevation and the temperature changerate near the MNPrsquos surface and into the tissue surroundingit are simulated Moreover the influence the core radius hason the maximal temperature change rate and on the maximaltemperature elevation was also explored in order to find the

Table 1 Thermal tissue properties for the phantom muscle cell at atemperature of 31015 K [48]

cm (J kgminus1Cminus1) ρm (kg mminus3) km (W mminus1Cminus1)

3500 1047 0518

Table 2 The magnetic parameters of the MNP [14]

a (nm) Vhyd (m3) Ku (kJmminus3) Ms (kAmminus1) γ (kμsminus1Aminus1m) α

10 334 middot 10minus26 96 300 202 01

optimal value that must be used for each case as suggested byRosensweig [4] and Kappiyoor et al [47] Furthermore theECHD that is defined by the point the temperature reacheseminus1 of the maximal value has also been explored for eachcase defining the confining heat region and can be comparedwith the thickness of a cell membrane which varies between3ndash10 nm [53ndash56]

3 Results

The mathematical expressions of the TTP were simulated inthis section using MATLAB for a single MNP immobilizedinside a uniform and isotropic phantom medium havingthe same biological thermal properties as a muscular tissue(Table 1) Moreover the MNPrsquos magnetic parameters aresummarized in Table 2

For Case 1 the mathematical expression of the temporaland spatial temperature increment (9) is presented inFigure 3 for Ts = 25μs

It can be seen from Figure 3 that the temperature changesperiodically with a time period of 125 μs that is equivalentto a frequency oscillation of 800 kHz which is twice thefrequency of the external applied MF as predicted by (9)This can be explained by the multiplication of the magneticfield and the magnetic induction both being a function of f0or ω0

Moreover the temperature increment reached its max-imum value after 0 μs reaching ΔTmax = 21 nK on thesurface of the MNP As expected the hottest spots are on thesurface of the MNP and as the point of view gets further fromthe surface the temperature declines as (9) predicted Thisvalue causes only a low-temperature gradient in the thermalproperties of the surrounding medium therefore the thermalparameters of the phantom cell can be considered constantsas assumed

According to Figure 3 the temperature profile has aldquoDCrdquo level that can be found from calculating the firstterm of (9) making the temperature increment to be alwayshigher than the initial temperature as expected because themagnetic losses inside the MNP irradiate heat to the mediumsurrounding it at all times [59 60]

Furthermore at a distance of 12 nm apart from theMNPrsquos surface the temperature maximal value equals 08 nKthat is equivalent to eminus1 of the absolute maximal valuedefining the ECHD or δ = 12 nm

In order to have a unique quantity to be compared witheach case and does not depend on time we averaged the TTP

00 05 1 15 2 25 3 35 4 45 5

Time (μs)

r = 0 nmr = 2 nmr = 4 nm

r = 6 nmr = 8 nmr = 10 nm

Figure 3 The temperature profile for a cosine MFP plotted as afunction of time and as a function of the observation point locatedat a distance r from the surface of the MNP r ranging from 0 nmto 10 nm for a core radius of 10 nm

over one cycle In Case 1 the averaged value over one timeperiod equals 055 nK

Next we explored the maximal temperature rise rate as afunction of the core radius using (9) and (10) receiving thedata in Figure 4

As can be seen from Figure 4 the maximal temperaturerate change equals 0011Ksminus1 and the maximal temperaturerise equals 47 nK both received for a core radius of 93 nmThe temperature rate rise and the maximal temperature areconsiderably small due to the relaxation time that dependson the volume of the particle making this MFP to be safe touse for MH treatments [47] For radii larger or smaller than93 nm the magnetic heat dissipation start to decrease as themagnetic relaxation time gets bigger or smaller respectivelyreducing the denominator or numerator in (9) and (10)

Equations (9) and (10) enable us to understand that thechanges in the temperature depend on many parameters suchas the magnetic field strength the magnetic field frequencythe magnetic properties of the material and the core radiusConsequently in order to optimize the heat losses we mustselect the most effective radius for a specific type of MNP

For Case 2 the mathematical expression of the temporaland spatial temperature increment are plotted in Figure 5 forthe summation of 25 indexes (not to exceed 10 MHz [16])and Δ = 02Ts where Ts = 25μs

For convenience Figure 5 describes the temperatureprofile for the first two cycles as given by (12) This equationshows that the characteristic behavior of the temperaturerepeats itself every Ts = 25μs that is the cycle of themagnetic field therefore one can limit the study to only afinal number of cycles

As can be seen from Figure 5 the temperature buildsup very fast due to the steep elevation of the magnetic fieldcaused by the Heaviside-shaped MP and reaches a maximalvalue of ΔTmax = 88 nK on the MNPrsquos surface after 045 μsThen the temperature begins to drop after 01 μs from the

05 6 7 8 9 10 11 12 13 14 15

partTm

axpartt

partTmaxpartt (Ks)

The core radius (nm)

ΔTmax (nK)

Figure 4 The maximal temperature rise rate and the maximaltemperature for a cosine MFP plotted as a function of the coreradius a ranging from 5 nm to 15 nm where the observation pointsare on the surface of the MNP

time the MF was turned on reaching a minimal value of6 nK near the surface of the MNP From that point onthe temperature profile temporal behavior is defined by thesummation of the total numbers of harmonics composingthe MF until the MF is turned off again as can be found from(12) Furthermore the temperature reaches its maximumvalue close to the surface of the MNP and decreases withdistance reaching a maximal value of 3 nK 12 nm apartfrom the MNPrsquos surface

For this case the maximal value is 4 times higher thanthe one received in Case 1 making it a preferable MFP to beused in MH as Morgan and Victora suggested [44]

Again the thermal parameters can be considered con-stant and not dependent on temperature near the MNPrsquossurface because the temperature rise is less than 1K

From Figure 5 the ECHD can be found as δ = 12 nmwhich is the same as the value received in Case 1 meaningthat the temperature decreases as fast as the cosine case andis confined to a specific area near the MNPrsquos surface

Moreover in order to have a unique quantity to becompared in each case that does not depend on time weaveraged the total temperature rise over one cycle In this casethe averaged temperature elevation was 13 nK after beennormalized to the time period This value is about 24 timeshigher than the value received in Case 1 making this MFP abetter candidate for MH treatments

Next we explored the maximal temperature rise rate as afunction of the core radius For Case 2 we can use (11) and(12) receiving the data in Figure 6

As seen from Figure 6 the absolute maximal temperatureelevation equals 0032 μK received for a core radius of83 nm and the maximal temperature derivative 101 Ksminus1

is received for a core radius of 82 nm Because this MFPproduces temperature changes that are too rapid to be safefor inducing MH [47] the radius that we chosen for asafer treatment is in consistence with Case 1 and equals10 nm Consequently the NP size plays an important role in

0 05 1 15 2 25 3 35 4 45 5

Time (μs)

r = 0 nmr = 2 nmr = 4 nm

r = 6 nmr = 8 nmr = 10 nm

minus1

Figure 5 The temperature rise for a periodic pulse-shaped MFPhaving a pulse width of 02Ts plotted as a function of theobservation point located at a distance r from the surface of theMNP r ranging from 0 nm to 10 nm for a core radius of 10 nm andthe number of indexed summed is 25

5 6 7 8 9 10 11 12

partTm

axpartt

partTmaxpartt (Ks)

ΔTmax (μK)

Figure 6 The absolute maximal temperature rise rate and themaximal temperature for a periodic pulse-shaped MFP having apulsed width of 02Ts plotted as a function of the core radiusa ranging from 5 nm to 15 nm the observation point are on thesurface of the MNP and the number of indexed summed is 25

determining the amount of heating that an MFH treatmentcan provide as Kappiyoor et al [47] already mentioned

Again the maximal temperature rate rise and the maxi-mal temperature are considerably small due to the relaxationtime that depends on the volume of the particle For radiilarger or smaller than 84 nm the magnetic heat dissipationstarts to decrease as the magnetic relaxation time gets biggeror smaller respectively due to its affect on the relaxationtime reducing the denominator or numerator in (11) and(12)

By comparing Case 2 to Case 1 we see that for the sameMNP radius (10 nm) having the same magnetic materialproperties (given by Table 2) the maximal temperature risereceived is about 4 times higher for Case 2 than in Case 1 and

r = 0 nmr = 2 nmr = 4 nm

r = 6 nmr = 8 nmr = 10 nm

Time (μs)

0 5 10 15 20 25 30

Figure 7 The temperature profile for a periodic discontinuouscosine MFP having a pulse width of 02Ts plotted as a function oftime and as a function of the observation point located at a distancer from the surface of the MNP r ranging from 0 nm to 10 nm for acore radius of 10 nm and the number of indexed summed is 25

the maximal temperature derivative for this case is 40 timeshigher than Case 1 making the periodic pulse-shaped MFPa better magnetic field source for MH treatments

For Case 3 the analytic expression for the TTP can beplotted for Δ = 02Ts and Ts = 15μs and are shown inFigure 7 for the summation of 25 indexes (not to exceed10 MHz [16]) The cosine MF time period that multipliesthe Heaviside function equals 25 μs and is equivalent to afrequency of 400 KHz

As Figure 7 shows the changes in the temperature profileresult from the MNP reaction to two different MFPs thecosine profile and the periodic rectangular pulse profile Thelast is responsible for switching on and off the MF

The influence that the periodic rectangular-pulse-shapedMF has on the temperature gradient can be seen by the steeptemperature elevation at the beginning and at the end ofevery cycle caused by the derivative of Heaviside functioncomposing the magnetic flux density B(t) and the influencethat the cosine MFP has on the temperature gradient can beseen as the cosine ldquoripplerdquo that is added This ldquoripplerdquo has3 peaks that are separated 125 μs apart which is twice thecosine MF frequency received in Case 1 On the MNP surfacethe maximal temperature gradient reaches ΔTmax = 23 nKafter 2 ns from the time the MF was turned on and repeatsitself every 15 μs which is equivalent to the time period ofthe signal This value is higher than the value received for thecosine-shaped MF but lower than the one received in Case2 However after the highest peak the maximal value of thecosine ldquoripplerdquo reaches the same one as in the cosine case or21 nK as expected

For this case the ECHD equals δ = 12 nm that is thesame as for the other two cases where the temperature changereaches a value of 08 nK After 02Ts the temperatureelevation becomes insignificant as the MF is turned off

5 6 7 8 9 10 11 12 13 14 15

partTm

axpartt

partTmaxpartt (Ks)ΔTmax (nK)

Figure 8 The absolute maximal temperature rise rate and themaximal temperature rise for a periodic discontinuous cosine MFPhaving a pulse width a pulsed width of 02Ts plotted as a functionof the core radius a ranging from 5 nm to 15 nm the observationpoint are on the surface of the MNP and the number of indexedsummed is 25

Again in order to have a unique quantity to be comparedin each case that does not depend on time we can average thetotal temperature rise over one cycle In this case the averagetemperature elevation equals 0235 nK after we normalizeit to the time period This value is about 25 times lowerthan the value received in Case 1 making this MFP a lesspreferable candidate for MH treatments

Next we explored the maximal temperature rise rate as afunction of the core radius For Case 3 we used (15) and (16)receiving the data in Figure 8

As seen from Figure 8 the absolute maximal temperaturerate elevation equals 54 nK received for a core radius of92 nm The maximal temperature derivative 0024Ksminus1 isreceived for a core radius of 91 nm Again the maximaltemperature rate rise and the maximal temperature areconsiderably small due to the relaxation time that dependson the volume of the particle For radii larger or smaller than92 nm the magnetic heat dissipation starts to decrease as themagnetic relaxation time gets bigger or smaller respectivelyreducing the denominator or the numerator in (15) and (16)

By comparing Case 3 to Case 1 for a core radius of10 nm and the same magnetic material properties (given byTable 2) the maximal temperature rise received for Case 3is about two times higher than Case 1 and the maximaltemperature derivative for this case is 22 times higher thanCase 1 However due to the long period for which the MFis turned off and consequently the lower heat released fromthe MNP over one cycle this MFP is less preferable than thecosine MFP for MH treatments

In order to make it easier to understand the differencesbetween the three cases analyzed in this paper Table 3 isadded that summarizes the most significant parameters

Moreover a summarizing figure Figure 9 describing thetemperature rise as a function of time is also added for aparticle with a core radius of 10 nm when the observation

Table 3 Summary of the most significant parameters received from all the simulations

MATLAB simulation for a single MNP

The maximaltemperature for a coreradius of 10 nm (nK)

ECHD(nm)

The temperature 12 nmapart from the surface

of the MNP (nK)

The most effectiveradius for maximal

temp rise (nm)

The maximaltemperature derivativeat the optimal radius

The averagetemperature over onecycle for a core radius

of 10 nm (nK)

Case 1 21 12 08 93 0011 055

Case 2 88 12 3 83 11 136

Case 3 23 12 08 92 0024 0235

Time (μs)

0 5 10 15 20 25 30

minus1

Case IIICase IICase I

Figure 9 The temperature rise profiles plotted as a function oftime for a core radius of 10 nm and an observation point locatedon surface of the MNP for all the cases explored in this paper Case1 (red) the cosine MFP Case 2 (green) the periodic pulse-shapedMFP with a duty cycle of 02 and N = 25 Case 3 (blue) the periodicdiscontinuous cosine MFP with a duty cycle of 02 and N = 25

point is on the MNP surface in order to easily evaluate thedifferences in the three cases studied in this article

4 Discussion

An analytical analysis of the TTP was preformed for threeMFPs The mathematical models were received by solving theDHFE for different BC matching each MFP using the FTs

Major work have been done in the past to solve theDHFE equation for a cosine-MF source as can be found in[36 37 49 59] Keblinski et al [38] found that a laser sourcehaving a constant power of 14middot10minus8 W heating a single MNPwith a radius of 65 nm can cause a temperature change of006 K at the particle surface Moreover for a cosine-MF heatsource the local temperature was found to be even lowercausing a maximum change in temperature of 01 mK for aparticle having a radius of 50 nm at a frequency of 2 MHZ[48] Both results are negligible from the point of view ofbiological applications as expected

However Keblinski et al [38] and others [4 20] solvedthe DHFE equation only for a constant heat flux havingthe average power of a cosine-MF without exploring the

temperature temporal behavior In addition until now therehas not been an explicit mathematical formulation thatsolves the DHFE equation for other periodic MFPs that canbe used as radiation sources for MH treatments Morgan andVictora [44] showed that the use of an incident square waveas opposed to the usual sine wave increases the normalizedpower heat by at least 50 however this conclusion wasbased on calculating only the Poynting vector and notbased the solving the DHFE in order to find the explicittemperature change

In consequence to the above we should explore theinfluences that different magnetic irradiation profiles haveon the induced temperature gradients inside tumor cellsfor the same physical and thermal MNPrsquos parameters inorder to verify what Morgan and Victora [44] suggestedFurthermore optimizing the heat power is of great impor-tance from biological point of view A typical cell having adiameter ranging from 2ndash10 μm [61] can uptake a maximalquantity of anionic MNPs that varies between 28 middot 105 and72 middot 106 per cell consequently limiting the total amountof magnetic material per cell Moreover high concentrationof MNPs with different types of coatings can cause atoxic reaction to the central nervous system [62] or maycause cellular perturbations [63] therefore it is importantto reduce the MNPrsquos concentration Nevertheless reachingthese quantities in vivo proves to be a very difficult task alltypes of cancerous cells [64 65] Hence one must optimizeother parameters such as the profile of the MF in order touse lower magnetic concentration in order to reach the sametemperature gradient values

Consequently this paper focuses on the influences thatthree different MFPs have on the temperature surrounding asingle MNP as mentioned in Section 1 when being exposedto it analytically proving to be the most effective onein causing the highest temperature rise using the samemagnetic and thermal parameters

For all three cases the MATLAB (in Section 3) andCOMSOL (in Appendix B) simulations results showed thatthe maximum temperature rise for a given core radius of10 nm ranges between 21 nK and 88 nK depending on theMFP

Similar results were received by Keblinski et al [38] andRabin [31] for a constant heat flux and an MNPrsquos havingapproximately the same physical and magnetic propertiesThe very low absolute change in temperature caused by asingle MNP can be explained by its low magnetic suscepti-bility χ0 and by the effective relaxation time that changes

drastically with the MNPrsquos volume [15] Therefore a singleMNP can release only a small amount of heat causing a verysmall change in the temperature surrounding it

However for larger magnetic concentration occupyinga single cell such as 1 ng of Fe3O4 per human cell that isequivalent to 5middot107 MNPs per cell (for a particle radius 10 nmand cell radius 5 μm) Linh et al [66] and Balivada et al [67]showed that a local temperature elevation of several degreescan be reached making MH treatments effective In additionthese quantities were also proposed by Vera and Bayazitoglu[59] Chan et al [61] Huang [64] and Melancon et al [65]who proved their efficiency in inducing MH Consequentlyone can produce a significant global temperature incrementinside a cell even if the local temperature increment of eachparticle is negligible as long as we heat many particles in thesame volume of interest

For the multiparticle case Rabin [31] and Keblin-ski et al [38] calculated the temperature rise inside aspherical region with radius R (m) consisting of manyrandomly dispersed heat sources using ΔTglobal(t) =(R2ΔTnano(t)rp2)(4π3)rp3ρN where ρN (mminus3) is the num-ber of MNPs per unit volume k (Wmminus1Cminus1) is the thermalconductivity of the medium rp (m) is the radius of MNPsand ΔTnano(t) is the temperature gradient caused by a singleMNP

For ρN = 5 middot 1021 (mminus3) and an average tumor cell radiusof rcell = 7μm [52] the number of MNPs inside a single cellcan be calculated to equal 8 middot 106 that fits the concentrationsfound by Linh et al [66] and Balivada et al [67] From ρN wecalculated the distances between two neighboring particlesthat is approximately 58 nm This means that the volumefraction of the MNPs inside the cell is about 002

By choosing a solid volume fraction of φ = 0032the calculated distance between two neighboring particles isabout 50 nm fitting a concentration of ρN = 8 middot 1021 mminus3that is in the toxicity safety range for a tumor cell having anaverage radius of 7 μm [52 66]

Due to the large distances between the particles weassumed that the interparticle interactions are negligibleso the relaxation time and magnetic susceptibility can becalculated using the same expression as (5) [68]

The total temperature increment for the three casesanalyzed in this paper can be found by substituting thereceived values for the single-MNP case (9) (12) and (15)into ΔTglobal(t) when average tumor radius of R = 4 mm wasassumed in consistence with magnitudes of cancer tumors[31 38 67]

For the cosine MFP an average value of ΔTglobal cos(t) =29K over one cycle is received near the MNPrsquos surfaceThis means that the MNPrsquos concentration is not sufficient togive increment to a dramatic temperature gradient under theparameters summarized in Tables 1 and 2

In order to receive a 6 increment that is needed for MHin the temperature near the MNPrsquos surface a larger amountthan the proposed of particles is required

For the PMP an average value of ΔTglobal pulsed(t) =72K over one cycle is received for the same parameterssummarized in Tables 1 and 2 that is sufficient to induce MH

For the discontinuous pulse-shaped MF a maximumpeak of ΔTglobal pulsed cos(t) = 123K over one cycle isreceived meaning that the MNPrsquos concentration in this caseas in Case 1 is not sufficient to ensure that MH can occur

The comparison between the three temperature gradientsreceived for each case shows that the preferable MFP forMH is the PMP one compared to Case 1 and Case 3For Case 2 the temperature gradient at the surface ofthe MNP is sufficient to cause damage to biologic cells[58 69 70] Therefore using a periodic pulse MFP canreduce the necessary amount of MNPs by a factor or evenmore allowing a wider range of markers to be used forhyperthermia treatments and simplifying the biologicalprocesses to conjugate them to a cell

In addition we also explored the influence that theMNPrsquos radius has on the maximal temperature gradientan on its rate rise As seen from Figures 4 6 and 8 theNP size has a great influence on determining the amountof heat released from the MFPrsquos surface effecting both thetemperature gradient as well as the temperature rise rate asprevious works showed [50 51 70]

For the first Case 1 studied the optimal core radius wasfound as 93 nm where the maximal temperature reaches47 nK and the temperature change rate equals 0011Ksminus1

(Figure 4) This optimal radius was also received by Kap-piyoor et al [47] for almost the same MF properties andmagnetic material properties However because the equationsolved by Kappiyoor et al [47] is different than (1) themaximal value is slightly lower that the values received byRosensweig [4] and Kappiyoor et al [47] Moreover themaximal value is also affected by the parameters chosen todescribe the magnetic properties of the MNP as demon-strated by Kappiyoor et al [47] Our magnetic parameters areslightly different than the ones used by Rosensweig [4] andKappiyoor et al [47] which may account for the differencesin the maximal values in this study as Kappiyoor et al [47]showed

For Case 2 studied the optimal core radius was foundas 83 nm where the maximal temperature gradient reaches32 nK and the temperature change rate equals 11Ksminus1

for a summation of 25 indexes (Figure 6) As can be seenby comparison there is a benefit in using a PMP ratherthan a cosine MF due to the higher temperature gradientreceived in the MNPrsquos surrounding and the sufficient averagetemperature gradient received per cycle that is about 25times higher in Case 2 than in Case 1

Although for a total summation of 25 indexes thetemperature change rate is approximately 1Ksminus1 (suggestedto be less safe [47]) one can reduce the received value bylimiting the number of the summation indexes composingthe MF to a lower number such as N = 10 instead of N =25 making the treatment safer but also maintaining highertemperature values that in Case 1 (Figure 14) Furthermorewhen looking at the results of multiplying each coefficientrsquosamplitude with its matched harmonic the limitation for thetreatment to be biologically noninvasive remains valid aslong as Aeff f0 le 5 middot 109 Ammiddots as mathematically justified in(A29) and (A30) limiting the total summation index to avalue lower than N = 25

For the Case 3 studied the optimal core radius was foundas 9 nm where the maximal temperature reaches 064 μKand the temperature change rate equals 273Ksminus1 (Figure 8)Although the values are higher than the ones received for thefirst case the average temperature elevation was found to belower after normalizing it to the time period making thistype of MFP less preferable

As we can see for all the three cases analyzed in this paperthe optimal radius depends very much on the magneticmaterial properties [47] and on the profile of the magneticfield as we have proven in Section 2 Therefore for eachcase studied and for each magnetic material the equationsdeveloped for Cases 1ndash3 must be solved separately in orderto optimize the MH treatment

Another interesting finding driven from the mathemat-ical equations is the confinement of the temperature to anarea having an average radius of 12 nm from the MNPrsquossurface for all three cases This means that most of theheat dissipation occurs in the vicinity of the heat sourcesconfining the temperature increment in the proximity oftumor cells alone unaffecting the healthy cells

The importance of this paper lies in the fact that untilnow there was no explicit mathematical formulation thatsolves the DHFE equation for other types of periodic MFPsused as excitation sources for MH treatments As we foundout changing the profile of the MF radiation can inducehigher temperature gradients in tumor cells for the samephysical and thermal parameters enabling reduction of theMNPs concentration per cell This is of great importancebecause a typical cell has a maximal quantity of MNPsthat it can uptake and because high concentration of MNPswith different types of coatings can cause a toxic reactionto the central nervous system [62] Therefore lowering themagnetic concentration per cell but still receiving the sametemperature gradients may be of great use

With the outcome of this paper we are moving forwardto in vitro studies in order to verify the theoretical resultsreceived in this paper experimentally

5 Conclusions

This study investigates the effects of different heat-fluxprofiles on a single MNP immobilized inside a phantomhaving the same thermal properties as a muscle tissue Theexact solution of DHFE was solved for different boundariesconditions using FTs According to the analytic solutions thePMP profile was found to be the more effective in rising thetemperature of the medium surrounding the MNP than thecosine profile making it a better candidate for hyperthermiatreatments rather than the conventional cosine MP

Moreover in order to reach a significant temperaturegradient for all cases studied (a) a cosine profile (b) a PMPprofile and (c) a discontinuous cosine profile there is aneed for a larger number of MNPs to be immobilized insidethe cell medium as Rabin [31] and Keblinski et al [38]previously suggested Using their techniques a significanttemperature rise was achieved for the periodic pulse-shapedMF in comparison to the other two cases studies

In order to understand the influences that a denser clusterhas on the temperature gradient other studies should bedone investigating the interparticle interactions affecting thetemperature increment and its derivative

Appendices

A Methods

In this appendix we are deducing the equations for thetemperature profiles introduced in Section 2 step by stepFor simplicity new variables are used to solve (1) where

R =radic

r αm = kmρmcm

θ(R t) = km(Tm(R t)minus Tb)R R0 =radic

Therefore by substituting the new variables from (A1) intothe left part of (1) we receive the following

kmnabla2Tm(r t) = kmr

part2rTm(r t)part2r

∣∣∣∣∣Tm(rt)rarr θ(Rt)kmRrrarrR

radicαm

= 1αm

part2θ(R t)Rpart2R

And by substituting the new variables from (A1) into theright part of (1) we receive that

ρmcmpartTm(r t)

partt= ρmcm

partθ(R t)Rpartt

∣∣∣∣αmequivkm(ρmcm)

= 1αm

partθ(R t)Rpartt

So (1) can be rewritten as follows

part2θ(R t)part2R

= partθ(R t)partt

The same procedure can be done to the BC substituting thenew variables from (A1) into the left part of (2) to receivethe following

minuskmnabla(Tm(r t)

)∣∣∣∣r=a= minus km partTm(r t)

∣∣∣∣Tm(rt)rarr θ(Rt)kmRrrarrR

radicαm

= minus 1radicαm

(θ(R t)R

= minus 1radicαmnablaR

(θ(R t)R

)∣∣∣∣∣R=R0

And (2) can be rewritten as follows

minusnabla(θ(R t)R

)∣∣∣∣R=R0

= qprimeprimes (t)radicαm (A6)

By taking the FT of (A4) (defined as in (4)) one receives thetransformation in the frequency domain so

0 = iωθ(Rω)minus part2θ(Rω)part2R

The general solution of (A7) can be found as follows

θ(Rω) = c2(ω)eminusradiciωR (A8)

Substituting (A8) into the LT of (A6) the BC can be writtenas follows

c2aradiciω +

radicαm

a2= qprimeprimes (ω)e

radiciωR0 (A9)

θ(Rω) = a2

radicαm((aradicαm)radiciω + 1

) qprimeprimes (ω)eminusradiciω(RminusR0)

R0radiciω + 1

qprimeprimes (ω)eminusradiciω(RminusR0)

Using technical computing software (Maple orand Wolfram

Mathematica) the inverse FT of θ(Rω)qprimeprimes (ω) = φ(Rω) fort gt 0 can be found by substituting iω rarr s in (A10) andtaking the inverse Laplace transform of the received equationso that

φ(R t)

=a(eminus(RminusR0)24tradicπt

minus erfc((RminusR0)2

radict+radictR0

)e(RminusR0)R0+tR0

This function converges to 0 for t rarr infin orand for RminusR0 rarrinfin

So the changes in the temperature can be found using(A1) and (A10) as follows

radicαm (A12)

Equation (A12) slightly differs than the one received byKeblinski et al [38] due to the BC that define the heatflux coming from the surface of the MNP defining the heatcreated by the magnetic losses inside it whereas Keblinski etal [38] suggested that the heat sources are inside the mediumof interest and that the heat-power density is constant intime In order to analytically calculate (3) or (A12) thegeneral expression of qprimeprimes (t) (Wmminus2) must be found for eachcase

Case 2 (a cosine MFP) For Case 1 the magnetization M(t)can be found in the time domain after substituting (5) andthe MF in (4) and taking the inverse FT of it that results in

M(t) = χ(t)lowastH(t)

= Aχ0

(cos(ω0t)

τ+ ω0 sin(ω0t)

(1τ)2 + ω02

Substituting (A13) into the magnetic induction [5] results in

B(t) = μ0H(t) + μ0M(t) (A14)

Further substituting the received magnetic inductiondescribed in (A14) into (7) one can calculate the conversionof the magnetic energy into heat losses resulting in

H(t) middot partB(t)partt

=intintdω dωprimeH(rωprime)

(iωμ0μ(ω)

) middotH(rω)ei(ωminusωprime)t

= μ0A2 cos(ω0t)

middotintdω(iω(

1 +χ0

1 + iωτ

times δ(ω minus ω0) + δ(ω + ω0)2

= μ0A2 cos(ω0t)ω0

times(

ω0χ0τ

1 + ω02τ2

cos(ω0t)minus sin(ω0t)[

1 +χ0

1 + ω02τ2

= PLoss(t) +partU

PLoss(t) = μ0A2 cos(ω0t)ω0

ω0χ0τ

1 + ω02τ2

cos(ω0t) (A16)

Because PLoss(t) is only a function of time between 0 lt r lta (isotropic and homogeneous material) then the outwardheat flux at r = a can be calculated as follows

qprimeprimes (r = a t)4πa2 = 4πa3

3PLoss(t) (A17)

qprimeprimes (r = a t) = aμ0A2ω0

(ω0χ0τ

1 + (ω0τ)2 (cos(2ω0t) + 1)

Taking the FT of (A18) and substituting it in (A10) one cancalculate the FT of θ(R t) to receive the following

θ(Rω) = aμ0A2ω0

times(

ω0χ0τ

1+(ω0τ)2

(δ(ωminus2ω0)+δ(ω+2ω0)

2+δ(ω)

middot aR0

R0radiciω + 1

exp(minusradiciωR

Case 3 (a PMP profile) The PMP (11) can be decomposedusing the theory of Fourierrsquos series into its harmonics toreceive [71 72] the following

H(t) = 2A middot Δ

Ts+infinsumn=1

sin(nπΔ

)cos(nω0t) (A20)

By substituting (A20) into (A14) and then using them in(7) we can calculate the total heat dissipation for this case asfollows

(iωμ0μ(ω)

(2A middot Δ

Ts+sum 4A

πmsin(mπΔ

)cos(mω0t)

middotsum 4A

πnsin(nπΔ

)intdω(iω(

1 +χ0

1 + iωτ

times δ(ω minus nω0) + δ(ω + nω0)2

(2A middot Δ

Ts+sum 4A

πmsin(mπΔ

)cos(mω0t)

middotsum 4A

πsin(nπΔ

times(

nω0χ0τ

1 + (nω0)2τ2cos(nω0t)

minus sin(nω0t)

1 + (nω0)2τ2

= PLoss(t) +partU

Therefore we can find that the heat losses equal to

PLoss(t) = μ0

(2A middot Δ

Ts+sum 4A

πmsin(mπΔ

)cos(mω0t)

middotsum 4A

πsin(nπΔ

(nω0χ0τ

And for Δ lt t lt Ts

= 0 (A23)

Using (A17) and (A22) we can calculate the heat flux at thesurface of the MNP

qprimeprimes (r = a t)

= PLoss(t)a

(2A middot Δ

Ts+sum 4A

πmsin(mπΔ

)cos(mω0t)

middotsum 4A

πsin(nπΔ

(nω0χ0τ

By taking the FT of the resulted heat flux and substituting itin (A10) one can receive

θ(Rω)

38ω0μ0A

2 middot Δ

sumsin(nπΔ

)nω0χ0τ

1 + (nω0)2τ2

times(δ(ωminus nω0) + δ(ω + nω0)

R0radiciω + 1

exp(minusradiciωR

38μ0A2ω0

sin(nπΔ

times nω0χ0τ

1 + (nω0)2τ2

middot(δ(ωminus (n +m)ω0) + δ(ω + (n +m)ω0)

+δ(ω minus (mminus n)ω0) + δ(ω + (mminus n)ω0)

times aR0

R0radiciω + 1

exp(minusradiciωR

When looking at (A25) the multiplication of eachcoefficientrsquos amplitude with its matched harmonic mustmeet the biologically noninvasive limitation Am middot fm le 5 middot109 Ammiddots The mathematical justification is deduced next

Looking at the eddy currents that evolve in the body [73]

E(ω) = minusiωr2

J(ω) = minus iωrσ2

Bz minusrarr E(t) = minus r2partBz(t)partt

= minusμ0r

2partH(t)partt

J(t) = minus rσ2partBz(t)partt

= minusμ0rσ

2partH(t)partt

They can be written using (A20) as follows

E(t) = minusμ0r

2partH(t)partt

= μ0r

sum 4Aπ

sin(mπΔ

)sin(mω0t)

J(t) = minusμ0rσ

2partH(t)partt

= μ0rσ

sum 4Aπ

sin(nπΔ

)sin(nω0t)

So the eddy losses inside the body can be found usingPoynting theory [5] as follows

Paverage = 1Ts

intP(t)dt = 1

intE(t)J(t)dt

(2rω0μ0A

)2 int sumsin(mπΔ

)sin(mω0t)

timessum

sin(nπΔ

)sin(nω0t)dt

(πr f0μ0

)2sumsin2

(mπΔ

timesint

sin2(mω0t)dt

= σ(πr f0μ0

sumsin2

(mπΔ

= Aeff2σ(πr f0μ0

The last expression is the same as the one received byAtkinson et al [16] Therefore the limitation on the MFS andthe frequency can be summarized as follows [11 16 24]

Aeff f0 = f0

(4Aπradic

)radicsumsin2

(mπΔ

)le 5 middot 109 Am middot s

ForN = 25 and a duty cycle of d = ΔTs = 02 the treatmentis safe as long as

Aeff f0 = f0

(4Aπradic

)radicsumsin2

(mπΔ

= A f031 le 5 middot 109 Am middot s

Consequently as long as (A30) is valid the treatmentis safe Choosing other maximal summation index valuessuch as N = 20 will result in a new constraint overthe frequency and the MFS that must fulfill Aeff f0 =f0(4Aπ

radic2)radicsum

sin2(mπΔTs) asymp A f028 le 5 middot109 Ammiddots andso on

Moreover for frequencies lower than 10 MHz there isessentially no attenuation of the MFS within cylinders ofmuscle-equivalent material therefore the maximal harmonicfrequency should not exceed 10 MHz [16]

Case 4 (a discontinuous cosine MFP) As for the previouscase we decompose the MF using the theory of Fourierrsquosseries into its harmonics for 0 le t le Δ to receive [71 72]

=infinsumn=1

[sin((Δ2)(ω0 +nω1))

ω0 +nω1+

sin((Δ2)(ω0minusnω1))ω0minusnω1

times cos(nω1t)(A31)

For Δ le t le Ts the magnetic power losses are zerobecause the MF dose not exists

By substituting (A31) into (A14) and then using themin (7) we can calculate the total heat dissipation for this caseas follows

(iωμ0μ(ω)

middot(sum[ sin((Δ2)(ω0 +mω1))

ω0 +mω1

ω0 minusmω1

times cos(mω0t))

middotsum[ sin((Δ2)(ω0 + nω1))

ω0 + nω1+

sin((Δ2)(ω0 minus nω1))ω0 minus nω1

timesintdω(iω(

1 +χ0

1 + iωτ

times δ(ωminus nω1) + δ(ω + nω1)2

((2ATs

times[

sin((Δ2)(ω0 +mω1))ω0 +mω1

ω0 minusmω1

times cos(mω1t))

middotsum( sin((Δ2)(ω0 + nω1))

ω0 + nω1+

times(

(nω1)2χ0τ

minus sin(nω1t)

1 + (nω1)2τ2

= PLoss(t) +partU

PLoss(t)

((2ATs

times[

ω0minusmω1

times cos(mω1t))

ω0 + nω1+

times(

(nω1)2χ0τ

= PLoss(t)a

((2ATs

)2sum[ sin((Δ2)(ω0 +mω1))ω0 +mω1

ω0 minusmω1

]cos(mω1t)

ω0 + nω1+

times(

(nω1)2χ0τ

θ(Rω)

= 2a3μ0

)2 aR0

R0radiciω + 1

exp(minusradiciωR

ω0 +mω1

ω0 minusmω1

times(

sin((Δ2)(ω0 + nω1))ω0 + nω1

ω0 minus nω1

middot (nω1)2χ0τ

1 + (nω1)2τ2

times(δ(ωminus (n +m)ω1) + δ(ω + (n +m)ω1)

+δ(ωminus(mminusn)ω1)+δ(ω+(mminusn)ω1)

When looking at (A35) the multiplication of each coeffi-cientrsquos amplitude with its matched harmonic must meet thebiologically noninvasive limitation Am middot fm le 5 middot 109 AmmiddotsThe mathematical justification is deduced next

For f0 gt f1 and Ts = 1 f1 we find that (A26) becomes

E(t) = minus μ0r

2partH(t)partt

= μ0r

sum n2ATs

[sin((Δ2)(ω0 + nω1))

ω0 + nω1

ω0 minus nω1

]sin(nω1t)

J(t) = minus μ0rσ

2partH(t)partt

= μ0rσ

timessum m2A

[sin((Δ2)(ω0 +mω1))

ω0 +mω1

ω0 minusmω1

]sin(mω1t)

Consequently (A28) becomes

Pavr = 1Ts

intP(t)dt = 1

intE(t)J(t)dt

(2rω1μ0A

middotint sum

sin((Δ2)(ω0 + nω1))ω0 + nnω1

ω0 minus nω1

]sin(nω1t)

timessumm[

sin((Δ2)(ω0 + nω1))ω0 + nω1

ω0 minus nω1

]sin(nω1t)dt

(2rω1μ0A

timessumn2[

sin((Δ2)(ω0 + nω1))ω0 + nω1

ω0 minus nω1

timesint

sin2(nω1t)dt

= σ(πr f1μ0

times 12

sumn2[

sin((Δ2)(ω0 + nω1))ω0 + nω1

ω0 minus nω1

= Aeff2σ(πr f1μ0

The last expression is the same as the one received byAtkinson et al [16] Therefore the limitation on the MFSand the frequency can be summarized as follows [11 16 24]

Aeff f1

(4ATsradic

timesradicradicradicradicsumn2

[sin((Δ2)(ω0 +nω1))

ω0 +nω1+

le 5 middot 109 Am middot s(A38)

For N = 25 a duty cycle of d = ΔTs = 02 and ω0 = 6ω1the treatment is safe as long as

Aeff f1

(4ATsradic

[sin((Δ2)(ω0 +nω1))

ω0 +nω1+

= A f146 lt A f0 le 5 middot 109 Am middot s(A39)

Consequently as long as (A39) the treatment will besafe Moreover for frequencies lower than 10 MHz there isessentially no attenuation of the MFS within cylinders ofmuscle-equivalent material therefore the maximal harmonicfrequency should not exceed 10 MHz [16]

B COMSOL Results

In order to validate the analytic solutions and the MATLABsimulations a numerical simulation was performed usingCOMSOL for the same thermal and magnetic propertiesgiven in Tables 1 and 2 The simulation results can be seenfor each case studied in Methods and Results parts in thisAppendix

For Case 1 the mathematical expression of the temper-ature increment (9) was plotted as a function of time andspace where the results are given in Figure 10 for Ts = 25μs

The maximal temperature elevation in Figure 10 reacheda value of 225 nK on the surface of the MNP whichis 015 nK higher than the one received for the analyticsimulation Figure 3

At 2 nm apart from the surface of the MNP surface thetemperature elevation reached a value of 205 nK that is

r = 0 nm

Time (μs)

r = 2 nmr = 4 nm

r = 6 nmr = 8 nmr = 10 nm

00 05 1 15 2 25 3

02 nK higher than the one receive in Figure 3 Again thereis a small difference between both simulations results As theobservation point gets further from the surface of the MNPthe temperature differences get bigger reaching a value of04 nK at an observation point located 10 nm apart from thesurface

This may be caused by the triangles constructing theCOMSOLrsquos numeric mesh which are used to solve numer-ically the heat problem that get larger and bigger asthe observation point gets further from the MNP surfacecontributing to the error

Comparing between Figures 3 and 10 we conclude thatthe numerical simulation fits the analytic solution

For Case 2 as in Case 1 in order to validate the analyticsolution a numerical simulation was also performed usingCOMSOL for the same thermal and magnetic properties(Tables 1 and 2) The simulation result can be seen inFigure 11

The maximal temperature elevation in Figure 11 reachesa value of 85 nK on the surface of the MNP which is 04 nKhigher than the one receive in Figure 5

At 2 nm apart from the surface of the MNP surfacethe temperature elevation reached a value of 75 nK thatis 02 nK higher than the one receive in Figure 5 Againit seems that there exists a small difference between thesimulations results As the observation point gets furtherfrom the surface of the MNP the differences gets biggerreaching a value of 08 nK at an observation point located10 nm a part from the surface

This may be caused by the bigger triangles in the meshthat are formed in the COMSOL software as the observationpoint gets further from the MNP surface contributing to theerror

r = 0 nm

minus10 1 2 3 4 5 6

Time (μs)

r = 2 nmr = 4 nm

r = 6 nmr = 8 nmr = 10 nm

Figure 11 The temperature profile for a PMP-shaped MFP plottedas a function of time and as a function of the observation pointlocated at a distance r from the surface of the MNP r ranging from0 nm to 10 nm for a core radius of 10 nm for N = 25

As can be seen from Figure 11 there are 5 peaks duringthe time that the MF is tuned on that fit the number ofpeaks in Figure 5 these peaks evolve due to the final numberof harmonics that form the PMP MF as given by (11)However there is a slightly difference in the temperatureprofiles between Figures 11 and 5 in Figure 11 the first peakis lower than the others in comparison to Figure 5 where thefirst peak is about the same high as the last peak

Again there are some small changes between bothsoftware simulations as expected however the results forboth simulations conclude that there is a benefit in usingthe PMPs instead of the cosine MFP due to the highertemperature rise values received for the same magneticparameters

For Case 3 we used again the numerical simulationCOMSOL in order to validate the analytic solution for thesame thermal and magnetic properties The simulation resultcan be seen in Figure 12

The maximal temperature elevation in Figure 12 reacheda value of 23 nK on the surface of the MNP which is thesame as the one receive in Figure 7

At 2 nm apart from the surface of the MNP surfacethe temperature elevation reached a value of 2 nK that is02 nK higher than the one received in Figure 7 Again thereis a small difference between the simulations results As theobservation point get further from the surface of the MNPthe differences gets bigger reaching a value of 03 nK at anobservation point located 10 nm a part from the surface

As explain before this may be caused by the biggertriangles in the mesh that are formed in the COMSOLsoftware as the observation point gets further from the MNPsurface contributing to the error Although there are somesmall changes between both simulations as expected themaximal temperature rise is almost the same as the cosineMFP

r = 0 nm

Time (μs)

r = 2 nmr = 4 nm

r = 6 nmr = 8 nmr = 10 nm

0 2 4 6 8 10 12 14 16 18 20 22

Figure 12 The temperature profile for a discontinuous cosine MFPplotted as a function of time and as a function of the observationpoint located at a distance r from the surface of the MNP r rangingfrom 0 nm to 10 nm for a core radius of 10 nm for N = 25

C The Effects the Maximal Number of IndexesHas on Cases 2 and 3 Results

In Appendix C we examined the influences that the maximalnumbers of indexes composing the MF signal have on thetemperature rise and on the temperature rate rise for Case 2and Case 3

The maximal index numbers for summation were chosenas N = 100 N = 25 N = 15 N = 10 and N = 1 AboveN = 25 the MF is practically absorbed in the tissue [14] butthis fact was not taken in consideration in the simulationsresults

Case 3 The temperature rise for Case 2 as a function of themaximal summation indexes can be seen in Figure 13

From Figure 13 we concluded that the maximal temper-ature rise depends on the number of harmonics composingthe MF signal For N = 100 the maximal temperature risereaches a value of 50 nK for a core radius of 77 nm that is16 times higher than the maximal value received forN = 25The summation of 100 indexes can be seen as the ideal PMPsshaped MFP

For N = 15 we receive a temperature rate of 28 nK fora core radius of 88 nm and for N = 10 we received a valueof 26 nK for a core radius of 9 nm Furthermore we can seethat the number of indexes composing the MF changes theoptimal radius as it gets smaller as the index number getsbigger

Now we examined the influences that the number ofmaximal summation indexes composing the MF signal hason the temperature rate rise The chosen numbers were N =100 N = 25 N = 15 N = 10 and N = 1

From Figure 14 we concluded that the maximal tem-perature rate rise depends on the number of harmonicscomposing the MF signal For N = 100 the maximal

5 6 7 8 9 10 11 12

N = 100N = 25N = 15

N = 10N = 1

Figure 13 The absolute maximal temperature rise was plotted as afunction of the number of harmonics summedN and as a functionof the core radius for a periodic pulse-shaped MFP having a pulsedwidth of 02Ts the maximal summation valueN ranges fromN = 1to N = 100

05 6 7 8 9 10 11 12

N = 100N = 25N = 15

N = 10N = 1

partTm

axpartt

Figure 14 The absolute maximal temperature rise rate was plottedas a function of the number of harmonics summed N and as afunction of the core radius for a periodic pulse-shaped MFP havinga pulsed width of 02Ts plotted as a function of the core radiusa ranging from 5 nm to 15 nm the observation points are on thesurface of the MNP

temperature rate reaches a value of 39Ksminus1 is received fora core radius 76 nm and is 39 times higher than the valuereceived for N = 25 For N = 15 we receive a temperaturerate of 05Ksminus1 for a core radius of 82 nm that is half thevalue received for N = 25 and for N = 10 we received avalue of 03Ksminus1 for a core radius of 85 nm

Case 4 Now we examined the influences that the maximalnumber of summation indexes composing the MF signal has

05 6 7 8 9 10 11 12 13 14 15

N = 100N = 25N = 15

N = 10N = 1

Figure 15 The absolute maximal temperature rise was plotted as afunction of the number of harmonics summedN and as a functionof the maximal summation value for a periodic discontinuouscosine MFP having a pulsed width of 02Ts N ranging from N = 1to N = 100 for a core radius of 10 nm the observation point are onthe surface of the MNP

on the temperature rise The chosen numbers were N = 100N = 25 N = 15 N = 10 and N = 1

From Figure 15 we concluded that the maximal temper-ature rise depends on the number of harmonics composingthe MF signal For N = 100 the maximal temperature risereaches a value of 5 nK for a core radius of 93 nm that is11 times higher than the maximal value received forN = 25For N = 15 we receive a temperature rate of 55 nK for acore radius of 92 nm and for N = 10 we received a valueof 61K for a core radius of 93 nm As already mentionedthere is a limitation to the highest frequency that can be usedfor MH and should not exceed 10 MHz [16] in our casethis limits the summation to 25 indexes that compose theMF signal Moreover we can see that the number of indexescomposing the MF changes the optimal radius it gets smalleras the index number gets higher Furthermore we can seethat the number of indexes composing the MF changes theoptimal radius by getting smaller as the index number getsbigger

Now we examined the influences that the number ofindexes composing the MF signal has on the temperature raterise The chosen numbers were N = 100 N = 25 N = 15N = 10 and N = 1

From Figure 16 we concluded that the maximal tem-perature rise rate depends on the number of harmonicscomposing the MF signal For N = 100 the maximaltemperature rate reaches a value of 009Ksminus1 is received fora core radius 84 nm and is 45 times higher than the valuereceived for N = 25 For N = 15 we receive a temperaturerate of 0016Ksminus1 for a core radius of 93 nm that is halfthe value received for N = 25 and for N = 10 we received avalue of 0015Ksminus1 for a core radius of 93 nm

5 6 7 8 9 10 11 12

partTm

axpartt

N = 100N = 25N = 15

N = 10N = 1

13 14 15

Figure 16 The absolute maximal temperature rise rate was plottedas a function of the number of harmonics summed N and as afunction of the core radius for a periodic discontinuous cosine MFPhaving a pulsed width of 02Ts amdashranging from 5 nm to 15 nm theobservation point are on the surface of the MNP

As already mentioned there is a limitation to the highestfrequency that can be used for MH and should not exceed10 MHz [16] in our case this limits the summation to 25indexes that compose the MF signal Moreover we can seethat the number of indexes composing the MF changes theoptimal radius and it gets smaller as the index number getshigher

References

[1] I M Gescheit M Ben-David and I Gannot ldquoA proposedmethod for thermal specific bioimaging and therapy tech-nique for diagnosis and treatment of malignant tumors byusing magnetic nanoparticlesrdquo Advances in Optical Technolo-gies vol 2008 Article ID 275080 7 pages 2008

[2] H G Bagaria and D T Johnson ldquoTransient solution to thebioheat equation and optimization for magnetic fluid hyper-thermia treatmentrdquo International Journal of Hyperthermia vol21 no 1 pp 57ndash75 2005

[3] M A Giordano G Gutierrez and C Rinaldi ldquoFundamentalsolutions to the bioheat equation and their application tomagnetic fluid hyperthermiardquo International Journal of Hyper-thermia vol 26 no 5 pp 475ndash484 2010

[4] R E Rosensweig ldquoHeating magnetic fluid with alternatingmagnetic fieldrdquo Journal of Magnetism and Magnetic Materialsvol 252 no 1ndash3 pp 370ndash374 2002

[5] J D Jackson Classical Electrodynamics John Wily amp Sons1998

[6] L D Landau L P Pitaevskii and E M Lifshitz Electrodynam-ics of Continuous Media vol 8 2nd edition 1984

[7] N Guskos E A Anagnostakis V Likodimos et al ldquoFerromag-netic resonance and ac conductivity of a polymer composite ofFe3 O4 and Fe3 C nanoparticles dispersed in a graphite matrixrdquoJournal of Applied Physics vol 97 no 2 Article ID 024304 6pages 2005

[8] X Zhang B Q Li and S S Pang ldquoA perturbational approachto magneto-thermal problems of a deformed sphere levitatedin a magnetic fieldrdquo Journal of Engineering Mathematics vol2-3 no 4 pp 337ndash355 1997

[9] A Jordan P Wust H Fahlin W John A Hinz and R FelixldquoInductive heating of ferrimagnetic particles and magneticfluids physical evaluation of their potential for hyperthermiardquoInternational Journal of Hyperthermia vol 9 no 1 pp 51ndash681993

[10] P C Fannin Y L Raikher A T Giannitsis and S W CharlesldquoInvestigation of the influence which material parametershave on the signal-to-noise ratio of nanoparticlesrdquo Journal ofMagnetism and Magnetic Materials vol 252 no 1ndash3 pp 114ndash116 2002

[11] Q A Pankhurst J Connolly S K Jonesand and J DobsonldquoApplications of magnetic nanoparticles in biomedicinerdquoJournal of Physics D vol 36 no 13 pp 167ndash181 2003

[12] J Weizenecker B Gleich J Rahmer and J Borgert ldquoParticledynamics of mono-domain particles in magnetic particleimagingrdquo in Proceedings of the 1st International Workshop onMagnetic Particle Imaging Magnetic Nanoparticles pp 3ndash15World Scientific 2010

[13] P C Fannin ldquoMagnetic spectroscopy as an aide in under-standing magnetic fluidsrdquo Journal of Magnetism and MagneticMaterials vol 252 no 1ndash3 pp 59ndash64 2002

[14] P C Fannin ldquoCharacterisation of magnetic fluidsrdquo Journal ofAlloys and Compounds vol 369 no 1-2 pp 43ndash51 2004

[15] P C Fannin and S W Charles ldquoOn the calculation of theNeel relaxation time in uniaxial single-domain ferromagneticparticlesrdquo Journal of Physics D vol 27 no 2 pp 185ndash1881994

[16] W J Atkinson I A Brezovich and D P ChakrabortyldquoUsable frequencies in hyperthermia with thermal seedsrdquoIEEE Transactions on Biomedical Engineering vol 31 no 1 pp70ndash75 1984

[17] S Mornet S Vasseur F Grasset and E Duguet ldquoMagneticnanoparticle design for medical diagnosis and therapyrdquoJournal of Materials Chemistry vol 14 no 14 pp 2161ndash21752004

[18] C H Moran S M Wainerdi T K Cherukuri et alldquoSize-dependent joule heating of gold nanoparticles usingcapacitively coupled radiofrequency fieldsrdquo Nano Researchvol 2 no 5 pp 400ndash405 2009

[19] V P Torchilin ldquoTargeted pharmaceutical nanocarriers forcancer therapy and imagingrdquo The AAPS Journal vol 9 no 2pp E128ndashE147 2007

[20] T R Sathe Integrated magnetic and optical nanotechnology forearly cancer detection and monitoring [PhD thesis] GeorgiaInstitute of Technology 2007

[21] N Gigel ldquoMagnetic nanoparticles impact on tumoral cells inthe treatment by magnetic fluid hyperthermiardquo Digest Journalof Nanomaterials and Biostructures vol 3 no 3 pp 103ndash1072008

[22] F Matsuoka M Shinkai H Honda T Kubo T Sugitaand T Kobayashi ldquoHyperthermia using magnetite cationicliposomes for hamster osteosarcomardquo BioMagnetic Researchand Technology vol 2 no 3 pp 1ndash6 2004

[23] Q A Pankhurst ldquoNanomagnetic medical sensors and treat-ment methodologiesrdquo BT Technology Journal vol 24 no 3pp 33ndash38 2006

[24] E Kita T Oda T Kayano et al ldquoFerromagnetic nanoparticlesfor magnetic hyperthermia and thermoablation therapyrdquo

Journal of Physics D vol 43 no 47 Article ID 474011 9 pages2010

[25] L Pilon and K M Katika ldquoModified method of characteristicsfor simulating microscale energy transportrdquo Journal of HeatTransfer vol 126 no 5 pp 735ndash743 2004

[26] G Chen R Yang and X Chen ldquoNanoscale heat transfer andthermal-electric energy conversionrdquo Journal de Physique IVvol 125 no 1 pp 499ndash504 2005

[27] G Chen ldquoNon local and nonequilibrium heat conduction inthe vicinity of nanoparticlesrdquo Journal of Heat Transfer vol118 no 3 pp 539ndash546 1996

[28] C Kittel Introduction to Solid-State Physics John Wiley ampSons New York NY USA 1996

[29] R Rohlsberger W Sturhahn T S Toellner et al ldquoPhonondamping in thin films of Ferdquo Journal of Applied Physics vol86 no 1 pp 584ndash592 1999

[30] K E Goodson and M I Flik ldquoElectron and phonon thermalconduction in epitaxial high-Tc superconducting filmsrdquo Jour-nal of Heat Transfer vol 115 no 1 pp 17ndash25 1993

[31] Y Rabin ldquoIs intracellular hyperthermia superior to extracellu-lar hyperthermia in the thermal senserdquo International Journalof Hyperthermia vol 18 no 3 pp 194ndash202 2002

[32] G Chen ldquoBallistic-diffusive heat-conduction equationsrdquoPhysical Review Letters vol 86 no 11 pp 1197ndash2300 2000

[33] E H Wissler ldquoPennesrsquo 1948 paper revisitedrdquo Journal ofApplied Physiology vol 85 no 1 pp 35ndash41 1998

[34] T C Shih P Yuan W L Lin and H S Kou ldquoAnalytical analy-sis of the Pennes bioheat transfer equation with sinusoidal heatflux condition on skin surfacerdquo Medical Engineering amp Physicsvol 29 no 9 pp 946ndash953 2007

[35] P Yuan H E Liu C W Chen and H S Kou ldquoTemperatureresponse in biological tissue by alternating heating andcooling modalities with sinusoidal temperature oscillation onthe skinrdquo International Communications in Heat and MassTransfer vol 35 no 9 pp 1091ndash1096 2008

[36] J Liu and L X Xu ldquoEstimation of blood perfusion using phaseshift in temperature response to sinusoidal heating at the skinsurfacerdquo IEEE Transactions on Biomedical Engineering vol 46no 9 pp 1037ndash1043 1999

[37] I K Tjahjono An analytical model for near-infrared lightheating of a slab by embedded gold nanoshells [PhD thesis]Rice University 2006

[38] P Keblinski D G Cahill A Bodapati C R Sullivan and TA Taton ldquoLimits of localized heating by electromagneticallyexcited nanoparticlesrdquo Journal of Applied Physics vol 100 no5 Article ID 054305 5 pages 2006

[39] E Gescheidtova R Kubasek and K Bartusek ldquoQuality ofgradient magnetic fields estimationrdquo Journal of EE vol 57 no8 pp 54ndash57 2006

[40] M Squibb ldquoA guide to experimental exposure of biologicaltissue to pulsed magnetic fieldsrdquo PEMF Usage Guide 2007

[41] G C Goats ldquoPulsed electromagnetic (short-wave) energytherapyrdquo British Journal of Sports Medicine vol 23 no 4 pp213ndash216 1989

[42] T Niwa Y Takemura N Aida H Kurihara and T HisaldquoImplant hyperthermia resonant circuit produces heat inresponse to MRI unit radiofrequency pulsesrdquo The BritishJournal of Radiology vol 81 no 961 pp 69ndash72 2008

[43] P Cantillon-Murphy L L Wald M Zahn and E Adalsteins-son ldquoProposing magnetic nanoparticle hyperthermia in low-field MRIrdquo Concepts in Magnetic Resonance A vol 36 no 1pp 36ndash47 2010

[44] S M Morgan and R H Victora ldquoUse of square waves incidenton magnetic nanoparticles to induce magnetic hyperthermiafor therapeutic cancer treatmentrdquo Applied Physics Letters vol97 no 9 Article ID 093705 3 pages 2010

[45] K E Oughstun Electromagnetic and Optical Pulse Propaga-tion Springer 2006

[46] L R Squire and J A Zouzounis ldquoECT and memory briefpulse versus sine waverdquo The American Journal of Psychiatryvol 143 no 5 pp 596ndash601 1986

[47] R Kappiyoor M Liangruksa R Ganguly and I K PurildquoThe effects of magnetic nanoparticle properties on magneticfluid hyperthermiardquo Journal of Applied Physics vol 108 no 9Article ID 094702 8 pages 2010

[48] A Trakic F Liu and S Crozier ldquoTransient temperature rise ina mouse due to low-frequency regional hyperthermiardquo Physicsin Medicine and Biology vol 51 no 7 pp 1673ndash1691 2006

[49] A O Govorov W Zhang T Skeini H Richardson E J Leeand N A Kotov ldquoGold nanoparticle ensembles as heatersand actuators melting and collective plasmon resonancesrdquoNanoscale Research Letters vol 1 no 1 pp 84ndash90 2006

[50] O N Strand ldquoA method for the computation of the errorfunction of a complex variablerdquo Mathematics of Computationvol 19 pp 127ndash129 1965

[51] J Kestin and L N Persen ldquoOn the error function of a complexargumentrdquo Zeitschrift fur Angewandte Mathematik und Physikvol 7 no 1 pp 33ndash40 1956

[52] M Kettering J Winter M Zeisberger et al ldquoMagneticnanoparticles as bimodal tools in magnetically inducedlabelling and magnetic heating of tumour cells an in vitrostudyrdquo Nanotechnology vol 18 no 17 Article ID 175101 9pages 2007

[53] R Hergt S Dutz R Muller and M Zeisberger ldquoMagneticparticle hyperthermia nanoparticle magnetism and materialsdevelopment for cancer therapyrdquo Journal of Physics vol 18 no38 pp S2919ndashS2934 2006

[54] I Hilger R Hergt and W A Kaiser ldquoUse of magneticnanoparticle heating in the treatment of breast cancerrdquo IEEProceedings-Nanobiotechnology vol 152 no 1 pp 33ndash392005

[55] A A Velayati P Farnia and T A Ibrahim ldquoDifferences incell wall thickness between resistant and nonresistant strainsof Mycobacterium tuberculosis using transmission electronmicroscopyrdquo Chemotherapy vol 55 no 5 pp 303ndash307 2009

[56] V Dupres Y F Dufreene and J J Heinisch ldquoMeasuringcell wall thickness in living yeast cells using single molecularrulersrdquo American Chemical Society Nano vol 4 no 9 pp5498ndash5504 2010

[57] ldquoWhat is the thickness of the cell membranerdquo httpwwwweizmannacilplantsMiloimagesmembraneThickness110109RMpdf

[58] M Kaiser J Heintz I Kandela and R Albrecht ldquoTumor celldeath induced by membrane melting via immunotargetedinductively heated coreshell nanoparticlesrdquo Microscopy andMicroanalysis vol 13 supplement 2 pp 18ndash19 2007

[59] J Vera and Y Bayazitoglu ldquoGold nanoshell density variationwith laser power for induced hyperthermiardquo InternationalJournal of Heat and Mass Transfer vol 52 no 3-4 pp 564ndash573 2009

[60] E I Gabrielle Biology The Easy Way Barronrsquos EducationalSeries New York NY USA 1990

[61] B Chan B D Chithrani A A Ghazani and C W WarrenldquoDetermining the size and shape dependence of gold nanopar-ticle uptake into mammalian cellsrdquo Nano Letters vol 6 no 4pp 662ndash668 2006

[62] Z Yang Z W Liu R P Allaker et al ldquoA review of nanoparticlefunctionality and toxicity on the central nervous systemrdquoJournal of the Royal Society Interface vol 7 no 4 pp S411ndashS422 2010

[63] N Singha G J S Jenkinsa R Asadib and S H DoakaldquoPotential toxicity of superparamagnetic iron oxide nanopar-ticlesrdquo Nano Reviews vol 1 pp 1ndash15 2010

[64] H Huang Magnetic nanoparticles based magnetophresis forefficient separation of foodborne pathogenes [MS thesis]University of Arkansa 2009

[65] M P Melancon W Lu Z Yang et al ldquoIn vitro and in vivotargeting of hollow gold nanoshells directed at epidermalgrowth factor receptor for photothermal ablation therapyrdquoMolecular Cancer Therapeutics vol 7 no 6 pp 1730ndash17392008

[66] P H Linh N C Thuan N A Tuan et al ldquoInvitro toxicity testand searching the possibility of cancer cell line exterminationby magnetic heating with using Fe3O4 magnetic fluidrdquo Journalof Physics vol 187 no 1 Article ID 012008 9 pages 2009

[67] S Balivada R S Rachakatla H Wang et al ldquoAC magnetichyperthermia of melanoma mediated by iron(0)iron oxidecoreshell magnetic nanoparticles a mouse studyrdquo Bio MedCenteral Cancer vol 10 article 119 9 pages 2010

[68] S Bedanta and W Kleemann ldquoSupermagnetismrdquo Journal ofPhysics D vol 42 no 1 Article ID 013001 28 pages 2009

[69] M Lewin N Carlesso C H Tung et al ldquoTat peptide-derivatized magnetic nanoparticles allow in vivo tracking andrecovery of progenitor cellsrdquo Nature Biotechnology vol 18 no4 pp 410ndash414 2000

[70] S Purushotham and R V Ramanujan ldquoModeling the per-formance of magnetic nanoparticles in multimodal cancertherapyrdquo Journal of Applied Physics vol 107 no 11 ArticleID 114701 9 pages 2010

[71] J S Walker Encyclopedia of Physical Science and TechnologyElsevier Science 3th edition 2003

[72] S W Smith The Scientist and Engineerrsquos Guide to Digital SignalProcessing chapter 13 California Technical Publishing 1997

[73] J Pellicer-Porres R Lacomba-Perales J Ruiz-Fuertes DMartınez-Garcıa and M V Andres ldquoForce characterizationof eddy currentsrdquo American Journal of Physics vol 74 no 4pp 267ndash271 2006

Helmholtz Bright Spatial Solitons and Surface Waves atPower-Law Optical Interfaces

J M Christian1 2 E A McCoy1 G S McDonald1

J Sanchez-Curto2 and P Chamorro-Posada2

1 Joule Physics Laboratory School of Computing Science and Engineering Materials amp Physics Research Centre University of SalfordGreater Manchester M5 4WT UK

2 Departamento de Teorıa de la Senal y Comunicaciones e Ingenierıa Telematica Universidad de Valladolid ETSI TelecomunicacionCampus Miguel Delibes Paseo Belen 15 E-47011 Valladolid Spain

Correspondence should be addressed to J M Christian jchristiansalfordacuk

Academic Editor Alan Migdall

We consider arbitrary angle interactions between spatial solitons and the planar boundary between two optical materials witha single power-law nonlinear refractive index Extensive analysis has uncovered a wide range of new qualitative phenomena innon-Kerr regimes A universal Helmholtz-Snell law describing soliton refraction is derived using exact solutions to the governingequation as a nonlinear basis New predictions are tested through exhaustive computations which have uncovered substantiallyenhanced Goos-Hanchen shifts at some non-Kerr interfaces Helmholtz nonlinear surface waves are analyzed theoretically andtheir stability properties are investigated numerically for the first time Interactions between surface waves and obliquely incidentsolitons are also considered Novel solution behaviours have been uncovered which depend upon a complex interplay betweenincidence angle medium mismatch parameters and the power-law nonlinearity exponent

1 Introduction

A light beam impinging on the interface between twodissimilar dielectric materials is a fundamental opticalgeometry [1ndash12] After all the single-interface configurationis an elemental structure that facilitates more sophisticateddevice designs and architectures for a diverse range ofphotonic applications The seminal work of Aceves et al[6 7] some two decades ago considered perhaps the simplestscenario where a spatial soliton (ie a self-trapped andself-stabilizing optical beam) is incident on the boundarybetween two different Kerr-type materials Their intuitiveapproach reduced the full complexity of the electromag-netic interface problem to something far more tractablenamely the solution a scalar equation of the inhomogeneousnonlinear Schrodinger (NLS) type The development of anequivalent-particle theory [3ndash6] provided an enormous levelof insight into the behaviour of scalar solitons at material

boundaries The adiabatic perturbation technique developedby Aliev et al [13 14] provides another toolbox for analyzinginterface phenomena (eg light incident on the boundarybetween a linear and a nonlinear medium) Photorefractive[15] and quadratic [16] materials have also been considered

A recurrent feature of the waves at interfaces literature isthe appearance of the paraxial approximation which com-bines the assumptions of broad (predominantly transverse-polarized) beams and slowly varying envelopes [1ndash16]The adoption of this ubiquitous mathematical device canimpose some strong physical constraints that should beborne in mind when modelling precisely these types ofangular geometries Indeed the class of problem at handis inherently nonparaxial since impinging beams may bearbitrarily oblique with respect to the interface Externalrefraction (where the refracted beam deviates away from theinterface) provides a specific context where beam refractioncannot be described using conventional approaches Paraxial

wave optics must be applied with care since in potentiallyoff-axis regimes it holds true only where angles (in thelaboratory frame) of incidence reflection and refractionwith respect to the reference direction are negligibly (or near-negligibly) small

Recently we proposed the first scalar model of spatialsolitons at interfaces that is valid across the entire angularrange [17 18] By respecting the essential role played byHelmholtz diffraction [19ndash24] the angular restriction waslifted while retaining an intuitive and manageable envelopeequation Preliminary analyses considered bright [17 18]and dark [25 26] spatial solitons incident on the boundarybetween dissimilar Kerr-type materials They focused onestablishing and developing the propagation aspects of ourHelmholtz interfaces approach By enforcing appropriatecontinuity conditions at the interface a Snellrsquos law for Kerrspatial solitons was derived whose validity was tested andconfirmed by extensive numerical computations Here wetake the first steps in a systematic study of the materialsaspects of nonlinear beam-interface interactions The sim-plest non-Kerr system one might consider is a class of hostmedia whose refractive index nNL(E) has a generic power-law dependence on the (complex) electric field amplitude E[27ndash29]

nNL(E) = α

2n0|E|q (1)

where α is a positive coefficient n0 is the linear index (at theoptical frequency) and the exponent q lies within the range0 lt q lt 4 Typically the nonlinear response of the medium isassumed to be weak so that αE

q0n0 O(1) where E0 is the

peak field amplitudePower-law models have played a key role in the theory of

nonlinear waves for the past three decades [30 31] Indeed[32] provides an excellent review of the fundamental impor-tance of model (1) in photonics contexts Materials that fallinto this broad category include some semiconductors (egInSb [33] and GaAsGaAlAs [34]) doped filter glasses (egCsSxSexminus2 [35 36]) and liquid crystals [32] One expectsnon-Kerr regimes (where q deviates from the value of 2) togive rise to a diverse range of new quantitative and qualitativephenomena The physical basis for this expectation lies in theidealized nature of the Kerr response In a range of materialsone can often find higher-order nonlinear effects coming intoplay Perhaps the most obvious example of the breakdownof Kerr-type behaviour is optical saturation where therefractive index change becomes bleached in the presence ofsufficiently high-intensity illumination In such cases model(1) with q lt 1 can be used to describe generic leading-order corrections from a saturable (dispersive) nonlinearity[35 36]

In this paper a detailed account is presented of arbitrary-angle refraction of spatial solitons at the interface betweendissimilar power-law materials Also of intrinsic interest arenonlinear surface waves (ie localized modes that travelalong the boundary) This fundamental class of excitationhas been the subject of previous power-law studies involvinga single interface [35ndash39] and guided waves in multilayer

structures (eg slab waveguides) [40ndash43] Stability char-acteristics have been inferred from inspection of power-propagation constant solution branches However to the bestof our knowledge direct verification of such predictions[37ndash43] (eg through numerical solution of the underlyingnonlinear Helmholtz equation) has been absent from theliterature to date Rather computational studies of surfacewaves tend to have been in the limit of slowly varyingenvelopes and nonlinear Schrodinger-type models typicallyof the diffusive-Kerr [44 45] thermal [46] or saturable[47] type Here we investigate the stability of exact ana-lytical Helmholtz surface waves through direct numericalcalculations As a fairly stringent test of solution robustnesswe also report on some key findings concerning arbitrary-angle interactions between surface waves and solitons Inbeam-refraction and surface-wave contexts simulations haveuncovered strikingly distinct behaviours as the exponentq is varied and across a range of quasi-paraxial and fullynonparaxial angular regimes

The layout of this paper is as follows In Section 2 wepropose a governing equation for scalar optical fields intwo adjoining power-law materials with dissimilar mediumcoefficients Exact analytical bright solitons are presented forboth media and these solutions are used as a nonlinear basisto derive a generalized Helmholtz-Snell law In Section 3extensive computations test predictions of the new refractionlaw over a range of system parameters We also extendour first calculations of the Goos-Hanchen (GH) shifts[48] in the Helmholtz angular regime [49] with power-law nonlinearities Nonlinear surface waves are derived inSection 4 and simulations provide what appears to be thefirst full investigation of the stability properties of this newclass of Helmholtz solution We conclude in Section 5 withsome comments about the impact of our results

2 Helmholtz Model for ScalarSoliton Refraction

The formalism of Helmholtz soliton theory [23 24] is nowdeployed to develop a framework for describing refractionphenomena in wider classes of nonlinear optical materialsThis type of modelling approach is valid when the beamwaist w0 is much broader than its free-space carrier wave-length λ such that ε equiv λw0 O(1) Ultranarrow beamcorrections to the governing equation typically obtainedfrom single-parameter (ie ε-based) order-of-magnitudeanalyses of fully-nonlinear Maxwell equations [50ndash55] areunnecessary in such regimes

21 Governing Equation Within the scalar approximation[19ndash24] we consider an electric field of the form

E(x z t) = E(x z) exp(minusiωt) + Elowast(x z) exp(+iωt) (2)

which is time harmonic with angular frequency ω Thelaboratory space and time coordinates are (x z) and trespectively In medium j (where j = 1 and 2) it is well

47Helmholtz Bright Spatial Solitons and Surface Waves at Power-Law Optical Interfaces

known that the complex spatial field E(x z) satisfies theHelmholtz equation

part2E

partz2+part2E

partx2+ω2

c2n2j (E)E = 0 (3)

where c is the vacuum speed of light The refractiveindex distribution nj(E) on either side of the boundary isintroduced through n2

j (E) equiv n20 j + αj|E|q where n0 j is

the linear index at frequency ω and αj is a nonlinearitycoefficient To facilitate comparison with our earlier work[17 18 25 26] we look for travelling-wave solutions to(3) of the form E(x z) = E0u(x z) exp(ik1z) Here E0 is a(real) scale factor determining electric-field units u(x z) isthe dimensionless envelope and exp(ik1z) biases the solutionin the forward longitudinal direction (taken to be z) wherek1 equiv n01ωc is the (linear) propagation constant of thecarrier wave in medium 1 It then follows that u satisfies theinhomogeneous equation

part2u

partz2+ i2k1

partz+part2u

partx2+ω2

c2α1E

q0|u|qu

(1minus n2

c2α1E

(1minus α2

]h(x z)u

where h(x z) is a Heaviside function that is equal to zero(unity) in the domain of medium 1 (medium 2) Equation(4) may be normalized with respect to the parameters inmedium 1 in which case the following governing equationmay be derived without further approximation [17 18 5657]

κpart2u

partζ2+ i

partζ+

12part2u

partξ2+ |u|qu =

4κ+ (1minus α)|u|q

]h(ξ ζ)u

The dimensionless longitudinal and transverse coordinatesare ζ = zLD and ξ = 212xw0 respectively where LD =k1w

202 is the diffraction length of a reference (paraxial)

Gaussian beam The inverse beam width is quantified byκ = 1(k1w0)2 = ε24π2n2

01 O(1) where ε equiv λw0 and the

field amplitude scales with E0 = (2n201α1k1LD)

1q Model (5)

is supplemented by the mismatch parameters [17 18 25 26]

Δ equiv 1minus n202

α equiv α2

α1 (6b)

which determine how the linear and nonlinear refractiveproperties of the system change as one traverses the bound-ary

Equation (5) allows one access to material scenarioswhere Δ lt 0 (ie configurations with n02 gt n01) [17

18 25 26] By contrast the scalings deployed in classicparaxial theory [8 9] restrict those models to considerationof regimes with Δ gt 0 It is also apparent that setting κ asymp 0in an attempt to recover the paraxial model is going to leadto complications when handling the linear mismatch termΔ4κ The physical and mathematical difficulties of interpret-ing the paraxial approximation as the single-parameter limitκ asymp 0 have been discussed at length elsewhere [23 24] it isparticularly well illustrated by interface geometries

22 Solitons as a Nonlinear Basis When investigating the re-fraction of nonlinear light beams at material boundaries itis essential to have an appropriate set of basis functions withwhich to formulate the problem Such a basis is provided bythe underlying exact analytical Helmholtz solitons [56] Inthe following analysis we align the interface along the z axisso that it is located at transverse position x = 0 Medium 1(the domain of the incident beam where h = 0) is taken tooccupy the half-plane minusinfin le x lt 0 while medium 2 (thedomain of the refracted beam where h = +1) occupies 0 lex le +infin

In medium 1 the governing equation (5) becomes

κpart2u

partζ2+ i

partζ+

12part2u

partξ2+ |u|qu = 0 (7)

Sufficiently far from the interface (7) admits exact analyticalsolitons of the form [56]

u(ξ ζ) = η0sech2q

⎛⎝a ξ minusVincζradic

1 + 2κV 2inc

⎞⎠

times exp

[plusmni

radicradicradic 1 + 4κβ0

1 + 2κV 2inc

(Vincξ +

times exp(minusi

where η0 is the peak amplitude of the beam a = q[ηq0(2 +

q)]12 determines the (inverse) solution width and

β0 = 2ηq0

2 + q(8b)

quantifies nonlinear phase shift through the (typicallysmall) quantity 4κβ0 The plusmn sign flags evolution in theforwardbackward longitudinal direction The propagationangle of the beam in the laboratory (ie the (x z)) framedenoted by θinc and measured with respect to the z axisis related to the transverse velocity parameter Vinc throughtanθinc = (2κ)12Vinc [23 24] In medium 2 u satisfies

κpart2u

partζ2+ i

partζ+

12part2u

partξ2minus Δ

4κu + α|u|qu = 0 (9)

(n01α1)

(n02 α2)

(n01α1)

(n02 α2)

(n01α1)

(n02 α2)

Figure 1 Schematic diagram illustrating (a) internal (θref lt θinc) and (b) external (θref gt θinc) refraction in the laboratory frame Thetransparency condition (θref = θinc) is shown in part (c) External refraction regimes tend to be highly angular and cannot be adequatelydescribed by the paraxial approximation

and one may derive similar families of solitons

⎛⎝aradicα ξ minusVrefζradic

1 + 2κV 2ref

⎞⎠

times exp

[plusmni

radicradicradic1minus Δ + 4κβ0α

1 + 2κV 2ref

(Vrefξ +

times exp(minusi

Note that the connection between transverse velocity Vref

and propagation angle θref that is tanθref = (2κ)12Vref isunaffected by the (additional linear) term Δ4κ in (9) or bythe nonlinear coefficient α The geometry of these solitonsand their inherent stability against perturbations to the localbeam shape was explored in detail in [56]

23 Phase Continuity and Refraction In recent analyses wehave shown that arbitrary-angle refraction is well describedby anticipating that the phase distribution of the light becontinuous across the interface [17 18 25 26] Matchingthe phases of solutions (8a) and (10) at x = 0 leads to therequirement that

plusmnradicradicradic 1 + 4κβ0

1 + 2κV 2inc= plusmn

1 + 2κV 2ref

Hence continuity is possible if and only if the incident andrefracted solitons share a common longitudinal sense (ieboth must be in either the forward or backward directions)By rearranging (11) one can show that Vref is related to Vinc

through

V 2ref = V 2

inc minus1

(1 + 2κV 2

1 + 4κβ0

)[Δ + 4κβ0(1minus α)

] (12)

Expressed in this way (12) provides a helpful form ldquoV 2ref =

V 2inc + deviationrdquo where the sign of the deviation can be

analysed separately It is then instructive to define a netmismatch parameter δ as [17 18]

δ equiv Δ + 4κβ0(1minus α) (13)

This parameter can be interpreted as the sum of linear andnonlinear mismatches in material parameters Its sign fully

characterizes beam refraction When δ gt 0 one has thatV 2

ref lt V 2inc which is equivalent to θref lt θinc This regime

is referred to as internal refraction and it corresponds to thesituation where the beam in medium 2 is deviated towardthe interface (see Figure 1(a)) Conversely δ lt 0 implies thatV 2

ref gt V2inc or equivalently θref gt θinc This external refraction

regime corresponds to the beam in medium 2 being bentaway from the interface (see Figure 1(b)) The special caseof δ = 0 is the transparency condition where linear andnonlinear index mismatches oppose each other exactly sothat V 2

ref = V 2inc (or θref = θinc) The interface is thus

essentially transparent to the incident beam (see Figure 1(c))which experiences no net change in dielectric properties as itcrosses the boundary It is interesting to note that the absenceof an interface provides a parameter subset (ie Δ = 0 andα = 1) that satisfies the transparency condition identically

24 The Helmholtz-Snell Law for Spatial Solitons By recog-nizing the rotational symmetry inherent to Helmholtz spatialsolitons [23 24 56] it becomes clear that ldquoforwardrdquo andldquobackwardrdquo designations are arbitrary The only physicaldistinction between the two families is the propagationdirection relative to the observer By deploying trigonometricidentities to eliminate velocities Vinc and Vref the forwardand backward solutions in each medium may be written as

u(ξ ζ) = η0sech2q[a(ξ cos θinc minus ζradic

2κsin θinc

times exp

⎡⎣i

radic1 + 4κβ0

(ξ sin θinc +

ζradic2κ

cos θinc

)⎤⎦

times exp(minusi

u(ξ ζ) = η0sech2q[aradicα(ξ cos θref minus ζradic

2κsin θref

times exp

⎡⎣i

radic1minus Δ + 4κβ0α

times(ξ sin θref +

ζradic2κ

cos θref

times exp(minusi

In this representation the angles are bounded byminus180 lt θinc ref le +180 with respect to the z-axisBy matching the solution phase at ξ = 0 one canobtain a compact Helmholtz-Snell refraction law involvinglaboratory-frame angles

γn01 cos θinc = n02 cos θref (15a)

γ equiv[

1 + 4κβ0

1 + 4κβ0α(1minus Δ)minus1

It is worthwhile noting that (15a) has a form which isalmost exactly identical to that encountered when studyingthe classic electromagnetic problem of plane wave refractionat the boundary between different linear dielectrics Thusthe single correction factor γ captures the interplay betweenfinite-waist beams (through the appearance of κ) anddiscontinuities in both the linear and nonlinear propertiesof the adjoining media The exponent q appears implicitlythrough β0

When a beam encounters the boundary with a rarermedium there is little penetration of light across thatboundary until the incidence angle exceeds a critical valuedenoted by θcrit At criticality where θinc = θcrit the trajectoryof the incident beam is deviated so that in principle theoutgoing beam travels along the interface (ie θref = 0)Applying this condition to law (15a) and (15b) leads toan analytical prediction for θcrit in terms of the mismatchparameters Δ and α and also the solution parameter 4κβ0

tan θcrit =[Δ + 4κβ0(1minus α)1minus Δ + 4κβ0α

In practice one rarely finds the refracted soliton travellingalong the interface boundary since other effects tend toappear (we will return to this point later)

25 Universal versus Specific Representations There is clearlya universal flavour about (12) (13) (15a) (15b) and (16)For instance there is no explicit mention of the systemnonlinearity so that refraction is fully described by themismatch parameters Δ and α and the beam parameter4κβ0 These equations are in fact more general than theyfirst appear for instance laws of exactly the same structuregovern the refraction of plane waves in power-law materialsa wave with real amplitude u0 has β0 equiv u

q0 (it is noteworthy

that the refraction analysis for plane waves does not capturethe modulational instability of such solutions in the singlepower-law context [58])

The power-law nature of the problem becomes apparentafter one substitutes for β0 from (8b) The γ factor (cf (15b))then becomes

1 + 8κηq0

(2 + q

)minus1

1 + 8κηq0α(2 + q

)minus1(1minus Δ)minus1

while the relation for the critical angle (cf (16)) is given by

tan θcrit =[Δ + 8κη

(2 + q

)minus1(1minus α)

1minus Δ + 8κηq0α(2 + q

)minus1

and the net mismatch parameter (cf (13)) is δ = Δ +8κη

q0(1minus α)(2 + q)

3 Simulations of Solitons atPower-Law Interfaces

The Helmholtz type of off-axis nonparaxiality demands thatthe inequalities κ O(1) and 4κβ0 O(1) are alwaysmet which is equivalent to the simultaneous requirementsof broad beams with moderate intensities respectively [2324 56] Here attention is restricted to configurations wherethe mismatch parameters are relatively small typically α =O(1) and |Δ| O(1) We now proceed with a three-stage analysis The simplest case to consider first is that oflinear interfaces We then move on to investigate nonlinearinterfaces and conclude by noting the dependence of GHshifts [48 49] on the nonlinearity exponent q Stable solitonsof the homogeneous power-law Helmholtz model tend toexist in the continuous interval 0 lt q lt 4 [27 56] Fordefiniteness we consider here only three discrete values q =1 (sub-Kerr) 2 (Kerr) and 3 (super-Kerr)

31 Solitons at Linear Interfaces From (13) linear interfacesare defined by the inequality 4κβ0|1 minus α| |Δ| To isolatethe effects of a linear-index change alone we set α = 10so that δ = Δ One therefore finds the existence of acritical angle in regimes where Δ gt 0 (since n02 lt n01)The following simulations consider q = 1 Figure 2 showsgenerally good agreement between theoretical predictionsand full numerical calculations when κ = 25times10minus3 the levelof agreement is improved even further when κ = 10times 10minus4

The fact that smaller values of κ give rise to better theory-numerics agreement despite the increased magnitude of thelinear-interface perturbation term at Δ4κ invites commentWe suspect that one possible explanation may lie in theorigin of the Helmholtz-Snell law whereby one matchessolution phase (but not amplitude) at the boundary thematching condition thus takes no account of amplitudecurvature In the laboratory frame broader beams (iethose characterized by smaller κ values) tend to have loweramplitude curvature and the corresponding spatial solitons(which play the role of nonlinear basis functions) thus mapmuch more consistently onto the inherent assumptions ofthe analytical approach

Upon crossing the interface the refracted soliton maysuffer small oscillations (in its amplitude width and area)

0 2 4 6 8 10

θinc (degrees)

0 2 4 6 8 10

θinc (degrees)

|Δ| = 0001|Δ| = 00025

|Δ| = 0005|Δ| = 001

Figure 2 Comparison of the theoretical Snellrsquos law given by (15a)and (15b) (lines) against full numerical computations (points) fora unit-amplitude (η0 = 10) spatial soliton at a linear interface (α =10) with q = 1 and when (a) κ = 25times10minus3 and (b) κ = 10times10minus4Curves below (above) the θref = θinc line have Δ gt 0 (Δ lt 0) so thatthe refraction is internal (external)

reminiscent of those reported in previous studies [56] and beaccompanied by a radiation pattern Computations [59] haveverified the effective independence of the refraction angle θref

with respect to the incident amplitude η0 Accordingly thecurves in Figure 2 are essentially insensitive to q they arequantitatively very similar to those obtained for q = 2 [10]

and (when θinc is sufficiently above θcrit in internal-refractionregimes) for q = 3

Any interaction between a spatial soliton and an interfacegenerally involves three distinct components a reflectedbeam a refracted beam (sometimes more than one) andradiation The way in which the incident energy is distributedamongst these components depends on a complicatedinterplay between the interface and beam parameters andalso the incidence angle At very small angles (eg θinc lt1) the interaction can be highly inelastic and nonadiabatic(especially in external refraction regimes) Crucially thesingle refracted soliton (as predicted in Section 2) dominatesas θinc approaches even modest nonparaxial angles withreflected and radiation components hardly excited at all TheHelmholtz-Snell law embodied by (15a) and (15b) is ofcourse most valid in such regimes

32 Solitons at Nonlinear Interfaces Nonlinear interfaceeffects dominate beam refraction when 4κβ0|1 minus α| |Δ| Without loss of generality we isolate such effects bysetting Δ = 0 so that the net mismatch parameter is givenby δ = 4κβ0(1 minus α) = 8κη

q0(1 minus α)(2 + q) Refraction

thus becomes far more sensitive to κ in nonlinear regimes(compare this to linear regimes where δ = Δ is independentof κ) Theoretical predictions are shown in Figure 3 Whilethere is generally good agreement with numerics for bothκ = 25 times 10minus3 and κ = 10 times 10minus4 when α asymp 10 thefit becomes less reliable for α = 20 and α = 03 For suchparameters the nonlinear refractive index change across theboundary is no longer small one cannot expect to find sucha close match because of strong nonlinear effects (eg beamsplitting and radiation phenomena) While the fit is clearlybetter for broader beams (κ = 10 times 10minus4) the Helmholtz-Snell predictions for narrower beams (κ = 25 times 10minus3) arestill in good qualitative agreement

Detailed attention is first paid to regimes with α gt 1(external refraction since δ lt 0) where the nonlinearityis stronger in the second medium Since the width of therefracted soliton is proportional to αminus12 it follows that thebeam must become narrower as it crosses the interface Inthis type of material regime the incident soliton always hassufficient energy flow to excite a self-trapped soliton-likestate in medium 2

Simulations have shown that nonlinear external refrac-tion tends to induce stronger oscillations in the parameters(amplitude width and area = amplitude times width) of theoutgoing beam than in the linear case Such oscillationsare not captured by the adiabatic analysis in Section 2(which anticipates a stationary state) but one expects theirappearance intuitively Qualitatively different effects canappear at quasi-paraxial incidence angles as the exponentq is varied an illustrative example is shown in Figure 4for θinc = 3 when α = 20 A unit-amplitude solitonexhibits a pronounced splitting phenomenon when q = 1(see Figure 4(b)) whereby the field distribution in the secondmedium is shared between a dominant externally refractedbeam (as predicted by analysis) and a weaker internallyrefracted component (there is also a low-amplitude reflected

0 2 4 6 8θinc (degrees)

0 05 15 25

θinc (degrees)

α = 09α = 07α = 05α = 03

α = 11α = 13α = 15α = 2

Figure 3 Comparison of the theoretical Snellrsquos law given by (15a)and (15b) (lines) against full numerical computations (points) for aunit-amplitude (η0 = 10) spatial soliton at a nonlinear interface(Δ = 0) with q = 1 and when (a) κ = 25 times 10minus3 and (b)κ = 10times 10minus4 Curves below (above) the θref = θinc line are labelledby the right-hand (left-hand) legend and have α lt 1 (α gt 1)so that refraction is internal (external) Note that the numericaldatapoints for α = 03 and α = 05 are very close together in bothpanes

component in the form of radiation modes) Since theinternally refracted beam carries away some of the momen-tum of the incident beam it follows that the dominantrefracted beam travels at a smaller angle than that predictedby (15a) and (15b) This type of splitting is not presentfor unit-amplitude solitons with q = 2 (see Figure 4(b))though it may appear for incident solitons with higher peakintensities [60] In such cases the properties of the daughtersolitons may be quantified with recourse to inverse scatteringtechniques Splitting is also absent at q = 3 (see Figure 4(c))though one finds quite a complicated radiation ripple patternin the second medium

Refraction in nonparaxial regimes tends to be a muchcleaner process with little radiation generated by the beam-interface interaction in comparison with quasi-paraxialregimes Even at modest angles (eg θinc = 30) where theinterface perturbation is distributed over a relatively shortinteraction length the quantitative characteristics of theoutgoing beam depend crucially on the power-law exponentBoth the depth of modulation and (longitudinal spatial)frequency of the oscillations tend to increase with q as shownin Figure 5(a) When q = 2 the oscillations tend to vanish inζ for q = 1 and 3 they survive in the long-term evolution(this is also true for the oscillations shown in Figure 4(a))A more detailed comparison of how the q affects beamrefraction is presented in Figures 5(b)ndash5(d)

For material combinations with α lt 1 (internal refrac-tion since δ gt 0) the nonlinearity is weaker in thesecond medium In that case one should expect a criticalangle to exist (in accordance with (17b)) If the incidentsoliton survives the interaction with the interface then therefracted beam may be expected to undergo self-reshapingoscillations in its parameters with the overall trend beingtoward an increase in solution width Simulations haveconfirmed this to be the case with diffractive broadeninggenerally accompanied by a reduction in peak amplitude(see Figure 6(a))mdashthese oscillations are reminiscent of thoseuncovered previously for perturbed initial-value problems[56]

Computations have uncovered a range of q-dependenteffects an illustrative sample of which is shown in Figure 6for beams with κ = 25 times 10minus3 a nonparaxial incidenceangle θinc = 30 and a nonlinear mismatch of α = 05 The(longitudinal spatial) frequency of the reshaping oscillationstends to decrease with increasing q (cf the increase withq when α gt 1) Also at higher q values (eg for q = 3)a threshold phenomenon can appear whereby the energy-flow [56] of the incident soliton may not be great enoughto excite a refracted beam (if the energy flows associatedwith solutions (8a) and (10) are denoted by Winc and Wrefrespectively then it can be shown that Wref asymp Wincα12)This instability is shown in Figure 6(d) upon colliding withthe interface the beam breaks up into radiation (this scenariois also present at quasi-paraxial incidence angles above thecritical angle θcrit)

33 Snaking at Nonparaxial Angles Equations (15a) and(15b) show that at nonlinear interfaces the refraction

minus10 0 10 20 30 40 5008

q = 1q = 2

minus10 0 10 20 30 40 5015

minus5

minus10

minus15

minus10 0 10 20 30 40 5015

minus5

minus10

minus15

minus10 0 10 20 30 40 5015

minus5

minus10

minus15

Figure 4 External refraction of a unit-amplitude (η0 = 10) spatial soliton at a nonlinear interface with α = 20 and a quasi-paraxialincidence angle θinc = 3 when κ = 25 times 10minus3 (a) Evolution in ζ of the peak amplitude |u|m of the beam (b) (c) and (d) show the fullnumerical solution |u(ξ ζ)| of (5) when the nonlinearity exponent is q = 1 2 and 3 respectively

angle must depend on q (a prediction supported by simpleinspection of Figures 4 5 and 6) At this point it alsobecomes instructive to consider the trajectory of refractedbeams more carefully Detailed numerical calculations revealthat at quasi-paraxial incidence angles the beam in thesecond medium tends to follow a straightline path Such asimple notion of refraction founded upon intuition fromplane wave theory is illustrated in Figure 7(a) for a nonlinearinterface with α = 20 and a beam with θinc = 3

and κ = 25 times 10minus3 However if the incidence angle isincreased into the nonparaxial domain (eg θinc = 30) aqualitatively different picture emerges Now the straightlinepath ξ minus Vrefζ = 0 predicted by solution (10) defines anaverage trajectory about which the refracted beam ldquosnakesrdquoFigure 7(b) quantifies this snaking effect for the externalrefraction simulations shown in Figures 5(b)ndash5(d) Snakingis more apparent with sub-Kerr nonlinearities (ie whereq lt 2) and it increases for narrower beams (ie largervalues of κ) at a fixed amplitude (see Figure 8(a) whereη0 = 10) Beams with larger amplitudes also exhibit snaking

but oscillations tend to be more rapid in the longitudinaldirection (see Figure 8(b))

The snaking phenomenon is most pronounced inregimes with α gt 1 where the nonlinearity is stronger in thesecond medium There is also an intrinsic dependence on θinc

that can be seen in Figure 7 For small angles of incidencethe incoming soliton experiences an interface perturbationthat is distributed over a relatively long distance Therefracting beam is able to accommodate the inhomogeneityin the system since changes in focusing properties of thehost medium occur gradually in the longitudinal directionFor larger-incidence angles the effective beam-interfaceinteraction length may be much shorter Solitons impingingon the boundary then exhibit a sharp (rather than a gradual)perturbation whose action is to induce sustained oscillations

34 Goos-Hanchen Shifts at Power-Law Interfaces RecentlyGH shifts [48] have been investigated within the context ofHelmholtz spatial solitons at Kerr-type material interfaces[49] These shifts describe the translation in the trajectory

0 5 10 15 2008

minus5

q = 1q = 2q = 3

minus5 0 5 10 15 2010

minus10

minus5

minus5 0 5 10 15 2010

minus10

minus5

minus5 0 5 10 15 2010

minus10

minus5

Figure 5 External refraction of a unit-amplitude (η0 = 10) spatial soliton at a nonlinear interface with α = 20 and a nonparaxial incidenceangle θinc = 30 when κ = 25 times 10minus3 (a) Evolution in ζ of the peak amplitude |u|m of the beam (b) (c) and (d) show the full numericalsolution |u(ξ ζ)| of (5) when the nonlinearity exponent is q = 1 2 and 3 respectively

of a reflected beam relative to its position as predictedby geometrical optics Extensive numerical investigationsconsidered the interplay between incidence angle θinc mate-rial mismatches (Δα) and the nonparaxial parameter κRadiation-induced trapping was found to play a key rolein determining the magnitude of the shift Also uncoveredwere giant external GH shifts (in regimes with δ gt 0 butwhere the second medium has a weaker nonlinearity (ieα lt 1)) While a similar investigation of GH shifts in thepower-law context is certainly outside our current scope asmall selection of results will now be presented to illustratehow they depend upon the nonlinearity exponent q

We begin by considering linear interfaces and unit-amplitude incident solitons with κ = 25 times 10minus3 Accordingto (16) interfaces with Δ = 00025 have a theoretical criticalangle of θcrit asymp 286 (this value depends only very weakly onq) Figure 9(a) gives a representative set of results Inspectionshows that for any θinc the magnitude of the shift is generallygreater for systems with q = 1 than for q = 2 or q = 3The true critical angle (which can only be found throughfull simulations) is also slightly greater than that predictedby theory (for q = 1 and q = 2 θcrit asymp 3016 and θcrit asymp

3030 both angles exceed their theoretical values of θcrit asymp2857 and θcrit asymp 2859 respectively) While the qualitativebehaviour of systems with q = 1 and q = 2 is largely verysimilar strong qualitative differences have been uncovered inthe case of q = 3 As θinc approaches the theoretical criticalangle the incident soliton often becomes unstable against theinterface perturbation Large amounts of radiation tend to begenerated by the interaction (cf Figure 9(d)) so that thereis essentially no reflected or refracted beam and a GH shift isthus not easily quantifiable (or even meaningful) Howeverwhen θinc is sufficiently above θcrit the refraction angle is stillwell described by theory

GH shifts at nonlinear interfaces have also been analyzedresults are presented in Figure 10 for α = 07 and wheresystem nonlinearity has been augmented by consideringincident solitons with η0 = 20 Regimes with Δ =minus0001 and Δ = minus00025 are associated with linearexternal refraction while (13) shows that δ gt 0 (iefor these parameter sets net refraction is internal so thata critical angle should still exist) One general trend toemerge is that the true critical angle is slightly less thanthe theoretical value (cf linear interfaces of Figure 9 where

0 25 50 75 100 1250

q = 1q = 2q = 3

0 25 50 75 100 12520

minus10

minus20

0 25 50 75 100 12520

minus10

minus20

0 25 50 75 100 12520

minus10

minus20

Figure 6 Internal refraction of a unit-amplitude (η0 = 10) spatial soliton at a nonlinear interface with α = 05 and a nonparaxial incidenceangle θinc = 30 when κ = 25 times 10minus3 (a) Evolution in ζ of the peak amplitude |u|m of the beam (b) (c) and (d) show the full numericalsolution |u(ξ ζ)| of (5) when the nonlinearity exponent is q = 1 2 and 3 respectively

the true critical angle slightly exceeds theory) Howeverit is worth noting that the qualitative behaviour predictedby (16) namely that θcrit increases with q is supportedby numerics Close to the (true) critical angle simulationsshow that there is a strong divergence in the GH shift(which becomes highly sensitive to θinc) Two other gen-eral trends are that (i) GH shifts are larger (sometimesnotably) for q = 1 than for q = 3 (ii) in nonlinearregimes the GH shifts depend more strongly on q thanfor the case of linear interfaces (compare Figure 10 toFigure 9(a))

Figure 10(b) reveals new types of behaviour at power-law interfaces when q = 2 In particular for q = 3 oneenters a regime wherein the GH shift no longer increasesmonotonically with θinc instead there is a marked decreasein the shift before the divergence at θinc asymp θcrit sets inThese results provide clear evidence that one can quitereasonably expect to find new qualitative phenomena inmaterial regimes that deviate from the idealized Kerr-typeresponse

4 Helmholtz Nonlinear Surface Waves

Surface waves are well known in nonlinear photonicsbeing stationary localized light states that travel along theinterface between different media The transverse modeprofiles are typically asymmetric due to the differencesin dielectric properties defining the interface We nowderive the surface modes of model (5) using solitons(8a) and (10) as a nonlinear basis These new solu-tions are most conveniently parameterized by β which isrelated to the propagation constant in paraxial theory [2756]

41 Exact Analytical Solutions To proceed one seeks solu-tions to (5) of the form u(ξ ζ) = F(ξ minus ξj) exp(ikζζ)exp(minusiζ2κ) where kζ is the propagation constant and F(ξminusξj) is the (real) envelope profile that is centred on ξj Aftersubstituting for u and defining κk2

ζ minus 14κ equiv β it can beshown that in medium 1

minus10 0 10 20 30 40

minus5

0 4 8 12 16

q = 1q = 2q = 3

Figure 7 External refraction of a unit-amplitude (η0 = 10)spatial soliton at a nonlinear interface with α = 20 when theincidence angle is (a) quasi-paraxial (θinc = 3) and (b) nonparaxial(θinc = 30) for κ = 25 times 10minus3 In (a) the trajectory of thebeam in the second medium is essentially a straight line In (b)the trajectory oscillates (ldquosnakesrdquo) around the straight-line pathpredicted by the analysis in Section 2 Calculations of the beamcentre ξ0 were obtained by fitting the numerical solution at eachlongitudinal position to a trial function of the form ufit(ξ) =η(ζ)sech2qa(ζ)[ξ minus ξ0(ζ)] Black dashed lines best-fit trajectory

u(ξ ζ) =(

2β)1q

sech2q[qradic2β12(ξ minus ξ1)

times exp(plusmniradic

1 + 4κβζ

(minusi

while in medium 2

0 4 8 12 16 20

minus025

0 4 8 12 16 20

minus025

κ = 1 times 10minus4

κ = 1 times 10minus3κ = 25 times 10minus3

Figure 8 External refraction of spatial solitons at a nonlinearinterface with α = 20 for a nonparaxial angle θinc = 30 for q = 1and different values of κ The peak amplitude of the incident beamin each case is (a) η0 = 10 and (b) η0 = 20

u(ξ ζ) =[(

times sech2q

⎡⎣ qradic

(ξ minus ξ2)

⎤⎦

1 + 4κβζ

(minusi

For a nonlinearity exponent q the surface waves associatedwith any given interface are parameterized solely by βThe displacements ξ1 and ξ2 as yet undetermined can befound by considering the auxiliary equations that arise fromrespecting continuity of u and its normal derivative (here

0 1 2 3minus40

θinc (degrees)

04 1 16 22

minus10

q = 2q = 3

minus30 0 30 60 90 120 15040

minus20

minus40

minus30 0 30 60 90 120 15040

minus20

minus40

minus30 0 30 60 90 120 15040

minus20

minus40

Figure 9 Demonstration of the GH shift for a unit-amplitude (η0 = 10) spatial soliton at a linear interface with Δ = 00025 and whenκ = 25times 10minus3 (a) Variation of the GH shift with changing nonlinearity exponent q (the q = 3 results (inset) closely follow those for q = 2until radiation effects come into play more strongly) (b) (c) and (d) show the full numerical solution |u(ξ ζ)| of (5) when q = 1 2 and3 respectively (note that over longer propagation lengths the solution in (d) breaks up into radiation) The incidence angle in (b) (c) and(d) is θinc = 3016 which exceeds the (almost q-independent) critical angle θcrit asymp 286

partupartξ or equivalently dFdξ) across the interface Theseconditions lead to

sech2q(qradic2β12ξ1

times sech2q

⎡⎣ qradic

⎤⎦

tanh(qradic2β12ξ1

times tanh

⎡⎣ qradic

⎤⎦

respectively After some algebraic manipulation of (19a) and(19b) one finds that

ξ1 =radic

2qβminus12 ln

(1plusmnradic1minus δ2

)(20a)

ξ2 =radic

2qβminus12

)minus12

⎛⎝1plusmn

radic1minus μ2

⎞⎠ (20b)

where the parameters δ and μ are given by δ equiv [Δ4κβ(α minus1)]12 and μ equiv [(Δ4κβ)(1 + Δ4κβ)minus1 (1minus 1α)minus1]12

42 Surface Wave Existence Criterion For displacements ξ1

and ξ2 to be real it must be that 0 lt δ2 lt 1 and 0 ltμ2 lt 1 These two simultaneous requirements lead to a third

0 1 2 3

minus15

θinc (degrees)

0 1 2 3

minus15

θinc (degrees)

21 23 25minus20

minus10

q = 1q = 2q = 3

Figure 10 Numerical calculation of the GH shift for incidentspatial solitons with η0 = 20 at a nonlinear interface with α = 07(a) Δ = minus0001 and (b) Δ = minus00025 when κ = 25 times 10minus3 (insetshows the behaviour of the shift for q = 3 around the minimum)

inequality placed on the product 4κβ namely 4κβ gt 4κβminwhere

4κβmin = Δ

αminus 1(21)

(it is interesting to note that 4κβmin is independent ofq) Thus existence criterion (21) for Helmholtz surface

minus6 minus4 minus2 0 2 40

q = 1 (minus)

q = 1 (+)

q = 3 (minus)q = 3 (+)

q = 1 (+) q = 3 (+)

q = 1 (minus) q = 3 (minus)

Figure 11 Nonlinear surface wave profiles for κ = 25times 10minus3 in (a)regime 1 (with Δ = 0005 and α = 20) and (b) regime 2 (with Δ =minus0005 and α = 05) From (21) one has that 4κβmin = 0005 andhence βmin = 05 for the solutions in (a) while 4κβmin = 001 andhence βmin = 10 in (b) In these profiles β = 20 so that β gt βmin

in each case The + and minus signs in the legends refer to the choice ofsign solution in (20a) and (20b)

waves explicitly involves the (inverse) beam size through theappearance of κ Since 4κβ must remain positive it followsthat surface modes are supported in two distinct parameterregimes (i) regime 1 Δ gt 0 and α gt 1 (ie n2

02 lt n201 and

α2 gt α1) and (ii) regime 2 Δ lt 0 and 0 lt α lt 1 (ien02 gt n01α2 lt α1) We mention in passing that (21) isreminiscent of the existence criterion derived by Aceves etal [8] it differs through the explicit appearance of κ Typicalsurface wave profiles are shown in Figure 11

0 05 1 15 2 250

q = 1 (minus)

q = 1 (+)

q = 3 (minus)q = 3 (+)

0 05 1 15 2 25

q = 1 (+) q = 3 (+)

Figure 12 Power curves as a function of the propagation constantβ obtained from (22) with κ = 25 times 10minus3 (a) Regime 1 withΔ = 0005 and α = 20 and (b) regime 2 with Δ = minus0005 andα = 05 The + and minus signs in the legends refer to the choiceof solution in (20a) and (20b) Lower (upper) solution branchesappear as red (blue) lines and each branch generally satisfies theVK stability criterion [61]

43 Solution Families and Wave Power For both forward-and backward-propagating surface waves there exist twosolution families The origin of this duality lies in solvingsimultaneous equations (19a) and (19b) where one iseventually obliged to find the roots of quadratic equationsFigure 11 reveals that for fixed (Δ α β) the profile dependsstrongly on the nonlinearity exponent q That is the peakamplitude width and area all decrease with increasingq The difference between the two peak amplitudes andthe distance of each solution peak from the interface alsodecrease with increasing nonlinearity exponent

Since the surface wave profiles differ it is plausible thatthe two families will not share the same stability propertiesWe begin an analysis of Helmholtz solutions (18a) and (18b)by considering the power P where

P(β q

) equivint +infin

minusinfindξ∣∣u(ξ ζ)

∣∣2 (22)

as a function of the free parameter β for different valuesof the nonlinearity exponent q The energy-flow invariantW [56] is related to P through W(β) = plusmn(1 + 4κβ)12P(β)where the plusmn sign here corresponds to forward- or backward-propagating envelopes (being distinct from the sign choicein (20a) and (20b)) A representative set of curves is shownin Figure 12 where it can be seen that P(β) comprises twobranches In regime 1 (where Δ gt 0 and α gt 1) the lower(upper) branch corresponds to the minus(+) sign in (20a) and(20b) This situation is reversed for regime 2 (where Δ lt0 0 lt α lt 1) in which the lower (upper) branch correspondsto the +(minus) sign (see Figure 11) We note that for lower-branch solutions the peak of the surface wave always residesin whichever medium has the lower linear refractive index

Global trends in the parameter dependence of the modesprofiles can be readily identified and discussed in the contextof the two solution branches For instance one mightfix Δ β and κ and consider the effect of varying α Inregime 1 one finds that upon increasing α the upper-branch solutions tend to retain their shape while the lower-branch solutions experience a decrease in amplitude widthand area The separation between the pair of solutions alsobecomes greater with each localized wave moving away fromthe interface As α is increased in regime 2 the lower-branchsolutions tend to retain their shape while the upper-branchsolution exhibits decreases in amplitude width and areaAlso the separation between the solutions tends to decreasewith increasing α (so that the solutions move toward theboundary)

44 Surface Wave Stability Except near the intersectionpoint (where β asymp βmin) both P(β) branches satisfy the classicVakhitov-Kolokolov (VK) criterion for stability namelydPdβ gt 0 [61] Extensive simulations have revealed thatlower-branch solutions always tend to remain self-trappedwithin the vicinity of the interface (so long as dPdβ gt0) evolving with a stationary profile over arbitrarily longdistances

Upper-branch solutions tend to display a spontaneousinstability in finite ζ A set of typical results is shownin Figure 13 for regime 1 with Δ = 0005 and α =20 where the input wave is localized predominantly inmedium 1 (compare with Figure 11(a)) The initial stages ofevolution appear to be stationary but instability sets in aftera finite propagation length The unstable solution deviatesspontaneously into medium 2 crossing the boundary andshedding radiation in the process The beam in medium2 undergoes narrowing since α gt 1 For fixed interfaceand solution parameters the instability growth rate clearlyincreases with q However the angular deviation of the

0 7 14 21 28 3515

q = 1q = 2q = 3

0 7 14 21 28 3510

minus5

minus10

0 7 14 21 28 3510

minus5

minus10

0 7 14 21 28 3510

minus5

minus10

Figure 13 Spontaneous instability of nonlinear surface waves lying on the upper solution branch of Figure 12(a) where κ = 25times 10minus3 andβ = 20 (interface mismatch parameters are Δ = 0005 and α = 20) (a) Evolution in ζ of the peak amplitude |u|m of the beam (b) (c) and(d) show the full numerical solution |u(ξ ζ)| of (5) when the nonlinearity exponent is q = 1 2 and 3 respectively Note that the profiles ofthe input waves in (b) and (d) correspond to the upper-branch solutions shown in Figure 11(a)

(reshaping) daughter beam relative to the interface is largelyinsensitive to q

Qualitatively different effects appear in regime 2 withΔ = minus0005 and α = 05 this time the input wave is localizedpredominantly in medium 2 (compare with Figure 11(b))After a finite propagation length the surface wave bendssmoothly away from the interface and is deflected deeperinto medium 2 There is relatively little radiation shed inthis process and the localized wave suffers only a very smallchange to its shape (largely because the beam remains alwayson the same side of the interface so does not encounterchanges in refractive index) In common with regime 1 theinstability growth rate increases with q

45 Interactions between Solitons and Surface Waves Thestability of lower-branch surface waves is now investigated byconsidering their resilience against interactions with spatial

solitons Only a brief summary is presented here since theprimary motivation is to uncover qualitatively new effectsthat depend upon the exponent q (detailed quantitativeanalyses are reserved for future works) For definitenesswe present simulation results for collisions between a unit-amplitude (η0 = 10) soliton and surface waves in regimes1 (Δ = 0005α = 20) and 2 (Δ = minus0005α = 05) withβ = 20 and κ = 25times 10minus3

Regime 1 is considered first for a quasi-paraxial incidenceangle of θinc = 3 (see Figure 14) When q = 1 the twodistinct beams persist after the interaction The path of theoutgoing soliton has been deflected relative to its ingoingtrajectory The surface wave on the other hand survives asa localized spatial structure but can no longer be interpretedas a ldquosurface waverdquo per se since it travels obliquely to (notalong) the interface This picture is qualitatively different forq = 2 and 3 there the interaction results in the coalescence

0 10 20 30 40 50 601

q = 1q = 2q = 3

0 10 20 30 40 50 6020

minus10

minus20

0 10 20 30 40 50 6020

minus10

minus20

0 10 20 30 40 50 6020

minus10

minus20

Figure 14 Quasi-paraxial interaction (θinc = 3) between a lower-branch nonlinear surface wave (with β = 20) and a unit-amplitude(η0 = 1) soliton in regime 1 (mismatch parameters Δ = 0005 and α = 20) with κ = 25 times 10minus3 (a) Evolution in ζ of the peak amplitude|u|m of the solution Parts (b) (c) and (d) show the full numerical solution |u(ξ ζ)| of (5) when the nonlinearity exponent is q = 1 2 and3 respectively

of the soliton and surface wave producing a single higher-intensity narrow filament travelling obliquely to the interface(narrowing is to be expected for medium combinations withα gt 1) It is noteworthy that the propagation angle of thefilament relative to the interface increases with q Alsoas one might expect nonlinear beams interacting at quasi-paraxial angles tend to shed a large amount of radiation

The qualitative behaviour can change dramatically atnonparaxial angles a representative set of simulations forθinc = 30 is shown Figure 15 We have not observedcoalescence phenomena instead of this individual beamsretain their separate identities and can be clearly resolvedWhile the soliton often survives intact (and experiences anarrowing effect due to α gt 1) the evolution of the surfacewave depends strongly on the nonlinearity exponent (i)for q = 1 it acquires slow modulations in its shape but

remains localized within the vicinity of the interface (ieit remains essentially a surface wave after the interaction)(ii) for q = 2 its path is deviated by the interaction sothat it no longer travels along the interface (this obliquely-evolving self-trapped structure is by definition not a surfacewave) (iii) for q = 3 the collision destroys it completelyIt is interesting to note the general trend that larger-interaction angles generate far less radiation than theirparaxial counterparts [62]

We now turn our attention to similar interaction sce-narios in regime 2 For a quasi-paraxial incidence angle of3 the behaviour is strikingly different from that uncoveredfor the same angle in regime 1 (compare Figures 16 and14 respectively) When q = 1 the soliton survives theinteraction and the surface wave remains quasi-bound to theinterface (but exhibiting a longitudinal ldquoskimmingrdquo effect)

0 2 4 6 8 10 12

q = 1q = 2

minus10

minus20

0 2 4 6 8 10 12

minus10

minus20

0 2 4 6 8 10 12

minus10

minus20

0 2 4 6 8 10 12

Figure 15 Nonparaxial interaction (θinc = 30) between a lower-branch nonlinear surface wave (with β = 20) and a unit-amplitude(η0 = 1) soliton in regime 1 (mismatch parameters Δ = 0005 and α = 20) with κ = 25 times 10minus3 (a) Evolution in ζ of the peak amplitude|u|m of the solution (b) (c) and (d) show the full numerical solution |u(ξ ζ)| of (5) when the nonlinearity exponent is q = 1 (surface wavefollows interface) 2 (surface wave deflected) and 3 (surface wave destroyed) respectively

For q = 2 and 3 the interaction deflects the surface waveaway from the boundary (ie the surface wave becomes anobliquely-evolving beam) However the behaviour of thesoliton is different for q = 2 and 3 it survives intact in theformer case and breaks up into radiation in the latter (thiseffect is related to the threshold phenomenon discussed inSection 32 and is not a consequence of the interaction withthe surface wave)

5 Conclusion

We have presented to the best of our knowledge thefirst investigation of the way spatial solitons behave atthe planar interface between dissimilar materials whoserefractive index has a power-law dependence on the electricfield amplitude This analysis has thus extended arbitrary

angle refraction considerations beyond the ubiquitous Kerr-type case [17 18 25 26] Exact analytical solitons have beendeployed as a nonlinear basis [56] permitting the derivationof a generalized Helmholtz-Snell law Extensive numericalcomputations have tested its predictions which are mostaccurate in regimes where only the linear refractive indexchanges across the boundary

A range of new quantitative and qualitative effects thatdepend strongly upon the exponent q has been identifiedFor example simulations have found that at linear interfaceswith Δ gt 0 and where q = 1 or 2 there is generallya well-defined transition (as θinc increases) from solitonreflection through GH shifting to soliton refraction Incontrast systems with q = 3 are often far more complex thereflection-to-refraction transition is generally obscured byradiation effects over a finite band of incidence angles aroundthe (theoretical) critical angle solitons interacting with

0 10 20 30 40 50 601

q = 1q = 2

0 40 80 120 160 20020

minus10

minus20

0 40 80 120 160 20020

minus10

minus20

0 40 80 120 160 20020

minus10

minus20

Figure 16 Quasi-paraxial interaction (θinc = 3) between a lower-branch nonlinear surface wave (with β = 20) and a unit-amplitude(η0 = 1) soliton in regime 2 (mismatch parameters Δ = minus0005 and α = 05) with κ = 25times 10minus3 (a) Evolution in ζ of the peak amplitude|u|m of the solution (b) (c) and (d) show the full numerical solution |u(ξ ζ)| of (5) when the nonlinearity exponent is q = 1 (surface waveldquoskimmingrdquo) 2 (deflection of the surface wave) and 3 (deflection of the surface wave and breakup of the soliton into radiation) respectively

the interface may collapse into low-amplitude diffractingwaves with GH shifts becoming difficult to interpret orquantify in the absence of a well-defined reflected beamHowever strong supporting evidence has been obtainedto confirm the validity of our Helmholtz-Snell modellingin arbitrary-angle non-Kerr regimes In this way the firststeps have been taken towards understanding how (fully2D) diffractionnonlinearity interplays govern spatial solitonrefraction in a much wider class of systems

Nonlinear surface waves of model (5) have been derivedand we have performed the first numerical analysis of thesetypes of solutions Simulations have addressed the stabilityproperties of the new surface waves which tend to lie onone of two possible branches of the classic (β P) curvesSolutions lying on the lower branch are predicted to behaveas stable robust entities while solutions on the upper branchare inherently unstable Extensive computations have lent

direct numerical support for this stability prediction in themore general Helmholtz context and the growth rate of theupper-branch instability has been found to increase with q

The stability properties of lower-branch Helmholtz sur-face waves have been further investigated by consideringcollisions with obliquely incident spatial solitons A richvariety of behaviours which depend crucially on both thenonlinearity exponent and the interaction angle has beendiscovered Finding analytical descriptions (eg througha perturbation theory [62]) of these phenomena seems aremote possibility since much of the behaviour is clearly non-adiabatic Hence computer simulations play a fundamentalrole in investigating solitons surface waves and theirinteractions in non-Kerr regimes

The research presented in this paper provides a clearindication that deviating from the ideal Kerr-type nonlin-earity (q = 2) can give rise to novel interesting and

potentially exploitable phenomena Each component of thispaper (testing the Helmholtz-Snell law calculating GH shiftsanalyzing surface wave stability and studying soliton-surfacewave interactions) is a problem for detailed investigation inits own right Our findings unpin analyses of other typesof optical (and nonoptical) contexts involving solitons andsurface waves where the power-law type of nonlinearity takescentre stage One can expect other distinct classes of surfacewave to exist when the interface comprises combinations offocusingdefocusing power-law nonlinearities [42 43 63]the stability properties of these waves can quite reasonablybe expected to differ from those reported here Furthermorethe validity of our Helmholtz-Snell modelling in power-law regimes suggests that it may also be applicable to othermaterial configurations for example to single- and multi-interface problems with cubic-quintic [64ndash67] and saturable[68ndash70] nonlinearities Research is currently underway thatinvestigates the generality of our findings in these other con-texts and preliminary results do suggest wider applicability

Acknowledgment

This work was supported by the Engineering and Phys-ical Sciences Research Council (EPSRC) grant numberEPH0115951

References

[1] P W Smith and W J Tomlinson ldquoNonlinear optical in-terfaces switching behaviourrdquo IEEE Journal of QuantumElectronics vol 20 no 1 pp 30ndash36 1984

[2] W J Tomlinson J P Gordon P W Smith and A E KaplanldquoReflection of a Gaussian beam at nonlinear interfacerdquo AppliedOptics vol 21 no 11 pp 2041ndash2051 1982

[3] P W Smith J P Hermann W J Tomlinson and P J MaloneyldquoOptical bistability at a nonlinear interfacerdquo Applied PhysicsLetters vol 35 no 11 pp 846ndash848 1979

[4] A E Kaplan ldquoTheory of hysteresis reflection and refractionof light by a boundary of a nonlinear mediumrdquo Soviet PhysicsJournal of Experimental and Theoretical Physics vol 45 no 1pp 896ndash905

[5] A E Kaplan ldquoHysteresis reflection and refraction by nonli-near boundary a new class of effects in nonlinear opticsrdquoJournal of Experimental and Theoretical Physics Letters vol 24no 1 pp 115ndash119 1976

[6] A B Aceves J V Moloney and A C Newell ldquoReflection andtransmission of self-focused channels at nonlinear dielectricinterfacesrdquo Optics Letters vol 13 no 11 pp 1002ndash1004 1988

[7] A B Aceves J V Moloney and A C Newell ldquoSnellrsquos laws atthe interface between nonlinear dielectricsrdquo Physics Letters Avol 129 no 4 pp 231ndash235 1988

[8] A B Aceves J V Moloney and A C Newell ldquoTheory oflight-beam propagation at nonlinear interfaces I Equivalent-particle theory for a single interfacerdquo Physical Review A vol39 no 4 pp 1809ndash1827 1989

[9] A B Aceves J V Moloney and A C Newell ldquoTheory of light-beam propagation at nonlinear interfaces II Multiple-particleand multiple-interface extensionsrdquo Physical Review A vol 39no 4 pp 1828ndash1840 1989

[10] A B Aceves P Varatharajah A C Newell et al ldquoParticlesaspects of collimated light channel propagation at nonlinearinterfaces and waveguidesrdquo Journal of the Optical Society ofAmerica B vol 7 no 6 pp 963ndash974 1990

[11] P Varatharajah A C Newell J V Moloney and A B AcevesldquoTransmission reflection and trapping of collimated lightbeams in diffusive Kerr-like nonlinear mediardquo Physical ReviewA vol 42 no 3 pp 1767ndash1774 1990

[12] A B Aceves and J V Moloney ldquoEffect of two-photonabsorption on bright spatial soliton switchesrdquo Optics Lettersvol 17 no 21 pp 1488ndash1490 1992

[13] Y M Aliev A D Boardman A I Smirnov K Xie andA A Zharov ldquoSpatial dynamics of solitonlike channels nearinterfaces between optically linear and nonlinear mediardquoPhysical Review E vol 53 no 5 pp 5409ndash5419 1996

[14] Y M Aliev A D Boardman K Xie and A A ZharovldquoConserved energy approximation to wave scattering by anonlinear interfacerdquo Physical Review E vol 49 no 2 pp1624ndash1633 1994

[15] A D Boardman P Bontemps W Ilecki and A A ZharovldquoTheoretical demonstration of beam scanning and switchingusing spatial solitons in a photorefractive crystalrdquo Journal ofModern Optics vol 47 no 11 pp 1941ndash1957 2000

[16] I V Shadrivov and A A Zharov ldquoDynamics of optical spa-tial solitons near the interface between two quadraticallynonlinear mediardquo Journal of the Optical Society of America Bvol 19 no 3 pp 596ndash602 2002

[17] J Sanchez-Curto P Chamorro-Posada and G S McDonaldldquoNonlinear interfaces intrinsically nonparaxial regimes andeffectsrdquo Journal of Optics A vol 11 no 5 Article ID 0540152009

[18] J Sanchez-Curto P Chamorro-Posada and G S McDonaldldquoHelmholtz solitons at nonlinear interfacesrdquo Optics Lettersvol 32 no 9 pp 1126ndash1128 2007

[19] T A Laine and A T Friberg ldquoSelf-guided waves and exactsolutions of the nonlinear Helmholtz equationrdquo Journal of theOptical Society of America B vol 17 no 5 pp 751ndash757 2000

[20] S Blair ldquoNonparaxial one-dimensional spatial solitonsrdquoChaos vol 10 no 3 pp 570ndash583 2000

[21] A P Sheppard and M Haelterman ldquoNonparaxiality stabilizesthree-dimensional soliton beams in Kerr mediardquo OpticsLetters vol 23 no 23 pp 1820ndash1822 1998

[22] M D Feit and J A Fleck ldquoBeam nonparaxiality filamentformation and beam breakup in the self-focusing of opticalbeamsrdquo Journal of the Optical Society of America B vol 5 no3 pp 633ndash640 1988

[23] P Chamorro-Posada G S McDonald and G H C NewldquoExact soliton solutions of the nonlinear Helmholtz equationcommunicationrdquo Journal of the Optical Society of America Bvol 19 no 5 pp 1216ndash1217 2002

[24] P Chamorro-Posada G S McDonald and G H C NewldquoNon-paraxial solitonsrdquo Journal of Modern Optics vol 45 no6 pp 1111ndash1121 1998

[25] J Sanchez-Curto P Chamorro-Posada and G S McDonaldldquoBlack and gray Helmholtz-Kerr soliton refractionrdquo PhysicalReview A vol 83 no 1 Article ID 013828 2011

[26] J Sanchez-Curto P Chamorro-Posada and G S McDonaldldquoDark solitons at nonlinear interfacesrdquo Optics Letters vol 35no 9 pp 1347ndash1349 2010

[27] A W Snyder and D J Mitchell ldquoSpatial solitons of the power-law nonlinearityrdquo Optics Letters vol 18 no 2 pp 101ndash1031993

[28] A Biswas ldquoPerturbation of solitons due to power law non-linearityrdquo Chaos Solitons and Fractals vol 12 no 3 pp 579ndash588 2001

[29] S Konar and A Biswas ldquoSoliton-soliton interaction withpower law nonlinearityrdquo Progress in Electromagnetics Researchvol 54 no 1 pp 95ndash108 2005

[30] E W Laedke and K H Spatschek ldquoLifetime of spikonsrdquoPhysics Letters A vol 74 no 3-4 pp 205ndash207 1979

[31] Y V Katyshev N V Makhaldiani and V G Makhankov ldquoOnthe stability of soliton solutions to the Schrodinger equationwith nonlinear term of the form ψ|ψ|νrdquo Physics Letters A vol66 no 6 pp 456ndash458 1978

[32] D Mihalache M Bertolotti and C Sibilia ldquoNonlinear wavepropagation in planar structuresrdquo Progress in Optics vol 27pp 229ndash313 1989

[33] J G H Mathew A K Kar N R Heckenberg and I GalbraithldquoTime resolved self-defocusing in InSb at room temperaturerdquoIEEE Journal of Quantum Electronics vol 21 no 1 pp 94ndash991985

[34] D S Chemla D A B Miller and P W Smith ldquoNonlinearoptical properties of GaAsGaAlAs multiple quantum wellmaterial Phenomena and applicationsrdquo Optical Engineeringvol 24 no 4 pp 556ndash564 1985

[35] R K Jain and R C Lind ldquoDegenerate four-wave mixing insemiconductor-doped glassesrdquo Journal of the Optical Society ofAmerica vol 73 no 5 pp 647ndash653 1983

[36] S S Yao C Karaguleff A Gabel R Fortenberry C T Seatonand G I Stegeman ldquoUltrafast carrier and grating lifetimes insemiconductor-doped glassesrdquo Applied Physics Letters vol 46no 9 pp 801ndash802 1985

[37] J G Ma ldquoNonlinear surface waves on the interface of two non-Kerr-like nonlinear mediardquo IEEE Transactions on MicrowaveTheory and Techniques vol 45 no 6 pp 924ndash930 1997

[38] A W Snyder and H T Tran ldquoSurface modes of power lawnonlinearitiesrdquo Optics Communications vol 98 no 4ndash6 pp309ndash312 1993

[39] G I Stegeman C T Seaton J Ariyasu T P Shen and J VMoloney ldquoSaturation and power law dependence of nonlinearwaves guided by a single interfacerdquo Optics Communicationsvol 56 no 5 pp 365ndash368 1986

[40] L Wu ldquoExamination of the core field uniformity for 3-layerpower-law nonlinear slab waveguidesrdquo Optics Communica-tions vol 224 no 1ndash3 pp 51ndash56 2003

[41] G I Stegeman E M Wright C T Seaton et al ldquoNonlinearslab-guided waves in non-Kerr-like mediardquo IEEE Journal ofQuantum Electronics vol 22 no 6 pp 977ndash983 1986

[42] J G Ma and I Wolff ldquoTE wave properties of slab dielectricguide bounded by nonlinear non-Kerr-like mediardquo IEEETransactions on Microwave Theory and Techniques vol 44 no5 pp 730ndash738 1996

[43] J G Ma and I Wolff ldquoPropagation characteristics of TE-waves guided by thin films bounded by nonlinear mediardquo IEEETransactions on Microwave Theory and Techniques vol 43 no4 pp 790ndash795 1995

[44] P Varatharajah A B Aceves J V Moloney and E M WrightldquoStationary nonlinear surface waves and their stability indiffusive Kerr-like nonlinear mediardquo Journal of the OpticalSociety of America B vol 7 no 2 pp 220ndash229 1990

[45] D R Andersen ldquoSurface-wave excitation at the interfacebetween diffusive Kerr-like nonlinear and linear mediardquoPhysical Review A vol 37 no 1 pp 189ndash193 1988

[46] Y V Kartashov F Ye V A Vysloukh and L Torner ldquoSurfacewaves in defocusing thermal mediardquo Optics Letters vol 32 no15 pp 2260ndash2262 2007

[47] P J Bradley and C De Angelis ldquoSoliton dynamics and surfacewaves at the interface between saturable nonlinear dielectricsrdquoOptics Communications vol 130 no 1ndash3 pp 205ndash218 1996

[48] F Goos and H Hanchen ldquoEin neuer und fundamentalerVersuch zur Totalreflexionrdquo Annalyen Der Physik vol 1 no1 pp 333ndash346 1947

[49] J Sanchez-Curto P Chamorro-Posada and G S McDonaldldquoGiant Goos-Hanchen shifts and radiation-induced trappingof Helmholtz solitons at nonlinear interfacesrdquo Optics Lettersvol 36 no 18 pp 3605ndash3607 2011

[50] S Chi and Q Guo ldquoVector theory of self-focusing of an opticalbeam in Kerr mediardquo Optics Letters vol 20 no 15 pp 1598ndash1600 1996

[51] M Lax W H Louisell and W B McKnight ldquoFrom Maxwellto paraxial wave opticsrdquo Physical Review A vol 11 no 4 pp1365ndash1370 1975

[52] A Ciattoni B Crosignani S Mookherjea and A Yariv ldquoNo-nparaxial dark solitons in optical Kerr mediardquo Optics Lettersvol 30 no 5 pp 516ndash518 2005

[53] B Crosignani A Yariv and S Mookherjea ldquoNonparaxialspatial solitons and propagation-invariant pattern solutions inoptical Kerr mediardquo Optics Letters vol 29 no 11 pp 1254ndash1256 2004

[54] A Ciattoni P Di Porto B Crosignani and A Yariv ldquoVec-torial nonparaxial propagation equation in the presence of atensorial refractive-index perturbationrdquo Journal of the OpticalSociety of America B vol 17 no 5 pp 809ndash819 2000

[55] B Crosignani P Di Porto and A Yariv ldquoNonparaxialequation for linear and nonlinear optical propagationrdquo OpticsLetters vol 22 no 11 pp 778ndash780 1997

[56] J M Christian G S McDonald R J Potton and PChamorro-Posada ldquoHelmholtz solitons in power-law opticalmaterialsrdquo Physical Review A vol 76 no 3 Article ID 0338342007

[57] J M Christian G S McDonald R J Potton and PChamorro-Posada ldquoErratum Helmholtz solitons in power-law optical materials (Physical Review A (2007) 76 (033834))rdquoPhysical Review A vol 76 no 4 Article ID 049905 2007

[58] J M Christian G S McDonald and P Chamorro-PosadaldquoHelmholtz bright and boundary solitonsrdquo Journal of PhysicsA vol 40 no 7 pp 1545ndash1560 2007

[59] P Chamorro-Posada G S McDonald and G H C NewldquoNon-paraxial beam propagation methodsrdquo Optics Commu-nications vol 192 no 1-2 pp 1ndash12 2001

[60] J Sanchez-Curto P Chamorro-Posada and G S McDonaldldquoHelmholtz bright and black soliton splitting at nonlinearinterfacesrdquo Physical Review A vol 85 no 1 Article ID 0138362012

[61] N G Vakhitov and A A Kolokolov ldquoStationary solutions ofthe wave equation in a medium with nonlinearity saturationrdquoRadiophysics and Quantum Electronics vol 16 no 7 pp 783ndash789 1975

[62] P Chamorro-Posada and G S McDonald ldquoSpatial Kerrsoliton collisions at arbitrary anglesrdquo Physical Review E vol74 no 3 Article ID 036609 2006

[63] Y Chen ldquoBright and dark surface waves at a nonlinearinterfacerdquo Physical Review A vol 45 no 7 pp 4974ndash49781992

[64] J M Christian G S McDonald and P Chamorro-PosadaldquoBistable Helmholtz solitons in cubic-quintic materialsrdquo Phys-ical Review A vol 76 no 3 Article ID 033833 2007

[65] D Mihalache D Mazilu M Bertolotti and C Sibilia ldquoExactsolutions for nonlinear thin-film guided waves in higher-ordernonlinear mediardquo Journal of the Optical Society of America Bvol 5 no 2 pp 565ndash570 1988

[66] D Mihalache and D Mazilu ldquoStability and instability ofnonlinear guided waves in saturable mediardquo Solid StateCommunications vol 63 no 3 pp 215ndash217 1987

[67] K I Pushkarov D I Pushkarov and I V Tomov ldquoSelf-actionof light beams in nonlinear media soliton solutionsrdquo Opticaland Quantum Electronics vol 11 no 6 pp 471ndash478 1979

[68] J M Christian G S McDonald and P Chamorro-PosadaldquoBistable Helmholtz bright solitons in saturable materialsrdquoJournal of the Optical Society of America B vol 26 no 12 pp2323ndash2330 2009

[69] D Mihalache and D Mazilu ldquoStability of nonlinear stationaryslab-guided waves in saturable media a numerical analysisrdquoPhysics Letters A vol 122 no 6-7 pp 381ndash384 1987

[70] D Mihalache and D Mazilu ldquoTM-polarized nonlinear slab-guided waves in saturable mediardquo Solid State Communicationsvol 60 no 4 pp 397ndash399 1986

The Effect of Nonnative Interactions on the Energy Landscapes ofFrustrated Model Proteins

Mark T Oakley1 David J Wales2 and Roy L Johnston1

1 School of Chemistry University of Birmingham Edgbaston Birmingham B15 2TT UK2 University Chemical Laboratories Lensfield Road Cambridge CB2 1EW UK

Correspondence should be addressed to Mark T Oakley mtoakleybhamacuk

Academic Editor Jan Petter Hansen

The 46- and 69-residue BLN model proteins both exhibit frustrated folding to β-barrel structures We study the effect of varying thestrength of nonnative interactions on the corresponding energy landscapes by introducing a parameter λ which scales the potentialbetween the BLN (λ = 1) and Go-like (λ = 0) limits We study the effect of varying λ on the efficiency of global optimisation usingbasin-hopping and genetic algorithms We also construct disconnectivity graphs for these proteins at selected values of λ Bothmethods indicate that the potential energy surface is frustrated for the original BLN potential but rapidly becomes less frustratedas λ decreases For values of λ le 09 the energy landscape is funnelled The fastest mean first encounter time for the globalminimum does not correspond to the Go model instead we observe a minimum when the favourable nonnative interactions arestill present to a small degree

1 Introduction

Proteins are biopolymers constructed from a sequence ofamino acid residues The potential energy landscapes of pro-teins have many degrees of freedom and include importantcontributions between pairs of residues that are distant insequence but close to each other in space Despite thiscomplexity many globular proteins fold to a well-definedthe native state According to the thermodynamic hypothesisthis structure is the global free energy minimum for a giv-en sequence [1] Frustration occurs when there are low-ly-ing structures separated by high barriers [2] All the favour-able interactions between pairs of residues cannot be accom-modated at the same time which can lead to energetic frus-tration where there are several low-lying structures withdifferent patterns of contacts Geometric frustration occurswhen the interconversion of two low-lying structures re-quires the breaking of several favourable contacts

A systematic way to simplify the potential energy surfacefor a protein is to include only attractive interactions bet-ween pairs of residues that are in contact in the native statewhich constitutes a Go model [3] Various on- and off-latticeGo models have been investigated by different authors to

study a range of different proteins In spite of the simplifiedpotential these models have proved capable of reproducingcertain aspects of protein dynamics and thermodynamics[4ndash11] Using a Go model tends to lead to funnelled ener-gy landscapes [12] with very little frustration For some pro-teins neglecting nonnative interactions can have a significantinfluence on the energy landscape [13]

United atom representations introduce a further level ofcoarse-graining which can speed up simulations significant-ly at the cost of atomistic detail The simplest coarse-grainedmodel is the HP model in which each protein residue is re-presented by a single hydrophobic (H) or polar (P) beadand is constrained to lie on a regular lattice [14 15] TheBLN model is an off-lattice generalisation of the HP modelwith three types of bead hydrophobic (B) hydrophilic (L)and neutral (N) The 46-residue sequence [12 16ndash33]B9N3(LB)4N3B9N3(LB)5L and the 69-residue sequence [34ndash38] B9N3(LB)4N3B9N3(LB)4N3B9N3(LB)5L were designedto exhibit frustrated folding and have several alternate β-barrel structures that are separated by large energy barriersDisconnectivity graphs [39] for both of these proteins exhibitenergy landscapes comprising several folding funnels [1238] Using a Go potential for these two proteins changes

0 02 04 06 08 1

Figure 1 Mean first encounter times (number of minimisations)for 100 global optimisation runs initiated from random startingpoints for the 46-residue scaled BLN protein The searches wererun using a genetic algorithm (red) basin-hopping starting fromrandom structures confined to a sphere (green) and basin-hoppingstarting from chain structures with randomised dihedral angles(blue) The error bars are the uncertainties calculated at the 95level

the nature of their energy landscapes and they both exhibitsingle funnels with very little frustration [12 38]

Intermediate potentials can be generated using a parame-ter λ which scales the strength of the nonnative interactionsbetween the Go (λ = 0) and BLN (λ = 1) limits Thefolding thermodynamics of the 46-residue BLN protein havebeen investigated using this scaled BLN potential [23 32 33]showing that most of the frustration is only present for valuesof λ ge 09 The introduction of salt bridges (gatekeepers)to the 46-residue protein also produces energy landscapes ofintermediate character [27 28]

In the present work we study the effect of varying λon the ease of global optimisation of the 46- and 69-residue BLN proteins using a basin-hopping algorithm and agenetic algorithm We also construct disconnectivity graphsto compare the energy landscapes of the proteins for differentvalues of λ

The protein structures were modelled using the followingBLN potential [12 21 26 28]

VBLN = 12Kr

Nminus1sumi=1

(Rii+1 minus Re

)2 +12Kθ

Nminus2sumi=1

(θi minus θe)2

+ εNminus3sumi=1

[Ai(1 + cosφi

(1 + 3 cosφi

+ 4εNminus2sumi=1

Nsumj=i+2

⎡⎣(σ

minusDij

)6⎤⎦

0 5000 10000 15000 20000 25000

Energy minimisations performed

Figure 2 Energy of the minima in the Markov chain for a BH runwhere trapping occurs for the 46-residue scaled BLN protein withλ = 0

Figure 3 The most stable misfolded structure which acts as a trapfor global optimisation of the 46-residue BLN protein illustratedusing the VMD program [40] with a colouring scheme for the beadsthat varies from red to blue (N-terminus to C-terminus)

where Rij is the distance between two beads i and j Thefirst term is a harmonic bond restraint with Kr = 2312εσminus2

and Re = σ The second term is a bond angle restraint withKθ = 20 radminus2 and θe = 18326 rad The third term involvestorsional angles φ defined by four successive beads If twoor more of these beads are N then A = 0 and B = 02 Forall other sequences A = B = 12 The final term introducespairwise nonbonded interactions If one residue is L and theother is L or B then C = 23 and D = minus1 If either of theresidues is N then C = 1 and D = 0 If both residues are BthenC = 1 but the value ofD depends on the presence of thecontact in the native state of the protein For native contactsD = 1 For nonnative contacts D = λ where 0 lt λ lt 1 Thecase where λ = 1 is the original BLN potential and λ = 0 isthe Go potential

1 08 06 04 02 0

Figure 4 The energies of the five most stable BLN-46 structuresrelative to the global minimum as a function of λ Also shown(orange) is the energy of the trap structure illustrated in Figure 3The steep decreases mark the points at which structures cease to belocal minima and collapse into the basin of attraction [41] of theglobal minimum

Table 1 Parameters used for the two optimisation strategies

BLN-46 BLN-69

kTε 23 34

Step sizeσ 065 070

Population size 140 200

Crossover rate 09 09

Mutation rate 005 005

Native contacts are defined as all pairs of residues whereRij is less than a fixed cut-off distance in the native state(global minimum) of the protein When λ = 1 the valueof this cut-off radius will influence the energy landscapeHere we use 1167σ for consistency with previous work[12 28 38]

Global optimisation was performed using the basin-hopping approach [42ndash44] and a Lamarckian genetic algo-rithm [38 45] which are both implemented in the GMINprogram [46] Each algorithm involves local energy minimi-sation after each structural perturbation This minimisationtransforms the potential energy surface into the basins ofattraction of local minima [47] and removes downhill bar-riers The search parameters for both algorithms were opti-mised in previous work for BLN proteins [38] and theseparameters were used without adjustment for all searchespresented here (Table 1) The GMIN input files used for thesesearches are included in the supplementary data (see Sup-plementary Material available online at doi1011552012192613)

The genetic algorithm represents each structure with agenome consisting of the torsion angles in the backboneof the protein Offspring structures are generated by one-point crossover from two parent structures Mutants are

generated by making a copy of an existing structure (parentor offspring) and replacing one of the torsion angles Toprevent stagnation of the genetic algorithm searches a restartoperator was used If an entire generation of offspring con-tains no solutions that are fitter than any of the parent struc-tures a new epoch is started with a new random populationFor the 69-residue protein the fittest structure from eachepoch survives into the next epoch

All conformational searches were run until the globalminimum structure was found We report the mean timetaken to encounter this structure in conformational searchesfrom randomised starting points to compare the explorationof the energy landscape as a a function of λ Searches wereperformed for values of λ between 0 and 1 in steps of 01with additional points at λ = 095 and λ = 099 Theinitial structures for this benchmarking were generated usingtwo alternative methods either random placement of theresidues inside a sphere of radius 3σ or random assignmentof the backbone dihedral angles Full details of all of theglobal optimisation runs are available as supplementary data

The disconnectivity graphs for the model proteins wereconstructed from databases of stationary points generatedusing the PATHSAMPLE program [48] which organises inde-pendent pathway searches using OPTIM [49] All the tran-sition state searches in OPTIM were conducted in Cartesiancoordinates [50] using a quasicontinuous interpolationscheme to avoid chain crossings with local maxima accu-rately refined to transition states by hybrid eigenvector-fol-lowing [51ndash53] Successive pairs of local minima were select-ed for connection attempts within OPTIM using the missingconnection algorithm [54] Disconnectivity graphs [39] willbe illustrated for both the 46- and 69-residue scaled BLNproteins with λ values of 0 05 09 and 1

We also study the effect of λ on key structures of the BLNproteins These structures were reminimised using values ofλ between 0 and 1 in steps of 01 Pathways between pairs ofinteresting minima were studied by Dijkstra analysis [55] inPATHSAMPLE [48] with the discrete paths [56] that make thelargest contribution to the steady-state rate constant [56 57]presented here

With a few exceptions all of the stationary points ofthe BLN model proteins are chiral However the BLN po-tential includes no chiral terms so each structure has an en-antiomer with the same energy When evaluating the opti-misation algorithms we accept convergence to either of theenantiomers of the global minimum When looking at thepathways it is important to use the same chirality for bothstructures otherwise much longer paths result For some ofthe trapped structures pathways to both enantiomers of theglobal minimum can be viable

3 Results

31 BLN-46 Searches for λ = 0 (Go potential) find theglobal minimum much more rapidly than when λ = 1 (BLNpotential) as one would expect for a more funnelled energylandscape [2 58ndash61] However the number of steps requiredvaries nonlinearly between these two extremes and behaves

69The Effect of Nonnative Interactions on the Energy Landscapes of Frustrated Model Proteins

λ = 0

λ = 05

λ = 09

λ = 1

Figure 5 Disconnectivity graphs [39] showing the most stable minima accessible by transition states lower than 7ε from the globalminimum of the 46-residue scaled BLN proteins For λ ge 09 only the 1000 most stable minima are shown The structures of selectedminima are illustrated close to the bottoms of the corresponding branches

differently for each search algorithm When optimising withthe GA the mean first encounter time decreases rapidly fromλ = 1 to λ = 09 and then more slowly to a minimumat λ = 05 (Figure 1) After this minimum there is a smallincrease in the required time as λ decreases to 0 This result isconsistent with previous observations that the introductionof some nonnative interactions can assist the folding of someproteins [62] Below λ = 09 almost all searches find theglobal minimum within the first epoch of the GA For largervalues of λ several searches require two or more epochsleading to much more variation in the first encounter timeThe choice of the random starting configurations for theinitial population of the GA makes little difference to themean first encounter time

In basin-hopping searches the choice of starting struc-tures makes a large difference to the efficiency of the opti-misation When starting from residues randomly distributedinside a sphere for values of λ lt 07 95 of the searches

find the global minimum rapidly The remaining searchesbecome trapped and require several thousand attemptedMonte Carlo moves to escape (Figure 2) In this trap the firstthird and fourth strands are correctly packed but the secondis wrapped around the outside of the protein (Figure 3)Searches with larger values of λ do not become trapped inthis basin which suggests that the nonnative interactionsare important in stabilising the intermediates between thisstructure and the global minimum

The trap configuration lies 124ε above the globalminimum when λ = 0 and becomes more unfavourable forlarger values of λ (Table 2) The fastest escape route fromthis trap involves unthreading of the N-terminus from theloop made by the second strand (Table 2) The energy ofthe highest transition state on this pathway relative to thetrapped state increases from λ = 0 to λ = 09 before levellingoff The highest transition state on this pathway lies abovethe barrier to interconversion of the two enantiomers of

00 02 04 06 08 1

the global minimum For searches starting from a randomset of torsion angles this trapping is much less frequentand is only seen in 3 of the 700 searches performed where0 le λ le 06 By retaining some notion of connectivitythese initial structures cover less of the configurational spacethan the entirely random starting points However thecomplete coverage of conformational space comes at the costof including more unstable structures such as the trap seenhere

The five lowest minima in the BLN-46 protein spanan energy range of less than ε (Figure 4) The two moststable minima are in the same basin and both have all ofthe BB contacts from the native state Across the range ofλ the relative energies of these minima are within 01ε ofeach other with the second-best minimum becoming slightlymore stable as λ decreases and moving below the formerglobal minimum when λ lt 03 [12] The next three minimaare stabilised by some nonnative contacts and become lessstable relative to the global minimum as λ decreases In theregion around λ = 05 these structures cease to be minimaand fall into the basins of attraction [41] of the two lowestenergy structures

The disconnectivity graphs within 7ε of the globalminimum for λ = 0 and λ = 05 are funnelled andalmost indistinguishable (Figure 5) When λ = 09 somefrustration appears in the low-energy regions of the energylandscape but it is still mostly funnelled Almost all of thefrustration is introduced between λ = 09 and λ = 1where several alternate β-barrel structures are separated bybarriers of 4 to 5ε This organisation is consistent with theincrease in the mean first encounter times seen for globaloptimisation with λ gt 09 and agrees with previous studiesof the thermodynamics of the 46-residue protein [32 33]

(a) (b)

(c) (d)

Figure 7 Side and top views of the global minimum (left) andtrapped (right) structures of the 69-residue BLN protein illustratedusing the VMD program [40] with a colouring scheme for the beadsthat varies from red to blue (N-terminus to C-terminus)

1 08 06 04 02 0

Figure 8 The energies of the five most stable BLN-69 structuresrelative to the global minimum as a function of λ Also shown(orange) is the trap structure from Figure 7 The steep decreasesin energy mark the points at which structures cease to be localminima and collapse into the basin of attraction [41] of the globalminimum

where λ = 0 and λ = 05 were found to be good foldersλ = 09 an intermediate folder and λ = 1 a poor folder

32 BLN-69 The behaviour of the GA for the 69-residueprotein is similar to that for the 46-residue protein with thefastest search time found at λ = 05 When optimising withbasin-hopping on the 69-residue protein there are several

λ = 0

λ = 05

λ = 09

λ = 1

Figure 9 Disconnectivity graphs [39] showing the minima accessible by transition states lower than 8ε from the global minimum of the69-residue scaled BLN proteins For λ ge 05 only the 1000 most stable minima are shown The structures of selected minima are illustratedclose to the bottoms of the corresponding branches

Table 2 Energies of the trapped minimum and transition state forescape from the principal kinetic trap in the 46-residue scaled BLNprotein All energies are in units of ε and measured relative to theglobal minimum

λ Etrap Euntrap

00 124 229

05 140 271

09 153 322

10 156 322

slow searches between λ = 04 and λ = 08 (Figure 6)There are multiple trap structures and the one that is seenmost frequently which is responsible for the slowest searchesis formed from three strands from the left-handed barreland three strands from right-handed barrel (Figure 7) This

structure is a six-stranded β-barrel similar to the globalminimum but with two sets of interstrand contacts swapped(1ndash6 and 3-4 in the global minimum compared to 1ndash4 and3ndash6 in the trap)

Conversion from the above structure to the global mini-mum proceeds either by inversion of the three strands atthe N-terminus or of the three strands at the C-terminusThe barriers to these two mechanisms are different and varywith λ (Table 3) The barrier for the fastest pathway forinversion at the C-terminus becomes larger with increasing λHowever the barrier for inversion of the N-terminus variesmuch less with λ In the region where 05 le λ le 07 thebarriers to both routes out of the trap are relatively highwhich is a possible explanation for the slow basin-hoppingoptimisation for these values of λ This is doubtless an over-simplification when we consider that there are multiple trapstructures

Table 3 Energies of the trapped minimum and transition states forescape from the principal kinetic trap by inversion of the N- andC-termini in the 69-residue scaled BLN protein All energies are inunits of ε and measured relative to the global minimum

λ Etrap Euntrap-C Euntrap-N

00 176 302 382

05 167 438 332

07 138 302 311

09 84 257 269

10 51 258 239

For the 69-residue BLN protein the energies of the fivelowest minima span less than 04ε (Figure 8) One structurelies in the same funnel as the global minimum and its relativeenergy increases from 02ε to 16ε when λ decreases from 1to 0 The other three structures occupy different funnels fromthe global minimum with several nonnative contacts andtheir stability decreases steeply with decreasing λ Unlike the46-residue protein the global minimum structure remainsthe same for all values of λ The low-energy region ofdisconnectivity graphs for values of λ between 0 and 09 aremostly funnelled (Figure 9) Almost all of the frustration inthis region of the potential energy surface appears for λ gt 09

4 Conclusions

Much of the energetic frustration in the BLN proteins isremoved once the potential contains a 10 contributionfrom the Go function When looking at geometric frustrationin higher-energy traps the effect of λ is less predictable Theremoval of nonnative interactions can stabilise or destabilisethe transition states that must be crossed to escape fromthese traps Measures of the landscape complexity [30]could provide a useful way to understand the influence ofnonnative interactions and will be considered in future work

Acknowledgments

The authors acknowledge the Engineering and PhysicalSciences Research Council UK (EPSRC) for funding underProgramme Grant EPI0013521 The calculations describedin this paper were performed using the University of Birm-inghamrsquos BlueBEAR HPC service which was purchasedthrough HEFCE SRIF-3 funds (see httpwwwbearbhamacuk)

References

[1] C B Anfinsen ldquoPrinciples that govern the folding of proteinchainsrdquo Science vol 181 no 4096 pp 223ndash230 1973

[2] J D Bryngelson J N Onuchic N D Socci and P G WolynesldquoFunnels pathways and the energy landscape of protein fold-ing a synthesisrdquo Proteins vol 21 no 3 pp 167ndash195 1995

[3] Y Ueda H Taketomi and N Go ldquoStudies on protein foldingunfolding and fluctuations by computer simulation II Athree-dimensional lattice model of lysozymerdquo Biopolymersvol 17 no 6 pp 1531ndash1548 1978

[4] C Micheletti F Seno and A Maritan ldquoPolymer principles ofprotein calorimetric two-state cooperativityrdquo Proteins vol 40no 4 pp 637ndash661 2000

[5] C Clementi H Nymeyer and J N Onuchic ldquoTopologicaland energetic factors what determines the structural details ofthe transition state ensemble and ldquoen-routerdquo intermediates forprotein folding An investigation for small globular proteinsrdquoJournal of Molecular Biology vol 298 no 5 pp 937ndash953 2000

[6] J W H Schymkowitz F Rousseau and L Serrano ldquoSurfingon protein folding energy landscapesrdquo Proceedings of the Nat-ional Academy of Sciences of the United States of America vol99 no 25 pp 15846ndash15848 2002

[7] P Das C J Wilson G Fossati P Wittung-Stafshede K SMatthews and C Clementi ldquoCharacterization of the foldinglandscape of monomeric lactose repressor quantitative com-parison of theory and experimentrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 102no 41 pp 14569ndash14574 2005

[8] A R Lam J M Borreguero F Ding et al ldquoParallel foldingpathways in the SH3 domain proteinrdquo Journal of MolecularBiology vol 373 no 5 pp 1348ndash1360 2007

[9] P F N Faısca R D M Travasso R C Ball and E IShakhnovich ldquoIdentifying critical residues in protein foldinginsights from φ-value and Pfold analysisrdquo Journal of ChemicalPhysics vol 129 no 9 Article ID 095108 2008

[10] R D Hills and C L Brooks ldquoInsights from coarse-grainedgo models for protein folding and dynamicsrdquo InternationalJournal of Molecular Sciences vol 10 no 3 pp 889ndash905 2009

[11] P O Craig J Latzer P Weinkam et al ldquoPrediction of native-state hydrogen exchange from perfectly funneled energy land-scapesrdquo American Chemical Society vol 133 no 43 pp17463ndash17472 2011

[12] M A Miller and D J Wales ldquoEnergy landscape of a modelproteinrdquo Journal of Chemical Physics vol 111 no 14 pp6610ndash6616 1999

[13] L Sutto J Latzer J A Hegler D U Ferreiro and PG Wolynes ldquoConsequences of localized frustration for thefolding mechanism of the IM7 proteinrdquo Proceedings of the Na-tional Academy of Sciences of the United States of America vol104 no 50 pp 19825ndash19830 2007

[14] K F Lau and K A Dill ldquoA lattice statistical mechanicsmodel of the conformational and sequence spaces of proteinsrdquoMacromolecules vol 22 no 10 pp 3986ndash3997 1989

[15] K A Dill S Bromberg K Yue et al ldquoPrinciples of proteinfoldingmdasha perspective from simple exact modelsrdquo Protein Sci-ence vol 4 no 4 pp 561ndash602 1995

[16] J D Honeycutt and D Thirumalai ldquoMetastability of the fold-ed states of globular proteinsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 87 no9 pp 3526ndash3529 1990

[17] J D Honeycutt and D Thirumalai ldquoThe nature of fold-ed states of globular proteinsrdquo Biopolymers vol 32 no 6 pp695ndash709 1992

[18] Z Guo and D Thirumalai ldquoNucleation mechanism forprotein folding and theoretical predictions for hydrogen-exchange labeling experimentsrdquo Biopolymers vol 35 no 1 pp137ndash140 1995

[19] Z Guo and D Thirumalai ldquoKinetics and thermodynamicsof folding of a de novo designed four-helix bundle proteinrdquoJournal of Molecular Biology vol 263 no 2 pp 323ndash343 1996

[20] Z Guo and C L Brooks III ldquoThermodynamics of proteinfolding a statistical mechanical study of a small all-β proteinrdquoBiopolymers vol 42 no 7 pp 745ndash757 1997

[21] R S Berry N Elmaci J P Rose and B Vekhter ldquoLinkingtopography of its potential surface with the dynamics of fold-ing of a protein modelrdquo Proceedings of the National Academyof Sciences of the United States of America vol 94 no 18 pp9520ndash9524 1997

[22] H Nymeyer A E Garcıa and J N Onuchic ldquoFolding funnelsand frustration in off-lattice minimalist protein landscapesrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 95 no 11 pp 5921ndash5928 1998

[23] J E Shea Y D Nochomovitz Z Guo and C L BrooksldquoExploring the space of protein folding Hamiltonians thebalance of forces in a minimalist β-barrel modelrdquo Journal ofChemical Physics vol 109 no 7 pp 2895ndash2903 1998

[24] N Elmaci and R S Berry ldquoPrincipal coordinate analysis on aprotein modelrdquo Journal of Chemical Physics vol 110 no 21pp 10606ndash10622 1999

[25] J E Shea J N Onuchic and C L Brooks ldquoEnergetic frustra-tion and the nature of the transition state in protein foldingrdquoJournal of Chemical Physics vol 113 no 17 pp 7663ndash76712000

[26] D A Evans and D J Wales ldquoFree energy landscapes of modelpeptides and proteinsrdquo Journal of Chemical Physics vol 118no 8 pp 3891ndash3897 2003

[27] A D Stoycheva J N Onuchic and C L Brooks ldquoEffect ofgatekeepers on the early folding kinetics of a model β-barrelproteinrdquo Journal of Chemical Physics vol 119 no 11 pp5722ndash5729 2003

[28] D J Wales and P E J Dewsbury ldquoEffect of salt bridges onthe energy landscape of a model proteinrdquo Journal of ChemicalPhysics vol 121 no 20 pp 10284ndash10290 2004

[29] T Komatsuzaki K Hoshino Y Matsunaga G J Rylance RL Johnston and D J Wales ldquoHow many dimensions arerequired to approximate the potential energy landscape of amodel proteinrdquo Journal of Chemical Physics vol 122 no 8Article ID 084714 pp 1ndash9 2005

[30] G J Rylance R L Johnston Y Matsunaga C-B LiA Baba and T Komatsuzaki ldquoTopographical complexityof multidimensional energy landscapesrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 103 no 49 pp 18551ndash18555 2006

[31] J Kim and T Keyes ldquoInherent structure analysis of proteinfoldingrdquo Journal of Physical Chemistry B vol 111 no 10 pp2647ndash2657 2007

[32] J Kim and T Keyes ldquoInfluence of go-like interactions onglobal shapes of energy landscapes inβ-barrel forming mod-el proteins inherent structure analysis and statistical tempera-ture molecular dynamics simulationrdquo Journal of PhysicalChemistry B vol 112 no 3 pp 954ndash966 2008

[33] J Kim T Keyes and J E Straub ldquoRelationship betweenprotein folding thermodynamics and the energy landscaperdquoPhysical Review E vol 79 no 3 Article ID 030902 2009

[34] S A Larrass L M Pegram H L Gordon and S MRothstein ldquoEfficient generation of low-energy folded states ofa model protein II Automated histogram filteringrdquo Journal ofChemical Physics vol 119 no 24 pp 13149ndash13158 2003

[35] P W Pan H L Gordon and S M Rothstein ldquoLocal-structural diversity and protein folding application to all-betaoff-lattice protein modelsrdquo The Journal of Chemical Physicsvol 124 no 2 p 024905 2006

[36] J Kim J E Straub and T Keyes ldquoStatistical temperaturemolecular dynamics application to coarse-grained β-barrel-forming protein modelsrdquo Journal of Chemical Physics vol 126no 13 Article ID 135101 2007

[37] S-Y Kim ldquoAn off-lattice frustrated model protein with a six-stranded β-barrel structurerdquo Journal of Chemical Physics vol133 no 13 Article ID 135102 2010

[38] M T Oakley D J Wales and R L Johnston ldquoEnergy land-scape and global optimization for a frustrated model proteinrdquoJournal of Physical Chemistry B vol 115 no 39 pp 11525ndash11529 2011

[39] O M Becker and M Karplus ldquoThe topology of multidimen-sional potential energy surfaces theory and application topeptide structure and kineticsrdquo Journal of Chemical Physicsvol 106 no 4 pp 1495ndash1517 1997

[40] W Humphrey A Dalke and K Schulten ldquoVMD visual mol-ecular dynamicsrdquo Journal of Molecular Graphics vol 14 no 1pp 33ndash38 1996

[41] P G Mezey Potential Energy Hypersurfaces Elsevier Amster-dam The Netherlands 1987

[42] Z Li and H A Scheraga ldquoMonte Carlo-minimizationapproach to the multiple-minima problem in protein foldingrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 84 no 19 pp 6611ndash6615 1987

[43] D J Wales and H A Scheraga ldquoGlobal optimization of clust-ers crystals and biomoleculesrdquo Science vol 285 no 5432 pp1368ndash1372 1999

[44] D J Wales and J P K Doye ldquoGlobal optimization by basin-hopping and the lowest energy structures of Lennard-Jonesclusters containing up to 110 atomsrdquo Journal of Physical Chem-istry A vol 101 no 28 pp 5111ndash5116 1997

[45] R L Johnston ldquoEvolving better nanoparticles genetic algo-rithms for optimising cluster geometriesrdquo Dalton Transactionsno 22 pp 4193ndash4207 2003

[46] D J Wales ldquoGMIN A program for finding global minima andcalculating thermodynamic properties from basin-samplingrdquohttpwww-waleschcamacukGMIN

[47] P G Mezey ldquoCatchment region partitioning of energy hyper-surfaces Irdquo Theoretica Chimica Acta vol 58 no 4 pp 309ndash330 1981

[48] D J Wales ldquoPATHSAMPLE A program for refining andanalysing kinetic transition networksrdquo httpwww-waleschcamacukOPTIM

[49] D J Wales ldquoOPTIM A program for characterising station-ary points and reaction pathwaysrdquo httpwww-waleschcamacukPATHSAMPLE

[50] D J Wales ldquoLocating stationary points for clusters in car-tesian coordinatesrdquo Journal of the Chemical Society FaradayTransactions vol 89 no 9 pp 1305ndash1313 1993

[51] L J Munro and D J Wales ldquoDefect migration in crystallinesiliconrdquo Physical Review B vol 59 no 6 pp 3969ndash3980 1999

[52] G Henkelman and H Jonsson ldquoA dimer method for findingsaddle points on high dimensional potential surfaces usingonly first derivativesrdquo Journal of Chemical Physics vol 111 no15 pp 7010ndash7022 1999

[53] Y Kumeda L J Munro and D J Wales ldquoTransition statesand rearrangement mechanisms from hybrid eigenvector-following and density functional theory application to C10H10

and defect migration in crystalline siliconrdquo Chemical PhysicsLetters vol 341 no 1-2 pp 185ndash194 2001

[54] J M Carr S A Trygubenko and D J Wales ldquoFindingpathways between distant local minimardquo Journal of ChemicalPhysics vol 122 no 23 Article ID 234903 pp 1ndash7 2005

[55] E W Dijkstra ldquoA note on two problems in connexion withgraphsrdquo Numerische Mathematik vol 1 no 1 pp 269ndash2711959

[56] D J Wales ldquoDiscrete path samplingrdquo Molecular Physics vol100 no 20 pp 3285ndash3305 2002

[57] D J Wales ldquoEnergy landscapes calculating pathways andratesrdquo International Reviews in Physical Chemistry vol 25 no1-2 pp 237ndash282 2006

[58] J D Bryngelson and P G Wolynes ldquoSpin glasses and thestatistical mechanics of protein foldingrdquo Proceedings of the Na-tional Academy of Sciences of the United States of America vol84 no 21 pp 7524ndash7528 1987

[59] J N Onuchic P G Wolynes Z Luthey-Schulten and N DSocci ldquoToward an outline of the topography of a realistic pro-tein-folding funnelrdquo Proceedings of the National Academy ofSciences of the United States of America vol 92 no 8 pp 3626ndash3630 1995

[60] M Karplus and A Sali ldquoTheoretical studies of protein foldingand unfoldingrdquo Current Opinion in Structural Biology vol 5no 1 pp 58ndash73 1995

[61] J N Onuchic H Nymeyer A E Garcıa J Chahine and ND Socci ldquoThe energy landscape theory of protein folding in-sights into folding mechanisms and scenariosrdquo Advances inProtein Chemistry vol 53 pp 87ndash152 2000

[62] C Clementi and S S Plotkin ldquoThe effects of nonnative in-teractions on protein folding rates theory and simulationrdquoProtein Science vol 13 no 7 pp 1750ndash1766 2004

Proton Transfer Equilibria and Critical Behavior of H-Bonding

L Sobczyk B Czarnik-Matusewicz M Rospenk and M Obrzud

Faculty of Chemistry University of Wrocław Joliot-Curie 14 50-383 Wrocław Poland

Correspondence should be addressed to L Sobczyk lucjansobczykchemuniwrocpl

Academic Editor Marek J Wojcik

The aim of the present paper is an analysis of the hydrogen bond properties for the acid-base systems depending on the ability to theproton transfer in the formulation of the Bronsted approach After definition of the proton transfer equilibrium expressed by usingthe equation logKPT = ξΔpKN various examples of different physical properties such as dipole moments IR spectra and nuclearmagnetic resonances are presented which correlate with the ΔpKN value In such a way a critical state of hydrogen bonding can bedefined that corresponds to the potential of the proton motion for either single minimum or double minimum with low barrierA particular attention in this paper found electronic spectra which have not been analysed so far and the quantitative analysis ofthe vibrational polarizability which can reach very high values of the order of electronic polarizability

1 Introduction

The subject of our interest in the present review is hydrogenbonds which can be expressed as AndashHmiddot middot middotB It is an acid-basesystem in the Bronsted formulation when the AndashH group istreated as an acid while the B atom or group of atoms as pro-ton acceptor (base) The potential energy curves for the pro-ton motion can reach various shapes as shown in Figure 1

The extreme curves (1) and (6) correspond to stateseither without proton transfer (1) or to the complete ioniza-tion when the proton is attached to B while atom A isnegatively charged (6) Among the intermediate states takeplace those when the proton is located in the central positioneither with two minima (3) and a low barrier or with onesingle minimum (4)

There is a rich literature [1ndash16] with various approachesto the hydrogen bonding corresponding to different defini-tions showing an increase of systems analyzed with compre-hensive theoretical treatments and containing different richchemical characteristic features of hydrogen bonds Mostactual comprehensive review was recently published by GGilli and P Gilli [16]

From the point of view of the approach based onthe acid-base interaction the substantial parameter is theproton transfer degree which evokes changes of further

physico-chemical parameters The main quantity is theΔpKa value which can be expressed in the form

ΔpKa = pKB+H minus pKAH (1)

This quantity was introduced by Huyskens and Zeegers-Huyskens [17] We introduced normalized parameterdefined as

ΔpKN = ΔpKa minus ΔpKa (crit) (2)

where ΔpKa(crit) is related to ΔpKa region when the protontransfer degree reaches 50 [18]

The dependence of proton transfer degree on the ΔpKNvalue needs a correction connected with ldquosoftnesshardnessrdquoof interaction by using parameter ξ lt 1 [19] The value of thisparameter is the higher the harder is the interaction reachingmaximal value equal to unity As will be seen this quantityis well correlated with the polarizability in the transitionstate of hydrogen bonds The general equation presenting thedependence of proton transfer degree on ΔpKN possesses theform

logKPT = ξΔpKN (3)

One should remember that physicochemical parametersmeasured depending on ΔpKN and connected with the

(1) (2)

(3) (4)

(5) (6)

Proton motion Proton motion

Figure 1 Postulated potential energy curves for the proton motionstarting from nonproton-transfer state (1) up to fully ionized state(6)

PT equilibria

Figure 2 Three regions of physical properties depending on ΔpKN HB-related to nonproton-transfer states PT-related to protontransfer state and HB + PT proton transfer equilibrium

softness of interaction are related not only to ΔpKN as hasbeen shown in Figure 2

There exist three regions the central one with theequilibria of the proton transfer and side regions withoutproton transfer (HB) and with full ionization (PT)

Finally as will be shown it is necessary to mention therole of medium such as electric permittivity of the solvent

Figure 3 Proton transfer degree from NQR measurements forcomplexes composed of chlorine containing proton donors plot-ted versus ΔpKN (1) CCl3COOH complexes (ξ = 012) (2)CHCl2COOH complexes (ξ = 042) and (3) C6Cl5OH complexes(ξ = 074) [20]

and specific interaction between the solute and solventmolecules

For characterization of the role of the ξ parameter wepresent in Figure 3 dependencies of the proton transferdegree deduced from the measurements of nuclear quadru-ple resonance (NQR) for complexes of CCl3COOH (1) (ξ =012) CHCl2COOH (2) (ξ = 042) and C6Cl5OH (3) (ξ =074) [20] It is well seen the property of the curves in thecritical region when approaching to ΔpKN = 0

It is justified to mention in the introduction that curvesexpressing dependencies of physicochemical parameters onΔpKN possess various shapes [18] One can distinguishtwo types of correlations between the physical quantity andΔpKN namely of the sigma and delta type The examples ofsuch correlations will be presented in the next chapter

2 Examples of Correlation betweenPhysicochemical Parameters and theΔpKN Quantity

So far a most precisely investigated phenomenon is thedependence of the increase of dipole moment Δμ forcomplexes of phenols with N-bases In Figure 4 we presentcorrelation between Δμ and ΔpKN obtained for a number ofsystems in nonpolar solvents particularly in benzene [18]The experimental points are adjusted to the equation [21]

Δμ = ΔμHB + bHBΔpKN

1 + exp(

2303ξΔpKN

(ΔμPT + bPTΔpKN

)middot exp

(2303ξΔpKN

)1 + exp

(2303ξΔpKN

77Proton Transfer Equilibria and Critical Behavior of H-Bonding

0minus15 minus10 minus5 0 5 10

Figure 4 The increase of dipole moment Δμ plotted versusnormalized parameter ΔpKN [18]

where ΔμHB and ΔμPT mean the increase of the dipolemoment without proton transfer (HB) and after the protontransfer (PT) These quantities depend nearly linearly onΔpKN with coefficients bHB and bPT When approaching thecritical region around ΔpKN = 0 a stepwise change of thedipole moment connected with the increase of the protondegree takes place The proton transfer degree xPT defines theequilibrium

XPT = exp(2303ξΔpKN

)1 + exp

(2303ξΔpKN

To obtain the agreement with the experiment it isnecessary to introduce the coefficient ξ which as has beenformulated characterizes softnesshardness of interactionsIt can be on the other hand connected with the barrierheight for the proton transfer The value of the ξ coefficientfor the case of the situation in Figure 4 equals 065

Very similar run of the dependence on ΔpKN showsthe value of the 15N resonance chemical shift with the ξvalue equal to 056 [22] However one should rememberthat the results are related to markedly different experimentalconditions Thus the results obtained for 15N chemicalshift were obtained for complexes of carboxylic acids withpyridine in liquefied freons

Sigmoidal type of the relationship of physical quantity onΔpKN is also observed for complexes of pentachlorophenolwith amines by using the nuclear quadrupole resonance(NQR) [23] that is presented in Figure 5 In addition toexperimental points there are indicated values correspond-ing to neat pentachlorophenol H-bis-phenolate as wellas to Na+ and tributylamine salts One should rememberthat NQR measurements are performed for solid state thatreflects observed behavior

The similar shape of the plot with that in Figure 5 isobserved between geometrical parameters of complexes andΔpKN and particularly between CndashO bond length and ΔpKN[25]

An example of correlation between the measured quan-tity and ΔpKN of the delta type relates first of all to theproton magnetic resonance δ1H It is presented for the

36minus4 minus2 0 2 4 6 8

Phenol

H-bis-phenolate

Na+ salt

TBA+ salt

Figure 5 The dependence of average NQR 35Cl frequency uponΔpKN for complexes of pentachlorophenol [24]

minus6 minus4 minus2 0 2

Figure 6 The dependence of δ1H for complexes of carboxylic acidswith pyridine in liquid freon [22]

systems analogues to the δ15N resonance [18] The experi-mental points of δ1H presented in Figure 6 were obtained inthe same conditions as for δ15N The value of the ξ parameteris however somewhat lower (046) that we are not able toexplain From already done numerous experiments it followsthat methods applied do not possess marked influence on theξ value

In the analysis of the correlation plots exhibiting anextremum in the critical region as in the case of δ1H amodified approach can be used Thus for the descriptionof the dependence of given physical property Q showing anextremum the following simple procedure can be employedThe reference value of a given physical property Q isits extremum that is maximum or minimum In thecase of δ1H for the systems composed of carboxylic acidsand pyridine in liquid freons the maximum value equals215 ppm The delta type correlation can be transformedto the sigmoidal one by assuming that Q(crit) = 0 whileΔQHB lt 0 and ΔQPT gt 0 as has been done in Figure 7

minus5

Figure 7 Correlation between δ1H and ΔpKN for complexes ofcarboxylic acids with pyridine in liquid freon according (6)

The correlation between ΔQ and ΔpKN is presented in thefollowing equation [18]

ΔQ = ΔQHB + ΔQPT exp(2303ξΔpKN

)1 + exp

(2303ξΔpKN

The parameters for best fitting are Qmax = 215 ppm ΔQHB =minus83 ppm while ΔQPT = 44 ppm and ξ = 046 as has beenalready mentioned

The properties of infra-red spectra are commonlyaccepted for the hydrogen bonded systems This relates firstof all to the absorption band ascribed to the stretchingvibrations of either AH group (HB state) or BH+ group(PT state) The evolution of broad absorption ascribed tothe ν(AH) or ν(NH+) vibrations is illustrated in Figure 8taking as an example complexes of pentachlorophenol withamines [26] In the infra-red spectra the correlated quantityis the center of gravity of protonic vibrations (νcg) versusthe ΔpKN value Figure 9 represents numerous data relatedto νcg collected for various OndashHmiddot middot middotN hydrogen bridges[27] The scattering of experimental points is very largethat seems to be understandable taking into account variousexperimental conditions and differences in the acid-baseinteraction for various components One of the reasons ofscattering is a difficulty connected with precise assessmentof the position of broad bands As follows from the resultscollected by Albrecht and Zundel [28] for the complexesof phenols with octylamine the maximal absorbance inthe range of continuous absorption corresponds to 50 ofproton transfer that is shown in Figure 10

3 Electronic Spectra andthe Proton Transfer Degree

The UV-Vis spectroscopy is a very useful method of studieson the proton transfer degree in the Bronsted acid-base

1000 2000 3000Wavenumber (cmminus1)

ngΔpK

Critical region

Figure 8 The evolution of infra-red absorption ascribed to ν(OH)when increasing ΔpKN for complexes of pentachlorophenol withamines [26]

minus4 0 4 8

(cmminus1

Figure 9 The center of gravity νcg for protonic vibrations as afunction of ΔpKN for various complexes of carboxylic acids [27]

system for the diluted solutions The majority of quantitativedata related to the proton transfer equilibria relates mainly tothe complexes between phenols and amines [24 29ndash35] Inthe UV spectra the tautomeric equilibrium is characterizedby appearance of a new band corresponding to the π rarr πlowast

transition in the phenolate ion After careful quantitativeseparation of the HB and PT bands the proton transferequilibrium cPTcHB can be evaluated As an example of theUV spectra with the proton transfer equilibrium we use

minus3 minus2 minus1 0 1 2 3ΔpKN

2middotm

olminus1

Figure 10 The proton transfer degree (a) and intensity of continuous absorption (b) for complexes of phenols with octylamine [28]

the system of 246-trichlorophenol in tributylamine (TBA)[29] presented in Figure 11 which shows the overlapping ofHP and PT bands From the equilibrium constant otherthermodynamic parameters can be determined according toequation

lnK = ΔS

Rminus ΔH

RT (7)

where K is calculated by using intensities of bands and molarabsorption coefficients of corresponding forms

K =(IPT

)(εHB

The first quantitative studies by using the electronicabsorption spectra were performed by Baba et al [30]for complex of 4-nitrophenol with triethylamine in 12-dichloroethane who found ΔH = minus13 kJmiddotmolminus1 and ΔS =minus498 Jmiddotmolminus1middotKminus1 Similarly Crooks and Robinson [31]investigated complexes of bromophenol with methyl deriva-tives of pyridine in chlorobenzene The obtained data corre-spond to minusΔH in the range 12ndash38 kJmiddotmolminus1 and minusΔS inthe range 29ndash55 Jmiddotmolminus1middotKminus1 The values of thermodynamicparameters for the complexes of chlorophenols with TBA[29] are comparable with those of nitrophenol

From the studies [29 37ndash44] it follows that the con-centration of the PT form independently of the H-bondingtype increases with an increase of ΔpKa value of interactingcomponents as well as with increase of the solvent activityand the drop of temperature

For the systems with negative or close to zero ΔpKavalues it was not possible to find traces of the PT bandeven in the most active solvents at temperatures as low asbelow minus190C [38] Thus for observation in UV spectrumparticipation of the PT form even in favorable conditions(low temperature and high polarity of solvent) someboundary ΔpKa value is necessary

36000 32000 28000

Wavenumber (cmminus1)

Figure 11 The plot of the absorbance versus wavenumber for246-trichlorophenol in tributylamine at room temperature c =5 middot 10minus4 molmiddotdmminus3 d = 5 mm

Figure 12 shows the UV spectra for the series ofcomplexes formed by TBA with various chlorophenols ofincreasing acidity It can be seen that 24-dichlorophenoland 245-trichlorophenol do not show any contributionsof PT species only 26-dichlorophenol shows traces of theionic PT form For 246-trichlorophenol a considerableamount (ca 25) of the PT form was estimated fromthe UV spectrum Pentachlorophenol appears entirely inthe zwitterionic state whereas in a case of 26-dichloro-and 245-trichloro derivatives characterized by almost thesame ΔpKa values some contribution of the PT state showsonly the former one The ΔpKa value is not howevera completely satisfactory measure of the proton donor-acceptor properties in nonaqueous media

In several papers for example [45ndash48] one considersthe attention that one should apply another scale of protondonor and acceptor properties for defining the proton

36000 32000 28000 24000Wavenumber (cmminus1)

36000 32000 28000 24000

Figure 12 UV spectra of chlorophenols (a) 24-dichlorophenol(b) 245-trichlorophenol (c) 26-dichlorophenol (d) 246-tri-chlorophenol (e) pentachlorophenol in TBA at room temperaturec = 5 middot 10minus4 molmiddotdmminus3 d = 5 mm [29]

36000 34000 32000 30000 28000 26000

Figure 13 UV spectra of 24-dichlorophenol in TBA as a functionof temperature 298 K (1) 223 K (2) 203 K (3) 186 K (4) 165K(5) 143 K (6) 128 K (7) C = 4 times 10minus4 mol dmminus3 d = 5 mmwavenumber of PT formsim= 32160 cmminus1 and HB formsim= 33840 cmminus1

150 200 250

ET (kJmol)

minusΔH

Figure 14 Comparison of the ΔHPT with ET parameters forMannich bases (A) 2-(NN-dimethylaminomethyl)-46-dibro- mo-phenol (B) 2-(NN-diethylaminomethyl)-4-nitro-phenol (C) 2-(NN-diethylaminomethyl)-346-trichlorophenol (D) 2-(NN-di-ethylaminomethyl)-3456-tetrachlorophenol (E) 2-(NN-dieth-ylaminomethyl)-4-nitronaphthol-1 in 12-dichloroethane (1) di-chloromethane (2) n-butylchloride (3) chloroform (4) 14-diox-ane (5) isopropylbenzene (6) squalane (7) methanol (8) ethanol(9) butan-1-ol (10) propan-1-ol (11) acetonitrile (12) and NN-dimethylformamide (13) [36]

position in hydrogen-bonded complexes In the analysis onetakes into account the proton affinity and deprotonationenthalpy based on calculations by using DFT methodsHowever in the present article we limited our considerationsto experimental methods leading to evaluation of the pKavalues

A strong influence of cooling on the increase of con-centration of the PT form indicates on negative change ofenthalpy effect on the proton transfer process In Figure 13the UV spectra of 24-dichlorophenol in TBA are shown asa function of temperature [29] The 24-dichlorophenolmdashTBA system at room temperature does not show any

contribution of the PT state Similar to other systems ofthis type we observe a very strong influence of cooling onthe contribution of the PT state At the temperatures 203186 and 165 K the values of KPT are 033 082 and 570respectively The complete proton-transfer state is reached atabout 143 K and further cooling does not affect the intensityof the phenolate band

By using electronic spectroscopy in the UV range thePT equilibrium constants have been measured as a functionof temperature in various solvents for various H-bondedsystems They allowed to determine the thermodynamicparameters of the PT process and correlate with variousempirical parameters of the solvent activity The results forMannich bases [36 39 40] correlated with the Dimroth-Reichardt ET parameter [49 50] are presented in Figure 14These correlations present individual straight lines withsimilar slope for particular Mannich bases Such cleardifferentiation shows that the differences in the protonaffinity of particular acid-base centers contribute essentiallyto the stabilization of both forms The observed effect ofsolvent activity shows that the proton transfer process ischaracterized by two factors Simultaneously with previousΔpKa effect that can be classified as inter one an additionalfactor called an external takes place which correlates withthe solvent activity expressed by the ET parameter Formallyone can express

ΔHPT = ΔHint

(ΔpKa

)+ ΔHext(ET) (9)

however quantitative estimation of both components is notan easy task

The attempt has been undertaken to correlate the ΔHPTvalues with other parameters characterized the solventactivity but the best correlation was obtained with ET Thus the external factor contains two effects that is theelectrostatic stabilization of the ionic form and the donor-acceptor interaction of solvent molecules with the freeelectron pair of the phenolate oxygen atom So far no protontransfer equilibrium was observed in the gas phase thatprooves decisive role of the solvent for observation of theproton transfer This is confirmed by relatively high valuesof entropy effect ΔSPT from minus30 up to minus70 J Kminus1 molminus1

[36 39 40] that confirms a considerable redistribution ofmolecules and high increase of ordering of solvent moleculesunder influence of intramolecular proton transfer

The UV spectra were used to locate the position of 50proton transfer in chloranilic acid-amine complexes thesimilar result was deduced from IR and NMR studies [51]Chranina et al [52] studied the proton transfer equilibriabetween hydroxyanthraquinone dyes and aliphatic amines inlow-polarity solvents by UV spectroscopy The shift of thisequilibrium in an external electrical field has been observedby the method of electrochromism in the visible region Alsothe mechanism of proton transfer reactions between variousacids and amines was studied kinetically by applying UVspectroscopy when the order and the isotopic ratio effectwere discussed [53 54]

4 Vibrational Polarization ofHydrogen Bonded Systems

It has been broadly postulated by Zundel [55] that forthe characteristic dependences of the important physicalparameters on ΔpKN with the anomalous behavior in thecritical region the large proton polarizability of the hydrogenbonds is responsible The extraordinary increase in protonpolarizability with increased strength of the hydrogen bondsin heteroconjugated systems was the aim of detailed infraredstudies conducted by Hawranekrsquos group For six systemsof pentachlorophenol (PCPh) dissolved in different basisthe molar vibrational polarization (Pvib called also atomicpolarization as it arises from atomic motions) and molarelectronic polarization were determined according to theprocedure sketched below Names of the basis are given inTable 1 The PCPh-base complexes were studied in binarysolutions that is the proton donor (PCPh) was directlydissolved in an excess of the proton acceptor Such condi-tions facilitated accurate determination of optical quantitiesnecessary for calculations of the Pvib values according to thefollowing scheme

Table 1 shows the Pvib2 values along with the position

(νmax) and the half width (Δν12) of the νs(OH) bandThe spectral parameters were obtained only for H-bondedsystems related to the nonproton-transfer state their valuescannot be estimated with a sufficient accuracy for systemscorresponding to other two states (see Figure 2) The plot ofthe Pvib

2 values versus ΔpKa shown in Figure 15 possess thedelta type character with a maximum

It has to be mentioned here that the measurementsin binary system have many advantages that facilitate theused procedure of determination of the molar vibrationalpolarization However there is also one disadvantage thePvib values are obtained for H-bonded systems differentlypolarized by their environment The PCPh-base complexesare immersed in various media that have different macro-scopic parameters and more or less strongly polarize thehydrogen bonds For each system the ξ and ΔpKa (crit)parameters should be determined whenever the ΔpKa valuesare subjected to the normalization procedure Due to the lackof such data the Pvib values on Figure 15 are plotted againstΔpKa parameter We can guess that the normalizationand the different influence of solvents on the vibrationalpolarization should not meaningfully change the delta-typerelation between Pvib and strength of the hydrogen-bondedsystems

According to Table 1 the molar vibrational polarizationincreases from a very small value for TMPh in inert CCl4solution to a slightly larger for the OH group involved ina weak OHmiddot middot middotCl intramolecular hydrogen bond in PCPhNoticeable increase is observed for OH group engaged in aweak intermolecular hydrogen bonds in the PCPh-CH3CNand PCPh-dioxane systems Their Pvib values compared withthat for the 246-TMPh-CCl4 indicate on the 17- and 20-foldincrease The changes are strictly correlated with the typicalspectral features of H-bond formation that is the shift ofνs(OH) bands towards lower frequencies and the increase inits bandwidth In relation to the system with intramolecular

Table 1 Spectral parameters related to the νs(OH) band and Pvib of the H-bond complexes of PCPh

Acceptor νmax (cmminus1) Δν12 (cmminus1) Pvib (cm3 molminus1) Reference

CCl4 3525 216 0048 [56]

CH3CN 3322 2755 0294 [57]

Dioxane-D8 3162 3165 0333 [58]

3-Chloropyridine 2737 945 1182 [59]

Pyridine mdash mdash 125 [60]

246-Trimethylpyridine mdash mdash 17 8 [61]

Tri-n-octylamine mdash mdash 90 [62]

246-TMPh-CCl4 3622 0017 [56]

0minus10 minus8 minus6 minus4 minus2 0 2 4 6

Pentachlorophenol

m3middotm

olminus1

Figure 15 Pvib plotted versus ΔpKa of complexes formed by PCPhwith various proton acceptors

hydrogen bonds (PCPh-CCl4) the increase is 6-fold for thePCPh-CH3CN and 7-fold for the PCPh-dioxane complexIt reveals that formation even rather weak intermolecularH-bond when the proton is located in a relatively narrowsingle-minimum proton potential near the acid (Figure 1(1)) leads to a drastic increase in Pvib of the OH group

The PCPh-3-chloropyridine system with still relativelyasymmetrical hydrogen bond is close to a border betweenthe HB and the PT equilibrium states (see Figure 2)However its Pvib value compared with that obtained forthe system with intramolecular H-bonded shows almost25- and 70-fold increase in comparison with the free OH-group in the 246-TMPh-CCl4 system Despite this themolar vibrational polarization of the PCPh-3-chloropyridinesystem is still markedly less than its molar electronicpolarization

The complex of PCPh with pyridine with symmetricalOmiddot middot middotHmiddot middot middotN hydrogen bond is classified to the protontransfer state The molar vibrational polarization of theOH group rises to 125 cm3middotmolminus1 This value comparedwith that obtained for free (246-TMPh-CCl4) and for theintramolecularly bonded (PCPh-CCl4) OH group showsalmost 600- and 200-fold increase respectively

According to [28] the complex of PCPh with 245-trimethylpyridine is close to the border between the PTequilibrium and the PT states Its molar vibrational polar-ization is more than 370 and 1000 times higher than in

the PCPh-CCl4 and 246-TMPh-CCl4 system respectivelyFor the PCPh-246-trimethylpyridine complex hydrogenbond possess largest proton polarizability The last complexof PCPh with tri-n-octylamine belongs to the PT stateAccording to Figure 15 its Pvib value drops almost twicewhen compared with the previous system For such largechange of Pvib a characteristic evolution of the infraredspectra corresponding to the PT state shown in Figure 8 isresponsible

Summing up the very large Pvib values determined forPCPh complexes with pyridine and 3-chloropyridine areexcellent confirmation of the extraordinary properties ofhydrogen bonds from the transition region with symmetricalpotential Moreover they confirm very well Zundelrsquos conceptthat an extreme broadening of the OH band occurs forhydrogen bonds showing the largest proton polarizability[55]

References

[1] D Hadzi and H W Thompson Eds Hydrogen Bonding Per-gamon Press London UK 1959

[2] L Pauling The Nature of the Chemical Bond and the Structureof Molecules and Crystals An Introduction to Modern StructuralChemistry Cornell University Press Ithaca NY USA 1960

[3] G C Pimentel and A L McClellan The Hydrogen Bond WH Freeman San Francisco Calif USA 1960

[4] S N Vinogradov and R H Linnel Hydrogen Bonding VanNostrand-Reinhold New York NY USA 1971

[5] M D Joesten and L J Schaad Hydrogen Bonding MarcelDekker New York NY USA 1974

[6] P Schuster G Zundel and C Sandorfy Eds The HydrogenBond Recent Developments in Theory and Experiments vol 1ndash3 North Holland Amsterdam The Netherlands 1976

[7] H Ratajczak and W J Orwille-Thomas Eds MolecularInteractions John Wiley amp Sons New York NY USA 1980

[8] P L Huyskens W A P Luck and Th Zeegers-Huyskens EdsIntermolecular Forces An Introduction to Modern Methods andResults Springer Heidelberg Germany 1991

[9] S Scheiner Ed Hydrogen Bonding A Theoretical PerspectiveOxford University Press Oxford UK 1997

[10] G A Jeffrey Introduction to Hydrogen Bonding Oxford Uni-versity Press Oxford UK 1997

[11] D Hadzi Ed Theoretical Treatments of Hydrogen BondingOxford University Press Oxford UK 1997

[12] G R Desiraju and T Steiner The Weak Hydrogen Bond inStructural Chemistry and Biology Oxford University PressOxford UK 1999

[13] Th Elsaesser and H J Bakker Eds Ultrafast Hydrogen Bond-ing Dynamics and Proton Transfer Processes in the CondensedPhase Kluwer Academic Publishers Dordrecht The Nether-lands 2002

[14] S J Grabowski Ed Hydrogen BondingmdashNew InsightsSpringer Dordrecht The Netherlands 2006

[15] Y Marechal The Hydrogen Bond and the Water MoleculeThe Physics and Chemistry of Water Aqueous and Bio-MediaElsevier Amsterdam The Netherlands 2007

[16] G Gilli and P Gilli The Nature of The Hydrogen Bond Outlineof a Comprehensive Hydrogen Bond Theory Oxford UniversityPress Oxford UK 2009

[17] P L Huyskens and Th Zeegers-Huyskens ldquoAssociations mol-eculaires et equilibres acide-baserdquo Journal de Chimie Physiquevol 61 aticle 84 1964

[18] P Huyskens L Sobczyk and I Majerz ldquoOn a hardsoft hydro-gen bond interactionrdquo Journal of Molecular Structure vol 615no 1ndash3 pp 61ndash72 2002

[19] L Sobczyk ldquoSoftness of hydrogen bond interactionrdquo Khimich-eskaya Fizika vol 24 article 31 2005

[20] L Sobczyk ldquoQuasi-symmetric OndashHmiddot middot middotN hydrogen bonds insolid staterdquo Molecular Physics Reports vol 14 pp 19ndash31 1996

[21] R Nouwen and P Huyskens ldquoDipole moments and structureof the complexes of phenols with pyridinesrdquo Journal ofMolecular Structure vol 16 no 3 pp 459ndash471 1973

[22] S N Smirnov N S Golubev G S Denisov H BenedictP Schah-Mohammedi and H H Limbach ldquoHydrogendeu-terium isotope effects on the NMR chemical shifts andgeometries of intermolecular low-barrier hydrogen-bondedcomplexesrdquo Journal of the American Chemical Society vol 118no 17 pp 4094ndash4101 1996

[23] E Grech J Kalenik and L Sobczyk ldquo35Cl nuclear quadrupoleresonance studies of pentachlorophenol-amine hydrogen-bonded complexesrdquo Journal of the Chemical Society FaradayTransactions 1 vol 75 pp 1587ndash1592 1979

[24] J P Castaneda G S Denisov and V M Schreiber ldquoStructureof 1 1 and 1 2 complexes formed by aromatic NH and OHproton donors with aliphatic amines Possibility of homo-conjugated NHN+ cation formationrdquo Journal of MolecularStructure vol 560 no 1ndash3 pp 151ndash159 2001

[25] I Majerz Z Malarski and L Sobczyk ldquoProton transfer andcorrelations between the CndashO OndashH NndashH and Omiddot middot middotN bondlengths in amine phenolatesrdquo Chemical Physics Letters vol274 no 4 pp 361ndash364 1997

[26] Z Malarski M Roepenk E Grech and L Sobczyk ldquoDielectricand spectroscopic studies of pentachlorophenol-amine com-plexesrdquo Journal of Physical Chemistry vol 86 no 3 pp 401ndash406 1982

[27] J Kalenik I Majerz L Sobczyk E Grech and M M MHabeeb ldquoInfra-red and 35Cl nuclear quadrupole resonancestudies of hydrogen bonded adducts of 2-chlorobenzoic acidderivativesrdquo Collection of Czechoslovak Chemical Communica-tions vol 55 no 1 pp 80ndash90 1990

[28] G Albrecht and G Zundel ldquoPhenolndashamine hydrogen bondswith large proton polarizabilities Position of the OHmiddot middot middotN Ominusminus middot middot middotH+N equilibrium as a function of the donor andacceptorrdquo Journal of the Chemical Society Faraday Transactions1 vol 80 no 3 pp 553ndash561 1984

[29] V M Schreiber A Kulbida M Rospenk L Sobczyk ARabold and G Zundel ldquoTemperature effect on proton-transfer equilibrium and IR spectra of chlorophenol-tribu-tylamine systemsrdquo Journal of the Chemical Society FaradayTransactions vol 92 no 14 pp 2555ndash2561 1996

[30] H Baba A Matsuyama and H Kokubun ldquoProton transferin p-nitrophenol-triethylamine system in aprotic solventsrdquoSpectrochimica Acta Part A vol 25 no 10 pp 1709ndash17221969

[31] J E Crooks and B H Robinson ldquoHydrogen-bonded and ion-pair complexes in aprotic solventsrdquo Faraday Symposia of theChemical Society vol 10 pp 29ndash40 1975

[32] H Romanowski and L Sobczyk ldquoUltraviolet spectra andproton-transfer equilibria in 26-dichloro-4-nitrophenol-amine systemsrdquo Journal of Physical Chemistry vol 79 no 23pp 2535ndash2542 1975

[33] M M Habeeb and M A Kharaba ldquoIntermolecular hydrogenbonds and proton transfer equilibrium in some nitro cresols-aliphatic amines-acetonitrile or methanol systemsrdquo Journal ofMolecular Liquids vol 107 no 1ndash3 pp 205ndash219 2003

[34] M M Habeeb and R M Alghanmi ldquoSpectrophotometricstudy of intermolecular hydrogen bonds and proton transfercomplexes between 12-dihydroxyanthraquinone and somealiphatic amines in methanol and acetonitrilerdquo Journal ofChemical and Engineering Data vol 55 no 2 pp 930ndash9362010

[35] Z Dega-Szafran E Dulewicz and M Szafran ldquoSpectroscopicstudies of N-methylpiperidine betaine complexes with phe-nolsrdquo Journal of Molecular Structure vol 704 no 1ndash3 pp 155ndash161 2004

[36] M Rospenk ldquoThe influence of steric effects of proton-transferequilibrium in intramolecular hydrogen bondsrdquo Journal ofMolecular Structure vol 221 pp 109ndash114 1990

[37] V M Schreiber M Rospenk A I Kulbida and L SobczykldquoShaping of broad IR absorption in proton transfer equili-brating OHmiddot middot middotN hydrogen bonded systemsrdquo SpectrochimicaActamdashPart A vol 53 no 12 pp 2067ndash2078 1997

[38] V M Schreiber A Koll and L Sobczyk ldquoEffect of temperatureon the proton transfer equilibrium in the intramolecularhydrogen bond hydroxylmiddot middot middotnitrogenrdquo Bulletin de lrsquoAcademiePolonaise des Sciences Serie des Sciences Chimiques vol 26article 651 1978

[39] A Koll M Rospenk and L Sobczyk ldquoThermodynamic para-meters for the proton-transfer reaction in Mannich basesrdquoJournal of the Chemical Society Faraday Transactions 1 vol 77no 10 pp 2309ndash2314 1981

[40] M Rospenk I G Ruminskaya and V M Schreiber ldquoElek-tronnye spektri i wnutrimolekularnij perekhod protona vosnovanyakh Mannikha v zhidkikh i tverdikh stekloobraznikhrastvorakhrdquo Journal of Applied Spectroscopy vol 36 article756 1982

[41] M Rospenk L Sobczyk A Rabold and G Zundel ldquoLow tem-perature studies on ultraviolet and infrared spectra of orthoMannich basesrdquo Spectrochimica ActamdashPart A vol 55 no 4pp 855ndash860 1999

[42] I Krol-Starzomska M Rospenk Z Rozwadowski and TDziembowska ldquoUV-visible absorption spectroscopic studiesof intramolecular proton transfer in N-(R-salicylidene)-alky-laminesrdquo Polish Journal of Chemistry vol 74 no 10 pp 1441ndash1446 2000

[43] M Rospenk I Krol-Starzomska A Filarowski and A KollldquoProton transfer and self-association of sterically modifiedSchiff basesrdquo Chemical Physics vol 287 no 1-2 pp 113ndash1242003

[44] A Koll M Rospenk L Sobczyk and T Glowiak ldquoPropertiesof a strong intramolecular OHO hydrogen bond in 2-(NN-diethylamino-N-oxymethyl)-46-dichlorophenolrdquo CanadianJournal of Chemistry vol 64 no 9 pp 1850ndash1854 1986

[45] S Kong I G Shenderovich and M V Vener ldquoDensity func-tional study of the proton transfer effect on vibrationsof strong (short) intermolecular OndashHmiddot middot middotNOminus middot middot middotHndashN+

hydrogen bonds in aprotic solventsrdquo Journal of Physical Chem-istry A vol 114 no 6 pp 2393ndash2399 2010

[46] T Lankau and C H Yu ldquoSolubility of methane in waterThe significance of the methane-water interaction potentialrdquoChemical Physics Letters vol 424 article 264 2006

[47] P Gilli L Pretto and G Gilli ldquoPApKa equalization andthe prediction of the hydrogen-bond strength a synergismof classical thermodynamics and structural crystallographyrdquoJournal of Molecular Structure vol 844-845 pp 328ndash3392007

[48] T Lankau and C H Yu ldquoCorrelated proton motion in hydro-gen bonded systems tuning proton affinitiesrdquo Physical Chem-istry Chemical Physics vol 9 no 2 pp 299ndash310 2007

[49] C Reichardt ldquoEmpirical parameters of the polarity of sol-ventsrdquo Angewandte Chemie International Edition in Englishvol 4 no 1 pp 29ndash40 1965

[50] C Reichardt and K Dimroth ldquoSolvents and empirical param-eters for characterization of their polarityrdquo Fortschritte derChemischen Forschung vol 11 article 1 1968

[51] M Habeeb H Alwakil A El-Dissouky and H Abdel-FattahldquoSpectroscopic studies of 11 chloranilic acid-amine com-plexesrdquo Polish Journal of Chemistry vol 69 article 1428 1995

[52] O V Chranina F P Czerniakowski and G S DenisovldquoUV-vis electrochromism due to proton transferrdquo Journal ofMolecular Structure vol 177 pp 309ndash315 1988

[53] W Galezowski and A Jarczewski ldquoKinetics isotope effectsof the reaction of 1-(4-nitrophenyl)-1-nitroalkanes with DBUin tetrahydrofuran and chlorobenzene solventsrdquo CanadianJournal of Chemistry vol 68 no 12 pp 2242ndash2248 1990

[54] A Jarczewski G Schroeder and K T Leffek ldquoThe protontransfer reaction between bis(24-dinitrophenyl)methane andnitrogen bases in dimethyl sulfoxide and toluene solventsrdquoCanadian Journal of Chemistry vol 69 no 3 pp 468ndash4731991

[55] G Zundel ldquoHydrogen bonds with large proton polarizabilityand proton transfer processes in electrochemistry and biol-ogyrdquo Advances in Chemical Physics vol 111 2000

[56] J P Hawranek and B Czarnik-Matusewicz ldquoInfrared disper-sion of H-bonded systems The dielectric function for weakcomplexesrdquo Chemical Physics Letters vol 109 no 2 pp 166ndash169 1984

[57] J P Hawranek and B Czarnik-Matusewicz ldquoInfrared disper-sion of the H-bonded pentachlorophenol-acetonitrile com-plexrdquo Chemical Physics Letters vol 138 no 5 pp 397ndash4001987

[58] J P Hawranek and B Czarnik-Matusewicz ldquoIR dispersionof hydrogen bonded systems III Pentachlorphenolmdashdioxane-D8 complexrdquo Journal of Molecular Structure vol 143 no Cpp 337ndash340 1986

[59] B Czarnik-Matusewicz and J P Hawranek ldquoInfrared disper-sion of the hydrogen-bonded pentachlorophenolmdash3-chloro-pyridine complexrdquo Journal of Molecular Structure vol 219 pp221ndash226 1990

[60] J P Hawranek B Czarnik-Matusewicz and W WrzeszczldquoInfrared dispersion of the hydrogen-bonded pentachloro-phenol-pyridine complexrdquo Journal of Molecular Structure vol322 pp 181ndash186 1994

[61] J P Hawranek J Z Flejszar-Olszewska and A S MuszynskildquoInfrared dispersion of the pentachlorophenol-sym-collidinecomplexrdquo Journal of Molecular Structure vol 448 no 2-3 pp149ndash159 1998

[62] J P Hawranek and A S Muszynski ldquoInfrared dispersion of thepentachlorophenol-trioctylamine complexrdquo Journal of Molec-ular Structure vol 552 no 1ndash3 pp 205ndash212 2000

Polymorphism Hydrogen Bond Properties and VibrationalStructure of 1H-Pyrrolo[32-h]Quinoline Dimers

Alexandr Gorski1 Sylwester Gawinkowski1 Roman Luboradzki1 Marek Tkacz1

Randolph P Thummel2 and Jacek Waluk1

1 Institute of Physical Chemistry Polish Academy of Sciences Kasprzaka 4452 01-224 Warsaw Poland2 Department of Chemistry University of Houston Houston TX 77204-5003 USA

Correspondence should be addressed to Jacek Waluk walukichfedupl

Academic Editor Paul Blaise

Two forms of cyclic doubly hydrogen-bonded dimers are discovered for crystalline 1H-pyrrolo[32-h]quinoline a bifunctionalmolecule possessing both hydrogen bond donor and acceptor groups One of the forms is planar the other is twisted Analysis ofIR and Raman spectra combined with DFT calculations allows one to assign the observed vibrations and to single out vibrationaltransitions which can serve as markers of hydrogen bond formation and dimer structure Raman spectra measured for samplessubmitted to high pressure indicate a transition from the planar towards the twisted structure Formation of intermolecularhydrogen bonds leads to a large increase of the Raman intensity of the NH stretching band it can be readily observed for thedimer but is absent in the monomer spectrum

1 Introduction

In studies of the intermolecular hydrogen bond (HB) animportant class of model compounds consists of moleculeswhich can form both H-bonded dimers and complexeswith water or alcohols [1] Such molecules are usuallycharacterized by the simultaneous presence of HB donor andacceptor groups Whether the strength of the intermolecularHBs is greater for dimers or complexes depends on therelative positions of the donor and acceptor in the molecularframe Interestingly different structures and stoichiometriesare often encountered for the same molecule A well-knownexample is 7-azaindole (7AI Figure 1) which forms doublyhydrogen bonded dimers in solution [2] while the X-raydata reveal a tetrameric structure in the crystalline state [3]Different stoichiometries and structures are possible for thecomplexes of 7AI with methanol and water 1 1 1 2 and1 3 species have been detected [4ndash9]

The crystal structure of multiply H-bonded dimersoligomers seems to be determined by the interplay of H-bonding and longer range intermolecular interactions For

instance 1-azacarbazole (1AC) a molecule closely relatedto 7AI exists in the crystal in the form of planar doublyhydrogen bonded dimers [10] (Figure 2) While there is nodoubt that 1AC also forms dimers in solutions variouspossible structures have been discussed [11ndash14]

1H-pyrrolo[32-h]quinoline (PQ Scheme 1) can be con-sidered a counterpart to 7AI with regard to intermolecularHB characteristics The NH group of PQ (HB-donor) andthe pyridine nitrogen (HB-acceptor) are positioned threebonds apart whereas in 7AI these groups are separatedby two bonds This change results in completely differentexcited state behaviour of complexes with water or alcohols[15ndash20] Rapid photoinduced double proton transfer isobserved for PQ in complexes of 1 1 stoichiometry Theprocess occurs on the time scale of single picoseconds andis not stopped by lowering of temperature or by increasingthe viscosity of the medium On the contrary the reaction isslower and viscosity-dependent in 7AI complexes [21] sinceit requires a solvent rearrangement around an excited chro-mophore [22ndash27] These different phototautomerization

Figure 1 Various motifs of intermolecular HB formed by 7-azaindole (a) Dimers and tetramers (b) complexes with water

Scheme 1

characteristics reflect different intermolecular HB strengthsimposed by molecular structure

The HB characteristics and in consequence tautomer-ization abilities in the dimeric species are expected to becomereversed in PQ and 7AI For the latter a planar dimeric struc-ture reveals two strong linear equivalent HBs Thereforeit is not quite surprising that photoinduced double protontransfer in 7AI dimers has been observed at temperaturesas low as 4 K [28] In contrast PQ dimers are predictedby theory to be nonplanar This has been confirmed by X-ray studies which reported an angle of 226 between thetwo monomeric units [29] Our previous work on a similarstructure dipyrido[23-a32-i]carbazole [30] demonstratedthat in the crystalline phase this molecule forms cyclicbut strongly nonplanar doubly hydrogen-bonded dimers(Figure 3) No tautomeric fluorescence has been observedfor such a dimer but it could be readily detected when thecrystalline sample was exposed to water vapor prepared ona hydrophilic support or embedded in a polymer containinghydroxyl groups A general conclusion from this study wasthat HB-donor-acceptor molecules which readily form flatdimers should have a weak tendency for the formation ofcyclic complexes and vice versa

In this work we analyze structure and vibrational spectraof crystalline PQ dimers Somewhat unexpectedly our X-ray

Figure 2 The structure of dimers of 1-azacarbazole in the solidphase

Figure 3 The X-ray structure of dipyrido[23-a32-i]carbazole

measurements of PQ reveal the existence of planar doublyhydrogen-bonded dimeric species and thus a structure verydifferent than the one reported previously [29] (Figure 4)We analyze the experimental and theoretically predictedvibrational patterns with particular interest regarding thevibrations involved in intermolecular hydrogen bondsFinally we show the influence of high pressure upon the HBstrength manifested by spectral shifts observed in the Ramanspectra

2 Experimental and Theoretical Details

Synthesis and purification of PQ have been described before[31]

The IR spectra were recorded on a Nicolet Magna560 FTIR spectrometer equipped with MCTB liquid-nitrogen-cooled detector with 1 cmminus1 resolution For themeasurements of infrared spectra thin polycrystalline PQfilms were prepared on KBr or ZnSe windows by quickevaporation from a concentrated solution The monomer IR

87Polymorphism Hydrogen Bond Properties and Vibrational Structure of 1H-Pyrrolo[32-h]Quinoline Dimers

2930 31

Figure 4 (a) X-ray structure of PQ reported in [29] thegeometrical positions of hydrogen atoms were inserted (b) Our X-ray structure and atom numbering

spectra have been recorded for PQ isolated in argon matricesusing a closed-cycle helium cryostat (CSW-202N AdvancedResearch Systems) The compound contained in a glass tubewas heated to 350 K and codeposited with argon at a ratioof about 1 1000 onto a cold (20 K) KBr window mountedin a cryostat with 10minus6 Torr background pressure Duringspectral measurements the matrix temperature was kept at10 K

Renishaw inVia microscopic system was used for themeasurements of Raman spectra Ar+ 5145 nm (Stellar ProModu-Laser LLC) laser line and a diode laser (HPNIR785)emitting 785 nm line were used as the excitation sourcesWith atimes100 microscope objective the laser light was focusedon a sample the laser power at the sample being 5 mWor less The Raman scattered light was collected by thesame objective through a cut-off filter to block out Rayleighscattering Gratings of 1800 and 1200 groovesmm were usedfor 5145 and 785 nm laser lines respectively The resolutionwas 5 cmminus1 with the wavenumber accuracy of 2 cmminus1 bothcalibrated with the Rayleigh line and the 5206 cmminus1 line ofsilicon The Raman scattered light was recorded by a 1024 times256 pixel Peltier-cooled RenCam CCD detector

High pressure experiments have been performed inTakemura type of diamond anvil cell [32] The diameter ofthe diamond culet was 600 μm and a gasket made of stainlesssteel was used with 300 μm centrally drilled hole Samplepowder was loaded into the gasket hole without any pressuretransmitting medium Pressures were measured by recordingthe fluorescence spectrum of a small ruby chip embedded inthe sample and converting the shift of the wavelength of theR1 line to pressure according to the scale proposed by Mao[33]

The samples of different polymorphs were preparedby quick crystallization by evaporation from concentratedPQ solutions in dichloromethane diethyl ether methanolcyclohexane and toluene

For the X-ray studies a colorless PQ crystal of approx-imate dimensions of 01 times 02 times 02 mm3 was usedDiffraction data were collected at 100 K using a Bruker KappaCCD diffractometer with graphite monochromated Mo Kαradiation Structure was solved by direct methods (SHELXS-97) and refined on F2 by full-matrix least-squares method(SHELXL-97) [34] Formula is C11H8N2 monoclinic spacegroup P21c a = 90104(4) b = 47302(1) c = 193117(9) Aβ = 1031825(17) R1 = 00449 (I gt 2σ(I)) wR2 = 01144for all data

Unit cell parameters (but not the whole data) werealso measured at room temperature showing no significantdifferences compared with 100 K data (a = 913 b = 487c = 1942 A β = 10254 parameters not refined)

The crystallographic data have been deposited with theCambridge Crystallographic Data Centre as a supplementarypublication no CCDC 868707 The data can be obtainedfree of charge at httpwwwccdccamacuk or from theCambridge Crystallographic Data Centre 12 Union RoadCambridge CB2 1EZ UK

Geometry optimizations were performed using densityfunctional theory (DFT) with B3LYP functional and cc-PVTZ basis set as implemented in Gaussian 09 This choiceof functionalbasis set was guided by extensive calculationsfor the PQ monomer which resulted in reliable assignmentsof nearly all of the vibrations

In order to simulate the structure of PQ dimers inthe crystalline environment DFT-based quantum chemicalcalculations were performed using the CASTEP (Cam-bridge Serial Total Energy Package) computer code [35] inthe framework of the generalized gradient approximation(GGA) as proposed by Perdew et al [36] in combinationwith Vanderbilt ultrasoft pseudopotentials [37] The planewave basis set was truncated at a kinetic energy of 240 eVComputations were performed over a range of k-pointswithin the Brillouin zone as generated by the full Monkhorst-Pack scheme [38] with a 2 times 2 times 1 mesh A furtherincrease of the cutoff energy and the number of k-pointsresulted in negligibly small changes in structure energiesindicating that the energy values are well converged Twoinitial geometries of planar and twisted PQ dimers weretaken from the X-ray data In every case a slab including16 molecules of PQ was constructed and repeated periodi-cally

3 Results

31 Dimer Structure Geometry optimization performed forthe PQ dimer yields a nonplanar doubly H-bonded struc-ture The calculated nonplanar geometry agrees qualitativelywith the X-ray data published in 1991 (Figure 4(a)) How-ever the quantitative differences are quite significant Thecalculated twisting angle between the monomeric moieties456 is much larger than the experimental one 226For the separation of the H-bonded nitrogen atoms thesame value 298 A is computed for both pairs while thereported X-ray distances are very different 292 and 299 AThe calculations yield nearly planar monomeric units inthe dimer whereas the experiment clearly shows distortionsFor instance the experimental NCCN angles are 38 and54 while the calculations yield the same smaller value of25 These results suggest that intermolecular interactionsin the crystal may affect the dimer structure We havetherefore repeated the X-ray measurements performingexperiments both at 293 K and at lower temperaturesSurprisingly a different structure than previously reportedwas obtained (Figure 4(b)) consisting of two doubly H-bonded PQ monomeric units in a planar arrangement Inorder to obtain a theoretical model for the planar dimerwe imposed the planarity in the optimization procedureThis resulted in one negative frequency in the optimizedstructure the vibration corresponding to mutual twisting ofthe planar moieties Computationally the planar geometryshows the NN distance of 312 A whereas the experimentalvalue is 301 A

One can conclude that PQ forms polymorphs in thecrystal which differ in the structure of dimers especiallywith regard to parameters usually considered important forthe strength of intermolecular hydrogen bond Thereforeit seemed interesting to carry out vibrational spectroscopystudies in order to (i) determine how does the formation ofa doubly H-bonded dimer affect the vibrational pattern and(ii) probe the possible differences in the vibrational structurebetween planar and nonplanar (but both doubly H-bonded)dimers

32 IR Measurements Figure 5 presents the IR spectrarecorded for the monomeric PQ isolated in an Ar matrixand the spectra of polycrystalline PQ corresponding tothe planar dimeric structure measured on a KBr windowThe experimental data are compared with the results ofcalculations performed for the monomer and for the twoforms of the dimer a fully-optimized non-planar structureand a form with imposed planar geometry

The spectra of monomeric PQ are very well reproducedby calculations with regard to both band positions andintensities They will be treated in detail in a separate workin which the combination of theoretical modelling IRRaman and high resolution fluorescence spectra obtainedfor supersonic jet-isolated PQ allowed reliable assignmentsof nearly all of 57 vibrations of monomeric PQ Here wefocus on the dimer using the monomer vibrations as astarting point Figure 5 shows that while the general patternof the IR spectrum of dimeric PQ roughly resembles that

800 1200 1600 3200 3600

Figure 5 (a) IR spectrum of the monomer in Ar matrix at15 K (b) simulated monomer spectrum (c) dimeric polycrystallinePQ at 293 K results of calculations performed for the planar (d)and twisted (e) dimer The scaling factor of 09682 was used incalculations

of the monomer significant differences are observed inspecific regions The largest difference is observed for theNH stretching mode The monomer peak observed around3500 cmminus1 (the observed triplet is due to argon site structure)disappears in the crystalline sample where a broad bandis detected centered at 3210 cmminus1 This red shift of almost300 cmminus1 is characteristic for the formation of fairly strongNHmiddot middot middotN intermolecular hydrogen bonds The calculationspredict the shifts of 320 and 220 cmminus1 for the twisted andplanar forms respectively As expected the larger shift iscomputed for a structure with a shorter NndashN distance andthus a stronger hydrogen bond The better agreement withexperiment for the larger value is somewhat misleadingsince the X-ray measurements demonstrated that the samplecorresponded to a planar dimer Further arguments areprovided by the analysis of the IR spectrum in the energyregion corresponding to out-of-plane vibrations For themonomeric PQ calculations yield two modes that containsignificant NH out-of-plane contributions They can bereadily identified in the experimental spectrum as the bandsat 491 and 527 cmminus1 In the IR spectrum of a dimer thesebands are still observed but in addition a broad bandappears at 743 cmminus1 in nice agreement with calculationswhich predict for a planar structure a transition at 734 cmminus1For the twisted dimer structure there no longer exist pureldquoout-of-planerdquo modes The mode which still retains much ofthat character is predicted to lie at 807 cmminus1 and to have anintensity twice that of the planar structure Comparison ofthe experimental and simulated IR spectra in the region of650ndash950 cmminus1 leaves no doubt that the observed spectrumoriginates from a planar species The value of the blueshift of the NH out-of-plane bending mode which exceeds200 cmminus1 again points to a strong intermolecular HB indimeric PQ

There is no single particular vibration in the monomerwhich could be assigned to a pure NH in-plane bendingmode This is also true for the dimer The IR transitions

800 1200 1600 3200 3600

Raman shift (cmminus1)

Figure 6 (a) Raman spectrum of the monomer in Ar matrix at15 K (b) simulated monomer spectrum (c) dimeric polycrystallinePQ measured at 293 K results of calculations performed for theplanar (d) and twisted (e) dimer The scaling factor of 09682 wasused in calculations

computed for the planar dimer consist of symmetric andantisymmetric combinations of the monomer modes Onlythe latter are IR-active A very similar pattern of IR tran-sitions is obtained for the nonplanar dimer Figure 5 showsthat in the region above 1000 cmminus1 the predicted IR spectrafor both planar and twisted dimer are almost identical

The analysis of the IR spectra demonstrates that bothNH stretching and out-of-plane bending modes are efficientmarkers for the HB formation However only the latter canbe used to indicate the planar structure of the H-bondeddimer

33 Raman Spectra Comparison of Raman spectra sim-ulated and measured for monomeric and dimeric PQ ispresented in Figure 6 Contrary to the case of the IR spectrathe calculations now predict differences between monomericand dimeric species in the region above 1200 cmminus1 Belowthat value the simulated spectra are very similar for the threespecies But even above 1200 cmminus1 the Raman spectra com-puted for planar and twisted dimers resemble each other verystrongly excluding their use for structure determination

The calculations predict that the Raman activity of theNH stretching mode should be drastically increased about15 times upon HB formation This increase was confirmedby experiment No band corresponding to the NH stretchwas observed for monomeric PQ but it could be readilydetected at 3200 cmminus1 for the crystalline sample Thusformation of the intermolecularly H-bonded dimer enhancesthe polarizability to a degree that enables observation of avibrational feature characteristic of the hydrogen bond

34 Vibrational Assignments Based on IR and Raman spec-tra and the results of calculations we present in Table 1the tentative assignments for the vibrations of dimeric PQThe experimental data given in the Table correspond to the

planar structure whereas the calculations are given for bothplanar and twisted forms Since the planarity was artificiallyimposed in the calculations one might expect that the resultsin this case are less reliable Still as can be seen from Figures5 and 6 the calculated vibrational patterns are very similarboth for IR and Raman spectra The largest differences areobserved for the NH stretching and out-of-plane bendingmodes which were specifically discussed above

35 Obtaining Different Polymorphic Forms As alreadymentioned the crystalline samples of PQ which we haveexamined by X-ray IR and Raman techniques correspondedto planar dimers and thus to a different polymorphic formthan observed previously [29] We have tried to obtainboth forms by crystallization from different solvents andthen using Raman spectroscopy as a tool for structuredetermination A trial and error approach was adopted sinceno information about crystallization details was given in thework reporting the twisted structure [29] Figure 7 presentsthe Raman spectra measured for samples crystallized fromfive different solvents The spectra are similar but significantdifferences can be detected in two regions A peak of weakintensity appears at 738 cmminus1 for PQ crystallized fromcyclohexane toluene and methanol but not from diethylether and dichloromethane The second region correspondsto two fairly strong peaks observed at 1062 and 1074 cmminus1Their relative intensity patterns (a more intense feature lyingat higher energy) are the same for the samples revealingthe 738 cmminus1 transition For two other samples which lackthe 738 cmminus1 peak the intensity ratio changes now thelower energy peak becomes higher Such behavior stronglysuggests that the PQ samples obtained from cyclohexanetoluene and methanol correspond to planar dimers whereasthose crystallized from diethyl ether and dichloromethaneto the nonplanar ones This is confirmed by the resultsof calculations which predict exactly such reversal of therelative intensity pattern for the 1062 and 1074 cmminus1 peaksupon going from a planar to a twisted dimeric form (seeFigure 6)

36 High-Pressure Experiments The idea behind spectralmeasurements for samples submitted to high pressures wasto observe pressure-induced changes in the strength andpossibly also of the structure of the intermolecular hydrogenbond Figure 8 shows the Raman spectra recorded for PQdimers under normal and elevated pressures Nearly all peaksobserved below 1700 cmminus1 evolve in a similar way withincreasing pressure the maxima shift to the blue by 5ndash8 cmminus1 Much larger shifts towards higher transition energiesare detected for the CH stretching bands which shift by30 cmminus1 or more A reversal of the relative intensities isobserved for the bands at 3114 and 3137 cmminus1 All thesechanges are reversible as shown by comparison of the spectrarecorded for the same sample before and after going throughthe high pressure cycle

The effects most relevant to this work are related tochanges in the HB strength and structure Figure 8 shows thatthe NH stretching band observed at 3200 cmminus1 moves to

Table 1 Comparison of the experimental IR and Raman spectra with the vibrational frequencies calculated for the twisted and planar formsof the PQ dimer

Calculateda ObservedAssignmentd

frequencye

(cmminus1)frequencyf

(cmminus1)IR intensityg

(kmmol)Raman activityg

(A4amu)IRb Ramanc

ν1 32189 32188 257838 (180254) 2841 (0) 3180h s as NH str

ν2 32013 32112 4828 (001) 112668 (103201) 3193h s s NH str

ν3 31513 30653 497 (727) 4950 (5358) as CH str

ν4 31513 30653 052 (143) 28136 (27292) 3137 w s CH str

ν5 31330 30393 075 (064) 16425 (23743) 3112 w s CH str

ν6 31330 30393 502 (687) 3947 (2230) as CH str

ν7 30914 29967 488 (014) 45354 (55504) 3067 w s CH str

ν8 30914 29966 2973 (3636) 9536 (232) as CH str

ν9 30783 29832 161 (001) 50164 (51594) 3052 w s CH str

ν10 30782 29831 6342 (6051) 2711 (005) as CH str

ν11 30624 29702 017 (1758) 30247 (12190) 3040 w s CH str

ν12 30624 29699 1975 (1735) 2436 (12543) as CH str

ν13 30597 29666 218 (851) 232 (11318) as CH str

ν14 30596 29664 033 (659) 1254 (14011) 3019 sh s CH str

ν15 30392 29645 487 (027) 14824 (3094) 3000 s CH str

ν16 30391 29645 1573 (038) 2875 (4260) as CH str

1660 m

1632 m

ν17 16028 15514 625 (0) 5040 (5335) 1620 m NH s b CC str cr

ν18 15987 15484 3668 (2779) 1029 (0) 1615 s NH as bCC str cr

ν19 15811 15342 2787 (3495) 1316 (007) 1594 m (CC CN) as str pyridine

ν20 15803 15334 433 (003) 6613 (8174) 1595 m (CC CN) s str pyridine

ν21 15499 14998 099 (001) 2848 (3040) 1562 m NH CH s b pyridine

ν22 15459 14977 5703 (4342) 611 (001) 1560 m NH CH as b pyridine

ν23 15150 14681 414 (0) 2116 (2748) 1528 w NH CH18204139 s b

ν24 15125 14666 9078 (9524) 573 (0) 1524 s NH CH18204139 as b

ν25 14880 14413 4020 (4288) 1742 (0) 1497 m NH as b CC str pyr

ν26 14843 14364 455 (0) 11120 (17284) 1500 s NH s b CC str pyr

ν27 14698 14235 132 (0) 24381 (22553) 1484 vs CH20 s b skel def CC

ν28 14681 14220 996 (703) 5654 (0) 1482 w CH20 as b skel def CC

ν29 14279 13854 077 (0) 3740 (3230) 1440 w NH CH20214142 s b skel def pyr

ν30 14272 13794 925 (1049) 691 (0) 1435 w NH CH20214142 as b skel def pyr

ν31 14149 13691 496 (627) 097 (0) 1428 w CH1718192038394041 as b CC str cr

ν32 14147 13669 084 (0) 932 (1332) 1430 m CH1718192038394041 s b CC str cr

ν33 13946 13452 004 (0) 15239 (13761) 1407 m NH CH2142 s b CC str pyr

ν34 13887 13436 3368 (3734) 1726 (001) 1403 m NH CH2142 as b CC str pyr

ν35 13637 13187 12483 (14601) 2236 (0) 1386 s skel def CH as b

ν36 13617 13159 2055 (0) 11109 (15734) 1386 s skel def CH s b

ν37 13241 12806 601 (0) 7848 (8413) 1341 m skel def CH s b

ν38 13202 12778 8762 (8548) 1037 (0) 1333 m skel def CH as b

ν39 12919 12494 358 (130) 3285 (026) 1301 vw skel def CH as b

ν40 12913 12488 077 (0) 11986 (12570) 1302 m skel def CH s b

ν41 12638 12256 151 (0) 1398 (2415) 1275 mCH s b C7N11 C28N32 C8C9 C29C30str

ν42 12634 12227 3725 (3235) 174 (0) 1268 mCH as b C7N11 C28N32 C8C9C29C30 str

Table 1 Continued

frequencye

(A4amu)IRb Ramanc

ν43 12390 11999 123 (2433) 228 (002) 1251 w NH and CH s b

ν44 12365 11998 2470 (011) 043 (481) 1243 m NH and CH as b

ν45 11996 11603 167 (057) 025 (0) 1210 w CH as b CC str cr

ν46 11995 11597 003 (0) 127 (108) 1210 w CH s b CC str cr

ν47 11807 11407 058 (1659) 252 (0) 1193 w CH s b CC str

ν48 11806 11388 1528 (0) 009 (207) 1193 m CH s b CC str

ν49 11237 10892 881 (1081) 066 (001) 1133 m CH17181920 and CH38394041 as b

ν50 11235 10891 171 (003) 224 (215) 1132 w CH17181920 and CH38394041 s b

ν51 11123 10746 050 (2590) 101 (0)CH15171820 and CH36383941 andNH s b

ν52 11106 10741 2251 (0) 023 (496) 1121 mCH15171820 and CH36383941 andNH as b

ν53 10756 10417 014 (0) 342 (806) 1090 w skel def CH s b

ν54 10710 10353 5141 (6984) 035 (0) 1082 s skel def CH as b

ν55 10568 10267 002 (0) 3098 (6948) 1075 s CH15163637 s b

ν56 10559 10246 403 (350) 200 (0) 1065 w CH15163637 s b

ν57 10478 10159 086 (0) 5349 (2134) 1062 m skel def CH as b

ν58 10464 10123 531 (1168) 232 (0) 1058 w skel def CH as b

ν59 10089 9785 564 (669) 318 (0) 1025 w skel def

ν60 10086 9755 060 (0) 2442 (1784) 1019 w skel def

ν61 9686 9318 125 (127) 015 (0) 972 vw 973 vw CH19-21 and CH40-42 s ldquoooprdquo twisting

ν62 9686 9317 032 (0) 004 (024) 965 vw 969 vw CH19-21 and CH40-42 as ldquoooprdquo twisting

ν63 9442 9094 003 (0) 026 (060) 951 wCH171821 and CH383942 as ldquoooprdquowag

ν64 9441 9094 011 (038) 013 (0)CH171821 and CH383942 as ldquoooprdquowag

ν65 9389 8875 086 (171) 040 (011) 946 wCH17-1921 and CH38-4042 as ldquoooprdquotwisting

ν67 8884 8571 5113 (4491) 284 (0) 899 m as skel def pyr (N11-C12-C13)

ν68 8850 8514 1647 (0) 2678 (2525) 890 m s skel def pyr (N11-C12-C13)

ν69 8729 8418 1051 (1727) 486 (0) 882 m skel def NH s twisting

ν70 8713 8398 1674 (0) 146 (2052) skel def NH as twisting

ν71 8609 8090 907 (0) 1306 (003) 853 w CH15163637 and s ldquoooprdquo wag

ν72 8572 8088 282 (2139) 083 (0) 860 vw CH15163637 and s ldquoooprdquo wag

ν73 8291 7931 131 (4369) 063 (0) s skel ldquoooprdquo def NH CH wag

ν74 8238 7896 518 (0) 065 (124) as skel ldquoooprdquo def NH CH wag

ν75 8146 7740 10369 (7652) 272 (0) 823 m 826 w s NH ldquoooprdquo

ν76 8050 7718 908 (0) 045 (073) as cr ldquoooprdquo def CH as wag

ν77 8043 7350 2908 (0) 076 (029) 801 s s cr ldquoooprdquo def CH s wag

ν78 7829 7335 592 (3311) 003 (0) as NH ldquoooprdquo

ν79 7642 7282 535 (0) 023 (3285) CH17-2138-42 s ldquoooprdquo wag

ν81 7512 7168 101 (5755) 3636 (0) 763 m s ldquoiprdquo skel def

ν82 7508 6991 133 (0) 339 (026) 773 m 770 sh as ldquoiprdquo skel def

ν83 7278 6883 1714 (195) 014 (0) 738 m 739 w CH15-18 and CH36-39 ldquoooprdquo s wag

ν84 7263 6697 258 (6515) 002 (0) CH15-18 and CH36-39 ldquoooprdquo as wag

ν85 6962 6683 6118 (0) 187 (088) s ldquoooprdquo skel def CH s wag

ν86 6961 6569 1335 (0) 005 (071) as ldquoooprdquo skel def CH s wag

Table 1 Continued

frequencye

(A4amu)IRb Ramanc

ν87 6786 6534 003 (0) 6470 (7042) 684 vw 686 ss cr and pyridine ring b (sym alongN1-C4 C7-C10 axis)

ν88 6752 6511 1594 (1461) 187 (0) 696 m 696 shas cr and pyridine ring b (sym alongN1-C4 C7-C10 axis)

ν89 6106 5891 528 (430) 001 (0)as cr and pyridine ring b (asym alongN1-C4 C7-C10 axis)

ν90 6070 5878 115 (0) 613 (811) 615 vw 614 ws cr and pyridine ring b (asym alongN1-C4 C7-C10 axis)

ν91 6053 5759 089 (1468) 058 (0) as ldquoooprdquo skel def pyr

ν92 6045 5722 617 (0) 214 (068) 602 w s ldquoooprdquo skel def pyr

ν93 5737 5524 075 (239) 053 (0) as ldquoooprdquo skel def pyridine and cr

ν94 5735 5476 055 (0) 098 (204) 571 w s ldquoooprdquo skel def pyridine and cr

ν95 5162 5029 597 (719) 151 (0) as ldquoiprdquo skel def pyridine and cr

ν96 5161 5006 074 (0) 804 (1089) 524 m s ldquoiprdquo skel def pyridine and cr

ν99 4728 4550 828 (0) 182 (913) as ldquoiprdquo skel def cr

ν100 4715 4535 017 (839) 811 (0) 479 m s ldquoiprdquo skel def cr

ν101 4301 4145 054 (803) 1161 (0) 438 m s ldquoiprdquo skel def cr CH17193840 ldquoooprdquo

ν102 4299 4143 754 (0) 259 (1589) 432 sh as ldquoiprdquo skel def cr CH17193840 ldquoooprdquo

ν103 4263 4114 052 (0) 164 (225) as ldquoooprdquo skel def cr CH17193840 ldquoooprdquo

ν104 4251 4103 058 (200) 449 (0) 424 w s ldquoooprdquo skel def cr CH17193840 ldquoooprdquo

ν105 2844 2733 189 (193) 081 (0) as ldquoooprdquo pyridine and cr rock

ν106 2835 2709 137 (0) 134 (206) 300 w s ldquoooprdquo pyridine and cr rock

ν107 2528 2447 065 (673) 004 (0) as ldquoooprdquo pyr and cr rock

ν108 2495 2380 620 (629) 017 (0) s ldquoooprdquo pyr and cr rock

ν109 2488 2375 353 (0) 052 (682) 267 w as pyr and pyridine rings ldquoiprdquo bend

ν110 2433 2318 007 (0) 518 (005) 251 m s pyr and pyridine rings ldquoiprdquo bend

ν111 1630 1418 001 (004) 027 (0) 169 w as pyr and pyridine tor

ν112 1504 1391 003 (0) 043 (124) 154 m s pyr and pyridine tor

ν113 1223 1170 627 (0) 023 (244) as ldquoooprdquo pyridine and pyr rock

ν114 1219 1123 370 (600) 091 (0) s ldquoooprdquo pyridine and pyr rock

ν115 802 759 358 (102) 354 (0) dim rock

ν116 782 714 002 (0) 154 (180) dim b

ν117 670 596 000 (0) 115 (312) dim b

ν118 434 213 018 (012) 982 (0) dim rock

ν119 261 92 027 (0) 496 (1509) dim rock

ν120 213 minus353 004 (0) 1169 (0) dim toraB3LYPcc-pVTZ C2 symmetry group scaling factor = 09682 as recommended in the literature [39]bPolycrystalline sample 293 KcPolycrystalline sample 293 K 785 nm laser (633 nm was used in the NH region)dAbbreviations s symmetric as antisymmetric str stretch b bend ip in-plane oop out-of-plane skel def skeletal deformation tor torsion pyridpyridine pyr pyrrole cr central ringeTwisted dimerfPlanar dimergIn parentheses values computed for the planar dimerhVery broad (sim200 cmminus1)

800 1200 1600 3200 3600

Figure 7 Raman spectra measured for samples crystallized fromfive different solvents cyclohexane (a) methanol (b) toluene(c) dichloromethane (d) diethyl ether (e) Dashed vertical linesindicate regions with structure-sensitive transitions (see text)

200 400 600 800 1000 1200 1400 3000 3200

Figure 8 Raman spectra of crystalline PQ as a function of pressurenormal pressure 1 atm (a) 2 times 103 atm (b) 22 times 103 atm (c) 35 times103 atm (d) 1 atm (e) at the end of pressure cycle The low and highfrequency regions are normalized separately to their highest bandsA region between 1300ndash1370 cmminus1 exhibiting a strong Raman peakfrom diamond culets was removed

the red with increasing pressure Such behavior is oppositeto that of other modes and indicates the increase of theHB strength most probably due to a shorter NHmiddot middot middotNdistance Unfortunately the exact amount of the shift cannotbe determined as the band becomes buried under thetransitions corresponding to CH stretches Experimentsare planned with either N- or C-deuterated PQ to avoidinterferences of NHND vibrations with other modes

The second effect is the change in the relative intensitypattern with increasing pressure observed for the peaks at1062 and 1074 cmminus1 As discussed above such behaviour canindicate a transition from a cyclic toward a twisted structureFor another mode diagnostic in this respect 738 cmminus1 weobserve decreasing intensity However it can still be detectedat the highest pressures applied It may be that what is

Figure 9 PQ dimer surrounded by identical neighbors (taken fromX-ray data) The dimer in the middle was being distorted along thetwisting coordinate and then the whole structure was optimized

observed is gradual twisting not necessarily leading to thesame angle between the monomeric units as observed for thenonplanar polymorph under normal pressure More detailedinvestigations are planned once both planar and twisteddimeric samples are available The experiments described inthe previous section bode well for such studies

37 Simulations of Polymorphic Structures The existence ofboth planar and twisted dimers leads to the question ofthe energy barrier separating the two phases Theoreticalsimulations have been carried out in order to check the localminimum character of each structure and to estimate theirrelative stabilities In this procedure a dimer surrounded by14 identical neighbours (Figure 9) was distorted towards thestructure of the other polymorph (twisted for the initiallyplanar form and vice versa) The whole ensemble wasthen optimized Both planar and twisted structures relaxedback to the initial form showing that they correspondto the minimum and providing additional independentconfirmation of the existence of two crystal polymorphicforms of the PQ dimer These results indicate that a collectiverather than local distortion of the crystal is required for thephase change in PQ

In agreement with the high pressure experiments com-parison of energies calculated for the slab consisting of 16molecules for both planar and twisted dimers revealed alower energy for the latter

4 Summary and Conclusions

A combination of X-ray IR and Raman spectroscopy highpressure techniques and quantum chemical calculationsresulted in the detection of two polymorphic forms ofdimeric PQ Both types of dimer reveal a cyclic doubly

hydrogen-bonded structure but differ in the planar versustwisted arrangement of the monomeric units The calcula-tions predict a twisted dimer structure whereas imposingplanarity results in one negative vibrational frequencycorresponding to the twisting coordinate These results showthat the isolated dimer should be nonplanar and thus thepolymorphism is due to the interplay of interactions betweenthe two monomeric units forming the hydrogen bond anddimer-dimer interactions in the crystal The experimentsindicate that upon applying pressure the planar form can beconverted into the twisted one

The NH stretching and out-of-plane bending modesobserved in the IR spectra were shown to be clear indicatorsof the HB formation The analysis of the position of thelatter could be used to determine the structure of the H-bonded dimer With respect to the influence of HB formationon the Raman spectra a large increase of the intensitywas observed for the NH stretching band in the H-bondeddimers indicating increase of polarizability The Ramanspectra were also diagnostic for structural assignments eventhough the spectra are quite similar the intensity ratio of twopeaks observed at 1062 and 1074 cmminus1 provides informationwhether the PQ dimer is planar or not

Our future plans include testing a possibility of photoin-duced double proton transfer in both forms of crystallinePQ Both kinetics and thermodynamics of such a processshould be strongly structure-sensitive Moreover we haveselected PQ as one of the objects in the investigations ofthe influence of plasmonic structures on the spectral andphotophysical characteristics of chromophores located in thevicinity of metallic environments The results of vibrationaland structural analysis presented in this work will providea starting point for experiments in which monomers anddimers of PQ will be placed on or close to metal surfaces

Acknowledgments

The work was supported by the Grant 3550BH03201140from the Polish National Science Centre The authorsacknowledge the computing grant G17-14 from the Inter-disciplinary Centre for Mathematical and ComputationalModeling of the Warsaw University They would like tothank bwGRiD (httpwwwbw-gridde) member of theGerman D-Grid initiative funded by the Ministry forEducation and Research (Bundesministerium fur Bildungund Forschung) and the Ministry for Science Research andArts Baden-Wurttemberg (Ministerium fur WissenschaftForschung und Kunst Baden-Wurttemberg) for providingthe opportunity to use parallel computing facilities andperform quantum chemical calculations R P Thummelthanks the Robert A Welch Foundation (E-621) and theNational Science Foundation (CHE-0714751)

References

[1] J Waluk ldquoHydrogen-bonding-induced phenomena in bifunc-tional heteroazaaromaticsrdquo Accounts of Chemical Research vol36 no 11 pp 832ndash838 2003

[2] J A Walmsley ldquoSelf-association of 7-azaindole in nonpolarsolventsrdquo The Journal of Physical Chemistry vol 85 no 21pp 3181ndash3187 1981

[3] P Dufour Y Dartiguenave M Dartiguenave et al ldquoCrys-tal structures of 7-azaindole an unusual hydrogen-bondedtetramer and of two of its methylmercury(II)complexesrdquoCanadian Journal of Chemistry vol 68 no 1 pp 193ndash2011990

[4] H Yokoyama H Watanabe T Omi S I Ishiuchi and MFujii ldquoStructure of hydrogen-bonded clusters of 7-azaindolestudied by IR dip spectroscopy and ab initio molecular orbitalcalculationrdquo Journal of Physical Chemistry A vol 105 no 41pp 9366ndash9374 2001

[5] K Sakota Y Kageura and H Sekiya ldquoCooperativity ofhydrogen-bonded networks in 7-azaindole(CH3OH)n (n =2 3)clusters evidenced by IR-UV ion-dip spectroscopy andnatural bond orbital analysisrdquo Journal of Chemical Physics vol129 no 5 Article ID 054303 2008

[6] K Sakota Y Komure W Ishikawa and H SekiyaldquoSpectroscopic study on the structural isomers of 7-azaindole(ethanol)n (n = 1minus 3) and multiple-proton transferreactions in the gas phaserdquo Journal of Chemical Physics vol130 no 22 Article ID 224307 2009

[7] T B C Vu I Kalkman W L Meerts Y N Svartsov CJacoby and M Schmitt ldquoRotationally resolved electronicspectroscopy of water clusters of 7-azaindolerdquo Journal ofChemical Physics vol 128 no 21 Article ID 214311 2008

[8] G A Pino I Alata C Dedonder C Jouvet K Sakota andH Sekiya ldquoPhoton induced isomerization in the first excitedstate of the 7-azaindole-(H2O)3 clusterrdquo Physical ChemistryChemical Physics vol 13 no 13 pp 6325ndash6331 2011

[9] K Sakota C Jouvet C Dedonder M Fujii and H SekiyaldquoExcited-state triple-proton transfer in 7-azaindole(H2O)2

and reaction path studied by electronic spectroscopy in the gasphase and quantum chemical calculationsrdquo Journal of PhysicalChemistry A vol 114 no 42 pp 11161ndash11166 2010

[10] K Suwinska ldquoCrystal structure communicationsrdquo Acta Crys-tallographica C vol 41 pp 973ndash975 1985

[11] J Waluk and B Pakuła ldquoViscosity and temperature effectsin excited state double proton transfer iuminescence of 1-azacarbazole dimers in solid state and solutionrdquo Journal ofMolecular Structure vol 114 pp 359ndash362 1984

[12] J Waluk A Grabowska B Pakuła and J Sepioł ldquoViscosityvs temperature effects in excited-state double proton transferComparison of 1-azacarbazole with 7-azaindolerdquo The Journalof Physical Chemistry vol 88 no 6 pp 1160ndash1162 1984

[13] J Waluk J Herbich D Oelkrug and S Uhl ldquoExcited-statedouble proton transfer in the solid state the dimers of 1-azacarbazolerdquo Journal of Physical Chemistry vol 90 no 17pp 3866ndash3868 1986

[14] J Catalan ldquoPhotophysics of 1-azacarbazole dimers a reap-praisalrdquo The Journal of Physical Chemistry A vol 111 no 36pp 8774ndash8779 2007

[15] D Marks H Zhang P Borowicz J Waluk and M Glasbeekldquo(Sub)picosecond fluorescence upconversion studies of inter-molecular proton transfer of dipyrido[23-a3prime2prime-i]carbazoleand related compoundsrdquo Journal of Physical Chemistry A vol104 no 31 pp 7167ndash7175 2000

[16] A Kyrychenko J Herbich M Izydorzak F Wu R PThummel and J Waluk ldquoRole of ground state structure inphotoinduced tautomerization in bifunctional proton donor-acceptor molecules 1H-pyrrolo[32-h]quinoline and related

compoundsrdquo Journal of the American Chemical Society vol121 no 48 pp 11179ndash11188 1999

[17] A Kyrychenko and J Waluk ldquoExcited-state proton transferthrough water bridges and structure of hydrogen-bondedcomplexes in 1H-pyrrolo[32-h]quinoline adiabatic time-dependent density functional theory studyrdquo The Journal ofPhysical Chemistry A vol 110 no 43 pp 11958ndash11967 2006

[18] Y Nosenko M Kunitski R P Thummel et al ldquoDetectionand structural characterization of clusters with ultrashort-lived electronically excited states IR absorption detected byfemtosecond multiphoton ionizationrdquo Journal of the AmericanChemical Society vol 128 no 31 pp 10000ndash10001 2006

[19] Y Nosenko A Kyrychenko R P Thummel J Waluk BBrutschy and J Herbich ldquoFluorescence quenching in cyclichydrogen-bonded complexes of 1H-pyrrolo[32-h]quinolinewith methanol cluster size effectrdquo Physical Chemistry Chem-ical Physics vol 9 no 25 pp 3276ndash3285 2007

[20] Y Nosenko M Kunitski C Riehn et al ldquoSeparation ofdifferent hydrogen-bonded clusters by femtosecond UV-ionization-detected infrared spectroscopy 1H-pyrrolo[32-h]quinolinemiddot(H2O)n=12 complexesrdquo Journal of Physical Chem-istry A vol 112 no 6 pp 1150ndash1156 2008

[21] J Herbich J Sepioł and J Waluk ldquoDetermination of theenergy barrier origin of the excited state double proton trans-fer in 7-azaindole alcohol complexesrdquo Journal of MolecularStructure vol 114 pp 329ndash332 1984

[22] D McMorrow and T J Aartsma ldquoSolvent-mediated protontransfer The roles of solvent structure and dynamics onthe excited-state tautomerization of 7-azaindolealcohol com-plexesrdquo Chemical Physics Letters vol 125 no 5-6 pp 581ndash585 1986

[23] J Konijnenberg A H Huizer and C A G O Varma ldquoSolute-solvent interaction in the photoinduced tautomerization of 7-azaindole in various alcohols and in mixtures of cyclohexaneand ethanolrdquo Journal of the Chemical Society Faraday Transac-tions 2 vol 84 no 8 pp 1163ndash1175 1988

[24] R S Moog S C Bovino and J D Simon ldquoSolvent relaxationand excited-state proton transfer 7-azaindole in ethanolrdquoJournal of Physical Chemistry vol 92 no 23 pp 6545ndash65471988

[25] R S Moog and M Maroncelli ldquo7-Azaindole in alcoholssolvation dynamics and proton transferrdquo Journal of PhysicalChemistry vol 95 no 25 pp 10359ndash10369 1991

[26] A V Smirnov D S English R L Rich et al ldquoPhotophysicsand biological applications of 7-azaindole and its analogsrdquoJournal of Physical Chemistry B vol 101 no 15 pp 2758ndash2769 1997

[27] S Mente and M Maroncelli ldquoSolvation and the excited-statetautomerization of 7-azaindole and 1-azacarbazole computersimulations in water and alcohol solventsrdquo Journal of PhysicalChemistry A vol 102 no 22 pp 3860ndash3876 1998

[28] K C Ingham M Abu-Elgheit and M Ashraf El-BayoumildquoConfirmation of biprotonic phototautomerism in 7-azaindole hydrogen-bonded dimersrdquo Journal of the AmericanChemical Society vol 93 no 20 pp 5023ndash5025 1971

[29] S N Krasnokutskii L N Kurkovskaya T A Shibanova andV P Shabunova ldquoStructure of 1H-pyrrolo[32-h]quinolinerdquoZhurnal Strukturnoi Khimii vol 32 p 131 1991

[30] J Herbich M Kijak R Luboradzki et al ldquoIn search for pho-totautomerization in solid dipyrido[23-a3prime2prime-i]carbazole rdquoJournal of Photochemistry and Photobiology A vol 154 no 1pp 61ndash68 2002

[31] F Wu C M Chamchoumis and R P Thummel ldquoBidentateligands that contain pyrrole in place of pyridinerdquo InorganicChemistry vol 39 no 3 pp 584ndash590 2000

[32] K Takemura S Minomura O Shimomura and Y FujiildquoObservation of molecular dissociation of iodine at highpressure by X-ray diffractionrdquo Physical Review Letters vol 45no 23 pp 1881ndash1884 1980

[33] H K Mao P M Bell J W Shaner and D J SteibergldquoSpecific volume measurements of Cu Mo Pd and Ag andcalibration of the ruby R1 fluorescence pressure gauge from006 to 1 Mbarrdquo Journal of Applied Physics vol 49 no 6 pp3276ndash3283 1978

[34] G M Sheldrick ldquoFoundations of crystallographyrdquo ActaCrystallographica A vol 64 pp 112ndash122 2008

[35] M D Segall P J D Lindan M J Probert et al ldquoFirst-principles simulation ideas illustrations and the CASTEPcoderdquo Journal of Physics Condensed Matter vol 14 no 11 pp2717ndash2744 2002

[36] J P Perdew J A Chevary S H Vosko et al ldquoAtomsmolecules solids and surfaces applications of the generalizedgradient approximation for exchange and correlationrdquo Physi-cal Review B vol 46 no 11 pp 6671ndash6687 1992

[37] D Vanderbilt ldquoSoft self-consistent pseudopotentials in ageneralized eigenvalue formalismrdquo Physical Review B vol 41no 11 pp 7892ndash7895 1990

[38] H J Monkhorst and J D Pack ldquoSpecial points for Brillouin-zone integrationsrdquo Physical Review B vol 13 no 12 pp 5188ndash5192 1976

[39] J P Merrick D Moran and L Radom ldquoAn evaluationof harmonic vibrational frequency scale factorsrdquo Journal ofPhysical Chemistry A vol 111 no 45 pp 11683ndash11700 2007

Effective Potential for Ultracold Atoms at the Zero Crossing ofa Feshbach Resonance

N T Zinner1 2

1 Department of Physics Harvard University Cambridge MA 02138 USA2 Department of Physics and Astronomy University of Aarhus 8000 Aarhus Denmark

Correspondence should be addressed to N T Zinner zinnerphysaudk

Academic Editor Ali Hussain Reshak

We consider finite-range effects when the scattering length goes to zero near a magnetically controlled Feshbach resonanceThe traditional effective-range expansion is badly behaved at this point and we therefore introduce an effective potential thatreproduces the full T-matrix To lowest order the effective potential goes as momentum squared times a factor that is well definedas the scattering length goes to zero The potential turns out to be proportional to the background scattering length squared timesthe background effective range for the resonance We proceed to estimate the applicability and relative importance of this potentialfor Bose-Einstein condensates and for two-component Fermi gases where the attractive nature of the effective potential can leadto collapse above a critical particle number or induce instability toward pairing and superfluidity For broad Feshbach resonancesthe higher order effect is completely negligible However for narrow resonances in tightly confined samples signatures might beexperimentally accessible This could be relevant for suboptical wavelength microstructured traps at the interface of cold atomsand solid-state surfaces

1 Introduction

Cold atomic gases have enjoyed many great successes sincethe first realizations of Bose-Einstein condensates in themid nineties [1] Ensembles of ultracold atomic gases canbe manipulated in magnetic or optical trap geometriesand in lattice setups effectively mimicking the structure ofreal materials and teaching us about their properties Inparticular extreme control can be exercised over the atom-atom interactions through the use of Feshbach resonance[2] Tuning the system into the regime of resonant two-bodyinteractions provides a controlled way of studying stronglycorrelated dynamics which is believed to be crucial for mate-rial properties such as high-temperature superconductivityor giant magnetoresistance

Recently there has been extended interest in weaklyinteracting Bose-Einstein condensates for use as an atomicinterferometer [3] and also to probe magnetic dipolarinteractions in condensates [4] This work was based on39K atoms where a broad Feshbach resonance exists at amagnetic field strength of B0 = 4024 G [5] which allows

a large tunability of the atomic interaction in experiments[6] Similar tunability has also been reported in a condensateof 7Li [7] The atomic interaction can be reduced by tuningthe scattering length a to zero also known as zero crossingIn a Gross-Pitaevskii mean-field picture we can thus neglectthe usual nonlinear term proportional to a The question isthen what other interactions are relevant As shown in [4]the magnetic dipole will contribute here

In the Gross-Pitaevskii picture we might also ask whetherhigher order terms in the interaction can contribute aroundzero crossing Recently it was shown that effective-rangecorrections can in fact influence the stability of condensatesaround zero crossing [8ndash10] The Feshbach resonances usedthus far in experiments have typically been very broad andas a result the effective range re will be small renderingthe higher order terms negligible However around narrowresonances this is not necessarily the case and finite-rangecorrections are not necessarily negligible

For the two-component Fermi gas there has beenincreased interest in producing a cold atom analog of thecelebrated Stoner model of ferromagnetism [11] which

applies to repulsively interacting fermions Theoretical pro-posals indicate that this should be possible [12ndash19] and anMIT experiment subsequently announced indications of theferromagnetic transition [20] The results caused controversysince the spin domains were not resolved [21ndash24] A laterexperiment in the same group did not find evidence ofthe ferromagnetic transition [25] However these studiesconsider broad Feshbach resonances and the situation withnarrow resonances is less clear One can imagine that finite-range corrections could play a role in driving the phasetransition In fact a recent experiment in Innsbruck [26] hasfound increased lifetimes of the repulsive gas in the stronglyimbalanced case providing hope that decay into moleculescan be controlled and ferromagnetism can be studied

The systematic inclusion of finite-range effects throughderivative terms in zero-range models was begun in thestudy of nuclear matter decades ago [27 28] Later on theintricacies of the cut-off problems that arise in this respectwere considered by many authors both for the relativistic andnonrelativistic case (see [29] for discussion and references)In the context of cold atoms and Feshbach resonances weneed to use a two-channel model [30] in order to take thelowest order finite-range term into account Similar modelswere already introduced in [31] and denoted resonancemodels (see fx [32] for a comprehensive review of scatteringmodels for ultracold atoms) We note that whereas resonancemodels treat the closed-channel molecular state as a pointboson the model of [30] treats the molecule more naturallyas a composite object of two atoms In the end the parametersof the two models turn out to be similarly related to thephysical parameters of Feshbach resonances (see for instancethe discussion of resonance models in [32])

In Figure 1 we show calculations of scattering length andeffective range for the Feshbach resonance at B = 2021 Gin 40K in both a coupled-channel model [33] and in thezero-range model discussed here We see the effective rangebeing roughly constant at resonance and then start to divergeat zero crossing The zero-range model provides a goodapproximation to the full calculations and for many-bodypurposes it is preferable due to its simplicity

Whereas the earlier work of [31] considered the regimeclose to the resonance we will be exclusively concerned withzero crossing To our knowledge the intricacies of this regionhave not been addressed in the literature in the context ofFeshbach resonances Around zero crossing the Feshbachmodel turns out to have a badly behaved effective-rangeexpansion The parameters obtained from the effective-rangeexpansion should therefore be used with extreme caution asthe series is divergent at this point However as we showin this paper the finite-range corrections obtained from thefull T-matrix at low momenta via an effective potential turnout to be the same as one would naively expect based onthe effective-range expansion After introducing the effectivepotential we consider its applicability and importance in thecase of Bose-Einstein condensates and for two-componentFermi gases where the attractive nature of the effectiveinteraction at zero crossing could lead to collapse abovea certain critical particle number or to pairing instabilityand superfluidity In general we find that tight external

180 190 200 210 220

minus500

minus1000

minus1500

B (Gauss)

a(B) zero rangea(B) numerical

re zero rangere numerical

Figure 1 Scattering length and effective range for the s-wavescattering of fermionic 40K atoms around the Feshbach resonanceat B0 = 2021 G demonstrating the divergence in a coupled-channelcalculation (symbols) [33] and in a zero-range model (full lines)The difference in the zero-range and coupled-channel models iscaused by the presence of a bound state close to threshold in theopen channel

confinement is a necessary condition for the higher ordereffects to dominate the magnetic dipole interaction and beexperimentally observable

2 Two-Channel Model

We consider a two-channel s-wave Feshbach model withzero-range interactions [30] for which the on-shell open-open channel T-matrix as a function of magnetic field Bis

Too(B) =(4π2m

1 + ΔμΔB(2q2mminus Δμ(B minus B0)

))minus1 + iabgq

where Δμ is the difference between the magnetic moments inthe open and closed channel q is the relative momentum ofthe atoms of mass m abg is the scattering length away fromthe resonance at magnetic field B0 and ΔB is the width ofthe resonance We can compare this to the standard vacuumexpression for the T-matrix in terms of the phase-shift givenby

Tν =(4π2m

minusqa cot δ(q)

+ iaq (2)

Typically one has the low-energy expression minusq cot δ(q)rarr minus1a which implies that

Tν minusrarr(4π2m

1 + iaqminusrarr 0 (3)

However as we now discuss for the realistic two-channelT-matrix for Feshbach resonances the quantity minusq cot δ(q)is not well defined and the conclusion that the T-matrixvanishes at zero crossing is only true for zero momentumq = 0 as we now discuss

From (1) and (2) we obtain the relation for the phase-shift

q cot δ(q) = minus1

ΔμΔB(2q2mminus Δμ(B minus B0)

))minus1

We now expand the right-hand side in powers of q as isusually done in an effective-range expansion This yields

+infinsumn=1

minus1abg

[minusabgre02

]n[ abga(B)

minus 1]n+1

where a(B) = abg (1 minus ΔB(B minus B0)) is the commonparametrization from single-channel models and re0 =minus22(mΔBΔμabg) is the background value of the effectiverange around the resonance From (5) we can now read offall coefficients in an effective-range expansion with their fullB-field dependence For instance the effective range is givensimply by re = re0[(abga) minus 1]2 which is divergent whena(B) rarr 0 We also clearly see that all the other coefficientsare divergent in that limit This is signaled also before doingthe full expansion in q as the first term in (5) diverges at zerocrossing However in effective potentials derived from theT-matrix these problems are not transparent as the lowestorder coefficient is proportional to a(B) (see (12)) Belowwe will discuss what kind of constraints this introduces onthe applicability of the effective-range expansion near zerocrossing We note that similar issues were briefly discussed ina different context in [34] where an equivalent to (7) belowwas obtained

Let us first consider the low-q limit and compare the fullT-matrix with the effective-range expansion as zero crossingis approached Taking the low-q limit of (4) at zero crossingwhere ΔB(B minus B0) = 1 we find

q cot δ(q) minusrarr minus1

abgminus ΔμΔB

2q2m (6)

which diverges as qminus2 Therefore the coefficients of theexpansion in (5) must necessarily diverge in order to retainany hope of describing the low-q behavior Furthermoresince the expansion is an alternating series and thereforeslowly converged we also conclude that many terms must beretained for a fair approximation at very small but nonzeroq The same conclusion can be reached by considering theradius of convergence of (5) which we find by locating thepole in (4) at 2q2m = Δμ(BminusB0minusΔB) This radius indeedgoes to zero at zero crossing We are thus forced to concludethat the effective-range expansion breaks down near zerocrossing

21 Effective Potential at Zero Crossing Since the effective-range expansion is insufficient we consider the full T-matrixin the low-q limit at zero crossing To lowest order we have

Too(B = B0 + ΔB) = minus4π2abgm

mΔμΔB+O

(q4) (7)

Using the expression for re0 this can be written

a2bgre0

2q2 (8)

Knowing the T-matrix at low q we can now proceed tofind an effective low-q potential through the Lippmann-Schwinger equation

V = T minus TG0V (9)

where G0 = (EminusH0 + iδ)minus1 is the free space Greenrsquos function[35] This equation can be solved for T(q qprime)prop q2 + qprime2 (thesymmetrized version of the full T-matrix) in an explicit cut-off approach [29 35] and then be expanded to order q2 forconsistence with the input T-matrix In the long-wavelengthlimit we can take the cut-off to zero [35] and for the on-shelleffective potential we then obtain the obvious answer

V(q) = 4π2

a2bgre0

2q2 (10)

in momentum space The effective potential in real-space isnow easily found by canonical substitution (q rarr minusinabla) andappropriate symmetrization [36] We have

V(r) = minus4π2

a2bgre0

[larrnabla2

rδ(r) + δ(r)nabla2r

] (11)

Notice that the Lippmann-Schwinger approach is nonper-turbative as opposed to the perturbative energy shift method[36 37]

22 Comparison to Effective-Range Expansion and Energy-Shift Method Away from zero crossing one can easilyrelate the effective-range expansion to an effective potentialthrough the perturbative energy shift method [18 25 26] Tosecond order the s-wave effective potential is

V(r) = 4π2a

[δ(r) +

(larrnabla2

)] (12)

where the first term is the effective interaction usuallyemployed in mean-field theories of cold atoms [35] In termsof a and re we have g2 = a23minus are2 [36 37] with the field-dependent a = a(B) and re = re(B)

At zero-crossing the first term in (12) vanishes and onemight expect the second term to vanish as well Howeverin the naive effective-range expansion of the two-channelmodel discussed above we saw that re diverges as aminus2 and wetherefore have

limararr 0

ag2 = minusa2bgre0

2 (13)

99Effective Potential for Ultracold Atoms at the Zero Crossing of a Feshbach Resonance

In particular if we for a moment ignore q4 terms inthe effective-range expansion we recover exactly the sameeffective potential as in (11) at zero crossing The finitelimiting result in (13) shows that the potential in (12) is welldefined as a rarr 0 provided that appropriate regularizationand renormalization are performed Equation (12) thusapplies equally well at resonance (a rarr infin) where thegradient terms are small and at zero crossing where the lowestorder delta function term is unimportant It is thus a well-defined effective potential over the entire range of a Feshbachresonance

We therefore see that even though the effective-rangeexpansion has divergent coefficients at zero crossing thelowest order does in fact give the same effective potential asthe full T-matrix if we apply it naively The effective-rangeexpansion should thus be viewed as an asymptotic seriesHowever we cannot use the effective-range expansion toestimate the validity of the second-order effective potentialsince the radius of convergence goes to zero at zero crossingas discussed above

The two-channel model in (1) compares well with acoupled-channel calculation [33] as shown in Figure 1 Italso compares well to other scattering models [38 39] thatinclude finite-range effects In fact the model used herecompares well with the analytical models of [38] when a(B)and re(B) have the field-dependence introduced above Thiscan be seen for instance in Figure 12 of [38] although adifference is that our a(B) and re(B) are parametrizationand not taken from coupled-channels values as in [38] (ourFigure 1 quantifies the difference which is largest on re(B))However here we are concerned with the behavior whena(B) rarr 0 in the context of Feshbach resonances whichis not addressed in [38 39] We note that the resonancemodels of [31] and the two-channel and resonance modelsin [32] are very similar to the model employed here butagain those references do not consider the specific problemsarising when a(B) rarr 0 In addition and in contrast toprevious discussions here we construct appropriate zero-range pseudo potentials that work around zero crossing

3 Relation to Experiments

Above we only retained terms of order q2 in the fullT-matrixWe now estimate the energy regime in which this expressionis valid Demanding that the q4 term be smaller than the q2

term gives the criterion

m∣∣∣abgre0∣∣∣ (14)

We relate this condition to recent experiments with bosoniccondensates of 39K working around zero crossing [3] Theresonance used there is very broad (ΔB = minus52 G) withabg = minus29a0 and re0 = minus58a0 (a0 is the Bohr radius)The right-hand side of (14) is 23 middot 10minus7 eV correspondingto a temperature of about 3 mK Since the experiments areperformed at much lower temperatures the approximationabove is certainly valid However as abg and particularlyre0 are small the front factor in (11) is also small The

relevant scale of comparison is the outer trap parameter b[9] which is typically of order 1μm yielding a vanishingratio |a2

bgre0|b3 sim 10minus9 For broad Feshbach resonances thehigher order interactions can thus be safely ignored For verynarrow resonances the situation potentially changes as re0can be very large and make the potential in (11) importantAs an example we consider the narrow resonance in 39K atB0 = 2585 G with ΔB = 047 G abg = minus33a0 and re0 =minus5687a0 [5] The right-hand side of (14) is now 2 middot 10minus9 eVcorresponding to 24 μK This is again much higher thanexperimental temperatures A more careful argument canbe made from the energy per particle of the noncondensedcloud Ignoring the trap we have EN = 0770kBTc(TTc)

(Tc is the critical temperature) [35] For a sample of 3 middot 104

a critical temperature of 100 nK was reported in [6] Usingthis Tc we find that T 900 nK for (14) to holdAgain this is within the experimental regime The effectivepotential approach should therefore be applicable aroundzero crossing for narrow resonances However even with thisnarrow resonance we find |a2

bgre0|b3 sim 10minus7 and the effectis still completely negligible

In order to increase the relevance of the higher orderterm we now consider some very narrow resonances thathave been found in 87Rb In particular the resonance at B0 =913 G [40] which was recently utilized in nonlinear atominterferometry [41] We have ΔB = 0015 G abg = 998a0and Δμ = 200μB [42] which gives re0 = minus198 middot 103a0

and a ratio |a2bgre0|b3 = 292 middot 10minus5(1μmb)3 A trap length

of b sim 05μm as used in [41] would thus yield 10minus4 anddemonstrates that higher order corrections can safely beneglected For a ratio of 1 we need b sim 003μm whichis unrealistically small in current traps or optical latticesHowever a resonance of width ΔB = 00004 G is known inthe same system at B0 = 4062 G [43] with abg = 100a0 andΔμ = 201μB [42] In this case we find re0 = minus74 middot 105a0 anda much more favorable ratio of |a2

bgre0|b3 = 0001(1μmb)3Here we see that a ratio of 1 is achieved already for b sim01μm which not far off from tight traps or optical latticedimensions In terms of temperature we still have to be inthe ultralow regime of T 30 nK according to (14) for thelatter resonance

Consider now a fermionic two-component system wheres-wave interactions are dominant Since we have re0 lt 0 forall Feshbach resonances [42] the effective potential in (10)is attractive and the system could potentially be unstabletoward a paired state or become unstable to collapse abovea critical particle number For simplicity we will use thesemiclassical Thomas-Fermi approach to describe a gas withequal population of the two components and estimate thecritical particle number Assuming an isotropic trappingpotential with length scale b = radicmω where ω is the trapfrequency the ground-state density ρ(x) can be found byminimization and satisfies[

ωminus 1

)2]= 1

2(kF(x)b)2 minus 4

30πα(kF(x)b)5 (15)

where ρ(x) = kF(x)6π2 and α = a2bg|re0|b3 The maximum

allowed momentum and chemical potential μ is found by

solving for the turning point of the right-hand side of (15)which gives

kmaxb =[

3π2α

μmax = 310

ω(kmaxb)2 (16)

We can now compare this kmax to the value obtainedfrom the noninteracting density within the Thomas-Fermiapproximation at the center of the trap In terms of thenumber of particles in each component N at the center ofthe trap we have kF(0)b asymp 1906N16 [35] By equating thesetwo expression we obtain an estimate for the critical numberof particles Nmax Inserting the relevant units we have

Nmax = 2 middot 1025

where a0 is the Bohr radius We note that the scaling Nmax propαminus2 can also be obtained by considering the point at whichthe monopole mode becomes unstable

Typical numbers for common fermionic species 6Li or40K in the lowest hyperfine states [42] lead toNmax sim 1012 forb = 1μm This is of course a huge number and experimentsare well within this limit Even if one reduced the traplength by a factor of ten and made the presumably unrealisticassumption that the particle number remains the same westill have N Nmax The reason is that the s-wave Feshbachresonances utilized in the two-component gases are generallybroad in order to study the universal regime If we considerthe narrow resonance at B0 = 54325 G in 6Li [44] withΔB = 01 G abg = 60a0 and Δμ = 200μB [42] we haveNmax sim 2 middot 1013(b1μm)6 This is somewhat better but westill need b sim 006μm to get to an experimentally relevantNmax sim 106 We have to conclude that higher order s-waveinteractions are highly unlikely to be observable throughmonopole instabilities In light of this it seems better toconsider p-wave resonances which are much more narrow ingeneral However also here extremely small trap sizes appearnecessary [45]

The instability toward Cooper pairing around zerocrossing can also be estimated in simple terms In general thecritical temperature is Tc sim TF exp(minus1N0|U|) where N0 =mkF(0)2π22 is the density of states at the Fermi energy inthe trap center and U lt 0 is a measure of the attraction Forthe latter we use the effective potential in momentum spacefrom (10) and make the assumption that q sim kF(0) Usingthe expression for kF(0) in terms of N above we find

1N0|U| =

15 middot 1012radicN

|re0| (18)

For broad resonances in 6Li or 40K this exponent is of order103 and Tc is thus vanishingly small However the scalingwith trap size can help and if we imagine reducing to b =01μm we find Tc 05TF for N = 106 atoms For thenarrow resonance in 6Li discussed above we find that Tc sim05TF with N = 106 can be achieved for b sim 05μm andTc sim 01TF for N = 105 Thus there may be a possibility toreach the pairing instability near zero crossing if high particle

numbers can be cooled in tight traps and narrow resonancesare used

While the suboptical wavelength trapping sizes neededfor the above effects to be large are not achievable withtypical optical or magnetic traps or optical lattice setupsthey could potentially be reached via hybrid setups whereatoms are trapped near a surface Inspired by surfaceplasmon subwavelength optics [46] nanoscale trapping forneutral atoms has been studied [47 48] and micropotentialtraps with width less than 100 nanometer (lt01 μm) arewithin reach [49] In these very tightly confined systemsit is very likely that finite-range effects could be enhancedDevices that provide an interface between atoms and solid-state systems are under intense study at the moment andour considerations here imply that finite-range correctionsshould be considered when the scattering length is tunedclose to zero

31 Dipole-Dipole Interactions The discussion above ignoresthe dipole-dipole interaction discussed in the introductionwhich will compete against the higher order effective poten-tial from the Feshbach resonance A simple estimate can bemade along the lines of the discussion in [35] The externaltrapping potential is the characteristic scale of spatialvariations and we thus find a ratio r of magnetic dipole-dipole Umd to higher-order s-wave zero-range interactionstrength U2 which can be written as

r = Umd

U2= a0b2

a2bg|re0|

]2[100a0

]21000a0

For r lt 1 the higher order interaction term will there-fore dominate the magnetic dipole term For the case ofnarrow resonances in 87Rb discussed above we find r sim011(b1μm)2 for the resonance at B0 = 913 G and r sim005(b1μm)2 for the one at B0 = 4062 G For the narrowresonance in 6Li at B0 = 54325 G we find r sim 14(b1μm)2These ratios clearly indicate that magnetic dipole-dipoleinteractions can be suppressed relative to higher order zero-range terms for narrow Feshbach resonances and standardtrap sizes This dominance becomes even stronger forthe tight traps needed for the realization of the effectsdiscussed above and we thus conclude that interference ofthe magnetic dipole-dipole term is not a major concern

4 Conclusions and Outlook

In this paper we have discussed the effective potential arounda Feshbach resonances as the scattering length is tuned tozero and finite-range corrections become important Weshowed that the effective-range expansion is badly behavedand the effective potential most be defined from the T-matrix We have demonstrated that the low momentaeffective potential obtained from the full T-matrix agreeswith one obtained naively from the effective-range expansionwhen the scattering length goes to zero Thus even though theeffective-range expansion has divergent coefficients at zero-crossing the first terms of the associated effective potential

yield consistent results We then estimated the effects of theterms on different condensates Since the effective potentialat zero crossing is attractive it may induce various instabilitieswhich we considered for the case of a two-component Fermigas under harmonic confinement

For the broad Feshbach resonances used in current exper-iments the effective potential discussed here are negligibleand the dipole-dipole interaction dominates completely atzero crossing However for narrow resonances in very tightlyconfined systems some of the effects might be detectable Inparticular future generations of microtraps with subopticalwavelength trap sizes using surface plasmons could besmall enough to make finite-range effects important Thecompeting dipole interaction is small for narrow resonancesin tight confinement However it is conceivable that effects ofspherically symmetric higher order terms could be separatedfrom dipolar effects which change with system geometry [4]

Small trapped Fermi systems have recently become anexperimental reality with particle numbers ranging fromtwo to ten [50] For two atomic fermions with differentinternal states the system turns out to be well describedby the analytic zero-range model of Busch et al [51ndash56]and similarly for three fermions [57 58] Effective-rangecorrections to these results have also been studied [59ndash62]Mesoscopic Fermi systems (less than about 50 particles)have been studied in harmonic traps using a number ofnumerical methods [63ndash79] with particular emphasis on theunitary regime where the scattering length diverges It wouldbe interesting to investigate the situation also around zero-crossing of a narrow resonance where the effective range issizable A preliminary study along this line for three bosonsis discussed in [80]

Another interesting direction of future work is the studyof the contact introduced by Tan [81ndash90] to describe theuniversal behavior of strongly interacting quantum gases at abroad resonance where the range corrections are negligiblefor instance through the tail of the momentum distributionwhich is predicted to behave as Ck4 where C is the contactand k the momentum of a single particle The relationsfound by Tan [81 82] have subsequently been confirmedexperimentally in three dimensions [91ndash93] While thecontact originally pertains to two-body correlations sig-natures of three-body physics in momentum distributionshave also been studied both theoretically [86 94ndash98] andexperimentally [99] While a few studies have consideredthe universal behavior when including the effective rangeterm [100 101] it would be very interesting to consider theregime around zero crossing for a narrow resonance wherethe background effective range parameter

Acknowledgments

The author would like to thank Martin Thoslashgersen forvery fruitful collaborations Correspondence with GeorgBruun about two-channel models is highly appreciatedThe author is grateful to Nicolai Nygaard for discussionsand for producing Figure 1 The author acknowledges thehospitality of the Niels Bohr Institute Blegdamsvej 17 2100

Copenhagen Oslash Denmark This work was supported by theVillum Kann Rasmussen foundation

References

[1] I Bloch J Dalibard and W Zwerger ldquoMany-body physicswith ultracold gasesrdquo Reviews of Modern Physics vol 80 no3 pp 885ndash964 2008

[2] C Chin R Grimm P Julienne and E Tiesinga ldquoFeshbachresonances in ultracold gasesrdquo Reviews of Modern Physics vol82 no 2 pp 1225ndash1286 2010

[3] M Fattori C DrsquoErrico G Roati et al ldquoAtom interferometrywith a weakly interacting bose-Einstein condensaterdquo PhysicalReview Letters vol 100 no 8 Article ID 080405 4 pages2008

[4] M Fattori G Roati B Deissler et al ldquoMagnetic dipolarinteraction in a Bose-Einstein condensate atomic interfer-ometerrdquo Physical Review Letters vol 101 no 19 Article ID190405 4 pages 2008

[5] C DrsquoErrico M Zaccanti M Fattori et al ldquoFeshbachresonances in ultracold 39Krdquo New Journal of Physics vol 9article 223 2007

[6] G Roati M Zaccanti C DrsquoErrico et al ldquo39K bose-Einsteincondensate with tunable interactionsrdquo Physical Review Let-ters vol 99 no 1 Article ID 010403 4 pages 2007

[7] S E Pollack D Dries M Junker Y P Chen T A Corcovilosand R G Hulet ldquoExtreme tunability of interactions in a 7LiBose-Einstein condensaterdquo Physical Review Letters vol 102no 9 Article ID 090402 4 pages 2009

[8] H Fu Y Wang and B Gao ldquoBeyond the Fermi pseudopoten-tial a modified Gross-Pitaevskii equationrdquo Physical Reviewvol 67 no 5 Article ID 053612 6 pages 2003

[9] N T Zinner and M Thoslashgersen ldquoStability of a Bose-Einsteincondensate with higher-order interactions near a Feshbachresonancerdquo Physical Review vol 80 no 2 Article ID 0236074 pages 2009

[10] M Thoslashgersen N T Zinner and A S Jensen ldquoThomas-Fermi approximation for a condensate with higher-orderinteractionsrdquo Physical Review A vol 80 no 4 Article ID043625 8 pages 2009

[11] E Stoner ldquoLXXX Atomic moments in ferromagnetic metalsand alloys with non-ferromagnetic elementsrdquo PhilosophicalMagazine vol 15 no 101 pp 1018ndash1034 1933

[12] M Houbiers R Ferwerda H T C Stoof W I McAlexanderC A Sackett and R G Hulet ldquoSuperfluid stateof atomic6Li in a magnetic traprdquo Physical Review A vol 56 no 6 pp4864ndash4878 1997

[13] Y Zhang and S Das Sarma ldquoExchange instabilities in elec-tron systems bloch versus Stoner ferromagnetismrdquo PhysicalReview B vol 72 no 11 Article ID 115317 9 pages 2005

[14] R A Duine and A H MacDonald ldquoItinerant ferromag-netism in an ultracold atom Fermi gasrdquo Physical ReviewLetters vol 95 no 23 Article ID 230403 4 pages 2005

[15] G J Conduit and B D Simons ldquoItinerant ferromagnetismin an atomic Fermi gas influence of population imbalancerdquoPhysical Review A vol 79 no 5 Article ID 053606 9 pages2009

[16] J Conduit A G Green and B D Simons ldquoInhomogeneousphase formation on the border of itinerant ferromagnetismrdquoPhysical Review Letters vol 103 no 20 Article ID 207201 4pages 2009

[17] G J Conduit and B D Simons ldquoRepulsive atomic gas in aharmonic trap on the border of itinerant ferromagnetismrdquo

Physical Review Letters vol 103 no 20 Article ID 200403 4pages 2009

[18] L J Leblanc J H Thywissen A A Burkov and AParamekanti ldquoRepulsive Fermi gas in a harmonic trapferromagnetism and spin texturesrdquo Physical Review A vol80 no 1 Article ID 013607 2009

[19] S Zhang H-H Hung and C Wu ldquoProposed realization ofitinerant ferromagnetism in optical latticesrdquo Physical Reviewvol 82 no 5 Article ID 053618 5 pages 2010

[20] G B Jo Y R Lee J H Choi et al ldquoItinerant ferromagnetismin a fermi gas of ultracold atomsrdquo Science vol 325 no 5947pp 1521ndash1524 2009

[21] H Zhai ldquoCorrelated versus ferromagnetic state in repulsivelyinteracting two-component Fermi gasesrdquo Physical Review Avol 80 no 5 Article ID 051605 4 pages 2009

[22] X Cui and H Zhai ldquoStability of a fully magnetizedferromagnetic state in repulsively interacting ultracold Fermigasesrdquo Physical Review A vol 81 no 4 Article ID 041602 4pages 2010

[23] D Pekker M Babadi R Sensarma et al ldquoCompetitionbetween pairing and ferromagnetic instabilities in ultracoldFermi gases near Feshbach resonancesrdquo Physical ReviewLetters vol 106 no 5 Article ID 050402 4 pages 2011

[24] V B Shenoy and T-L Ho ldquoNature and properties of arepulsive Fermi gas in the upper branch of the energyspectrumrdquo Physical Review Letters vol 107 no 21 ArticleID 210401 5 pages 2011

[25] C Sanner E J Su W Huang A Keshet J Gillen and WKetterle ldquoCorrelations and pair formation in a repulsivelyinteracting Fermi gasrdquo Physical Review Letters vol 108 no24 Article ID 240404 5 pages 2012

[26] C Kohstall M Zaccanti M Jag et al ldquoMetastability andcoherence of repulsive polarons in a strongly interactingFermi mixturerdquo Nature vol 485 pp 615ndash618 2012

[27] T H R Skyrme ldquoCVII The nuclear surfacerdquo PhilosophicalMagazine vol 1 no 11 pp 1043ndash1054 1956

[28] T H R Skyrme ldquoThe effective nuclear potentialrdquo NuclearPhysics vol 9 no 4 pp 615ndash634 1959

[29] D R Phillips S R Beane and T D Cohen ldquoNonpertur-bative regularization and renormalization simple examplesfrom nonrelativistic quantum mechanicsrdquo Annals of Physicsvol 263 no 2 pp 255ndash275 1998

[30] G M Bruun A D Jackson and E E Kolomeitsev ldquoMulti-channel scattering and Feshbach resonances effective theoryphenomenology and many-body effectsrdquo Physical Review Avol 71 no 5 Article ID 052713 10 pages 2005

[31] S J J M F Kokkelmans J N Milstein M L ChiofaloR Walser and M J Holland ldquoResonance superfluidityrsenormalization of resonance scattering theoryrdquo PhysicalReview A vol 65 no 5 Article ID 536171 4 pages 2002

[32] E Braaten M Kusunoki and D Zhang ldquoScattering modelsfor ultracold atomsrdquo Annals of Physics vol 323 no 7 pp1770ndash1815 2008

[33] N Nygaard B I Schneider and P S Julienne ldquoTwo-channelR-matrix analysis of magnetic-field-induced Feshbach reso-nancesrdquo Physical Review A vol 73 no 4 Article ID 04270510 pages 2006

[34] P Massignan and Y Castin ldquoThree-dimensional stronglocalization of matter waves by scattering from atoms ina lattice with a confinement-induced resonancerdquo PhysicalReview A vol 74 no 1 Article ID 013616 2006

[35] C J Pethick and H Smith Bose-Einstein Condensation inDilute Gases Cambridge University Press Cambridge MassUSA 2002

[36] R Roth and H Feldmeier ldquoEffective s- and p-wave contactinteractions in trapped degenerate Fermi gasesrdquo PhysicalReview A vol 64 no 4 Article ID 043603 17 pages 2001

[37] A Collin P Massignan and C J Pethick ldquoEnergy-dependent effective interactions for dilute many-body sys-temsrdquo Physical Review A vol 75 no 1 Article ID 0136152007

[38] B Marcelis E G M van Kempen B J Verhaar and S J J MF Kokkelmans ldquoFeshbach resonances with large backgroundscattering length interplay with open-channel resonancesrdquoPhysical Review A vol 70 no 1 Article ID 012701 15 pages2004

[39] B Marcelis and S Kokkelmans ldquoFermionic superfluiditywith positive scattering lengthrdquo Physical Review A vol 74no 2 Article ID 023606 2006

[40] A Widera O Mandel M Greiner S Kreim T W Hanschand I Bloch ldquoEntanglement interferometry for precisionmeasurement of atomic scattering propertiesrdquo PhysicalReview Letters vol 92 no 16 Article ID 160406 2004

[41] C Gross T Zibold E Nicklas J Esteve and M KOberthaler ldquoNonlinear atom interferometer surpasses clas-sical precision limitrdquo Nature vol 464 no 7292 pp 1165ndash1169 2010

[43] A Marte T Volz J Schuster et al ldquoFeshbach resonances inrubidium 87 precision measurement and analysisrdquo PhysicalReview Letters vol 89 no 28 Article ID 283202 4 pages2002

[44] K E Strecker G B Partridge and R G Hulet ldquoConversionof an atomic Fermi gas to a long-lived molecular bose gasrdquoPhysical Review Letters vol 91 no 8 Article ID 080406 4pages 2003

[45] N T Zinner ldquoStability of a fully polarized ultracold Fermigas near zero-crossing of a p-wave Feshbach resonancerdquoEuropean Physical Journal D vol 57 no 2 pp 235ndash240 2010

[46] W L Barnes A Dereux and T W Ebbesen ldquoSurfaceplasmon subwavelength opticsrdquo Nature vol 424 no 6950pp 824ndash830 2003

[47] B Murphy and L V Hau ldquoElectro-optical nanotraps forneutral atomsrdquo Physical Review Letters vol 102 no 3 ArticleID 033003 4 pages 2009

[48] D E Chang J D Thompson H Park et al ldquoTrapping andmanipulation of isolated atoms using nanoscale plasmonicstructuresrdquo Physical Review Letters vol 103 no 12 ArticleID 123004 4 pages 2009

[49] C Stehle H Bender C Zimmermann D Kern M Fleischerand S Slama ldquoPlasmonically tailored micropotentials forultracold atomsrdquo Nature Photonics vol 5 no 8 pp 494ndash4982011

[50] F Serwane G Zurn T Lompe T B Ottenstein A N Wenzand S Jochim ldquoDeterministic preparation of a tunable few-fermion systemrdquo Science vol 332 no 6027 pp 336ndash3382011

[51] T Busch B G Englert K Rzazewski and M Wilkens ldquoTwocold atoms in a harmonic traprdquo Foundations of Physics vol28 no 4 pp 549ndash559 1998

[52] T Stoferle H Moritz K Gunter M Kohl and T EsslingerldquoMolecules of fermionic atoms in an optical latticerdquo PhysicalReview Letters vol 96 no 3 Article ID 030401 4 pages 2006

[53] T Volz N Syassen D M Bauer E Hansis S Durr and GRempe ldquoPreparation of a quantum state with one molecule

at each site of an optical latticerdquo Nature Physics vol 2 no 10pp 692ndash695 2006

[54] G Thalhammer K Winkler F Lang S Schmid R Grimmand J Hecker Denschlag ldquoLong-lived Feshbach molecules ina three-dimensional optical latticerdquo Physical Review Lettersvol 96 no 5 Article ID 050402 4 pages 2006

[55] C Ospelkaus S Ospelkaus L Humbert P Ernst KSengstock and K Bongs ldquoUltracold heteronuclear moleculesin a 3D optical latticerdquo Physical Review Letters vol 97 no 12Article ID 120402 4 pages 2006

[56] D Blume and C H Greene ldquoFermi pseudopotential approx-imation two particles under external confinementrdquo PhysicalReview A vol 65 no 4 Article ID 043613 6 pages 2002

[57] F Werner and Y Castin ldquoUnitary quantum three-bodyproblem in a Harmonic traprdquo Physical Review Letters vol97 Article ID 150401 2006

[58] F Werner and Y Castin ldquoUnitary gas in an isotropic har-monic trap symmetry properties and applicationsrdquo PhysicalReview A vol 74 no 5 Article ID 053604 2006

[59] Z Idziaszek and T Calarco ldquoAnalytical solutions for thedynamics of two trapped interacting ultracold atomsrdquo Physi-cal Review A vol 74 no 2 Article ID 022712 2006

[60] A Suzuki Y Liang and R K Bhaduri ldquoTwo-atom energyspectrum in a harmonic trap near a Feshbach resonance athigher partial wavesrdquo Physical Review A vol 80 no 3 ArticleID 033601 6 pages 2009

[61] S G Peng S Q Li P D Drummond and X J Liu ldquoHigh-temperature thermodynamics of strongly interacting s-waveand p-wave Fermi gases in a harmonic traprdquo Physical ReviewA vol 83 no 6 Article ID 063618 10 pages 2011

[62] N T Zinner ldquoUniversal two-body spectra of ultracoldharmonically trapped atoms in two and three dimensionsrdquoJournal of Physics A vol 45 no 20 Article ID 205302 2012

[63] J L DuBois and H R Glyde ldquoBose-Einstein condensation intrapped bosons a variational Monte Carlo analysisrdquo PhysicalReview A vol 63 no 2 Article ID 023602 2001

[64] J Carlson S Y Chang V R Pandharipande and KE Schmidt ldquoSuperfluid Fermi gases with large scatteringlengthrdquo Physical Review Letters vol 91 no 5 Article ID050401 4 pages 2003

[65] S Y Chang and G F Bertsch ldquoUnitary Fermi gas in aharmonic traprdquo Physical Review A vol 76 no 2 Article ID021603 2007

[66] D Blume J Von Stecher and C H Greene ldquoUniversal prop-erties of a trapped two-component fermi gas at unitarityrdquoPhysical Review Letters vol 99 no 23 Article ID 2332012007

[67] M Thoslashgersen D V Fedorov and A S Jensen ldquoTrappedBose gases with large positive scattering lengthrdquo vol 79 no4 Article ID 40002 6 pages 2007

[68] J von Stecher C H Greene and D Blume ldquoEnergeticsand structural properties of trapped two-component Fermigasesrdquo Physical Review A vol 77 no 4 Article ID 043619 20pages 2008

[69] D Lee ldquoLattice simulations for few- and many-body sys-temsrdquo Progress in Particle and Nuclear Physics vol 63 no 1pp 117ndash154 2009

[70] W C Haxton and T Luu ldquoPerturbative effective theory inan oscillator basisrdquo Physical Review Letters vol 89 no 18Article ID 182503 4 pages 2002

[71] I Stetcu B R Barrett U van Kolck and J P Vary ldquoEffectivetheory for trapped few-fermion systemsrdquo Physical Review Avol 76 no 6 Article ID 063613 7 pages 2007

[72] Y Alhassid G F Bertsch and L Fang ldquoNew effectiveinteraction for the trapped fermi gasrdquo Physical Review Lettersvol 100 no 23 Article ID 230401 2008

[73] N T Zinner K M Moslashlmer C Ozen D J Dean andK Langanke ldquoShell-model Monte Carlo simulations of theBCS-BEC crossover in few-fermion systemsrdquo Physical ReviewA vol 80 no 1 Article ID 013613 5 pages 2009

[74] I Stetcu J Rotureau B R Barrett and U van Kolck ldquoAneffective field theory approach to two trapped particlesrdquoAnnals of Physics vol 325 no 8 pp 1644ndash1666 2010

[75] T Luu M J Savage A Schwenk and J P Vary ldquoNucleon-nucleon scattering in a harmonic potentialrdquo Physical ReviewC vol 82 no 3 Article ID 034003 2010

[76] J Rotureau I Stetcu B R Barrett M C Birse and U VanKolck ldquoThree and four harmonically trapped particles in aneffective-field-theory frameworkrdquo Physical Review A vol 82no 3 Article ID 032711 2010

[77] J R Armstrong N T Zinner D V Fedorov and A S JensenldquoAnalytic harmonic approach to the N-body problemrdquoJournal of Physics B vol 44 no 5 Article ID 055303 2011

[78] J R Armstrong N T Zinner D V Fedorov and A S JensenldquoQuantum statistics and thermodynamics in the harmonicapproximationrdquo Physical Review E vol 85 no 2 Article ID021117 10 pages 2012

[79] J R Armstrong N T Zinner D V Fedorov and A S JensenldquoVirial expansion coefficients in the harmonicapproxima-tionrdquo httparxivorgabs12052574

[80] N T Zinner ldquoEfimov trimers near the zero-crossing of aFeshbach resonancerdquo httparxivorgabs11126358

[81] S Tan ldquoEnergetics of a strongly correlated Fermi gasrdquo Annalsof Physics vol 323 no 12 pp 2952ndash2970 2008

[82] S Tan ldquoLarge momentum part of a strongly correlated Fermigasrdquo Annals of Physics vol 323 no 12 pp 2971ndash2986 2008

[83] E Braaten and L Platter ldquoExact relations for a stronglyinteracting fermi gas from the operator product expansionrdquoPhysical Review Letters vol 100 no 20 Article ID 2053012008

[84] S Zhang and A J Leggett ldquoUniversal properties of theultracold Fermi gasrdquo Physical Review A vol 79 no 2 ArticleID 023601 2009

[85] R Combescot F Alzetto and X Leyronas ldquoParticle distribu-tion tail and related energy formulardquo Physical Review A vol79 no 5 Article ID 053640 2009

[86] F Werner and Y Castin ldquoExact relations for quantum-mechanical few-body and many-body problems withshort-range interactions in two and three dimensionsrdquohttparxivorgabs10010774

[87] M Barth and W Zwerger ldquoTan relations in one dimensionrdquoAnnals of Physics vol 326 no 10 pp 2544ndash2565 2011

[88] M Valiente N T Zinner and K M Moslashlmer ldquoUniversalrelations for the two-dimensional spin-12 Fermi gas withcontact interactionsrdquo Physical Review A vol 84 no 6 ArticleID 063626 4 pages 2011

[89] M Valiente ldquoTanrsquos distributions and Fermi-Huang pseu-dopotential in momentum spacerdquo Physical Review A vol 85no 1 Article ID 014701 4 pages 2012

[90] M Valiente N T Zinner and K M Moslashlmer ldquoUniver-sal properties of Fermi gases in arbitrary dimensionsrdquohttparxivorgabs12056388

[91] J T Stewart J P Gaebler T E Drake and D S JinldquoRification of universal relations in a strongly interactingfermi gasrdquo Physical Review Letters vol 104 no 23 ArticleID 235301 2010

[92] E D Kuhnle H Hu X J Liu et al ldquoUniversal behavior ofpair correlations in a strongly interacting fermi gasrdquo PhysicalReview Letters vol 105 no 7 Article ID 070402 2010

[93] E D Kuhnle S Hoinka P Dyke H Hu P Hannaford andC J Vale ldquoTemperature dependence of the universal contactparameter in a unitary Fermi gasrdquo Physical Review Lettersvol 106 no 17 Article ID 170402 2011

[94] E Braaten D Kang and L Platter ldquoUniversal relations foridentical bosons from three-body physicsrdquo Physical ReviewLetters vol 106 no 15 Article ID 153005 2011

[95] Y Castin and F Werner ldquoSingle-particle momentum distri-bution of an Efimov trimerrdquo Physical Review A vol 83 no 6Article ID 063614 2011

[96] K Helfrich and H W Hammer ldquoResonant three-bodyphysics in two spatial dimensionsrdquo Physical Review A vol 83no 5 Article ID 052703 7 pages 2011

[97] F F Bellotti T Frederico M T Yamashita D V FedorovA S Jensen and N T Zinner ldquoScaling and universality intwo dimensions three-body bound states with short-rangedinteractionsrdquo Journal of Physics B vol 44 no 20 Article ID205302 2011

[98] F F Bellotti T Frederico M T Yamashita D V FedorovA S Jensen and N T Zinner ldquoSupercircle descriptionof universal three-body states in two dimensionsrdquo PhysicalReview A vol 85 no 2 Article ID 025601 5 pages 2012

[99] R J Wild P Makotyn J M Pino E A Cornell and D S JinldquoMeasurements of Tanrsquos contact in an atomic bose-Einsteincondensaterdquo Physical Review Letters vol 108 no 14 ArticleID 145305 5 pages 2012

[100] E Braaten D Kang and L Platter ldquoUniversal relations fora strongly interacting Fermi gas near a Feshbach resonancerdquoPhysical Review A vol 78 no 5 Article ID 053606 2008

[101] F Werner ldquoVirial theorems for trapped cold atomsrdquo PhysicalReview A vol 78 no 2 Article ID 025601 4 pages 2008

Transition Parameters for Doubly Ionized Lanthanum

Betul Karacoban and Leyla Ozdemir

Department of Physics Sakarya University 54187 Sakarya Turkey

Correspondence should be addressed to Betul Karacoban bkaracobansakaryaedutr

The transition parameters such as the wavelengths weighted oscillator strengths and transition probabilities (or rates) for the nd(n = 5minus9)minusnf (n = 4minus8) nd (n = 5minus9)minusnp (n = 6minus9) np (n = 6minus9)minusns (n = 6minus10) and ng (n = 5minus8)minusnf (n = 4minus8) electricdipole (E1) transitions of doubly ionized lanthanum (La IIIZ = 57) have been calculated using the relativistic Hartree-Fock (HFR)method In this method configuration interaction and relativistic effects have been included in the computations combined witha least squares fitting of the Hamiltonian eigenvalues to the observed energy levels We have compared the results obtained fromthis work with the previously available calculations and experiments in literature We have also reported new transitions with theweighted transition probabilities greater than or equal to 105

1 Introduction

The radiative properties of the lanthanides and their ionshave been rather little considered This can be explained bythe fact that these atoms or ions are characterized by com-plex electronic structures with an unfilled 4f subshell whichmakes the calculations very difficult and that the laboratoryanalyses are still extremely fragmentary or even missing formany ions Owing to the importance of rare earth elementsin astrophysics especially in relation to nucleosynthesisand star formation (notably the lanthanides in chemicallypeculiar (CP) stars) [7] there is a growing need for accuratespectroscopic data that is wavelengths radiative transitionrates oscillator strengths branching fractions radiative life-times hyperfine structure and isotope shift data for lan-thanide atoms and ions

The lanthanum atom is the first member of the rare earthelements Doubly ionized lanthanum (La III) is characterizedby a simple atomic structure with core [Xe] and only oneouter electron There is substantial spectroscopic literatureconcerning La III though less than the neutral or singlyionized species The available theoretical and experimentalworks on energy levels radiative lifetimes and transitionparameters for La III can be found in the literature [1ndash3 56 8ndash13] These works were reported in our previous workin detail [14]

Up till now the wavelengths oscillator strengths andtransition probabilities available for La III were obtained byexperimental semiempirical or pure theoretical approachesSixty-five spectral lines of La III in the 2000ndash12000 A intervalwere reported by Odabasi [2] Sugar and Kaufman [13]observed forty-five La III spectral lines in the interval from700 to 2000 A Johansson and Litzen [5] recorded wave-lengths of 5dndash4f lines of La III Relativistic single-configu-ration Hartree-Fock oscillator strengths for 6sndash6p transitionsin La III were reported by Migdalek and Baylis [4] Migdalekand Wyrozumska [3] have calculated oscillator strengthsobtained using the relativistic model-potential approach inthere different versions a model-potential without valence-core electron exchange but with core-polarization included(RMP + CP) with semiclassical exchange and core-polari-zation (RMP + SCE + CP) and with empirically adjustedexchange and core-polarization (RMP + EX + CP) for the 6sndash6p 5dndash6p 5dndash4f 5dndash5f 5dndash6f 6pndash6d and 6pndash7d transitionarrays The single-configuration relativistic Hartree-Fockionization potentials of La III were computed by Migdalekand Bojara [9] Biemont et al [1] have performed oscillatorstrengths and transition probabilities in La III by relativisticHartree-Fock method with core-polarization

Our aim here is to determine the transition parameterssuch as the wavelengths oscillator strengths and transition

Table 1 Wavelengths λ(A) weighted oscillator strengths gf and weighted transition probabilities gAki (sminus1) for electric dipole (E1) transi-tions in La III

Transition λ g f gAki

Lower level Upper level This work Other works This work Other works This work Other works

6p 2Po12 7s 2S12 247941AB 247866a 0489A 0475a 531 times 108 A 516 times 108 a

2478652c 0463B 503 times 108 B

2684757c 0855B 791 times 108 B

6p 2Po12 6d 2D32 247736A 247660a 2474A 2365a 269 times 109 A 257 times 109 a

247735B 2476599c 2651B 2142d 288 times 109 B 227 times 109 b

0064B 200 times 108 B

0123B 348 times 108 B

6p 2Po12 7d 2D32 145945AB 145945a 0158A 0156a 496 times 108 A 488 times 108 a

0216B 0137d 676 times 108 B 426 times 108 b

0372B 0211d 1070 times 108 B 602 times 108 b

0041B 0022d 1180 times 107 B 623 times 107 b

0023B 106 times 108 B

0045B 189 times 108 B

0065B 295 times 108 B

0112B 476 times 108 B

0012B 527 times 107 B

0012B 639 times 107 B

0022B 115 times 108 B

0029B 1610 times 107 B

0050B 261 times 108 B

0006B 289 times 107 B

6d 2D32 5f 2Fo52 992670A 992404a 2549A 2370a 173 times 108 A 160 times 108 a

992674B 9923989c 2574B 174 times 108 B

6d 2D52 5f 2Fo72 1028759AB 10284790c 3515A mdash 221 times 108 A mdash

3548B 224 times 108 B

6d 2D52 5f 2Fo52 1037315A 10370335c 0174A mdash 108 times 107 A mdash

1037312B 0176B 109 times 107 B

6d 2D32 7p 2Po32 827767AB 827541a 0243A 0240a 237 times 107 A 234 times 107 a

8275388c 0250B 243 times 107 B

6d 2D52 7p 2Po32 858581A 858342a 2115A 2081a 191 times 108 A 188 times 108 a

858576B 8583453c 2165B 196 times 108 B

107Transition Parameters for Doubly Ionized Lanthanum

Table 1 Continued

6d 2D32 7p 2P o12 921520A 921268a 1094A 1077a 859 times 107 A 846 times 107 a

921523B 9212628c 1121B 880 times 107 B

307606B 3075173c 0790B 557 times 108 B

6d 2D52 6f 2Fo72 311288AB 311197a 1037A 1161a 714 times 108 A 799 times 108 a

3111969c 1116B 768 times 108 B

3116738c 0056B 382 times 107 B

295464B 0008B 618 times 106 B

2992098c 0072B 535 times 107 B

301011B 3009223c 0040B 292 times 107 B

223905B 0356B 474 times 108 B

2258609c 0504B 659 times 108 B

2260295c 0025B 329 times 107 B

6d 2D32 9p 2Po32 219518A 219450a 0003AB 0002a 415 times 106 A 328 times 106 a

219519B 358 times 106B

0013B 175 times 107 B

0023B 313 times 107 B

192334B 0188B 339 times 108 B

0267B 473 times 108 B

6d 2D52 8f 2Fo52 193951AB 193951a 0013AB 0013a 224 times 107 A 232 times 107 a

236 times 107 B

6f 2Fo52 6g 2G72 829018A 828776a 8903A 8527a 864 times 108 AB 828 times 108 a

829016B 8287752c 8904B

832334B 8321107c 11496B

6f 2Fo72 6g 2G72 832398A 832163a 0328AB 0315a 316 times 107 AB 303 times 107 a

832397B

6f 2Fo52 7g 2G72 514717AB 514572a 1282A 1239a 323 times 108 AB 312 times 108 a

5145729c 1283B

6f 2Fo72 7g 2G92 515984A 515839a 1658A 1602a 415 times 108 A 401 times 108 a

515982B 5158410c 1659B 416 times 108 B

516019B

6f 2Fo52 8g 2G72 413043AB 412924a 0411AB 0394a 161 times 108 AB 154 times 108 a

413858B 4137428c 0532B

413881B

6f 2Fo52 9d 2D32 551977A 551819a 0132A 0130a 289 times 107 A 285 times 107 a

551975B 5518187c 0128B 279 times 107 B

Table 1 Continued

6f 2F52 9d 2D52 549845A 549688a 0009AB 0009a 208 times 106 A 206 times 106 a

549843B 202 times 106B

6f 2Fo72 9d 2D52 551329AB 551176a 0188A 0186a 414 times 107 A 409 times 107 a

5511721c 0182B 401 times 107 B

6s 2S12 6p 2Po32 317260AB 317169a 1673A 1527a 111 times 109 A 101 times 109 a

3171735c 1935B 1418d 128 times 109 B 940 times 108 b

6s 2S12 6p 2Po12 351816A 351716a 0754A 0689a 406 times 108 A 371 times 108 a

0001B 027 times 107 B

00003B 013 times 107 B

7s 2S12 7p 2Po32 825485A 825253a 2418A 2279a 237 times 108 AB 223 times 108 a

825477B 8252603c 2424B

918692B 9184380c 1089B 861 times 107 B

7s 2S12 8p 2Po32 295172AB 2950843c 0002AB mdash 181 times 106 A mdash

154 times 106B

7s 2S12 8p 2Po12 300707A 3006186c 0001AB mdash 857 times 105 A mdash

300708B 727 times 105 B

7p 2Po12 8s 2S12 589023A 588863a 0716A 0718a 138 times 108 A 138 times 108 a

589025B 5888620c 0714B 137 times 108 B

634997B 6348213c 1324B 219 times 108 B

577971B 5778138c 3045B 608 times 108 B

614371B 6141987c 5157B 911 times 108 B

622169B 6219999c 0566B 975 times 107 B

7p 2Po12 9s 2S12 319777A 319685a 0089AB 0092a 577 times 107 AB 598 times 107 a

319778B 3196844c

332861B 3327655c

317361B 3172689c 0284B 189 times 108 B

329007B 3289110c 0494B 305 times 108 B

330243B 3301481c 0055B 335 times 107 B

7p 2Po12 10s 2S12 252474AB 252398a 0031AB 0033a 325 times 107 A 344 times 107 a

327 times 107 B

7p 2Po32 10s 2S12 260559A 260482a 0060AB 0064a 591 times 107 A 625 times 107 a

260560B 2604827c 594 times 107 B

2513432c 0088B 927 times 107 B

2588867c 0153B 153 times 108 B

0017B 169 times 107 B

Table 1 Continued

605756B 6055838c 0578B 105 times 108 B

612100B 6119254c 0818B 146 times 108 B

613335B 0041B 724 times 106B

419472B 0283B 107 times 108 B

422629B 0402B 150 times 108 B

423092B 0020B 747 times 106B

574690B 0007B 148 times 106B

581507B 5813447c 0065B 129 times 107 B

587731B 5875632c 0036B 692 times 106B

829335B

831511B

831604B

546935B 5467812c 0035B 777 times 106B

553107B 5529542c 0482B 105 times 108 B

549343B 5491902c 0693B 153 times 108 B

448425B 4482967c 8889B

450034B 4499050c 11482B

450043B

5f 2Fo52 8d 2D52 308628A 308538a 0004AB 0005a 298 times 106 A 313 times 106 a

308629B 3085379c 280 times 106B

309716B 3096255c 0056B 388 times 107 B

309394B 3093028c 0080B 556 times 107 B

289874B 2897875c

290541B 2904576c

5f 2Fo52 9d 2D52 246170AB 246095a 0001AB 0002a 151 times 106 A 167 times 106 a

141 times 106B

5f 2F o52 9d 2D32 246597AB 246522a 0019A 0021a 211 times 107 A 233 times 107 a

0018B 196 times 107 B

Table 1 Continued

5f 2F o72 9d 2D52 246659A 246582a 0027A 0030a 301 times 107 A 333 times 107 a

253557B 0025B 280 times 107 B

238872B 2387988c

239322B 2392492c

5f 2F o72 8g 2G92 214744AB 214677a 0115AB 0096a 167 times 108 AB 139 times 108 a

5g 2G92 7f 2Fo72 811678A 811448a 0018AB 0021a 179 times 106 AB 211 times 106 a

811675B 8114415c

813819B 8135964c

5g 2G92 8f 2Fo72 509054A 508912a 0003AB 0003a 683 times 105 A 782 times 105 a

509051B 686 times 105B

509711B 527 times 105B

5d 2D32 4f 2Fo52 1389850A 1389447f 0072A 0031d 251 times 106 A mdash

1389806B 0074B 254 times 106B

1410019B 0104B 348 times 106B

5d 2D52 4f 2Fo52 1788204A 1787809f 0004AB 0002d 841 times 104 A mdash

1788369B 852 times 104 B

229844g

238010g

1640B 1604d 935 times 109 B 906 times 109 b

5d 2D52 5f 2Fo72 109973AB 109973a 2317A 1935a 128 times 1010A 107 times 1010 a

2304B 2325d 127 times 1010B 128 times 1010 b

0119d 634 times 108 B 649 times 108 b

5d 2D32 7p 2Po32 105863AB 105863a 0013AB 0010a 784 times 107 A 572 times 107 a

802 times 107 B

386 times 108 B

0119B 686 times 108 B

0480d 540 times 109 B 421 times 109 b

0696d 741 times 109 B 595 times 109 b

0035d 370 times 108 B 299 times 108 b

436 times 107 B

Table 1 Continued

215 times 108 B

377 times 108 B

0305B 328 times 109 B

5d 2D52 7f 2Fo72 79699AB 79699a 0430AB 0286a 451 times 109 AB 300 times 109 a

5d 2D52 7f 2F o52 79720AB 79720a 0021AB 0014a 226 times 108 AB 150 times 108 a

262 times 107 B

1301 times 107 B

227 times 108 B

5d 2D52 8f 2Fo72 75303AB 75303a 0249A 0155a 292 times 109 AB 183 times 109 a

017 times 106B

023 times 107 B

316 times 106B

4f 2Fo52 5g 2G72 92972AB 92972a 0058A 0040a 449 times 108 A 306 times 108 a

0060B 463 times 108 B

0077B 576 times 108 B

4f 2Fo72 5g 2G72 94287AB 94287a 0002AB 0001a 160 times 107 A 109 times 107 a

165 times 107 B

0048B 458 times 108 B

0061B 572 times 108 B

163 times 107 B

0034B 368 times 108 B

0044B 460 times 108 B

0024B 282 times 108 B

0031B 354 times 108 B

0004B 934 times 105B

8s 2S12 9p 2Po12 576724A 576563a 0002AB 0001a 476 times 105 A 261 times 105 a

576725B 437 times 105B

8p 2Po12 8d 2D32 1094083A 10937898c 3508A mdash 195 times 108 A mdash

1094086B 3439B 192 times 108 B

574566B 5744088c 0347B 701 times 107 B

Table 1 Continued

5932706c 0604B 114 times 108 B

595921B 0067B 126 times 107 B

580107B 0113B 224 times 107 B

601884B 6017114c 0218B 401 times 107 B

aReference [1] breference (in [1]) creference [2] dreference [3 RMP + EX + CP ] ereference [4 RHF + CP (a)] freference [5] greference [6]

probabilities for electric dipole transitions (E1) in La III(Z = 57) These calculations have been performed by usingcode [15] developed Cowan for relativistic Hartree-Fock(HFR) [16] calculations This code considers the correlationeffects and relativistic corrections These effects contributeimportantly to the physical and chemical properties of atomsor ions especially lanthanides The ground-state level ofdoubly ionized lanthanum is [Xe] 5d 2D32 We have takeninto account 5p6nd 5p6ng (n = 5ndash10) 5p6ns (n = 6ndash10)5p56s6p 5p56s4f 5p55d6p 5p6nf (n = 4ndash10) 5p6np (n = 6ndash10) 5p54f2 and 5p56p2 configurations outside the core [Cd]and nd ng (n = 5ndash25) ns (n = 6ndash24) nf (n = 4ndash22) andnp (n = 6ndash25) configurations outside the core [Xe] in La IIIThe configuration sets that we used have been denoted byA and B respectively and are given in tables and text Wepresented the energies the Lande g-factors and the lifetimesfor nd ng (n = 5ndash25) ns (n = 6ndash24) nf (n = 4ndash22) and np(n = 6ndash25) excited levels of La III [14] In addition we havereported various atomic structure calculations such as energylevels transition energies hyperfine structure lifetimes andelectric dipole transitions for some lanthanides (La IminusIII LuIminusIII and Yb IminusIII) [17ndash27]

2 Calculation Method

An electromagnetic transition between two states is charac-terized by the angular momentum and the parity of the cor-responding photon If the emitted or absorbed photon hasangular momentum k and parity π = (minus1)k then the tran-sition is an electric multipole transition (Ek) However if thephoton has parity π = (minus1)k+1 the transition is a magneticmultipole transition (Mk)

According to HFR method [16] the total transitionprobability from a state γprimeJ primeMprime to all states M levels of γJis given by

A = 64π4e2a20σ

3hSsumMq

(J 1 J prime

minusM q Mprime

= 64π4e2a20σ

3h(2J prime + 1)S

and absorption oscillator strength is given by

fi j = 8π2mca20σ

3h(2J + 1)S =

(Ej minus Ei

)3(2J + 1)

where σ = [(EjminusEi)hc] has units of kaysers (cmminus1) and S =|〈γJP(1)γprimeJ prime〉|2 is the electric dipole line strength in atomicunits of e2a2

0 The strongest transition rate (or probability)is electric dipole (E1) radiation For this reason the E1transitions are understood as being ldquoallowedrdquo whereas high-order transitions are understood as being ldquoforbiddenrdquo

In HFR method for anN electron atom of nuclear chargeZ0 the Hamiltonian is expanded as

H = minussumi

nabla2i minus

ri+sumi gt j

ζi(ri)li middot si (3)

in atomic units with ri the distance of the ith electron fromthe nucleus and ri j = |ri minus r j| ζi(R) = (α22)(1r)(partVpartr) isthe spin-orbit term with α being the fine structure constantand V the mean potential field due to the nucleus and otherelectrons

In this method one calculates single-configuration radialfunctions for a spherically symmetrised atom (center-of-gravity energy of the configuration) based on Hartree-Fockmethod The radial wave functions are also used to obtain theatomrsquos total energy (Eav) including approximate relativisticand correlation energy corrections Relativistic terms in thepotential function give approximate relativistic correctionsto the radial functions as well as improved relativistic energycorrections in heavy atoms In addition a correlation termis included to make the potential function more negativethereby helping to bind negative ions These radial functionsare also used to calculate Coulomb integrals Fk and Gk

and spin-orbit integrals ζnl After radial functions have beenobtained based on Hartree-Fock model the wave function|γJM〉 of the M sublevel of a level labeled γJ is expressed interms of LS basis states |αLSJM〉 by the formula∣∣γJMrang =sum

|αLSJM〉langαLSJ | γJrang (4)

If determinant wave functions are used for the atom thetotal binding energy is given by

E =sumi

⎛⎝Eik + Ein +

sumj lt i

⎞⎠ (5)

where Eik is the kinetic energy Ein is the electron-nuclearCoulomb energy and Ei j is the Coulomb interaction energy

Table 2 New λ(A) g f and gAki(sminus1) for electric dipole (E1) transitions in La III

Transitionλ gf gAki

Lower level Upper level

5d 2D32 12f 2Fo52 68299B 0039B 562 times 108 B

5d 2D52 12f 2Fo72 69053B 0056B 777 times 108 B

5d 2D52 12f 2Fo52 69055B 0003B 389 times 107 B

5d 2D32 11f 2Fo52 69098B 0053B 746 times 108 B

5d 2D52 11f 2Fo72 69869B 0075B 103 times 109 B

5d 2D52 11f 2Fo52 69872B 0004B 515 times 107 B

5d 2D32 12p 2Po32 70116B 0001B 821 times 106 B

5d 2D32 12p 2Po12 70160B 0003B 410 times 107 B

5d 2D32 10f 2Fo52 70210AB 0074A 100 times 109 A

0075B 102 times 109 B

5d 2D52 12p 2Po32 70913B 0005B 714 times 107 B

5d 2D52 10f 2Fo72 71005AB 0104A 138 times 109 A

0106B 141 times 109 B

5d 2D52 10f 2Fo52 71009AB 0005AB 692 times 107 A

703 times 107 B

5d 2D32 11p 2Po32 71683B 0001B 115 times 107 B

5d 2D32 11p 2Po12 71749B 0004B 576 times 107 B

5d 2D52 11p 2Po32 72517B 0008B 100 times 108 B

4f 2Fo52 10g 2G72 72902A 0013AB 157 times 108 A

72797B 167 times 108 B

4f 2Fo72 10g 2G72 73709A 0001AB 570 times 106A

73601B 597 times 106 B

4f 2Fo72 10g 2G92 73709A 0016A 197 times 108 A

73601B 0017B 209 times 108 B

4f 2Fo52 9g 2G72 74258A 0017A 206 times 108 A

74062B 0018B 215 times 108 B

5d 2D32 10p 2Po32 74094A 0001AB 160 times 107 A

74093B 170 times 107 B

5d 2D32 10p 2Po12 74200A 0007AB 799 times 107 A

74197B 844 times 107 B

4f 2Fo72 9g 2G72 75095A 0001AB 744 times 106 A

74894B 772 times 106 B

4f 2Fo72 9g 2G92 75095A 0022A 258 times 108 A

74894B 0023B 270 times 108 B

5d 2D52 10p 2Po32 74986A 0012AB 139 times 108 A

74984B 147 times 108 B

6p 2Po12 12d 2D32 97838B 0006B 454 times 107 B

6p 2Po12 11d 2D32 100277B 0010B 650 times 107 B

6p 2Po32 12d 2D52 100875B 0011B 746 times 107 B

6p 2Po32 12d 2D32 100894B 0001B 829 times 106 B

6p 2Po32 11d 2D52 103462B 0017B 106 times 108 B

6p 2Po32 11d 2D32 103489B 0002B 118 times 107 B

6p 2Po12 10d 2D32 106180A 0008A 459 times 107 A

103929B 0016B 983 times 107 B

6p 2Po32 10d 2D52 109743A 0014A 783 times 107 A

107342B 0028B 161 times 108 B

6p 2Po32 10d 2D32 109789A 0002A 855 times 106A

107384B 0003B 178 times 107 B

6d 2D32 12f 2Fo52 156165B 0036B 996 times 107 B

6d 2D52 12f 2Fo72 157218B 0052B 139 times 108 B

6d 2D52 12f 2Fo52 157229B 0002B 697 times 106B

Table 2 Continued

6d 2D32 11f 2Fo52 160409B 0050B 131 times 108 B

6d 2D52 11f 2Fo72 161516B 0072B 183 times 108 B

6d 2D52 11f 2Fo52 161532B 0004B 915 times 106 B

6d 2D32 12p 2Po12 166251B 0002B 562 times 106 B

6d 2D32 10f 2Fo52 166531A 0080A 193 times 108 A

166537B 0073B 176 times 108 B

6d 2D52 12p 2Po32 167206B 0004B 994 times 106B

6d 2D52 10f 2Fo72 167717AB 0114A 270 times 108 A

0104B 246 times 108 B

6d 2D52 10f 2Fo52 167742AB 0006A 135 times 107 A

0005B 123 times 107 B

6d 2D32 11p 2Po12 175458B 0004B 785 times 106 B

6d 2D52 11p 2Po32 176404B 0006B 139 times 107 B

6d 2D32 10p 2Po32 190187A 0002A 294 times 106A

190169B 0001B 231 times 106 B

6d 2D32 10p 2Po12 190876A 0008A 145 times 107 A

190857B 0006B 114 times 107 B

6d 2D52 10p 2Po32 191768A 0014A 257 times 107 A

191749B 0011B 203 times 107 B

5f 2Fo52 10g 2G72 192641A 0025AB 450 times 107 A

191909B 453 times 107 B

5f 2Fo72 10g 2G72 192939A 0001AB 166 times 106 A

192205B 167 times 106 B

5f 2Fo72 10g 2G92 192205A 0032AB 585 times 107 A

192939B 580 times 107 B

5f 2Fo52 12d 2D32 193160B 0003B 503 times 106 B

5f 2Fo72 12d 2D52 193392B 0004B 716 times 106 B

7p 2Po12 12d 2D32 196107B 0014B 238 times 107 B

7p 2Po32 12d 2D52 200877B 0024B 399 times 107 B

7p 2Po32 12d 2D32 200950B 0003B 443 times 106 B

5f 2Fo52 9g 2G72 202406A 0044AB 709 times 107 A

200957B 726 times 107 B

5f 2Fo72 9g 2G72 202735A 0002AB 261 times 106 A

201281B 267 times 106 B

5f 2Fo72 9g 2G92 202734A 0056A 915 times 107 A

201281B 0057B 936 times 107 B

5f 2Fo52 11d 2D32 202902B 0004B 736 times 106 B

5f 2Fo72 11d 2D52 203129B 0006B 105 times 107 B

7p 2Po12 11d 2D32 206156B 0022B 347 times 107 B

7p 2Po32 11d 2D52 211403B 0039B 580 times 107 B

7p 2Po32 11d 2D32 211516B 0004B 643 times 106 B

5f 2Fo52 10d 2D52 228424A 0001AB 740 times 105 A

218260B 822 times 105 B

5f 2Fo52 10d 2D32 228623A 0008AB 103 times 107 A

218434B 115 times 107 B

5f 2Fo72 10d 2D52 228843A 0012AB 147 times 107 A

218643B 163 times 107 B

7p 2Po12 10d 2D32 232762A 0034A 421 times 107 A

222209B 0040B 541 times 107 B

7p 2Po32 10d 2D52 239398A 0060A 701 times 107 A

228258B 0070B 898 times 107 B

7p 2Po32 10d 2D32 239617A 0007A 773 times 106 A

Table 2 Continued

228449B 0008B 995 times 106 B

7d 2D32 12f 2Fo52 278697B 0045B 386 times 107 B

7d 2D52 12f 2Fo72 280254B 0064B 542 times 107 B

7d 2D52 12f 2Fo52 280290B 0003B 271 times 106 B

7d 2D32 11f 2Fo52 292508B 0064B 498 times 107 B

7d 2D52 11f 2Fo72 294209B 0091B 700 times 107 B

7d 2D52 11f 2Fo52 294264B 0004B 350 times 106 B

7d 2D32 12p 2Po32 311657B 0001B 410 times 105 B

7d 2D32 12p 2Po12 312535B 0003B 203 times 106 B

7d 2D32 10f 2Fo52 313524AB 0103A 699 times 107 A

0096B 653 times 107 B

7d 2D52 12p 2Po32 313651B 0005B 362 times 106B

7d 2D52 10f 2Fo72 315457A 0146A 978 times 107 A

315455B 0137B 916 times 107 B

7d 2D52 10f 2Fo52 315545A 0007AB 489 times 106 A

315542B 458 times 106 B

6f 2Fo52 10g 2G72 339275A 0098A 570 times 107 A

337011B 0099B 582 times 107 B

6f 2Fo72 10g 2G72 339840A 0004AB 210 times 106 A

337569B 214 times 106 B

6f 2Fo72 10g 2G92 339841A 0127A 735 times 107 A

337569B 0128B 750 times 107 B

6f 2Fo52 12d 2D52 340676B 0001B 458 times 105 B

6f 2Fo52 12d 2D32 340888B 0011B 640 times 106 B

6f 2Fo72 12d 2D52 341246B 0016B 911 times 106 B

7d 2D32 11p 2Po32 345208B 0001B 578 times 105 B

7d 2D32 11p 2Po12 346738B 0005B 285 times 106 B

7d 2D52 11p 2Po32 347657B 0009B 509 times 106 B

8p 2Po12 12d 2D32 349371B 0028B 154 times 107 B

8p 2Po32 12d 2D52 356921B 0050B 260 times 107 B

8p 2Po32 12d 2D32 357154B 0005B 288 times 106 B

6f 2Fo52 9g 2G72 370780A 0181A 876 times 107 A

365944B 0183B 913 times 107 B

6f 2Fo72 9g 2G72 371455A 0007AB 323 times 106 A

366601B 336 times 106 B

6f 2Fo72 9g 2G92 371449A 0234A 113 times 108 A

366601B 0237B 118 times 108 B

6f 2Fo52 11d 2D52 372097B 0001B 683 times 105 B

6f 2Fo52 11d 2D32 372447B 0020B 954 times 106 B

6f 2Fo72 11d 2D52 372777B 0028B 136 times 107 B

8p 2Po12 11d 2D32 382597B 0050B 230 times 107 B

8p 2Po32 11d 2D52 391563B 0089B 386 times 107 B

8p 2Po32 11d 2D32 391950B 0010B 428 times 106 B

7d 2D32 10p 2Po32 409395A 0003A 105 times 106 A

409309B 0002B 864 times 105 B

7d 2D32 10p 2Po12 412600A 0013A 511 times 106 A

412512B 0011B 422 times 106 B

7d 2D52 10p 2Po32 412847A 0023A 913 times 106 A

412755B 0019B 759 times 106 B

6f 2Fo52 10d 2D52 468540A 0003AB 822 times 105 A

427687B 110 times 106 B

6f 2Fo52 10d 2D32 469378A 0038A 114 times 107 A

Table 2 Continued

428355B 0042B 154 times 107 B

6f 2Fo72 10d 2D52 469617A 0054A 163 times 107 A

428585B 0060B 219 times 107 B

8p 2Po12 10d 2D32 485614A 0091A 257 times 107 A

441837B 0110B 374 times 107 B

8p 2Po32 10d 2D52 499826A 0160A 426 times 107 A

453605B 0192B 622 times 107 B

8p 2Po32 10d 2D32 500781A 0018A 469 times 106 A

454357B 0021B 688 times 106 B

8d 2D32 12f 2Fo52 461396B 0058B 182 times 107 B

8d 2D52 12f 2Fo72 463733B 0083B 257 times 107 B

8d 2D52 12f 2Fo52 463830B 0004B 128 times 106 B

8d 2D32 11f 2Fo52 500522B 0086B 230 times 107 B

8d 2D52 11f 2Fo72 503228B 0123B 323 times 107 B

8d 2D52 11f 2Fo52 503388B 0006B 161 times 106 B

6g 2G92 12f 2Fo72 513694B 0001B 175 times 105 B

6g 2G72 12f 2Fo52 513790B 0001B 135 times 105 B

8d 2D32 12p 2Po32 559328B 0001B 196 times 105 B

8d 2D32 12p 2Po12 562162B 0005B 965 times 105 B

6g 2G92 11f 2Fo72 562607B 0001B 271 times 105 B

6g 2G72 11f 2Fo52 562778B 0001B 209 times 105 B

8d 2D52 12p 2Po32 562909B 0008B 173 times 106 B

8d 2D32 10f 2Fo52 565376A 0144A 302 times 107 A

565369B 0138B 287 times 107 B

8d 2D52 10f 2Fo72 568753A 0203A 423 times 107 A

568746B 0196B 403 times 107 B

8d 2D52 10f 2Fo52 569036A 0010AB 211 times 106 A

569029B 201 times 106 B

7f 2Fo52 10g 2G72 577305A 0247A 495 times 107 A

570779B 0250B 513 times 107 B

7f 2Fo72 10g 2G72 578403A 0009AB 182 times 106 A

571853B 189 times 106 B

7f 2Fo72 10g 2G92 578405A 0320A 638 times 107 A

571853B 0324B 661 times 107 B

7f 2Fo52 12d 2D52 581372B 0002B 469 times 105 B

7f 2Fo52 12d 2D32 581989B 0033B 655 times 106 B

7f 2Fo72 12d 2D52 582486B 0047B 933 times 106 B

9p 2Po12 12d 2D32 599659B 0061B 112 times 107 B

9p 2Po32 12d 2D52 613185B 0107B 189 times 107 B

9p 2Po32 12d 2D32 613872B 0012B 210 times 106 B

6g 2G92 10f 2Fo72 645777A 0003AB 451 times 105 A

645776B 459 times 105 B

6g 2G72 10f 2Fo52 646109A 0002AB 348 times 105 A

646103B 353 times 105 B

7f 2Fo52 9g 2G72 674879A 0512A 750 times 107 A

659026B 0525B 807 times 107 B

7f 2Fo72 9g 2G72 676380A 0019AB 276 times 106 A

660458B 297 times 106 B

7f 2Fo72 9g 2G92 676363A 0663A 966 times 107 A

660458B 0679B 104 times 108 B

Table 2 Continued

8d 2D32 11p 2Po32 677504B 0002B 286 times 105 B

7f 2Fo52 11d 2D52 679255B 0005B 723 times 105 B

7f 2Fo52 11d 2D32 680423B 0070B 101 times 107 B

7f 2Fo72 11d 2D52 680777B 0100B 144 times 107 B

8d 2D52 11p 2Po32 682766B 0017B 252 times 106 B

8d 2D32 11p 2Po12 683422B 0010B 139 times 106 B

9p 2Po12 11d 2D32 704699B 0130B 175 times 107 B

9p 2Po32 11d 2D52 723087B 0228B 291 times 107 B

9p 2Po32 11d 2D32 724409B 0025B 321 times 106 B

9d 2D32 12f 2Fo52 745770B 0079B 947 times 106 B

9d 2D52 12f 2Fo72 749442B 0112B 133 times 107 B

9d 2D52 12f 2Fo52 749698B 0006B 666 times 105 B

7g 2G92 12f 2Fo72 826395B 0003B 286 times 105 B

7g 2G72 12f 2Fo52 826614B 0002B 221 times 105 B

9d 2D32 11f 2Fo52 853625B 0124B 114 times 107 B

9d 2D52 11f 2Fo72 858310B 0176B 160 times 107 B

9d 2D52 11f 2Fo52 858774B 0009B 797 times 105 B

between electrons i and j averaged over all possible magneticquantum numbers

In this method relativistic corrections have been limitedto calculations to the mass-velocity and the Darwin cor-rections by using the relativistic correction to total bindingenergy The total binding energy can be given in by formulas(757) (758) and (759) in [16]

We calculated the radiative parameters (wavelengths oscilla-tor strengths and transition probabilities) for electric dipole(E1) transitions in La III (Z = 57) using HFR code [15] Wehave taken into account 5p6nd 5p6ng (n = 5ndash10) 5p6ns (n =6ndash10) 5p56s6p 5p56s4f 5p55d6p 5p6nf (n = 4ndash10) 5p6np(n = 6ndash10) 5p54f2 and 5p56p2 configurations outside thecore [Cd] for calculation A and nd ng (n = 5ndash25) ns (n =6ndash24) nf (n = 4ndash22) and np (n = 6ndash25) configurationsoutside the core [Xe] for calculation B Table 1 shows thewavelengths λ (in A) the weighted oscillator strengths gf the weighted transition rates (or probabilities) gAki (in sminus1)for nd (n = 5ndash9)ndashnf (n = 4ndash8) nd (n = 5ndash9)ndashnp (n = 6ndash9)np (n = 6ndash9)ndashns (n = 6ndash10) and ng (n = 5ndash8)ndashnf (n = 4ndash8) electric dipole (E1) transitions The data obtained are toomuch For this reason we have here presented just a part ofthe results The comparing values for these exist in literatureTherefore it is also made a comparison with other calcu-lations and experiments in Table 1 We have also reportedthe wavelengths the weighted oscillator strengths and theweighted transition probabilities that are greater than orequal to 105 for some new transitions (680 A le λ le 8600 A)in Table 2 References for other comparison values are

indicated below the tables with a lowercase superscript odd-parity states are indicated by the superscript ldquo ordquo

Electron correlation effects and relativistic effects play animportant role in the spectra of heavy elements To accuratelypredict the radiative atomic properties for heavy atoms suchas La III complex configuration interactions and relativis-tic effects must be considered simultaneously AlthoughCowanrsquos approach is based on Schrodingerrsquos equation itincludes the most important relativistic effects like mass-velocity corrections and Darwin contributions Also forcomplex atoms it is important to allow for spin-orbit inter-action which represents the magnetic interaction energybetween electronrsquos spin magnetic moment and the magneticfield that the electron sees due to its orbital motion throughthe electric field of the nucleus These contributions areconsidered as perturbations Thus to solve the Schrodingerequation with this Hamiltonian we define a new angularmomentum operator in an intermediate coupling scheme

In calculations the eigenvalues of Hamiltonian wereoptimized to the observed energy levels via a least-squaresfitting procedure using experimentally determined energylevels specifically all of the levels from the NIST compilation[28] The scaling factors of the Slater parameters (Fk and Gk)and of configuration interaction integrals (Rk) not opti-mized in the least-squares fitting were chosen equal to 085while the spin-orbit parameters were left at their initial val-ues This low value of the scaling factors has been suggestedby Cowan for neutral heavy elements [15 16]

We obtained 7785 and 4278 possible E1 transitionsbetween odd- and even-parity levels in the calculations Aand B respectively The results obtained are in excellentagreement with those of other works except some transi-tions For some transitions although the agreement is less in

the weighted oscillator strengths and the weighted transitionprobabilities it is very good in the wavelengths Most ofresults related to low-lying levels obtained from this work arein agreement with literature [1ndash6] The differences betweenour HFR results and other works for gf and gAki have beenfound in the 0ndash10 range for the transitions np (n = 6ndash8)ndashns(n = 6ndash10) nd (n = 6ndash9) in the 05ndash9 range for the tran-sitions nd (n = 6 7)ndashnf (n = 5ndash8) np (n = 7ndash9) and in the15ndash20 range for the transitions nf (n = 5ndash8)ndashnd (n = 7ndash9)ng (n = 5ndash8) But the agreement is less in the weightedoscillator strengths and the weighted transition probabilitiesfor 5d and 4f transitions In fact except the transitions 6p2Po

32ndash9d 2D32 5d 2D32ndash9p 2Po12 4f 2Fo

72ndash5g 2G72 4f 2Fo72ndash

6g 2G72 and 4f 2Fo52ndash8g 2G7292 we found the values 1064

(in calculation A) and 1078 (in calculation B) for the meanratio gf (this work)gf [1] Except the transitions 5dndash9p 4fndash7g 8g 8sndash9p and 4fndash6d we found also the values 1084 (incalculation A) and 1126 (in calculation B) for the mean ratiogAki (this work)gAki [1] The transition results obtainedfrom the calculation A agree with other works This calcula-tion includes core correlation (including excitation from 5pshell in core) These results obtained from HFR calculationsmay be better in case that the increasing number ofconfigurations including the excitations from core It is notedthat there are no exist the works especially experimental onLa III recently in available literature A detailed comparisonneeds new experimental works Most of our results areexcellent in agreement expect the transition results to 4f and5d levels (for gf and gAki results) generally It is well knownthat these levels interact strongly with core

In conclusion the main purpose of this paper was toperform HFR calculations for obtaining the description of LaIII spectrum Accurate atomic structure data is an essentialingredient for a wide range of research fields Areas fromplasma research applications in nuclear fusion to lightingresearch as well as astrophysics and cosmology dependon such data In spectrum synthesis works particularly forCP stars accurate data for transition probabilities (rates)and oscillator strengths for lanthanide atoms are needed toestablish reliable abundances for these species The agree-ment is excellent especially for wavelengths when our HFRresults are compared with other available works in literaturefor the radiative transitions for La III So we may men-tion that new results presented in Table 2 for the transitionsbetween some highly levels in this work are also reliableThere are a few experimental or theoretical radiative transi-tion data for La III in literature Consequently we hope thatour results especially the new results in Table 2 which areobtained using the HFR method will be useful for researchfields technological applications and other works in thefuture for La III spectra

Acknowledgment

The authors are very grateful to the anonymous reviewerfor stimulating comments and valuable suggestions whichresulted in improvements to this paper

References

[1] E Biemont Z S Li P Palmeri and P Quinet ldquoRadiativelifetimes in La III and oscillator strengths in La III and Lu IIIrdquoJournal of Physics B vol 32 no 14 pp 3409ndash3419 1999

[2] H Odabasi ldquoSpectrum of doubly ionized lanthanum (La III)rdquoJournal of the Optical Society of America vol 57 no 12 pp1459ndash1463 1967

[3] J Migdalek and M Wyrozumska ldquoRelativistic oscillatorstrengths for the Cs isoelectronic sequence and collapse of fand d orbitalsrdquo Journal of Quantitative Spectroscopy andRadiative Transfer vol 37 no 6 pp 581ndash589 1987

[4] J Migdalek and W E Baylis ldquoRelativistic Hartree-Fockoscillator strengths for the lowest srarr p transitions in the firstfew members of the Rb(I) and Cs(I) isoelectronic sequenceswith allowance for core polarizationrdquo Journal of QuantitativeSpectroscopy and Radiative Transfer vol 22 no 2 pp 127ndash1341979

[5] S Johansson and U Litzen ldquoResonance lines of La IIIrdquo Journalof the Optical Society of America vol 61 no 10 pp 1427ndash14281971

[6] Z S Li and J Zhankui ldquoLifetime measurements in La II andLa III using time-resolved laser spectroscopyrdquo Physica Scriptavol 60 no 5 pp 414ndash417 1999

[7] E Biemont and P Quinet ldquoRecent advances in the study oflanthanide atoms and ionsrdquo Physica Scripta vol T105 pp 38ndash54 2003

[8] J S Badami ldquoThe spectrum of trebly-ionized cerium (Ce IV)rdquoProceedings of the Physical Society vol 43 no 1 pp 53ndash581931

[9] J Migdalek and A Bojara ldquoRelativistic effects core polarisa-tion and relaxation in ionisation potentials along Rb and Csisoelectronic sequencesrdquo Journal of Physics B vol 17 no 10pp 1943ndash1951 1984

[10] P Quinet and E Biemont ldquoLande g-factors for experimentallydetermined energy levels in doubly ionized lanthanidesrdquoAtomic Data and Nuclear Data Tables vol 87 no 2 pp 207ndash230 2004

[11] R C Gibbs and H E White ldquoRelations between doublets ofstripped atoms in five periods of the periodic tablerdquo PhysicalReview vol 33 no 2 pp 157ndash162 1929

[12] H N Russell and W F Meggers ldquoAn analysis of lanthanumspectra (La I La II La III)rdquo Journal of Research of the NationalBureau of Standards vol 9 no 5 pp 625ndash668 1932

[13] J Sugar and V Kaufman ldquoSpectrum of doubly ionizedlanthanum (La III)rdquo Journal of the Optical Society of Americavol 55 no 10 pp 1283ndash1285 1965

[14] B Karacoban and L Ozdemir ldquoEnergies Lande g-factors andlifetimes for some excited levels of doubly ionized lanthanumrdquoCentral European Journal of Physics vol 10 no 1 pp 124ndash1312012

[15] httpwwwtcdiePhysicsPeopleCormacMcGuinnessCowan[16] R D Cowan The Theory of Atomic Structure and Spectra

California USA 1981[17] B Karacoban and L Ozdemir ldquoEnergies and lifetimes for

some excited levels in La Irdquo Acta Physica Polonica A vol 113no 6 pp 1609ndash1618 2008

[18] B Karacoban and L Ozdemir ldquoElectric dipole transitionsfor La I (Z = 57)rdquo Journal of Quantitative Spectroscopy andRadiative Transfer vol 109 no 11 pp 1968ndash1985 2008

[19] B Karacoban and L Ozdemir ldquoThe hyperfine structure cal-culations of some excited levels for (139)La Irdquo Acta PhysicaPolonica A vol 115 no 5 pp 864ndash872 2009

[20] B Karacoban and L Ozdemir ldquoTransition energies of neutraland singly ionized lanthanumrdquo Indian Journal of Physics vol84 no 3 pp 223ndash230 2010

[21] B Karacoban and L Ozdemir ldquoElectric dipole transitions forLu I (Z = 71)rdquo Arabian Journal for Science and Engineeringvol 36 no 4 pp 635ndash648 2011

[22] B Karacoban and L Ozdemir ldquoEnergies and Lande factors forsome excited levels in Lu I (Z = 71)rdquo Central European Journalof Physics vol 9 no 3 pp 800ndash806 2011

[23] B Karacoban and L Ozdemir ldquoEnergies Lande factors andlifetimes for some excited levels of neutral ytterbium (Z =70)rdquo Acta Physica Polonica A vol 119 no 3 pp 342ndash3532011

[24] B Karacoban and L Ozdemir ldquoElectric dipole transitions forneutral ytterbium (Z = 70)rdquo Journal of the Korean PhysicalSociety vol 58 no 3 pp 417ndash428 2011

[25] B Karacoban and L Ozdemir ldquoTransition energies of ytter-bium (Z = 70)rdquo Zeitschrift fur Naturforschung A vol 66 pp543ndash551 2011

[26] B Karacoban and L Ozdemir ldquoThe level structure of atomiclutetium (Z = 71) a relativistic Hartree-Fock calculationrdquoIndian Journal of Physics vol 85 no 5 pp 683ndash702 2011

[27] B Karacoban and L Ozdemir ldquoTransition energies oflutetiumrdquo Chinese Journal of Physics vol 50 no 1 pp 40ndash492012

[28] NIST httpwwwnistgovpmldataasdcfm

Relativistic Time-Dependent Density Functional Theory andExcited States Calculations for the Zinc Dimer

Ossama Kullie

Laboratoire de Chimie Quantique Institute de Chimie de Strasbourg CNRS et Universite de Strasbourg 4 rue Blaise Pascal67070 Strasbourg France

Correspondence should be addressed to Ossama Kullie ossamakullieunistrafr

I present a time-dependent density functional study of the 20 low-lying excited states as well the ground states of the zinc dimer Zn2analyze its spectrum obtained from all electrons calculations performed using time-depended density functional with a relativistic4-component and relativistic spin-free Hamiltonian as implemented in Dirac-Package and show a comparison of the resultsobtained from different well-known and newly developed density functional approximations a comparison with the literatureand experimental values as far as available The results are very encouraging especially for the lowest excited states of this dimerHowever the results show that long-range corrected functionals such as CAMB3LYP gives the correct asymptotic behavior forthe higher states and for which the best result is obtained A comparable result is obtained from PBE0 functional Spin-freeHamiltonian is shown to be very efficient for relativistic systems such as Zn2

1 Introduction

Zinc dimer Zn2 is the first member of the group 12 (IIB)(Zn2 Cd2 Hg2 and Cn2) and has a representative characterof these dimers The interest in the dimers of the group IIB(12) is in part due to the possibility of laser applicationsin analogy with the rare gas dimers A second point isthe importance of the metallic complexes similar to thetransition metal complexes [1ndash4] and some importantapplication like the solar cell and renewable energy [5 6] aswell as electric battery for new cars technology [7 8] Zn2Cd2 and Hg2 are exciter with a shallow predominantly Vander Waals ground state and low-lying covalent bound excitedstates They are also interesting from a theoretical pointof view due to the different character of the ground andexcited states and consequently the different methodologicaldemands for an accurate theoretical description of thespectrum The dimer of group 12 has been studied bothexperimentally and theoretically Relevant reviews havebeen provided by Morse [9] and more recently by Koperski[10 11] The covalent contributions to the ground statebonding in the group 12 dimers have been investigated in[12] it was concluded that the bond is a mixture of 34 Van

der Waals and 14 covalent interactions Bucinisky et al [13]provides spectroscopic constants using the coupled clustermethod (CCSD(T)) and different level of the theory 4-component relativistic Hamiltonian using Dirac-CoulombHamiltonian relativistic spin-free Hamiltonian and nonrel-ativistic (NR) Hamiltonian Furthermore they investigatedthe relativistic effects and found to be about 5 8 19 ofthe binding energies for Zn2 Cd2 and Hg2 respectivelyFinally the last member of the group Cn2 copernicium hasan academic interest [14ndash16] due the chemical character ofthe bonding in comparison to Hg2 (and the lighter dimers ofthe group) and the influence of the relativistic effects on theatomic orbitals providing a change of the boding characterin the dimer to more covalent or Van der Waals type

The paper presents all-electron calculations on thelowest-lying excited states as well as the ground state Thefirst 8 lowest exited states are discussed with a comparison toexperimental and literature values and several other higherexcited states are presented and discussed Earlier worksinvestigated the lowest 8 excited states using different wavefunction methods Ellingsen et al [17] showed ab initioresults for the ground and lowest 8 excited states of Zn2 theyperformed all electron calculations and present NR as well as

relativistic spin-free Douglas-Kroll result the spin-orbit cou-pling was accounted perturbatively The ground state is stud-ied at ACPTF (averaged coupled pair functional CCSD(T)and CASPT2 (complete active space second-order perturba-tion theory) level and the excited states are studied at MR-ACPF (multireference ACPF) and CASPT2 level Czuchaj etal [18ndash20] performed their computations for Zn2 (later forCd2 and Hg2) using (NR) pseudopotential approach andMRCI (multireference configuration interaction) and thespin-orbit coupling was taken only approximately

In this work we use a relativistic spin-free Hamiltonian(SFH) without spin-orbit coupling with a comparisonto a relativistic 4-component Dirac-Coulomb Hamiltonian(DCH) spin-orbit coupling included in the framework oftime-dependent density functional theory (TDDFT) andits linear-response approximation (LRA) The calculationsare performed using Dirac-Package (program for atomicand molecular direct iterative relativistic all-electron calcula-tions) [21] The relativistic effects for Zn2 (and even for Cd2)are small but visible and in some respects not negligible Tomy experience generally around zinc (Z = 30) the relativisticeffects started to become important for chemical propertiesFor Hg2 they are large enough (for Cn2 expected to be verylarge) to make it necessary to incorporate them into anyproperties that are sensitive to the potential [13] This ispredominantly due to the contraction of 6s orbital a well-known and important relativistic effects in heavy atoms [22ndash25] We will follow this issue in future works on the group 12(IIB)

The paper is organized as follows Section 2 is devoted tothe theory and method We briefly introduce in Section 21the key concepts of the static density functional (DFT) anddiscuss its extension to the relativistic domain In Section 22we introduce the key concepts of time-dependent densityfunctional (TDDFT) and the linear response approximationSection 3 is devoted to the computational details andSection 4 to the result and discussion and finally we give aconclusion in Section 5 Some useful (well-known) notationsused in this paper are collected in Table 1

2 Theory and Methods

Time-dependent density functional theory (TDDFT) cur-rently has a growing impact and intensive use in physics andchemistry of atoms small and large molecules biomoleculesfinite systems and solidstate For excited states resultingfrom a single excitation that present a single jump from theground state to an excited state I used in this work the LRAas implemented in Dirac-Package [26ndash28] and well-knownapproximations of density functionals like LDA (SVWN5correlation) [29 30] PBE [31] PB86 [32ndash34] BPW91(Becke exchange [32] and Perdew-Wang correlation [35])long-range corrected PBE0 [36] and its gradient correctedfunctional GRAC-PBE0 [37 38] BLYB and B3LYP [32 39ndash41] or newly developed range-separated functionals such asCAMB3LYP [42] Todayrsquos available DFT cannot describe theground state of the group IIB dimers accurately due to a largecontribution of dispersion in the bonding [12] despite this

Table 1 Some of the acronyms used in this work

HF Hartree Fock method

NR Nonrelativistic

DHF Dirac or relativistic HF

DCH Dirac-Coulomb Hamiltonian

MP2 Moslashller-Plesst 2nd-order perturbation theory

CCSD(T) Coupled cluster singles-doubles (triples)

SFH Relativistic spin-free Hamiltonian

(TD)DFT (Time-depended) density functional theory

xc Exchange-correlation

LR(A) Linearresponse (approximation)

ALR Adiabatic LR

srLDAMP2 Short-range LDA long-range MP2

when calculating the covalently well-bound excited states theerror is reduced considerably quite possible accompaniedwith error cancellations

The ground state of the group 12 dimer has a (closed-shell) valence orbitals configuration (ns2 + ns2) σ2

g σ2u n =

4 5 6 for Zn2-Hg2 This configuration essentially arisingfrom the interaction of atomic (ns) orbitals It is weakly cova-lent and preponderantly dispersion interaction well knownespecially in the rare gas dimers [43] The potential curvedisplays a shallow van der Waals type of minimum Excitingelectrons from σ2

g or σ2u to the lowest set of molecular orbitals

spanned by the atomic orbitals Atom(ns2) + Atom(nsnp) orAtom(ns2) + Atom(ns(n+ 1)s) or Atom(ns2) + Atom(ns(n+1)p) gives rise to a manifold of states (see Table 2) amongthem states which strongly have covalent contributions aswe will see in Section 4 Results and Discussion This makesTDDFT using LRA and well-known functional approxima-tions adequate to describe these states [26]

We will discuss the lowest 20 excited states dissociatingto the atomic asymptotes (NR notation) given in Table 2resulting from exciting one electron from the ground state(4s2 1S+ 4s2 1S)1Σ+

g The concern will be in the first place onthe 8 lowest excited states corresponding to the asymptoteAtom(ns2) + Atom(nsnp) States corresponding to the higherasymptotes Atom(ns2) + Atom(ns(n + 1)s) and Atom(ns2) +Atom(ns(n + 1)p) are computed and some of them are well-bound states we will discuss their quality in view of the limitof the validity of the known DFT approximations yieldinginaccurate potential curves and causing a disturbance nearthe avoiding crossing with states of the same symmetry (seeSection 4) To my best knowledge there is no experimentalor theoretical values from DFT or wave function methodsavailable for the higher states to compare with this makesit difficult to judge the result of the present work It isexpected that the result of the lowest states will show anexcellent agreement with the experimental data [10 11](and the references therein) whereas for the higher states asatisfactory result is expected showing the important featuresof these states The comparison between spin-free and 4-component results shows clearly the capability of SFH to dealwith the computation of the properties of the Zn2 dimer

Table 2 Lowest excited states and the corresponding asymptotes

Equation (1) ((ns2) 1S + (nsnp) 3P) 3Πg 3Πu3Σ+g 3Σ+

Equation (3) ((ns2) 1S + (ns(n + 1)s) 3S) 3Σ+g 3Σ+

Equation (5) ((ns2) 1S + (ns(n + 1)p) 3P) 3Πg 3Πu3Σ+g 3Σ+

or similar systems We also emphasize its importance forheavier relativistic systems [13] although spin-orbit effect isexpected to be larger for Cd2 Hg2 and Cn2 Pyper et al [22]pointed out that the relativistic ground-state potential welldepth of Hg2 is 45 of the NR one and clearly it is strongerfor Cn2

21 Density Functional Theory Density functional theory[44ndash46] has become recently a very large popularity asa good compromise between accuracy and computationalexpediency The Hohenberg-Kohn theorem [44] proves theexistence of an unique (up to an additive constant) externalpotential vext(r) for a given nondegenerate density n(r) ofinteracting Fermions The key point behind this scheme isthe very useful simplification namely the transformation ofthe many-body quantum problem to a set of equations ofone-particle Schrodinger (or Dirac) type of a noninteractingreference system with the density as a central ingredientquantity to carry all the relevant information of the systemunder consideration instead of the many-body quantumwave function in which all the information of the system isstored

Hφi(r) = E[n(r)]φi(r) (1)

H = T +Veff[n(r)] =sumi

t(ri) + veff(ri)[n(r)] (2)

veff(ri) = vext(ri) + vH(ri) + vxc(ri) + vnn (3)

n(r) =Nsumi=1

∥∥φi(r)∥∥2 (4)

where n(r) is the total density of the system and the sum isover N that is all occupied orbitals φi(r) t(ri) is the one-particle kinetic energy operator veff(ri) is the one-particleeffective potential (also called Kohn-Sham potential veff(ri) equivvKS(ri)) with vext(ri) is the Coulombic interaction of theelectron i with all the nuclei called the external potentialvH(ri) is the Hartree and vxc(ri) exchange-correlation poten-tial And vnn is the classical Coulombic repulsion of the nucleiin the system vH(ri) is given by the usual expression but thecrucial part vxc(ri) in this scheme is the explicitly unknownvxc(ri)

vH(ri) =intd3r

n(r)|ri minus r|

vxc(ri) = partExc[n(r)]partn(ri)

for which an appropriate good approximation must befound Experiences in DFT (and TDDFT) over the pastdecades shows that the density of atoms molecules finitesystems and solids have very complicated structures [47]To find a good mathematical functionality form betweenthe density (and its gradients) and an exchange-correlationpotential with widely physical applications success is oneof the most challenging problems in quantum physics andchemistry Moreover most of the problems arise whenevaluating the results of the calculating systems can betracked back to the limits of the validity of the todayrsquosknown and employed approximations specially the long-range behavior leaving quite a room for improvementsOne should note that that in many applications the usualapproximations are quite reliable and give good results andacceptable accuracies The present work is not an exceptionas we will see when analyzing the results of the ground stateand excited states of the Zn2 dimer

211 Density Functional Theory in the Relativistic DomainIn the relativistic Dirac theory in absence of electromagneticfield the DCH has the same generic form as the NRHamiltonian (for molecules) [26 48]

HDC =Nsumi

hD(i) +12

Nsumi = j

gCoul(i j) +Msum

K =K primeVnnK K prime

hD(i) =(c2β + c α middot p(i)minus c2 middot I4

)+ I4 middot

MsumK=1

V extK (i)

α j =(

0 σjσj 0

) j = x y z β =

(I2 00 minusI2

where hD(i) is the one-particle DCH and c is the speed oflight in atomic units (atomic units are used throughout thiswork unless otherwise noted) Vnn is the classical nucleus-nucleus repulsion and V ext

K (i) = minusZKriK is the externalCoulombic interaction of the electron i with the nucleus K and the sum is over all nuclei M I2 and I4 are the 2times 2- and4 times 4-unity matrix and the term c2 middot I4 is a shift to align the

relativistic and NR energy scales β and α = (αxαy αz) arethe Dirac matrices with the well-known Pauli matrices σ primesThe generic term

gCoul(i j) = I4 times I4

ri j(7)

is the Coulombic instantaneous two-electron i j interactionoperator it contains in the relativistic theory the spin-own orbit interaction The DCH approximation reduces thedensity functional theory in the relativistic domain to theusual density functional theory with the density as the centralingredient and there is no need to introduce the currentdensity [48] A density functional theory in the relativisticdomain can be constructed on the the basis of (1)ndash(4) with

123Relativistic Time-Dependent Density Functional Theory and Excited States Calculations for the Zinc Dimer

the density is constructed from the relativistic 4-componentwave function The total energy of the system is given by

E[n] =Nsumi

εi minus EJ[n] + Exc[n]minusintd3r vxc(rn)n(r) + Enn

where εi are the electronic eigenvalues of the system andare calculated iteratively in a self-consistent manner (SCFiterations) in an effective many-body potential veff given in(3) Enn is the nuclear-nuclear repulsion energy EJ[n] is theHartree energy equation (9) and Exc[n] is the exchange-correlation energy it can be further divided into exchangeand correlation parts Exc[n] = Ex[n] + Ec[n] At the(single determinant) Hartree-Fock (HF) level which in therelativistic calculations is usually called Dirac-Hartree-Fock(DHF) the two-particle interaction the Hartree and exactexchange are given by (9) and (10) as follows

EJ[n] = 12

int intd3r1d

3r2n(r1)n(r2)|r1 minus r2| (9)

Ex = minus14

Nsumi j

int intd3r1d

φdaggeri (r1)φdaggerj (r2)φj(r1)φi(r2)

|r1 minus r2| (10)

where r1 and r2 denote the coordinates of the electronone and two respectively EJ[n] is a classical interactionbetween two one-particle densities n(r1) and n(r2) whereasEx is a quantum mechanical nonlocal part of many-particleinteraction The φ(r)s are the electronic one-particle HF-orbitals and the sum is over all the occupied orbitals N Awell-known approximation for the Hartree-Fock exchangeenergy is the (α-)Slater approximation [29] with remarkableperformance for covalent bonding in covalently boundmolecules with heavy atoms [49 50]

Eαx [n] = minus32αCx

intd3rn43(r) (11)

where Cx = (34)(3π)13 is a constant in the Slaterapproximation the parameter α = 07 is chosen Themissing of the correlation made the Slater approximationunpopular for chemical calculations In the DFT the exactExc[n] is unknown as a functional of the density (andits gradients) Many approximations exist with differentperformance and accuracy depending on their applicationarea In LDA one assumes a slowly varying local densitydependence hence the Dirac-formula [51] of the exchangeenergy for an uniform electronic gas equation (11) withα = 23 is applied and the Vosko-Wilk-Nusair correlationformula [29 30] for the correlation energy (we use SVWN5)LDA depends only on the density whereas in the generalizedgradient approximation (GGA) the density and its gradientare involved meta GGAs [52] include higher gradients thissystematic improvements is known in the DFT communityunder the term ldquoJacobrsquos ladderrdquo In hybrid functional forexample BLYP and B3LYP [32 39ndash41] one add a (fixed)suitable fraction of exact (Hartree-Fock) exchange (10)to the approximate x-energy part which often improves

the performance of the DFT approximation whereas inthe range-separated density functional [53] a parametricfraction of exchange (and possibly correlation) from wavefunction methods are added to the DFT exchange energywith the parameter dictate the amount of exchange to beadded like CAMB3LYP [42] or of exchange-correlation likesrLDAMP2 (see [43 54ndash56] and the references therein) thisimproves the results considerably unfortunately it is foundthat the optimum parameter value depends on the specificproperty of the system

212 The Relativistic 4-Component and SFH The Diracequation with the Dirac-Coulomb Hamiltonian (DCH)describes the important relativistic effects for chemicalcalculation which become large for systems with large Z Itis a firs-order differential equation(s) hence nonvariationalldquovariational collapserdquo in contrast to the second-order differ-ential Schrodinger equation in the NR case The solutionsto the Dirac equation describe both positrons (the ldquonegativeenergyrdquo states) and electrons (the ldquopositive energyrdquo states) aswell as both spin orientations and a four-component wavefunction is involved called Dirac spinors

∣∣ψrang =(ΨL

) ΨL =

) ΨS =

) (12)

where ΨL is called the large and ΨS the small componentThis notation originally comes from the well-known kineticbalance approximation and is justified by the relationsim 1c between them from which it follows the NR limitlimcrarrinfinΨS = 0 and one identify ΨL with the 2-componentvector (spin up down) of the Schrodinger equation The fullrelativistic 4-component DCH is computationally demand-ing therefore it is desirable to reduce the computationaleffort in relativistic calculations by reducing the dimen-sion of the involved quantities normally by reducing ortransforming the Hamiltonian to a new from so that thecalculations involving operators acting only on the largecomponents and requiring a moderate computational effortby keeping the main physical features of the results Therelativistic SFH implemented in Dirac-Package uses theDyallrsquos formulation [57] to obtain results without spin-orbitcoupling for the four-component Hamiltonian in the defaultrestricted kinetic balance scheme In Section 4 we show thatthe results obtained for the excited states of Zn2 based on(relativistic) SFH are accurate similar and well comparableto those obtained from the 4-component DCH For thederiving of this Hamiltonian we kindly refer the reader to[57] see also [58] with advanced description in framework ofsecond quantization formalism The relativistic SFH permitsfactorization of the spin as in NR calculations so thatstandard NR post-SCF methods can be used for inclusionof electron correlation The extension and implementationof relativistic SFH for many-body system or molecularcalculation is straightforward see [21]

22 TDDFT and Linear Response In this section we brieflyintroduce TDDFT formulation with a special emphasis onthe linear density-response function and its connection

to the electronic excitation spectrum a more extensivederivations and wide discussions can be found in refs [4759ndash78] and the references therein TDDFT was pioneeredby a work of Zangwill and Soven [78] but the fundamentalstep was done later by Runge and Gross [60 61] the Runge-Gross theorem is a rigorous foundation for the formallyextension of the Hohenberg-Kohn theorem [44] to the time-dependent phenomena It results in a time-dependent Kohn-Sham equation

[T + vextσ[n](rt) + vH[n](rt) + vxcσ[n](rt)

]ψjσ(rt)

= ipart

parttψjσ(rt)

where T is the kinetic energy vextσ(rt) vH(rt) are vxcσ(rt)are the time-dependent external Hartree and exchange-correlation potential respectively and we adopt the notation(rt) equiv (r t) ψjσ(rt) is the wave function of a particle j witha spin σ The external potential is unique determined via thetotal density

n(rt) =sumσ

nσ(rt) =sumσ

Nσsumj

∥∥∥ψjσ(rt)∥∥∥ (14)

of the interacting system where the sum is taken over alloccupied spin-orbitals Nσ of a spin possibility σ

221 Linear Response In the special case of the response ofthe ground-state density to a weak external field that is thecase in the most optical applications the slightly perturbedsystem which can be written in a series expansion vext =v0

ext + v1ext + middot middot middot asymp v0

ext + δvext see [72] starts its evolutionslowly from its ground-state density n0 corresponding tothe ground-state external potential v0

ext The xc can beexpressed in terms of the states of (unperturbed) systemand thus as a functional of the ground-state density Theinteracting real system and the Kohn-Sham fictitious systemare connected via the same infinitesimal density changeδn(rt) The infinitesimal change in the Hartree-xc-potentialδvHxc = δvH + δvxc due to the infinitesimal change in thedensity can be expressed in its functional derivative

δvHxc(rt) =intd3rprimedtprime fHxc(rrprime t minus tprime)δn(rprimetprime) (15)

where fHxc is called the Hartree-xc-kernel and is given in LRregime by

fHxc[n0](rrprime t minus tprime) = δ(t minus tprime)|rminus rprime| +

δvxc[n](rt)δn(rprimetprime)

∣∣∣∣n=n0(r)

where δ(t minus tprime) is the Dirac-delta function The first termin (16) is the Hartree contribution it is instantaneous orlocal in time The second term in (16) fxc[n0] called the xc-kernel is much simpler than vxc[n](rt) since it is a functionalof the ground-state density n0 it is nonlocal in space andtime [70]

In the adiabatic approximation which is the most com-mon in TDDFT one ignores all time-dependencies in thepast and takes only the instantaneous density n(t) being localin time The adiabatic approach is a drastic simplificationand a priori only justified for systems with a weak time-dependence which are always locally close to equilibrium[72] In practice one takes a known ground-state functionalapproximation and insert n0(t) into it thus any ground-state approximation (LDA GGA ) provides an adiabaticapproximation for the TDDFT xc-functional The mostcommon one is the ALDA

3 Computational Details

The reported results in this paper have been performedusing a development version of the Dirac10-Package [21]based on the 4-component relativistic DCH and SFH Wewould like to stress though that the present implementationallows the use of all Hamiltonians implemented in theDirac-Package such as the eXact 2-component relativisticHamiltonian (X2C) [79] and the 4-component NR Levy-Leblond Hamiltonian [80] The nuclear charge distributionwas described by a Gaussian model using the recommendedvalues of [81]

The values of the spectroscopic constants Re ωe andDe were extracted from a Morse potential fit based on atleast ten equidistant points of step length 005 au aroundthe equilibrium distance a second fit using polynomial fitprocedure available in Dirac-Package is used too the com-parison between the two fits show that 5-order polynomialfit is rather equivalent to a Morse potential fit providedthat Morse potential fit is performed for small regionaround the minimum which is done throughout this workthe agreement between the two fits gives us an additionalcriterion for the safety and correctness of the calculatedspectroscopic constants reported in the present result

We employed the aug-cc-pVTZ (likewise aug-cc-pVQZ)Gaussian basis sets of Dunning and coworkers [82ndash84] Thisbasis set is widely used in the literature thus simplifying thecomparison between different works The small componentsbasis set for the 4-component relativistic calculations hasbeen generated using restricted kinetic balance imposed inthe canonical orthogonalization step [80] All basis sets areused in uncontracted form Test calculations with aug-cc-pVQZ basis sets indicate that the reported structures can beconsidered converged with respect to the chosen basis setssee Section 4 The potential curves are generated with a bout175 point densely chosen equidistant with of step length of005 au in the significant part of the potential curves 400ndash1000 au The asymptotic point is taken at 400 au the valueof this point is used to get the values (De(Ri)) at the point i

In this section we discuss our computational result based onour calculations with the linear response adiabatic TDDFTmodule in Dirac-Package Our main concern will be (besidethe correctness of our computational result) to compare the

behavior of different density functional approximations (andin comparison to other methods) to draw conclusions onthe performance the quality and the validity of the differentfunctional approximations also in regard to applications tosimilar systems and possibly enlighten improvements of theDFT approximations in future works The comparison withthe literature values is accompanying our discussion whereworks with different computational methods are availableand with experimental values as far as available to judge thequality of our result

41 Ground State As already mentioned the ground-statebond of Zn2 dimer is a mixture of 34 Van der Waals and14 covalent interactions [85] and the DFT can hardly dealwith it as seen in Table 3 where the spectroscopic constantsof the ground state are given for different density functionalapproximations We note that the effect of the basis setsize typically by DFT is very small clearly seen in Table3 from PBE values calculated with aug-cc-pVTZ and aug-cc-pVQZ basis set In Table 3 one sees that a comparableresult is obtained by MP2 and srLDAMP2 as expected [43]Similar to the rare-gas dimers [43] the range-separated DFTimproves the DFT result (here LDA) for Zn2 and suitablycure the lack of correct long-range behavior known by pureDFT approximations because the long-range part of theexchange (and the long-range correlation in srLDAMP2)is treated by a wave function method (MP2) However acrucial point is to determine a suitable value of the rage-separation parameter Generally a suitable range for thisparameter is 02ndash05 au for details and indepth discussionsee [43] and the references therein DFT approximationsand CAMB3LYP as well as srLDAMP2 do not yield asatisfactory result Looking at the LDA we see that thecorrection of the LDA by srLDA-MP2 is large howeverthe improvement gives no advantage over the MP2 as theyhave similar computational coast Dramatically behave thelong-range corrected PBE0 and the hybrid functionals BLYPand B3LYP (contain a fixed fraction of exact HF-exchangeonly) they yield a dissociative ground state BP86 is the onlyfunctional with accurate dissociation energy value but its Reand ωe are not helpful Although CAMB3LYP gives the bestRe value comparison to experiment this is not sufficient asthe bond energy and vibrational frequency are not helpful Itis worthwhile to mention at this point that CAMB3LYP givesthe correct asymptotic behavior for the excited states seeFigure 2 in contrast to pure (LDA PBE BPW91 BP86 )long-range corrected (PBE0GARC-PBE0) or hybrid (BLYPB3LYP) DFTs as seen in Figures 2 and 3 Whether this meansthat CAMB3LYP potential curves has a correct shape (in allregions) is difficult to say at the moment The shape of thepotential curve is an important feature for the DFT accuracyas noted by Gruning et al [38]

42 Excited States The excited states shown in the pw aregiven in Table 2 where n = 4 for Zn atom The results aregiven in the Tables 5ndash8 We first discuss the lowest 8 statesgiven in the Tables 5ndash8 then we proceed to discuss the higherstates given in Table 8

Table 3 Ground-state 1Σ+g of Zn2 dimer

Re (A) ωe (cmminus1) De (eV)

exp1 257 0034

exp2 419 259 0035

HF-MP2Q 3611 29 0049

srLDAMP2Q 3445 31 00459

PBEQ 3157 48 0678

PBE 3156 49 0683

PBE0 diss diss diss

BPW91 3225 41 00154

BP86 3181 46 0036

BLYP diss diss diss

B3LYP diss diss diss

GRAC-PBE0 3338 400 0045

CAMB3LYP 4219 11 0001

LDA 2846 85 0225a 3959 22 0024b 396 225 0030c1 403 204 00205c2 403 204 00205

pw using aug-cc-pVTZ basis set and SFH Qaug-cc-pVQZ basis set for PBEHF-MP2 and srLDAMP2 (NR with parameter μ = 05) see text 1[86]2[85] a[12] using CCSD(T) in pseudopotential b[17] using NR-CCSD(T)c1[13] CCSD(T) with 4-comp DCH c2[13] CCSD(T) with SFH

Table 4 Comparison between SFH (NR state assignment) and4-component DCH of the spectroscopic constant Above Re (A)middle ωe (cmminus1) and below De (cmminus1) calculated with PBEfunctional and aug-cc-pVTZ basis set For 4-component statesassignment gerade ungerade follow the symmetry of state in thefirst line

3Πg3Σ+

SFH 2347 2534 4795 479

4-c 0minusg 0minusu 2345 mdash 4874 mdash

4-c 0+g 0+

u 2345 mdash 4480 mdash

4-c 0u 0g mdash 2534 mdash 4553

4-c 1g(1u) 2347 2534 4625 4574

4-c 2g 2u 2349 mdash 4945 mdash

SFH 219 172 7 27

4-c 0+g 0+

4-c 1g(1u) 219 172 13 34

SFH 13097 10870 52 405

4-c 0+g 0+

4-c 1g(1u) 12906 10680 235 550

At first we compare for PBE functional a 4-componentand spin-free result for the four lowest states calculated in

Table 5 Bond lengths Re (A) of the lowest states corresponding to the lowest two asymptotes

Method 3Πg3Σ+

1Πg1Σ+

PQ 2345 2532 4254 4765 2350 2596 4735 2573

PT 2347 2534 4795 479 2351 2602 4744 2592

W91T 2343 2517 diss 4546 2347 2621 dis 5158

P0T 2358 2517 5046 4517 2351 2631 2715 2594

GP0T 2356 2522 diss 5806 2345 2780 2929 4755

CB3LT 2343 2489 diss diss 2327 2613 2637 2572

B3LT 2371 2566 diss 5525 2366 2655 2807 2624

BLT 2371 2587 diss 4882 2376 2648 diss 2639

B86T 2337 2534 diss 4583 2341 2611 4647 5370

LDAT 2265 2454 2764 4364 2267 2485 2702 5414

[17]a 233 248 399 diss 230 264 240 274

[17]b 235 250 411 diss 233 269 242 292

[87]c 241 270 diss diss 233 322 240 305

[19]d 238 259 436 diss 238 264 265f 265f

[88]d 253 274 diss mdash 251 297 264 307

[89]d 256 270 diss diss 248 292 264 mdash

[90]e 2372 253

exp mdash mdash 449g mdash mdash 30g mdash mdashT

Present work calculated with aug-cc-pVTZ and Qwith aug-cc-pVQZ basis set P W91 P0 GP0 B86 BL B3L and CB3L denote PBE BPW91 PBE0 GRAC-PBE0 BP86 BLYP B3LYP and CAMB3LYP respectively aWith DK-CASPT2 bWith DK-MRACPF cWith CI dWith MRCI eWith CCSD(T) f Value are cagFrom [85] for 3Πu [91] gives the value 330

Table 6 Vibrational frequencies ωe (cmminus1) of the lowest states corresponding to the lowest two asymptotes

Method 3Πg3Σ+

1Πg1Σ+

PQ 220 173 9 28 219 136 12 142

PT 219 172 7 27 219 135 13 135

W91T 218 177 diss 33 220 129 diss 21

P0T 215 182 9 11 223 135 116 146

GP0T 216 181 diss 11 226 106 78 38

CB3LT 220 189 diss diss 232 139 137 150

B3LT 211 167 diss 13 215 126 90 139

BLT 210 157 diss 27 207 122 diss 115

B86T 222 172 diss 37 222 131 14 29

LDAT 247 189 85 45 247 160 89 26

[17]a 231 200 23 diss 250 131 211 58

[17]b 220 208 32 diss 244 121 205 104

[87]c 211 169 diss diss 212 77 175 112

[88]d 192 175 mdash diss 210 134 178 mdash

[89]d 175 150 diss diss 202 107 166 104

exp 223plusmn 5e 161plusmn 5f 203plusmn 02g mdash mdash 122plusmn 10h 148plusmn 6i mdash

For the acronyms see Table 5 TQas in Table 5 aWith DK-CASPT2 bWith DK-MRACPF cWith CI dWith MRCI eFrom [92] f From [93] gFrom [85] hFrom[94] iFrom [95]

aug-cc-pVTZ basis set and demonstrate that SFH describesaccurately the main relevant contributions of the relativisticeffects As seen in Table 4 the difference between SFH and4-components DCH is rather small To see the differenceand the splitting in the 4 component precisely De is givenin cmminus1 The splitting is very small or negligible clearly seenin Figure 1 where we compare visually the 8 lowest states ofPBE functional using SFH and the corresponding 16 lowest

excited states using 4-component DCH We note that theCCSD(T) result of [13] for the ground state (see Table 3)using SFH and 4-components DCH confirms our result

In Figure 2 we show the 20 lowest excited states cor-responding to the 6 asymptotes given in Table 2 for theCAMB3LYP and B3LYP functionals The overall behaviorin Figure 2 for CAMB3LYP is satisfactory it shows a betterbehavior for all states and the states follow (at least)

Table 7 Dissociation energies (eV) of the lowest states corresponding to the lowest two asymptotes

Method 3Πg3Σ+

1Πg1Σ+

PQ 1626 1347 00065 0050 1703 0579 00180 0112lowast

PT 1624 1348 00065 0050 1698 0572 00175 008lowast

W91T 1423 123 diss 0031 1654 0541 diss 0027

P0T 1481 1332 00034 00031 2387 1247 0413 010

GP0T 143 1316 diss 00148 2385 1111 0270 0279

CB3LT 1436 1281 diss diss 2298 1126 0099 0292lowast

B3LT 145 1189 diss 00033 2226 1125 0393 0148lowast

BLT 1514 1181 diss 0468 1426 0361 diss 0542

B86T 1593 1312 diss 00673 1688 0546 0008 0058

LDAT 2119 1704 1456 1902 2089 0798 0145 0788

[17]a 1502 1225 0026 diss 2713 1189 0734 060

[17]b 1457 1204 0110 diss 2694 1292 0718 0204

[87]c 091 090 diss diss 235 071 mdash mdash

[19]d 121 095 0016 diss 226 112 063 032

[89]d 105 087 diss diss 242 106 083 044

[90]e 141 121 mdash mdash mdash mdash mdash mdash

exp mdash mdash 0027f mdash mdash 1117g mdash mdash

For the acronyms see Table 5 TQAs in Table 5 lowastSee text aWith DK-CASPT2 bWith DK-MRACPF cWith CI d With MRCI eWith CCSD(T) f From [96]gFrom [94] (1117plusmn 0025) whereas [91] gives the value 130

Table 8 Higher states corresponding to higher asymptotes see Table 2 and text

StateRe (A) ωe (cmminus1) De (eV)

CB3L P0 GP0 B3L W91 B86 CB3L P0 GP0 B3L W91 B86 CB3L P0 GP0 B3L W91 B863Σ+

u 2527 2546 2711 2578 2531 2532 168 164 115 150 163 160 0914 0938 0174 0636lowast 0555 0644lowast

3Σ+g 2737 2769 5772 2802 271 2714 185 196 23 168 193 186 0533 0728 0596 0421 0118 0094

1Σ+u 260 2630 2787 2679 2622 2605 149 142 92 120 134 140 0839 0677 0231 0583 0513 0539

1Σ+g 3444 3388 8434 3449 3256 321 174 146 19 118 131 139 0339 0333 0383 0097 0152 0153

3Πu 2919 3080 3162 3352 3323 3451 99 82 72 59 51 45 1416 095 090 0646lowast 0039 0040lowast

3Πg 2487 2504 4748 2524 2491 2485 178 174 41 163 171 172 1140 0434 0635 0213lowast 0143 0203Σ+

u 2519 2532 diss 2551 2546 2506 172 171 diss 158 166 164 0905 0270 diss 0515 0482 04803Σ+

g 2569 2583 diss 2603 2513 2563 153 150 diss 145 140 150 0247 0158lowast diss 0163lowast 0150 01571Πu 3650 5750 6209 9026 diss diss 123 14 22 12 diss diss 150 0483 0486 0274 diss diss1Πg 2459 2482 6317 2495 2472 2465 190 184 26 174 180 182 1417 0344 0482 046lowast 043 03931Σ+

u 2534 2555 diss 2585 2537 2533 169 167 diss 155 162 159 1125 0302lowast diss 050lowast 0561 05601Σ+

g 2704 2682 diss 4237 2616 2583 281 288 diss 296 244 210 0517lowast 0298 diss 0218lowast 0146 0165

All values with SFH and aug-cc-pVTZ basis set For the acronyms see Table 5 lowastSee text

qualitatively to the correct asymptotes In contrast to theB3LYP as seen in Figure 2(b) where similar result is obtainedfor all other functionals used in this work These functionalsshow an incorrect asymptotic limit and only for the lowest8 states give the correct (two) asymptotes whereas most ofthe higher states follow to a wrong asymptotic limit This issomehow unexpected since B3LYP includes a (fixed) fractionof exact exchange

In Figure 3 a second example is presented for PBE0 andGARC-PBE0 GARC-PBE0 is supposed to give a better resultthan PBE0 but for Zn2 dimer it does not show a correctdescription for the higher excited states Indeed it is wellknown that pure DFT has incorrect long-range behavior

which is the key point behind the range-separated DFT Itis clearly from this result that the separation of the two-electron interaction in short- and long-range parts as donein range-separated DFT like CAMB3LYP offers an advantageby treating the long-range part with a wave functionmethod incorporating a suitable parametric amount of exactexchange That only CAMB3LYP shows a better or a correctlong-range behavior does not mean generally that a range-separated functional describes the excited states better in theshort-range (or mid-range) region however its accuracy issatisfactory even it fails for the ground state (see Table 3)rather due to the lack of long-range correlation (in HF cor-relation is not present) important for dispersion interaction

PBE ground and lowest 8 excited stateswith sipn-free Hamiltonian

(8) 1sum+g

(7) 1produ

(6) 1sum+u

(5) 1prodg

(4) 3sum+g

(3) 3produ

(2) 3sum+u

(1) 3prodg

(0) 1sum+g

2 25 3 35 4 545

minus35917

minus35918

minus35919

minus3592

minus35921

Bond length (A)

PBE 4-component first 16 excited states correspondto the lowest 8 spin-free excited states

minus3592

minus35921

2 25 3 35 4 545

(8) 0+g

(6) 0+u

(2) 0+u

(7) 1u

(5) 1g

(4) 1g

(1) 1g

(1) 2g

(4) 0+g

(3) 0+u (1) 0+

(3) 2u

(3) 1u

(2) 1u

(3) 0minusu(1) 0minusg

Bond length (A)

Figure 1 (a) Zn2 PBE functional with SFH (left) ground state (lowest curve) and 8 lowest excited state (corresponding to the two asymptotes(4s2S1 + 4s4pP1) lower ones and (4s2S1 + 4s4pPv) upper ones And (b) accordingly the 16 excited states with the same asymptotes using4-component DCH Numbering in brackets shows the correspondence between states of (a) and (b)

CAMB3LYP lowest 20 excited statesspin-free Hamiltonian

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

minus359254

minus359258

minus359262

minus359266

minus35927

minus35925minus359252

minus359256

minus35926

minus359264

minus359268

minus359272

1sum+g

1sum+u

1prodg

1produ

3sum+g

3sum+u

3prodg

3produ

1sum+g

1sum+u

3sum+g

3sum+u

1sum+g

1sum+u

3sum+g

3produ

1prodg

1produ

3sum+u

3prodg

Bond length (A)

B3LYP lowest 20 excited states spin-free Hamiltonian

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

minus359218

minus359222

minus359226

minus35923

minus359234

minus359216

minus35922

minus359224

minus359228

minus359232

minus359236

1sum+g

1sum+u

1prodg

1produ

3sum+g

3sum+u

3prodg

3produ

1sum+g

1sum+u

3sum+g

3sum+u

1sum+g

1sum+u

3sum+g

3produ

1prodg

1produ

3sum+u

3prodg

Bond length (A)

Figure 2 Zn2 CAMB3LYP (a) and B3LYP (b) the 20 lowest states with SFH corresponding to the asymptotes (from below) 4s2 1S + 4s4p 3P4s2 1S + 4s4p 1P 4s2 1S + 4s5s 3S 4s2 1S + 4s5s 1S 4s2 1S + 4s5p 3P and 4s2 1S + 4s5p 1P respectively Note some of the upper curves of B3LYPshow incorrect asymptotes

PBE0 lowest 20 excited states spin-free Hamiltonian

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

minus359176

minus35918

minus359184

minus359188

minus359192

minus359196

minus359178

minus359182

minus359186

minus35919

minus359194

minus359198

1sum+g

1sum+u

1prodg

1produ

3sum+g

3sum+u

3prodg

3produ

1sum+g

1sum+u

3sum+g

3sum+u

1sum+g

1sum+u

3sum+g

3produ

1prodg

1produ

3sum+u

3prodg

Bond length (A)

GRAC-PBE0 lowest 20 excited statesspin-free Hamiltonian

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

minus359176

minus35918

minus359184

minus359188

minus359192

minus359196

minus359174

minus359178

minus359182

minus359186

minus35919

minus359194

minus359198

1sum+g

1sum+u

1prodg

1produ

3sum+g

3sum+u

3prodg

3produ

1sum+g

1sum+u

3sum+g

3sum+u

1sum+g

1sum+u

3sum+g

3produ

1prodg

1produ

3sum+u

3prodg

Bond length (A)

Figure 3 Zn2 PBE0 (a) and GRAC-PBE0 (b) the 20 lowest states with SFH corresponding to the same asymptotes as in Figure 2 Note thatsome of the upper curves show an incorrect asymptotes compare CAMB3LYP Figure 2 and see text

Obviously a crucial point in calculating the excited statesin TDDFT is that the most of the DFT approximations aresemilocal the long-range interaction is incorrectly describedconsequently a disturbed potential curves is obtained espe-cially near the avoiding crossing point where the disturbedcurves show enhanced effects This can be clearly seen forthe 1Σ+

g 3Σ+g and 1Π+

u in Figure 4 For CAMB3LYP we seeevery two states of the same symmetry push each other awayand later both follow to the correct limit For PBE0 as anexample the avoiding crossing is clear for 1Σ+

g and 3Σ+g

states but not for 1Π+u most likely because it is disturbed

by the incorrect long-range behavior Similar behavior toPBE0 was found in all other DFT approximations used inthis work that is an incorrect long-range behavior with (orleading to) an incorrect asymptotic limit (and a disturbedavoiding crossing) is responsible for incorrect description ofthe higher excited states We will discuss the accuracies indetail in the next sections

421 Lowest 8 Excited States In Tables 5ndash7 we give theevaluated spectroscopic constants for the lowest 8 excitedstates of Zn2 using TDDFT SFH and aug-cc-pVTZ basisset The lowest 8 excited states 3Πg 3Πu 3Σ+

g 3Σ+u and

1Πg 1Πu 1Σ+g 1Σ+

u are corresponding to the Atom((4s2) 1S)+ Atom((4s4p) 3P) and Atom((4s2) 1S) + Atom((4s4p) 1P)respectively

First we look at the PBE values using aug-cc-pVTZ basisset and aug-cc-pVQZ basis set As we see from Tables 3ndash5 thebasis effect is small and only about 2 lowast 10minus3 A for Re about1 unit for ωe and between 2ndash6 meV in De Following this

we conclude that the SFH (see Table 4) with aug-cc-pVTZbasis set enable us to calculate the excited states of zinc dimeraccurately Our result is sufficiently accurate to compare withexperimental values wave function methods and comparethe behavior of different functional approximations witheach other for this dimer

(a) The Lowest States 3Πg 3Πu 3Σ+g 3Σ+

u Looking atthe Tables 5ndash7 we see immediately that the best result isobtained for these states For the lowest two state 3Πg 3Σ+

u all functionals give excellent agreement with wavefunction results giving in the literature for example [17]or the experimental value of ωe although the agreement forthe first excited state 3Πg is more pronounced RecentlyDeterman et al [90] have published accurate result forthese two states using CCSD(T) and some density functionalapproximations the excellent agreement with our valuesconfirms our result This is not surprising since these statesare well bound and largely covalent in contrast to the groundstate moreover the most known DFT approximations aremore or less capable to describe (strong) covalent bondingdue to its largely localized character in the bond region Itis also noticeable that all DFTs show for the eight loweststates asymptotically a correct behavior and the correct (two)asymptote see Figures 2 and 3 For the lowest two states3Πg 3Σ+

u only LDA strongly underestimates the dissociationenergy and gives short bond lengths and large ωersquos PBEgives larger bond energy for both states likewise BP86 forthe first one BLYP and PBE0 give smaller values for ωeFor Re all these approximations give a similar result For the

2 25 3 35 4 545

CAMB3LYP spin-free Hamiltonian avoiding crossingbetween states of the same symmetry

minus35924

minus359244

minus359548

minus359252

minus359256

minus35926

minus359264

minus359268

1sum+g

3sum+g (4s5s)S3

3sum+g (4s4p)P3

(4s5p)P1

1produ(4s5p)P1

1produ(4s4p)P1

1sum+g (4s5s)S1

1sum+g (4s4p)P1

Bond length (A)

2 25 3 35 4 545

PBE0 spin-free Hamiltonian avoiding crossing betweenstates of the same symmetry

minus35916

minus359164

minus359168

minus359172

minus359176

minus35918

minus359184

minus359188

1sum+g (4s + 5p1)

3sum+g(4s + 5s)S3

3sum+g (4s4p)P3

1produ(4s5p)P1

1produ(4s4p)P1

1sum+g (4s5s)S1

1sum+g (4s4p)P1

Bond length (A)

Figure 4 Zn2 spinfree Hamiltonian avoiding crossing CAMB3LYP (a) and PBE0 (b) between the two 1Σ+g corresponding to the asymptotes

(4s2S1 + 4s4p 1P) and (4s2 1S + 4s5s 1S) the two 1πu corresponding to the asymptotes (4s2S1 + 4s4p 1P) and (4s2 S1 + 4s5p 1P) the two 3Σ+g

states corresponding to the asymptotes (4s2 S1 + 4s4p 3P) and (4s2 S1 + 4s5s 3S) The highest 1Σ+g is corresponding to the asymptotes (4s2 S1 +

4s5p 1P) see text

next lowest two states 3Πu 3Σ+g the situation is somehow

complicated For 3Πu the experimental value shows a weakbound state whereas wave function methods show differentresults likewise in the DFT PBE and PBE0 describe it as aweak bound state but apart from LDA all other DFTs givea dissociative state Whereas for the 3Σ+

u only CAMB3LYPshows a dissociative state in an agreement with the wavefunction methods This is a first hint that CAMB3LYP givesa better long-range behavior and correct asymptotic limitfor higher states than the other DFTs shown in the presentwork This can be attributed to the fact that for high-quality response properties it is of primary importance forthe potential curve to be accurate in the shape rather thanthe condition to be met of being a functional derivative of agiven density functional for the exchange-correlation energy[38] For higher states both the long-range behavior andthe asymptotic limit in pure DFTs are incorrect and thusthe shape of potential curves BLYP gives De asymp 047eV for3Σ+

g which somehow large comparing to other functionalThe state 3Σ+

g (Atom(4s+ 4s) 1S + Atom(4s+ 4p1) 3P) shows

a hump around 25 A clearly seen in Figure 4 due to anavoiding crossing with the higher state 3Σ+

g (Atom(4s+4s) 1S

+ Atom(4s + 5p1) 3P) the later is well bound (see Table 8)and shows a small hump around 22 A (hardly seen inFigure 4) presumably due to an avoiding crossing with amore higher state of the same symmetry

(b) The States 1Πg 1Πu 1Σ+g 1Σ+

u From Tables 5ndash7 we againsee a good agreement especially for 1Πg and 1Σ+

u between

our result and the results of the wave function methodswhere the agreement is less pronounced than the lowest twostates For 1Πg and 1Σ+

u bond lengths apart from LDAall functionals give comparable results For the vibrationalfrequencies BLYP and B3LYP give smaller values for 1Σ+

this is in excellent agreement with the experimental value of[94] or the value of [17] CAMB3LYP gives the largest valueof ωe For the dissociation energy De B3LYP CAMB3LYPPBE0 and GRAC-PBE0 give reasonable values with a goodagreement with the experiment for 1Σ+

u This remarkableresult could be a hint that these three functionals have acorrect mid-range behavior From the agreement with theexperiment and the wave function values one concludes thatthe values of 1Πg of B3LYP CAMB3LYP PBE0 and GRAC-PBE0 should be close to the experiment Next we look atthe two states 1Πu 1Σ+

g as mentioned above in Figure 4these two states have avoided crossing with higher lying statesof the same symmetry From the tables we now see a lessagreement with the wave function method and the lackof experimental values makes it more difficult to judge theresult If we take the values of [17] as a reference we see thatreasonable DFTs values show larger bond lengths smallervibrational frequencies for 1Πu and for 1Σ+

g vice versafor the most of the functionals For 1Πu the dissociationenergies are smaller than the reference value For 1Σ+

g theobtained bond energy values for some functionals denotingthe depth of the minimum (marked with ldquolowastrdquo) relating to theshallowest point after the minimum otherwise the incorrectasymptotic point will show a dissociative state which ofcourse an artifact of the (quantitatively) incorrect tail of the

potential curve We have seen in Figure 2 that CAMB3LYPhas asymptotically a correct behavior specially for the higherstates however it is quantitatively questionable and for somestates seems to be inaccurate In such cases the spectroscopicconstants are calculated relative to the shallowest point afterthe minimum and not to the asymptotic point This yieldsapproximately the same Re and ωe but the obtained valueDe will be definitely shallower than or approximately equalto a value De relating to the ldquocorrectrdquo asymptotic point Wenote that all values marked with an ldquolowastrdquo in Tables 7-8 areobtained this way For the 1Σ+

g state we see from Table 7that CAMB3LYP has a good agreement with [19] likewiseB3LYP with [17] whereas BLYP shows an agreement withDK-CASPT2 value of [17] but to conclude we see that theresult(s) of 1Σ+

g are widely distributed furthermore the lackof any experimental value makes the situation more difficult

422 Higher Excited States To deal with more higher excitedstates is difficult because of the above-mentioned reasonsAvailable approximations do not describe the long-rangebehavior correctly andor fail to offer the correct asymptoticlimit or predict it accurately [97] We will discuss thehigher molecular states given in Table 6 corresponding tothe last four asymptotes (3ndash6) in Table 2 The result isgiven for the functionals BPW91 and BP86 (pure) B3LYP(hybrid) CAMB3LYP (range-separated) PBE0 (long-rangecorrected) and its gradient corrected one GRAC-PBE0GRAC is an interpolation scheme it is an asymptoticcorrection and supposed to be able to deal with higherexcited states [37 38] The pw shows that the best result isobtained for CAMB3LYP and a comparable result is obtainedfor PBE0 Indeed strictly only CAMB3LYP was able to dealwith higher excited states it shows (at least qualitatively) thecorrect asymptotic as can be clearly seen in Figure 2 Otherfunctionals do not show a correct asymptotic behavior asexpected [37] including the ones for which no data shownin Table 8 B3LYP is given in Figure 2 as an example itmixes the asymptotic for higher states with lower states Ourconclusion based on analyzing the data of all functionalsand comparing them with each other It is clear that lackingto the correct long-range behavior is primarily the originof the problem CAMB3LYP is able to cure this althoughnot accurately the question is why other corrections likeGRAC does not have the expected improvement At oneside important is the nonlocal part of exact exchange whichimproves the situation considerably when the two-electroninteraction is separated in short- and long-range part suchas in CAMB3LYP and we notice that there is no long-range correlation present in CAMB3LYP because HF offersonly (nonlocal) exchange Another point is the wrong long-range behavior of the response function [72 77] caused bythe incorrect long-range behavior of the density functionalapproximation is more crucial than it might be believed Thisis supported by the fact that the spatial nonlocality of fxc

is strongly frequency-dependent [98] in [98] Tokatly andPankratov argued that not only any static approximation butalso any LDA-based dynamic approximation (including anygradient corrections) for fxc cannot provide consistent result

To my best knowledge there is no calculated or experimentalresult reported for any of the higher states given in Table8 this makes the situation more difficult to analyze andbe clarified In Table 8 surprisingly we see that PBE0 givesa better result for higher excited states than its asymptoticcorrected one GRAC-PBE0 and better than B3LYP BP86or BPW91 Furthermore it gives for all states a comparableresult to CAMB3LYP for Re and ωe This supports our viewand stress the importance of the long-range correction It isa clear evident that PBE0 has a correct shape in inner partof the potential curve and only its asymptotic part (tail ofthe potential curve) is incorrect unfortunately the appliedcorrection of GRAC is not good As seen in Table 8 ournext four states 3Σ+

u 3Σ+g and 1Σ+

u 1Σ+g corresponding

to Atom((4s2) 1S + Atom((4s5s) 3S) and Atom((4s2) 1S +Atom((4s5s) 1S) have more or less a similar result for allfunctionals only GRAC-PBE0 shows unexplainable resultsince it is supposed to show asymptotically a better behaviorWe think that the CAMB3LYP result is the most correctone although it might be not satisfactory accurate It isworthwhile to mention that states with avoiding crossing geta second shallow minimum after the avoiding crossing atlarge internuclear distances this is not reported and onlythe first minimum is presented Next we look to the states3πu 3πg 3Σ+

u 3Σ+g corresponding to the Atom((4s2) 1S +

Atom((4s5p) 3P) Here we see that the result is distributedBPW91 BP86 and B3LYP show similar results whereasGRAC-PBE0 differs considerably from all approximationsgiven in Table 8 PBE0 result is close to CAMB3LYP whenlooking to Re and ωe but its De values are different clearlydue to its incorrect asymptotic limit The last states treatedin this work 1πu 1πg 1Σ+

u 1Σ+g are corresponding to the

Atom((4s2) 1S + Atom((4s5p) 1P) The results of 1πu arepuzzling and presumably only the values of CAMB3LYP arereasonable whereas for 1πg all functional apart from GRAC-PBE0 give comparable values for ωe and Re which could be ahint that these values are reasonable 1Σ+

u and 1Σ+g follow

the general trend that PBE0 result is close to CAMB3LYPBPW91 BP86 and B3LYP show a similar result GRAC-PBE0shows unexplainable result

The general conclusion of this section is that CAMB3LYPgives the best result due to its better treatment of thelong-range part of the two-electron interaction and itsasymptotically better behavior (tail of the potential curve)apparently due to including a suitable amount of exactexchange PBE0 gives a comparable result the main problemhere is the tail of the potential curve BPW91 BP86 andB3LYP are less satisfactory but still show acceptable resultwhereas (most likely) the result of GRAC-PBE0 is not useful

5 Conclusion

In the present work we have studied the ground as well the20 lowest exited states of the zinc dimer in the frameworkof DFT and TDDFT using well-known and newly developedfunctional approximations We performed the calculationswith Dirac-Package using relativistic 4-component DCH andSFH First we showed that SFH is capable to achieve the same

accuracy as 4-components DCH and can describe quanti-tatively the main relevant contributions of the relativisticeffects In analyzing the results obtained from differentfunctional approximations comparing them with each otherwith literature and experimental values as far as availablewe drew some conclusions The results show that the linearresponse in the adiabatic approximation with the knownDFT approximations give good performance for the 8 lowestexcited states of Zn2 For higher excited states we foundsomehow as expected that most of DFT approximationsused in the pw did not show a correct long-range behaviorand the correct asymptotic limit to perform a fair accuracyfor these states where we have to stress that the lack ofexperimental or other theoretical results makes a judgmentdifficult Nevertheless we can say that the best result isobtained with the range-separated CAMB3LYP functionalwhich was the only one able (at least qualitatively) to showthe correct asymptotic behavior This can be led back to theseparation of the two-electron interaction in a suitable man-ner short- and long-range part where the former is handledby the DFT and the later by HF Showing that including asuitable (parametric) amount of the exact exchange improvesthe result considerably Moreover the (long-range corrected)PBE0 was able to give a comparable result to CAMB3LYPfor the higher states although it fails to give the correctasymptotes The comparison between CAMB3LY and otherfunctionals allows us to conclude that for higher statesthe lack of a correct long-range and a suitable amount ofexact exchange is responsible for incorrect result rather thanthe linear response approximation and the adiabatic limitIn addition it causes a wrong long-range behavior of theresponse function a crucial point for the long-range behaviorin TDDFT In future works we will be concerned with theheavier members of the group 12 Cd2 and Hg2 whererelativistic effects are expected to be more important than inzinc dimer Furthermore the superheavy dimer Cn2 is underconsideration where the bonding character of its ground andexcited states of academic interest due to the large relativisticeffects and its influence on the atomic levels and hence on themolecular ground and excited states of the dimer

Acknowledgments

The author gratefully acknowledges fruitful discussionswith Dr Trond Saue Laboratoire de Chimie et PhysiqueQuantique Universite de Toulouse (France) and the kindlysupport from him Dr Radovan Bast Tromsoslash University(Norway) is acknowledged for his kindly support and thekindly support from the Laboratoire de Chimie QuantiqueCNRS et Universite de Strasbourg

References

[1] K G Caulton and L G Hubert-Pfalzgraf ldquoSynthesis struc-tural principles and reactivity of heterometallic alkoxidesrdquoChemical Reviews vol 90 no 6 pp 969ndash995 1990

[2] M C Heitz K Finger and C Daniel ldquoPhotochemistry oforganometallics quantum chemistry and photodissociationdynamicsrdquo Coordination Chemistry Reviews vol 159 pp 171ndash193 1997

[3] L Huebner A Kornienko T J Emge and J G BrennanldquoHeterometallic lanthanide group 12 metal iodidesrdquo InorganicChemistry vol 43 no 18 pp 5659ndash5664 2004

[4] R Kobayashia and R D Amos ldquoThe application ofCAM-B3LYP to the charge-transfer band problem of thezincbacteriochlorin-bacteriochlorin complexrdquo ChemicalPhysics Letters vol 420 no 1ndash3 pp 106ndash109 2006

[5] G Hua Y Zhang J Zhang X Cao W Xu and L ZhangldquoFabrication of ZnO nanowire arrays by cycle growth insurfactantless aqueous solution and their applications on dye-sensitized solar cellsrdquo Materials Letters vol 62 no 25 pp4109ndash4111 2008

[6] J H Lee Y W Chun M H Hon and I C Leu ldquoDensity-controlled growth and field emission property of aligned ZnOnanorod arraysrdquo Applied Physics A vol 97 no 2 pp 403ndash4408 2009

[7] T Yamase H Gerischer M Lubke and B Pettinger ldquoSpectralsensitization of ZnO-electrodes by methylene bluerdquo Berichteder Bunsengesellschaft fur physikalische Chemie vol 83 no 7pp 658ndash6663 1979

[8] D K Roe L Wenzhao and H Gerischer ldquoElectrochemicaldeposition of cadmium sulfide from DMSO solutionrdquo Journalof Electroanalytical Chemistry vol 136 no 2 pp 323ndash3371982

[9] M D Morse ldquoClusters of transition-metal atomsrdquo ChemicalReviews vol 86 no 6 pp 1049ndash11109 1986

[10] J Koperski ldquoStudy of diatomic van der Waals complexes insupersonic beamsrdquo Physics Reports vol 369 no 3 pp 177ndash1326 2002

[11] J Koperski ldquoGroup-12 vdW dimers in free-jet supersonicbeams the legacy of Eugeniusz Czuchaj continuesrdquo Euro-physics Letters vol 144 pp 107ndash114 2007

[12] M Yu and M Dolg ldquoCovalent contributions to bonding ingroup 12 dimers M2 (Mn = Zn Cd Hg)rdquo Chemical PhysicsLetters vol 273 no 5-6 pp 329ndash3336 1997

[13] L Bucinisky S Biskupic M Ilcin V Lukes and V LauringldquoOn relativistic effects in ground state potential curves ofZn2 Cd2 and Hg2 dimers A CCSD(T) studyrdquo Journal ofComputational Chemistry vol 30 no 1 pp 65ndash674 2009

[14] R Eichler N V Aksenov A V Belozerov et al ldquoChemicalcharacterization of element 112rdquo Nature vol 447 no 7140pp 72ndash75 2007

[15] N Gaston I Opahle H W Goggeler and P Schwerdtfeger ldquoIsEka-Mercury (element 112) a group 12 metal rdquo AngewandteChemie International Edition vol 46 pp 1663ndash11666 2007

[16] V Pershina J Anton and T Jacob ldquoTheoretical predictionsof adsorption behavior of elements 112 and 114 and theirhomologs Hg and Pbrdquo Journal of Chemical Physics vol 131no 8 Article ID 084713 8 pages 2009

[17] K Ellingsen T Saue C Puchan and O Groupen ldquoAn Abinitio study of the electronic spectrum of Zn2 including spin-orbit couplingrdquo Chemical Physics vol 311 no 1-2 pp 35ndash344 2005

[18] E Czuchaj F Rebentrost H Stoll and H Preuss ldquoAdiabaticpotential curves for the Cd2 dimerrdquo Chemical Physics Lettersvol 225 no 1ndash3 pp 233ndash239 1994

[19] E Czuchaj F Rebentrost H Stoll and H Preuss ldquoPotentialenergy curves for the Zn2 dimerrdquo Chemical Physics Letters vol255 no 1ndash3 pp 203ndash209 1996

[20] E Czuchaj F Rebentrost H Stoll and H Preuss ldquoCalculationof ground- and excited-state potential energy curves forthe Hg2 molecule in a pseudopotential approachrdquo ChemicalPhysics vol 214 no 2-3 pp 277ndash289 1997

[21] T Saue L Visscher H J Aa Jensen et al DIRAC a relativisticAb initio electronic structure program Release DIRAC102010 httpdiracchemvunl

[22] N C Pyper I Grant and R Gerber ldquoRelativistic effectson interactions between heavy atoms the Hg Hg potentialrdquoChemical Physics Letters vol 49- pp 479ndash483 1977

[23] M Seth P Schwerdtfeger and M Dolg ldquoThe chemistry of thesuperheavy elements I Pseudopotentials for 111 and 112 andrelativistic coupled cluster calculations for (112)H+ (112)F2and (112)F4rdquo Journal of Chemical Physics vol 106 no 9 pp3623ndash3632 1997

[24] J Antona B Fricke and P Schwerdtfeger ldquoNon-collinearand collinear four-component relativistic molecular densityfunctional calculationsrdquo Chemical Physics vol 311 no 1-2pp 97ndash103 2005

[25] L Belpassi L Storchi H M Quineyb and F Taran-telli ldquoRecent advances and perspectives in four-componentDirac-Kohn-Sham calculationsrdquo Physical Chemistry ChemicalPhysics vol 13 pp 12368ndash12394 2011

[26] R Bast A Heszligelmann P Sałek T Helgaker and T SaueldquoStatic and frequency-dependent dipole-dipole polarizabili-ties of all closed-shell atoms up to radium a four-componentrelativistic DFT studyrdquo ChemPhysChem vol 9 no 3 pp 445ndash453 2008

[27] R Bast H J A A Jensen and T Saue ldquoRelativistic adiabatictime-dependent density functional theory using hybrid func-tionals and noncollinear spin magnetizationrdquo InternationalJournal of Quantum Chemistry vol 109 no 10 pp 2091ndash2112 2009

[28] T Saue and H J A Jensen ldquoLinear response at the 4-component relativistic level application to the frequency-dependent dipole polarizabilities of the coinage metal dimersrdquoJournal of Chemical Physics vol 118 no 2 pp 533ndash515 2003

[29] J C Slater ldquoA simplification of the Hartree-Fock methodrdquoPhysical Review vol 81 no 3 pp 385ndash390 1951

[30] S J Vosko L Wilk and M Nusair ldquoAccurate spin-dependentelectron liquid correlation energies for local spin densitycalculations a critical analysisrdquo Canadian Journal of Physicsvol 58 no 8 pp 1200ndash11211 1980

[31] J P Perdew K Burke and M Ernzerhof ldquoGeneralizedgradient approximation made simplerdquo Physical Review Lettersvol 77 no 18 pp 3865ndash3868 1996

[32] A D Becke ldquoDensity-functional exchange-energy approxima-tion with correct asymptotic behaviorrdquo Physical Review A vol38 no 6 pp 3098ndash3100 1988

[33] J P Perdew ldquoDensity-functional approximation for the cor-relation energy of the inhomogeneous electron gasrdquo PhysicalReview B vol 33 no 12 pp 8822ndash8824 1986

[34] J P Perdew ldquoDensity-functional approximation for the cor-relation energy of the inhomogeneous electron gasrdquo PhysicalReview B vol 34 no 10 article 7406 1986

[35] J P Perdew and Y Wang ldquoAccurate and simple analyticrepresentation of the electron-gas correlation energyrdquo PhysicalReview B vol 45 no 23 pp 13244ndash13249 1992

[36] M Ernzerhof and G E Scuseria ldquoAssessment of the Perdew-Burke-Ernzerhof exchange-correlation functionalrdquo Journal ofChemical Physics vol 110 no 11 pp 5029ndash5036 1999

[37] R van Leeuwen and E J Baerends ldquoExchange-correlationpotential with correct asymptotic behaviorrdquo Physical ReviewA vol 49 no 4 pp 2421ndash2431 1994

[38] M Gruning O V Gritsenko S J A van Gisbergen andE J Baerends ldquoShape corrections to exchange-correlationpotentials by gradient-regulated seamless connection of modelpotentials for inner and outer regionrdquo Journal of ChemicalPhysics vol 114 no 2 pp 652ndash660 2001

[39] C Lee W Yang and R G Parr ldquoDevelopment of the Colle-Salvetti correlation-energy formula into a functional of theelectron densityrdquo Physical Review B vol 37 no 2 pp 785ndash789 1988

[40] A D Becke ldquoDensity-functional thermochemistry III Therole of exact exchangerdquo Journal of Chemical Physics vol 98no 7 article 5648 5 pages 1993

[41] P J Stephens F J Devlin C F Chabalowski and M JFrisch ldquoAb initio calculation of vibrational absorption andcircular dichroism spectra using density functional forcefieldsrdquo Journal of Physical Chemistry vol 98 no 45 pp11623ndash11627 1994

[42] T Yanai D P Tew and N C Handy ldquoA new hybrid exchange-correlation functional using the Coulomb-attenuatingmethod (CAM-B3LYP)rdquo Chemical Physics Letters vol 393no 1ndash3 pp 51ndash57 2004

[43] O Kullie and T Saue ldquoRange-separated density functionaltheory a 4-component relativistic study of the rare gas dimersHe2 Ne2 Ar2 Kr2 Xe2 Rn2 and Uuo2rdquo Chemical Physics vol395 pp 54ndash62 2012

[44] P Hohenberg and W Kohn ldquoInhomogeneous electron gasrdquoPhysical Review vol 136 no 3B pp B864ndashB871 1964

[45] W Kohn and L J Sham ldquoSelf-consistent equations includingexchange and correlation effectsrdquo Physical Review vol 140 no4 pp A1133ndashA1138 1965

[46] W Kohn ldquoNobel lecture electronic structure of mattermdashwavefunctions and density functionalsrdquo Reviews of Modern Physicsvol 71 no 5 pp A1133ndashA1266 1999

[47] W Koch and M C Holthausen A Chemistrsquos Guide to DensityFunctional Theory Willy-VCH New York NY USA 2001

[48] T Saue and T Helgaker ldquoFour-component relativistic Kohn-Sham theoryrdquo Journal of Computational Chemistry vol 23 no8 pp 814ndash823 2002

[49] O Kullie H Zhang and D Kolb ldquoRelativistic and non-relativistic local-density functional benchmark results andinvestigation on the dimers Cu2Ag2Au2Rg2rdquo ChemicalPhysics vol 351 no 1ndash3 pp 106ndash110 2008

[50] O Kullie E Engel and D Kolb ldquoAccurate local densityfunctional calculations with relativistic two-spinor minimaxand finite element method for the alkali dimersrdquo Journal ofPhysics B vol 42 no 9 Article ID 095102 2009

[51] P A M Dirac ldquoNote on exchange phenomena in theThomas atomrdquo Mathematical Proceedings of the CambridgePhilosophical Society vol 26 no 3 pp 376ndash385 1930

[52] J P Perdew S Kurth A Zupan and P Blaha ldquoAccuratedensity functional with correct formal properties a stepbeyond the generalized gradient approximationrdquo PhysicalReview Letters vol 82 no 12 pp 2544ndash2547 1999

[53] A Savin in Recent Developments of Modern Density FunctionalTheory J M Seminario Ed pp 327ndash357 Elsevier Amster-dam The Netherlands 1996

[54] E Goll H J Werner and H Stoll ldquoA short-range gradient-corrected density functional in long-range coupled-clustercalculations for rare gas dimersrdquo Physical Chemistry ChemicalPhysics vol 7 pp 3917ndash3923 2005

[55] I C Gerber and J G Angyan ldquoPotential curves for alkaline-earth dimers by density functional theory with long-rangecorrelation correctionsrdquo Chemical Physics Letters vol 416 no4ndash6 pp 370ndash375 2005

[56] R Baer E Livshits and U Salzner ldquoTuned range-separatedhybrids in density functional theoryrdquo Annual Review ofPhysical Chemistry vol 61 pp 85ndash109 2010

[57] K G Dyall ldquoAn exact separation of the spinminusfree andspinminusdependent terms of the Dirac-Coulomb-Breit Hamilto-nianrdquo Journal of Chemical Physics vol 100 no 3 article 211810 pages 1994

[58] L Cheng and J Gauss ldquoAnalytical evaluation of first-orderelectrical properties based on the spin-free Dirac-CoulombHamiltonianrdquo Journal of Chemical Physics vol 134 no 24Article ID 244112 11 pages 2011

[59] M A L Marques C A Urlich F Nogueira A RubioK Burke and E K Gross Eds Time-Dependent DensityFunctional Theory Lecture Notes in Physics Springer NewYork NY USA 2006

[60] E Runge and E K U Gross ldquoDensity-functional theory fortime-dependent systemsrdquo Physical Review Letters vol 52 no12 pp 997ndash1000 84

[61] E Gross and W Kohn ldquoTime-dependent density-functionaltheoryrdquo Advances in Quantum Chemistry vol 21 pp 255ndash2911990

[62] M E Casida in Recent Advances in Density FunctionalMethods D P Chong Ed p 155 World Scientific Singapore1995

[63] E Gross J Dobson and M Petersilka ldquoDensity functionaltheory of time-dependent phenomenardquo Topics in CurrentChemistry vol 181 pp 81ndash172 1996

[64] M Casida ldquoTime-dependent density functional responsetheory of molecular systems theory computational methodsand functionalsrdquo in Recent Developments and Applicationsof Modern Density Functional Theory J M Seminario Edchapter 11 p 391 Elsevier Amsterdam The Netherlands1996

[65] K Burke and E K U Gross in Density Functionals Theoryand Applications D Joubert Ed vol 500 of Springer LectureNotes in Physics p 116 Springer New York NY USA 1998

[66] R van Leeuwen ldquoKey concepts in time-dependent density-functional theoryrdquo International Journal of Modern Physics Bvol 15 no 14 pp 1969ndash2023 2001

[67] M A L Marques and E K U Gross ldquoTime dependent densityfunctional theoryrdquo in A Primer in Density Functional TheoryM M C Fiolhais and F Nogueira Eds p 144 Springer NewYork NY USA 2003

[68] H Appel E K Gross and K Burke ldquoExcitations in time-dependent density-functional theoryrdquo Physical Review Lettersvol 90 no 4 Article ID 043005 4 pages 2003

[69] M A L Marques and E K U Gross ldquoTime-dependent den-sity functional theoryrdquo Annual Review of Physical Chemistryvol 55 pp 427ndash455 2004

[70] K Burke J Werschnik and E Gross ldquoTime-dependentdensity functional theory past present and futurerdquo Journal ofChemical Physics vol 123 Article ID 062206 12 pages 2005

[71] P Elliott F Furche and K Burke in Reviews in ComputationalChemistry K B Lipkowitz and T R Cundari Eds pp 91ndash165 Wiley Hoboken NJ USA 2009

[72] S Botti A Schindlmayr R Del Sole and L Reining ldquoTime-dependent density-functional theory for extended systemsrdquoReports on Progress in Physics vol 70 no 3 pp 357ndash407 2007

[73] O V Gritsenko and E J Baerends ldquoDouble excitation effect innon-adiabatictime-dependent density functional theory withan analytic construction of the exchange-correlation kernelin the common energy denominator approximationrdquo PhysicalChemistry Chemical Physics vol 11 pp 4640ndash4646 2009

[74] T Ziegler M Seth M Krykunov J Autschbach and FWangc ldquoIs charge transfer transitions really too difficult forstandard density functionals or are they just a problem fortime-dependent density functional theory based on a linearresponse approachrdquo Journal of Molecular Structure vol 914no 1ndash3 pp 106ndash109 2009

[75] M E Casida ldquoTime-dependent density-functional theoryfor molecules and molecular solidsrdquo Journal of MolecularStructure vol 914 no 1ndash3 pp 3ndash18 2009

[76] M E Casida and M Huix-Rotllant ldquoProgress in time-dependent density-functional theoryrdquo Annual Review of Phys-ical Chemistry vol 63 pp 287ndash323 2012

[77] G Onida R Reininger and A Rubio ldquoElectronic excita-tions density-functional versus many-body Greenrsquos-functionapproachesrdquo Reviews of Modern Physics vol 74 no 2 pp 601ndash659 2002

[78] A Zangwill and P Soven ldquoResonant photoemission in bariumand ceriumrdquo Physical Review Letters vol 45 no 3 pp 204ndash207 1980

[79] M Ilias and T Saue ldquoAn infinite-order two-componentrelativistic Hamiltonian by a simple one-step transformationrdquoJournal of Chemical Physics vol 126 no 6 Article ID 0641029 pages 2007

[80] L Visscher and T Saue ldquoApproximate relativistic electronicstructure methods based on the quaternion modified Diracequationrdquo Journal of Chemical Physics vol 113 no 10 pp3996ndash4002 2000

[81] L Visscher and K G Dyall ldquoDirac-fock atomic electronicstructure calculations using different nuclear charge distribu-tionsrdquo Atomic Data and Nuclear Data Tables vol 67 no 2 pp207ndash224 1997

[82] T Dunning ldquoGaussian basis sets for use in correlatedmolecular calculations I The atoms boron through neon andhydrogenrdquo Journal of Chemical Physics vol 90 no 2 article1007 17 pages 1989

[83] D Woon and T Dunning ldquoGaussian basis sets for use incorrelated molecular calculations III The atoms aluminumthrough argonrdquo Journal of Chemical Physics vol 98 no 2article 1358 14 pages 1993

[84] A K Wilson D E Woon K A Peterson and T HDunning ldquoGaussian basis sets for use in correlated molecularcalculations IX The atoms gallium through kryptonrdquo Journalof Chemical Physics vol 110 no 16 pp 7667ndash7676 1999

[85] M A Czajkkowski and J Koperski ldquoThe Cd2 and Zn2 vander Waals dimers revisited Correction for some molecularpotential parametersrdquo Spectrochimica Acta vol 55 no 11 pp2221ndash2229 1999

[86] R D Van Zee S C Blankespoor and T Z Zweir ldquoDirectspectroscopic determination of the Hg2 bond length and ananalysis of the 2540 A bandrdquo Journal of Chemical Physics vol88 no 8 article 4650 5 pages 1988

[87] A Aguado J de la Vega and B Miguel ldquoAb initio configura-tion interactioncalculations of ground state and lower excitedstates of Zn2 using optimized Slater-typewavefunctionsrdquoJournal of the Chemical Society Faraday Transactions vol 93no 1 pp 29ndash32 1997

[88] H Tatewaki M Tomonari and T Nakamura ldquoThe excitedstates of Zn2 and Zn3 Inclusion of the correlation effectsrdquo TheJournal of Chemical Physics vol 82 no 12 pp 5608ndash56151984

[89] P J Hay T H Dunning and R C Raffenetti ldquoElectronicstates of Zn2 Ab initio calculations of a prototype for Hg2rdquoThe Journal of Chemical Physics vol 65 no 7 pp 2679ndash26891976

[90] J J Determan M A Omary and A K Wilson ldquoModeling thephotophysics of Zn and Cd monomers metallophilic dimersand covalent excimersrdquo Journal of Physical Chemistry A vol115 no 4 pp 374ndash382 2011

[91] C H Su P K Liao Y Huang S Liou and R F Brebick ldquoAstudy of the symmetric charge transfer reaction H+

2 +H2 usingthe high resolution photoionization and crossed ion-neutralbeam methodsrdquo Journal of Chemical Physics vol 81 no 12article 5672 20 pages 1984

[92] W Kedzierski J B Atkinson and L Krause ldquoLaser-inducedfluorescence from the 3Πu (4 3P 4 3P) state of Zn2rdquo ChemicalPhysics Letters vol 215 no 1ndash3 pp 185ndash187 1993

[93] W Kedzierski J B Atkinson and L Krause ldquoThesum

g+ (43P

43P) larr sumu

+ (43P 41S) vibronic spectrum of Zn2rdquo ChemicalPhysics Letters vol 222 no 1-2 pp 146ndash148 1994

[94] G Rodriguez and J G Eden ldquoBoundrarr free emission spectraand photoassociation of 114Cd2 and 64Zn2rdquo Journal of Chemi-cal Physics vol 95 no 8 article 5539 14 pages 1991

[95] W Kedzierski J B Atkinson and L Krause ldquoLaser-inducedfluorescence of the Zn2 excimerrdquo Optics Letters vol 14 no 12pp 607ndash608 1989

[96] M Czajkkowski R Bobkowski and L Krause Physical ReviewA vol 200 p 103 1990

[97] T Bally and G N Sastry ldquoIncorrect dissociation behavior ofradical ions in density functional calculationsrdquo The Journal ofPhysical Chemistry A vol 101 no 43 pp 7423ndash7925 1997

[98] I Tokatly and O Pankratov ldquoMany-body diagrammaticexpansion in a Kohn-Sham basis implications for time-dependent density functional theory of excited statesrdquo PhysicalReview Letters vol 86 no 10 pp 2087ndash2081 2001

Millimeter-Wave Rotational Spectra oftrans-Acrolein (Propenal) (CH2CHCOH) A DC DischargeProduct of Allyl Alcohol (CH2CHCH2OH) Vapor andDFT Calculation

A I Jaman and Rangana Bhattacharya

Experimental Condensed Matter Physics Division Saha Institute of Nuclear Physics Sector 1 Block AF BidhannagarKolkata 700 064 India

Correspondence should be addressed to A I Jaman aismailjamansahaacin

Academic Editor Nigel J Mason

Millimeter-wave rotational spectrum of trans-acrolein (propenal) (CH2CHCOH) produced by applying a DC glow dischargethrough a low-pressure (sim10ndash20 mTorr) flow of allyl alcohol (CH2CHCH2OH) vapor has been observed in the ground and severalexcited torsional states in the frequency region 600ndash990 GHz A least-square analysis of the measured and previously reportedrotational transition frequencies has produced a set of rotational and centrifugal distortion constants for the ground as well asexcited torsional states Detailed DFT calculations were also carried out with various functional and basis sets to evaluate thespectroscopic constants dipole moment and various structural parameters of the trans conformer of propenal for the groundstate and compared with their corresponding experimental values A linear variation of the inertia defect values with torsionalquantum number (v = 0 1 2 3) demonstrates that the equilibrium configuration of trans-propenal is planar

1 Introduction

The trans form of propenal (CH2CHCOH) also knownas trans-acrolein has been detected largely in absorptiontoward the star-forming region Sagittarius B2(N) by Holliset al [2] through the observation of rotational transitionsusing 100 m Green Bank Telescope (GBT) operating in thefrequency range from 180 GHz to 260 GHz Spectroscopicmeasurements in the microwave [1 3] infrared [4] andnear ultraviolet region [5 6] have confirmed that the trans-form is the most abundant and stable conformer of acroleinThe first microwave study of trans-acrolein in the J = 2larr13larr2 and 4larr3 a-type R-branch transitions was reported byWagner et al [3] Later on Cherniak and Costain [1] havemeasured both a- and b-type transitions for J = 2larr1 andJ = 3larr2 First spectroscopic evidence of the existence of

the less abundant cis-conformer of acrolein in the gas phasewas found from studies of the near ultraviolet spectrum[7 8] Later on cis-acrolein were detected in argon matrices[9 10] and in the gas-phase Raman spectrum [11] Thefirst microwave detection of the cis form of acrolein in thegas phase was reported by Blom and Bauder [12] Theyhave reported the ground state rotational quartic centrifugaldistortion constants as well as dipole moment values Blomet al [13] also reported the complete substitution structuresof both trans and cis conformers The dipole moment valuesof the trans and cis-form of acrolein have been found to beμ = 3117 plusmn 0004 D [13] and μ = 2552 plusmn 0 003 D [12]respectively Winnewisser [14] have extended the analysisof the ground state of the trans-form of acrolein to themillimeter-wave region up to 1800 GHz which has yieldeda set of ground state rotational and centrifugal distortion

Table 1 Microwave and millimeter wave rotational transition frequencies of trans-propenal (CH2CHCOH) in the ground and excitedtorsional states (in MHz)

Transitions Torsional levels

J prime K primeminus1 K prime+1 J primeprime K primeprimeminus1 K primeprime+1v = 0 v = 1 v = 2 v = 3

Obs Freq Obs minus cal Obs Freq Obs minus cal Obs Freq Obs minus cal Obs Freq Obs minus cal

1 0 1 0 0 0 890219 minus01

2 0 2 1 0 1 1780128 minus06

2 1 1 1 1 0 1822112 minus07 1825818 minus12 1828986 minus01

3 0 3 2 0 2 2669435 minus01 2682488 01 2689518 05

3 0 3 2 1 2 2676543 minus09 2748770 06

3 1 2 2 1 1 2732973 minus07 2738558 08 2743289 02 2632250 minus07

3 2 1 2 2 0 2671870 minus16

3 2 2 2 2 1 2670676 04

4 0 4 3 0 3 3567316 minus09 3575272 17 3584664 15

4 1 4 3 1 3 3476896 minus11

7 0 7 6 0 6 6214450a 01 6231222a 01 6245230a minus11

7 1 7 6 1 6 6081660a 04 6101840a 02

7 2 6 6 2 5 6229080a minus06 6245680a minus03 6259720a 05 6276010a minus02

7 2 5 6 2 4 6246030a minus05 6262420a minus08 6276220a minus07 6292230a 03

7 3 4 6 3 3 6234120a minus01 6250650a minus08 6264890a 00

7 3 5 6 3 4 6233940a minus07 6250480a minus04 6264720a 03

7 4 3 6 4 2 6233110a 02 6249650a minus06

7 5 3 6 5 2 6249340a 03

8 0 8 7 0 7 7096180a 00 7115430a 10 7131520a 04 7150660a minus01

8 1 7 7 1 6 7282000a minus01 7296940a 03 7309620a 09 7324330a minus06

8 1 8 7 1 7 6948980a 06 6972040a minus12 6991160a 01 7013980a minus08

8 2 7 7 2 6 7117900a 07 7152920a 07

8 2 6 7 2 5 7143270a 12 7161940a 08 7177620a minus03 7195830a 04

8 3 6 7 3 5 7125150a 06 7144040a 05

8 3 5 7 3 4 7125500a 08 7144390a 07 7160610a minus17

8 4 4 7 4 3 7123910a 05 7142820a 09

8 5 4 7 5 3 7142310a 12

8 6 3 7 6 2 7142110a minus06

9 0 9 8 0 8 8037170a minus01

9 1 8 8 1 7 8190120 01 8206950a minus01 8221230a 06

9 2 8 8 2 7 8006280a 04 8027630a minus11 8045700a 08 8066710a 09

9 2 7 8 2 6 8042400a 07 8063330a 06 8080870a minus05 8101240a minus08

9 3 6 8 3 5 8017210a 05 8038450a 04

9 3 7 8 3 6 8016570a 03 8037820a 12

9 4 5 8 4 4 8014840a minus04 8036110a 04

9 5 5 8 5 4 8014070a 08 8035340a 06

9 6 4 8 6 3 8013750a minus01 8035040a 06

10 0 10 9 0 9 8852330a minus01 8876590a minus05 8921190a minus12

10 1 9 9 1 8 9097440a minus05

10 1 10 9 1 9 8681850a minus02

10 2 8 9 2 7 8943620a minus01 8966783a 03 8986160a minus02

10 2 9 9 2 8 8894180a minus01 8917948a 10 8938010a 07 8961380a minus02

10 3 7 9 3 6 8909320a minus01 8932907a minus01

10 3 8 9 3 7 8908220a minus08

10 4 6 9 4 5 8905935a minus09 8929547a minus09

10 5 5 9 5 4 8904830a minus01

Table 1 Continued

10 6 4 9 6 3 8904355a minus04

10 7 3 9 7 2 8904190a minus02

10 9 1 9 9 0 8904317a minus02

11 1 11 10 1 10 9547305a minus03 9579110a 01 9605440a 14 9636980a 04

11 2 9 10 2 8 9847083a minus01 9872430a minus11 9893600a 02

11 2 10 10 2 9 9781559a 03 9807710a 00 9855560a 08

11 3 8 10 3 7 9801904a minus02 9827840a 05 9850060a 03 9875270a minus03

11 3 9 10 3 8 9800132a 00 9826060a 01 9848290a minus06 9873530a minus01

11 4 8 10 4 7 9823170a minus08

11 5 7 10 5 6 9795700a 00 9821690a minus05 9845350a 04

11 6 5 10 6 4 9795028a 00 9845620a minus01

11 7 4 10 7 3 9794754a 01

11 8 3 10 8 2 9794705a 04

1 1 1 2 0 2 2489258 01

2 1 2 3 0 3 1558586 01

6 0 6 5 1 5 1344427 02

7 0 7 6 1 6 2345046 01

8 0 8 7 1 7 3359568 minus01aThis work rest are from [1]

constants Analysis of the far-infrared spectrum of transacrolein in the ν18 fundamental and (ν17 + ν18) minus ν18 hotbands were reported by McKellar et al [15] Very recently10 μm high-resolution rotational spectral analysis of the ν11ν16 ν14 and ν16 + ν18 minus ν18 bands of trans-acrolein werereported by Xu et al [16] In all the previous works trans-acrolein (propenal) was either procured commercially orprepared chemically

Production identification and spectroscopic character-ization of new stable and transient molecules by applyinga DC glow discharge through a low-pressure flow of gasor a mixture of gases inside an absorption cell havebecome a well-established area of research in the field ofmolecular spectroscopy [21] Recently Jaman et al havereported analysis of the millimeter-wave rotational spectraof propyne (CH3CCH) [22] and propynal (HCCCOH) [23]produced by DC glow discharge technique and carried outdetailed DFT calculations for both the molecules to evaluatethe spectroscopic constants and molecular parameters andcompared them with their respective experimental valuesIn the present communication we report the analysisof the ground state (v = 0) as well as several torsionalexcited states (v = 1 2 3) rotational spectra of trans-pro-penal produced by a DC glow discharge through a low-pressure flow of allyl alcohol (CH2CHCH2OH) vapor in thefrequency region 600ndash990 GHz Asymmetric-top Kminus1 K+1-structures of different J+1larrJ transitions which falls underthis frequency range have been observed and measuredThe measured rotational transition frequencies along withthe previously reported frequencies were fitted to standard

Table 2 Ground state rotational and centrifugal distortion con-stants of trans-propenal (CH2CHCOH)

ConstantsGlobal fit using microwave and

millimeter wave dataDFT calculation

A (MHz) 47353729plusmn 0009 47532149

B (MHz) 46594894plusmn 00004 4635391

C (MHz) 42427034plusmn 00004 4223524

DJ (kHz) 1031plusmn 0001 0983

DJK (kHz) minus8684plusmn 0006 minus9099

DK (kHz) 361949plusmn 0963 346316

d1 (kHz) minus01197plusmn 00002 minus0119

HJK (Hz) 0014plusmn 0012

HKJ (Hz) minus0490plusmn 0021

σb 0041

κc minus09806

Δd minus0018

N e 224bStandard deviation of the overall fit

cAsymmetry parameterdInertia defect Δ = Ic minus Ib minus IaeNumber of transitions used in the fit

asymmetric-top Hamiltonian to determine the rotationaland centrifugal distortion (CD) constants for the ground aswell as excited torsional states A detailed quantum chemical

139Millimeter-Wave Rotational Spectra of trans-Acrolein (Propenal) (CH2CHCOH) A DC Discharge Product of Allyl Alcohol (CH2CHCH2OH) Vapor and DFT Calculation

Table 3 Excited-state spectroscopic constants of trans-propenal (CH2CHCOH)

Constantsv = 1 v = 2 v = 3

This work Ref [3] This work Ref [3] This work Ref [3]

A (MHz) 45782822plusmn 3231 44727881plusmn 3873 43420393plusmn 5888

B (MHz) 4666210plusmn 0004 466619plusmn 02 4672056plusmn 0005 467210plusmn 002 4678661plusmn 0006 467869plusmn 002

C (MHz) 4259668plusmn 0005 425966plusmn 002 4273558plusmn 0006 427356plusmn 002 4290297plusmn 0007 429029plusmn 002

DJ (kHz) 1078plusmn 0012 1280plusmn 0017 1168plusmn 0026

DJK (kHz) minus8735plusmn 0075 minus46831plusmn 0112 minus28714plusmn 0631

σ f 0077 0085 0075

κg minus09804 minus09803 minus09802

Δh minus0702 minus1212 minus1861

Ni 43 28 20fStandard deviation of the overall fit

gAsymmetry parameterhInertia defect Δ = Ic minus Ib minus IaiNumber of transitions used in the fit

Table 4 Comparison of the observed inertia defect (ΔuA2) values

for the ground and excited torsional state (v) of trans-propenal withsome other molecules

MoleculesInertia defect (ΔuA

2) values

v = 0 v = 1 v = 2 v = 3

Trans-propenalj

minus0018 minus0702 minus1212 minus1861(CH2CHCHO)

o-cis 3-fluorobenzaldehydek

minus0078 minus0988 minus1876 minus2726(C6H5FCOH)

Nitrobenzenel

minus0481 minus1863 minus3186 minus4470(C6H5NO2)

Benzoyl fluoridem

minus0325 minus1528 minus2765 minus3963(C6H5COF)

2-Fluorpstyrenen

minus1215 minus2689 minus3341 minus4380(C6H4FC2H3)

jThis work k[17] l[18] m[19] n[20]

calculation was also carried out to evaluate the spectroscopicconstants dipole moment and the structural parameters ofthe trans conformer of propenal Finally the experimentallydetermined rotational and CD constants were comparedwith the best set of values obtained after a series of DFTcalculations

2 Experimental Details

The spectrometer used in the present work is basically a50 kHz source-modulated system combined with a free spaceglass discharge cell of 15 m in length and 10 cm in diameterThe cell is fitted with two Teflon lenses at each end Ahigh voltage DC regulated power supply (6 kV 1300 mA)procured from Glassman Japan was used to apply a DCvoltage through a flow of low pressure precursor gases Thecell is connected with a high vacuum pump at one end and tothe sample holder section through a glass port on the other

Klystrons and Gunn diodes followed by frequencydoubler (Millitech model MUD-15-H23F0 and MUD-10-LF000) have been used as radiation sources Millimeter waveradiation was fed into the cell by a waveguide horn andTeflon lens A similar horn and lens arrangement was usedto focus the millimeter-wave power onto the detector afterpropagating through the cell The output frequency of themillimeter wave radiation was frequency modulated by abidirectional square-wave of 50 kHz [24] and the signal fromthe detector (Millitech model DBT-15-RP000 and DXP-10-RPFW0) was amplified by a 100 kHz tuned preamplifierand detected by a phase-sensitive lock in amplifier in the 2fmode The output of the lock in amplifier was connectedto an oscilloscope or a chart recorder for signal displayThe spectrometer was calibrated by measuring standard OCSsignals in the entire frequency range After calibration theuncertainty in frequency measurement has been estimatedto be plusmn010 MHz A block diagram of the spectrometer isshown in Figure 1 Details of the spectrometer used havebeen described elsewhere [25 26]

Propenal (CH2CHCOH) was produced inside theabsorption cell by applying a DC glow discharge througha low pressure (sim5ndash10 mTorr) flow of allyl alcohol(CH2CHCH2OH) vapor The discharge current was main-tained at around 5 mA with an applied voltage of 10 kV Amechanical onoff type discharge was found to be suitable toobserve good signals of propenal Signals could be observedat room temperature However a controlled flow of liquidnitrogen vapor through the cell helps in improving the signalintensity The observed signals of propenal appeared as sharplines immediately after the DC discharge was applied butstarted losing intensity with time

3 Computational Method

Quantum chemical computations were performed usingGAUSSIAN 09W package [27] Density functional methodswith various functionals were used to calculate the structuralparameters dipole moment total energy (sum of electronic

Teflonlens Cathode

Solenoidcoil

Liquid N2 Anode

15-meterdischarge cell

Pump Copperjacket

Sampleinlet

Frequencydoubler Horn

Variableattenuator

Wavemeter

Isolator

Gunnoscillator

Gunn biassupply

Receiver

Frequencysynthesizer300 MHz to3000 MHz

Y input(A)

Y input(B)X input

Oscilloscope

Chartrecorder

Preamplifier

Lock-inamplifier

Bidirectional

generatorsquare wave

Liquid N2

Horn Detector

Teflonlens

Sweepgenerator

Figure 1 Block diagram of source-modulated millimeter wave spectrometer with DC discharge facility

and zero point energy) as well as the rotational and cen-trifugal distortion constants of trans-propenal The geometryoptimization was carried out using different functionals likeBecke 3-term correlation functional(B3LYP) with basis sets6-31+g(d p) to 6-311++g(d 2p) Becke three-Parameterhybrid functional and PerdewWang 91 nonlocal correlationfunctional (B3PW91) method with different basis sets from6-31 g to 6-311++g(d 2p) modified Perdew-Wang one-parameter hybrid model taking basis sets from 6ndash31 g to 6-311++g(d 2p) and Perdew Burke and Ernzerhof functional(PBEPBE) with the basis sets 6-311 g to 6-311++g(d 2p)The frequency calculation along with its anharmonicity wasdone on optimized geometry The objective of this DFTcalculation is to compare the structural parameters and rota-tional constants of trans-propenal with the experimentallyobserved values in its ground state The molecular drawingis done by using GAUSSVIEW50 [28]

4 Rotational Spectrum and Analysis

41 Ground State The ground state rotational spectrumof the trans conformer of propenal was predicted in thefrequency range 600-990 GHz using the rotational and cen-trifugal distortion constants reported earlier [14] J = 7larr6to J = 11larr10 series of transitions along with theirdifferent Kminus1 K+1 components falls within this frequencyrange Different components in each J+1larrJ series weremeasured The observed lines were found very close totheir predicted values Finally 224 a- and b-type R- andQ-branch transitions consisting of all previous microwave[1 3] millimeter-wave [14] and present data were usedto perform a kind of global fit to the semirigid rotorWatsonrsquos S-reduction Hamiltonian (Ir-representation) [29]to determine a set of three rotational five quartic and

two sextic centrifugal distortion constants The shifts infrequency of the absorption lines from their rigid rotorpositions due to centrifugal distortion effect were found to beless than that of propynal [23] The observed and measuredtransition frequencies by us corresponding to J = 7larr6to 11larr10 series are listed in Table 1 The ground statespectroscopic constants obtained for trans-propenal usingthe global fit are listed in Table 2 The small negative valueof the inertia defect (Δ = minus0018 uA2) demonstrates that theequilibrium configuration of trans-propenal is planar Theagreement between the derived set of spectroscopic constantsand those obtained earlier [1 3 14] with commercial samplesindicates that the newly assigned transition frequencies ofTable 1 definitely belong to trans-acrolein (trans-propenal)a discharge product of allyl alcohol vapor Figure 2 shows theobserved trace of the Kminus1 = 3 doublet of J = 9larr8 transitionimmediately after the DC discharge was applied The traceremained visible for a couple of minutes on the oscilloscopescreen with gradually diminishing intensity

42 Excited Torsional States From an analysis of the ultravi-olet [5] and far infrared spectrum [4] of acrolein vapor thefirst four excited torsional levels were found to lie around157 cmminus1 (v = 1) 312 cmminus1 (v = 2) 468 cmminus1 (v = 3)and 623 cmminus1 (v = 4) respectively Wagner et al [3] havereported a few low J transitions of trans- acrolein in the 180ndash360 GHz for the first three (v = 1 v = 2 and v = 3) excitedtorsional states and determined only the rotational constantsB and C for each of these excited states In this work we haveextended the analysis of rotational transitions in each of theabove three excited states up to 990 GHz which has resultedin the determination of three rotational and two quarticcentrifugal distortion constants for all the three torsional

Table 5 Calculated ground state rotational constants of trans-propenal (CH2CHCOH) with various models and basis sets

Model Basis set A0 (MHz) B0 (MHz) C0 (MHz)

B3LYP 6-31+g(d p) 47421368 4599276 4189265

B3LYP 6-31++g(d 2p) 47462833 4596304 4190461

B3LYP 6-311++g(d 2p) 47713427 4621512 4213560

B3PW91 6-31g 47562896 4564835 4164742

B3PW91 6-31++g(d 2p) 47445030 4617117 4207640

B3PW91 6-311++g(d 2p) 47640454 4642978 4230737

MPW1PW91 6-31g 47680556 4583545 4181241

MPW1PW91 6-31++g(d 2p) 47532149 4635391 4223524

MPW1PW91 6-311++g(d 2p) 47719894 4660612 4246017

PBEPBE 6-311g 46829581 4528525 4119901

PBEPBE 6-311++g(d 2p) 46855368 4594974 4175330

PBEPBE 6-31++g(d 2p) 46605708 4567810 4150825

Expto 47353729 46594894 42427034oThis work

Table 6 Comparison of the molecular bond lengths dipole moment and total energy of trans-propenal calculated by various methods andbasis sets with the experimental values

Models Basis setsBond lengths between

Dipole Moment (D) Energy (eV)1Cndash2H 1Cndash3H 1Cndash4C 4Cndash5H 4Cndash6C 6Cndash7H 6Cndash8O

6-31+g(d p) 1088 1085 1340 1087 1474 1112 1218 3515 minus5220511

B3LYP 6-31++g(d 2p) 1087 1084 1340 1086 1474 1111 1218 3502 minus5220755

6-311++g(d 2p) 1084 1081 1335 1083 1474 1109 1211 3464 minus5221979

6-31g 1088 1084 1341 1086 1465 1106 1239 3542 minus5216647

B3PW91 6-31++g(d 2p) 1085 1082 1334 1084 1471 1111 1208 3491 minus5218334

6-311+g(d 2p) 1085 1082 1334 1084 1417 1111 1208 3434 minus5219694

6-31g 1086 1083 1339 1085 1463 1104 1236 3564 minus5217218

MPW1PW91 6-31++g(d 2p) 1086 1083 1336 1085 1470 1110 1213 3481 minus5219204

6-311++g(d 2p) 1084 1081 1332 1083 1469 1109 1205 3443 minus5220238

6-311g 1093 1090 1348 1092 1467 1115 1250 3392 minus5213844

PBEPBE 6-31++g(d 2p) 1096 1092 1349 1095 1475 1123 1228 3394 minus5214252

6-311++g(d 2p) 1093 1089 1344 1092 1474 1121 1221 3367 minus5215531

Exptp 1089 1081 1341 1084 1468 1113 1215 3117pRef [13]

excited states The new assigned transitions along with thosereported earlier [3] are also shown in Table 1 along withthe ground state transitions The excited state data were alsoused to fit to the same semirigid rotor Watsonrsquos S-reductionHamiltonian (Ir-representation) [29] Three rotational andtwo quartic (DJ and DJK ) CD constants were used to fit thedata The contribution of other CD parameters was foundto be negligible while fitting the excited state data Thederived spectroscopic constants and inertia defect values forthe three torsional excited states are shown in Table 3 Themore negative inertia defect values for successive torsionalexcited states indicate that the excited state lines arise froman out-of-plane vibration in this case COH group torsionabout CndashC single bond The observed inertia defect valuesfor the ground and torsional excited states of trans-propenaland some other related molecules are compared in Table 4The plots of inertia defect values with torsional quantum

numbers for trans-propenal along with other molecules areshown in Figure 3 for comparison

43 Computational Results Propenal is a slightly asymmetricprolate top molecule (κ = minus09806) The optimization ofgeometry for the trans conformer of propenal was testedby employing various levels of theory and basis setsHowever the computed rotational and centrifugal distortionconstants and the structural parameters obtained with modelMPW1PW91 model with 6-31++g(d 2p) basis set werefound to be in good agreement with the observed valuesCalculated values of ground state rotational constants oftrans-propenal obtained with various models and basissets are shown in Table 5 Results obtained with DFTMPW1PW916-31++g(d 2p) have been compared withthe corresponding experimental values in Table 2 Foroptimized geometry of trans-propenal the calculated energy

Table 7 Comparison of the molecular bond angles of trans-propenal calculated by various methods and basis sets with the experimentalvalues

Models Basis setsBond angles between

H2ndashC1ndashH3 H2ndashC1ndashC4 H3ndashC1ndashC4 C1ndashC4ndashH5 C1ndashC4ndashH6 H5ndashC4ndashC6 C4ndashC6ndashH7 C4ndashC6ndashO8 H7ndashC6ndashO8

6-31+g(dp) 116799 121067 122133 122342 121089 116569 115154 124162 120684

B3LYP 6-31++g(d2p) 116834 121043 122123 122314 121153 116533 115123 124165 120712

6-311++g(d2p) 116849 120973 122177 122347 121069 116583 114830 124348 120822

6-31g 116488 121219 12222 122091 121314 116596 115381 123994 120626

B3PW91 6-31++g(d2p) 116699 121104 122196 122363 120926 116711 115126 124248 120625

6-311+g(d2p) 116922 120852 122226 122414 120821 116766 114689 124414 120897

6-31g 116516 121201 122284 122146 121259 116594 115443 123945 120612

MPW1PW91 6-31++g(d2p) 116938 120930 122131 122419 120864 116717 115023 124176 120800

6-311++g(d2p) 116964 120807 122227 122491 120714 116794 114691 124394 120914

6-311g 116573 121132 122295 122136 121724 116139 115404 123451 120604

PBEPBE 6-31++g(d2p) 116976 120764 122260 122329 120985 116685 114796 124334 120870

6-311++g(d2p) 116998 120664 122337 122394 120846 116762 114486 124549 120964

Exptq 1180 1198 1222 1224 1203 1173 1147 1239 1213qRef [13]

minus001

minus002

minus003

minus004

minus005

minus006

8016579(3 7) minus 8(3 6) 801721

9(3 6) minus 8(3 5)

Frequency (MHz)

Figure 2 Observed trace of the Kminus1 = 3 doublet of J = 9larr8 transi-tion of trans-propenal produced by DC discharge

is minus5219204 eV and the dipole moment is 3481 D Thenumber and labeling of atoms in propenal molecule asshown in Figure 4 Bond lengths and angles have beencomputed using different models and basis sets and areshown in Tables 6 and 7 respectively

5 Conclusion

An efficient method of generating trans-propenal (trans-acrolein) in the gas phase by applying a DC glow dischargethrough a low pressure vapor of allyl alcohol inside theabsorption cell has been presented The gas phase rotationalspectra of the trans conformer of propenal produced inthis way has been recorded and analyzed in the frequencyrange 600ndash990 GHz for the ground as well as three torsionalexcited states (v = 1 2 and 3) The asymmetric top Kminus1 K+1-components of different transitions having J values 6 to 10have been measured The observed transition frequencies

minus05

minus1

minus15

minus2

minus25

minus3

minus35

minus4

minus45

minus50 1 2 3

Torsinal levels (v)

Trans-propenal2-fluorostyrene3-fluorobenzaldehyde

NitrobenzeneBenzoyl fluoride

Figure 3 Plot of the variation of inertia defect values with tor-sional state quantum number for trans-propenal and some othermolecules

along with the previously reported data [1 3 14] werefitted to a standard asymmetric-top Watsonrsquos S-reductionHamiltonian (Ir-representation) to determine ground staterotational and centrifugal distortion constants Analysis ofthe rotational transitions for the three excited torsional stateshas been extended up to 990 GHz which has enabled us todetermine the three rotational and two centrifugal distortionconstants The small negative value of the inertia defect

(Δ = minus0018uA2) in the ground vibrational state (v = 0)

and the linear variation of the inertia defect values withtorsional quantum number (v = 1 2 3) demonstrate thatthe equilibrium configuration of trans-propenal is planar asnoticed in case of 3-fluorobenzaldehyde benzoyl fluoride

Figure 4 Optimized geometry of trans-propenal molecule and thenumbering of atoms

and nitrobenzene (Figure 4) The existence of a slightlybent or twistedndashCOH group would have resulted in a zig-zag behavior in the variation of inertia defect values withtorsional quantum number as observed in the case of2-fluorostyrene (Figure 4) To compare the experimentalresults with theory DFT calculations were performed usingvarious models and basis sets However it was found thatMPW1PW91 model with 6-31++g (d 2p) basis set producedthe best values of rotational and quartric centrifugal distor-tion constants which are close to the experimental values

Acknowledgment

The authors would like to thank Mr A K Bhattacharya forhis technical assistance during the course of this work

References

[1] E A Cherniak and C C Costain ldquoMicrowave spectrum andmolecular structure of trans-acroleinrdquo The Journal of ChemicalPhysics vol 45 no 1 pp 104ndash110 1966

[2] J M Hollis P R Jewell F J Lovas A Remijan and HMoslashllendal ldquoGreen bank telescope detection of new interstellaraldehydes propenal and propanalrdquo Astrophysical Journal vol610 no 1 pp L21ndashL24 2004

[3] R Wagner J Fine J W Simmons and J H GoldsteinldquoMicrowave spectrum structure and dipole moment of s-trans acroleinrdquo The Journal of Chemical Physics vol 26 no3 pp 634ndash637 1957

[4] R K Harris ldquoVibrational assignments for glyoxal acroleinand butadienerdquo Spectrochimica Acta vol 20 no 7 pp 1129ndash1141 1964

[5] J C D Brand and D G Williamson ldquoNear-ultra-violetspectrum of propenalrdquo Discussions of the Faraday Society vol35 pp 184ndash191 1963

[6] J M Hollas ldquoThe electronic absorption spectrum of acroleinvapourrdquo Spectrochimica Acta vol 19 no 9 pp 1425ndash1426E1ndashE2 1427ndash1441 1963

[7] A C P Alves J Christoffersen and J M Hollas ldquoNear ultra-violet spectra of the s-trans and a second rotamer of acroleinvapourrdquo Molecular Physics vol 20 no 4 pp 625ndash644 1971

[8] A C P Alves J Christoffersen and J M Hollas ldquoErratum toldquoNear ultra-violet spectra of the s-trans and a second rotamerof acrolein vapourrdquordquo Molecular Physics vol 21 no 2 p 3841971

[9] A Krantz T D Goldfarb and C Y Lin ldquoA simple methodfor assigning vibrational frequencies to rapidly equilibratingrotational isomersrdquo Journal of the American Chemical Societyvol 94 no 11 pp 4022ndash4024 1972

[10] C E Blom R P Miller and H Gunthard ldquoS-trans and S-cis acrolein trapping from thermal molecular beams and uv-induced isomerization in argon matricesrdquo Chemical PhysicsLetters vol 73 pp 483ndash486 1980

[11] L A Carreira ldquoRaman spectrum and torsional potentialfunction of acroleinrdquo Journal of Physical Chemistry vol 80 no11 pp 1149ndash1152 1976

[12] C E Blom and A Bauder ldquoMicrowave spectrum rotationalconstants and dipole moment of s-cis acroleinrdquo ChemicalPhysics Letters vol 88 no 1 pp 55ndash58 1982

[13] C E Blom G Grassi and A Bauder ldquoMolecular structureof s-cis- and s-trans-acrolein determined by microwave spec-troscopyrdquo Journal of the American Chemical Society vol 106no 24 pp 7427ndash7431 1984

[14] M Winnewisser G Winnewisser T Honda and E HiritaldquoGround state centrifugal distortion constants of trans-acrolein CH2=CH-CHO from the microwave and millimeterwave rotational spectrardquo Zitschrift Naturforsch vol 30 pp1001ndash1014 1975

[15] A R W McKellar D W Tokaryk and D R T AppadooldquoThe far-infrared spectrum of acrolein CH2CHCHO theν18 fundamental and (ν17 + ν18)mdashν18 hot bandsrdquo Journal ofMolecular Spectroscopy vol 244 no 2 pp 146ndash152 2007

[16] L-H Xu X Jiang H Shi et al ldquo10 μm High-resolutionspectrum of trans-acrolein rotational analysis of the ν11ν16 ν14 and ν16 + ν18mdashν18 bandsrdquo Journal of MolecularSpectroscopy vol 268 no 1-2 pp 136ndash146 2011

[17] J L Alonso and R M Villamanan ldquoRotational isomerismin monofluorobenzaldehydesrdquo Journal of the Chemical SocietyFaraday Transactions vol 85 no 2 pp 137ndash149 1989

[18] J H Hoslashg L Nygaard and G Ole Soslashrensen ldquoMicrowavespectrum and planarity of nitrobenzenerdquo Journal of MolecularStructure vol 7 no 1-2 pp 111ndash121 1971

[19] R K Kakar ldquoMicrowave spectrum of benzoyl fluoriderdquo TheJournal of Chemical Physics vol 56 no 3 pp 1246ndash12521972

[20] R M Villamanan J C Lopez and J L Alonso ldquoOn theplanarity of 2-fluorostyrenerdquo Journal of the American ChemicalSociety vol 111 no 17 pp 6487ndash6491 1989

[21] S Saito ldquoLaboratory microwave spectroscopy of interstellarmoleculesrdquo Applied Spectroscopy Reviews vol 25 pp 261ndash2961989

[22] A I Jaman P Hemant Kumar and P R Bangal ldquoRotationalspectrum of propyne observed in a DC glow discharge andDFT calculationrdquo Asian Journal of Spectroscopy Special Issuepp 43ndash48 2010

[23] A I Jaman R Bhattacharya D Mondal and A Kumar DasldquoMillimeterwave spectral studies of propynal (HCCCHO)produced by DC discharge and ab initio DFT calculationrdquo

Journal of Atomic Molecular and Optical Physics vol 2011Article ID 439019 8 pages 2011

[24] J F Verdieck and C D Cornwell ldquoRadio-frequency spectrom-eter with bidirectional square wave frequency modulationrdquoReview of Scientific Instruments vol 32 no 12 pp 1383ndash13861961

[25] A I Jaman ldquoMillimeterwave spectroscopy of transient mol-ecules produced in a DC dischargerdquo Pramana vol 61 no 1pp 85ndash91 2003

[26] A I Jaman ldquoMillimeter wave spectrum of ICN a transientmolecule of chemical and astrophysical interestrdquo Journal ofPhysics vol 80 no 1 Article ID 012006 2007

[27] M J Frisch G W Trucks and H B Schlegel Gaussian 09Revision A 1 Gaussian Wallingford Conn USA 2009

[28] GaussView 5 0[29] J K G Watson ldquoAspects of quartic and sextic centrifugal

effects on rotational energy levelsrdquo in Vibrational Spectra andStructure J R Durig Ed vol 6 pp 1ndash89 Mercel Dekker NewYork NY USA 1977

The Effect of Nanoparticle Size on Cellular Binding Probability

Vital Peretz1 Menachem Motiei2 Chaim N Sukenik1 and Rachela Popovtzer2

1 The Department of Chemistry The Institute of Nanotechnology and Advanced Materials Bar-Ilan UniversityRamat Gan 52900 Israel

2 Faculty of Engineering The Institute of Nanotechnology and Advanced Materials Bar-Ilan University Ramat Gan 52900 Israel

Correspondence should be addressed to Rachela Popovtzer rachelapopovtzerbiuacil

Academic Editor Zeev Zalevsky

Nanoparticle-based contrast agents are expected to play a major role in the future of molecular imaging due to their manyadvantages over the conventional contrast agents These advantages include prolonged blood circulation time controlled biologicalclearance pathways and specific molecular targeting capabilities Recent studies have provided strong evidence that molecularlytargeted nanoparticles can home selectively onto tumors and thereby increase the local accumulation of nanoparticles in tumorsites However there are almost no reports regarding the number of nanoparticles that bind per cell which is a key factor thatdetermines the diagnostic efficiency and sensitivity of the overall molecular imaging techniques Hence in this research we havequantitatively investigated the effect of the size of the nanoparticle on its binding probability and on the total amount of materialthat can selectively target tumors at a single cell level We found that 90 nm GNPs is the optimal size for cell targeting in termsof maximal Au mass and surface area per single cancer cell This finding should accelerate the development of general designprinciples for the optimal nanoparticle to be used as a targeted imaging contrast agent

1 Introduction

Imaging plays a critical role in overall cancer managementin diagnostics staging radiation planning and evaluationof treatment efficiency Conventional imaging technologiesfor cancer detection such as CT MRI and ultrasound canbe categorized as structural imaging modalities They areable to identify anatomical patterns and to provide basicinformation regarding tumor location size and spread basedon endogenous contrast However these imaging modalitiesare not efficient in detecting tumors and metastases thatare smaller than 05 cm [1] and they can barely distinguishbetween benign and cancerous tumors Molecular imaging isan emerging field that integrates molecular biology chem-istry physics and medicine in order to gain understandingregarding biological processes and to identify diseases basedon molecular markers which appear before the clinicalpresentation of the disease

Recently much research has focused on the develop-ment of targeted nanoparticles for use as contrast agentsfor molecular imaging These include superparamagneticnanoparticles for MRI [2ndash6] quantum dots for optical

imaging [7ndash9] and gold nanoparticles (GNPs) for opticalimaging [10 11] and CT [12ndash14]

GNPs are a class of contrast agents with unique opticalproperties They are well known for their strong interactionswith visible light through the resonant excitations of thecollective oscillations of the conduction electrons within theparticles [15] As a result local electromagnetic fields nearthe particle can be many orders of magnitude higher than theincident fields and the incident light around the resonant-peak wavelength is scattered very strongly The resonancecondition is determined from absorption and scatteringspectroscopy and is found to depend on the shape size anddielectric constants of both the metal and the surroundingmaterial This localized surface plasmon resonance (LSPR)has led to the development of a wide range of biochemicaldetection assays [16] and various nanoprobes for opticalimaging of cancer [17 18]

In order to study whether incubation of the cancer cellswith different sizes of GNPs can improve the coverage of thecellsrsquo surface 15 70 and 150 nm GNPs were incubated withhead and neck cancer cells (A431) in different incubationorders as illustrated in Table 2 We hypothesized that small

GNPs when introduced to the cells in a second cycle (afterlarger GNPs were incubated) will fill the gaps between thelarger GNPs on the cellsrsquo surfaces

GNPs are also an ideal CT contrast agents The abilityof CT to distinguish between different tissues is basedon the fact that different tissues provide different degreesof X-ray attenuation where the attenuation coefficient isdetermined by the atomic number and electron densityof the tissue the higher the atomic number and electrondensity the higher the attenuation coefficient The atomicnumber and electron density of gold (79 and 1932 gcm3resp) are much higher than those of the currently usediodine (53 and 49 gcm3) and therefore gold induces astrong X-ray attenuation When the gold particles are linkedto specific-targeting ligands such as monoclonal antibodiesor peptides these nanoparticles can selectively tag a widerange of medically important targets for example specificcancer cells with high affinity and specificity In additiongold nanoparticles proved to be nontoxic and biocompatiblein vivo [19 20]

Recently Hainfeld et al [21] showed that GNPs canenhance the visibility of millimeter-sized human breasttumors in mice and that active tumor targeting (with anti-Her2 antibodies) is more efficient than passive targetingThey also showed that the specific uptake of the targetedGNPs in the tumorrsquos periphery was 22-fold higher thanin surrounding muscle Another recent study demonstratedenhanced CT attenuation of bombesin-functionalized GNPsthat selectively targeted cancer receptor sites that are over-expressed in prostate breast and small-cell lung carcinoma[22] In our own research [23] we recently demonstratedthat a small tumor which is currently undetectable throughanatomical CT is enhanced and becomes clearly visible bythe molecularly-targeted GNPs We further showed that theCT number of molecularly targeted head and neck tumor isover five times higher than the corresponding CT numberof an identical but untargeted tumor and that active tumortargeting is more efficient and specific than passive targeting

These studies have provided strong evidence thatnanoparticles accumulate in vivo on the tumor Howeverthere are almost no reports regarding the number ofnanoparticles that bind per cell which is a key factorthat determines the diagnostic efficiency and sensitivity ofthe overall molecular imaging techniques Hence in thisresearch we have quantitatively investigated the effect of thesize of the nanoparticle on its binding probability and on thetotal amount of material that can selectively target tumorson a single cell level We have further investigated the abilityto increase the amount of contrast material that binds per cellby simultaneously targeting nanoparticles in different sizes orin consequent cycles

2 Methods

21 Gold Nanospheres Synthesis Conjugation and Char-acterization Gold nanospheres (10 15 and 30 nm) weresynthesized by citrate reduction according to the methoddescribed by Turkevitch et al [24] Larger GNPs (70 90 and150 nm) were synthesized using the seed mediated growth

method [25] Briefly gold seeds were synthesized using anaqueous HAuCl4 solution (025 mL of 005 M solution) andadding it to 50 mL H2O and boiling After boiling 175 mLof 1 citrate (175 mL of a solution that was 114 mg in10 mL H2O) were added to the solution and stirring wascontinued for 20 min The solution was cooled to roomtemperature and used directly for further experiments Thismethod produced gold nanospheres with a diameter of15 nm [26] In order to enlarge the nanoparticles 170 mLDD water along with 044 mL of 14 M HAuCl4 and 26 mLseed solution to make 90 nm GNPs or 31 mL seed solution tomake 70 nm GNPs was added to a 400 mL Erlenmeyer flaskThen 372 mL of 01 M 2-mercaptosuccinic acid (MSA) wasadded as the reducing agent The solution was stirred for1 min and left overnight According to this method 70 and90 nm gold nanospheres were synthesized

In order to prevent aggregation and to stabilize theparticles in physiological solutions a layer of polyethyleneglycol (mPEG-H) was absorbed onto the GNPs This layeralso provides the chemical groups that are required forantibody conjugation (SH-PEG-COOH) The SH-PEG layerconsisted of a mixture of 15 SH-PEG-COOH (Mw 3400)and 85 SH-PEG-methyl (Mw 5000) both obtained fromCreative PEGWorks Winston Salem NC The PEG solutionwas added to the GNPs solution and stirred for 3 hoursIt was then centrifuged in order to get rid of excess PEGcitrate and MSA The ratio of PEG molecules to GNPs wascalculated based on a footprint area of 035 nm2 [27]

211 Conjugation of Antibodies to GNPs In order to specif-ically bind to the SCC cancer cells anti-EGFR (epidermalgrowth factor receptor Erbitux Merck KGaA) antibodieswhich bind exclusively to the EGF receptor were conjugatedto the outer coating of the nanoparticles The interactionbetween the GNP and the antibody are based on electrostaticattraction between the negatively charged heterofunctionPEG (SH-PEG-COOH) and the positive segment of theantibody in pH 74

EGFR conjugation to 70 nm GNPs 131 mg of SH-PEG-COOH and 11 mg of SH-PEG were dissolved in 2 mL DDwater and added to the GNPs solution with stirring for 3hours The solution was centrifuged and 15 mL of 5 mgmLof anti-EGFR were added to the solution with stirring for 1hour in order to get rid of excess anti-EGFR The solutionwas kept at 4C The conjugation of anti-EGFR to allother GNPs sizes was achieved using the same method withdifferent quantities

22 In Vitro Cell Targeting Study Using GNPs In orderto study the effect of nanoparticle size on its bindingprobability 15 70 and 150 nm GNPs were incubated withhead and neck cancer cells (A431) (Group A with 15 nmGNPs group B with 70 nm GNPs and group C with 150 nmGNPs) A431 cells (15 times 106) in 5 mL DMEM mediumcontaining 5 FCS 05 penicillin and 05 glutaminewere incubated for a quantitative cell binding study (eachexperimental group was run in triplicate) Each group wasincubated 3 times with access amount of anti-EGFR-coatedGNPs for 30 minutes at 37C After incubation the medium

147The Effect of Nanoparticle Size on Cellular Binding Probability

Table 1 The effect of each additional incubation cycle on the amount of contrast material that binds per cell Each cell sample contained15 times 106 cells The cells were incubated with the different sized GNPs in consequent cycles for 30 min at 37C each cycle Each experimentalgroup was run in triplicate The GNPs were added in excess

The effect of each additional incubation cycle Cell type GNPs size (nm) Comment

A A431 90

The cells were incubated with the GNPs in consequent cyclesB A431 90 90

C A431 90 30

D A431 90 30 10

Table 2 The effect of incubation of different sized GNPs in consequent cycles experiment Each cell sample contained 15 times 106 cells Thecells were incubated with the different sized GNPs in consequent cycles for 30 min at 37C each cycle Each experimental group was run intriplicate The GNPs were added in excess amount

The effect of combination of different sizedGNPs in consequent cycles

Cell type GNPs size (nm) Comment

A A431 15 70 150 The cells were incubated with thedifferent size of GNPs inconsequent cycles

B A431 150 70 15

C A431 150 150 150

was washed twice with PBS followed by addition of 1 mL ofaqua-regia After evaporation of the acid the sediment wasdissolved in 5 mL of 005 M HCl The gold concentrationsof the samples were quantified by Flame Atomic AbsorptionSpectroscopy (SpectrAA 140 Agilent Technologies)

We have further investigated whether we are able toincrease the amount of contrast material that binds per cellby subsequent cycles of binding and how many GNPs canbind to a single cancer cell after each cycle of incubationTherefore 90 30 and 10 nm GNPs were incubated inconsequent cycles with the A431 cells (each cycle for 30 minat 37C) as illustrated in Table 1 After each incubation cyclethe amount of gold (Au masscell) was measured using FAASThen the number of GNPs that were bound in each cyclecould be calculated

31 Gold Nanospheres Synthesis Conjugation and Character-ization We have successfully synthesized GNPs in varioussizes ranging from 10 nm up to 150 nm Figure 1 showsTEM images of 10 30 70 and 150 nm GNPs As can beseen the small GNPs (sim10 nm) have a relatively large sizedistribution (25 Figure 1(a)) while particles that are largerthan 30 nm are more homogeneous with a very narrowsize distribution (Figures 1(b) 1(c) and 1(d)) The surfaceplasmon resonances of the various size GNPs are illustratedin Figure 2 It can be seen that when the nanospheres areenlarged there is a red shift in the surface plasmon resonanceof the particles (from 525 nm to 580 nm)

GNPs were successfully coated with PEG and anti-EGFRantibody The antibody conjugated GNPs were stable for upto three months confirmed by their maintenance of the sameplasmon resonance

32 Quantitative Investigation of the Size Effect of theNanoparticle on Its Binding Probability Before studying the

effect of nanoparticle size on its binding probability wehave evaluated the specificity of the interaction betweenthe antibody-coated GNPs and the A431 SCC cancer cells(which highly express the EGF receptor) Two types of GNPs(50 microL of 25 mgmL) were introduced to the SCC head andneck cancer cells (25 times 106 cells) The first was specificallycoated with anti-EGFR antibody while the second whichwas used as a negative control was coated with a nonspecificantibody (anti-Rabbit IgG) Atomic absorption spectroscopymeasurements quantitatively demonstrated that the activetumor targeting (anti-EGFR coated GNPs) was significantlymore specific than the control experiment (anti-Rabbit IgGcoated GNPs) The A431 cells took up 263 plusmn 23microg oftargeted GNPs (39 times 104 GNPs per A431 cell) whileparallel cells in the negative control experiment absorbedonly 02 plusmn 001microg of GNPs (34 times 103 GNPs per cell)Our results correlate well with previously published studieswhich report that head and neck SCC express from 2 times 104

to 2 times 106 EGFRscell [28 29]In order to quantitatively investigate the effect of

nanoparticle size on its binding probability (on a singlecell level) head and neck cancer cells were incubated withdifferent size GNPs (15 70 and 150 nm) for 30 min Formaximal binding the particles were incubated three timeswith the cancer cells Figure 3 shows the total amount ofgold (Au masscell) that binds per cell for nanoparticlesof different sizes (15 70 and 150 nm) The results clearlydemonstrate that larger particles produce larger amounts ofgold per cancer cell For the 15 nm GNPs only 00018 ng ofgold was bound to a single cancer cell (A431) while for thelargest particles 150 nm 0145 ng of gold was bound to asingle cancer cell

Once we have quantitatively measured (using FAAS) thetotal amount of gold that was bound to a single cancer cellthe exact number of nanoparticles and the GNPsrsquo surfacearea per cancer cell could then be calculated Table 3 showsthe total Au mass the number of GNPs of different sizes

1313 nm

848 nm

2801 nm

3284 nm

007 microm

(c) (d)

Figure 1 TEM images of different sized GNPs (a) 10 nm (b) 30 nm (c) 70 nm and (d) 150 nm

Table 3 Quantitative analysis number of GNPs of different sizesthat are bound to a single cancer cell total Au mass and the GNPsrsquosurface area per single cell

GNP size(nm)

Number ofGNP per cell

Aucell(ng)

Total surface area(m2)

15 54000 000186 38times 10minus11

30 39000 001 11times 10minus10

90 12000 0124 305times 10minus10

150 4200 0145 29times 10minus10

and the surface area of the GNPs that are bound to a singlecancer cell

These results clearly demonstrate that smaller particleshave a higher probability to bind to cancer cells (via antibody-antigen interaction) than larger particles (Figure 4(a)) Theprobability of 15 nm GNPs to bind to cancer cells is about 13

times more than the probability of 150 nm GNPs Howeverlarger particles produce larger amounts of Au mass per cellas well as larger surface area as illustrated in Figures 4(b)and 4(c) Nevertheless particles larger than 90 nm onlyslightly increased the Au masscell and the surface areacell

We have further investigated whether incubation of thecancer cells with different sizes of GNPs can improve thecoverage of the cellsrsquo surface We have hypothesized thatsmall GNPs when introduced to the cells in a second cycle(after larger GNPs were incubated) will fill the gaps betweenthe larger GNPs on the cellsrsquo surface However as can be seenin Figure 5 maximum coverage (or max Au masscell) wasobtained for the largest GNPs (column C 014 ngr Aucell) Ithas also been demonstrated that the order of the incubation(between the cells and the GNPs) is critical When 15 nmGNPs were introduced first to the cells overall a muchsmaller amount of gold was bound (column A)

400 500 600 700

Wavelength (nm)

10 nm30 nm

70 nm150 nm

Figure 2 UV-Vis spectroscopy of 10 30 70 and 150 nm goldnanospheres

0(A) 15 (B) 70 (C) 150

GNP size (nm)

Figure 3 Quantitative measurements using FAAS of Au masscellfor different sizes of GNPs Each cell sample contained 15 times106 cells and was incubated 3 times with the GNPs The GNPs wereadded in excess The error bars represent the standard deviation ofthree samples

We have further investigated whether we are able toincrease the amount of contrast material that binds per cellby consequent cycles of binding and how many GNPs werebound to a single cancer cell after each cycle of incubationAs seen in Figure 6 the first incubation is the most criticalAfter one incubation with 90 nm GNPs 01048 ngr of gold(=14258 GNPs) was bound to a single cancer cell In thesecond cycle of incubation with 90 nm GNPs (Figure 6column B) only a relatively small number of GNPs werebound (2742 90 nm GNPs (20)) Adding smaller GNPs (30and 15 nm Figure 6 columns C and D) barely influenced theamount of gold per cell (00015 ngr (5772 30 nm GNPs (1)for 30 nm GNPs and 0 ngr for the 15 nm GNPs) It has beenalso demonstrated that the first antibody antigen interaction(first incubation between the cells and the GNPs) is the mosteffective (Figure 6 column A)

4 Summery and Conclusions

In order to develop general design principles for nanopar-ticles to be used as in vivo imaging contrast agents we

0 50 100 150

GNP size (nm)

Number of GNP per single cancer cell

0 50 100 150

GNP size (nm)

Au mass per single cancer cell

0 50 100 150GNP size (nm)

Total surface area per single cancer cell

m2) 35Eminus10

3Eminus10

25Eminus10

2Eminus10

15Eminus10

1Eminus10

5Eminus110

Figure 4 correlation between GNPsrsquo sizes number mass andsurface area for a single cancer cell (a) number of GNP per cell(b) Au masscell (c) surface areacell

0(A) 15 70 150 (B) 150 70 15 (C) 150

GNP size (nm)

Figure 5 Atomic absorption measurement of gold per cell Eachcolumn shows 3 incubations differing in GNP size and order ofapplication

(A) 90 (B) 90 90 (C) 90 30 (D) 90 30 15

GNP size (nm)

014012

01008006004002

Figure 6 Atomic absorption measurement of gold concentrationper cell for various cycles and different sizes of GNPs

have quantitatively investigated the effect of the size of thenanoparticle on its binding probability and on the totalamount of material that can selectively target tumors on asingle cell level We found that 90 nm GNPs are the optimalsize for cell targeting both in terms of maximal Au mass andsurface area per single cell For in vivo applications 90 nm isin the right size range since the particles should be larger thansim15 nm to avoid rapid clearance by the kidneys or uptake inthe liver and smaller than sim150 nm to avoid filtration in thespleen [30] It has been demonstrated that smaller particlesdespite having higher binding probability produce a smalleramount of Au mass per cell as well as a smaller surface areaParticles that are larger than 90 nm only slightly increasedthe Au masscell and decreased the surface areacell It hasbeen also demonstrated that the first incubation is the mostcritical However subsequent incubation can increase theamount of contrast material by about 20 The resultsof this study should accelerate the development of generaldesign principles for the optimal nanoparticle to be used as atargeted imaging contrast agent

References

[1] H Rusinek D P Naidich G McGuinness et al ldquoPulmonarynodule detection low-dose versus conventional CTrdquo Radiol-ogy vol 209 no 1 pp 243ndash249 1998

[2] R Lamerichs ldquoMRI-based molecular imaging using nano-particlesrdquo Cellular Oncology vol 30 no 2 p 100 2008

[3] C Sun O Veiseh J Gunn et al ldquoIn vivo MRI detectionof gliomas by chlorotoxin-conjugated superparamagneticnanoprobesrdquo Small vol 4 no 3 pp 372ndash379 2008

[4] R Kopelman Y E Lee Koo M Philbert et al ldquoMultifunc-tional nanoparticle platforms for in vivo MRI enhancementand photodynamic therapy of a rat brain cancerrdquo Journal ofMagnetism and Magnetic Materials vol 293 no 1 pp 404ndash410 2005

[5] Y E L Koo G R Reddy M Bhojani et al ldquoBrain cancerdiagnosis and therapy with nanoplatformsrdquo Advanced DrugDelivery Reviews vol 58 no 14 pp 1556ndash1577 2006

[6] A M Neubauer H Sim P M Winter et al ldquoNanoparticlepharmacokinetic profiling in vivo using magnetic resonanceimagingrdquo Magnetic Resonance in Medicine vol 60 no 6 pp1353ndash1361 2008

[7] X H Gao and S M Nie ldquoLong-circulating QD probes for in-vivo tumor imagingrdquo Nanosensing Materials and Devices vol5593 pp 292ndash299 2004

[8] P Diagaradjane J M Orenstein-Cardona N E Colon-Casasnovas et al ldquoImaging epidermal growth factor recep-tor expression in vivo pharmacokinetic and biodistributioncharacterization of a bioconjugated quantum dot nanoproberdquoClinical Cancer Research vol 14 no 3 pp 731ndash741 2008

[9] Y Guo D Shi J Lian et al ldquoQuantum dot conjugated hydrox-ylapatite nanoparticles for in vivo imagingrdquo Nanotechnologyvol 19 no 17 Article ID 175102 2008

[10] C Loo A Lowery N Halas J West and R DrezekldquoImmunotargeted nanoshells for integrated cancer imagingand therapyrdquo Nano Letters vol 5 no 4 pp 709ndash711 2005

[11] A M Gobin M H Lee N J Halas W D James R ADrezek and J L West ldquoNear-infrared resonant nanoshells forcombined optical imaging and photothermal cancer therapyrdquoNano Letters vol 7 no 7 pp 1929ndash1934 2007

[12] R Popovtzer A Agrawal N A Kotov et al ldquoTargeted goldnanoparticles enable molecular CT imaging of cancerrdquo NanoLetters vol 8 no 12 pp 4593ndash4596 2008

[13] D Kim S Park H L Jae Y J Yong and S Jon ldquoAntibiofoul-ing polymer-coated gold nanoparticles as a contrast agent forin vivo X-ray computed tomography imagingrdquo Journal of theAmerican Chemical Society vol 129 no 24 pp 7661ndash76652007

[14] J F Hainfeld D N Slatkin T M Focella and H MSmilowitz ldquoGold nanoparticles a new X-ray contrast agentrdquoBritish Journal of Radiology vol 79 no 939 pp 248ndash2532006

[15] K H Su Q H Wei X Zhang J J Mock D R Smith and SSchultz ldquoInterparticle coupling effects on plasmon resonancesof nanogold particlesrdquo Nano Letters vol 3 no 8 pp 1087ndash1090 2003

[16] C T Campbell and G Kim ldquoSPR microscopy and itsapplications to high-throughput analyses of biomolecularbinding events and their kineticsrdquo Biomaterials vol 28 no15 pp 2380ndash2392 2007

[17] P K Jain I H ElSayed and M A El-Sayed ldquoAu nanoparticlestarget cancerrdquo Nano Today vol 2 no 1 pp 18ndash29 2007

[18] I H El-Sayed X Huang and M A El-Sayed ldquoSurfaceplasmon resonance scattering and absorption of anti-EGFRantibody conjugated gold nanoparticles in cancer diagnosticsapplications in oral cancerrdquo Nano Letters vol 5 no 5 pp 829ndash834 2005

[19] E E Connor J Mwamuka A Gole C J Murphy and M DWyatt ldquoGold nanoparticles are taken up by human cells but donot cause acute cytotoxicityrdquo Small vol 1 no 3 pp 325ndash3272005

[20] T S Hauck A A Ghazani and W C W Chan ldquoAssessing theeffect of surface chemistry on gold nanorod uptake toxicityand gene expression in mammalian cellsrdquo Small vol 4 no 1pp 153ndash159 2008

[21] J F Hainfeld M J OrsquoConnor F A Dilmanian D NSlatkin D J Adams and H M Smilowitz ldquoMicro-CTenables microlocalisation and quantification of Her2-targetedgold nanoparticles within tumour regionsrdquo British Journal ofRadiology vol 84 no 1002 pp 526ndash533 2011

[22] N Chanda V Kattumuri R Shukla et al ldquoBombesin func-tionalized gold nanoparticles show in vitro and in vivo cancerreceptor specificityrdquo Proceedings of the National Academy ofSciences of the United States of America vol 107 no 19 pp8760ndash8765 2010

[23] T Reuveni M Motiei Z Romman A Popovtzer and RPopovtzer ldquoTargeted gold nanoparticles enable molecular CTimaging of cancer an in vivo studyrdquo International Journal ofNanomedicine vol 6 pp 2859ndash2864 2011

[24] J Turkevich P C Stevenson and J Hillier ldquoA study of thenucleation and growth processes in the synthesis of colloidalgoldrdquo Discussions of the Faraday Society vol 11 pp 55ndash751951

[25] J Niu T Zhu and Z Liu ldquoOne-step seed-mediated growthof 30-150 nm quasispherical gold nanoparticles with 2-mercaptosuccinic acid as a new reducing agentrdquo Nanotechnol-ogy vol 18 no 32 Article ID 325607 2007

[26] G Frens ldquoControlled nucleation for regulation of particle-sizein monodisperse gold suspensionsrdquo Nature-Physical Sciencevol 241 no 105 pp 20ndash22 1973

[27] W P Wuelfing S M Gross D T Miles and R W MurrayldquoNanometer gold clusters protected by surface-bound mono-layers of thiolated poly(ethylene glycol) polymer electrolyterdquoJournal of the American Chemical Society vol 120 no 48 pp12696ndash12697 1998

[28] R Todd and D T W Wong ldquoEpidermal growth factorreceptor (EGFR) biology and human oral cancerrdquo Histologyand Histopathology vol 14 no 2 pp 491ndash500 1999

[29] P Stanton S Richards J Reeves et al ldquoEpidermal growth fac-tor receptor expression by human squamous cell carcinomasof the head and neck cell lines and xenograftsrdquo British Journalof Cancer vol 70 no 3 pp 427ndash433 1994

[30] F Hallouard N Anton P Choquet A Constantinesco and TVandamme ldquoIodinated blood pool contrast media for preclin-ical X-ray imaging applicationsmdasha reviewrdquo Biomaterials vol31 no 24 pp 6249ndash6268 2010

Electron-Pair Densities with Time-DependentQuantum Monte Carlo

Ivan P Christov

Physics Department Sofia University 1164 Sofia Bulgaria

Correspondence should be addressed to Ivan P Christov ipcphysuni-sofiabg

We use sets of de Broglie-Bohm trajectories to describe the quantum correlation effects which take place between the electronsin helium atom due to exchange and Coulomb interactions A short-range screening of the Coulomb potential is used to modifythe repulsion between the same spin electrons in physical space in order to comply with Paulirsquos exclusion principle By calculatingthe electron-pair density for orthohelium we found that the shape of the exchange hole can be controlled uniquely by a simplescreening parameter For parahelium the interelectronic distance hence the Coulomb hole results from the combined action ofthe Coulomb repulsion and the nonlocal quantum correlations In this way a robust and self-interaction-free approach is presentedto find both the ground state and the time evolution of nonrelativistic quantum systems

1 Introduction

The electronic many-body problem is of key importancefor the theoretical treatments of physics and chemistry Atypical manifestation of the quantum many-body effects isthe electron correlation which results from the Coulomband exchange interactions between the electrons combinedwith the underlying quantum nonlocality Since in generalthe electron correlation reshapes the probability density inconfiguration space it is difficult to elucidate this effect forhigher dimensionsTherefore to better understand the effectsof electron correlation in atoms and molecules one needsbesides one-particle quantities such as the electron densityfunction to consider also extensions which explicitly incor-porate many-body effects Such an appropriate quantity isthe electronic pair-density function which represents theprobability density of finding two electrons at distance u fromeach other [1 2]

119868 (u 119905) = ⟨Ψ (R 119905)10038161003816100381610038161003816100381610038161003816100381610038161003816

119894lt119895

120575 [(r119894minus r119895) minus u]

10038161003816100381610038161003816100381610038161003816100381610038161003816

Ψ (R 119905)⟩

where r119894is the position of the 119894th electron and the many-

body wave function Ψ(R 119905) resides in configuration space

with arguments being the instantaneous coordinates of allelectrons R = (r

1 r2 r

119873)

The importance of the electron-pair density also knownas electron position intracule comes from the fact that itcan be associated with experimental data obtained fromX-ray scattering and it can also be used to visualize thenotion of exchange and correlation holes which surroundthe quantum particles However the calculation of the many-body wave function in (1) is hampered by the computationalcost which scales exponentially with system dimensionalityTherefore different approximations have been employed inorder to calculate the electronic pair densities These includeHartree-Fock (HF) approximation as well as Hylleraas typeexplicitly correlated wave functions represented as productof HF function and pair-correlation factors [3ndash6] Other(eg quantum Monte Carlo [7]) approaches use appropriateSlater-Jastrow-typemany-bodywave functionswhich involvenumber of parameters which after optimization can be usedto calculate the average in (1)

Here we calculate the electron-pair densities for heliumatom in 2 1S and 2 3S states using the recently proposed time-dependent quantum Monte Carlo (TDQMC) method whichemploys sets of particles and quantum waves to describe theground state and the time evolution ofmany-electron systems[8ndash13] In TDQMC each electron is described statistically

as an ensemble of walkers which represent different replicasof that electron in position space where each walker isguided by a separate time-dependent de Broglie-Bohm pilotwave The correlated guiding waves obey a set of coupledtime-dependent Schrodinger equations (TDSE) where theelectron-electron interactions are accounted for using explicitnonlocal Coulomb potentials In the TDQMC algorithmthe preparation of the ground state of the quantum systeminvolves a few steps which include initialization of the MonteCarlo (MC) ensembles of walkers and guide waves followedby their concurrent propagation in complex time towardsteady state in the presence of random component in walkerrsquosmotion to account for the processes of quantum drift anddiffusion Once the ground state is established the real-time quantum dynamics can be studied for example theinteraction of atoms andmolecules with external electromag-netic fields The large speedup of the calculations when usingTDQMCcomes from the fact thatwalkerrsquos distribution repro-duces the amplitude (or modulus square) of the many-bodywave function while its phase is being disregarded as it is notneeded for most applications Also the TDQMCmethod canbe implemented very efficiently on parallel computers wheretens of thousands of coupled Schrodinger equations can besolved concurrently for affordable time

2 General Theory

The TDQMC is an ab initio method with respect to theelectron correlation in that it does not involve explicit pair-correlation factors which may become too complex whenused for larger systems For a system of 119873 electrons themany-body wave function obeys the Schrodinger equation

119894ℎ120597

120597119905Ψ (R 119905) = minus ℎ

2119898nabla2Ψ (R 119905) + 119881 (R) Ψ (R 119905) (2)

where nabla= (nabla1nabla2 nabla

119873) The potential 119881(R) in (2) is a sum

of electron-nuclear electron-electron and external poten-tials

119881 (r1 r

119873) = 119881119890minus119899(r1 r

119873) + 119881119890minus119890(r1 r

119873)

+ 119881ext (r1 r119873 119905) =119873

119896=1

119881119890minus119899(r119896)

119873

119896gt119897

119881119890minus119890(r119896minus r119897) + 119881ext (r1 r119873 119905)

For Hamiltonians with no explicit spin variables theexchange effects can be accounted for efficiently usingscreened Coulomb potentials as described in [10]The simpleidea behind this approach is that the short-range screenedCoulomb potential ensures full-scale Coulomb interactionbetween only electron replicas (MC walkers) which are nottoo close to each other in accordance with Paulirsquos exclusionprinciple The use of screened Coulomb potentials is benefi-cial in that it eliminates the need of using antisymmetrizedproducts of guiding waves in the Broglie-Bohm guiding

equation for the velocity of the walkers Instead the many-body wave function is replaced by a simple product

Ψ119896(r1 r2 r

119873 119905) =

119873

119894=1

120593119896

119894(r119894 119905) (4)

where 120593119896119894(r119894 119905) denote the individual time-dependent guide

waves with indexes 119894 and 119896 for the electrons and the walkersrespectivelyThen the guiding equations for theMonte Carlowalkers read

k (r119896119894) =

119898Im[ 1

120593119896

119894(r119894 119905)nabla119894120593119896

119894(r119894 119905)]

r119894=r119896119894 (119905) (5)

On the other side the guide waves obey a set of coupledTDSE

119894ℎ120597

120597119905120593119896

119894(r119894 119905) =[minus

2119898nabla2

119894+ 119881119890minus119899(r119894)

119873

119895 = 119894

119881eff119890minus119890[r119894minus r119896119895(119905)]

+119881ext (r119894 119905)] 120593119896

119894(r119894 119905)

where the effective electron-electron potential119881eff119890minus119890[r119894minus r119896119895(119905)]

is expressed as a Monte Carlo sum over the smoothed walkerdistribution [9]

119885119896

119895

119872

119897=1

119881scr119890minus119890[r119894minus r119897119895(119905)]119870(

10038161003816100381610038161003816r119897119895(119905) minus r119896

119895(119905)10038161003816100381610038161003816

120590119896

119895(r119896119895 119905)

119885119896

119895=

119872

119897=1

119870(

10038161003816100381610038161003816r119897119895(119905) minus r119896

119895(119905)10038161003816100381610038161003816

120590119896

119895(r119896119895 119905)

where119870 is a smoothing kernel and119885119896119895is the weighting factor

The width 120590119896119895(r119896119895 119905) of the kernel in (7) is a measure for the

characteristic length of nonlocal quantum correlationswithinthe ensemble of walkers which represent the 119895th electronIn practice the parameter 120590119896

119895(r119896119895 119905) is determined by varia-

tionally minimizing the ground state energy of the quantumsystem [13]

In our calculation a Coulomb potential screened by anerror function is used [10]

119881scr119890minus119890[r119894minus r119897119895(119905)] = 119881119890minus119890 [r119894 minus r

119897

119895(119905)] erf [

10038161003816100381610038161003816r119894minus r119897119895(119905)10038161003816100381610038161003816

119903119904

119895120575119904119894119904119895

where the Kronecker symbol 120575119904119894119904119895

restricts the screeningeffect to the repulsion between only the same-spin walkers

while the value of screening parameter 119903119904119894is estimated from

the Hartree-Fock approximationIn the approach outlined previously a self-interaction-

free dynamics in physical space is achieved where the sep-arate walkers do not share guiding waves which representdifferent distributions In order to calculate the many-bodyprobability distribution in configuration space a separateauxiliary set of walkers with primed coordinates r1015840119896

119894is intro-

duced which is guided by an antisymmetric wave function

k1015840 (r1015840119896119894)

119898Im[ 1

Ψ1015840119896 (r10158401 r1015840

119873 119905)nabla119894Ψ1015840119896(r10158401 r1015840

119873 119905)]

r1015840119895=r1015840119896j (119905)

whereΨ1015840119896(r10158401 r1015840

119873 119905) is an antisymmetrized product (Slater

determinant or a sum of Slater determinants) of the time-dependent guide waves 120593119896

119894(r119894 119905) of (6)

Ψ1015840119896(r10158401 r10158402 r1015840

119873 119905) = 119860

119873

119894=1

120593119896

119894(r1015840119894 119905) (11)

From (10) and (11) one can see that each walker withprimed coordinates samples the many-body wave functionand thus it belongs to all guide waves (ie it represents anindistinguishable electron) The distribution of these walkerscan be used to directly estimate the average in (1) by reducingit to (for states with spherical symmetry)

119868 (119906 119905) prop sum

119894

119870119894[

100381610038161003816100381610038161199031015840119894

12(119905) minus 119906

10038161003816100381610038161003816

120590119894

] (12)

where 119903101584011989412(119905) = |r1015840119894

1(119905)minus r1015840119894

2(119905)| In other words the pair-density

function can be simplified to a smoothed histogram (or akernel density estimation with kernel 119870

119894and bandwidth 120590119894

[14]) over the ensemble of the distances between the primedwalkers

3 Exchange and CoulombCorrelations in Helium

The two major sources of electron-electron correlation aredue to the symmetry of the quantum state and due to theCoulomb repulsion Here we consider first the effect of theexchange correlation on the pair-density function of heliumatom Although the electron-pair densities for helium havebeen analyzed by different techniques they have never tothe authorrsquos knowledge been studied using time-dependentmethods

In order to examine the electron correlation which isdue to the exchange interaction we consider the spin-tripletground state of helium (orthohelium)The preparation of theground state is described elsewhere [11 12] In the calculationhere we use up to 100 000 Monte Carlo walkers and the samenumber of guiding waves which are propagated over 2000complex time steps (see (5) through (10)) in the presence

Distance (au)2 4 6 8

minus4

minus8

Distance (au)0 4 8minus4minus8

Figure 1 Radial electron density for the ground state of orthohe-lium forMCwalkers guided in physical space (blue and green lines)and for MC walkers guided in configuration space (red line) Theinset shows the projection of the coordinates of the MC walkers inthe x-y plane

of random component in walkerrsquos motion such that eachwalker samples the distribution given by its own guidingwave In order to determine the screening parameter 119903119904

119894of

(9) we invoke the Hartree-Fock approximation where for120590119896

119895(r119896119895 119905) rarr infin the Coulomb potential in (7) reduces to a

simple (unweighted) sum of the Coulomb potentials due toall walkers Because of the spherical symmetry of the 2 3Sstate 119903119904

119894is being varied until minimizing the mean integrated

squared error of the walkerrsquos distribution against the prob-ability distribution obtained from an independent Hartree-Fock solution (eg in [15]) Figure 1 shows the probabilitydistributions obtained from TDQMC for the optimizingvalue of 119903119904

119894= 119903119904= 113 au in (9) The blue and the green

lines show the densities of the walkers guided in physicalspace (see (5) through (9)) respectively while the red linerepresents the radial distribution of the walkers guided inconfiguration space (see (10)) In these calculations a newaccurate algorithm for kernel density estimation was used[16] Notice that all probability distributions throughout thispaper are normalized to unity

The electron-pair density for the ground state was cal-culated very efficiently by simply performing kernel densityestimation over the ensemble of distances between theprimed walkers The result is shown in Figure 2(a) where theblue and the red lines present the cases with and withoutexchange interaction respectively The lack of exchange(119903119904119894rarr 0 in (9)) leads to a full (unscreened) Coulomb

repulsion which in the limit of infinite nonlocal correlationlength (120590119896

119895(r119896119895 119905) rarr infin) becomes equivalent to the Hartree

approximation Figure 2(b) shows the difference between thetwo curves in Figure 2(a) which in fact depicts the shapeof the exchange hole for the 2 3S state of helium (see alsoeg [5]) Note that the exchange hole in our calculationmay differ from other results because the distribution of theMonte Carlo walkers varies in radial direction as 11990321198772(119903)

155Electron-Pair Densities with Time-Dependent Quantum Monte Carlo

002 4 6 8 10

Interelectronic distance (au)

2 4 6 8 10

minus002

Figure 2 Electron-pair density as function of the interelectronic distance for the ground state of orthohelium (a) Red linemdashno screening(no exchange) blue linemdashshort-range screened Coulomb potentials Exchange hole (b) for screened Coulomb potentials (black) and forHartree-Fock exchange (green)

minus4

minus8

Figure 3 Radial electron density for the ground state of paraheliumfor MC walkers guided in physical space (red line) and fromthe Hartree-Fock approximation (blue line) The inset shows theprojection of the coordinates of the MC walkers in the x-y plane

instead of as 1198772(119903) where 119877(119903) is the radial wave functionThe green line in Figure 2(b) shows the exchange holeobtained from an independentHartree-Fock calculationwithno potential screening It is seen that the two curves areclose where the deviations for larger interelectronic distancesare mainly due to the fast decrease of the walkerrsquos densityaway from the core As the screening parameter 119903119904

119894tends

to zero both the height and the width of the exchange holedecrease until the two curves in Figure 2(b) become veryclose with the only remaining difference being a result ofpurely Coulomb correlations

For the ground state of the 2 1S (para)helium thequantity of interest is the Coulomb hole which occurs due tothe repulsion of the closely spaced walkers Figure 3 showsthe probability distribution of the ground state walkers ascompared to the Hartree-Fock calculation while Figure 4(a)depicts the corresponding interelectronic distances for the

two cases The Coulomb hole calculated as the differencebetween the two curves is presented in Figure 4(b) which isclose to previous results by othermethods [3] As the nonlocalcorrelation length 120590119896

119895(r119896119895 119905) tends to infinity both the height

and the width of the Coulomb hole decrease until the twocurves in Figure 4(b) coincide Thus in our approach wherethe exchange and the Coulomb correlations are accountedfor by solely modifying the potential of electron-electroninteraction in physical space the two parameters 119903119904

119894and

120590119896

119895(r119896119895 119905)may ensure a smooth transition between theHartree

the Hartree-Fock and the fully correlated approximations tothe electron-electron interaction It is important to point outthat in the ℎ119898 rarr 0 limit the quantum drift in (6) vanishesand so does the width of the quantumwave packetThereforefor an isolated atom the quantum correlation length 120590119896

119895(r119896119895 119905)

tends to zero in this limit and if there are no exchange effects(119903119904119894rarr 0) the ensemble of quantum particles governed by (5)

and (6) transforms to an ensemble of classical particles withthe only force being due to the standard Coulomb repulsionbetween these particles

4 Conclusions

In this paper it has been shown that for charged particles thequantum correlation effects which occur due to the exchangeand Coulomb correlations can adequately be described bysets of de Broglie-Bohm walkers within the time-dependentquantum Monte Carlo framework A short-range screeningof the Coulomb potential ensures that each replica of agiven electron interacts with only those replicas of the restof the same spin electrons which are sufficiently apart torespect Paulirsquos exclusion principle in space On the otherhand the electron-electron interaction is modified by thequantum nonlocality which demands that each replica ofa given electron interacts with the replicas of the otherelectrons which are within the range of the nonlocal quantumcorrelation length This concept allows one to build a robustself-consistent and self-interaction-free approach to find

02Prob

2 31Interelectronic distance (au)

minus008

minus004

Figure 4 Electron-pair density as function of the interelectronic distance for the ground state of parahelium (a) Red linemdashcorrelated resultblue linemdashHartree-Fock approximation The Coulomb hole (b)

both the ground state and the time evolution of quantumsystems It is demonstrated here that the otherwise awkwardprocedure for calculating the pair distribution functions ofpara- and orthohelium atom can be simplified to the levelof finding the ground state probability distributions of thecorresponding Monte Carlo walkers

Besides the relative ease of its implementation anotheradvantage of using TDQMC is the affordable time scalingit offers which is almost linear with the system dimension-ality This is especially valid when using multicore parallelcomputers where little communication overhead between thedifferent processes can be achieved thus utilizing the inher-ent parallelism of the Monte Carlo methods This nears theTDQMC to other efficient procedures for treating many-body quantum dynamics such as the time-dependent densityfunctional approximation which however suffers systematicself-interaction problems due to the semiempirical characterof the exchange-correlation potentials

Acknowledgments

The author gratefully acknowledges support from theNational Science Fund of Bulgaria under Grant DCVP 021(SuperCA++) Computational resources from the NationalSupercomputer Center (Sofia) are gratefully appreciated

References

[1] A J Coleman ldquoDensitymatrices in the quantum theory ofmat-ter energy intracules and extraculesrdquo International Journal ofQuantum Chemistry vol 1 supplement 1 pp 457ndash464 1967

[2] A J Thakkar ldquoExtracules intracules correlation holes poten-tials coefficients and all thatrdquo in Density Matrices and DensityFunctionals R Erdahl and V H Smith Jr Eds pp 553ndash581Reidel New York NY USA 1987

[3] C A Coulson and A H Neilson ldquoElectron correlation in theground state of heliumrdquo Proceedings of the Physical Society vol78 no 5 p 831 1961

[4] R J Boyd andC A Coulson ldquoTheFermi hole in atomsrdquo Journalof Physics B vol 7 no 14 pp 1805ndash1816 1974

[5] N Moiseyev J Katriel and R J Boyd ldquoOn the Fermi hole inatomsrdquo Journal of Physics B vol 8 no 8 pp L130ndashL133 1975

[6] P M W Gill D OrsquoNeill and N A Besley ldquoTwo-electron dis-tribution functions and intraculesrdquo Theoretical ChemistryAccounts vol 109 no 5 pp 241ndash250 2003

[7] B M Austin D Y Zubarev andW A Lester ldquoQuantummontecarlo and related approachesrdquo Chemical Reviews vol 112 no 1pp 263ndash288 2012

[8] I P Christov ldquoCorrelated non-perturbative electron dynamicswith quantum trajectoriesrdquo Optics Express vol 14 no 15 pp6906ndash6911 2006

[9] I P Christov ldquoDynamic correlations with time-dependentquantumMonte Carlordquo Journal of Chemical Physics vol 128 no24 Article ID 244106 2008

[10] I P Christov ldquoPolynomial-time-scaling quantum dynamicswith time-dependent quantum Monte Carlordquo The Journal ofPhysical Chemistry A vol 113 pp 6016ndash6021 2009

[11] I P Christov ldquoCorrelated electron dynamics with time-depend-ent quantum Monte Carlo three-dimensional heliumrdquo Journalof Chemical Physics vol 135 no 4 Article ID 044120 2011

[12] I P Christov ldquoErratum ldquoCorrelated electron dynamics withtime-dependent quantum Monte Carlo three-dimensional he-liumrdquordquo Journal of Chemical Physics vol 135 no 14 Article ID149902 2011

[13] I P Christov ldquoExploring quantumnon-locality with de Broglie-Bohm trajectoriesrdquo Journal of Chemical Physics vol 136 no 3Article ID 034116 2012

[14] BW SilvermanDensity Estimation for Statistics andDataAnal-ysisMonographs on Statistics andApplied Probability Chapmanand Hall London UK 1986

[15] S E Koonin and D C Meredith Computational PhysicsAddison-Wesley 1990

[16] Z I Botev J F Grotowski and D P Kroese ldquoKernel densityestimation via diffusionrdquoThe Annals of Statistics vol 38 no 5pp 2916ndash2957 2010

Multispark Discharge in Water as a Method ofEnvironmental Sustainability Problems Solution

E M Barkhudarov1 I A Kossyi1 Yu N Kozlov2 S M Temchin1

M I Taktakishvili1 and Nick Christofi3

1 AM Prokhorov General Physics Institute of RAS (GPI RAS) Vavilov Street 38 Moscow 119991 Russia2 Semenov Institute of Chemical Physics of RAS Kosygin Street 4 Moscow 119991 Russia3 Edinburgh University Edinburgh EH9 3JF UK

Correspondence should be addressed to I A Kossyi kossyifplgpiru

Academic Editor Elena Tatarova

Multispark discharge excited in water is described and its useful physical and chemical properties are discussed in the light ofsome environmental issues Discharge of such a type generates hot and dense plasmoids producing intense biologically activeUV radiation and chemically active radicals atoms and molecules Simultaneously discharge creates strong hydrodynamicperturbations and cavitation bubbles Particular attention is given to factors influencing on water purity with special reference todischarge application for effective sterilization of water and its cleaning of harmful chemicals The gas discharges of this type showconsiderable promise as a means for solving some actual plasma-chemical problems The above-mentioned discharge propertieshave been demonstrated in a series of laboratory experiments which proved the efficiency of disinfection of potable and wastewater water cleaning of pesticide (herbicide) contaminations and conversion (recovery) of natural methane

1 Introduction

High voltage electric discharge inwater [1 2] has been consid-ered as a potential method of water treatment to kill microor-ganisms and to clean it of harmful contaminations negatingthe use of chemicals that leads to by-products which mayadditionally compromise human health [3ndash5] Factors favor-ing their use include the generation of UV radiation acous-tic shock waves chemically active substances cavitationprocesses pyrolysis and hydrolysis There are also possiblesynergetic effects following physical and chemical reactions

Among the differentmeans of in-liquid electric dischargea novel method involvesmultielectrode (multispark) slipping(gliding) discharges (SSDs) [6] which may have some advan-tages over the two-electrode systems generally used at present[1 7]

The present work describes the construction of a multi-spark discharger and discusses results of experimental inves-tigation of SSD-basedmethods ofwater disinfection and theirapplication in plasma-chemical technology for solving some

of environmental problems such as conversion (recovery) ofmethane (as well as other natural hydrocarbons) and watercleaning of pesticide (herbicide) contamination

2 Treatment System

The apparatus used to treat liquids is shown schematically inFigures 1 and 2 The basic components were a chamber filledwith water a multielectrode system for exciting of slippingsurface discharge and high voltage power supply (Figure 1)The multielectrode discharge system (Figure 2) was similarin design to that previously described in [6 8 9] Thedischarger consisted of a set of annular electrodes mountedon a dielectric tube surrounding a back-current conductorA gas (air argon oxygen etc) was injected through a set ofholes into water between the electrodes producing fine gasbubbles Discharge in each interelectrode gap was producedthroughout the system including the metal electrodes adielectric substrate a gas bubble and water

Air or O2

bubbles

Microbiallycontaminated

Clean water

Figure 1 Scheme of multispark discharge disinfection of water (1) Chamber (2)multispark discharger (3) generator of high voltage pulses(4) cleaning water (5) plasma of gliding discharge

Air O2

Figure 2 Multielectrode gliding surface discharge facility (1) Electrodes (2 3) dielectric tube (4) back-current rod (5) discharge plasma(6) gas bubbles

The initial plasma channel may be thought as originatingin ordinary gas discharge in a gas bubble if the electric fieldtherein is higher than the gas breakdown threshold [10 11]But in actual fact a large (sometimes dominant) part in theinterelectrode plasma formation could be played by a glidingdischarge along the dielectric surface with the subsequentinteraction of discharge plasma with electrodes and explosivemicroplasma production on their surface [12] (see Figure 3)There are just these processes that have been considered tobe operative in the case when multispark discharger works inthe gas medium [13]

When the high voltage pulse is applied to the immersedin the aqueous medium discharger (shown in Figure 2)plasma bunches (plasmoids) appear almost simultaneouslybetween electrodes Reasoning from their characteristicsthese plasmoids can be classified (in accordance with the

recently adopted terminology) as ldquomicroplasmardquo formationsinvolved in various applications [14] According to the resultsof previously performed experiments the electron density inplasmoids attains 1017 cmminus3 and the gas temperature 4000ndash5000K [15] According to [16] explosive metallic plasma is asource of intensive hard UV radiation

A typical photograph of the operating system is shown inFigure 4

The principal advantage of the multispark system lies inthe following peculiarities of their construction

(i) The area of the surface of all electrodes contactingwater in the multielectrode version can be minimizedby introducing insulating dielectric screens ensuringthe SSD operation in high-conducting water (upto conductivities of 104 120583S cmminus1) without substantial

159Multispark Discharge in Water as a Method of Environmental Sustainability Problems Solution

GasGas

6GasGas

Figure 3 Two consecutive phases of plasma production in each interelectrode gap (1) Electrodes (2) dielectric tube (3) back-current rod(4) gliding surface discharge (5)metallic plasma (6) unipolar arc

Figure 4 Typical photograph of multispark discharger operating inwater

reduction of the efficiency of energy supply to thedischarge region

(ii) Thedischarger has no pointed electrodes theworkingsurface of the electrodes (unprotected by dielectricscreens) is developed and is either a part of cylindricalsurface of tubular electrodes or the plane surfaceat the exit sections of the tube Thus the principaladvantage of SSD system lies in the decrease in thedischarge load of each electrode (thereby enhancingthe erosion resistance on the system as a whole)which ultimately substantially increases the lifetimeof the system

(iii) The dischargers can affect the aqueous (liquid)medium through several simultaneously actingmechanisms among them the direct influenceof discharge plasma the action of UV radiationgenerated by microscopic discharges the chemicalaction of chemically active radicals atoms andmolecules produced in discharges and the hydro-dynamic action through microscopic cavitationbubbles

(iv) Cleansing action and bactericidal effect of a multi-spark discharge in the water medium unessentiallydepend on electrode material Nevertheless among

the tested metals (Fe Mo Cu Ti etc) just stainlesssteel and titanium have been selected as materialsexhibiting the most promise for working as a detailof multispark discharger Just these two metals havebeen used in electrodischarge systems applied in theGeneral Physics Institute (GPI RAS) for solution ofwater purification problems

(v) The discharge gaps could be distributed in such a wayas to increase the efficiency of the discharge action onliquids in particular by focusing the shock waves andUV radiation flux [17]

The experiments were conducted using the high volt-age multichannel (5 channels) generator with the followingparameters high voltage amplitude 119880 le 20 kV pulserepetition frequency119891 le 100Hz capacitive storage energy ofone channel119882 le 2 J andpulse duration 120591 asymp 5 120583sThe circuitof the output stage of each channel is shown in Figure 5 Eachmultispark discharger was powered from one channel of amultichannel generator The discharge current and voltagewere measured with the aid of a Rogowski coil and voltagedividerThe signals shownon the Figure 6were recordedwithan oscilloscope (TDS 3012) These measurements allowedthe determination of the energy density (J cmminus3) released inliquid

3 Multispark Electric Discharge in Wateras a Source of UV Radiation Ozone andHydrogen Peroxide

Figure 7 shows a schematic of the experiment intended toinvestigate a multispark SSD in water as a source of UVradiation ozone and hydrogen peroxide Multielectrodedischarger (2) is positioned in a cell (1) with water Ahigh voltage pulse produces a plasma channel between theelectrodes The gas leaving the reactor (as a working gasair or oxygen has been applied) flows into a quartz cell (3)intended for determining the ozone content by the methodof absorption spectroscopy In the course of the experimentsthe production ofH

2O2was alsomeasured UV radiationwas

measured in the presence and absence of water in the reactorchamber

Figure 5 Output stage of one channel of the high voltage pulsespower supply (1)Rogowski coil (2) voltage divider119877

11198772-resistors

119862-capacitor 119871-inductor

5 120583s

Figure 6 Typical oscillograph trace of SSD current and voltage

The discharge emission spectrum in the region 230 lt

120582119889lt 300 nm was measured with the help of an MUM-1

monochromator ((8) Figure 7) and with an FEU-142 photo-multiplier Typical spectra of UV emission from the dischargeare shown in Figure 8

Chemical (actinometric) measurements have been usedas well In this case the UV intensity was deduced fromphotolysis of an irradiated K

3Fe(C2O4)3solution with a

phenanthroline admixtureThis techniquewas described in [18] and successfully used

in [13] to study the multispark discharge in gaseous (Ar)medium

To measure the O3content in the gas flowing from the

reactor we used both spectroscopic and chemical methodsThe scheme of measurements of the O

3content in O

shown in Figure 7 From attenuation of the UV radiationpassing through the cell the O

3density in the gas was

determined by the absorption method The spectral intervalused to determine the ozone content corresponded to theHartley absorption band with the maximum near 120582

119889cong

2555 nm

Air O2 + O3

Air O2

Figure 7 Experimental layout (1) Vessel filled with water (2)multispark discharger (3) diagnostic quartz cell (4) deuteriumlamp (5) discharge plasma (6) MDR-3 monochromator (7) gasbubbles (8)MUM-1 monochromator and (9) quartz window

240 250 260 270 280 290120582 (nm)

Figure 8 Spectrum of soft UV radiation frommultispark dischargein the water

In the case of application of air as working gas theO3content was determined by the chemical method from

the reaction between O3and potassium iodide in the water

solution [19]Figure 9 shows the ozone density in the diagnostic cell as

a function of the repetition frequency of high voltage pulses(119891) for a discharge in water (for various oxygen flow rates)Restriction of 119891 values by amounts of the order of 100Hzis not critical and appears explicable only on the basis ofimproper technical equipment of laboratory

In the experiments when the oxygen flow rate throughthe interelectrode gaps and the water-filled reactor was 119908 cong

15 Lmin the ozone density in the oxygen flow was equal to119899O3 cong (1-2) 1015 cmminus3

The H2O2

content in water treated by the electricdischarge was measured by the iodide-molybdate method

0 20 40 60 80 100f (Hz)

18161412

108060402

O3)times10

mminus3)

Figure 9 Ozone density in the diagnostic cell as a function ofthe repetition frequency of multispark discharge in the tap waterfor various flow rates of O

2 ◼-119908 = 10 Lminminus1 -15 Lminminus1 998771-

20 Lminminus1

described in [20] andused in [13] to determine the intensity ofhard UV radiation of the gliding surface discharge in argon

Themeasurements of hydrogen peroxide production thatwere carried out in a discharge in water with injected argonshowed that a series of discharges for 6-7 minutes in 250 cm3of water produced H

2O2with a mean density of 119899H2O2 cong 2 sdot

10minus3mol Lminus1 cong 12 sdot 1018 cmminus3 The energy cost of production

of one H2O2molecule in this case is ℎH2O2 le 15 sdot 10

2 eVmolThe performed experiments demonstrated that for the

SSD in the water-gas mixture at least two factors are real-ized among the factors that are usually invoked to explainthe sterilization effect of electric discharges These are thegeneration ofUV radiation and the production of biologicallyactive ozone and hydrogen peroxide

It is possible to estimate using the results of measure-ments the effectiveness of these two factors in the degra-dation of microorganisms during operation of the electric-discharge systems under study

Examining the UV radiation from the discharge wehave to take into consideration that according to [21] thestrongest bactericidal effect is produced by ultraviolet rayswith wavelengths from 295 to 220 nm (the ldquobactericidalrdquospectral region)

Measurements performed in our work (see [9]) showedthat the radiation spectrum of the multispark discharge inwater contains the biologically active component and theintensity of this component increases substantially as thepulse energy increases

Based on the results of absolute measurements of UVradiation by the actinometric method we estimate the inten-sity of the flux of bactericidal rays per pulse discharge as119875UV(119894) asymp 3 sdot 10

6120583Wcm2 [9]

Given this intensity in turn the effectiveness of the actionof radiation on E coli bacteria can be estimated from theknown relation [21]

119899119887cong 1198991198870exp(

minus119875UV119905119886119896119887

where 119899119887is the number of bacteria in a unit volume that

remain living after bactericidal irradiation (cmminus3) 1198991198870is the

initial number of bacteria in a unit volume (cmminus3) 119875UV isthe mean intensity of the flux of bactericidal rays (120583Wcmminus2)119905119886is the irradiation time (s) and 119896

119887= 2500 is the bacterial

tolerance factorFor the case of repetitive discharge expression (1) can be

rewritten in the form

119899119887cong 1198991198870exp(

minus119875UV(119894)120591119891119905119886

119896119887

where 120591 is pulse duration (s) and119891 is the repetition frequencyof high voltage pulses (Hz)

It is easy to see that for 119875UV(119894) sim 3 sdot 106120583Wcmminus2 120591 = 5 120583s

and 119891 = 100Hz the exposure time equal to a few secondsis sufficient to decrease the number of bacteria in water bya factor of ten This means that the energy cost of treatingwater by bactericidal UV rays is of the order of 120585UV asymp (1-2) 10minus4 kWhLminus1

Under the experimental arrangement shown in Figure 7ozone generated in the discharge has no time to dissolve inwater and is almost completely removed by the air (oxygen)flow into the space over the water reactor In principle it ispossible to construct a reactor such that the produced ozonewill be completely ldquoentrappedrdquo in the water being treated Letus estimate how effective the role of ozone in the sterilizationaction of discharge may be in this case

As follows from the data presented in [19] the effect ofozone dissolved in water on microorganisms becomes signif-icantly stronger when the O

3content reaches the threshold

level [119899O3]th cong 8 sdot 1016 cmminus3 Over [119899O3]th the E coli bacteria

content decreases by more than four orders of magnitudeIt is easy to see that the bactericidal treatment capacity of

ozone can be as high as

119908O3 cong119899O3119908119887

[119899O3]th

cong 25 L hminus1 (3)

where 119908O3 is the water-treatment rate (L hminus1) and 119908119887is the

air flow rate through the discharge facility (L hminus1) Then theenergy cost of water treatment by ozone generated in thedischarge (assuming that it is completely dissolved in water)can reach 120585O3 cong 3 sdot 10

minus4 kWhLminus1 which is comparable withthe energy cost of sterilization by UV radiation

Finally we estimate the effectiveness of a possible bac-tericidal action of the multispark discharge in water due tothe production of hydrogen peroxide Specialmicrobiologicalstudies carried out by us showed that an addition of hydrogenperoxide as a level of 119899H2O2 sim 10

17 cmminus3 to tap water allowsthe number of E coli bacteria to be reduced by one order ofmagnitudeThismeans that the experimentallymeasured rateofH2O2production ensures the energy cost of water steriliza-

tion at the level 120585H2O2 sim 10minus4 kWhLminus1 which is close to the

energy cost of sterilization by ozone production in dischargeHence the performed direct measurements of UV radia-

tion and chemically active products evidence that describedbelow multispark slipping surface discharge (SSD) in waterwith air as an working gas is promising for water sterilizationsince two effects only examined in our work can ensure the

energy cost as low as 120585 cong 10minus4 kWhLminus1 for reducing the Ecoli bacteria content by one order of magnitude (ie with agenerator with a mean power of 1 kW it is possible to reach awater treatment rate of the order of 10m3 hminus1)

It should be pointed out that possibility to apply formultispark discharger excitation of practically every gas orgaseous mixtures offers great opportunities for action on amicrobiological component through the different chemicallyactive atoms and radicals However in this work authors haverestricted for water sterilization by the application only of airor oxygen taking into account that based on application ofthese gases discharger will be simplex and cheapest

4 Multispark Electric Discharge Disinfectionof Microbially Contaminated Liquids

As a step of our activity experimental investigation of effec-tiveness of disinfection action of multispark discharge on thewater containing Escherichia coli and its viruses (coliphages)has been carried out [22]

The apparatus used to treat liquids is the same asshown schematically in Figure 1 The discharge devicemdashmultispark dischargermdashwas situated in the treatment cham-ber through which water contaminated withmicroorganismswas pumped Water contaminated with E coli or viruses(somatic coliphages) can be used to test the killing efficienciesof the discharge system Samples of water for microbiologicalanalyses were taken via a sampling port triplicate samples in10 mL sterile bottles being removed for analysis

Escherichia coli (NCIMB 86 ATCC 4157) was grownovernight in nutrient broth (oxoid) at 37∘CThe cultures werediluted to population densities of approximately 106 cfumLminus1with tap water and placed in treatment chamber containingthe multispark discharger

Water samples treated by the electric discharges wereremoved from the system at varying time intervals and bacte-rial killing assessed using spread plate countingmethodologyEscherichia coli was determined by spreading 100 120583L aliquotsof diluted samples onto nutrient agar plates OccasionallyMacConcey agar (HMSO 1994) and a spiral platter wereutilized Replicate plates were incubated at 37∘C for 24 hColiphages were estimated by a plaque assay utilizing E coliC (ATCC 13706) as the host bacterium Dilutions of treatedsamples were spread onto lawns of E coli C sensitive to abroad spectrum of coliphages and the number of plaquesformed after 24 h incubation counted

Figure 10 shows the effect of multispark discharges onmicroorganisms in the water The fraction of surviving bac-teria and viruses (119873119873

0) is plotted versus the energy density

(J cmminus3) released in water Each point in the plot presentsthe mean of three measurements Deviation from the meandid not exceed 15 Numerous experiments were carried outusing E coli and all showed a similar killing efficiency ofthe multispark discharge system Data of microbial killingin liquids containing tap water-microbe combinations and aconductivity of 100120583S cmminus1 are presented It is evident fromFigure 10 that the viruses were killed using a lower energyinput to the liquid Escherichia coli required an energy input

1Eminus3

1Eminus4

1Eminus50 05 1 15 2

J (cmminus3)

Figure 10 Changes in populations of Escherichia coli and viruses(119873) in treated water relative to the initial populations (119873

0) as a

function of specific energy release (J cmminus3) during the treatmentPotable water with a conductivity 120590 = 100 120583S cmminus1 was used 119891 =

10Hz The initial (1198730) concentration of E coli was asymp106 colony-

forming units mLminus1 and that of coliphages asymp107 plaque-formingunits mLminus1 (1) E coli (2) coliphages

of 03 J cmminus3 (approx 10minus4 kWhLminus1) to reduce the populationby a factor 10 (1 log reduction) while coliphages required anenergy input of 015 J cmminus3 for the same result

The usedmultispark discharger regimes are identical withthe regimes previously investigated [9] where an examina-tion was made of the generation of biologically active UVradiation ozone hydrogen peroxide and other active species(see preceding section of this paper) Measurements carriedout during the present study allowed calculation of energycosts of the disinfection action using multispark electric dis-chargers and these were as low as 10minus4 kWhLminus1 for bacteriaThese values verified the bacterial action of discharges in thewater predicted in the preceding section and confirmed thatthe main factors affecting microbial destruction in the waterwere UV radiation and the production of biologically activechemicals The latter are not involved in treatment systemsutilizing UV lamps which would be unable to generatehighly reactive chemical species Acoustic and shock wavesgenerated by multispark discharge also played a part inmicrobial disinfection but in addition they facilitated themixing of treated water delivering reactive chemical speciesto all parts of the treatment system

The possibility that disinfection using electric dischargesmight lead to the production of toxic by-products was testedby the input of energy as high as sim1 J cmminus3 into water Watersamples were analyzed for a range of substances and physicalappearance by the Certification Control-Analytical Center(Moscow State University Russia) The water was tested forcolor turbidity pH ammonium Fe Pb Cr fluorite chloritenitrate and sulphateThe quality of the treated water fulfilledthe necessary standards of the European Union (CouncilDirectives on the quality of water intended for human con-sumption 80778EEC and the new drinking water Directive9883EC adopted by the Council on 3 November 1998) The

results for Fe were particularly important as the electrodesused in the study were manufactured from stainless steelErosion of multispark discharger is small and does not affectoverall concentrations in water In addition incubationsof multispark discharge treated water with microorganismswere carried out to test whether the killing action con-tinued This could be due to the persistence of oxidizingspecies produced by the discharge but these were rapidlyquenched within the system following treatment There wereno increased effects on E coli added to system containingplasma treated compared with nonplasma-treated tap waterThis is contrary to results obtained with two-electrode dis-charges [1] and could be explained by quite low level ofoperated multispark discharger electrodes sputtering and asa result extremely low level (in comparison with the two-electrode system) of content of metallic clusters responsible(according to [1]) for prolonged action of discharge on amicrobial population It is of interest to note that amultisparkdischarge treatment of short duration could sterilize tapwatercontaining E coli and coliphage The duration was shortenough for the cost-effective treatment of water supplies(lt5min) contact time being in the region of minutes ratherthan the 30mins used in chlorination

This study concentrated on verifying the predictions ofmicrobial killing made originally in [9] and utilized E coliand coliphage as representative organisms No attempt hasbeen made at this stage to examine the effect of multisparkdischarge plasma on the other bacteria (Gram-positive or-negative types) viruses or spores (bacterial or fungal)Preliminary experiments have been performed to determineonly the effect of multispark plasma on the oocysts of Cryp-tosporidium (a protozoan parasite causing gastrointestinaldisorders) which are resistant to chlorination The micro-scopic examination of cysts after treatment showed cell walldegradation and an inability to induce excystation in theorganism

It is of interest to investigate the possibility of using themultispark system described to treat industrial and domesticwastewater The first attempt at such an application has beentaken in [8 23] Water treatment was carried out usingwastewater directly abstracted from final effluent stream atthe Livingston Wastewater Treatment Plant in West LothianScotlandUKThe scheme of system forwastewater treatmentis shown in Figure 11 Results of SSD action on a finaleffluent stream are presented in Figure 12 It was shownthat a specific energy of 125ndash15 J cmminus3 was required toachieve 1 log reduction in bacterial (faecal coliformstotalaerobic heterotrophs) content This study has demonstratedthe effectiveness of the multispark dischargers in microbialdisinfection of wastewater The system can be engineeredto eradicate microbial populations to levels governed bylegislation by increasing treatment time or energy input

5 Plasma-Chemical Converter of Methane onthe Basis of Multielectrode Discharger

One from the currently important ecological problem con-sists in utilization of gases accompanying oil recovery Yearly

more than billion cubic meters of associated gases areburning down worldwide Russian oil producing companiesfor compensation of an ecological harm are paying near 500rubles for each 1000m3 of burning petroleum gas

Presented work objective is the investigation of possi-bility of natural hydrocarbons (namely CH

4) recovery in

plasma-chemical reactor based on the SSD Traditional forGPI research multispark dischargers have been used withonly one key distinctive feature of their construction as adischarge formative gas methane (or any other utilizablenatural hydrocarbons) has been applied

The diagram of the experiment is shown schematically inFigure 13 A multielectrode discharger is introduced into thereaction chamber in the form of an organic glass vessel filledwith water (volume 119881 sim 025 L) When a high-voltage pulseis applied to the discharger a system of plasma formations(plasmoids) in which the decomposition of hydrocarbonstakes place is formed in bubbles of methane or methane-oxygen mixture in the gaps between the electrodes Thesource of high voltage pulseswas a generator producing singlepulses or operating in the pulse-periodic regime The pulse-repetition rate was 119891 le 50Hz the pulse duration was 120591

119901asymp

1 120583s and the pulse amplitude was 119880119901asymp 40 kV

We analyzed samples of the gas taken at the outlet ofthe reaction volume Analysis of the gas passing through thedischarger was carried out using the following techniques

(i) special ITT IKVP test tubes (OOO Impulrsquos) used fordetermining the contents of acetylene (C

2H2) carbon

dioxide (CO2) and carbon monoxide (CO)

(ii) SPECORD IR spectrograph used for determining theacetylene content

(iii) gas chromatograph used for determining the concen-tration of methane (CH

4) and hydrogen (H

Figure 14 shows the characteristic spectrograms obtainedon the SPECORD IR spectrographThemain absorption linesof CH

4 C2H2 and CO can be distinguished (in subsequent

analysis of the experimental results CO was disregarded)The lines of the nearest unsaturated hydrocarbon ethyleneC2H4are also very weak (at the noise level) In analysis of

the efficiency of the plasma-chemical conversion of methaneit is expedient (see [24]) to use such parameters as thedegree of conversion120572 expressed in fractions (in other wordsthe fraction of methane fed to the reactor and convertedinto a certain product at the output) and the energy value120576 of conversion (ie the energy value of transformationof methane molecules in eVmolecule) If we disregard forsimplicity the small amounts of ethylene formed as a result ofmethane treatment we can assume that mainly two reactionsoccur in the plasma-chemical reactor pyrolysis reaction

CH4997888rarr C + 2H

and the reaction of transformation ofmethane into acetylene

2CH4997888rarr C

2H2+ 3H2 (5)

It can be seen from simplified reaction formulas (4) and(5) that the volume of the reaction products exceeds the

High voltage powersupply and pulse generator

Electrical connections

Reactionchamber 2 withSSD electrodes

Treatedwastewater

Untreatedwastewater

Figure 11 Diagrammatic representation of continuous wastewater treatment using system of multispark dischargers

minus05

minus1

minus15

minus2

minus25

minus3

minus35

minus4

Specific energy J (cmminus3)0 1 2 3 4 5

Figure 12 Log bacterial population (1198731198730) changes versus specific

energy released inwater during themultispark discharger operation998771-Total aerobic heterotrophic bacteria (22∘C) ◼-faecal coliforms(37∘C)

volume of the primary mixture For this reason the mea-surements of concentration of methane and decompositionproducts at the reactor outlet cannot be directly used forestimating the degree of conversion

It can easily be shown [24 25] that the degree of con-version 120572

1of methane into carbon and hydrogen according

to reaction (4) and the degree of conversion 1205722of methane

into acetylene according to reaction (5) are connected withexperimentally determined concentrations 119862CH4 119862C2H2 and119862H2 by the relations

1205721=

4119862H2 minus 3 (1 minus 119862CH4)

1 + 119862CH4

1205722=

4 (1 minus 119862H2 minus 119862CH4)

1 + 119862CH4

1205720=

1 minus 119862CH41 + 119862CH4

Sampling

412356

Figure 13 Schematic of the experiment (1) Dielectric tube (2)annular electrodes (3) working gas (CH

4) bubbles (4) water (5)

plasma in the interelectrode gaps (6) reaction chamber

where 1205720= 1205721+ 1205722is the total degree of conversion of

methane over channels (4) and (5) which is determined inthe given experiment

The energy value of the reaction of decomposition of amethane molecule (in other words the value of formation ofproducts) is defined by the relation

120576119899=

119875

120572119899119902CH4

500 1000 1500 2000 2500 3000 3500 4000120582 (cmminus1)

618 94

2160 21

3864 3900

Figure 14 Characteristic adsorption IR spectrum of a working gassample taken at the reactor outlet

where 119899 = 0 1 2 is the power supplied to the reactor 119902CH4 isthe methane flow rate and 119875 is the average microwave power

The dependences of flow rate 119902CH4 of methane and ofthe energy value on its decomposition and the formation ofproducts on the degree of conversion of methane are shownin Figures 15 and 16

The dependence of the degree of conversion of methaneon its flow rate shown in Figure 15 closely fits to the inverseproportionality function

1205720=

119860

119902CH4 (8)

Using iterations we find that 119860 = 002809 Lmin The factthat experimental points fit well to functional dependence(8) suggests that this dependence is preserved in a certaininterval of 119902CH4 beyond the range of the values studiedexperimentally This in turn raises hopes that if we couldimplement a regime with the methane flow rate on the orderof 01 Lmin the degree of conversion would increase to sim

28 The same results could be obtained by increasing therepetition rate of discharge pulses to 1 kHz for a methaneflow rate of 1 Lmin By increasing the pulse repetition rate to3 kHz for the samemethane flow rate we could reach a degreeof conversion as high as 84The implementation of basicallyattainable degrees of conversion involves modernization ofthe generator of high voltage pulses and the design ofthe discharger which will form the basis of subsequentexperimental investigation It is evident that without specialjustification these increased degrees of conversion are lookingrather as a wishful thinking

It can be seen from Figure 16 that the energy valueof the conversion is almost independent of the methaneflow rate and amounts to approximately 5 eVmolecule Suchenergy value is close to record-low values for the atmosphericpressure (see eg [25])

The fact that the energy value of conversion is almostindependent of the methane flow rate in the entire range ofits variation in the experiment is an additional argument infavor of the possibility of a substantial increase in the degreeof conversion due to passage to small values of 119902CH4

0 1 2 3 4

1205900

qCH4(1min)

Figure 15 Dependence of the total degree of conversion of methaneon its flow rate

qCH4(1min)

0 05 1 15 2 25 3 35 4

st120576 0

Figure 16 Dependence of the energy value of conversion ofmethane on its flow rate

Analysis made in [26] shows that a high efficiency ofmethane conversion processes characteristic of the describedtechnology is due to peculiarities contained just in dischargeslocalized in an interelectrode gaps Fast heating (up to4000ndash5000K) of the gas propagating between the electrodesthrough the area occupied by microplasma leads to theeffective decomposition of hydrocarbon At the same timefast cooling of the gas penetrating into the surrounding wateris followed by quenching phenomena and the level of theparent-gas decomposition does not change

The low energy value of methane decomposition andthe possibility of elevating the degree of conversion justifythe application of the method of plasma-chemical action forsolving the topical problem of recovery of natural blowouts ofhydrocarbons In this connection the role of pyrolysis in themethane decomposition is of interest in its own right If thecontribution from reaction (4) is significant it is expedientto determine the form and efficiency of the production ofcarbon accompanying the decomposition of methane

The experiments performed in accordance with the dia-gram in Figure 13 have shown that if themultispark dischargeis initiated in water using CH

4as the bubble-forming gas

most carbon particles appearing in water as a result ofplasma-chemical decomposition of methane precipitate

Analysis of the precipitate shows that its main partis nanosize carbon Figure 17 shows the characteristic sizedistribution of carbon particles as a function of the timeof electric-discharge treatment of water which was deter-mined using Fotokor dynamic scattering spectrometer Thetypical photograph of nanocarbon produced in the courseof methane recovery by multispark discharge in the water isshown on the Figure 18

The rate of production of nanosized particles in thedischarge which was determined by evaporation of the SSD-processed liquid and weighing the precipitate was about35mgh This means that the energy value of productionof nanocarbon upon decomposition of methane in the SSDis 03 kWhg The measured value is close to that obtainedfor arc discharges with carbon electrodes in water in whichcarbon is formed in the liquid as a result of destruction of theelectrodes [27 28]

The structure of the precipitate was determined using aLAB RAM HR 800 Raman spectrometer from the Ramanshift Fractions of disordered graphite and carbon weredetected

6 Water Cleaning of24-Dichlorophenoxyacetic Additive

Polychlorinated biphenyls (PCBs) among man-made pollu-tions deserve particular attention These compounds weresynthesized in 1920 s and with their advent new materialswith unique thermophysical and electrical insulating prop-erties became available

However in spite of the presence of a number of uniqueproperties these compoundswerewithdrawn from industrialprocesses already in 1970 s This is due to the fact that PCBswere implicated in a number of incidents in different coun-tries by causing mass intoxication and exerting a detrimentaleffect on the health of humans on a large scale

The PCBs are no longer manufactured but remain in theenvironment so that the search for ways of their destructionis one of the urgent problems of the day At the GeneralPhysics Institute of RAS experiments were carried out toexamine possibility of electric discharge (SSD) in wateras an efficient and inexpensive method for cleaning themanufacturing water of PCBs Instead of a toxic PCB inour experiments we used a 24-D dichlorophenoxyaceticacid (24-D) This material was chosen for plasmachemi-cal decomposition because the configuration of the 24-D molecule somewhat resembles PCB More exactly the24-D molecule like the PCB congeners contains a doublychlorine-substituted benzene ring with attached acetic acid

The experimental procedure was as follows Two solidparticles of 24-D (97) of weight 40mg were preliminarilydissolved in 10mL of alcohol The solution was poured intoa polyethylene container with 5 L of distilled water The acid

minus10010 15 20 25 30

t (min)

Figure 17 Dependence of the average size of carbon particlesproduced in the reactor on the time of electric discharge processingof methane

Figure 18 The typical photograph of nanocarbon produced in thecourse of methane recovery by multispark discharge in the water

concentration in the container was estimated at sim8120583g cmminus3that is about 300 times larger than the maximum allowable(ldquopermissiblerdquo) concentration

Decomposition of the acidic additive was accomplishedusing amultispark dischargermounted in a plexiglass reactorchamber of volume 119881 = 15 times 6 times 45 cm3 The multisparkdischarger which was placed on inside of the reactor coverproduced a discharge in water The working gas passedthrough the discharger was oxygen

Analysis of the SSD-processed solutionswas conducted inthe Laboratory of Analytic Environmental Toxicology at theSevertsov Institute of Ecology and Evolution of the RussianAcademy of Sciences

GCMS (Gas chromatographyMass spectrometry) anal-ysis of solutions was performed by using a Finnigan TRACEGCUltra gas chromatograph coupled with a Finnigan PolarisQmass spectrometer (ion trap)This GCMS system possess-ing ultra-high sensitivity allows detection of 24-D compoundand its possible organic products of fragmentation withsensitivity sim10minus9 g cmminus3

All our experiments were conducted at fixed values ofthe initial 24-D concentration 119873 = 8120583g cmminus3 and solutionvolume119881 = 250 cm3 In all experiments a sample of solution

was taken from the reactor before processing in order that theinitial 24-D concentration will be accurately known

Data of GCMS measurements ensure complete decom-position of 24-D (at the level of sensitivity 10minus9 g cmminus3) in allof the experiments when the processing time was longer than150 s and the mean power of the high-voltage generator wassim20WThese experiments give a conservative estimate of theefficiency of plasmachemical decomposition of the organic24-D compound by the use of a multielectrode systemexcited electric discharge (SSD) in water A characteristicdependence of the 24-D concentration on the duration ofSSD processing is presented in Figure 19

Almost complete (sim100) decomposition of 24-D ahigh-concentrated solution shows that the SSD processingwill outperform the traditional reactors From the experi-ments itmight be inferred that SSDworking in thewater con-taining about 300 maximum allowable concentration of 24-D provides almost complete decomposition of liquid solutionwith expenditure of energy as low as sim 2 sdot 10

minus3 kWhLAccordingly with a power source sim1 kW it is possible to cleanmore than 05m3 of water per hour

We do not have a clear notion of what mechanism isdominant in the technological process ofwater cleaning of the24-D additive Special experiments have yet to be performedto construct a physicochemical model for electric-dischargedestruction of the acid (and its decomposition fragments)However we have good reason to believe that a leading part indestruction is played by plasma-chemical reactions occurringin SSD with the resulting formation of chemically activeradicals and molecules

7 Conclusion

A new electric-discharge system which has been developedand tested at the GPI RAS has a multitude of potentialuses Examples can be found in the present paper A plasma-chemical reactor of simple design using a multielectrode(multispark) discharger operating in aqueous medium mayserve for efficient disinfection of microbially contaminatedpotable and waste water conversion (recovery) of methanedestruction of acidic 24-D pollutant

The SSD-based electrode system is capable of produc-ing multiple microplasma formations in liquid mediumat relatively low electrode voltages Physical and chemicalproperties peculiar to this type of discharges have beenstudied experimentally It is shown that these properties arecontrolled by the following four factors simultaneously actingupon the liquid (aqueous) medium

(i) direct influence of electric-discharge plasma pos-sessing a high electron density and relatively hightemperatures of the gas and electron component

(ii) exposure to intense UV radiation emitted bymicroplasma formations

(iii) chemical action of chemically active radicals atomsandmolecules produced in discharges and penetratedthe water

(iv) hydrodynamic action through cavitation bubbles

t (120583

Time (s)0 150 300 450 600

Figure 19 24-D content as a function of time of water treatment bymeans of multispark discharge

For each concrete application the electric-discharge sys-tem may be modified in design so as to increase one or theother of these factors

The experiments demonstrated high efficiency of multi-spark discharge inwater for solving diversified environmentalproblems listed above Note that the dominant mechanism insterilization of potable and waste water was the biologicallyactive UV radiation and generation of chemically activemolecules (ozone hydrogen peroxide) The achievement ofencouraging results in conversion of natural hydrocarbons iscredited to the immediate action of microplasma formationon the gas being treated The success in the accomplishmentof water cleaning of 24-D is attributed to plasmochemicalmechanism of generating chemically active substances

In conclusion the multispark discharge in water is beingused more and more Thus the action of SSD on the organicpollutions has been investigated in [29] Decomposition ofdissolved pentachlorophenol and parachlorophenol undermultispark discharge action has been measured Efficiency ofreforming these phenols was as good as 1-2 kJmg

References

[1] V L Goryachev F G Rutberg and V N FedyukovichldquoElectric-discharge method of water treatment Status of theproblem and prospectsrdquoApplied Energy vol 36 pp 35ndash49 1998

[2] L A Yutkin Electrohydroulic Effect and Industrial ApplicationMashinostroenie Leningrad Russia 1986

[3] J Sketchell H-G Peterson and N Christofi ldquoDisinfection by-product formation after biologically assisted GAC treatment ofwater supplies with different bromide andDOC contentrdquoWaterResearch vol 29 no 12 pp 2635ndash2642 1995

[4] F X R Van Leeuwen ldquoSafe drinking water the toxicologistrsquosapproachrdquo Food and Chemical Toxicology vol 38 pp 851ndash8582000

[5] U Von Gunten A Driedger H Gallard and E Salhi ldquoBy-products formation during drinking water disinfection a toolto assess disinfection efficiencyrdquoWater Research vol 35 no 8pp 2095ndash2099 2001

[6] PCT Treatment of Liquid International Patent Application noPCTGB9900755 1999

[7] L A Kulrsquoskii O S Savchuk and E Yu Deinega Influence ofElectron Field on Process of Water Sterilization Nauk DumkaKiev Ukraine 1980

[8] EM Barkhudarov I A KossyiM I Taktakishvili N Christofiand V Zadiraka Yu ldquoMultispark generation of plasma in liquidsand its utilization in waste water treatmentrdquo in Proceedingsof the 13th International Conference on Gas Discharges andtheir Applications vol 2 pp 680ndash683 Strathclyde UniversityGlasgow UK 2000

[9] A M Anpilov E M Barkhudarov Y B Bark et al ldquoElectricdischarge in water as a source of UV radiation ozone andhydrogen peroxiderdquo Journal of Physics D vol 34 no 6 pp 993ndash999 2001

[10] S M Korobeinikov and E V Yashin ldquoBubble model forbreakdown in water at pulsed voltage Electric discharge inliquid and its industrial application part 1 Nikolaev Russiardquo1988

[11] V L Goryachev A A Ufimtsev and A M KhodakovskiildquoMechanism of electrode erosion in pulsed discharges in waterwith a pulse energy ofsim1 Jrdquo Technical Physics Letters vol 23 no5 pp 386ndash387 1997

[12] A M Anpilov E M Barkhudarov N K Berezhetskaya et alldquoSource of a dense metal plasmardquo Plasma Sources Science andTechnology vol 7 no 2 pp 141ndash148 1998

[13] Y B Bark E M Barkhudarov Y N Kozlov et al ldquoSlippingsurface discharge as a source of hard UV radiationrdquo Journal ofPhysics D vol 33 no 7 pp 859ndash863 2000

[14] K H Becker K H Schoenbach and J G Eden ldquoMicroplasmasand applicationsrdquo Journal of Physics D vol 39 no 3 pp R55ndashR70 2006

[15] A M Anpilov N K Berezhetskaya V A Koprsquoev et alldquoExplosive-emissive source of a carbon plasmardquo Plasma PhysicsReports vol 23 no 5 pp 422ndash428 1997

[16] N K Berezhetskaya V A Koprsquoev I A Kossyi I I Kutuzovand B M Tiit ldquoExplosive emission phenomena on a metal-hotplasma interfacerdquo Zhurnal Tekhnicheskoi Fizikiv vol 61 no 2pp 179ndash184 1991 (Russian)

[17] E M Barkhudarov I A Kossyi and M I TaktakishvilildquoDistributed plasma generation in liquidsrdquo in Proceedings of13th International Conference on Gas Discharges and their Appli-cations vol 2 pp 340ndash342 Strathclyde University GlasgowUK 2000

[18] C G Hatchard and C A Parker ldquoA new sensitive chemicalactinometer II Potassium ferrioxalate as a standard chemicalactinometerrdquo Proceedings of the Royal Society A vol 235 no1203 pp 518ndash536 1956

[19] V V Lunin M P Popovich and S N Tkachenko PhysicalChemistry of Ozone Moscow State University Press MoscowRussia 1998

[20] J H Baxeudale ldquoThe flash photolysis of water and aqueoussolutionsrdquo Radiation Research vol 17 no 3 pp 312ndash326 1962

[21] B N Frog and A P Levchenko Preparation of Water MoscowState University Press Moscow Russia 1996

[22] A M Anpilov E M Barkhudarov N Christofi et al ldquoPulsedhigh voltage electric discharge disinfection of microbially con-taminated liquidsrdquo Letters in Applied Microbiology vol 35 no1 pp 90ndash94 2002

[23] A M Anpilov E M Barkhudarov N Christofi et al ldquoTheeffectiveness of a multi-spark electric discharge system inthe destruction of microorganisms in domestic and industrialwastewatersrdquo Journal of Water and Health vol 2 no 4 pp 267ndash277 2004

[24] A I Babaritskii S A Demkin V K Zhivotov et al Plasma-chemistry-91 (INKhS AN SSSRv) vol 2 pp 286ndash303 1991

[25] S I Gritsinin P A Gushchin A M Davydov E V Ivanov IA Kossyi and M A Misakyan ldquoConversion of methane in acoaxial microwave torchrdquo Plasma Physics Reports vol 35 no11 pp 933ndash940 2009

[26] A M Anpilov E M Barkhudarov N K Berezhetskaya et alldquoMethane conversion in a multielectrode slipping surface dis-charge in the two-phase water-gas mediumrdquo Technical Physicsvol 56 no 11 pp 1588ndash1592 2011

[27] N Parkansky O Goldstein B Alterkop et al ldquoFeatures ofmicro and nano-particles produced by pulsed arc submerged inethanolrdquo Powder Technology vol 161 no 3 pp 215ndash219 2006

[28] N Sano ldquoLow-cost synthesis of single-walled carbon nano-horns using the arc in water method with gas injectionrdquo Journalof Physics D vol 37 no 8 p L17 2004

[29] V M Shmelev N V Evtyukhin Y N Kozlov and E MBarkhudarov ldquoAction of pulsed surface discharge on organiccontaminants in waterrdquo Khimicheskaya Fizika vol 23 no 9 pp77ndash85 2004 (Russian)

The Advantages of Not Entangling MacroscopicDiamonds at Room Temperature

Mark E Brezinski1 2 3

1 Center for Optical Coherence Tomography and Modern Physics Department of Orthopedic SurgeryBrigham and Womenrsquos Hospital 75 Francis Street MRB-114 Boston MA 02115 USA

2 Center for Optical Coherence Tomography and Modern Physics Department of Orthopedic SurgeryHarvard Medical School 25 Shattuck Street Boston MA 02115 USA

3 Department of Electrical Engineering and Computer Science Massachusetts Institute of TechnologyRoom 36-360 77 Massachusetts Avenue Cambridge MA 02139 USA

Correspondence should be addressed to Mark E Brezinski mebrezinmitedu

The recent paper entitled by K C Lee et al (2011) establishes nonlocal macroscopic quantum correlations which they termldquoentanglementrdquo under ambient conditions Photon(s)-phonon entanglements are established within each interferometer armHowever our analysis demonstrates the phonon fields between arms become correlated as a result of single-photon wavepacketpath indistinguishability not true nonlocal entanglement We also note that a coherence expansion (as opposed to decoherence)resulted from local entanglement which was not recognized It occurred from nearly identical Raman scattering in each arm(importantly not meeting the Born and Markovian approximations) The ability to establish nonlocal macroscopic quantumcorrelations through path indistinguishability rather than entanglement offers the opportunity to greatly expand quantummacroscopic theory and application even though it was not true nonlocal entanglement

1 Introduction

The ability to observe and control nonlocal macroscopicquantum coherencecorrelations under ambient conditionswould likely have a powerful influence across a wide rangeof fields This was achieved recently by Lee et al in Scienceestablishing phonon field quantum correlations in twospatially separated diamonds [1 2] The paper was entitledentitled ldquoEntangling Macroscopic Diamonds at Room Tem-peraturerdquo Two other studies nonlocally correlating reflectors(by our group) and a cesium gas respectfully support theresults [3 4] However we will demonstrate on severalgrounds that while quantum correlations are establishedbetween the diamonds they are not true entanglement

The work in the Lee et al paper is essentially a two-arm extension of the DLCZ (Duan Lukin Cirac and Zoller)experiments [5ndash9] Figure 1 is a schematic of the key compo-nents of the Lee experiment but a more detailed schematiccan be found in Figure 1 of the original paper An ultrashort

pulsed source is used whose outputs can be represented bya collection of single photon wavepackets (each wave packetcan only interfere with itself) as they are neither entangledphotons nor significant biphoton wavepackets An MZIinterferometer is used where diamonds are present in eacharm which contain nearly identical Raman scatterers Thediamonds are 15 cm apart making any interaction betweenthem macroscopic The optical phonon modes of the dia-mond allow relatively low decoherence at room temperaturebecause they have very high oscillatory frequencies (40 THz)so are not readily disturbed by thermal energies A pumppulse is sent through the interferometer of sufficient intensityto entangle with and stimulate the Raman scatterers A Stokesphoton is then emitted with the diamond and Stokes photonentangled until detection The extra energy remaining inthe diamond (lost from the photon) is in the form ofincreased phonon field energy levels If the detector registersone Stokes photon it could have come from either of thediamond crystals in which one phonon was excited This

will be discussed in more detail below but because the pathsare indistinguishable the system behaves as if the photonat the beam splitter came from both arms ParaphrasingDirac a single photon wavepacket can only interfere withitself Therefore prior to the pump photon being detectedboth phonon fields are stimulated To confirm these resultsa probe photon is introduced into the interferometer thatinteracts with the diamonds producing the anti-Stokesphotons The probe photon must interact with the diamondprior to the Stokes photon being detected The nature ofthe detection scheme for the anti-Stokes photon allowsdetermination if one or both phonon fields are stimulatedIf we were only looking at one arm prior to the Stokesphoton detection there is an entanglement between theStokes photon phonon field and anti-Stokes photon Thisis somewhat analogous to the nonlocal entanglements inthe well-known studies performed by Brune et al describedbelow which we will use to support our conclusions about theLee paper [10ndash12] The key point of the Lee et al paper madebelow is that the two phonon fields are quantum correlatedbut not truly entangled as stated in the original paper

Our analysis is that Leersquos explanation in the Sciencepaper for the quantum correlations generated betweendiamonds (resulting from the pump photons) is unlikelyrepresentative of the actual situation They postulated anonlocal entanglement between the diamonds While weagree that quantum correlations are established we do notbelieve that the data or analysis of the experimental designsupports true entanglement The essential points will bemade here but the remainder of the paper will expand onthese points First our examination supports that thesenonlocal quantum correlations occur from a combination ofpaths indistinguishability (for a single photon wavepacket)plus nearly identical local entanglements (Raman scatterers)in each path [13ndash19] The source is coherent so building thepulses up from single photon wavepackets (a photon canonly interfere with itself) is a useful approach for illustratingthe physics The correlations between diamond phonons donot fit definitions of entanglement laid out for example byvon Neumann EPR-B or GHZ [20ndash23]

Second the pump photondiamond interactions donot (and must not) meet the Born (system-environmentcoupling weak) or Markovian (memory effects of theenvironment are negligible) approximations of decoherencetheory [15 17] This occurs largely from the high frequencyof the optical phonons and the strong coupling associatedwith the Raman scatterers The results then of the pumpphotondiamond interactions are more analogous to singlephoton wavepacket decoherence theory than nonlocal entan-glement (point 1) Environmental interactions are occurringwith indistinguishable paths but in the case of the Sciencepaper coherence is expanded rather than lost (point 2) [15ndash17] This demonstrates perhaps the most important pointof the paper that the diamonds can lead to either deco-herence (distinct local entanglements and meeting Born-Markov approximations) or coherence expansion (nearlyidentical local entanglements and not meeting Born-Markovapproximations) depending on the setup

In the next several paragraphs the topics addressed willbe as follows First nonlocal correlations will be examinedwhich can be represented by entangled states or statesgenerated by indistinguishable paths Second we review thegeneral definition of entanglement demonstrating why thenonlocal phonon field correlations in the Lee study are notaccurately described as being entangled Third we discussthat path indistinguishability and the quantum correlationsthat can be generated This and the previous paragraphsdraw heavily from the work by pioneers that include vonNeumman Mandel and Shih as well as insights from recentdecoherence theory by Zurcek and Zeh [15ndash17] Decoher-ence theory is particularly useful in illustrating the point ofthis paper as indistinguishable paths lead to coherence whiletypical environmental entanglements generally lead to deco-herence (with this paper representing an exception) Finallywe will also discuss how the authors represented visibilityconcurrence density operators and statistical significance(particularly the correlation coefficient) and how these arecompletely consistent with nonlocal correlations from eitherindistinguishable paths or entanglement We do not believethere is a basis to employ a two-mode squeeze state asdiscussed by Julsgaard et al for the nonlocal correlations [4]The appendix will speculate on the role misunderstandingtype II SPDC sources and Dirac notation play in the misuseof the term ldquoentanglementrdquo

2 Nonseparable States (Unfactorizable)and Quantum Correlations

In order to discuss quantum correlations including entan-gled states and those from path indistinguishability densityoperators and their nonseparability will be discussed Thedensity operator is a Hermitian operator acting on Hilbertspace with nonnegative eigenvalues whose sum is 1 (itis not a classical statistical operator) It should not beconfused with a classical statistical matrix and it has itsgreatest value in calculating expectation values of physicalproperties [24] A density operator does not specify aunique microscopic configuration which is not surprisingbased on its definition and contains the information aboutsuperpositions between subsystems Quantum correlationsimply unfactorizable density operators between multipleentities with quantum entanglement being one type Theydemonstrate correlations that exceed those describable byclassical mechanics They can be local or nonlocal with thelatter used extensively in decoherence theory For simplicityin this paper we will approximately describe the coherentportion of the system as the principal and everything elseas the environment We describe the principal as beingrepresented by a pure state density operator a single vectorin Hilbert space (there is no loss of generality as a mixed statecan be modeled using purification) [15] In the Lee systemfor clarity the phonon fields are part of the principal and canbe viewed as pure But as the phonon fields are part of thediamond the diamond itself is of low purity as the principalonly makes up a small portion of the diamond

Described more formally below a state describing apair of nonlocal quantum correlated entities (photons or

171The Advantages of Not Entangling Macroscopic Diamonds at Room Temperature

Coherent source ObjectRaman 1

ObjectRaman 2

Polarizer

Probe pulse

Pump pulse

Figure 1 This diagram is a simplified version of the interferometer used in the Lee et al experiments Components have been removedwhich are needed for practical application but not for understanding the physical principles

phonons) has an unfactorizable density operator for thepair that progresses forward in time via linear unitaryoperators But in performing the trace operation to obtainthe subsystems (eg a given diamond phonon field) thesesubsystems are represented by reduced density operatorsthat move forward in time unlike the true principal vianonlinear unitary operators (ie the trace gives informationon the subsystem statistical averages but is not the completedescription of the subsystems) [15 25] So for the Leesystem the principal contains both phonon fields that haveinseparable density operators

3 Entanglement

Entanglement a type of quantum correlation is a functionof superposition and the linearity of Schrodingerrsquos equationbut not generally path indistinguishability (which will bedealt with in a subsequent section) [22] Here we willlimit the discussion to complete entanglement and partialentanglement can be extrapolated from the discussionDemonstrating interference with entangled photons thoughdoes require path indistinguishable (see the Appendix) Theentanglement process is described by (as per von Neumman)[26] as follows

∣∣ψrang|ar〉 =⎛⎝sum

ci|si〉⎞⎠|ar〉 minusrarr |Ψ〉 =sum

ci|si〉|ai〉 (1)

This is a form which would be used to describe deco-herence (or a one-arm Lee experiment) where the principalis given by the wavefunction (ψ) [15 16] For two-particleentanglement the wavefunction is simply replaced by aparticle symbol The arrow describes the unitary transformThe principal is represented in terms of the basis si whilethe basis for the environment is given by ar Entanglementrepresents pairing of the eigenstates It can be stated in anequivalent form that their conjugate pairs (eg positionmomentum) are completely correlated So we have two

points (1) with two entangled particles the two basestates si and ai develop a constant relationship this is thecore to entanglement Measuring one of an entanglementpairs establishes the eigenvalue of both exactly from thesuperposition (2) This point will be more clear from thepath indistinguishability discussion but without furtherinteractions entangled particles continue to have inseparabledensity operators This is not true for quantum correlationsfrom path indistinguishability where the inseparability isdependent on such factors as detector time and wavepacketwidth We will use the phrase ldquoconditionally inseparablerdquo(3) The initial entanglement generally requires local inter-action between atomicsubatomic particles but can becomenonlocal with entanglement swapping for which we use forillustration the well-known-Brune studies described below[10 12] This local-to-nonlocal entanglement can be foundboth in the Lee and Brune papers

Equation (1) in the Lee paper (which is a DCLZ equationor one arm of the Lee interferometer) presents the initiallocal type of entanglement in the annihilation operatorform This form was introduced by Dirac and expandedupon by Glauber for the quantum theory of light [27 28]The equation is

|ΨS〉 asymp [1 + εSs+(lS)b+(lS)]|vac〉 (2)

The equation is described in detail in the Lee paperThe essential point is that for the potential annihilationoperators for the Stokes and phonon modes are in aninseparable product form It will be seen that this is incontrast to (2) from that paper which is a superposition(below)

As noted in addition to the Lee study being an extensionof the DLCZ experiments it is analogous to the pioneeringexperiments by Brune entangling atoms with fields (andthen a second atom) [10ndash12] These studies are more usefulthan the Lee study for understanding the physics of entan-glement and entanglement swapping because of the complexdesign of the Lee study Its analogy is to a single arm of the

Lee experiment Rubidium atoms in a Rydberg state werepassed through an EM field in the large Q cavity The atomand field become and remained entangled even after theRubidium atom exited the system (ie until a measurementis made at the output of the device) One can then onlyspeak of the combined Rubidium atom-Q cavity field systemas a pure state which is non-local (this is analogous tothe pump-phonon entanglement in the Lee experiment)The nonlocality can be extended even further by sending asecond atom after the first (analogous to the probe photon inthe Lee study) Here the second atom becomes non-locallyentangled with the first atom (which had already passedthrough) with perfect correlation (inseparable biparticlewave packet) The second atom non-local entanglementrepresents entanglement swapping with the field whichis no longer entangled This demonstrates true nonlocalentanglement of the two atoms as the eigenstates of eacheven though passing through the cavity at different timesexactly correlate The two atoms of course are analogous tothe Stokes and anti-Stokes photons in the Lee study and theEM field to the phonon field except only one arm is used

For a more formal description of entanglement and itssubsystems we will provide the mathematical framework forone EPR-B particle state There are two observers of theseparticles A and B separated by a large distance One of thesetwo entangled qubits is directed at each observer The specificpaths of each are inconsequential as long as no measurementhas occurred Neither does the order of detection nor thetimes between detection (as opposed to correlations fromindistinguishable paths) for these entangled states The Bellstate used here is given by (let them be spin 12 particles withtwo states 0 and 1)

∣∣Φ+rang = |Ψ〉 = 1radic2

(|0〉A otimes |0〉B + |1〉A otimes |1〉B) (3)

(Analogous Bell states with entangled energy and spingenerated by a SPDC source type II and the limitations arediscussed in the appendix) Equation (3) is a true entangledstate (spin superposition) in that the result of one observerexactly correlates with the results obtained with the secondobserver (irrespective of what spin axis is measured) theinformation of the system is complete The density operatoris given by

ρT = |Ψ〉〈Ψ| = ρA otimes ρB

= |00〉〈00| + |11〉〈00| + |00〉〈11| + |11〉〈11|2

The density operator product is nonfactorizable If weexamine a subsystem it is an inseparable state as the trace

operation of each observer (here observer B) yields lessinformation than the whole

ρA = Tr(ρ)

= TrB(|00〉〈00|) + TrB(|11〉〈00|)2

+TrB(|00〉〈11|) + TrB(|11〉〈11|)

= |0〉〈0|〈0 | 0〉 + |1〉〈0|〈1 | 0〉2

+|0〉〈1|〈0 | 1〉 + |1〉〈1|〈1 | 1〉

= |0〉〈0| + |1〉〈1|2

A reduced density operator is generated by the traceoperation representing an improper mixed state losinginformation about coherences It is an expectation valueTo paraphrase Schrodinger the best possible knowledge ofa whole does not include the best possible knowledge ofits parts (if that knowledge is even available) [22] In otherwords the principal is inseparable as any description ofthe subsystem is incomplete as demonstrated by (5) Wewill contrast this true entanglement with correlations fromindistinguishable paths where they are inseparable withincertain experimental limits (eg path lengths and detectorintegration time)

4 Path Distinguishability and First-OrderCorrelations

Path indistinguishability can lead to nonlocal macroscopiccorrelations but generally not entanglement A more com-plete discussion of coherence and indistinguishability can befound in the pioneering work of Mandel [20] reviewed byShih (for both single- and two-photon (boson) correlations)[13 19] The topic will be addressed here briefly It shouldalso be noted that our group in a previous paper alsoestablished nonlocal macroscopic correlations Correlationswere produced between two reflector arms with pathindistinguishability using a thermal source under ambientconditions [3]

We begin looking at path indistinguishability for asingle photon entering a beam splitter with the two armsas exit ports (essentially equivalent to the pump photonin the Lee paper) All first-order interference is a single-photon wavepacket interference (as per Dirac) no matterwhat the intensity along indistinguishable paths Second-order correlations are generally the interference of biphotonwavepackets and are reviewed elsewhere [3 13 19] First-order coherence (single-photon wavepacket interference) hasa wavefunction given by∣∣ψrang = α|1〉1|0〉2 + β|0〉1|1〉2 (6)

Here the subscripts 1 and 2 are the two paths and thevalue in the ket represents occupation number The alpha

and beta terms take into account beam splitter ratios Notethat this is the form of (2) of the Lee paper and is not anentangled state Equation (2) in the Lee paper was

E|ΨS〉 =[b+L (lS) + eminusiϕxb+

R(lS)]|vacvib〉 (7)

Again the specifics can be found in the original paper butunlike (7) the annihilation operators of the potential are nowin a summation form rather than a product form

Returning to (6) the density operator (in its expandedform) is given by

ρ = |α|2|1〉1|0〉22〈0|1〈1| +∣∣β∣∣2|0〉1|1〉22〈1|1〈0|

+[αβ lowast |1〉1|0〉22〈1|1〈0| + hc

The first two terms the diagonal terms are the DCterms that reduce fringe visibility to a maximum of 50unless they can be removed (for true entanglement thereare no DC terms and maximum visibility is 100) Whenpaths are distinguishable these are the only nonzero termsThe third and fourth terms represent indistinguishable pathsand generate interference (hc is the Hermitian conjugateor adjoint) (see Figure 3 in the Lee paper as off-diagonalelements are not exclusive to entanglement as suggested)These off-diagonal elements are complex It is importantto note that the density operator is inseparable only withinthe constraints of path indistinguishable (eg wavepacketwidth detector time path lengths etc) Coherence time isan example For an optical pulse delay times must be withinthe coherence time In contrast for most entangled statescoherence time is not an issue except when demonstratinginterference

Youngrsquos interferometer is useful for illustrating theconcepts of path indistinguishability We will use diamondssimilar to the Lee experiment before each slit in theYoungrsquos interferometer Examining the Youngrsquos interferom-eter (Figure 2) if one or the other slit is blocked the photonsare registered on the screen with no interference pattern(NI) If both slits are open classically it is easy to appreciatewhen waves pass through the apparatus and an interferencepattern will develop on the screen (I) The sinusoidal peaksin the Youngrsquos design are position-dependent interferenceon the screen (I) due to varying phase relationships TheYoungrsquos experiment results hold for a high intensity photonbeam but the interference is still single-photon wavepacketinterference Even when only one photon (or other particle)is coming from the source at a time a first order interferencepattern develops on the detection screen which is predictednaturally from quantum mechanics but is unexplainable byclassical mechanics (which would predict the NI pattern)[13 14] This is because quantum mechanics is predictingthe interference of potentials (along indistinguishable paths)and not intensities as long as no measurement is made priorto the screen There is no measurement of the pump photonsin the Science paper until after the second beam splitter sopaths are still indistinguishable (in spite of the frequencyshifts from the Raman scattering) So two-pump ldquobeamsrdquodo not actually interfere as in the classical description of

Figure 2 Illustration path indistinguishability and the influenceenvironmental entanglements (diamonds) with Youngrsquos interfer-ometer The I is an interference pattern and the NI is no interferencepattern E1 and E2 are the diamonds

interference after the second beam splitter it is a single-photon interference Interference of indistinguishable pathspotentials (of single-photon wave packets) leads to the inter-ference Interference is possible when these single photonpotential paths are identical with respect to the diamondinteractions as is more formally described in the nextparagraph Quantum correlations are established betweenthe diamonds because they are part of each indistinguishablepath that led to the single photon interference

Now we extend (8) beyond one photon (increase inten-sity) and include interactions with the environment E (dia-monds) in the form of an inner product This is a relativelycommon procedure for describing basic decoherence [1517] where the relevance to the diamond experiment willbecome apparent (though coherence is expanded rather thanreduced) The interference pattern at the screen (of theYoungrsquos interferometer) is described by the cross terms (off-diagonal) in the density operator (it is in the expandedmatrix form) as

ρ = 12

∣∣ψ1ranglangψ1∣∣ +

∣∣ψrang2

langψ∣∣

+∣∣ψ1

ranglangψ2∣∣〈E2 | E1〉 +

∣∣ψ2ranglangψ1∣∣〈E2 | E1〉

where ∣∣ψ1ranglangψ1∣∣ = ρ11

∣∣ψ2ranglangψ2∣∣ = ρ22∣∣ψrang1

langψ∣∣

2 = ρ12∣∣ψrang2

langψ∣∣

1 = ρ21(10)

The first two terms are again DC terms and the secondtwo represent interference terms The wavefunction (inthe bras-kets) incorporates all properties of the photons(polarization bandwidth photon numbers etc) now andnot just occupation number As can be seen from the densityoperator the interference pattern is independent of whetherthe photons come individually or at high intensity (if one ofthe wavefunctions was zero interference would still occur)In the density operator equation 1 and 2 correspond tothe two potential paths the photon can take The densityoperator contains an inner product (E) in the last two terms

that represents the diamonds which can be identical ordistinct The event that occurs at the screen is analogous todetection at D3 in the Science paper

To illustrate the counter-intuitive interaction of thephotons and phonons leading to indistinguishable paths andcoherence Youngrsquos experiment will be examined by varyingthe Raman scattering As a basic rule of quantum mechanicswhich can be found in any introductory quantum mechanicstextbook until a measurement is made potentials are addedthen squared but once a measurement occurs intensities(squared potentials) are added If we initially ignore theE terms (environmental entanglementsdiamonds) the pat-tern on the screen demonstrates interference that comes fromthe last two terms (off-diagonal) of the density operator(again even if one photon is coming through at a time)Now if E1 and E2 are substantially different terms (innerproduct near zero) such as when the Stokes photons are ofdifferent frequencies the third and fourth terms disappearas the paths become distinguishable Interference is lost inthis simple example of environmentally induced decoherenceby Raman scattering [15ndash17] The similarity of the Ramanscattering in each arm affects the degree to which coherence(and interference) is lost (fringe visibility) If E1 and E2

are similar (inner product 1) such that Stokes photonsare identical from the prospective of detection the pathsare indistinguishable even though the interaction with thediamonds occurred (and changed the frequency) and theinterference pattern is maintained The key point is thatindistinguishablity is needed at the time of measurement (thedetector)

But another critical point is that the Born and Markovianapproximations are not met hence decoherence will resultThe Born approximation is that the diamond-principalinteraction is sufficiently weak and environment (diamond)large such that the principal does not significantly changethe diamond Obviously the coupling is strong (Ramanscattering) and the diamond changes significantly (change inphonon frequency) The Markovian approximation havingno memory effects means that self-correlations withinthe diamondenvironment decay for all practical purposesinstantly into the environment If these two are not met(along with the diamond interactions being identical) thenthe diamonds become part of the coherent system ratherthan a source of decoherence Together the indistinguishablepaths of single-photon coherence near identical nature ofRaman scattering and not meeting the BornMarkovianapproximations resulted in expansion of the coherence (thetwo diamonds become part of the principal resulting inquantum correlations) This describes why the two phononfields become correlated and why it does not require (orinclude) an explanation of true non-local entanglementbetween arms

We suggest that confusion over the distinction betweenquantum correlations due to entanglement versus pathindistinguishability has arisen at least in part over a mis-understanding of the type II spontaneous parametric down-conversion (SPDC) source and overextending interpretationsof Dirac notation which is presented in the Appendix Thisspeculative topic is addressed in the Appendix

5 General Results of the Lee et al Paper

So to summarize in the Lee paper the state when using asingle armdiamond is initially a Stokes-phonon(s) entan-glement then Stokes-phonon(s)-anti-Stokes entanglementarising from and remaining consistent with (1) It is anentanglement in the von Neumann sense as measurementof one subsystem exactly determines the state of the othersubsystems When two paths are used the photon(s) andphonons are then entangled within a given path but notentanglement of phonons between paths However the twodiamonds are quantum correlated through path indistin-guishability The use of a coherent pulsed source allows theargument to be built up from single photon wavepacketinterference

As pointed out the coherence expansion that resultsrequires very specific conditions with respect to the dia-monds First the high phonon frequency minimizes thermaldecoherence Second the generated Stokes photons mustbe essentially identical with respect to detection Thirdthe Born and Markovian approximations must not be metTogether along with the path indistinguishability this resultsin quantum correlations between the diamond phonons

6 Notes on the Probe Photons

Just briefly discussing the probe photons what is beingmeasured is second-order correlation between detectorsDa+ and Daminus generated from phonon fields in the twoarms in a superposition In general we agree with theauthorrsquos interpretations of the physical principles of theprobe photons which will not be reviewed here becauseof space limitations [13 18 25] A quantitative descriptionof these second-order correlations from both entangledphotons and indistinguishable paths is best described interms of the correlation functions electric field operatorsand annihilation operators These are discussed elsewhere indetail for those interested [13 19 27]

7 Notes on the Quantitative Results

The four-quantitativequalitative results for discussion fromthe Science paper are the density operators presentedconcurrence confidence intervals and visibility (1) Thedensity operators in Figures 3 and 4 of the Lee paperdescribe a coherent state as demonstrated by the off-diagonal coherences which is not unique to entanglement(2) There was some confusion in editorialscommentarieson the article that there was 98 concurrence There wasactually a 98 confidence interval that the concordancewas positive (which as the reader is aware could mean itwas 98 confidence the concurrence was extremely smallbut positive) The concurrence was positive and somewherebelow 35 values consistent with quantum correlations thatare not exclusive to an entangled state [29] (3) The visibilitygraph (Figure 2 of the Science paper) demonstrates two mainpoints (A) The second order correlations are phase sensitivewith opposite signs due to the beam splitter which is knownfor second-order correlations (B) The correlations between

the pump and probe can exceed coincidence rates of classicalcorrelations These results demonstrate quantum correla-tions but are not sufficient for specifically demonstratingquantum entanglement This statement is also consistentwith the experimental design analysis described above

True entanglement between the phonon fields neitherneeds to be elicited as an explanation for the resultsnor leads to be proven in the paper Though the phraseldquoentanglement of diamondsrdquo attracts considerable attentionwe believe that the establishment of quantum correla-tionscoherence between two macroscopic objects using pathindistinguishability without nonlocal entanglements is farmore important to the field We point out that we havealso achieved this with two macroscopic distant reflectors[3] Path indistinguishability under the local entanglementconditions described above leads to quantum correlationsThis approach required that no quantum source could bedone under ambient conditions and potentially opens thedoor to a much larger number of applications than straightentanglement

8 Conclusion

The recent paper in Science entitled ldquoEntangling MacroscopicDiamonds at Room Temperaturerdquo by C Lee et al establishesnonlocal macroscopic quantum correlations between twodiamonds However while the authors claim the correlationsbetween diamonds represent entanglement we present whya different underlying mechanism exists which explain theresults The quantum correlations are generated by pathindistinguishability of first order correlations (single-photonwavepackets) in combination with essentially identical localentanglement in each arm Irrespective the results are ofconsiderable importance They offer a mechanism for gen-erating macroscopic nonlocal quantum correlations underambient conditions which could represent a substantialadvance to a wide range of applications

Appendix

Unfortunately many examples exist in the literature thattreat quantum correlations from path indistinguishabilityand entanglement as essentially identical an obstruction tothe field and in part likely due to misunderstanding of thewidely used SPDC II source (spontaneous parametric down-conversion) and misuse of Diracrsquos notation Two prominentexamples are a 2008 Nature review on entanglement and therecent study claiming entanglement between two diamondsin Science [1 30] A brief review of the SPDC may illustratethe point

SPDC sources generally use a CW pumped nonlinearcrystal to produce two energy entangled photon pairs(including entanglement of uncertainty) [31] They wereinitially pursed to test EPR-B Due to energy conservationphoton pairsrsquo angular frequency and wave number are

entangled According to the standard theory of parametricdownconversion the two-photon state can be written as

|Ψ〉 =intdωPA(ωP)

timesintdω1dω2δ(ω1 + ω2 minus ωP)a+(ω1)a+(ω2)|0〉

where ω represents the angular frequency of the signal (1)idler (2) and pump (p) of the downconversion The a+

represents the respective annihilation operators The deltafunction represents perfect frequency phase matching of thedownconversion (ie entanglement) A(ω) is related to thewavepacket extent and is not critical to the discussion here(but it is when interfering entangled photons) This is atype I SPDC source (no fixed polarization relationship) notethat the equation does not require path indistinguishabilityWith a type II SPDC source the signal and the idler haveorthogonal polarization states (ie the energy entangledphotons are associated with perpendicular polarizations)The state is given by [32]

timesintdω1dω2δ(ω1 + ω2 minus ωP)a+

o (ω1)a+e (ω2)|0〉

The subscripts on the signal and idler represent differentpolarization states associated with the entangled energystates (o and e) Again the energy states are entangled(and thereby the polarization states) without any use ofindistinguishable paths

Now using a SPDC II source with an interferometer(Figure 3) illustrates both entanglement and path indistin-guishability In this setup prior to the beam splitter thephotons are both entangled by energy and polarizationAfter the beam splitter indistinguishable paths are usedto generate interference Under the correct setup of thepolarizers (P1 and P2) in each arm Bell states can begenerated which can be used to test for example EPR-BThe path indistinguishability after the beam splitter does notcause the entanglement but rather it is used to generateBell states from the already entangled states Authors oftenabbreviate the wave function for these Bell states (entangledphotons grouped by indistinguishable paths) for exampleas (12)(|HV〉+ |VH〉) This representation as seen in theNature review can be misleading because it drops theenergypolarization entanglement that exists without thebeam splitter as well as the wavepacket for the biphoton(basically just using the e and o from (A2) and givingthe impression that they are being entangled by the beamsplitter) [37] Interfering light from the SPDC II sourcein the Nature paper a common yet incorrect statement inFigure 2 of that paper is made ldquoHowever in the regionswhere the two cones overlap the state of the photons willbe |HV〉+ |VH〉 It is around these points that entangledphotons are generatedrdquo This abbreviated representation of

Signal

Figure 3 An SPDC type II source using a beam splitter used togenerate Bell states

the state ignores the already entangled energypolarizationin areas outside the overlap (9) as well as the space-timeprobability density This leads to a misunderstanding of thephysics Entanglement exists in the areas outside the overlapnot just Bell states Similarly path indistinguishability didnot lead to entanglement of diamonds in the Lee experiment

The example also illustrates the misuse of Dirac notationwhich seems particularly common in the quantum commu-nication and computer fields Dirac notation is a powerfulshorthand technique for describing quantum informationflow But it is frequently treated as representing the state ofa system which it generally does not do If we represent avacuum and photon by |01〉+ |10〉 this neither tells us forexample about the state of the vacuum nor the bandwidth ofthe photon But this is how it is often interpreted leading toerroneous conclusions

Acknowledgments

This paper is sponsored by the National Institutes of HealthContracts R01-AR44812 R01-EB000419 R01 AR46996R01- HL55686 R21 EB015851-01 and R01-EB002638

References

[1] K C Lee M R Sprague B J Sussman et al ldquoEntanglingmacroscopic diamonds at room temperaturerdquo Science vol334 no 6060 pp 1253ndash1256 2011

[2] L M Duan ldquoQuantum correlation between distant dia-mondsrdquo Science vol 334 no 6060 pp 1213ndash1214 2011

[3] M E Brezinski and B Liu ldquoNonlocal quantum macroscopicsuperposition in a high-thermal low-purity staterdquo PhysicalReview A vol 78 no 6 Article ID 063824 13 pages 2008

[4] B Julsgaard A Kozhekin and E S Polzik ldquoExperimentallong-lived entanglement of two macroscopic objectsrdquo Naturevol 413 no 6854 pp 400ndash403 2001

[5] L M Duan M D Lukin J I Cirac and P Zoller ldquoLong-distance quantum communication with atomic ensembles andlinear opticsrdquo Nature vol 414 no 6862 pp 413ndash418 2001

[6] D N Matsukevich and A Kuzmich ldquoQuantum state transferbetween matter and lightrdquo Science vol 306 no 5696 pp 663ndash666 2004

[7] K S Choi H Deng J Laurat and H J Kimble ldquoMappingphotonic entanglement into and out of a quantum memoryrdquoNature vol 452 no 7183 pp 67ndash71 2008

[8] C W Chou H De Riedmatten D Felinto S V Polyakov SJ Van Enk and H J Kimble ldquoMeasurement-induced entan-glement for excitation stored in remote atomic ensemblesrdquoNature vol 438 no 7069 pp 828ndash832 2005

[9] T Chaneliere D N Matsukevich S D Jenkins S Y Lan T AB Kennedy and A Kuzmich ldquoStorage and retrieval of singlephotons transmitted between remote quantum memoriesrdquoNature vol 438 no 7069 pp 833ndash836 2005

[10] M Brune E Hagley J Dreyer et al ldquoObserving the progres-sive decoherence of the ldquometerrdquo in a quantum measurementrdquoPhysical Review Letters vol 77 no 24 pp 4887ndash4890 1996

[11] M Brune S Haroche J M Raimond L Davidovich and NZagury ldquoManipulation of photons in a cavity by dispersiveatom-field coupling quantum-nondemolition measurementsand generation of Schrodinger cat statesrdquo Physical Review Avol 45 no 7 pp 5193ndash5214 1992

[12] T Meunier S Gleyzes P Maioli et al ldquoRabi oscillationsrevival induced by time reversal a test of mesoscopic quantumcoherencerdquo Physical Review Letters vol 94 no 1 Article ID010401 4 pages 2005

[13] Y Shin An Introduction to Quantum Optics Photon andBiphoton Physics CRC Press New York NY USA 2011

[14] L Mandel ldquoCoherence and indistinguishabilityrdquo Optics Let-ters vol 16 no 23 pp 1882ndash1883 1991

[15] M Schlosshuaser Decoherence and the Quantum to ClassicalTransition Springer Melbourne Australia 2007

[16] W H Zurek ldquoDecoherence and the transition from quantumto classicalrdquo Physics Today vol 44 no 10 pp 36ndash44 1991

[17] M A Neilson and I L Chuang Quantum Computer andQuantum Information Cambridge University Press Cam-bridge UK 2007

[18] L Mandel ldquoQuantum effects in one-photon and two-photoninterferencerdquo Reviews of Modern Physics vol 71 no 2 ppS274ndashS282 1999

[19] H Chen T Peng S Karmakar Z Xie and Y Shih ldquoObser-vations of anti-correlations in incoherent thermal light fieldsrdquoPhysical Review A vol 84 Article ID 033835 2011

[20] A Einstein B Podolsky and N Rosen ldquoCan quantum-mechanical description of physical reality be consideredcompleterdquo Physical Review vol 47 no 10 pp 777ndash780 1935

[21] D Greenberger M Horne A Shimony and A ZeilingerldquoBellrsquos theorem without inequalitiesrdquo American Journal ofPhysics vol 58 p 1131 1990

[22] E Schrodinger ldquoDie gegenwartige situation in der quanten-mechanikrdquo Naturwissenschaften vol 23 no 807 pp 823ndash8441935

[23] G Jarger Entanglement Information and the Interpretation ofQuantum Mechanics Springer New York NY USA 2009

[24] K Blum Density Matrix Theory and Applications PlenumPress New York NY USA 1996

[25] C K Hong Z Y Ou and L Mandel ldquoMeasurementof subpicosecond time intervals between two photons byinterferencerdquo Physical Review Letters vol 59 pp 2044ndash20461987

[26] J von Neumann Mathematical Foundations of QuantumMechanics chapter 4 Princeton University Press PrincetonNJ USA 1955

[27] R J Glauber ldquoThe quantum theory of optical coherencerdquoPhysical Review vol 130 no 6 pp 2529ndash2539 1963

[28] P A M Dirac ldquoThe question theory of the emission andabsorption of radiationrdquo Proceedings of the Royal Society ofLondon A vol 114 no 767 pp 243ndash265 1927

[29] R Hildebrand ldquoConcurrence revistedrdquo Journal of Mathemati-cal Physics vol 48 no 10 Article ID 102108 23 pages 2007

[30] V Vedral ldquoQuantifying entanglement in macroscopic sys-temsrdquo Nature vol 453 no 7198 pp 1004ndash1007 2008

[31] P G Kwiat K Mattle H Weinfurter A Zeilinger AV Sergienko and Y Shih ldquoNew high-intensity source ofpolarization-entangled photon pairsrdquo Physical Review Lettersvol 75 no 24 pp 4337ndash4341 1995

[32] Y Shih ldquoEntangled Photonsrdquo IEEE Journal on Selected Topicsin Quantum Electronics vol 9 no 6 pp 1455ndash1467 2003

Energies Fine Structures and Hyperfine Structures ofthe 1s22snp 3

P (n = 2ndash4) States for the Beryllium Atom

Chao Chen

School of Physics Beijing Institute of Technology Beijing 100081 China

Correspondence should be addressed to Chao Chen chen chaotsinghuaorgcn

Academic Editor Derrick S F Crothers

Energies and wave functions of the 1s22snp 3P (n = 2ndash4) states for the beryllium atom are calculated with the full-core plus

correlation wave functions Fine structures and hyperfine structures are calculated with the first-order perturbation theory Forthe 1s22s2p 3P state the calculated energies fine structure and hyperfine structure parameters are in good agreement with thelatest theoretical and experimental data in the literature it is shown that atomic parameters of the low-lying excited states forthe beryllium atom can be calculated accurately using this theoretical method For the 1s22snp 3P (n = 3 4) states the presentcalculations may provide valuable reference data for future theoretical calculations and experimental measurements

1 Introduction

In recent years studies of energies fine structures andhyperfine structures of the low-lying excited states for theberyllium atom [1ndash10] have been of great interest to spectro-scopists because there are many strong optical transitionssuitable for spectral and hyperfine structure measurementsOn the other hand studies of the low-lying excited statesfor the beryllium atom play an important role in developingthe excited state theory of multielectron atoms and betterunderstanding the complicated correlation effects betweenelectrons The fine structure comes from the spin-orbitspin-other-orbit and spin-spin interactions The hyperfinestructure of atomic energy levels is caused by the interactionbetween the electrons and the electromagnetic multipolemoments of the nucleus The leading terms of this inter-action are the magnetic dipole and electric-quadrupolemoments The fine and hyperfine structure is sensitive to thecorrelation effects among electrons Experimentally someproperties of the atomic nucleus can be obtained by investi-gating the hyperfine structure of the atomic energy levelsThe nuclear electric-quadrupole moment which is difficultto measure directly with nuclear physics techniques can bedetermined using the measured hyperfine structure and theaccurate theoretical results

The 1s22s2p 3P state of the beryllium atom is of interest

since it is the lowest excited state in which hyperfine effects

can occur and the ground state has no hyperfine splittingbecause it is J = 0 It is generally a very demanding task tocalculate hyperfine structure accurately Polarization of theclosed shells in the 1s2 core due to the Coulomb interactionwith open shells can have a large effect on the hyperfinestructure Up till now the most sophisticated theoreticalcalculations of the hyperfine structure parameters for the1s22s2p 3

P state of the Be atom have been carried out usinglinked-cluster many-body perturbation (LC MBPT) theory[5 6] Hartree-Fock and CI allowing all SD excitations tocorrelation orbitals of Slater type by Beck and Nicolaides [7]as well as multiconfiguration Hartree-Fock (MCHF) method[8 9] Experimentally the magnetic dipole and electric-quadrupole hyperfine constants have been determined veryaccurately with the atomic-beam magnetic-resonance tech-nique [10] for the 1s22s2p 3

P state in beryllium To the bestof our knowledge few results on energies fine structuresand hyperfine structures have been investigated for the1s22snp 3

P (n ge 3) states of the beryllium atom due to therestriction of resolution from experiments and the numericalunsteadiness in theoretical calculations

An elegant and complete variation approach namelythe full core plus correlation (FCPC) method has beendeveloped by Chung [11 12] This method has been suc-cessfully applied to three- and four-electron systems withthe 1s2 core Many elaborate calculations especially for

the dipole polarizabilities [13] quadrupole and octupolepolarizabilities [14] and total atomic scattering factors [15]show that FCPC wave functions have a reasonable behav-ior over the whole configuration space for three-electronsystems This method has also been used to calculate thehyperfine structure of the 1s2ns 2S and 1s2np

2P states (n =

2ndash5) for the lithium isoelectronic sequence the results arein good agreement with the Hylleraas calculations and withthe experiment data [16] As is well known theoreticalcalculations of the hyperfine structure parameters dependsensitively on the behavior of the wave function in the prox-imity of the nucleus In addition core polarization effectsfor the low l states need to be included in the nonrelativisticwave function It would be interesting to find out whetherthe FCPC wave function can also be successful for calcu-lating hyperfine structure parameters of low-lying excitedstates for the beryllium atom In this work the FCPC wavefunctions are carried out on the 1s22snp 3

P (n = 2ndash4) statesof the beryllium atom The energies fine structures andhyperfine structures are calculated and compared with thedata available in the literature The purpose of this work is toexplore the capacity of the FCPC wave function to calculatethe atomic parameters of the low-lying excited states for theberyllium atom and provide more reliable theoretical data tostimulate further experimental measurements

2 Theory

According to the FCPC method [11 12] the wave functionfor the four-electron 1s22snp 3P state can be written as

Ψ(1 2 3 4)

⎡⎣Φ1s1s(1 2)Φ2snp(3 4)

CiΦn(i)l(i)(1 2 3 4)

⎤⎦

where A is an antisymmetrization operator Φ1s1s is apredetermined 1s2-core wave function which is representedby a CI basis set

Φ1s1s(1 2) = Asumknl

Cknlrk1 r

n2 exp

(minusβlr1 minus ρlr2)Yl(1 2)χ(1 2)

the angular part is

Yl(1 2) =summ

〈lm lminusm | 0 0〉Ylm(θ1ϕ1

)Ylminusm

(θ2ϕ2

χ (12) is a two-electron singlet spin function The linear andnonlinear parameters in (2) are determined by optimizingthe energy of the two-electron core The factor Φ2snp(3 4)

represents the wave function of the two outer electrons whichis given by

Φ2snp(3 4) = Asumknl

dknlrk3 r

n4 exp

(minusλlr3 minus ηlr4)Yl(3 4)χ(3 4)

the angular part is

Yl(3 4) =summ

〈lm l + 1minusm | 0 0〉Ylm(θ3ϕ3

)Yl+1minusm

(θ4ϕ4

The latter wave function of (1) describes the core relaxationand the intrashell electron correlation in the four-electronsystem It is given by

Φn(i)l(i)(1 2 3 4) = ϕn(i)l(i)(R)YLMl(i) (Ω)χSSZ (6)

ϕn(i)l(i)(R) =4prodj=1

j exp(minusαjr j

A different set of αj is used for each l(i) The angular part is

YLMl(i)

(R)=summj

〈l1l2m1m2 | l12m12〉

times 〈l12l3m12m3 | l123m123〉

times 〈l123l4m123m4 | LM〉4prodj=1

To simplify notation this angular function is simplydenoted as

l(i) = [(l1 l2)l12 l3]l123 l4 (9)

with the understanding that l123 and l4 couple into L thetotal orbital angular momentum There are three possiblespin functions for the 1s22snp 3

P state namely

χ1 = [(s1 s2)0 s3]12

χ2 = [(s1 s2)1 s3]12

χ3 = [(s1 s2)1 s3]32

For the radial basis functions of each angular-spin com-ponent a set of linear and nonlinear parameters is chosenThese parameters are determined in the energy optimiza-tion process For each set of l1 l2 l3 and l4 we try all possiblel(i) and χ and keep the ones which make significant contri-bution to the energy in (1)

The fine structure perturbation operators [1 2] are givenby

HFS = Hso + Hsoo + Hss (11)

where the spin-orbit spin-other-orbit and spin-spin opera-tors are

HSO = Z

4sumi=1

l i middot sir3i

HSOO = minus 12c2

4sumi j=1i = j

[1r3i j

(ri minus

rj)timespi

]middot[si + 2

HSS = 1c2

4sumi j=1ilt j

1r3i j

⎡⎢⎣si middot sj minus 3

(si middot ri j

)(sj middot ri j

)r2i j

⎤⎥⎦

To calculate the fine structure splitting the LSJ couplingscheme is used

ΨLSJJZ =sumMSZ

〈LSMSZ | JJZ〉ΦLSMSZ (13)

The fine structure energy levels are calculated by first-order perturbation theory

(ΔEFS)J =langΦLSJJZ

∣∣∣Hso + Hsoo + Hss

∣∣∣ΦLSJJZ

rang (14)

For an N-electron system the hyperfine interactionHamiltonian can be represented as follows [17 18]

Hh f s =sumk=1

T(k) middotM(k) (15)

where T(k) and M(k) are spherical tensor operators of rankk in the electronic and nuclear spaces respectively The k =1 term represents the magnetic-dipole interaction betweenthe magnetic field generated by the electrons and nuclearmagnetic dipole moments and the k = 2 term the electricquadrupole interaction between the electric-field gradientfrom the electrons and the nonspherical charge distributionof the nucleus The contributions from higher-order termsare much smaller and can often be neglected

In the nonrelativistic framework the electronic tensoroperators in atomic units can be written as

T(1) = α2

4sumi=1

[2glrminus3

i l(1)i minus

radic10gs

s(1)i C(2)

(1)rminus3i

+8π3gss

(1)i δ(ri)

T(2) = minus4sumi=1

rminus3i C(2)

where gl = (1 minus meM) is the orbital electron g factor andgs = 20023193 is the electron spin g factor M is the nuclear

mass The tensorC(2)i is connected to the spherical harmonics

Ylm(i) by

C(l)m =

radic4π2l

+ 1Ylm (17)

The hyperfine interaction couples the electronic angularmomenta J and the nuclear angular momenta I to a totalangular momentum F = I + J The uncoupling and couplinghyperfine constants are defined in atomic units as [17 18]

aC =langγLSMLMS

∣∣∣∣∣∣Nsumi=1

8πδ3(ri)s0(i)

∣∣∣∣∣∣γLSMLMS

(Fermi contact)

aSD =langγLSMLMS

∣∣∣∣∣∣Nsumi=1

2C(2)0 (i)s0(i)rminus3

(Spin dipolar

al =langγLSMLMS

∣∣∣∣∣∣Nsumi=1

l0(i)rminus3i

(orbital)

bq =langγLSMLMS

∣∣∣∣∣∣Nsumi=1

2C(2)0 (i)rminus3

(electric quadrupole

AJ = μII

[J(J + 1)(2J + 1)]12

langγJ∥∥∥T(1)

∥∥∥γJrang

AJminus1J = μII

[J(2J minus 1)(2J + 1)]12

langγJ minus 1

∥∥∥T(1)∥∥∥γJrang

BJ = 2Q[

2J(2J minus 1)(2J + 1)(2J + 2)(2J + 3)

]12langγJ∥∥∥T(2)

∥∥∥γJrang

where ML = L and MS = S In these expressions μI isthe nuclear magnetic moment and Q is the nuclear electricquadrupole moment I is the nuclear spin and J is the atomicelectronic angular moment

3 Results and Discussions

In order to achieve accurate calculation results for variousproperties of the low-lying excited states for the berylliumatom the choice of basis function with sufficiently highquality is critical and it is our major concern The sevenl components (00) (11) (22) (33) (44) (55) (66)altogether 159 terms are used for the 1s2 core The Φ2snp

in (1) has four angular components l is summed from 0to 3 with the angular components (01) (12) (23) and(34) and the number of terms in Φ2snp ranges from 36 to15 Most of the other correlation effects are included in (6)which accounts for the intershell as well as the intrashellcorrelations Many relevant angular and spin couplings areimportant for the energy these basis functions are triedto include in Φn(i)l(i) (1 2 3 4) with significant energycontribution For each set of orbital angular momenta l1 l2l3 and l4 there could be several ways to couple this set intothe desired total orbital angular momentum In this work

181Energies Fine Structures and Hyperfine Structures of the 1s22snp 3P (n = 2ndash4) States for the Beryllium Atom

Table 1 Nonrelativistic energies of the 1s22snp 3P (n = 2ndash4) states for the beryllium atom (in au)

This work Hibberta Weissb

1s22s2p 3P minus1456637 minus145184 minus1451844

1s22s4p 3P minus1436248 minus1431530aReference [1]

bReference [2]

Table 2 Fine structure splittings νJndashJ of the 1s22snp 3PJ (n = 2ndash4) states for the beryllium atom (in cmminus1)

1s22s2p 3PJ 1s22s3p 3PJ 1s22s4p 3PJ

ν2ndash1 236 035 013

Experimenta 235 (2)

Other theoryb 253

ν1ndash0 064 0092 0034

Experimenta 064 (1)

Other theoryb 071aReference [3]

bReference [4]

for 1s22snp 3P states the important angular series (l1 l2 l3l4) are (0 0 l (l+1)) (0 1 l l) (l l 0 1) and so forth In bothcases the value of l is from 0 to 6 as the energy contributionfrom set with l gt 6 is small and negligible In order to getthe high-quality wave function the number of angular-spincomponents in the Φn(i)l(i) wave functions ranges from 15 to66 and the number of terms in the Φn(i)l(i) of (6) is about790 The linear and nonlinear parameters are individuallyoptimized in the energy minimization process Using theRayleigh-Ritz variational method the basic wave function Ψand the corresponding eigenvalue E are determined

Nonrelativistic energies of the 1s22snp 3P (n = 2ndash4)

states for the beryllium atom are given in Table 1 As Table 1shows for the 1s22snp (n = 2 3) 3P states the nonrelativisticenergies in this work are lower and better than those ofHibbert and Weiss [1 2] the improvement ranging from00479 au to 00473 au Hibbert and Weiss reported a setof large-scale configuration interaction (CI) calculations forthe 1s22snp (n = 2 3) 3P states which can give an accurateapproximation for each state but it may tend to obscure theglobal picture of the spectrum which is so transparent inthe other approach The work of Hibbert and of Weiss didnot include any intrashell correlation in the 1s shell as thecalculations were of transitions in the outer subshells Thecorrelation energy of the 1s shell is almost independent of thenuclear charge and also of the number of additional electronsoutside the 1s shell For Be it is about 00457 au and thisaccounts for the main difference between earlier work andthe present more accurate results are presented in Table 1 Ofcourse for the calculation of hyperfine parameters correla-tion within the 1s shell is crucial in obtaining accurate hyper-fine parameters and this has been achieved in the presentwork For the 1s22s4p 3

P state the present calculation fromthe FCPC method is also lower than the result of Weiss [2]

If including the effects of the spin-orbit spin-other-orbitand spin-spin interactions the energies of the fine structure

resolved J levels are obtained In this work the fine structuresplittings of the triplet states are calculated with theHsoHsooand Hss operators using the first-order perturbation theoryTable 2 gives the fine-structure splittings of the 1s22snp 3PJ(n = 2ndash4) states for the beryllium atom The experimentalBe 2s2p 3PJ splittings are 235 (J = 2 rarr 1) and 064(J = 1 rarr 0) cmminus1 [3] They agree with our prediction236 and 064 cmminus1 Although many theoretical studies havebeen done on the BeI excited systems the published theore-tical fine structure results are scarce One exception is Laugh-lin Constantinides and Victor [4] They use a model poten-tial calculation and predict the splittings to be 253 and071 cmminus1 for the 1s22s2p 3

PJ state which should be con-sidered as quite good in view of the simplicity in theircomputation and fall in experimental uncertainties Pre-sent calculations for this state are more accurate due to cor-relation effect well described in this method For the experi-ment the splitting of 1s22s3p 3

PJ (J = 1 0) is not resolvedBut the splitting from the J = 2 state to the J = 1 0 is deter-mined to be 037 cmminus1 In this work the calculated splittingsare 035 (J = 2 rarr 1) and 0092 (J = 1 rarr 0) cmminus1 Thisimplies that the predicted splitting from J = 2 to the centerof gravity of J = 1 and 0 should be 0373 cmminus1 It agrees withthe experiment The good agreement with experiment couldbe used as the indication of the accuracy of the wave functionconstructed here For the 1s22s4p 3PJ state our calculatedsplittings are hoped to offer reference for further experi-mental measurements

The hyperfine structure parameters of the 1s22snp3P(n = 2ndash4) states for the beryllium atom are calculated inthis work Fermi contact ac the spin dipolar aSD the orbitalal and the electric quadrupole bq In the present calculationQ = 00530b μI = minus1177492 nm I = 15 for Be are takenfrom [19] The hyperfine interaction in the 1s22s2p 3

P statefor the beryllium atom is of interest since it is the lowestexcited state in which hyperfine effects can occur which has

Table 3 The hyperfine structure parameters (in au) and coupling constants (in MHz) of the 1s22s2p 3P state for the beryllium atom

Method ac asd al bq A2 A1 Reference

LC MBPTa 92319 minus006490 030478 minus01156 minus12421 [5 6]

HF + SDCIb 92738 minus006656 030014 minus01097 minus12476 minus13977 [7]

FE MCHFc 92349 minus006564 030261 minus01150 [8]

MCHF 92416 minus006587 030329 minus011570 minus12450 minus13935 [9]

This work 92436 minus006523 030201 minus011588 minus12451 minus13936

Experiment minus1245368 minus139373 [10]aLinked-cluster many-body perturbation theory

bHartree-Fock and CI allowing all SD excitations to correlation orbitals of Slater typecFinite-element multiconfiguration Hartree-Fock

Table 4 The hyperfine structure parameters (in au) and coupling constants (in MHz) of the 1s22snp 3P (n = 3 4) states for the berylliumatom

ac asd al bq A2 A1

1s22s3p 3P 12029 minus000898 004276 minus001788 minus15159 minus15361

been studied over the past four decades [5ndash10] Table 3 givesthe hyperfine structure parameters of the 1s22s2p 3

P statefor the beryllium atom through the FCPC wave function tocompare with data in the literature As can be seen fromTable 3 the present results for hyperfine structure para-meters are better than the earlier theoretical results [5ndash7] inwhole The present calculations also agree with the results byFE MCHF (finite-element multiconfiguration Hartree-Fock)method [8] to two significant figures The calculated Fermicontact term ac in this work differs from the results from thelatest calculation through MCHF method [9] by only 007and the differences for the other terms are on the order of afew parts in a thousand This means that the wave functionused in the present work is reasonable and accurate in thefull configuration space The hyperfine coupling constantsAJ are also listed in Table 3 to compare with results fromother calculations and experiments Our calculated hyperfinecoupling constants agree perfectly with the experimentalvalue [10] to four significant figures That is also true for theMCHF calculation of [9] It is shown that hyperfine structureparameters of the low-lying excited states for the berylliumatom can be calculated accurately using the present FCPCwave function For the 1s22snp 3

P (n = 3 4) states to the bestof our knowledge there is no report on hyperfine structureparameters in the literature The present predictions for thehyperfine structure parameters and coupling constants arelisted in Table 4 which may provide valuable reference datafor other theoretical calculations and experimental measure-ments in future

4 Summary

In this work energies fine-structure splittings and hyperfinestructure parameters of the 1s22snp 3

P (n = 2ndash4) statesfor the beryllium atom are calculated with the FCPC wavefunctions The obtained nonrelativistic energies are muchlower than the previous published theoretical values The cal-culated fine structure splittings are in good agreement with

experiment For the 1s22s2p 3P state the calculated hyperfine

structure parameters are in good agreement with the latesttheoretical and experimental data in the literature it is shownthat hyperfine constants of the low-lying excited states forthe beryllium atom can be calculated accurately using thiskind of wave function For other states the present predictedhyperfine structure parameters may provide valuable refer-ence data for future theoretical calculations and experimentalmeasurements

Acknowledgments

The author is grateful to Dr Kwong T Chung for his com-puter code The work is supported by National Natural Sci-ence Foundation of China and the Basic Research Founda-tion of Beijing Institute of Technology

References

[1] A Hibbert ldquoOscillator strengths of transitions involving2s3l3L states in the beryllium sequencerdquo Journal of Physics Bvol 9 no 16 pp 2805ndash2811 1976

[2] A W Weiss ldquoCalculations of the 2sns1S and 2p3p31P Levelsof Be Irdquo Physical Review A vol 6 no 4 pp 1261ndash1266 1972

[3] L Johansson ldquoThe spectrum of the neutral beryllium atomrdquoArkiv For Fysik vol 23 pp 119ndash128 1962

[4] C Laughlin E R Constantinides and G A Victor ldquoTwo-valence-electron model-potential studies of the Be I isoelec-tronic sequencerdquo Journal of Physics B vol 11 no 13 pp 2243ndash2250 1978

[5] S N Ray T Lee and T P Das ldquoMany-body theory of themagnetic hyperfine interaction in the excited state (1s22s2p3P) of the beryllium atomrdquo Physical Review A vol 7 no 5pp 1469ndash1479 1973

[6] S N Ray T Lee and T P Das ldquoStudy of the nuclearquadrupole interaction in the excited (2 3P) state of theberyllium atom by many-body perturbation theoryrdquo PhysicalReview A vol 8 no 4 pp 1748ndash1752 1973

[7] D R Beck and C A Nicolaides ldquoFine and hyperfine struc-ture of the two lowest bound states of Be- and their first

two ionization thresholdsrdquo International Journal of QuantumChemistry vol 26 supplement 18 pp 467ndash481 1984

[8] D Sundholm and J Olsen ldquoLarge MCHF calculations onthe hyperfine structure of Be(3PO) the nuclear quadrupolemoment of 9Berdquo Chemical Physics Letters vol 177 no 1 pp91ndash97 1991

[9] P Jonsson and C F Fischer ldquoLarge-scale multiconfigurationHartree-Fock calculations of hyperfine-interaction constantsfor low-lying states in beryllium boron and carbonrdquo PhysicalReview A vol 48 no 6 pp 4113ndash4123 1993

[10] A G Blachman and A Lurio ldquoHyperfine structure of themetastable (1s22s2p) 3P states of 4Be9 and the nuclear electricquadrupole momentrdquo Physical Review vol 153 no 1 pp164ndash176 1967

[11] K T Chung ldquoIonization potential of the lithiumlike 1s22sstates from lithium to neonrdquo Physical Review A vol 44 no9 pp 5421ndash5433 1991

[12] K T Chung X W Zhu and Z W Wang ldquoIonization potentialfor ground states of berylliumlike systemsrdquo Physical Review Avol 47 no 3 pp 1740ndash1751 1993

[13] Z W Wang and K T Chung ldquoDipole polarizabilities for theground states of lithium-like systems from Z = 3 to 50rdquoJournal of Physics B vol 27 no 5 pp 855ndash864 1994

[14] C Chen and Z W Wang ldquoQuadrupole and octupole polar-izabilities for the ground states of lithiumlike systems fromZ = 3 to 20rdquo The Journal of Chemical Physics vol 121 no9 pp 4171ndash4174 2004

[15] C Chen and Z W Wang ldquoTotal atomic scattering factors forthe ground states of the lithium isoelectronic sequence fromNa8+ to Ca17+rdquo The Journal of Chemical Physics vol 122 no 2Article ID 024305 5 pages 2005

[16] X X Guan and Z W Wang ldquoThe hyperfine structure of the1s2ns2S and 1s2np 2P states (n = 2 3 4 and 5) for the lithiumisoelectronic sequencerdquo The European Physical Journal D vol2 no 1 pp 21ndash27 1998

[17] J Carlsson P Jonsson and C Froese Fischer ldquoLarge multi-configurational Hartree-Fock calculations on the hyperfine-structure constants of the 7Li 2s 2S and 2p 2P statesrdquo PhysicalReview A vol 46 no 5 pp 2420ndash2425 1992

[18] A Hibbert ldquoDevelopments in atomic structure calculationsrdquoReports on Progress in Physics vol 38 no 11 pp 1217ndash13381975

[19] P Raghavan ldquoTable of nuclear momentsrdquo Atomic Data andNuclear Data Tables vol 42 no 2 pp 189ndash291 1989

Statistical Complexity of Low- and High-Dimensional Systems

Vladimir Ryabov1 and Dmitry Nerukh2

1 Department of Complex System School of Systems Information Science Future University Hakodate 116-2 Kamedanakano-ChoHakodate-Shi Hakodate Hokkaido 041-8655 Japan

2 Non-Linearity and Complexity Research Group Aston University Birmingham B4 7ET UK

Correspondence should be addressed to Dmitry Nerukh dnerukhastonacuk

We suggest a new method for the analysis of experimental time series that can distinguish high-dimensional dynamics fromstochastic motion It is based on the idea of statistical complexity that is the Shannon entropy of the so-called ε-machine(a Markov-type model of the observed time series) This approach has been recently demonstrated to be efficient for makinga distinction between a molecular trajectory in water and noise In this paper we analyse the difference between chaos andnoise using the Chirikov-Taylor standard map as an example in order to elucidate the basic mechanism that makes the valueof complexity in deterministic systems high In particular we show that the value of statistical complexity is high for the case ofchaos and attains zero value for the case of stochastic noise We further study the Markov property of the data generated by thestandard map to clarify the role of long-time memory in differentiating the cases of deterministic systems and stochastic motion

1 Introduction

Statistical complexity is a measure that had been introducedby Crutchfield and Young in 1989 [1] It has been provenuseful for describing various complex systems includingthose with hundreds of degrees of freedom [2] Accordingto our earlier paper [3] the statistical complexity of high-dimensional trajectories generated by the dynamics of anensemble of water molecules grows up to the time scaleof 1 microsecond that is an extremely long-time intervalfor a typical molecular dynamics simulation Moreover thisproperty is much less pronounced for so-called surrogatetime series that have exactly the same power spectrum andhence autocorrelation function as the original time series

For example in Figure 1 we plot the dependence ofstatistical complexity on the length of the time series forthe symbolic data obtained from a Poincare the section of3D velocities describing the motion of a hydrogen atomin an ensemble of 392 water molecules [3] The details ofcomputing the atomic trajectories as well as the method usedfor partitioning the phase space and obtaining a symbolicstring from the initially floating point data can be found in[4] In the same figure we draw the curves calculated for

so-called phase-shuffled surrogate time series [5] the datahaving identical autocorrelation function and hence powerspectrum as the original velocity trajectories One can noticesignificant differences between the statistical complexity ofthe physical and the artificially generated data

We then put forward a hypothesis that this propertythat is a high value of statistical complexity can be used fordistinguishing between deterministic and stochastic systems(see also [6]) The phenomenon of the complexity growthwith the length of time series that ensures the differencebetween the cases of deterministic and stochastic behaviourremains still unexplored In order to elucidate the mecha-nism that makes the value of complexity high we performednumerical experiments with the standard map (known alsoas the Chirikov-Taylor map) [7] one of the most studiedparadigmatic models in nonlinear dynamics We observedthat statistical complexity was high indeed in the case of thestandard map and it had much lower value for the surrogatetime series being close to zero for the case of noncorrelatednoise from a random number generator

For the purpose of estimating statistical complexity for asymbolic time series we utilize the CSSR algorithm [8] that

102412816

log2t (ns)

Figure 1 Statistical complexity versus the (log of) length of theanalysis interval for the hydrogen velocity time series (top curve)and four surrogate time series (bottom curves) three independentrealisations of the phase-shuffling algorithm (red green and blue)and single time series of a white noise passed through a low-passlinear filter (black) (from [3]) Note that the value of statisticalcomplexity for the data obtained from simple random numbergenerator is close to zero and does not depend on the length of timeseries for large enough value of the latter (not shown)

had been reported as an efficient reliable and easy to usesoftware The algorithm constructs an ε-machine a Markov-chain with l-step memory which constitutes a probabilisticmodel for the analysed data series Statistical complexitymeasures an information content of the ε-machine viaits Shannon entropy Our analysis shows however thatalthough the CSSR algorithm always converges well andproduces a finite value of complexity in some cases theapproximation of data with a Markov-chain-type modelis inadequate making the complexity value dependent onthe length of the analysed data Finally we came to theconclusion that at least in the case of standard map themain reason for the growth of complexity is the propertyof stickiness of periodic islands in the chaotic sea a genericphenomenon in Hamiltonian systems [9] It has been notedin [10] that due to the sticking property of the regularcomponent in a subcritical domain (K lt 09716) thedynamics of the standard map is subdiffusive that can bewell approximated with a continuous time random walkmodel Anomalous properties of the temporal behaviourof nonextensive entropy a generalization of the usualBoltzmann-Gibbs entropy have been also analysed in [11]

In the present work we mainly study the domain ofK 1 where the area occupied by periodic islands is smalland the chaotic motion can be expected to be strongly mixingand ergodic Nevertheless as our results show the presenceof stickiness is still an important factor defining the long-term statistical measures In terms of the CSSR algorithmthe property of stickiness breaks the independence of the datapoints separated by a history long-time interval thus makingthe Markov-chain approximation invalid

Finally we discuss a conjecture that the property ofthe non-Markovianity of the ε-machine and growth of

statistical complexity can be used in a constructive wayfor distinguishing deterministic and stochastic behavioursThe problem of detecting determinism in a noise lookingchaotic time series is a long standing one An extensive reviewof the issues related to the difference between chaos andnoise and to inherent difficulties encountered in the high-dimensional cases can be found in [12] We suggest that thereis a significant difference between the statistical complexitiesof Hamiltonian chaos and coloured noise with identicalpower spectrum the main reason for which consists in thepresence of the long-time memory in time series obtainedfrom Hamiltonian systems This property originates from thestickiness of periodic islands that are abundant in the chaoticsea due to multiple resonances that occur in the phase space

We would also like to note that since the phase space ofHamiltonian systems has a complicated structure of chaoticareas intermingled with periodic islands this leads in somecases to the necessity of distinguishing between chaos andcomplex quasiperiodic motion A measure called orbitalcomplexity had been introduced for this purpose in thecontext of analysing the orbital motions of planets [13ndash15]This measure although being based on the calculation ofthe Shannon entropy (but in the spectral domain) hasquite different meaning purpose and scope of applicabilitycompared to statistical complexity

2 Systems and Method of Analysis

The standard map is defined as

pn+1 = pn + K sin θn mod 2π

θn+1 = θn + pn+1 mod 2π(1)

where K is a single parameter defining the dynamics ofthis system In all the calculations below the value of theparameter K has been chosen at K = 6908745

First at the step called ldquosymbolizationrdquo the original real-valued time series is transformed to a symbolic sequenceby introducing a suitable partitioning of the phase space(Figure 2(a))

At the next stage the sequence of symbols is transformedto the sequence of histories the l-symbol strings representinga refinement of the partitioning in the phase space [3]ε-machine reconstruction requires a grouping of historiesto ldquocausal statesrdquo based on the analysis of the predictiveproperties of each history by one step forward in timeFinally the statistical complexity is calculated from the ε-machine as Shannon entropy of the probability distributionof the causal states

Cμ = minusNcsumi=1

pi log pi (2)

here pi are probabilities of the causal states in the ε-machineand Nc is the total number of the causal states

3 Numerical Experiments

We have calculated the statistical complexity using thealgorithm called CSSR [8] for the standard map and plotted

minus3 minus2 minus1 0 1 2 3

minus1

minus2

minus3

minus15 minus14 minus13 minus12 minus11 minus1 minus09 minus08

minus02

minus04

minus06

Figure 2 The standard map Symbolization with a three-symbolalphabet (a) Two periodic islands are embedded into the chaoticsea One of them zoomed is shown in (b)

the graphs of complexity versus the amount of data (thelength of symbolic sequence) We have also studied how thecalculated values depend on the method of partitioning themap initial conditions and the parameter K of the system

A typical plot of statistical complexity Cμ and the numberof causal states for the history length l = 2 middot middot middot 9 are shownin Figure 3

The results for the surrogate data generated using thesame trajectory of the standard map are shown in Figure 4Changing the initial conditions the type of partitioning thephase space at the stage of symbolization andor value ofthe parameter K brings qualitatively the same results that isthere is a significant difference between the complexity valuescalculated for the data obtained from the map and those forthe surrogate time series It should be noted though that thecomplexity value moderately increases with parameter K asshown in Figure 5 This behaviour is similar to that of other

219 221 223 225 227

Number of data points

Figure 3 The values of Cμ and the number of causal states forvarious history lengths (from bottom to top l = 2 middot middot middot 9) for thestandard map trajectory as a function of the data length

219 221 223 225 227

219 221 223 225 2270

Figure 4 Same as in Figure 3 but for the random surrogate

187Statistical Complexity of Low- and High-Dimensional Systems

0 2 4 6

Figure 5 Dependence of statistical complexity on the parameter K(the number of data points is 108)

00 02 04 06 08 1

Figure 6 Conditional distributions of the next symbol for allhistories at history length l = 8 Every point represents a historyThe total number of points (histories) is 38

characteristics used in nonlinear dynamics like Lyapunovexponents or measure of the chaotic area reported in [16]

4 A Hypothesis on Markov Property

In this section we would like to demonstrate that the largecomplexity values observed in the case of the standard mapare caused by the presence of certain segments in the chaotictrajectory (which become histories after symbolization) thatdo not possess a property necessary for building a Markov-chain from the data Consider the stage when the symbolicstring has been converted to the sequence of histories that issymbolic words of length l The Markov-chain (ε-machine)can be built from such a sequence if the conditionalprobability distribution of the next symbol in the symbolicsequence depends only on the l-symbol string preceding thesymbol and it is independent on the previous symbol that

P(1|larrminus s

P(0|larrminuss )

Figure 7 Conditional probability distribution for the history withthe largest deviation from the Markov property (black triangle)Adding a symbol to the history changes drastically the position ofthe point in the diagram Three circles correspond to adding ldquo0rdquoldquo1rdquo or ldquo2rdquo to the history l = 8 (crosses) The red triangle representsa randomly chosen history with clear Markov property that is thedistribution of probabilities does not depend on the added symbol

is the symbol that occurred l + 1 time steps before In otherwords if we consider the conditional probability distributionfor a given history it should not change (in statistical sense)if we increase the length of a history by one symbol to thepast

In Figure 6 we present a scatter diagram that demon-strates the distribution of the conditional probabilities foreach history at the history length l = 8 Every point in thediagram corresponds to a single history The large spreadaround the point with coordinates (13 13) evidencessignificant difference compared to the case of uniformdistribution The change from l = 8 to l = 9 does not changethe overall pattern of point distribution shown in Figure 6However the analysis of the conditional probabilities forindividual histories reveals huge changes in the position ofpoints depending on the extra symbol added at the beginningof the history In Figure 7 we depict the conditional distri-bution for the next symbol for two histories one that showsstrong deviation from Markov property and a ldquonormalrdquo onethat is a randomly chosen history The large deviation in thedistribution of conditional probabilities can be concludedfrom a big distance between the vertexes of the upper triangle(distribution of the conditional probabilities at l = 9) and thecross corresponding to the conditional probability at l = 8The probabilities at l = 9 are computed by adding one of thethree symbols (012) at the beginning of the history of l = 8

Finally we would like to show that the segments ofchaotic trajectories that correspond to the history with largedeviation from the Markov property are located in the areas

P(1|larrminus s

P(0|larrminuss )

minus3

minus2

minus1

3210minus3 minus2 minus1

Figure 8 Parts of chaotic trajectory corresponding to the historywith large deviation from the Markov-chain property Only pointscorresponding to the central three symbols in the history are shownApparently the history includes one of the two periodic islandsshown in Figure 2

of the phase space close to periodic islands For this purposewe plotted in Figure 8 only the points that correspond tothe history with large deviation from the Markov-chainproperty A comparison to Figure 2 suggests that the historywith large deviation in the distributions is located close tothe periodic islands Therefore we suppose that the breakingof Markovianity can be interpreted as a manifestation of thewell-known phenomenon of ldquostickinessrdquo [9] of trajectoriesin the areas close to periodic islands Prolonged wanderingof a trajectory around the island is equivalent to existingof long-time memory in the corresponding segments ofthe chaotic time series Figure 8 should be also comparedto Figure 9 which presents the segments of the chaotictrajectory corresponding to a history possessing the Markovproperty Apparently it has no relation to periodic islandsSuch histories represent a vast majority in the ensemble of3l histories the non-Markovian histories constituting only afraction of percent

P(1|larrminus s

P(0|larrminuss )

minus3

minus2

minus1

Figure 9 Parts of chaotic trajectory corresponding to the historywith no deviation from the Markov-chain property Only pointscorresponding to the central three symbols in the history are shownApparently the history does not include any of the two periodicislands shown in Figure 2

5 Discussion

It has been demonstrated in this paper that statisticalcomplexity appears to be a useful measure for distinguishingHamiltonian chaos in low- and high-dimensional systemsfrom correlated noise with identical autocorrelation func-tion Its value for the symbolic time series calculated fromthe dynamics of Hamiltonian systems is substantially largerthan that for a white noise time series or the time seriesobtained from the phase shuffling surrogate algorithm Ourexplanation of the origin of this phenomenon in termsof Markov-chain theory consists in breaking down theMarkov property by the symbolic sequences obtained fromHamiltonian systems

We believe that the large value of complexity observedin our numerical experiments is defined by the presenceof periodic islands with sticky borders in the phase spaceof Hamiltonian systems The stickiness of certain areas inthe phase space leads to long-time memory effects that are

responsible for breaking down the statistical independenceof the future states from the past ones This in turn makesthe procedure of grouping the histories into causal statesconstituting the core of CSSR algorithm unstable As a resultthe algorithm finds more and more causal states necessaryfor building the ε-machine as a Markov chain and the valueof complexity grows with the number of causal states

References

[1] J P Crutchfield and K Young ldquoInferring statistical complex-ityrdquo Physical Review Letters vol 63 no 2 pp 105ndash108 1989

[2] D P Feldman C S McTague J P Crutch-field et al ldquoTheorganization of intrinsic computation complexity-entropydiagrams and the diversity of natural information processingrdquoChaos vol 18 no 4 Article ID 043106 15 pages 2008

[3] D Nerukh and V Ryabov ldquoComputational mechanics ofmolecular systemsrdquo in Computational Mechanics ResearchTrends Computer Science Technology and Applications H PBerger Ed Nova Science 2010

[4] D Nerukh V Ryabov and R C Glen ldquoComplex temporalpatterns in molecular dynamics a direct measure of thephase-space exploration by the trajectory at macroscopic timescalesrdquo Physical Review E vol 77 no 3 Article ID 0362252008

[5] J Theiler S Eubank A Longtin B Galdrikian and J DoyneFarmer ldquoTesting for nonlinearity in time series the method ofsurrogate datardquo Physica D vol 58 no 1-4 pp 77ndash94 1992

[6] J M Amigo S Zambrano and M A F Sanjuan ldquoCombina-torial detection of determinism in noisy time seriesrdquo EPL vol83 no 6 Article ID 60005 2008

[7] B V Chirikov ldquoA universal instability of many-dimensionaloscillator systemsrdquo Physics Reports vol 52 no 5 pp 263ndash3791979

[8] C R Shalizi and K L Shalizi ldquoBlind construction of optimalnonlinear recursive predictors for discrete sequencesrdquo inProceedings of the Uncertainty in Artificial Intelligence 20thConference M Chickering and J Halpern Eds pp 504ndash511AUAI Press 2004

[9] G M Zaslavsky ldquoChaos fractional kinetics and anomaloustransportrdquo Physics Report vol 371 no 6 pp 461ndash580 2002

[10] J H Misguich J-D Reuss Y Elskens and R Balescu ldquoMotionin a stochastic layer described by symbolic dynamicsrdquo Chaosvol 8 pp 248ndash256 1998

[11] F Baldovin C Tsallis and B Schulze ldquoNonstandard entropyproduction in the standard maprdquo Physica A vol 320 pp 184ndash192 2003

[12] G Boffetta M Cencini M Falcioni and A Vulpiani ldquoPre-dictability a way to characterize complexityrdquo Physics Reportsvol 356 no 6 pp 367ndash474 2002

[13] N T Faber C M Boily and S Portegies Zwart ldquoOn time-dependent orbital complexity in gravitational N-body simula-tionsrdquo Monthly Notices of the Royal Astronomical Society vol386 no 1 pp 425ndash439 2008

[14] H E Kandrup B L Eckstein and B O Bradley ldquoChaos com-plexity and short time Lyapunov exponents two alternativecharacterisations of chaotic orbit segmentsrdquo Astronomy andAstrophysics vol 320 no 1 pp 65ndash73 1997

[15] I V Sideris and H E Kandrup ldquoChaos and the continuumlimit in the gravitational N-body problem II Nonintegrable

potentialsrdquo Physical Review E vol 65 no 6 Article ID 0662032002

[16] I I Shevchenko ldquoIsentropic perturbations of a chaoticdomainrdquo Physics Letters A vol 333 no 5-6 pp 408ndash414 2004

A First-Principles-Based Potential for the Description ofAlkaline Earth Metals

Johannes M Dieterich Sebastian Gerke and Ricardo A Mata

Institut fur Physikalische Chemie Universitat Gottingen Tammannstrasse 6 37077 Gottingen Germany

Correspondence should be addressed to Johannes M Dieterich jdietergwdgde

Academic Editor David Wales

We present a set of Gupta potentials fitted against highest-level ab initio data for interactions of the alkaline earth metals berylliummagnesium and calcium Reference potential energy curves have been computed for both pure and mixed dimers with thecoupled cluster method including corrections for basic set incompleteness and relativistic effects To demonstrate their usabilityfor the efficient description of high-dimensional complex energy landscapes the obtained potentials have been used for the globaloptimization of 38- and 42-atom clusters Both pure and mixed compositions (binary and ternary clusters) were investigatedDistinctive trends in the structure of the latter are discussed

1 Introduction

Metallic clusters have become over the years a subject ofintense study both theoretical as well as experimental [1]Interest stems from the distinct properties they reveal whencompared to the bulk phase and how these may changeas a function of the cluster size Different compositions(in binary ternary and higher mixtures) can also lead tonew chemical and physical phenomena Nanoalloys are aprime example of how both factors can be combined formaterial design and application in catalysis [2 3] Thecomputational study of their structures is a challenging taskfor two interlacing reasons On the one hand the numberof local minima is considered to scale exponentially withthe cluster size making the search for the global minimumNP-hard [4] This property reflects back on all algorithmsdesigned to explore the energy landscape of such systemsOn the other hand a suitable theoretical description ofthe interactions in play is required It needs to be accurateenough to properly describe the energy landscape for awide range of bonding patterns It should also be amenableto computation meaning that the computation of severalhundred many-body interactions can be carried out in asensible time frame This is even more important sincemultiple thousands of these computations are required fora proper sampling of the energy landscape

One of the most successful approaches to the study ofmetallic clusters has been the combination of fitted potentialswith global optimization algorithms [5ndash8] The former areusually obtained by fitting experimental data or electronicstructure results to an analytic expression The brute forceuse of quantum mechanical methods is impractical due tothe computational cost particularly linked to its scalingrelative to the system size Even semiempirical methods maybe too costly as the prefactors are high enough to hinder aproper sampling of conformational space

In this work we have made use of correlated wavefunction methods to calculate the two-body interactionpotential of alkaline earth metals (Be Mg and Ca) Emphasishas been placed on obtaining converged energy profilesrelative to basic set relativistic and electronic correlationeffects The high-level reference data thus obtained wasmapped to a two-body Gupta-type potential [9] which inturn could be used to explore the structure of pure binaryand ternary clusters A few comments should be made aboutthis choice of approach First of all it follows a bottom-to-toprationale that no information about nano- or macroscopicmaterials is used in the fit It is purely based on first principlesresults that no empirical information (aside from the form ofthe chosen potential) has been included This can certainlybe seen as an advantage since it allows us to improve the

description in a systematic way However since the referencedata has been computed with computationally demandingmethods it is not possible to benchmark the fit by repeatingcalculations for a selected test set of clusters In fact someof the terms included in the energy expression would behard to obtain even for a 3-atom system The advantages anddisadvantages of our choice are later discussed in the text

2 Methods and Techniques

Both for the cluster structure optimization as well as thepotential fit the OGOLEM framework for global opti-mization was used Its features have been introduced in aseries of publications [10ndash12] Therefore we will restrictourselves to a brief discussion of the relevant features TheOGOLEM framework is loosely based on genetic algorithmsas described in [13] replacing the generation-based schemewith the more efficient genetic pool scheme While standardgeneration-based schemes feature serial bottlenecks at theend of every generation a pool-based scheme removes thisconstraint through constant updates of a genetic popula-tion allowing for a more efficient parallelization of thealgorithm As a side effect elitism is a built-in featureof any genetic pool scheme therefore removing the needto define additional criteria for it Since the genetic poolcontains all current solution candidates parent individualsare chosen from it (father based on ranked fitness motherrandomly) and subject to the usual genetic operationscrossover and mutation The crossover operator used forthe global potential fit is a one-point genotype operatoraccompanied by a genotype mutation (probability 5) Forthe cluster structure optimization our implementation ofa phenotype operator [11] is used again accompanied bya genotype mutation (probability 5) It should be notedhere that no explicit exchange mutation (as eg proposedin [6] and applied in [11]) was used The phenotypeimplementation already includes some internal exchangewhich proved effective enough for lightly mixed clusters astargeted in this study

In the case of cluster structure optimization the solutioncandidates are then subject to a graph-based collision anddissociation detection Should a candidate structure showeither it will be rejected and does not enter the subsequentlocal optimization In the case of the potential fits nosuch restriction is applied Finally it is attempted to addthe fitter of the two locally optimized individuals to thegenetic pool This operation is only successful if it does notviolate the fitness-based diversity criterion After a definednumber of these iterations a converged solution pool isobtained containing the global minimum candidate In thecase of cluster structure optimizations such candidate is onlyaccepted if four independent runs yield the same individual

3 Global Fit of Potentials

All two-body interactions of beryllium magnesium andcalcium have been fitted against highest-level ab initio dataThe numerical data will be published elsewhere [27] To

obtain the highest possible accuracy at a still affordablecomputational footprint different levels of theory based onwave function methods are combined as follows

Einter = EinfinHF + ΔEinfinCCSD(T) + ΔErel + ΔEQ (1)

where EinfinHF is the CBS[3 4 5]-extrapolated HFaug-cc-pCVXZ [14ndash16] energy as proposed by Feller [17] ΔEinfinCCSD(T)is the CBS[4 5]-extrapolated correlation energy usingthe CCSD(T)aug-cc-pCVXZ (X = Q 5) level of theorywith the X minus 3 formula ΔErel is a relativistic correctionusing a Douglas-Kroll Hamiltonian at the CCSD(T)aug-cc-pCVTZ-DK [15] level of theory and ΔEQ is the quadru-ples contribution to the correlation energy obtained withCCSDT(Q)aug-cc-pVTZ with the frozen core approxima-tion in place All calculations were performed with theMolpro20101 program package [18] The CCSDT(Q) runswere carried out by the MRCC program [19 20] interfacedto the latter

The quality of this data set is high enough to reproducethe experimental dissociation energy of 111 kJ mol minus1 andequilibrium distance of 245 A for the beryllium dimer [21]and can be expected to be of similar quality for the otherinteractions Additionally it provides a consistent data setfor all pairs For the latter property the inclusion of allelectrons in the ΔEinfinCCSD(T) term calculation and the inclusionof relativistic effects are of particular importance

The Gupta potential [9] used is of the regular form

E(a b) = A(a b) middot exp[minusp(a b)

r0(a b)minus 1)]

minusradicχ(a b)2 middot exp

[minus2 middot q(a b)

r0(a b)minus 1)]

where rab is the distance between atoms a and b and A(a b)p(a b) r0(a b) χ(a b) and q(a b) are the parameters to befitted against the reference

Due to the rigid nature of the Gupta potential aweighting of data points was necessary to guarantee a goodfit This weighting followed the rationale that an exactreproduction of the depth and position of the minimum ismost important A good reproduction of the attractive partof the potential was the second target and less focus wasplaced on reproducing the repulsive part We consider theseto be reasonable design principles reflecting the standarddemands on potentials Used weighting factors are tabulatedin the supplementary information (see Supplementary Tablein the Supplementary Material available online at doi1011552012648386)

The derived potentials are depicted in Figures 1 and 2with the numerical values of the parameters to four digitsprecision available in Table 2 Perhaps one of the most strik-ing features upon inspection of the figures is the difficulty indescribing the weak-bonding regime Some of the potentialcurves show a close to linear profile on approaching theminimum This is the case for the Be-Be interaction and lessdrastically for the Be-Mg interactions In the former case aclear platteau is visible Under the constraints of the potential

minus2

minus4

minus6

minus8

minus10

minus122 4 6 8 10 12 14 16

r (BendashBe) [a0]

(kJmiddotm

olminus1

minus1

minus2

minus3

minus4

minus5

minus64 6 8 10 12 14 16 18

(kJmiddotm

olminus1

r (BendashMg) [a0]

minus2

minus4

minus6

minus84 6 8 10 12 14 16 18

(kJmiddotm

olminus1

r (BendashCa) [a0]

Figure 1 Derived Gupta potentials for Be-Be Be-Mg and Be-Ca interactions

Table 1 Quality of the fit for all alkaline earth interactionsDistance regime rAB in A and absolute and average deviation inkJmol Absolute deviation includes weights average deviation isweight-free

Pair rAB Nref Abs dev Avg dev

Be-Be 20rarr 100 81 4743 059

Be-Mg 25rarr 150 126 2571 020

Be-Ca 27rarr 150 124 2580 021

Mg-Mg 30rarr 150 121 1379 011

Mg-Ca 32rarr 150 119 1142 010

Ca-Ca 35rarr 150 116 1660 014

form chosen it is not possible to correctly reproduce thisbehavior without significantly affecting the description ofthe minimum Nevertheless all fitted potentials accuratelydescribe the position and depths of the minimum correctlyand are in overall good agreement with the reference TheMg-Ca and Ca-Ca fits reproduce extremely well the referencedata Numerical information on the fitting quality can beobtained from Table 1 It should be noted though that thedepth of the potential needs to be taken into account Theaverage deviation of 014 kJmiddotmol minus1 for the Ca-Ca interaction(minimum depth approximately 11 kJmiddotmolminus1) is less severe

Table 2 Numerical values to four digits precision for the fittedGupta potentials All values in atomic units

Parameter Be-Be Be-Mg Be-Ca Mg-Mg Mg-Ca Ca-Ca

A 17943 21964 13955 27232 25646 08815

p 42656 20473 35205 28024 20231 33835

r0 20323 14388 23759 18582 17852 39567

χ minus04088 02066 00913 minus00221 minus01453 minus05749

q 27536 11548 16065 07203 09789 28282

than the average deviation of 011 kJmiddotmol minus1 for the Mg-Mginteraction (minimum depth approximately 5 kJmiddotmol minus1)Further enhancements in the description would ultimatelyrequire another potential type either another rigid potentialmore suitable for these interactions or a more flexiblepotential form Both Morse potentials and LJ-type potentialswere found to be unsuitable to overcome this principleproblem In a recent study by Li et al [22] the Tang-Toennies potential model was used to fit experimental dataof homogeneous alkaline earth dimers The attractive part ofthe Be-Be interaction could not be perfectly described in thiscase either

Further enhancement to the potential would also bepossible through parametrization of three-body terms These

193A First-Principles-Based Potential for the Description of Alkaline Earth Metals

minus1

minus2

minus3

minus4

minus5

minus66 8 10 12 14 16 18 20

(kJmiddotm

olminus1

r (MgndashMg) [a0]

minus1

minus2

minus3

minus4

minus5

minus6

minus7

6 8 10 12 14 16 18 20

r (MgndashCa) [a0]

(kJmiddotm

olminus1

minus2

minus4

minus6

minus8

minus10

minus126 8 10 12 14 16 18 20

r (CandashCa) [a0]

(kJmiddotm

olminus1

Figure 2 Derived Gupta potentials for Mg-Mg Mg-Ca and Ca-Cainteractions

would have to be computed at a lower level of theory dueto the large number of points needed and the size increasein the system The computation of quadruple excitations isparticularly costly and would be hard to perform in systemsother than dimers A possible approach would be to addan effective 3-body term in agreement with experimentalstructural data or by using simulation results at a lowerlevel Caution should be taken in computing such a term

minus2

minus4

minus6

6 8 10 12 14 16 18 20

CCSD(T)AVTZCCSD(T)CBS

(kJmiddotm

olminus1

r (MgndashMg) [a0]

Figure 3 Mg-Mg interaction energy curves obtained at theCCSD(T)CBS and CCSD(T)AVTZ levels of theory

from three-atom systems for two reasons First of all it isexpected that basic set superposition effects (BSSEBSIE) cancontaminate the potential Most importantly we note thatmany-body stabilization is overestimated when consideringonly 3-body interactions [25] To illustrate the BSSEBSIEproblem we compare in Figure 3 the energy profile for theMg dimer computed at the CCSD(T)aug-cc-pCVTZ level(CCSD(T)AVTZ) and the energy obtained from the firsttwo terms in (1) (CCSD(T)CBS) The difference betweenthe two sets of data is exclusively due to differences in thebasic set The use of a triple-zeta quality basic set leads to aclear overestimation of the well depth The CCSD(T)AVTZlevel of theory predicts the equilibrium distance at 74 a0

with a dissociation energy of 51 kJmiddotmol minus1 in contrast tothe CCSD(T)CBS prediction of 76 a0 and 40 kJmiddotmol minus1respectively This amounts to an error of approximately 20in the dissociation energy If one were to estimate three-bodyterms with the triple-zeta basis an overestimation will beexpected The basic functions of a third atom can contributeto the basic space of the neighboring dimer resulting in abiased potential Only close-to-CBS values could be used forcorrectly estimating 3-body contributions

In general we expect that the inclusion of 3-body termsshould amount to an overall compression of the structurewhich would in turn induce local structural changes [25]This could however be balanced by even higher-order termsin the many-body expansion Work in this direction isunderway

4 Cluster Structure Optimization

To demonstrate the real-world applicability of the derivedpotentials they have been used in the global optimization ofmedium-sized alkaline earth clusters We focused on clustersof 38 alkaline earth atoms since this size typically exhibitsthe most interesting structural behaviour in the medium sizeregime [8] To check whether the observed structural trends

(a) 3800 (b) 4200 (c) 0380 (d) 0420

(e) 0038 (f) 0042 (g) 19190 (h) 21210

(i) 19019 (j) 21021 (k) 01919 (l) 02121

(m) 2990 (n) 9290 (o) 2909 (p) 9029

(q) 0299 (r) 0929

Figure 4 Global minimum candidate structures of homogenous and binary clusters of the alkaline earth metals beryllium (indigo)magnesium (yellow) and calcium (red) The caption XYZ denotes the number of beryllium atoms X of magnesium atoms Y and calciumatoms Z All graphics are obtained with Jmol [23] and POV-ray [24]

are specific to this cluster size similar compositions in 42atom clusters have been optimized The structural data willbe available from the Cambridge Cluster Database [26] afterpublication All global minimum candidate structures aredepicted in Figures 4 and 5

The homogeneous clusters show icosahedral structuralmotifs Depending on the atom in play the structure variesslightly While Be38 Ca38 and Ca42 possess mirror planesymmetry and seem to be magic numbers Be42 Mg38 andMg42 lack a number of atoms in defined positions which is

(a) 131312 (b) 131213 (c) 121313 (d) 141414

(e) 131510 (f) 13619 (g) 13916 (h) 16616

(i) 61616 (j) 20202 (k) 20220 (l) 22020

(m) 19181 (n) 18191 (o) 19118 (p) 18119

(q) 11918 (r) 11819

Figure 5 Global minimum candidate structures of ternary clusters of the alkaline earth metals beryllium (indigo) magnesium (yellow)and calcium (red) The caption XYZ denotes the number of beryllium atoms X of magnesium atoms Y and calcium atoms Z All graphicsare obtained with Jmol [23] and POV-ray [24]

clear through visual inspection It should be noted that nostable fcc structure could be located for any of the alkalineearth metals

The same principle motifs hold true for thebinary compositions Common features are icosahedralsubstructures and real or pseudo mirror plane symmetry

Additional structural motifs can be observed for all binaryclusters First of all a segregation of atom types can beobserved in the form of the well-known core-shell structures[8] for all clusters containing beryllium Beryllium formsan icosahedral core which can be easily explained with thepotential profiles The Be-Be interaction exhibits a deep

and narrow minimum at a short distance In contrarythe Mg-Mg and Ca-Ca interactions are both either notas deep (magnesium) or not as narrow (both magnesiumand calcium) The formation of core-shell structures is alsosupported by the shape of the Be-Mg and Be-Ca potentialsIn both cases the minimum is located at longer distancesthan the Be-Be equilibrium distance and is not as deep asthe Be-Be one Obviously the system must maximize thenumber of Be-Be contacts for an energetically low structurewhich is only the case for a small icosahedral beryllium core

A segregation of atom types can also be observed forthe MgCa binary compositions albeit not in the form ofcore-shell structures Again the potentials provide evidencefor this behaviour The Ca-Ca interactions possess a deeperminimum than the Mg-Ca interaction which in turn isslightly deeper than the Mg-Mg interaction The system musttherefore maximize the number of Ca-Ca contacts followedby the number of Mg-Ca contacts Since the equilibriumdistance of the Ca-Ca is longer than the Mg-Ca and Mg-Mg one a core-shell structure would require a very highMgCa ratio As can be seen from Figure 4(q) even a 29 9ratio is not sufficiently high for such behaviour In anyother ratio calcium forms the icosahedral backbone of thestructure with the magnesium atoms literally melting onthat backbone as can be seen for example in Figures 4(l)and 4(r) The resulting structures may probably be bestdescribed as Janus particles [8] possessing both magnesiumand calcium character on the surface Closely related is theball-and-cup structure found for example in Figure 4(l)

The same design principles hold true when movingto ternary compositions In the most simple case whensubstituting single atoms the binary cluster structure isslightly distorted but remains overall unchanged This canbe for example clearly seen in the transformation from thebinary Be21 Mg21 (Figure 4(h)) to the ternary Be20 Mg20 Ca2

(Figure 5(j)) cluster Once the composition contains moreatoms of the third species the cluster structure is againsubject to the principle rules that have been formulatedearlier Beryllium forms a small icosahedral core with mag-nesium and calcium segregating around it This behaviouris most pronounced in the Be13 Mg15 Ca10 (Figure 5(e)) andBe6 Mg16 Ca16 (Figure 5(i)) cluster structures In the earliercluster the beryllium core is large enough in comparisonto the number of magnesium and calcium atoms to allowforming two half-shells around the core In the latter thecore is small enough so that the calcium atoms form the shelland magnesium atoms remain at the surface This ordering isdue to the dissociation energy of the Be-Ca interaction beinghigher than the one of the Be-Mg interaction

It is possible to conclude that alkaline earth clusters in thestudied size regime seem to obey well-defined and rationalbuilding rules when using the Gupta model A possiblefault and one which will be addressed in later work [27]is the problematic description of the beryllium atom It isunclear how the deviations in the fit can influence the clusterstructures This however requires a more flexible functionalform than the Gupta potential

5 Conclusions

Gupta potentials for all bimetallic interactions involvingberyllium magnesium and calcium are derived fromhighest-level ab initio data using global optimization tech-niques All potentials reproduce the position and depths ofthe minimum correctly The potentials have been subse-quently used for the global optimization of medium-sizedcluster structures namely up to ternary 42 atom clusters

The structures obtained reveal several systematic trendsClusters containing beryllium will form beryllium coressurrounded by a shell of the other atoms in play Magnesiumand calcium segregate forming a calcium backbone withmagnesium on the surface

Acknowledgment

The authors gratefully acknowledge the financial supportfrom the German Excellence Initiative through the FreeFloater Research Group program of the University ofGottingen

References

[1] R Ferrando J Jellinek and R L Johnston ldquoNanoalloys fromtheory to applications of alloy clusters and nanoparticlesrdquoChemical Reviews vol 108 no 3 pp 845ndash910 2008

[2] B F G Johnson ldquoFrom clusters to nanoparticles and cataly-sisrdquo Coordination Chemistry Reviews vol 190ndash192 pp 1269ndash1285 1999

[3] C L Bracey P R Ellis and G J Hutchings ldquoApplicationof copper-gold alloys in catalysis current status and futureperspectivesrdquo Chemical Society Reviews vol 38 no 8 pp2231ndash2243 2009

[4] L T Wille and J Vennik ldquoComputational complexity of theground-state determination of atomic clustersrdquo Journal ofPhysics A vol 18 no 8 pp L419ndashL422 1985

[5] B Hartke ldquoGlobal optimizationrdquo WIREs ComputationalMolecular Science vol 1 no 6 pp 879ndash887 2011

[7] L O Paz-Borbon T V Mortimer-Jones R L Johnston APosada-Amarillas G Barcaro and A Fortunelli ldquoStructuresand energetics of 98 atom Pd-Pt nanoalloys potential stabilityof the Leary tetrahedron for bimetallic nanoparticlesrdquo PhysicalChemistry Chemical Physics vol 9 no 38 pp 5202ndash52082007

[8] L O Paz-Borbon R L Johnston G Barcaro and AFortunelli ldquoStructural motifs mixing and segregation effectsin 38-atom binary clustersrdquo Journal of Chemical Physics vol128 no 13 Article ID 134517 2008

[9] R P Gupta ldquoLattice relaxation at a metal surfacerdquo PhysicalReview B vol 23 no 12 pp 6265ndash6270 1981

[10] J M Dieterich and B Hartke ldquoOGOLEM global cluster struc-ture optimisation for arbitrary mixtures of flexible moleculesA multiscaling object-oriented approachrdquo Molecular Physicsvol 108 no 3-4 pp 279ndash291 2010

[11] J M Dieterich and B Hartke ldquoComposition-induced struc-tural transitions in mixed Lennard-Jones clusters globalreparametrization and optimizationrdquo Journal of Computa-tional Chemistry vol 32 no 7 pp 1377ndash1385 2011

[12] N Carstensen J M Dieterich and B Hartke ldquoDesignof optimally switchable molecules by genetic algorithmsrdquoPhysical Chemistry Chemical Physics vol 13 no 7 pp 2903ndash2910 2011

[13] D E Goldberg Genetic Algorithms in Search Optimization andMachine Learning Kluwer Academic Publishers 1989

[14] D E Woon and T H Dunning ldquoGaussian basis sets for usein correlated molecular calculations V Core-valence basis setsfor boron through neonrdquo Journal of Chemical Physics vol 103no 11 pp 4572ndash4585 1995

[15] B P Prascher D E Woon K A Peterson T H Dunningand A K Wilson ldquoGaussian basis sets for use in correlatedmolecular calculations VII Valence core-valence and scalarrelativistic basis sets for Li Be Na and Mgrdquo TheoreticalChemistry Accounts vol 128 no 1 pp 69ndash82 2011

[16] J Koput and K A Peterson ldquoAb initio potential energysurface and vibrational-rotational energy levels of Xrdquo Journalof Physical Chemistry A vol 106 no 41 pp 9595ndash9599 2002

[17] D Feller ldquoApplication of systematic sequences of wave func-tions to the water dimerrdquo Journal of Chemical Physics vol 96no 8 pp 6104ndash6114 1992

[18] H-J Werner P J Knowles R Lindh et al Molpro ver-sion 20101 a package of ab initio programs 2010 httpwwwmolpronet

[19] Mrcc a string-based quantum chemical program suite writtenby M Kallay see also M Kallay P R Surjan Journal of Chem-ical Physics vol115 pp 2945 2001 httpwwwmrcchu

[20] M Kallay and J Gauss ldquoApproximate treatment of higherexcitations in coupled-cluster theoryrdquo Journal of ChemicalPhysics vol 123 no 21 Article ID 214105 2005

[21] J M Merritt V E Bondybey and M C Heaven ldquoBerylliumdimer-caught in the act of bondingrdquo Science vol 324 no5934 pp 1548ndash1551 2009

[22] P Li J Ren N Niu and K T Tang ldquoCorresponding statesprinciple for the alkaline earth dimers and the van der waalspotential of Ba2rdquo Journal of Physical Chemistry A vol 115 no25 pp 6927ndash6935 2011

[23] Jmol an open-source java viewer for chemical structures in3d httpwwwjmolorg

[24] Pov-raymdashthe persistence of vision raytracer httpwwwpovrayorg

[25] E Blaisten-Barojas and S N Khanna ldquoDevelopment of afirst-principles many-body potential for berylliumrdquo PhysicalReview Letters vol 61 no 13 pp 1477ndash1480 1988

[26] The cambridge cluster database httpwww-waleschcamacukCCDhtml

[27] J M Dieterich S Fischmann and R A Mata In preparation

Uniformly Immobilizing Gold Nanorods on a Glass Substrate

Hadas Weinrib Amihai Meiri Hamootal Duadi and Dror Fixler

Faculty of Engineering and The Institute of Nanotechnology and Advanced Materials Bar-Ilan UniversityRamat Gan 52900 Israel

Correspondence should be addressed to Dror Fixler drorfixlerbiuacil

Academic Editor Rachela Popovtzer

The goal of this paper is to immobilize gold nanoparticles uniformly on a glass substrate In order to attach gold-nanorods (GNRs)to an area of a few squared microns surface of glass substrate without preliminary coating of the GNR 3-(Mercaptopro-pyl)trimethoxysilane molecules were used as linker while using different methods These methods included placing the glass slideand the GNR (1) inside a tube without any motion (2) inside a shaker (3) in a fan setup The fan setup included a tube containingthe GNR solution and the glass slide at a vertical position when the fan blows above the tube producing turbulations in theliquid Each method was evaluated according to the density and the homogeneousness of the GNR monolayer on the surfaceThe uniformity of the monolayer was demonstrated using AFM images of different areas on the slides and the effectiveness ofthe protocol was demonstrated by calculating the average density of the GNR on the surface using image processing and analysissoftware It was found that while both the shaker and the fan setups improved the monolayer density the fan setup improved thedensity by a factor of more than two than the density found using the shaker

1 Introduction

Nanoparticles play a significant role in an increasing numberof researches and variety of applications Recently goldnanoparticles (GNPs) have gained popularity and they serveas promising agents due to their favorable optical propertiessuch as an enhanced absorption cross-section [1] and scatter-ing properties [2] biocompatibility [3] and well-developedbioconjugation protocols [4] The increment in the numberof applications using GNP has led to an ongoing demandfor developing new techniques for immobilization of GNP tothe substrate surface Immobilization of GNP to substrates isrequired due to their applicability in various fields amongthem optical sensing using effects like surface plasmonresonance (SPR) and surface-enhanced Raman scattering(SERS) using nanoparticles as immobilizers for biomole-cules electron transfer enhancement and labeling of bio-molecules [5 6]

Immobilization of the particles on the surface can beachieved by using various methods which may be dividedinto two groups physical attachment [7 8] which is mainlyused to obtain a multilayer of particles or a thin film and

chemical attachment [9 10] which demands plating thesubstrate with linker molecules and is used to obtain a high-quality monolayer or structure of organized layers Whilesome of the chemical attachment methods take advantage ofthe ionic nature of the gold as well as its affinity toward thiolothers require the use of linker molecules [5]

Most of the common methods for chemical attachmentbetween GNP and a substrate use self-assembled monolayersthat contain organic groups especially amine and thiol whilethe use of charged polymer as a linker medium serves as analternative [11]

The principle of using linker molecules is based on theability of these molecules to self-arrange to what is called aself-assembled monolayer The linker molecules are in factbifunctional molecules where one end binds to the substrateand the other is ready to bind to gold nanoparticles

Previous studies [12ndash15] have shown that gold colloidscan be self-assembled from solution onto a functionalizedglass surface The self-assembled monolayer is stabilizedby attractive electrostatic interactions Aminopropyltrime-thoxy silane (APTMS or APS) has been commonly used toobtain amine-functionalised nonmetal substrate surface for

SH SH SH S S S

Figure 1 Chemical attachment between SndashH and gold (Au) on the left the original thiol moleculse and on the right the gold particleschemicaly bounded to the thiol molecules

AuAuAu

Silane

Figure 2 Scheme of the attachment between the GNR and the glass substrate by silane head and thiol tail

the attachment of gold colloidal nanoparticles [10 12 1415] This utilization has been used for a variance of purposesmainly for application using SPR [12 14 15] and SERSeffects [10 13]

Usually utilization of this method would be successfulwith gold nanospheres (GNSs) rather than gold nanorods(GNRs) because GNSs are usually produced in a negativelycharged citrate medium while GNR are usually producedusing a positively charged CTAB medium [16]

The positive charge of the top amine group of APTMS is amore suitable instrument in attracting the negatively chargedGNS Although one could stripe the solvent and achievenatural GNP [17] other options like negatively chargedcoupling agents are preferable due to the difficulty in the totalremoval of all the remnants of the medium Among theseagents we can state negatively charged polymer [11] or thiolgroups

Using thiol molecules to link GNP to substrates has beenextensively studied not only because of their unique physicaland chemical properties but also because of their easy prepa-ration and good performance

SndashH head groups are used on noble metal substrates dueto the strong affinity of sulfur in these metals (Figure 1)

The sulfur gold interaction is semicovalent and has astrength of approximately 45 kcalmol In addition gold isan inert and biocompatible material that can withstand harshchemical cleaning treatments These traits make thiol molec-ules attractive candidates for a wide range of applications[18ndash20] Among these applications are technologies for thebiosensor industry [21] and building of nanostructures forfabrication of nanodevices [9]

In most cases thiol molecules are used as linkers formetal-metal binding by using dithiol [22ndash25] or as a GNP

coating in order to prevent self-aggregation [24] Howeverthiol molecules could be useful for binding metal to othersubstrates such as glass Glass-metal binding is accomplishedby using molecules with thiol tail groups and head groupswhich are substrate-specific In case of a glass substrate anoptional functional group is silane [26]

Upon working with GNR a common problem of aggre-gation arise as GNR tend to self-aggregate in the solution oron the surface causing a disorder in the layer on the surfaceThe prevalent solution is an additional step of preliminarycoating the GNR by dithiols

To our knowledge there are no simple routine methodsfor coating uniformly a relatively wide area with high-qualitydensely packed monolayer of GNR without preliminary coat-ing the GNR In this work we used thiosilanes as bifunctionalmolecules in order to attach the GNR to a glass substratewhen silane head groups are attached to the glass substrateand the GNR are chemically bound to the thiol tail groups asshown in Figure 2

In order to simplify the process and avoid preliminarycoating of the GNR we used a shaker or a central processingunit (CPU) fan to prevent self aggregation in the solution andon the substrate and to increase the probability of a singleGNR to chemically bound to linker molecules on the glasswhat improves the quality of the monolayer

2 Materials and Methods

GNRs were synthesized using the seed-mediated growthmethod [21] Their size shape and uniformity were charac-terized using transmission electron microscopy (TEM) andthe resultant shape was 25 nm times 65 nm with a narrow sizedistribution (10) [27] (Figure 3)

400 500 600 700 800

Wavelength (nm)

Figure 3 TEM image of gold nanorods and corresponding absorption spectrum

Figure 4 Digital image of the custom-made tube

Glass slideinside the

GNR solutionthe special

glass tube

Powersupplier

CPU fan

Figure 5 The setup of the system including a CPU fan of top of thetube contains the glass slide inside the GNR solution

Figure 6 Scheme of dispersion of the GNR in the fan setup

3-(Mercaptopropyl)trimethoxysilane was purchased fromSigma-Aldrich (St Louis MO USA)

Cover glass slides (22 cm times 12 cm) with a diameter of013ndash017 mm were cleaned in a piranha solution (3 1 (vv)H2SO4H2O2) that causes vigorous oxidation for 90 min

A special glass tube was custom produced by us (Figure 4)The tube was composed of a flat part that enables verticalpositioning of the slide and a cylindrical part for possiblefuture use of a magnetic stirrer Vertical positioning of theglass slide is important for obtaining homogeneous bindingof the GNR to the slide

201Uniformly Immobilizing Gold Nanorods on a Glass Substrate

(a) (b) (c)

(d) (e) (f)

Figure 7 AFM images of the GNR on the glass slides (a)ndash(c) are images of different areas of a slide that was coated using a shaker (d)ndash(f)are images of different areas of a slide that was coated using a CPU fan

After cleaning the piranha was removed and the slideswere rinsed repeatedly a few times with water Subsequentlythe slides were rinsed repeatedly with ethanol and then incu-bated in thiosilane solutionmdashan MPTMS 1 ethanol solu-tion for 25 hours The slides were double-rinsed withethanol and sonicated for 10 min after each rinse Prior tofurther modification the slides were dried at 70C for 5minutes AFM measurements and imaging were carried outusing a ICON scanning probe microscope (Bruker AXSSanta Barbara CA USA) All images were obtained using thetapping mode with a single TESP silicon probe (force con-stant of 20ndash80 Nm Bruker Camarillo CA USA) The res-onance frequency of this cantilever was approximately 307ndash375 kHz The scan angle was maintained at 0 degrees andthe images were captured in the retrace direction with a scanrate of 15 Hz (resp for the scan size was 3000 times 3000 nm)The aspect ratio was 1 1 and image resolution was 1024samplesline Analysis of the image was done using the Nano-Scope software

The data was processed using ImageJ software ImageJ(image processing and analysis) is a public domain Java-based image processing program developed at the NationalInstitutes of Health [28]

Our first goal was to create a monolayer of the GNRWhen rinsing the glass vertically in the GNR solution

without any intervention produced no attachment to theglass was seen In order to improve the results differentmanipulations were tried First the slides were induced in theGNR colloid for 2 hours in a shaker in order to have maximalcontact of the GNR with the surface and try to avoid self-aggregation Afterwards the slides were rinsed in ethanolsonicated for 5 min and dried at 70C for 5 minutes Theresults can be seen in Figures 7(a)ndash7(c)

Placing the tube in a shaker improved the attachment ofthe GNR to the glass but further improvement was required

Next we sought to improve the density uniformity andhomogeneousness of the monolayer without adding steps ofpreliminary process such us prior coating of the GNR

In order to improve the quality of the monolayer theshaker was replaced by a CPU fan The fan was placed on topof the tube that contained the slides in the GNR solution Theintensity of the blow was controlled by the power supplier asshown in Figure 5

Both methods shaker and fan despite their differenceare based on the assumption that the motion of the GNR

solution increases the probability of a single GNR to chemi-cally bound to linker molecules on the glass A scheme ofmovement of the GNR in the fan setup is shown in Figure 6

In order to compare the density of the monolayerobtained by using the different methods AFM images weretaken To confirm the uniformity of the monolayer imagesfrom different areas of the slide were taken for every sample

While the use of a shaker produced a relatively uniformmonolayer (see Figures 7(a)ndash7(c)) the introduction of thefan to the protocol resulted in a denser layers can be seen inFigures 7(d)ndash7(f)

In order to quantify the improvement in density betweenthe methods we used a Java image processing program calledImageJ A particle analysis facility was used on the imageswhich occupied an area of 2 microm times 2 microm on the glass slideWe converted the images into 8-bit images and adjusted thelower threshold to 90 while the upper threshold was kept at255 This choice of threshold ensured the exclusion of thesurface and the binding layer from the count

The improvement in the quality of the monolayer isclearly demonstrated where the use of the fan increased theparticle density significantly to 18016 plusmn 39 particles perframe (2 microm times 2 microm) in comparison to 802 plusmn 32 particlesper frame for the shaker

4 Conclusions

The results reported in this work indicate that using thiosi-lane molecules combined with utilization of a fan in the setupapplies a uniform high-quality monolayer of GNR on a fewmicron squared area of glass substrate which is a relativelywide field

In order to improve the density and the homogeneous-ness of GNR monolayer and yet avoid preliminary coatingof the GNR two methods were tested In the first methodthe slides were rinsed in the GNR solution and deposited ina shaker In the second method the slides were placed in thesetup blown with a fan from above Both methods improvedthe attachment of the GNR to the glass but the latterproduced significantly better results by increasing the qualityof the monolayer The AFM images clearly demonstrate thatthe density of the monolayer using the fan is higher thanusing a shaker In addition the monolayer that was obtainedwas denser than the one that was achieved by Niidomeet al [11] using a negatively charged polymer Comparisonbetween AFM images of random different areas of the slideindicates the uniformity of the monolayer rather than oneimage of smaller area

As was demonstrated using the fan setup improved thequality of the GNR monolayer on the glass This suggestedthat the method enables simplifying the process of attachingGNR to glass substrate Such a process may serve as a firststep towards the development of a novel super resolutionmethod based on GNR attached to an observed object

References

[1] M A El-Sayed ldquoSome interesting properties of metals con-fined in time and nanometer space of different shapesrdquo

Accounts of Chemical Research vol 34 no 4 pp 257ndash2642001

[2] P K Jain K S Lee I H El-Sayed and M A El-Sayed ldquoCal-culated absorption and scattering properties of gold nanopar-ticles of different size shape and composition applicationsin biological imaging and biomedicinerdquo Journal of PhysicalChemistry B vol 110 no 14 pp 7238ndash7248 2006

[4] S Kumar J Aaron and K Sokolov ldquoDirectional conjugationof antibodies to nanoparticles for synthesis of multiplexedoptical contrast agents with both delivery and targetingmoietiesrdquo Nature Protocols vol 3 no 2 pp 314ndash320 2008

[5] X Luo A Morrin A J Killard and M R Smyth ldquoApplicationof nanoparticles in electrochemical sensors and biosensorsrdquoElectroanalysis vol 18 no 4 pp 319ndash326 2006

[6] R Ankri A Meiri S I Lau M Motiei R Popovtzer andD Fixler ldquoSurface plasmonresonance coupling and diffusionreflection measurements for real-time cancer detectionrdquo Jour-nal of Biophotonics In press

[7] X Xu M Stevens and M B Cortie ldquoIn situ precipitation ofgold nanoparticles onto glass for potential architectural appli-cationsrdquo Chemistry of Materials vol 16 no 11 pp 2259ndash22662004

[8] L Wang W Mao D Ni J Di Y Wu and Y Tu ldquoDirect elec-trodeposition of gold nanoparticles onto indiumtin oxidefilm coated glass and its application for electrochemical bio-sensorrdquo Electrochemistry Communications vol 10 no 4 pp673ndash676 2008

[9] H X He H Zhang Q G Li T Zhu S F Y Li and Z F LiuldquoFabrication of designed architectures of Au nanoparticleson solid substrate with printed self-assembled monolayers astemplatesrdquo Langmuir vol 16 no 8 pp 3846ndash3851 2000

[10] O Seitz M M Chehimi E Cabet-Deliry et al ldquoPreparationand characterisation of gold nanoparticle assemblies onsilanised glass platesrdquo Colloids and Surfaces A vol 218 no 1ndash3 pp 225ndash239 2003

[11] Y Niidome H Takahashi S Urakawa K Nishioka and SYamada ldquoImmobilization of gold nanorods on the glass sub-strate by the electrostatic interactions for localized plasmonsensingrdquo Chemistry Letters vol 33 no 4 pp 454ndash455 2004

[12] N Nath and A Chilkoti ldquoLabel-free biosensing by surfaceplasmon resonance of nanoparticles on glass optimization ofnanoparticle sizerdquo Analytical Chemistry vol 76 no 18 pp5370ndash5378 2004

[13] E J Bjerneld F Svedberg and M Kall ldquoLaser induced growthand deposition of noble-metal nanoparticles for surface-enhanced Raman scatteringrdquo Nano Letters vol 3 no 5 pp593ndash596 2003

[14] N Nath and A Chilkoti ldquoA colorimetric gold nanoparticlesensor to interrogate biomolecular interactions in real time ona surfacerdquo Analytical Chemistry vol 74 no 3 pp 504ndash5092002

[15] T Okamoto I Yamaguchi and T Kobayashi ldquoLocal plasmonsensor with gold colloid monolayers deposited upon glasssubstratesrdquo Optics Letters vol 25 no 6 pp 372ndash374 2000

[16] J Perez-Juste I Pastoriza-Santos L M Liz-Marzan and PMulvaney ldquoGold nanorods synthesis characterization andapplicationsrdquo Coordination Chemistry Reviews vol 249 no17-18 pp 1870ndash1901 2005

[17] X Xu T H Gibbons and M B Cortie ldquoSpectrally-selectivegold nanorod coatings for window glassrdquo Gold Bulletin vol39 no 4 pp 156ndash165 2006

[18] G Schmid S Peschel and T Sawitowski ldquoTwo-dimensionalarrangements of gold clusters and gold colloids on varioussurfacesrdquo Zeitschrift fur Anorganische und Allgemeine Chemievol 623 no 5 pp 719ndash723 1997

[19] T Ohgi H Y Sheng and H Nejoh ldquoAu particle depositiononto self-assembled monolayers of thiol and dithiol molec-ulesrdquo Applied Surface Science vol 130-132 pp 919ndash924 1998

[20] A Doron E Joselevich A Schlittner and I Willner ldquoAFMcharacterization of the structure of Au-colloid monolayers andtheir chemical etchingrdquo Thin Solid Films vol 340 no 1 pp183ndash188 1999

[21] B Nikoobakht and M A El-Sayed ldquoPreparation and growthmechanism of gold nanorods (NRs) using seed-mediatedgrowth methodrdquo Chemistry of Materials vol 15 no 10 pp1957ndash1962 2003

[22] E Hutter S Cha J F Liu et al ldquoRole of substrate metal in goldnanoparticle enhanced surface plasmon resonance imagingrdquoJournal of Physical Chemistry B vol 105 no 1 pp 8ndash12 2000

[23] M D Musick C D Keating L A Lyon et al ldquoMetal filmsprepared by stepwise assembly 2 Construction and charac-terization of colloidal Au and Ag multilayersrdquo Chemistry ofMaterials vol 12 no 10 pp 2869ndash2881 2000

[24] C N R Rao G U Kulkarni P J Thomas and P P EdwardsldquoMetal nanoparticles and their assembliesrdquo Chemical SocietyReviews vol 29 no 1 pp 27ndash35 2000

[25] Z M Qi I Honma M Ichihara and H Zhou ldquoLayer-by-layerfabrication and characterization of gold-nanoparticle myo-globin nanocomposite filmsrdquo Advanced Functional Materialsvol 16 no 3 pp 377ndash386 2006

[26] Y Wang L Q Chen Y F Li X J Zhao L Peng and C ZHuang ldquoA one-pot strategy for biomimetic synthesis and self-assembly of gold nanoparticlesrdquo Nanotechnology vol 21 no30 Article ID 305601 2010

[27] R Ankri V Peretz M Motiei R Popovtzer and D Fixler ldquoAnew method for cancer detection based on diffusion reflectionmeasurements of targeted gold nanorodsrdquo International Jour-nal of Nanomedicine vol 7 pp 449ndash455 2012

[28] T J Collins ldquoImageJ for microscopyrdquo BioTechniques vol 43no 1 pp 25ndash30 2007

Permissions

The contributors of this book come from diverse backgrounds making this book a truly international effort This book will bring forth new frontiers with its revolutionizing research information and detailed analysis of the nascent developments around the world

We would like to thank all the contributing authors for lending their expertise to make the book truly unique They have played a crucial role in the development of this book Without their invaluable contributions this book wouldnrsquot have been possible They have made vital efforts to compile up to date information on the varied aspects of this subject to make this book a valuable addition to the collection of many professionals and students

This book was conceptualized with the vision of imparting up-to-date information and advanced data in this field To ensure the same a matchless editorial board was set up Every individual on the board went through rigorous rounds of assessment to prove their worth After which they invested a large part of their time researching and compiling the most relevant data for our readers Conferences and sessions were held from time to time between the editorial board and the contributing authors to present the data in the most comprehensible form The editorial team has worked tirelessly to provide valuable and valid information to help people across the globe

Every chapter published in this book has been scrutinized by our experts Their significance has been extensively debated The topics covered herein carry significant findings which will fuel the growth of the discipline They may even be implemented as practical applications or may be referred to as a beginning point for another development Chapters in this book were first published by Hindawi Publishing Corporation hereby published with permission under the Creative Commons Attribution License or equivalent

The editorial board has been involved in producing this book since its inception They have spent rigorous hours researching and exploring the diverse topics which have resulted in the successful publishing of this book They have passed on their knowledge of decades through this book To expedite this challenging task the publisher supported the team at every step A small team of assistant editors was also appointed to further simplify the editing procedure and attain best results for the readers

Our editorial team has been hand-picked from every corner of the world Their multi-ethnicity adds dynamic inputs to the discussions which result in innovative outcomes These outcomes are then further discussed with the researchers and contributors who give their valuable feedback and opinion regarding the same The feedback is then collaborated with the researches and they are edited in a comprehensive manner to aid the understanding of the subject

Apart from the editorial board the designing team has also invested a significant amount of their time in understanding the subject and creating the most relevant covers They scrutinized every image to scout for the most suitable representation of the subject and create an appropriate cover for the book

The publishing team has been involved in this book since its early stages They were actively engaged in every process be it collecting the data connecting with the contributors or procuring relevant information The team has been an ardent support to the editorial designing and production team Their endless efforts to recruit the best for this project has resulted in the accomplishment of this book They are a veteran in the field of academics and their pool of knowledge is as vast as their experience in printing Their expertise and guidance has proved useful at every step Their uncompromising quality standards have made this book an exceptional effort Their encouragement from time to time has been an inspiration for everyone

The publisher and the editorial board hope that this book will prove to be a valuable piece of knowledge for researchers students practitioners and scholars across the globe

Yao Xu Ramachandran Gnanasekaran and David M LeitnerDepartment of Chemistry and Chemical Physics Program University of Nevada Reno NV 89557 USA

Henryk T Flakus and Anna Jarczyk-JędrykaInstitute of Chemistry University of Silesia 9 Szkolna Street 40-006 Katowice Poland

Brenda DanaDepartment of Electrical Engineering Faculty of Engineering Tel-Aviv University 69978 Tel-Aviv Israel

Israel GannotDepartment of Biomedical Engineering Faculty of Engineering Tel-Aviv University 69978 Tel-Aviv Israel

E A McCoy and G S McDonaldJoule Physics Laboratory School of Computing Science and Engineering Materials amp Physics Research Centre University of Salford Greater Manchester M5 4WT UK

J Saacutenchez-Curto and P Chamorro-PosadaDepartamento de Teoracuteıa de la Se˜nal y Comunicaciones e Ingenieracuteıa Telemacuteatica Universidad de Valladolid ETSI Telecomunicaciacuteon Campus Miguel Delibes Paseo Belacuteen 15 E-47011 Valladolid Spain

JM ChristianJoule Physics Laboratory School of Computing Science and Engineering Materials amp Physics Research Centre University of Salford Greater Manchester M5 4WT UKDepartamento de Teoracuteıa de la Se˜nal y Comunicaciones e Ingenieracuteıa Telemacuteatica Universidad de Valladolid ETSI Telecomunicaciacuteon Campus Miguel Delibes Paseo Belacuteen 15 E-47011 Valladolid Spain

Mark T Oakley and Roy L JohnstonSchool of Chemistry University of Birmingham Edgbaston Birmingham B15 2TT UK

David J WalesUniversity Chemical Laboratories Lensfield Road Cambridge CB2 1EW UK

L Sobczyk B Czarnik-Matusewicz M Rospenk and M ObrzudFaculty of Chemistry University of Wrocław Joliot-Curie 14 50-383 Wrocław Poland

Alexandr Gorski Sylwester Gawinkowski Roman Luboradzki Marek Tkacz and Jacek WalukInstitute of Physical Chemistry Polish Academy of Sciences Kasprzaka 4452 01-224 Warsaw Poland

Randolph P ThummelDepartment of Chemistry University of Houston Houston TX 77204-5003 USA

N T ZinnerDepartment of Physics Harvard University Cambridge MA 02138 USADepartment of Physics and Astronomy University of Aarhus 8000 Aarhus Denmark

Betuumll Karaccediloban and Leyla OumlzdemirDepartment of Physics Sakarya University 54187 Sakarya Turkey

Ossama KullieLaboratoire de Chimie Quantique Institute de Chimie de Strasbourg CNRS et Universitacutee de Strasbourg 4 rue Blaise Pascal 67070 Strasbourg France

A I Jaman and Rangana BhattacharyaExperimental Condensed Matter Physics Division Saha Institute of Nuclear Physics Sector 1 Block AF Bidhannagar Kolkata 700 064 India

Vital Peretz and Chaim N SukenikThe Department of Chemistry The Institute of Nanotechnology and Advanced Materials Bar-Ilan University Ramat Gan 52900 Israel

Menachem Motiei and Rachela PopovtzerFaculty of Engineering The Institute of Nanotechnology and Advanced Materials Bar-Ilan University Ramat Gan 52900 Israel

Ivan P ChristovPhysics Department Sofia University 1164 Sofia Bulgaria

E M Barkhudarov I A Kossyi S M Temchin and M I TaktakishviliAM Prokhorov General Physics Institute of RAS (GPI RAS) Vavilov Street 38 Moscow 119991 Russia

Yu N KozlovSemenov Institute of Chemical Physics of RAS Kosygin Street 4 Moscow 119991 Russia

Nick ChristofiEdinburgh University Edinburgh EH9 3JF UK

Mark E BrezinskiCenter for Optical Coherence Tomography and Modern Physics Department of Orthopedic Surgery Brigham and Womenrsquos Hospital 75 Francis Street MRB-114 Boston MA 02115 USACenter for Optical Coherence Tomography and Modern Physics Department of Orthopedic Surgery Harvard Medical School 25 Shattuck Street Boston MA 02115 USADepartment of Electrical Engineering and Computer Science Massachusetts Institute of TechnologyRoom 36-360 77 Massachusetts Avenue Cambridge MA 02139 USA

Chao ChenSchool of Physics Beijing Institute of Technology Beijing 100081 China

Vladimir RyabovDepartment of Complex System School of Systems Information Science Future University Hakodate 116-2 Kamedanakano-Cho Hakodate-Shi Hakodate Hokkaido 041-8655 Japan

Dmitry NerukhNon-Linearity and Complexity Research Group Aston University Birmingham B4 7ET UK

Johannes M Dieterich Sebastian Gerke and Ricardo A MataInstitut fumlur Physikalische Chemie Universitumlat Gumlottingen Tammannstrasse 6 37077 Gumlottingen Germany

Hadas Weinrib Amihai Meiri Hamootal Duadi and Dror FixlerFaculty of Engineering and The Institute of Nanotechnology and Advanced Materials Bar-Ilan UniversityRamat Gan 52900 Israel

List of Contributors 207

Contents

Preface

Chapter 1 Analysis of Water and Hydrogen Bond Dynamics at the Surface of an Antifreeze Protein

Chapter 2 Temperature and HD Isotopic Effects in the IR Spectra of the Hydrogen Bond in Solid-State 2-Furanacetic Acid and 2-Furanacrylic Acid

Chapter 3 An Analytic Analysis of the Diffusive-Heat-Flow Equation for Different Magnetic Field Profiles for a Single Magnetic Nanoparticle

Chapter 4 Helmholtz Bright Spatial Solitons and Surface Waves at Power-Law Optical Interfaces

Chapter 5 The Effect of Nonnative Interactions on the Energy Landscapes of Frustrated Model Proteins

Chapter 6 Proton Transfer Equilibria and Critical Behavior of H-Bonding

Chapter 7 Polymorphism Hydrogen Bond Properties and Vibrational Structure of 1H-Pyrrolo[32-h]Quinoline Dimers

Chapter 8 Effective Potential for Ultracold Atoms at the Zero Crossing of a Feshbach Resonance

Chapter 9 Transition Parameters for Doubly Ionized Lanthanum

Chapter 10 Relativistic Time-Dependent Density Functional Theory and Excited States Calculations for the Zinc Dimer

Chapter 11 Millimeter-Wave Rotational Spectra of trans-Acrolein (Propenal) (CH2CHCOH) A DC Discharge Product of Allyl Alcohol (CH2CHCH2OH) Vapor and DFT Calculation

Chapter 12 The Effect of Nanoparticle Size on Cellular Binding Probability

Chapter 13 Electron-Pair Densities with Time-Dependent Quantum Monte Carlo

Chapter 14 Multispark Discharge in Water as a Method of Environmental Sustainability Problems Solution

Chapter 15 The Advantages of Not Entangling Macroscopic Diamonds at Room Temperature

Chapter 16 Energies Fine Structures and Hyperfine Structures of the 1s22snp 3P (n = 2ndash4) States for the Beryllium Atom

Chapter 17 Statistical Complexity of Low- and High-Dimensional Systems

Chapter 18 A First-Principles-Based Potential for the Description of Alkaline Earth Metals

Chapter 19 Uniformly Immobilizing Gold Nanorods on a Glass Substrate