Vibrational Spectroscopy Vibrational Spectroscopy of Large Molecules:of Large Molecules:Anharmonic Algorithms Anharmonic Algorithms and Applicationsand Applications

R. Benny GerberThe Hebrew University of Jerusalem, Israeland University of California at Irvine, USA

Importance:Importance:

Vibrational spectroscopy is a major tool for studying properties of large molecules.

In some cases, it is a unique tool.

Determination of Structure by Determination of Structure by Spectroscopy : Spectroscopy : (SO(SO44--22))--(H(H22O)O)nn

Miller, Y.; Chaban, G.M.; Zhou, J.;Asmis,K.R.; Neumark,D.M.; Gerber,R.B., J.C.P. 127, 9 (2007).

Determination of Structure by Determination of Structure by Spectroscopy : Spectroscopy : The complex GThe complex G--C of C of nucleic acid basesnucleic acid bases

B.Brauer, R. B. Gerber, M. Kabeláč, P. Hobza, J. M. Bakker, A. G. Abo Riziq, M. S. de Vries, JPC A, 109, 6924, (2005).

Identification of intermediates in Identification of intermediates in processes : processes : The photocycle of Photoactive The photocycle of Photoactive Yellow Protein (PYP)Yellow Protein (PYP)

AA Adesokan, DH Pan, E AA Adesokan, DH Pan, E Fredj, RA Mathies, RB Gerber J. Am. Chem. Soc.(2007).

Hellingwerf et al. J Phys Chem A 2003.

Testing potential functions (Force Testing potential functions (Force Fields) for large moleculesFields) for large molecules

(A) The native state globular structure of ubiquitin. (B) The A state conformation, stable at low pH methanol water stable at low pH methanol water solution.

Moil MD simulation package with the AMBER force field

Segev, E; Wyttenbach, T; Bowers, MT, Gerber RB. P.C.C.P. 10, 21 (2008)

Outline of LectureOutline of Lecture

I. The challenges

II. Principles of the method

III. Algorithms, and how they scale with N (no. of atoms)atoms)

IV. Applications

V. Structures of sugars

VI. Spectra of long chains of hydrocarbons and imaging of Lipids

VII. Conclusions and future directions

The Harmonic ApproximationThe Harmonic Approximation

W

21 2

1),...,( jj

N

joN qkWqqW ∑+= jq - Normal modes

...........)2

1()

2

1( 2211.....1 ++++=Ε vvnvv ωω hh

q1

q2

The Challenges of Anharmonic The Challenges of Anharmonic CalculationsCalculations

A Quantum Mechanical Problem for Many Interacting Degrees of Freedom

� Find method to solve it

� Develop algorithm that scales well with N

� Apply for realistic potentials

VSCFVSCF, , First Step: Separable First Step: Separable ApproximationApproximation

Assume :

),...,( 1 NxxΨ - Vibrational wavefunction

)()....()(),...,( 22111 NNN xxxxx ϕϕϕ=Ψ )()....()(),...,( 22111 NNN xxxxx ϕϕϕ=ΨJ. M. Bowman, J.Chem.Phys. (1978)G. D. Carney, L. Sprandel, C.W. Kern, Adv.Chem.Phys. (1978)M. Cohen, S. Gretia, R.M. McEachran, Chem.Phys.Lett. (1979)R. B. Gerber, M. A. Ratner, Chem.Phys.Lett. (1979)

Key question: Which coordinates to use? Key question: Which coordinates to use?

R.B. Gerber, M. A. Ratner, Adv.Chem.Phys 70, 92 (1988)

Vibrational SelfVibrational Self--Consistent Consistent Field in Normal CoordinatesField in Normal Coordinates

nnn ΨΕ=ΗΨ

To solve:

(1)

),...,(2 12

2

1

2

Nj

N

j j

qqVqm+

∂∂

−=Η ∑=

h

)(),...,(1

1 j

N

jjN qqq ∏

=

=Ψ ϕ

VSCF (or Hartree) Approximation:

(2)

(3)

)()()(2 2

22

jjjjjjjj

qqqVqm

ϕεϕ =

+

∂∂

−h

(4)

(5)

SCF equations:

(4) , (5) are solved by iteration.

Total Energy is:

(6)

VSCFVSCF--PTPT22: Correcting SCF by : Correcting SCF by Perturbation TheoryPerturbation Theory

J. O. Jung and R. B. Gerber JCP 105,10332 (1996)

),...,( 1 NSCF qqV∆+Η=Η (7)

)(qVqqVV jN

jjN ∑

=

−=∆1

1 ),...,(

Treat ∆V as a small perturbation

(8)

Ab initio Anharmonic Ab initio Anharmonic SpectroscopySpectroscopy

Direct VSCF, VSCF – PT2 Calculations Using Electronic

Dr. G.M. Chaban

Dr. J.O. Jung

Structure Codes

G.M. Chaban, J.O. Jung, R.B. Gerber J.Chem.Phys. 111, 1823, (1999)

The Pairwise Coupling The Pairwise Coupling ApproximationApproximation

),q(qW)(qVV),...,qV(q jii ij

coupijj

N

j

diagjN ∑∑∑

>=

++=1

01i ijj >=1

Satisfactory in most cases for our system

Scaling of VSCFScaling of VSCF--PTPT2 2 with Nwith N

N = 3 Natoms -6(no. of vibrational modes)

Liat Pele Dr. Brina Brauer

L. Pele, B. Brauer, and R.B. Gerber, Theor. Chem. Acc. (2007)

VSCF-PT2 computing time is of O(N3).

Liat Pele Dr. Brina Brauer

Scaling of VSCFScaling of VSCF--PTPT2 2 with Nwith N

N = 3 Natoms -6(no. of vibrational modes)

Liat Pele Dr. Brina Brauer

L. Pele, B. Brauer, and R.B. Gerber, Theor. Chem. Acc. (2007)

New version reduces VSCF-PT2 computing time from O(N6) to O(N3).

Liat Pele Dr. Brina Brauer

How Many Pairwise Couplings How Many Pairwise Couplings Do We need ?Do We need ?

Numerical experiments of series of peptides:

Liat Pele

of series of peptides:

NlogN couplings Wij(Qi,Qj) suffice for good accuracy !

Pele and Gerber J. Chem. Phys. (2008)

Numerical Experiments: Mean Numerical Experiments: Mean Accuracy of (separable) VSCFAccuracy of (separable) VSCF

Improves with N!Improves with N!

Liat Pele

Tests for: Di- , Tri and Tetra – Peptides.

Deviations of VSCF from VSCF-PT2 are computed.

Pele and Gerber J. Phys. Chem. C (2010)

computed.

Deviations decrease as logN.

The Protonated ImidazoleThe Protonated Imidazole--Water Water Cluster, (ImHCluster, (ImH++))--HH22OO

CC-VSCF calculations with MP2/DZP and IRPD

Experiments for (ImH+)-H2O-N2:

AA Adesokan, GM Chaban, O Dopfer and RB GerberIR Photodissociation experiments by Prof. O. Dopfer

Yemi Adesokan

(ImH+)(H2O)-N2. Assignment and comparison of theoreticalharmonic and anharmonic frequencies in wave numbers tothose obtained from experiments.

NH stretch (towards water)289728903217

Water asym stretch372336964011

Water sym stretch3639 36483884

MODE DESCRIPTIONIRPDCC-VSCF(Ab initio)

Harmonic(Ab initio)

Ab initio = MP2/DZP

Mean VSCF- experiment deviation : 0.6%

Mean harmonic – experiment deviation: 7.1%

C5-H stretchapp 317232213392

C3-H stretchapp 317231653372

C1-H stretch, C2-H stretchapp 317232053380

NH stretch (towards N2)340433883587

NH stretch (towards water)289728903217

Approach:

Improving “Low Level” Improving “Low Level” Potentials for Potentials for

Spectroscopy CalculationsSpectroscopy CalculationsDr. BrinaBrauer

Dr. G.M. Chaban

Modify the PM3 potential by:

VNEW(q1,...,qn)=VPM3(λ1q1 ,... λnq)

( )

( 3)

harmonicj

j harmonicj

abinitio

PM

ωλ

ω=

Where

The Structures of The Structures of αα –– Glucose in a MatrixGlucose in a Matrix

Agreement with IR experiments supports the structures

MP2/TZP Energies

Dr. B. Brauer

Expt Calcd "A" Expt Calcd "B" Expt Calcd "X" Conf A G+g- Conf B G-g+ Conf X T

3632 3644 3634 3640 3604 35603634 3657 3636 3645 3627 36373641 3660 3640 3665 3640 36493645 3682 3645 3671 3645 3655

OH Stretch Vibrational Frequencies in cm-1

PhenylPhenyl--ββ--Glucose in the Gas PhaseGlucose in the Gas Phase

Average Deviation from Experiment < 1%!

Prof. J. P. Simons Dr. B. Brauer

Vibrational Spectroscopy and Imaging of Biological Membranes

in vivo tissue imaging of a mouse skin

Dodecane Dodecane –– Model Molecule for Model Molecule for Lipid MembranesLipid Membranes

Liat Pele Dr. Jiří Šebek

Prof. Eric O. Potma

Modeling Temperature and Modeling Temperature and Environment EffectsEnvironment Effects

Dodecane - VSCF (scaled PM3 level)

0.8

1R

elat

ive

Ram

an In

ten

sity smoothed spectrum

VSCF lines

2600 2800 3000 32000

0.2

0.4

0.6

Wavenumber [cm-1]

Rel

ativ

e R

aman

Inte

nsi

ty

VSCF lines

Spectroscopy of DodecaneSpectroscopy of DodecaneH-Dodecane - VSCF (scaled PM3 level)

0.2

0.4

0.6

0.8

1

Rel

ativ

e R

aman

Inte

nsi

ty

experimentVSCF

2600 2800 3000 32000

Wavenumber [cm-1]D-Dodecane - VSCF (scaled PM3 level)

1800 2000 2200 24000

0.2

0.4

0.6

0.8

1

Wavenumber [cm-1]

Rel

ativ

e R

aman

Inte

nsi

ty

experimentVSCF

Summary: State of the ArtSummary: State of the Art

1. Many successful applications for systems up to ~50 atoms

2. Method is most suitable for: isolated molecules, low temperatures

Collaborations with Experimental Collaborations with Experimental GroupsGroups

Prof. J.P. Prof. M.S. de Prof. G.Meijer and Prof. J.P. Simons

Prof. M.S. de Vries

Prof. G.Meijer and Prof. G. Von Helden

Prof. T. Rizzo

Prof. I.BarProf. D. M. Neumark

Prof. O. Dopfer

Prof. R. A. Mathies

The Present FrontierThe Present Frontier

1) Methods and algorithm for 103 atoms and more (e.g. for protein structure by spectroscopy )

2) Systems at room temperature, interacting with environment : Can VSCF be used ? How extensive is the applicability ?