Theoretical Actinide Chemistry - DiVA portal398617/FULLTEXT02.pdf · CASSI Complete Active Space State Interaction CC Coupled Cluster CCSD Coupled Cluster Singles and Doubles CCSD(T)

Theoretical Actinide Chemistry– Methods and Models

Theoretical Actinide Chemistry– Methods and Models

Pernilla Wåhlin

c! Pernilla Wåhlin, Stockholm 2011ISBN 978-91-7447-232-5

Printed in Sweden by US-AB, Stockholm 2011Distributor: Department of Physics, Stockholm University

Abstract

The chemistry of actinides in aqueous solution is important, and it is essential tobuild adequate conceptual models and develop methods applicable for actinide sys-tems. The complex electronic structure makes benchmarking necessary. In the thesisa prototype reaction of the water exchange reaction for uranyl(VI), for both groundand luminescent states, described with a six-water model, was used to study theapplicability of density functional methods on actinides and di!erent solvation mod-els. An excellent agreement between the wave function methods CCSD(T) and MP2was obtained in the ground state, implying that near-minimal CASPT2 can be usedwith confidence for the reaction in the luminescent state of uranyl(VI), while densityfunctionals are not suited to describe energetics for this type of reaction.

There was an ambiguity concerning the position of the waters in the secondhydration sphere. This issue was resolved by investigating a larger model, and prop-erly used the six-water model was found to adequately describe the water exchangereaction. The e!ect of solvation was investigated by comparing the results fromconductor-like polarizable continuum models using two cavity models. Scatterednumbers made it di"cult to determine which solvation model to use. The final con-clusion was that the water exchange reaction in the luminescent state of uranyl(VI)should be addressed with near-minimal CASPT2 and a solvation model without ex-plicit cavities for hydrogens. Finally it was shown that no new chemistry appears inthe luminescent state for this reaction.

The thesis includes a methodological investigation of a multi-reference densityfunctional method based on a range separation of the two-electron interaction. Themethod depends on a universal parameter, which has been determined for lighterelements. It is shown here that the same parameter could be used for actinides, aprerequisite for further development of the method. The results are in that sensepromising.

List of publications

1. An Investigation of the Accuracy of Di!erent DFT Functionals onthe Water Exchange Reaction in Hydrated Uranyl(VI) in the GroundState and the First Excited State.Pernilla Wåhlin, Cécile Danilo, Valérie Vallet, Florent Réal, Jean-Pierre Fla-ment, and Ulf Wahlgren J. Chem. Theory Comput., 4, 569–577, (2008)

2. On the combined use of discrete solvent models and continuum de-scriptions of solvent e!ects in ligand exchange reactions: a case studyof the uranyl(VI) aquo ion.Pernilla Wåhlin, Bernd Schimmelpfennig, Ulf Wahlgren, Ingmar Grenthe andValérie Vallet, Theor. Chem. Acc., 124, 377–384, (2009)

3. Water Exchange Mechanism in the First Excited State of HydratedUranyl(VI).Pernilla Wåhlin,Valérie Vallet, Ulf Wahlgren and Ingmar Grenthe, Inorg. Chem.,48, 11310–11313, (2009)

4. On the universality of the long-/short-range separation in multi-reference density-functional theory. II. Investigating f0 actinide species.Emmanuel Fromager, Florent Réal, Pernilla Wåhlin, Ulf Wahlgren, and HansJørgen Aa. Jensen J. Chem. Phys. 131, 054107–054118, (2009)

Reprints are made with permission from the publishers.

The author’s contribution to the papers

Paper 1Took active part in the planning of the investigation and carried out most of thecalculations. Prepared the manuscript together with the other authors.

Paper 2The contribution involved the design of the project together with Professor UlfWahlgren and Dr Bernd Schimmelpfennig. Performed all calculations and preparedthe manuscript together with the other authors.

Paper 3Actively involved in the planning of the project, performed all calculations and wrotethe first draft of the manuscript.

Paper 4Carried out all calculations for the UN2 complex and the calculations at the all-electron level for all actinide complexes except the CUO molecule. Dr EmmanuelFromager and Dr Florent Réal prepared the manuscript.

Unpublished investigations in this thesis

Chapter 8 includes an investigation regarding the amount of HF-exchange in theB3LYP functional. The study was carried out due to a proposition concerning pos-sible improvement of the functional by an increased amount of HF-exchange. Thehypothesis was the functional applicability on actinide complexes would increasewith more HF-exchange. The investigation concerning solvation models discussed inPaper 2 was further explored and eventual factors that can influence the solvatione!ect are presented in the second part of Chapter 10. The author planed this inves-tigation with support from Professor Ulf Wahlgren and Dr Bernd Schimmelpfennigand performed all calculations in this project. Also included in this solvation studyis the re-investigation of the water exchange mechanism for uranyl(V). The resultof this reaction mechanism is later in the thesis been included in a unpublishedcomparison with the reaction mechanism for the ground and first excited states ofuranyl(VI) presented in Chapter 11. In Chapter 12, the interpretation of the resultsconcerning the static correlation is modified in comparison to the interpretation inPaper 4.

List of Abbreviations

ACPF Averaged Coupled-Pair FunctionalAE All Electron

AIMP Ab Initio Model PotentialAMFI Atomic Mean Field Integrals

ANO-RCC Atomic Natural Orbital-Relativistic Correlation ConsistentAO Atomic Orbital

AQCC Averaged Quadratic Coupled ClusterBD Block Diagonal

BHLYP Becke’s half-and-half + LYPBLYP B88+LYPBSSE Basis Set Superposition Errors

B3LYP Becke 3-parameter + LYPB88 Becke 88CAS Complete Active Space

CASPT2 Complete Active Space 2:nd order PerturbationCASSCF Complete Active Space Self Consistent Field

CASSI Complete Active Space State InteractionCC Coupled Cluster

CCSD Coupled Cluster Singles and DoublesCCSD(T) CCSD incl. triples excitations by perturbation

CI Configuration InteractionCOSMO Conductor-like Screening ModelCPCM Conductor-like Polarizable Continuum ModelCPMD Car-Parrinello Molecular Dynamics

CSF Configuration State FunctionDFT Density Functional Theory

DFT-MRCI DFT Multi-Reference Configuration InteractionDKH Douglas-Kroll-HessECP E!ective Core Potential

EPCISO E!ective Polarized Configuration Interaction Spin-OrbitFW Foldy-Wouthuysen

GGA Generalized Gradient Approximations

HF Hartree-FockIC Individual CavityKS Kohn-Sham

LCAO Linear Combinations Atomic OrbitalLDA Local Density ApproximationLYP Lee, Yang and Parr

MCSCF Multi-Configurational Self Consistent FieldMCSCF-DFT Multi-Configurational Self Consistent Field DFT

MO Molecular OrbitalMP2 Møller–Plesset 2:nd order perturbation

MRCI Multi-Reference Configuration InteractionPBE Perdew, Burke and Ernzerhof

RECP Relativistic E!ective Core PotentialRHF Restricted Hartree-Fock

SAOP Statistical Average of Orbital PotentialsSCF Self Consistent FieldSD Slater Determinant

SDCI Single and Double Configuration InteractionSF Spin FreeSO Spin-Orbit

srDFT short-range Density Functional TheoryTD-DFT Time-Dependent DFT

TPSS Tao, Perdew, Staroverov and StaroverovTZVP Triple Zeta Valence Polarization

UA United AtomUA0 United Atom topological model

U-DFT Unrestricted DFTUFF Individual Cavity topological modelUHF Unrestricted Hartree-Fock

VWN Vosko, Wilk and NusairXC Exchange-Correlation

ZORA Zeroth-Order Regular Approximation

Sammanfattning

En ökad förståelse av aktinidkemi i vattenlösning är viktigt ur många perspektiv,men kanske speciellt dess betydelse för möjliga miljökonsekvenser. Det gör det nöd-vändigt att ta fram både konceptuella modeller och att utveckla metoder som ärtillämpbara på aktinider. De komplexa elektronstrukturerna gör det dessutom nöd-vändigt att verifiera nya teoretiska resultat mot kända metoder med hög noggrannhetoch om möjligt mot experiment.

I avhandlingen har vattenutbytesreaktion för uranyl(VI), i både grundtillstån-det och det första exciterad (luminescenta) tillståndet används som testreaktion.Reaktionen, som beskrivs med en kemisk modell som omfattar sex vattenmolekyler,har använts för att undersöka dels om metoder baserade på täthetsfunktionalteoriär tillämpbara på aktinider och dels olika lösningsmedelsmodeller. En utmärkt öv-erensstämmelse mellan de två vågfunktionsbaserade metoderna CCSD(T) och MP2erhölls för grundtillståndet. Däremot visade det sig att täthetsfunktionaler inte ärlämpade för att studera denna typ av reaktion.

Det finns en ambivalens angående placeringen av vattenmolekyler i den kemiskamodellens yttre hydreringssfär. Genom att jämföra med resultat från en större modellkonstaterades att rätt konstruerad kan en sex-vatten modell ge en fullgod beskrivn-ing av vattenutbytesreaktion för uranyl(VI). Vilken lösningsmedelsmodell som ärbäst för actinidreaktioner studerades genom att jämföra resultat från polariserbarakontinuummodeller (conductor-like polarizable continuum models) med hjälp tvåkavitetsmodeller. Resultaten spretade, vilket gjorde det svårt att avgöra vilken lös-ningsmedelsmodell som är bäst att använda. Sammanfattningsvis är slutsatsen attreaktionsenergierna för vattenutbytesreaktion för uranyl(VI) i det luminescenta till-ståndet bör beräknas med CASPT2 med ett litet aktivt rum samt att använda enlösningsmedelsmodell där en kavitet i stället för tre används för att innesluta vatten-molekyler. Resultatet visade att inga ny kemi förekom i det luminescenta tillståndetför denna reaktion.

I avhandlingen ingår även en metodundersökning av en täthetsfunktional baseradmetod som bör ha potential att beskriva multi-referense!ekter. Metoden grundarsig på en uppdelning av två-elektronväxelverkan i en del som primärt beskriverstatisk korrelation och en som beskriver dynamisk korrelation. Uppdelningen byggerpå en universell parameter, fastställd för en grupp lättare element. Det visade sigatt samma parameter kan användas för aktinider, vilket är en förutsättning förvidareutveckling av metoden. Resultaten är i denna mening lovande.

”A child of five would understand this. Send someoneto fetch a child of five.” – Groucho Marx

Contents

1 Introduction 1

2 Introduction Quantum Chemistry 32.1 The Schrödinger equation in quantum chemistry . . . . . . . . . . . 3

2.1.1 The Born-Oppenheimer approximation . . . . . . . . . . . . . 32.1.2 The independent particle approximation . . . . . . . . . . . . 42.1.3 Basis sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Quantum Chemical Methods 73.1 Wave function theory . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1.1 The Hartree-Fock method . . . . . . . . . . . . . . . . . . . . 83.1.2 Electron correlation and post Hartree-Fock methods . . . . . 93.1.3 The Multi-Configurational Self-Consistent Field method . . . 93.1.4 The Configuration Interaction method . . . . . . . . . . . . . 103.1.5 The Møller–Plesset 2nd order perturbation method . . . . . . 113.1.6 The Complete Active Space 2nd order Perturbation method 113.1.7 Coupled Cluster theory . . . . . . . . . . . . . . . . . . . . . 123.1.8 The Averaged Coupled-Pair Functional and Averaged Quadratic

Coupled Cluster methods . . . . . . . . . . . . . . . . . . . . 133.2 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Kohn-Sham Density Functional Theory . . . . . . . . . . . . 133.2.2 Exchange-correlation functionals . . . . . . . . . . . . . . . . 153.2.3 The generalized gradient approximation . . . . . . . . . . . . 153.2.4 The meta-generalized gradient approximation . . . . . . . . . 163.2.5 Hybrid functionals . . . . . . . . . . . . . . . . . . . . . . . . 163.2.6 Time-Dependent Density functional theory . . . . . . . . . . 16

3.3 Combined DFT and wave function based methods . . . . . . . . . . 183.3.1 Multi-reference density functional theory – MCSCF-srDFT . 183.3.2 Multi-reference density functional theory – DFT-MRCI . . . 19

4 Relativistic e!ects 214.1 The Dirac Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 The Foldy-Wouthuysen transformation . . . . . . . . . . . . . . . . . 234.3 The Douglas-Kroll-Hess transformation . . . . . . . . . . . . . . . . 244.4 Scalar-relativistic Hamiltonian and spin-orbit e!ects . . . . . . . . . 25

5 E!ective Core Potentials 275.1 E!ective core potentials . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Solvent e!ects 296.1 Continuum Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.1.1 The conductor-like polarizable continuum model (CPCM) . . 306.1.2 The molecular cavity . . . . . . . . . . . . . . . . . . . . . . . 31

7 Introducing the applications 337.1 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . 35

8 Wave function versus density functional theory 378.1 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.2 Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.3 Reaction energies – the ground state . . . . . . . . . . . . . . . . . . 408.4 Reaction energies – the luminescent state . . . . . . . . . . . . . . . 418.5 Increase of HF-Exchange in B3LYP . . . . . . . . . . . . . . . . . . . 438.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

9 Chemical models 459.1 The chemical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

9.1.1 The six-water model . . . . . . . . . . . . . . . . . . . . . . . 469.1.2 A comparison of the six- and ten-water models . . . . . . . . 47

9.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

10 Solvation models 5110.1 Solvation models from Paper 2 . . . . . . . . . . . . . . . . . . . . . 52

10.1.1 Wave function vs. density functional theory . . . . . . . . . . 5210.1.2 Reaction enthalpies . . . . . . . . . . . . . . . . . . . . . . . . 53

10.2 Conclusion from Paper 2 . . . . . . . . . . . . . . . . . . . . . . . . . 5410.3 Factors that can contribute to the obtained di!erence . . . . . . . . 55

10.3.1 The topological model — sphere radii . . . . . . . . . . . . . 5610.3.2 The water exchange reaction mechanism for uranyl(V) . . . . 5810.3.3 The central ion . . . . . . . . . . . . . . . . . . . . . . . . . . 5810.3.4 The charge on the central ion . . . . . . . . . . . . . . . . . . 59

10.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

11 Water exchange mechanism in the luminescent state of uranyl(VI) 6311.1 The exchange mechanism . . . . . . . . . . . . . . . . . . . . . . . . 6311.2 Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6411.3 Reaction energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

11.3.1 Spin-orbit corrections . . . . . . . . . . . . . . . . . . . . . . 6511.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

11.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

12 Multi-reference short-range DFT 6712.1 Chemical compounds and active space . . . . . . . . . . . . . . . . . 6712.2 The optimal µ for ThO2, PaO+

2 , UO2+2 , UN2 and CUO . . . . . . . . 68

12.2.1 ECP versus AE . . . . . . . . . . . . . . . . . . . . . . . . . . 6912.2.2 Natural orbital occupation . . . . . . . . . . . . . . . . . . . 70

12.3 The optimal µ for NpO3+2 . . . . . . . . . . . . . . . . . . . . . . . . 71

12.3.1 Natural orbital occupation . . . . . . . . . . . . . . . . . . . 7112.4 Equilibrium geometries . . . . . . . . . . . . . . . . . . . . . . . . . . 7212.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

13 Concluding remarks 7513.1 Erratum Paper 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7613.2 Erratum paper 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Chapter 1

IntroductionThe chemistry of actinides in aqueous solution is important and challenging frommany perspectives but maybe particularly in view of their importance in environmen-tal chemistry. Uranium, for example, is more soluble in acidic and alkaline waters andcan therefore be transported from mine sites to the groundwater. The managementof nuclear waste is another potential source for future groundwater contamination.The actinyl ions are highly soluble in water and they can be found in di!erent ox-idation states. Especially the early actinides are likely to undergo redox reactionsand change their oxidation states resulting in many distinct molecular species. Anoverview of actinide interactions in the environment can for example be found in thework by W. Runde.1

There are a number of problems connected with experimental actinide chemistry,in particular for plutonium and beyond. While uranium-238 is rather harmless, plu-tonium is radioactive and toxic, and special security laboratories are needed whenhandling it. The di"culty to perform experiments makes theoretical studies an at-tractive alternative. However, actinides are also a theoretical challenge. They areheavy, which makes it necessary to include relativistic e!ects, and the outer coreorbitals are very flexible, which makes it necessary to correlate many electrons. It isthus necessary to benchmark the theoretical models and methods. The uranyl(VI)ion, UO2+

2 , is the most common species in aqueous (and solid state) chemistry ofuranium and therefore the subject of a large number of investigations. It has beenstudied both experimentally and theoretically and can therefore be used to bench-mark methods to be applied on the heavier actinides that are produced in the burntfuel and radioactive waste, such as Neptunium, Plutonium, Americium and Curium.

In this thesis the water exchange reactions for uranyl(VI) have primarily beenstudied with an emphasis on methodological issues and reactions in the first ex-cited state. The water exchange reaction is a prototype reaction for ligand exchangereactions in general, the latter being of great importance both in environmentalchemistry and in recycling.

A better understanding of the properties of actinide complexes will increase thepossibilities of including information from theoretical calculations and increasingthe knowledge of the behavior of the actinides in both the laboratory and in theenvironment. The investigations in this thesis involve treatment of relativistic e!ectsand complexes where correlation e!ects are essential to take into account. The high

1

2 1 Introduction

solubility of actinide complexes in water shows that it is important to be able tomodel the water reactions with as high accuracy as possible. The work presented inthis thesis is certainly not applied, but in order to understand actinide chemistryand to build viable conceptual models a detailed knowledge of reaction mechanismsis needed.

Chapter 2

Introduction QuantumChemistryComputational chemistry uses quantum mechanical based methods in order to calcu-late di!erent properties of a chemical system. It can be used to determine propertiesthat are inaccessible experimentally or to interprete experimental data. Calculatingelectron and charge distributions, molecular geometries or potential energy surfacesfor di!erent electronic states are some examples of possible applications.

2.1 The Schrödinger equation in quantum chemistryThe time-dependent Schrödinger equation2 is a partial di!erential equation of thetotal energy operator, the Hamiltonian H.

i!!!!t

= H! (2.1)

For a closed system the conservation of energy makes a separation of time and spatialcoordinates possible. The time-independent Schrödinger equation can be formulatedas an eigenvalue equation,

H(r,R)!(r,R) = E!(r,R), (2.2)

where the Hamiltonian for a molecule can then be written (in atomic units) as

H = !!

i

1

2!2

i

" #$ %kinetic energy

electrons

!!

A

1

2MA!2

A

" #$ %kinetic energy

nuclei

!!

i,A

ZA

riA" #$ %

coulomb attr.electron-nucleus

+1

2

!

i<j

1

rij" #$ %

electron-electronrepulsion

+1

2

!

A<B

ZAZB

RAB

" #$ %nuclear

repulsion

,(2.3)

where i,j stand for electrons and A,B for nuclei.

2.1.1 The Born-Oppenheimer approximationThe large di!erence in mass and velocity between electrons and a heavy nuclei opensfor further approximations, and the exact wave function !(r,R) can be approxi-mated as a product of an electron and a nuclear part !(r,R) = "el(r)"nuclei(R).Fast response of electrons to movements of the nucleus lead to an approximation of

3

4 2 Introduction Quantum Chemistry

fixed positions of the nuclei – the electrons move in a constant potential field. Theterm in equation (2.3), describing the nuclear kinetic energy, can then be neglected,leading to the adiabatic approximation (see chapter 41 in ref.3). The electronic partof equation (2.3) including the nuclear repulsion can then be written (in atomicunits) as

H = !!

i

1

2!2

i !!

i,A

ZA

riA+

1

2

!

i<j

1

rij+

1

2

!

A<B

ZAZB

RAB. (2.4)

Within the Born-Oppenheimer approximation4,5 the nuclei move on a potentialenergy surface obtained by solving the electronic Hamiltonian. The approximationis well justified as long as the energy gap between ground and excited electronic statesis large compared to the energy scale of the nuclear motion. The Born-Oppenheimerapproximation breaks down for cases where the velocity of nuclei increases as forhighly excited vibrational states, or when two or more potential energy surfacesapproach or cross each other. In the following only the electronic Hamiltonian willbe considered and the nuclear repulsion term will be omitted.

2.1.2 The independent particle approximationA useful approximation when solving the Schrödinger equation for a n-electron sys-tem, is the independent particle approximation, where the n-electron wave function! is described as a product of n one-electron functions. The description of one elec-tron moving in an average field generated by the rest of the (n-1 ) electrons is calledthe mean field approximation. Since the electrons are fermions with spin 1/2, theelectronic wave function needs to fulfill the requirement of being anti-symmetricunder particle interchange and the Pauli exclusion principle, stating that two iden-tical fermions can not occupy the same quantum state simultaneously. A trial wavefunction that satisfy both requirements is the Slater Determinant (SD), a singledeterminant built up from orthonormal spin-orbitalsa.

!(r1, r2, . . . , rN ) =1"N !

&&&&&&&&&

#1(r1) #2(r1) . . .#N (r1)#1(r2) #2(r2) . . .#N (r2)

......

...#1(rN ) #2(rN ) . . .#N (rN )

&&&&&&&&&

The limitation of the single determinant model is that it does not adequately describethe global electronic structure and the e!ect on this structure from instantaneousrepulsion between the electrons. The energy that is not recovered is called correlationenergy, see section 3.1.2.

2.1.3 Basis setsIn practice, problems in quantum chemistry are in general solved by using basis setexpansions. An atomic orbital (AO) can be described by a combination of gaussian-and spherical harmonic functions, !(r) = R(r)e"!r2

Ylml(!,"). A linear combina-tions of AOs (LCAO)6can be used to represent a molecular orbital (MO).

A minimal basis set that only contains one basis function to each occupied atomicorbital, is defined as a basis set of Single-$ quality. This minimal LCAO description

aAn one-electron wave function including electron spin.

2.1 The Schrödinger equation in quantum chemistry 5

of the molecular orbital is in reality inadequate and needs to be improved. The basiscan be split and if all basis functions are doubled i.e., two basis functions per atomicorbital, the basis set has a Double-$ quality. The Triple- and Quadruple-$ typesof basis sets consist of additional splittings. Further improvement of the basis set isto add polarization functions that consist of higher angular momentum functions.Polarization functions are essential to describe the electron correlation. For moleculeswhere the charge distribution is more di!use as for example in negatively chargedmolecules, di!use basis functions are needed for a better modeling of the electrondistribution, and they are important for accurate description of excited states.

The accuracy of a quantum chemical calculation depends not only on the levelof theory but also on the quality of the finite set of basis functions used for theexpansion of the orbitals. The greater the number of basis functions the better theresulting MOs and thus the mean field wave function.

Chapter 3

Quantum Chemical MethodsIn quantum chemistry two main groups of methods exist for calculating propertiesfor a chemical system, wave function based methods and density functional theory(DFT) based methods. In the former, the methods provide a direct (but approxi-mate) solution to the Schrödinger equation. In the latter category the energy is afunctional of the electron density. Even if DFT formally is defined for ground stateproperties, the excitation energies can be calculated in the time-dependent DFT ap-proach. In the last section of this chapter two methods that combine wave functionmethods with DFT are briefly presented.

The choice of method depends on the character of the chemical problem. Someexamples of criteria that might be of importance when a method is considered are:

- Size-extensive – the energy has a correct linear scaling with the number ofelectrons. A size-extensive method makes it possible to compare calculationsinvolving di!erent numbers of electrons, as for example when describing ioniza-tion processes or when calculations with di!erent numbers of electrons need tobe compared. For a homogeneous system, the solutions for an isolated subsys-tem can be used to construct the solution for the compound system since theenergy has the proper scaling.7,8 A commonly used, less well defined, term issize-consistent, where the calculated energy scales correctly for a molecular sys-tem that consists of two separated, non-interacting subsystem. Size-consistencyusually implies correct dissociation into fragments.

- Variational – the energy from a variational method is an upper bound to thecorrect ground state energy.

- E!cient – useful/applicable for reasonably sized systems with proper basissets.

- Accurate – in the sense that it should give an adequate approximation to thebest available method together with a high quality basis set expansion.

3.1 Wave function theoryA wave function based method derives the electronic structure and the correspondingenergy for a system with n electrons, by solving the Schrödinger equation.

7

8 3 Quantum Chemical Methods

3.1.1 The Hartree-Fock methodThe expectation value of the electronic Hamiltonian in equation (2.4) can be writtenas

Eel =#!|H|!$#!|!$ ,

and the lowest possible energy for a system is obtained by varying Eel with respectto the orbitals using for example Lagrange’s method of undetermined multipliers. Ifa Slater determinant is used as a trial wave function, pseudo-eigenvalue equationsfor the molecular orbitals, the Hartree-Fock equations are derived.

F! = %!. (3.1)

At the Hartree-Fock (HF) level of theory,9–11 the Hamiltonian in equation (2.4)includes the one-electron operator, hi describing the kinetic energy of electron i inthe field of the nuclei and the two-electron operator describing the electron-electronrepulsion. The contribution from the 1/rij term is the Coulomb interaction, Jij ,and the exchange interaction, Kij , where the former describes the classical repulsionbetween electrons. The Fock operator,

Fi = hi +N!

j

(Jj ! Kj), (3.2)

depends on all occupied spin-orbitals. A specific orbital can only be determined ifall other occupied orbitals are known. In the iterative Self-Consistent Field (SCF)approach, the initial set of orbitals are used to generate a new set, the process is thenrepeated until an energy convergence criteria is fulfilled. A Hartree-Fock Hamiltoniancan be defined as

HHF =n!

i

Fi =n!

i

'hi +

n!

j

(Jij ! Kij)(. (3.3)

Using equations (3.1) to (3.3) the HF energy can be written as

EHF =N!

i=1

hii +1

2

N!

i=1

N!

j=1

(Jij ! Kij) =N!

i=1

'%i +

1

2

N!

j=1

(Jij ! Kij)(. (3.4)

Unrestricted open shell Hartree-FockFor systems with one or more open shell electrons, spin-orbital with individual spatialcoordinates for each spin function can be used,

#i(r) =

)#!j (r)&(')

#"j (r)(('),

where &(') and ((') corresponds to spin-up and spin-down respectively, and 'is an unspecified spin variable. The unrestricted wave function is normally not aneigenfunction of the total spin S2 and the method can su!er from e!ects due tospin contamination. It is an artifact due to a mixing of higher spin states in the

3.1 Wave function theory 9

obtained wave function. A high spin contamination can strongly a!ect the result ofa geometry optimization or reaction energies. The UHF method is size-extensive andvariational.

Restricted closed shell Hartree-FockFor a closed shell system the spin-orbitals can be described with a common spatialpart for both the &(') and ((') spin.

#i(r) =

)#j(r)&(')

#j(r)((')

The restricted HF energy will, in most cases, be the same as if derived with theunrestricted formalism. The RHF method is size-extensive and variational.

3.1.2 Electron correlation and post Hartree-Fock methodsIn the Hartree-Fock method the energy is calculated in the mean field approximationand this approach usually recovers more than 95 basis set. It can be su"cient forcertain situations but for many applications the remaining part is crucial in orderto get an accurate description of the properties of the system. At the Hartree-Focklevel of theory, the motion of the electrons is said to be uncorrelated. The correlationenergy Ecorr is defined as the di!erence between the total exact energy Etot of thesystem and the Hartree-Fock energy, EHF ,

Ecorr = Etot ! EHF .

The correlation can be divided into a dynamic part and a non-dynamic part wherethe latter arising from near-degenerate energy levels.

In correlated methods (post Hartree-Fock methods), MOs are frequently dividedinto occupied orbitals, from where excitation can take place up to the initially unoc-cupied orbitals. Another commonly used term is the Configuration State Function(CSF), a spin and symmetry-adapted linear combination of Slater determinants withdi!erent occupied orbitals, that can be used as a basis in some of the correlatedmethods.

3.1.3 The Multi-Configurational Self-Consistent Field methodSome chemical systems can not be correctly described with a single reference wavefunction. Low-lying excited states or bond dissociation processes are examples wherea multi-reference wave function is needed for a correct description. A standardapproach is to take linear combinations of Slater determinants or configurationstate functions into account as in the multi-configurational self-consistent field (MC-SCF)11 method.

|"MCSCF $ =!

I

cI |!I$, (3.5)

where |!I$ are Slater determinants or CSFs.Both the wave function and the orbital coe"cients, {ci}, are variationalyl de-

termined. The multi-reference wave function is capable of handling non-dynamiccorrelation e!ects, the recovered amount of dynamic correlation depends on the sizeof the active space.


In the complete active space self-consistent field (CASSCF)12 method, the or-bitals are divided into three categories, inactive doubly occupied orbitals that keepthe occupancy, active partially occupied orbitals, and virtual non-occupied orbitals.All possible configurations generated by distributing electrons in the active spaceare included in the calculation. The active orbitals define a subspace of the totalorbital space, in which the configuration expansion is complete. The calculation iscarried out as a sequence of iterations, where the first step is an eigenvalue equationsolved for the CIa problem within the active space. This is then followed by an or-bitals optimization within inactive and active space in order to determine the bestpossible multi-reference wave function. The amount of dynamic correlation that isrecovered clearly depends on the choice of active space. The method is size-extensiveand variational.13

3.1.4 The Configuration Interaction methodA Configuration Interaction (CI)11 wave function is a linear combination of Slaterdeterminants or spin-adapted configuration state functions generated by excitingelectrons from the reference. The reference wave function can be a sum of Slaterdeterminants or CSFs and the CI problem is solved variationally in the subspace ofSDs or CSFs.

|"CI$ = c0|!0$+!

ra

a†raa|!0$+!

a<br<s

a†ra†saaab|!0$+ . . . (3.6)

= c0|!0$+!

ra

cra|!ra$+

!

a<br<s

crsab|!rsab$+

!

a<b<cr<s<t

crstabc|!rstabc$+ . . . , (3.7)

=!

I

cI |!I$ (3.8)

The creation (a†) and annihilation (a) operators generate all excitations from a setof occupied orbitals (a, b, . . . ) to the set of unoccupied orbitals (r, s, . . . ). A full CIis exact in a complete basis and the accuracy is high if a large basis set is used.The method is size-extensive and variational, but has a limited use in practicalapplications.

The common approach is to use a truncated CI which can be applied to largersystems. A single and double CI (SDCI) includes as the name indicate, only singleand double excitations.

|"SDCI$ = c0|!0$+!

ra

cra|!ra$+

!

a<br<s

crsab|!rsab$ (3.9)

= !0 +!S +!D = !0 +!C (3.10)

The truncated CI wave function gives results that are often reasonably accurate butnot size-extensive14 since it does not include any contributions from higher thandouble excitations. The size-extensivity is strongly dependent on contributions fromfourth order excitations, see ref.15,16 and reference therein. A dominating part of thequadruple excitations can be recovered by contributions from two double excitations

aSee section 3.1.4.

3.1 Wave function theory 11

taking place simultanius, and based on this, corrections to the size-extensive errorcan be obtained.

One among the possible corrections is the Davidson correction.17 For a singlereference wave function the correction is written as

#Davidson = (1! c20)ESDCI ,

where ESDCI is the correlation energy

ESDCI =#*

i ci!i|H ! E0|*

i ci!i$*i c

2i

. (3.11)

E0 is the energy for the reference wave function. For a multi-reference wave functionthe Davidson correction can be expressed as

#Davidson ='1!

!

i#ref

c2i((EMRCI ! EMR), (3.12)

where EMRCI is the energy for the truncated MRCI and EMR is the energy for thereference wave function.

3.1.5 The Møller–Plesset 2nd order perturbation methodFor systems where one single determinant is a good reference wave function, themissing dynamic correlation can be obtained in a perturbative manner by adding aperturbation H $ corresponding to the interaction energy between the electrons, tothe zeroth-order Hamiltonian, H0. In the Møller–Plesset second order perturbation(MP2) method,18 the zeroth-order Hamiltonian is the same as in the Hartree-Fockmethod, see equation (3.3),

H = H0 + H $. (3.13)

By expanding both the Hamiltonian, H, and the wave function |!(0)$ = |!HF $ in apower series, the energy expression can be written as

EMP2 =!

i,j,a,b

#"i(1)"j(2)|r"112 |"a(1)"b(2)$ %

2#"a(1)"b(2)|r"112 |"i(1)"j(2)$ ! #"a(1)"b(2)|r"1

12 |"j(1)"i(2)$%i + %j ! %a ! %b

,

where "i,"j are occupied orbitals, "a,"b are virtual orbitals and %i, . . . , %b the cor-responding orbital energies. The MP2 energy is size-extensive and the method isreasonably accurate for certain applications but the non-dynamic part of the corre-lation can not be recovered due to the single-reference.

3.1.6 The Complete Active Space 2nd order Perturbation methodThe Complete Active Space second-order perturbation (CASPT2)19–22 method takesnon-dynamic correlation into account through an initial multi-reference CASSCFstep. The expression for the second-order energy has the same form as the usual


Møller-Plesset energy, the di!erence being that the active orbital energies are re-placed by a more complex expression involving a weighted sum over all active or-bitals.

A description of the method can be found in ref.19,21 The CASPT2 method isapproximatively size-extensive due to the proceeding CASSCF step and the accuracyis good with an adequate active space.

3.1.7 Coupled Cluster theoryA single-reference and non-variational approach that recover dynamic correlationenergy is based on Coupled Cluster (CC) theory.23–25 In equation (3.6) the totalwave function depends on the expansion coe"cients of single, double, triple excita-tions and so forth. By introducing the cluster operators, T = T1 + T2 + T3 + . . . ,corresponding to the excitation levels in the CI wave function, the CC wave functioncan be written as

|CC$ = eT |HF $.|HF $ = |!0$, and where T1 and T2 are the cluster operators that generate singleand double excitations respectively. {) ra} and {) rsab} are the expansion coe"cients orcluster amplitudes and {a†raa} and {a†a†saaab} are second quantized sets of singleand double excitation operators.

T1 =!

ra

) raa†raa T2 =

!

r>sa>b

) rsaba†ra

†saaab,

where a, b and r, s are orbitals defined as in equations (3.6) and (3.7).

The Coupled Cluster Singles and Doubles methodA truncation of T leads to approximate CC methods, G. D. Purvis and R. J.Bartlett,26 suggested a CC method with T = T1 + T2, the coupled cluster sin-gles and doubles (CCSD) model. By expressing eT as a power expansion, the wavefunction for CCSD can be written as

|CCSD$ = (1 + T1 + T2 +T 21

2+ T1T2 +

T 22

2+ . . . )|HF $.

The wave function has a similar structure as the full CI expansion in equation(3.6), since all the higher order excitations are products of double excitations. Itcan be shown26 that whatever the truncation of the CC wave function is, the powerexpansion assures that the approximation is size-extensive, see section 3.1.4. TheCCSD energy is obtained starting from the Schrödinger eigenvalue equation, and itcan be expressed as

ECCSD = #HF |e"T HeT |HF $. (3.14)

The amplitudes are generated by projecting equation (3.14) onto a space of exciteddeterminants, |!$,

#!|HeT |HF $ = ECCSD#!|HeT |HF $.

The CCSD energy is computed starting from a single-reference determinant and de-pends directly on the single and double amplitudes, with indirect contributions from

3.2 Density Functional Theory 13

higher excitations. Since the energy depends on the amplitudes and the amplitudedepend on the energy, the process is iterative.

A further improvement of the CCSD approximation is to include the triplesexcitations in a perturbative manner, CCSD(T).27

3.1.8 The Averaged Coupled-Pair Functional and Averaged QuadraticCoupled Cluster methods

The averaged coupled-pair functional (ACPF) and averaged quadratic coupled clus-ter(AQCC) methods can be related to the truncated SDCI. The wave function canbe written such that it has the same structure as the SDCI, and therefore expressedin short form according to equation (3.10).

!I = !0 +!C ,

where I is the space spanned by the reference wave function, !0, and all generatedsingle and double excitations.

The correlation wave function can be obtained as varying a functional FC forboth APCF28,29 and AQCC.30

FC =#!0 +!C |H ! E0|!0 +!C$

#!0|!0$+ g#!C |!C$The correlation energy can be obtained from FC . The renormalization factor g, thatdepends on the number of electrons, is the only di!erence between the two methods.For g=1 the SDCI method is recovered. Both methods have approximative size-extensive energies.

3.2 Density Functional TheoryIn density functional theory (DFT) the electron density, and therefore the energy,only depends on the three spatial coordinates. The two fundamental theorems inDFT are the Hohenberg-Kohn theorems,31 stating that the ground state electrondensity, *(r), uniquely determines the external potential v(r) and that the groundstate energy can be obtained variationally.

The energy as a functional of the density *(r), can then be written as

E[*(r)] =

+*(r)v(r)dr + T [*(r)] + Eee[*(r)] (3.15)

=

+*(r)v(r)dr + F [*(r)], (3.16)

where T [*(r)] is the kinetic energy and Eee[*(r)] the electron-electron interactionenergy including the Coulomb interaction, J [*(r)] and the non-classical interaction.Together, T [*(r)] and Eee[*(r)] define the density dependent functional, F[*(r)]

3.2.1 Kohn-Sham Density Functional TheoryIn the Kohn-Sham (KS) DFT, introduced by W. Kohn and L.J. Sham 1965,32 theelectron density for a system of interacting electrons is postulated to be the same asfor a system of non-interacting KS electronsb.

bFor more information about DFT several text books are available, see for example references.33,34


The kinetic energy and density for the non-interacting system is defined as

Ts[*(r)] =N!

i

#"i|!1

2&2|"i$, (3.17)

*(r) =N!

i

|"i(r)|2. (3.18)

The KS orbitals are determined from a set of pseudo-eigenvalue equations, in analogywith the Hartree-Fock equations (3.1)

hKSs "i = %i"i, (3.19)

where the one-electron Hamiltonian for the non-interacting system is

hKSs = !1

2&2

i + veff (r). (3.20)

An e!ective potential is defined by

veff [(r)] = v(r) +

+*(r!)

|r " r!|dr + vxc(r), (3.21)

andN!

i

%i = Ts[*(r)] +

+*(r)veff ,

and the energy33 for the Kohn-Sham system can then be written as

EKS =N!

i

%i !1

2

+*(r)*(r!)

|r " r!| drdr$ + Exc[*]!

+vxc(r)*(r)dr. (3.22)

By combining equations (3.15) and (3.22) the exchange-correlation energy, Exc, isformulated as

Exc[*(r)] = T [*(r)]! Ts[*(r)] + Eee[*(r)]! J [*(r)]. (3.23)

The exchange-correlation energy includes both the di!erence in kinetic energy be-tween a system of interacting electrons and non-interacting KS electrons, and thedi!erence between the electron-electron interaction in quantum mechanic and theclassical Coulomb interaction (exchange and correlation).

The exchange-correlation potential in equation (3.21) is

vxc(r) =+Exc[*(r)]

+*(r), (3.24)

and the density dependent functional, F[*(r)] in equation (3.16) can be expressedas

F [*(r)] = Ts[*(r)] + Exc[*(r)]. (3.25)The KS orbitals will result in the exact density and energy of the system, given thatthe exact exchange-correlation functional is known. It can be noted that in DFT, anexcitation energy is described as the di!erence in orbital energies since both occupiedand virtual orbitals see the same potential and the same number of electrons.


3.2.2 Exchange-correlation functionalsThe exchange-correlation energy can be viewed as the energy resulting from the inter-electronic repulsion interaction between an electron and its exchange-correlationhole. Each electron creates a hole in the electron density around itself as a directconsequence of exchange-correlation e!ects, i.e. the probability of finding two elec-trons extremely close to each other is close to zero. The exchange-correlation energycan be decomposed into an exchange and a correlation part,

Exc = Ex + Ec.

The exchange and correlation energy can be expressed in terms of exchange- andcorrelation energy per particle.

Ex[*] =

+*(r),x([*]; r)dr

Ec[*] =

+*(r),c([*]; r)dr

The first approximation for the exchange-correlation energy was the local densityapproximation (LDA) method, where the energy is a local function of the energydensity, two examples of LDA functionals can be found in ref.35–37 A uniform electrongas system has a constant electron density and the exchange energy is proportionalto the electron density,

ELDAx [*] = !3

2

' 3

4-

( 13

+*#(r)

43 d3r.

Binding energies of molecules obtained at the LDA level are often overestimated andthe bond lengths underestimated.

The e!ect of the exchange-correlation hole can be estimated by replacing it by auniform density inside a small sphere and zero elsewhere, neglecting corrections tothe exchange-correlation energy due to inhomogeneities in the electron density. Theself-correlation or self-interaction error arises from an interaction of a single electronwith its own density.

3.2.3 The generalized gradient approximationLDA might be suitable for solid state systems where the electron density can be con-sidered as slowly varying but for most chemical systems the description is not su"-cient. An improvement of the exchange-correlation energy can be found with gradientcorrected functionals, the generalized gradient approximations (GGA). The spher-ical approximation of the exchange-correlation hole is improved by a non-sphericalhole, the correlation energy is derived by a function of the local density and thedensity gradient.

EGGAxc [*] =

+*,xc

GGA(*,&*)dr

Another weakness with LDA is that the asymptotic behavior of the exchange isinadequately described. One example of a exchange correction functional is the B88


functional suggested by Becke38–40 including one parameter that is determined bya fit to the exact exchange energies of the five atoms He, Ne, Ar, Kr, and Xe.

Lee, Yang and Parr,41,42 suggested a correction of the correlation energy, theLYP correlation functional includes four parameters fitted so that the result for ahelium atom is correct. The combination of B88 and LYP gives the commonly usedBLYP functional38,41–43

A further example of GGA functionals is the PBE, which resembles the B88 func-tional but the two parameters are determined from properties derived from the linearresponse of the uniform electron gas simulations where the density is slowly varied,suggested by J. P. Perdew, K. Burke, and M. Ernzerhof 1996.44,45 BP8635,38–40,46

is a combination including the B88 exchange functional and correlation correctionssuggested by J. P. Perdew 1986,

3.2.4 The meta-generalized gradient approximationThe generalized gradient approximations can be expanded to include informationabout the electron density and gradient in terms of the Laplacian and non-interactingkinetic energy, leading to the so-called meta-GGA functionals.

TPSS36,39,40,47 is one example of a meta-GGA functional by J. M. Tao, J. P.Perdew, V. N. Staroverov, and G. E. Scuseria.47 It is based on the PBE functionaland includes self-correlation corrections and correction of the gradient including areduced Laplacian of the spin densities. The parameter is determined from a fit ofthe atomization energies of 20 small molecules.

3.2.5 Hybrid functionalsAn empirical approach to improving the exchange-correlation energy is the hybridfunctionals where GGA functionals are combined with explicit Hartree-Fock ex-change. One of the most commonly used hybrid functionals is B3LYP where theB88 and LYP functionals are combined with Hartree-Fock exchange.

EB3LYPxc = (1! a)ES

x + aEHFx + bEB88

x + cELYPc + (1! c)EVWN

c ,

where ESx is the Slater-Dirac exchange functional,39,40 EHF

x , the Hartree-Fock ex-change38–41 and EVWN

c is a local correlation function by Vosko, Wilk and Nusair.35The exchange energy is a linear combination of the HF and B88 contribution whereasthe correlation energy arise from the LYP and VWN functionals. The parametersa=0.20, b=0.72 and c=0.8148 were originally determined by a fit to the set of at-omization energies, ionization potentials, proton a"nities and total atomic energiesfrom systems in the so called G1 set.49

Another example is the BHLYP38–41,50 where the amount of Hartree-Fock ex-change is larger, 0.5 compared to the 0.2 for the B3LYP functional.

3.2.6 Time-Dependent Density functional theoryThe time-dependent Kohn-Sham equation can be written as

i!

!t!(r, t) =

'! 1

2&2 + v(r, t) +

+*(r, t)

|r " r!|dr + v#xc(r, t)(!(r, t).


It is a time-dependent analogue of the Hamiltonian in the time-independent DFT,see equation (3.20), and the energy can be expressed as

E[*(r, t)] = T [*(r, t)] + Eee[*(r, t]) +

+v(r, t)*(r, t)d3r, (3.26)

in analogy with equation (3.15), showing that the theory outlined in section 3.2apply to the time-dependent DFT as well.

In time-dependent response theory it is possible to calculate excited state energiesdue to the response of the ground state when a harmonic perturbation in terms ofan electric field is applied. Excitation energies and oscillator strengths can then beobtained from the poles and residues of the dynamic dipole polarizability, &(') where' is the frequency. The mean polarizability, &('), can be written,

&(') =!

I

fI('2

I ! '2), (3.27)

where 'I that represent the excitation energies are residues of the mean polarizabil-ity and fI is the set of corresponding oscillator strength. The calculated result isdependent on the functional used.

In general the excitation energies are obtained by solving the pseudo-eigenvalueequation

$#i = '2i #i, (3.28)

where $ is a matrix describing the transitions from occupied to unoccupied orbitals,see reference.51 For more detailed information, see for example references.51–53

Some functionals are derived especially to be used within a time-dependentDFT approach, and one example is the Statistical Average of Orbital Potentials(SAOP),54,55 where a LDA-dependent model potential with correct asymptotic be-havior for the highest occupied KS orbital is used together with a PBE-dependentmodel potential for remaining occupied orbitals. The functional can be applied tocalculate excitation energies and static and frequency-dependent multipole polariz-abilities.


3.3 Combined DFT and wave function based methodsThe need for methods that handle both static and dynamic correlation with rea-sonable accuracy has increased, especially if the method is to be applied to largersystems. Some wave function based methods lack an important part of the dynamiccorrelation, but this can be recovered by applying a perturbative method. Theircomputational complexity make them unworkable for large scale calculations. In thisthesis two methods are outlined where the wave function based theory is combinedwith DFT.

3.3.1 Multi-reference density functional theory – MCSCF-srDFTA long- and short-range separation of the two-electron Coulomb interaction 1/rijopens the possibility to enable a multi-reference description at the DFT level. Theseparation of the two-electron interaction is obtained by introducing an error func-tion (erf),

1

rij=

1! erf(µrij)rij

+erf(µrij)

rij. (3.29)

The two terms on the rhs of equation (3.29) describe the short- and long-rangepart of the Coulomb interaction respectively. For µ ' ( the short-range term inequation (3.29) vanish whereas for µ ' 0 the long-range term erf(µrij)/rij ' 0.This separation was first suggested by A Savin56 and the long- and short-rangeseparation the two-electron Coulomb interaction can be written as

Wee = W lr,µee (rij) +W sr,µ

ee (rij), (3.30)

W lr,µee (rij) =

1

2

!

i %=j

erf(µrij)

rij. (3.31)

Fromager et al.57 defined the criterion for the long-/short-range separation for thismulti-reference srDFT approach such that the static correlation and dispersion inter-action e!ects are assigned to the long-range interaction and the dynamic correlatione!ects to the short-range interaction. In principal any wave function based methodscan be adapted to this srDFT method which is interfaced with the Dalton58 andMolpro59 packages. The optimal value of the parameter µ (in a.u.) is defined as thelargest value of µ for which the wave function can be considered to be well describedwith a single determinant in systems with no significant static correlation and dis-persion interaction e!ects. Fromager et al. used the closed shell systems He, Ne, H2,N2 and H2O as a test set to determine a value for µopt.

They used the following definition: a multi-reference wave function is consideredas a single determinant when the di!erence between the HF-srDFT and MCSCF-srDFT energy is less than 10"3 a.u. Using this criterion, the wave functions forthe complexes in the test set could be regarded a single determinant systems for aµ = 0.5. Two short range functionals were used in the calculations, the srLDA56,60

and the srPBE44,61,62 funtionals, both gave the same results. The single-referencecharacter of the wave function was further investigated by considering the naturalorbitals occupation in the MCSCF-srDFT calculations. For a µ = 0.4, no naturalorbital occupation outside the occupied space was larger than 10"4, with exceptionfor N2 where the accuracy was 10"3. The µopt value was therefore chosen to 0.4 a.u.

3.3 Combined DFT and wave function based methods 19

As a result of the separation of the two-electron interaction the functional definedin equation (3.16) can be expressed in a similar fashion.

F [*] = F lr,µ[*] + Esr,µH [*] + Esr,µ

xc [*], (3.32)F lr,µ[*] = min

$#!|T +W lr,µ

ee |!$, (3.33)

including the short-range Hartree energy Esr,µH [*] and the exchange- and correlation

energy Esr,µxc [n]. The ground state energy can then be expressed as

E0 = min!

,-#!|T + W lr,µ

ee |!$+Esr,µH [*!] +Esr,µ

xc [*!] +

+vne(r)*!(r)dr

./. (3.34)

This method is still under development.

3.3.2 Multi-reference density functional theory – DFT-MRCIThe DFT/MRCI method developed by Grimme and Waletzke,63 is a combination ofKS DFT and a truncated multi-reference configuration interaction (MRCI) takingsingle and double excitation into account. The configuration state functions in theCI are constructed by Kohn-Sham orbitals so a dominating part of the dynamicelectron correlation is taken into account by the use of hybrid BHLYP.

The usual HF-based MRCI formalism is formally retained with the di!erence thatfive empirical parameters are incorporated in order to take into account the problemwith the double counting of dynamic electron correlation. (The energy expression fordiagonal matrix elements, Coulomb and exchange integrals are scaled with a factorthat has been parametrized.) The five DFT/MRCI parameters have been fittedto experimental data for ten reference molecules. The parameters do not dependon the molecules within the set of reference molecules, they rather depend on themultiplicity of the desired state, the number of open shells of the CSF.

The method has been shown by Silva-Junior et al.64 to give good electronicspectra for a variety of organic molecules, it is important to notice that neither heavyelements nor transition metal compounds are included in the parameterization set.

Chapter 4

Relativistic e!ectsIn computational chemistry most questions concern how electrons move and interactin the field generated by much heavier nuclei. Some direct consequences of relativistice!ects on the electronic structure are a contraction and stabilization of s (and p)shells and the splitting of the p, d, f and higher shells. The indirect e!ect, whicharises from the contraction of the inner shells, results in a screening of the nuclearcharge for the outer shells – leading to a decreased e!ective nuclear charge andan expansion and destabilization of valence d, f, and higher shells. In moleculesconsisting of lighter atoms, the relativistic e!ects are small and can in many casesbe disregarded. For the heavier atoms the e!ect has a large impact on the results.Therefore it is important to apply a methodology for an accurate description of suchsystems. In actinide compounds the relativistic e!ects result in close-lying 5f, 6d and7s orbitals, influencing bond lengths, binding energies and vibrational constants. Therelativistic contribution due to spin-orbit coupling in molecules tend to stabilize theground state energy, this e!ect is particularly important for molecules with one ormore unpaired electrons.

The theory for relativistic quantum chemistry is based on the four-componentDirac equation, which combines quantum mechanics with the prerequisites fromspecial relativity that the laws of physics and the speed of light c are the same inall inertial frames. The four-component Dirac equation can be transformed into atwo-component equation that can be reduced to a scalar-relativistic Hamiltonian. Inthis thesis the relativistic e!ects have been included by using an explicitly scalar-relativistic second-order Douglas-Kroll-Hess Hamiltonian or a scalar-relativistic ef-fective core potential (RECP). The following sections includes a brief presentationof the Dirac Hamiltonian, and the transformations into a Foldy-Wouthuysen orDouglas-Kroll-Hess Hamiltonian.a

4.1 The Dirac HamiltonianStarting from the relativistic energy expression, E2 = p2c2+m2c4, and the operatorsfor energy and momentum,

E = i! !

!tand p =

!i&.

aFor detailed reviews of relativistic and non-relativistic quantum mechanics in general, some classicaltext books are available, for example3 and.65

21

22 4 Relativistic e!ects

O. Klein and W. Gordon66–68 suggested a relativistic analogue to the Schrödingerequation based on E2, the Klein-Gordon equation for a free particle,

'&2 ! !2

c2!t2!m2c2

("(r, t) = 0. (4.1)

The Klein-Gordon equation is a scalar equation of the second order in time andspatial coordinates, unfortunately it yields a time-dependent probability for the lo-cation of the particle. P. A. M. Dirac69–71 suggested an anzats linear in both !/!tand !/!r(&) based on the Klein-Gordon equation in (4.1).

'&2 ! %2

c2%t2 !m2c2(=

(ax%%x

+ ay%%y

+ az%%z

+ ic

%%t

+ (mc)(ax%%x

! ay%%y

! az%%z

! ic

%%t

! (mc)

The equality above hold if the introduced algebraic quantities & and (, eliminatethe cross terms. If the components of & anti-commute with each other and witha scalar (, the number of anti-commuting relations indicates that both algebraicquantities need to be four-dimensional. The Dirac equation can be written as thesecond brackets on the rhs of the equality above,

'i!

!t! c# · p+ (mc2

("(r, t) = 0. (4.2)

# =

00 $$ 0

1, ( =

0I 00 !I

1, (4.3)

where I is the two-dimensional identity matrix and $ is a compact form of the Paulispin matrices.

$x =

00 11 0

1,$y =

00 !ii 0

1,$z =

01 00 !1

1

The Dirac equation (4.2) describe a particle with spin 1/2 and can be shown to beLorentz invariant. The equation can be separated as described in section 2.1 and thetime-independent Dirac operator can be used to solve the eigenvalue equation for afree particle,

D! = E!,

D = (c# · p+ (mc2). (4.4)

The wave function ! is a four-component Dirac spinor with the corresponding energyeigenvalue E. The wave function is expressed as a four-component column spinor,

!(r) =

2

334

"L& (r)

"L' (r)

"S& (r)

"S' (r)

5

667 =

0!L(r)!S(r)

1, (4.5)

where L and S are the large and small components of the wave function describingthe positive and negative energy solutions respectively. ) and * symbols the spin.

4.2 The Foldy-Wouthuysen transformation 23

The four-component time-independent Dirac equation for an electron (E +mc2 > 0)in an external potential, V , can be written as

0V c$ · p

c$ · p !2mc2 + V

10"L

"S

1= E

0"L

"S

1. (4.6)

The one-electron Dirac Hamiltonian can be extended into a n-electron Hamilto-nian by adding the electron-electron interaction term 1/rij to the Dirac Hamiltionianfor a particle in an external potential.

The electron-electron interaction 1/rij is not Lorentz invariant and a more rig-orous form is obtained by adding the Gaunt- and Breit two-electron interactionterms.72,73 However, the so called n-electron Dirac-Coulomb-Breit Hamiltonian, isnot gauge invariant. Normally only 1/rij is used.

The Dirac and Dirac-Coulomb-Breit Hamiltonian can be expressed as a sum ofeven, E , and odd, O, operators with respect to intrinsic parity. The even operatorshave no matrix elements coupling the large and small component, whereas the oddoperators only consist of such matrix elements. Operators including ( and V areeven and operators including & are odd, c.f. the diagonal and non-diagonal elementsin equation (4.6).

H = (mc2 + E + O (4.7)

4.2 The Foldy-Wouthuysen transformationThe Foldy-Wouthuysen (FW) transformation74 decouple the electron- and positronpart of the four-component Hamiltonian. In presence of an external potential, thisis acquired through a sequence of unitary transformations. Since the Hamiltoniancan be written as in equation (4.7), the unitary transformation can be chosen suchthat it reduces the odd operators. For the first order FW transformation, the evenand odd terms in the expansion of the Hamiltonian need to fulfill

[E1,(1] = 0 and {O1,(1} = 0, (4.8)

and the FW transformation can then be expressed as

H1 = H + i8S,H

9+ . . . S =

!i(O2mc2

. (4.9)

S is chosen such that odd terms are removed, but unfortunately this approach in-troduces new odd terms of higher order, by applying further iterations of the trans-formation they can be removed.

The Pauli Hamiltonian is obtained by a first order FW transformation of theDirac-Coulomb-Breit Hamiltonian75,76 expanded in a power series in c"2, includingup to second order terms of c"2,

HPauli = V +p2

2m+

1

4m2c2'! p4

2m+

1

2&2V + . ·&V % p

(. (4.10)

The first two terms correspond to the non-relativistic Schrödinger equation, the thirdis the mass-velocity term and the fourth term, the Darwin correction represent thescalar relativistic corrections. The last term is the spin-orbit coupling term.


An alternative to the Pauli Hamiltonian is the zeroth-order regular approxi-mation (ZORA)77 Hamiltonian, derived through a slightly di!erent method, andwhere the power series expansion is carried out with respect to E/(2c2 ! V ). TheBreit-Pauli Hamiltonian is derived from the Dirac-Coulomb-Breit Hamiltonian fora two-electron system through the FW transformation and then generalized to nelectrons. This Hamiltonian includes in addition to the Pauli Hamiltonian also thetwo-electron Darwin term and the two-electron spin-orbit terms.

The negative mass–velocity term in the Pauli Hamiltonian gives a strongly at-tractive term for states with high momentum, p, which leads to a variational collapse.The Darwin degenerates to a highly singular term if the potential V is a point-likenucleus, and finally the spin–orbit coupling term leads to variational collapse, sinceit is not bounded from below. Higher order Foldy-Wouthuysen74 transformationsare connected with same behavior as the Pauli Hamiltonian, with the addition thatthe wave functions do not yield a well defined non-relativistic limit.78,79 The varia-tional collapse and singular functions can be avoided by using the external field Vas perturbation parameter and applying a first order perturbation approach.

The FW transformation can not be found in closed form except for the case ofthe free particle,

HBD = W †0 DW0 =

0h+ 00 h"

1, (4.11)

where W0 is unitary, W †0W0 = 1 and includes an operator X containing the relation

between the large and the small components such that the o!-diagonals block iseliminated,80–82

X ='mc2 +

:m2c4 + p2c2

("1c$ · p.

The free particle FW transformation is uninteresting from a chemical point of viewbut is the starting point for a di!erent transformation.

4.3 The Douglas-Kroll-Hess transformationAn alternative unitary transformation was first introduced by N. Douglas and N.M. Kroll83 1974 , and this transformation was further developed by B. Hess84 1986,resulting in the Douglas-Kroll-Hess (DKH) transformation. The advantage with theDKH transformation is that it succeeds in decoupling the four-component equationfor a particle in an external potential V += 0 such that the results become varia-tional stable and bounded from below. The DKH transformation starts with theFW transformation of the free particle and add a transformation of the potential V ,

H = U†0 DU0 = U †

0 D0U0 + U †0V U0, (4.12)

where U0 is the same as W0 in FW transformation. By introducing the quantities

Ep =:m2c4 + p2c2, Ap =

;Ep +mc2

2Ep, Rp =

c$ · pEp +mc2

, (4.13)

U0 can be expressed as,

U0 = Ap(1! (Rp), U"10 = (Rp( + 1)Ap, (4.14)

4.4 Scalar-relativistic Hamiltonian and spin-orbit e!ects 25

and equation (4.12) can be rewritten,

H1 = U†0 DU0 = (Ep ! c2

" #$ %E0

+Ap(V +RpV Rp)Ap" #$ %E1

+(Ap[Rp, V ]Ap" #$ %O1

. (4.15)

By choosing a suitable transformation the non-diagonal block part can be success-fully eliminated, and the hermitian operator U1 can be defined as

U1 = (1 +W 21 )

12 +W1, (4.16)

U†1U1 = 1 for any anti-hermitean operator W, W † = !W . The Hamiltonian, result-

ing from the unitary transformation, can be expressed as a sum of E and O operatorsto be more transparent.

O1 disappear if [(Ep,W1] = O1 and therefore W1 is chosen as an odd operator.O1 is an integral operator and the kernel of W1 is given by

W1(p,p$) = (

O1(p,p$)

Ep! + Ep. (4.17)

The remaining odd terms of equation (4.15) are of order V 2 or higher and can beeliminated by performing new transformations of the same kind. This transforma-tion which decoupled the four-component Dirac spinor and the transformed DiracHamiltonian is correct to the second order of the external field.

HDKH2 = (Ep + E1 ! ('W1EpW1 +

1

2[W 2

1 , Ep](

(4.18)

Higher order transformations can be derived by

Un = (1 +W 2n)

12 +Wn, (4.19)

and a n-electron Hamiltonian is derived by adding the 1/rij contribution afterwards.

4.4 Scalar-relativistic Hamiltonian and spin-orbit e!ectsA scalar relativistic Hamiltonian can be obtained from the DKH Hamiltonian inthe section above. The equation for the second order DKH Hamiltonian in equation(4.18) include spin-orbit interaction and a scalar-relativistic Hamiltonian85,86 canbe obtained by separating out and neglecting the spin-orbit interaction terms.86Thescalar relativistic e!ects are included by adding the corresponding terms of the DKHHamiltonian to the one-electron integrals.

In calculations starting from a scalar-relativistic Hamiltonian, the spin-orbit ef-fects can be taken into account by either a two-step spin-orbit (SO)-CI procedure orby applying a variation-perturbation method. An example of a SO-CI method is theE!ective Polarized CI-SO, implemented in the EPCISO program.87 In this methodthe first step is a scalar relativistic CAS calculation in a reference space which in-cludes all CSFs which need to be included in the spin-orbit calculation, and a secondstep where the spin-orbit Hamiltonian is constructed and diagonalized. If all singleexcitations from the partially occupied orbitals in the reference are included in theCI expansion and the following diagonalization of the Hamiltonian, this approach


ensures that both electronic orbital relaxation and di!erential spinor relaxation willbe accounted for in the final wave function.87,88 An e!ective Hamiltonian can bederived by calculating energies at a correlated level for all states in the reference,the correlated energies are then used to ”dress” the Hamiltonian87,89 that is the di-agonal elements of the spin-orbit Hamiltonian are shifted using spin-free correlatedenergies.

In the variation-perturbation method CASSI-SO90,91 (a module in the Molcaspackage92), the reference is obtained with CASSCF and the Hamiltonian is thendiagonalized. As for the two-step SO-CI method, dynamic electron correlation e!ectscan be added by the ”dressing” of the Hamiltonian.

Spin-orbit integrals were calculated in the mean field approximation85 using theAMFI93 code implemented in the MOLCAS suite of programs.

Chapter 5

E!ective Core PotentialsThe electrons in an atom, can approximatly be divided into two categories, thecore electrons, located closer to the nucleus and the valence electrons, located in theoutermost part and actively participating in chemical bonding. Since the core orbitalsare chemically inert, they can to high accuracy be frozen in a post Hartree-Fockcalculation. By simply replacing the core electrons with an e!ective core potential(ECP) the size of the basis set is reduced, an advantage especially for the heavieratoms. The ECPs can be divided into pseudo-potentials and model-potentials, andboth can be constructed such that relativistic e!ects are included. The ab initiomodel potential (AIMP)94 is an example of model-potentials. In this thesis, theacronym ECP is exclusively used for pseudo-potentials.

5.1 E!ective core potentialsECPs are constructed to model all-electron calculations at lower computational costwithout loosing too much accuracy. In general, the level of electron correlation treat-ment contributes to much larger errors than the di!erence between a valence-onlyand an all-electron Hamiltonian. Starting from for example the Hartree-Fock orDirac-Hartree-Fock method, the two categories core and valence orbitals make itpossible to construct ECPs either as shape-consistent ECPs, where orbital energiesand shapes are preserved, or energy-consistent ECPs using electron spectra etc todetermine the potential.

A potential term is added to the valence Hamiltionian,

Hv =Nv!

i

Hi +Nv!

i<j

1

rij+ Vecp, (5.1)

where Nv is the number of valence electrons and Vecp is the e!ective core potential.A semi-local95 ansatz for the e!ective core potential is where the radial part is

local whereas the angular part is non-local. The ECP (in one-component form) canthen be written as

Vecp = !Zeff

r+!

li

<!

k

alj ,krnlj

k"2

e!lj ,kr2=Plj (5.2)

Plj =l!

ml="l

&&lml(i)$#lml(i)&& (5.3)

27

28 5 E!ective Core Potentials

where Zeff = Z !Ncore, the parameters a, n and & depend on the angular momen-tum and are determined by a fitting procedure of a n-term polynomial.

Shape-consistent e"ective core potentialsThe construction of a shape-consistent ECP starts with an accurate correlatedall-electron calculation, including relativistic e!ects for heavier elements. The all-electron valence orbitals are replaced with smooth and nodeless pseudo-orbitalswhich agree with the atomic orbitals in the valence region. The ECP are constructedin such a way that the solutions of the e!ective HF equation generate the nodelessvalence orbitals.96,97

Energy-consistent e"ective core potentialsThe energy-consistent e!ective core potentials are constructed to reproduce atomicvalence-energy spectra and parameters of the ECP and the corresponding basisset are adjusted in agreement with representative data, such as electronic spectra,electron a"nities, ionization potentials etc.

In the most recent version of the energy-consistent e!ective core potential ap-proach the reference data is derived from all-electron multi-configuration Dirac-Hartree-Fock calculations based on the Dirac-Coulomb or Dirac-Coulomb-Breit Hamil-tonian. These calculations are performed for a multitude of electronic configurationstates both of neutral atoms and low-charged ions.

The energy-consistent e!ective core potentials are more commonly used mainlydue to the easy access to parameters of energy-consistent ab initio e!ective corepotential and corresponding valence basis sets. Almost all elements of the periodictable have been made available by Dolg, Preuss, Schwerdtfeger, Stoll and coworkers,see98 and reference therein.

Considerations for e"ective core potentialsThe exact relativistic all-electron Hamiltonian for a many-electron system is notknown and the various e!ective core potential merely model the existing approximateformulations. For many cases of chemical interest, as for example geometries, bindingenergies and reaction energetics, the di!erence in accuracy between an all-electronand a valence-only calculation is ignorable, since typical errors due to a finite basisset expansion or a limited correlation treatment are much larger. For very accuratecalculations of excitation energies, ionization potentials and electron a"nities, thedi!erence in accuracy between an all-electron and a valence-only calculation becomesmore important.

It should be pointed out that the size of the core not only determines the com-putational e!ort, but it also influences the accuracy of the results. Small-core andmedium-core potentials are usually safe to use for reaction energies or frequencycalculations. The large-core potentials can be quite useful for certain applications.As an example, a small-core ECP for lantinides typically includes 28 electrons withquantum number n=1-3 in the core whereas a large-core ECP includes electrons upto n=4 in the core. An actinide small-core ECP includes the 60 electrons in n=1-4whereas a large-core includes 78 or more electrons in the core. A large-core ECP isin many cases an acceptable choice for systems without too strong interactions andwhere the electronic configuration is kept during the calculations. So, for lanthanidecalculations a large-core ECP is useful in many applications due the the inert 4forbitals.99 For the early actinides a small-core ECP is essential for a correct de-scription of a chemical process, because of the extension of the 5f orbitals into thevalence region.

Chapter 6

Solvent e!ectsIn solution chemistry it is not realistic to describe solvation e!ects with a largenumber of explicit molecules surrounding the solute. Apart from that the size of thecalculations becomes too large, the number of possible minima and transitions statesincrease considerably with the number of solution molecules in the model. Solvente!ects can have a significant impact on reaction energetics and it is essential to takethose into account. Other options for estimating the solvent e!ects, including forexample di!erent types of molecular dynamic simulations, have not been consideredwithin this thesis.100,101

6.1 Continuum ModelsThe first simple cavity model was the Born model where the solute is enclosed ina spherical cavity but presently often other topologies with shape-adapted cavitiesare used. A standard approach for addressing the solvent e!ects are the apparentsolvation charge methods where the solute is placed in a cavity surrounded by a po-larizable dielectric medium, describing the solvent. The charge distribution * of thesolute inside the cavity will polarize the dielectric continuum with a given solvent-dependent permittivity, %, that in turn polarizes the solute charge distribution. Aweakness of the continuum model is the inadequacy of describing the chemical inter-action between the solvent and the solute. The interactions with the solvent (s) aretaken into account through a reaction potential V (S, s) that acts as a perturbationon the solute (S). The Hamiltonian in a continuum solvent model can be written

He!(S, s) = H(S) + V (S, s), (6.1)

where H(S) is the Hamiltonian in gas phase. The energy can then be written as thesum of the solute energy E(s) and E(S,s). If C is the space occupied by the cavity,% is the dielectric constant and V is the sum of the electrostatic potential generatedby the charge distribution * and the reaction potential V (S, s), then the Poissonequation can be written as,

!&2V (r) = 4-*(r) within the cavity!%&2V (r) = 0 outside the cavity,

with the boundary condition Vin = Vout on the surface of the cavity. The solventreaction potential V (S, s) can be expressed in terms of a charge density spread on

29

30 6 Solvent e!ects

the cavity surface, the apparent surface charge, and its interaction with the polar-izable dielectric continuum surrounding the cavity. This is achieved by expandingthe apparent surface charge that builds up at the solute-solvent interface in termsof spherical Gaussian functions located at each surface element in which the cavitysurface is discretized. This interaction can be described by a self-consistent process,numerically solved following an iterative procedure.

6.1.1 The conductor-like polarizable continuum model (CPCM)As mentioned, the cavity surface can be built by a set of finite elements ai, tesserae,small enough to consider the surface charge distribution, .(si), constant within eachtessera. The polarization vector, describing the solvent-solute interaction is given bythe gradient of the total potential V (r),

Pi(r) =%! 1

4-&V (r), (6.2)

where % is the dielectric constant. At the boundary of two regions i and j, there isan apparent surface charge distribution given by

.ij = (P j ! P i) · nij , (6.3)

where nij is the unit vector at the boundary surface pointing from medium i tomedium j. The apparent surface charge can be written as

.(s) =%! 1

4-

!

!n(V + V#), (6.4)

where n is the unit vector perpendicular to the cavity surface and pointing outwardand V (S, s) = V#. The apparent surface charge .(si) leads to an expression for theapparent point charge, qi.(si)ai. The interaction potential (conductor-like reactionfield) between the solute and the solvation charges, in equation (6.1), can then bewritten as a finite sum,

V#(r) =

+

"

.(s)

| r ! s |d2s ,

!

i

.(si)ai| ri ! si |

=!

i

qi| ri ! si |

, (6.5)

where % is the cavity surface.The apparent solvation charge can be found by solving the linear system

Dq = !b, (6.6)

where D is a matrix dependent on the solvents dielectric constant and the tesserae,b is the solute potential V (r) and q is the solvation charge. For conductor-likepolarizable continuum models,

Dii =%

%! 11.07

'4-ai

("1,

Dij =%

%! 1

1

| ri ! rj |.

6.1 Continuum Models 31

The interaction energy between solute and solvent can be written as

Eint =n!

i

biqi =n!

i

V (ri)qi, (6.7)

where Vi is the solute potential in tessera i = 1, . . . , n. For further informations, seefor example ref.102 and references therein.

The conductor-like screening modelThe conductor-like screening model (COSMO) is another continuum solvation model,where the solute forms a cavity within the dielectric continuum of permittivity % rep-resents the solvent. The response of the medium is described by the generation ofscreening charges, equivalent to the point apparent charge, qi in CPCM, on the cav-ity surface. The COSMO gives essentially the same result as the CPCM approachfor the same topological variables. For further informations, see for example ref.103and references therein.

6.1.2 The molecular cavityThe cavity is a basic concept in continuum models. Two commonly used topologicalmodels are the individual cavity (IC) model, where all atoms in the molecule are de-scribed with individual spheres,104 depicted in 6.1a, and the united atom topologicalmodel, UA104 where hydrogen atoms are included within the spheres of the atomthey are bonded to depicted in 6.1b. The small tesserae that build up the cavitysurface can be modified together with the cavity radii.

(a) The individ-ual cavity topo-logical model

(b) The unitedatom topologicalmodel

Figure 6.1: The two groups of topological models, the individual cavity topologicalmodel (IC) in (a) and the united atom topological model, (UA) in (b)

Conductor-like models reproduce the solute energies using the dielectric constantscharacteristic of polar solvents,105 and the CPCM102 has been compared with theCOSMO, first suggested by A. Klamt and G. Schüürmann.103,106 A too simplifiedcavity model can lead to large unrealistic e!ects.

Chapter 7

Introducing the applicationsThe work of this thesis has been carried out in the group of Wahlgren, Grentheand coworkers107–118 that have studied ligand exchange mechanism for the earlyactinyls during more than a decade. The first experimental study was published bySzabó et al.110 1996 and the first article that combined an experimental and com-putational investigation, concerning the water exchange of uranyl complexes, waspublished 2000 by Farkas et al.111 The geometry optimization steps in those inves-tigations111–118 were performed at the Hartree-Fock level followed by single pointenergy calculations, usually at the MP2 level of theory. Solvent e!ects were includedby applying a conductor-like polarizable continuum model using the standard UAtopological model in the Gaussian package.

Figure 7.1: The reaction mechanisms for the ligand exchange in hydrated uranyl(VI)used in preceding investigations.

The chemical model that was used for the water exchange mechanism is illus-trated in Figure 7.1 above, describing possible reaction paths. This model consistsof one actinyl and six water ligands, arranged to describe the reactant/product andtwo possible intermediate structures. The model reaction takes place through anexchange of a water molecule between the first and second coordination shell. The

33

34 7 Introducing the applications

preferred mechanism was selected by comparing the activation energy for the possi-ble dissociative (D), associative (A) and/or interchange (I) reaction paths.

One of the first research assignments within this thesis, was to continue the seriesof ligand exchange studies, with the exchange mechanism for hydrated uranyl(VI)in the first excited state. An excitation from the singlet ground state takes placeup to a higher singlet state and then the energy is lost either by fluorescent emis-sion (s0 + !/ ' s1 ' s0 + !/$) where the most of the absorbed energy is directlyemitted, or by phosphorescent emission (s0 + !/ ' s1 ' t1 ' s0 + !/$). The phos-phorescent emission takes place through an internal conversion from higher to lowerexcited states together with an intersystem crossing to the triplet state, i.e. when asinglet state non-radiatively transitions to a triplet state. The emission of light is de-layed in comparison to fluorescence. Luminescence, including both fluorescence andphosphorescence, originally referred to complexes that emitted light at low tempera-tures. The first excited (triplet) state of uranyl(VI) is also known as the luminescentstate, characterized by de-excitations generating green luminescence. The lumines-cence lifetime of the UO+2

2 (aq) in the lowest excited state is about 2 µs119 and thedominating levels are located in the region 470 – 588 nm.120

During the initial work with the photo-excited uranyl(VI), questions concerningthe DFT results arose when reaction energies at the B3LYP level where comparedwith the results obtained with near-minimal CASPT2. It was a rather large dif-ference that was found and density functional based methods seem not to be ableto describe the water exchange reaction properly for uranyl(VI) in the first excitedstate. The wave function based methods CCSD(T) and MP2/near-minimal CASPT2gives results that are in excellent agreement for the water exchange reaction. Dif-ferences of the same magnitude between wave function based methods and DFThave been noticed in other investigations and some examples are mentioned in theintroductional part of Chapter 8.

The initiative of an investigation where the accuracy of DFT applied on actinidesystems was given priority to the exchange reaction. The result was published as AnInvestigation of the Accuracy of Di"erent DFT Functionals on the Water ExchangeReaction in Hydrated Uranyl(VI) in the Ground State and the First Excited State inthe Journal of Chemical Theory and Computation,121 and is referred to as Paper 1.The conclusions about this investigation are presented in more detail in Chapter 8.

When this methodological project proceeded, new questions arose. Did the orig-inal chemical model in Figure 7.1 describe the water exchange reaction adequatelyor were other configurations of the second sphere water molecule better suited torepresent the geometries involved in the exchange mechanism? Was a small six-watermodel enough or did more water molecules have to be included? In the second pa-per, both the chemical model and the solvent are studied; On the combined use ofdiscrete solvent models and continuum descriptions of solvent e"ects in ligand ex-change reactions: a case study of the uranyl(VI) aquo ion, published in TheoreticalChemistry Accounts.122 The results and conclusions about the chemical model arepresented in Chapter 9, whereas conclusions about the aspects of solvation e!ects inPaper 2 are presented in Chapter 10. The concern was if the solvent e!ects were sat-isfactory addressed with the slightly simplified cavity model in the UA0 approachor if an individual cavity model was a better choice. Chapter 10 also includes anunpublished investigation that set out to explore the di!erence between the twotopological models in terms of sphere radii, electronic structure and charge on the

7.1 Computational details 35

central ion. This unpublished part also includes a re-investigation of the water ex-change reaction for uranyl(V), this system was part of the investigation concerningthe electronic structure on the central ion.

After completing the two methodological studies described in Paper 1 and 2,the original project, with the ligand exchange mechanism in the first excited state,was perused. The results of previous conclusions are presented in more detail inChapter 11. The six-water model described in Figure 11.1 was found, in Paper 2, tobe su"cient to describe the chemistry taking place, and therefore used to obtain thereaction energies. The first paper showed that density functional methods cannotbe trusted for reaction mechanisms for actinide systems so the exchange mechanismwas investigated by comparing reaction energetics calculated at the near-minimalCASPT2 level.

In the investigation of the significant di!erences found between the solvation ef-fect obtained with the two topological models investigated and discussed in Chapter10, it was not possible to appraise which of the models to use. In continuity with theprevious series of ligand exchange investigations, the solvent e!ects in the study ofthe Water Exchange Mechanism in the First Excited State of Hydrated Uranyl(VI),published in Inorganic Chemistry,123 were addressed with the UA0 model for rea-sons of comparison. This investigation is discussed in Chapter 11 together with anunpublished comparison with the water exchange reaction for uranyl(V) that hasbeen re-investigated and presented in the preceding chapter.

For the majority of actinides it is important to include multi-reference e!ects andthe last paper is a methodological test of the suggested MCSCF-DFT method de-scribed in section 3.3.1. The method is in this investigation applied to a set of actinidemolecules and the result, On the universality of the long-/short-range separation inmulti-reference density-functional theory. II. Investigating f0 actinide species 124 waspublished in the Journal of Chemical Physics. In a preceding article,125 some of theauthors of Paper 4, published results from the survey where a value of the param-eter µ, involved in the range separation, that would be acceptable for an arbitrarymolecule was found. For a set of light neutral compounds, the optimal µ-value wasdetermined to be 0.4 a.u. Chapter 12 describes the aim of this project, which was toinvestigate if the same µ-value could be used for actinides molecules, together withsome reinterpreted conclusions from Paper 4.

7.1 Computational detailsIn Paper 1–4 the relativistic e!ects in the uranium atom have been taken into ac-count by replacing the core electrons with an energy consistent relativistic ECP ofStuttgart-type126 together with the associated basis sets127 of triple-$ quality. In Pa-per 1 and 4, additional all-electron calculations were included, where the relativistice!ects were treated with a scalar-relativistic second order Douglas-Kroll-Hess (DKH)Hamiltonian.83,84 The all-electron basis set was an atomic natural orbital-relativisticcorrelation consistent (ANO-RCC)128 type of triple-$ quality. In general the lighterelements, like oxygen and hydrogen, were treated at the all-electron level, using theTZVP basis set suggested by Ahlrichs et al.129,130 The exception is the investiga-tion discussed in Paper 4, where carbon, nitrogen and oxygen were treated with arelativistic ECP. In Paper 4 the dominating part of the calculations was carried outwith the Dalton package.58

36 7 Introducing the applications

The spin-orbit e!ect has been calculated in an LS-coupled framework by using thespin-orbit configuration interaction program EPCISO,87 and sometime verified by avariation-perturbation method as implemented for example in the complete activespace interaction with spin-orbit, CASSI-SO90,91 program in the Molcas package.92The two methods are outlined in section 4.4. In the two-step EPCISO program thefirst step is a spin-free CASSCF calculation to obtain an orbital basis. A set ofdeterminants is generated in this orbital basis and used as a reference. If all singleexcitations from the partially occupied orbitals in the reference are included in theCI expansion and the following diagonalization of the Hamiltonian, this approachensures that both electronic orbital relaxation and di!erential spinor relaxation willbe accounted for in the final wave function.87,88 An e!ective Hamiltonian can bederived by calculating energies at a correlated level for all states in the reference,the correlated energies are then used to ”dress” the Hamiltonian.87,89 The CASSI-SO method can in a similar way be ”dressed”. The spin-orbit integrals, regardlessof method, were calculated in the mean-field approximation85,131 with the AMFIprogram.93

Another consideration is the large number of correlated electrons in actinidesystems. For ground state calculations on actinide molecules, the electrons in thedoubly occupied .u, .g, -u and -g orbitals plus the 6s and 6p actinide orbitals, arecorrelated at both the CCSD(T) and MP2/CASPT2 level of theory. The 5d orbitalshave a minor influence on the lowest excited states.132 For the lighter elements, allorbitals except in the 1s orbital, are correlated. Both perturbational and variationallevels of theory can be used to study the e!ect electron correlation has on couplingbetween LS and where the jj-coupling is important within spectroscopy of actinidecomplexes.88,132 Danilo et al.132 study for example the e!ect correlating the 5d, 6sand 6p orbitals has on the electronic spectra for U4+ and U5+.

Actinides with open f -shells can in principle be handled with a MCSCF methodwith a subsequent CASPT2 calculation to recover dynamic correlation. The draw-back is that calculations at such levels of theory where all valence excited statesare included in the active space, can only be applied on small systems. For examplethe bare uranyl(VI) ion with twelve active orbitals, the six doubly occupied bondingU-Oyl orbitals .g, -u, .u and -g and the six anti-bonding orbitals .(

g , -(u, .(

u and -(g ,

approaching the limit of 15 active orbitals for a complete CASSCF/CASPT2 treat-ment. Limitations that eventually could be circumvented by density functional basedmethods, but the applicability of DFT functionals, developed by using lighter ele-ments, is not obvious on actinide molecules. The majority of the published actinidestudies concerns the uranyl(VI) ion, for which the closed-shell structure essentiallyremoves the problems related to spin-orbit e!ects and open f -shells. Although thef -shell problem does not appear explicitly, the f -orbitals participate actively in thebonding, and the applicability of density functional based methods rely on the abil-ity of the functionals to treat the f -electrons properly. All together seven di!erentfunctionals have been used in the articles, six in Paper 1 and the LDA functionalwas included in the investigation presented in Paper 4. The most commonly usedfunctional in this thesis is the hybrid B3LYP functional, mainly since it is known togive acceptable geometries118,133,134 for actinide complexes. Most of the structuresin the research studies presented in this thesis have been optimizes with the B3LYPfunctional.

Chapter 8

Wave function versus densityfunctional theoryAs pointed out in previous chapters, computational actinide chemistry needs to dealwith both relativistic e!ects, a large number of correlated electrons and the e!ects ofdegenerate or near-degenerate electronic states. In spite of the last years advances incomputing power, the calculations needed to predict chemical and physical propertiesfor actinides are very time- and memory consuming. Density functional theory hasbecome increasingly popular in computational actinide chemistry, but it is essentialto keep in mind that the performance of DFT functionals depends on the chemicalsystem and what kind of properties that are investigated. The fact that functionalsare mainly developed for much lighter elements add to the uncertainties.

A number of investigations regarding uranyl complexes include results both fromdensity functional and wave function based methods, see for example135–137 but inmany cases no such comparison is done.138–144

There are some examples where the applicability of DFT for actinide complexescan be questioned. In the investigation concerning the structure and thermodynam-ics of five di!erent uranium(VI) reactions by Privalov et al.,145 the di!erences inreaction enthalpies between B3LYP and CCSD(T) were found to be in the rangeof 20–70 kJ mol"1. Similar di!erences were also found for Np and Pu complexesby Schimmelpfennig et al.146 The exchange mechanism for the uranyl(VI) groundstate also shows a large di!erence (35 kJ mol"1) between B3LYP and MP2 as de-scribed in a review article concerning actinide chemistry in solution by Vallet etal.118 Gutowski et al.147 have investigated the UO2(CH3CO2)2 and UO2(NO3)2complexes at both the B3LYP and MP2 level and found di!erences larger than 40kJ mol"1 between the methods, similar to the di!erence Gutowski and Dixon148

found in a water binding study.In this chapter the results from Paper 1 are presented, a project where the ac-

curacy of di!erent DFT functionals were investigate when applied on the waterexchange reaction for uranyl(VI) in the ground and luminescent states. The sixfunctionals included in this study are outlined in section 3.2, and are among themost commonly used functionals. In the end of this chapter an unpublished investi-gation concerning the e!ect of the HF-exchange is included. It was suggested that asmall increase of the HF-exchange in the B3LYP functional could possibly improvethe results for actinides.

37

38 8 Wave function versus density functional theory

The model system that is used in this investigation is the six-water model usedin previous research studies by Wahlgren, Grenthe and coworkers111,112,117,118 con-cerning the water exchange mechanism for actinyls. The reaction can be describedas

[UO2((OH2)(OH2)4]

2+(H2O) ! [UO2(OH2)5]2+(H(

2O). (8.1)

The water exchange reaction for the six-water model in reaction (8.1) is illustratedin Figure 7.1 and the reaction includes three geometries depicted in Figure 8.1 where8.1a is the reactant (and product), (b) is the associative intermediate and (c) thedissociative intermediate.

(a) (b) (c)

Figure 8.1: The ground state structures of the aqueous uranyl(VI) used to probethe DFT accuracy. The reactant (and product) is depicted in (a), the associativeintermediate in (b) and (c) shows the dissociative intermediate.

It was not possible to identify the transition state for the associative reaction,due to what most probably were numerical problems related to the implementationof the method. In a preceding investigation concerning the water exchange mecha-nism for the ground state of uranyl(VI)112 it was found that the energy di!erencesbetween the transition states and the intermediate structures were small. It is there-fore assumed that the intermediate structures could be used as approximations tothe transition states.

The model includes several challenges, for example the number of coordinatedwater molecules is di!erent in the three structures and by including the luminescentstate in the investigation, an open shell structure – an open f-shell and a hole ina bonding orbital – is also considered. The water exchange mechanism is thereforesuitable for an investigation of the accuracy of di!erent DFT functionals concerningligand exchange reaction. The results from the density functional based methodshave been compared with reaction energies derived with the wave function basedmethods MP2, CCSD(T) and CASPT2, outlined in section 3.1.

8.1 FunctionalsAside from the general applicability of DFT on actinide complexes, this investiga-tion also included a comparison between results obtained for the luminescent stateof uranyl(VI) with the time-dependent and unrestricted formalism. The set of func-tionals probed in this investigation includes the pure GGA functional TPSS, PBE,BLYP, BP86 and the hybrid functionals B3LYP and BHLYP, see section 3.2. Inaddition to the more general functionals, the SAOP functional, developed for time-dependent formalism was included in this investigation (at the AE level of theory),

8.2 Geometries 39

see section 3.2.6. The hybrid BHLYP functional was not only used within the time-dependent and unrestricted formalism, it was also used in the DFT-MRCI method,outlined in section 3.3.2. The DFT-MRCI method combine Kohn-Sham DFT andmulti-reference configuration interaction (MRCI) method and can be used for cal-culating excitation energies starting from a closed shell system, similarly to theTD-DFT formalism.

8.2 GeometriesB3LYP normally gives good geometries for actinide complexes118,133,134 and resultsfor the ground states optimization obtained with B3LYP are presented in Table 8.1together with the TD-B3LYP optimized geometries used for the luminescent state.

Table 8.1: Geometry information for the uranyl(VI) in the ground and luminescentstates of uranyl(V) obtained at the B3LYP and TD-B3LYP level. All distances inÅngström and n is number of coordinated water molecules in the sphere.

Complex First sphere Second sphere

d(U-Oyl) n d(U-OH2) n d(U-OH2)

UO2+2 (H2O)6 Reac. 1.75 5 2.50 1 4.26

A-Int 1.76 6 4)2.52, 2)2.65 -D-int 1.75 4 4)2.42 2 3.93

U#O2+2 (H2O)6 Reac. 1.80 5 2.50 1 4.37

A-Int 1.80 6 4) 2.53, 2)2.62 -D-int 1.79 4 4)2.43 2 3.97

As mentioned above, the first exited state allows a comparison of the unrestrictedDFT formalism with TD-DFT. The geometrical comparison show that U-B3LYPand TD-B3LYP geometries are similar even though the unrestricted formalism givesslightly longer U-Oyl bonds (0.01 Å). All reaction energies for the luminescent statehave been obtained using the geometries optimized at the TD-B3LYP level. Thegeometries for the ground and luminescent states are overall very similar as can beseen in Table 8.1. The U-Oyl bond for the luminescent state is slightly longer thanin the ground state due to the excitation from the triple U-Oyl bond to one open f &or f ' orbital.

Only small basis set superposition errors (BSSE) were found for the ground statereaction energies, see Table 8.2, the largest calculated e!ect was less than 3 kJ mol"1

at the MP2 level and a similar e!ect is to be expected for the first excited state aswell.

Bühl et al.142 reported a structure which was di!erent from those used in manyprevious studies concerning the position of the outer sphere water molecule for theD-intermediate structure. To compare the two geometrical options energetically re-action energies were calculated at the MP2 level and the latter was found to be morestable than the original configuration depicted in Figure 8.1c. However, the geometryof the configuration with single hydrogen bonds caused some doubts concerning therather large di!erence in bond lengths for the inner sphere water molecules. Since theaim was to investigate the performance of DFT and not a reaction mechanism, theoriginal model displayed in Figure 8.1 was deemed useful. The issue concerning the


bonding between the inner and outer coordination spheres was further investigatedin the study presented in Paper 2 and the result is discussed in Chapter 9.

8.3 Reaction energies – the ground stateThe CCSD(T) method is considered to be the most accurate method in this inves-tigation, and all reaction energies have therefore been compared to the CCSD(T)results. It is known that the CCSD(T) method is sensitive to the size of basis set,this was tested by adding g functions to the basis set at the AE level for the uraniumand the energy di!erence was found to be smaller than 2 kJ mol"1. The size of thebasis set for oxygen149 was investigated on a smaller system, and the e!ect on thereaction energies from using a (4s3p2d) contraction was found to be around 3 kJmol"1, which confirmed that the choice of basis set was satisfactory. The spin-orbitcorrection was obtained for the ground state system with the EPCISO programincluding only single excitations from the .u electrons in the reference, while theother electrons were kept uncorrelated. The corrections destabilized the associativereaction with less than 1 kJ mol"1 and stabilized the dissociative reactions with lessthan 3 kJ mol"1 as can be seen in Table 8.2.

Table 8.2: Reaction energies in kJ mol"1 for the associative and dissociative reactionsfor uranyl(VI) in the electronic ground state.

Associative reaction Dissociative reaction

ECP AE ECP AE

Method HFexchange no g BSSE no g with g no g BSSE no g with g

B3LYP 20% 44.9 44.8 41.6 42.1 20.1 17.1 21.8 21.6BHLYP 50% 42.7 41.3 41.7 29.3 31.0 30.7BP86 0% 42.7 13.5PBE 0% 40.3 13.1TPSS 0% 37.0 15.7BLYP 0% 41.9 13.5SAOP 0% 43.1 13.1MP2 35.6 37.1 34.1 35.5 34.4 32.8 35.2 33.6MP2 SO 36.1 31.7CCSD(T) 36.7 31.9 33.2 34.6 37.4 36.0DFT-MRCI 50% 39.7 25.4

In Figure 8.2 the reaction energies for the ground state, see Table 8.2, includingDFT-MRCI outlined in section 3.3.2, are depicted relative to the CCSD(T) energies(36.7 and 34.6 kJ mol"1 for associative and dissociative reaction paths respectively)to illustrate more clearly the behavior of the DFT functionals. The pure GGA func-tionals are plotted in red, the hybrids in blue and finally the MP2 result is plottedin dashed green.

For the associative reaction there is an excellent agreement between the two wavefunction based methods MP2 and CCSD(T) as shown in Figure 8.2. The result forthe TPSS functional is very similar to CCSD(T), and there is an acceptable agree-ment for the other functionals. The largest deviation occur for the hybrid B3LYPfunctional, with an overestimate of 8 kJ mol"1 relative to CCSD(T), which still is

8.4 Reaction energies – the luminescent state 41

Figure 8.2: The spin-free reaction energy for the ground state relative to CCSD(T) inkJ mol"1. The GGA functionals are plotted in red, the hybrids in blue (DFT-MRCIdashed blue) and the MP2 result is plotted dashed green. The CCSD(T) energycorresponds to the black line.

a satisfactory result. The BHLYP result is very close to DFT-MRCI which is notsurprising since the ground state is a closed shell system.

The dissociative reaction energy obtained with CCSD(T) and MP2 are again ontop of each other. The hybrid BHLYP gives an acceptable result of 5–8 kJ mol"1

relative to CCSD(T), the latter is obtained with DFT-MRCI. The pure functionalsperform poorly for the four-coordinated dissociative reaction with an underestima-tion between 19 and 22 kJ mol"1 relative to CCSD(T). The hybrid B3LYP functionalgive a result that is in between the pure functionals and BHLYP.

8.4 Reaction energies – the luminescent stateDue to technical di"culties calculations with CCSD(T) failed for the excited state.The good agreement between MP2 and CCSD(T) for the ground state, and the factthat a minimal CASPT2 calculation is essentially equivalent to an MP2 calculation,indicates that the minimal or near minimal CASPT2 result for the excited stateshould be reasonably accurate as well. Therefore all density functional based resultsare compared with the results that were obtained with near-minimal CASPT2 usingan active space including the .u and the two f +u orbitals. As mentioned earlier theoptimization was performed both with U-B3LYP and TD-B3LYP, and the di!erencein reaction energy was at most 3 kJ mol"1. The reaction energies obtained with U-DFT and TD-DFT were in nearly all cases identical and therefore only TD-DFTresults are shown in Table 8.3, the exception are the two hybrid functionals B3LYPand BHLYP where all results are included in the table.


Table 8.3: Reaction Energies in kJ mol"1 for the associative reaction and dissociativereaction computed for uranyl(VI) in the electronic first excited state.

Method HF-exchange Associative reaction Dissociative reaction

U-B3LYP 20% 45.6 24.5U-BHLYP 50% 42.0 32.3TD-B3LYP 20% 42.4 24.6TD-BHLYP 50% 42.0 32.3TD-BP86 0% 42.6 22.7TD-PBE 0% 40.6 23.0TD-TPSS 0% 41.5 22.3TD-BLYP 0% 42.4 23.7TD-SAOP 0% 47.9 13.9DFT-MRCI 50% 45.5 45.8CASPT2 CAS=3 35.6 36.6CASPT2 CAS=3 + SO 32.7 39.1CASPT2 CAS=5 34.9 36.9

Since the two +u f -orbitals are nearly degenerate the near-minimal active spaceconsists of the .u and the two +u as mentioned above. It is known that the accuracyof CASPT2 is sensitive to the choice of active space so a slightly larger active spaceincluding five orbitals, the .u, the two +u and two #u, was also investigated. Thedi!erence in reaction energies was found to be at most 1 kJ mol"1 for the two activespaces. The spin-orbit correction was obtained for the luminescent state with a refer-ence space that was the same as the larger reference (five orbitals) used for CASPT2including the single excitations from the active orbitals in the reference. The e!ectis a stabilization with 3 kJ mol"1 of the associative reaction and destabilization ofthe dissociative reactions with less than 3 kJ mol"1, see Table 8.3.

Table 8.3 includes the reaction energies for the first excited state. TD-DFT andU-DFT gave in most cases identical results so with exception for the two hybridfunctionals, the rest of the DFT energies are obtained with the TD-formalism. InFigure 8.3 all reaction energies are presented relative to the near minimal CASPT2result, 35.6 and 36.6 kJ mol"1 for the associative and dissociative reaction pathrespectively.

For the associative reaction the largest di!erence compared to the near-minimalCASPT2, 12 kJ mol"1 occur for the TD-SAOP. Figure 8.3 shows that the two hybridfunctionals B3LYP and BHLYP overestimate the reaction energy with 10 kJ mol"1

(U-B3LYP) and 8 kJ mol"1 compared to near-minimal CASPT2. The U-BHLYP andTD-BHLYP results di!er with less then 4 kJ mol"1 compared to the DFT-MRCImethod. In all cases, density functional based methods overestimate the associativereaction energy. The results are similar to those obtained for the ground state.

The dissociative reaction energies also resemble those in the ground state, evenif the underestimate of the pure functionals are slightly smaller compared to near-minimal CASPT2. The B3LYP functional results are close to those obtained withthe pure functionals (Figure 8.3). TD-SAOP has the largest underestimation relativeto near-minimal CASPT2, 23 kJ mol"1. The U-BHLYP and TD-BHLYP reactionenergies di!er more from the DFT-MRCI than they do in the ground state.

There are large di!erences in the absolute transition energy values obtained withdi!erent methods, but as can be seen in Table 8.4 all methods result in adiabatic

8.5 Increase of HF-Exchange in B3LYP 43

Figure 8.3: The spin-free reaction energy for the luminescent state relative to near-minimal CASPT2 in kJ mol"1. The TD-DFT results for the GGA functionals areplotted in red and the hybrids in blue (DFT-MRCI dashed blue). The near-minimalCASPT2 energy (35.6 and 36.6 kJ mol"1 for the A- and D-reaction paths respec-tively) corresponds to the black line. The B3LYP and BHLYP TD-DFT and un-restricted results are almost identical in dissociative reaction and the B3LYP* andBHLYP* refer to both unrestricted and time-dependent results.

transition energies to the luminescent state that are very similar, with 820 cm"1

(10 kJ mol"1) as the largest di!erence, obtained with the BLYP functional. The

Table 8.4: The adiabatic transitions, in cm"1 computed for uranyl(VI). For all func-tionals, except the U-B3LYP, the TD-DFT results have been used in this table forthe excited state.

B3LYP BHLYP BP86 PBE TPSS BLYP SAOP MP2/CASPT2

reactant 18704 21157 15903 15997 17179 15566 25008 30115A-reaction 18843 20950 15847 15972 17493 15574 25009 30115D-reaction 19021 21284 16622 16766 17700 16387 25409 30304

adiabatic energies that are presented in Table 8.4, indicate that the reaction potentialenergy surfaces for the ground state and first excited state are parallel and that theassumption of using the intermediate structures to mimic the activation energy isreasonable.

8.5 Increase of HF-Exchange in B3LYPThis small investigation was initiated on the proposition concerning a possible im-provement of the B3LYP functional applicability on actinide complexes by increasingthe amount of HF-exchange in the functional. No results were found confirming this


hypothesis when using the same basis set and ECPs as in the main investigationdiscussed above.

A modified B3LYP functional was applied on a subsequent step of the reductionof U(VI) to U(IV), see reference.134 The results of applying the modified B3LYP werecompared with energies obtained at the HF, CASSCF and near-minimal CASPT2level, the modified energies simply converged towards a HF energy when the HF-exchange was increased in small steps (starting from the original 20% HF-exchangeand increase it with 1% at a time up to 50%). The same trend was observed for ge-ometry optimizations, clearly showing that a hybrid functional needs to be carefullyadjusted to be applicable.

8.6 ConclusionsThe good agreement between the reaction energies obtained with TD-DFT and U-DFT respectively for the first excited state is satisfying. The geometries that wereobtained with TD-B3LYP and U-B3LYP are also in good agreement and geome-try optimizations for higher excited states can thus be accessible within the time-dependent formalism.

The DFT-MRCI overestimate the dissociative reaction energy in contrast to thetraditional DFT results, and it can be noted that DFT-MRCI overestimates both theassociative and dissociative reaction energies compared to near-minimal CASTP2 by10 kJ mol"1, indicating that this method can be useful for actinides.

The density functional based reaction energies for the associative and dissocia-tive mechanisms, when compared to wave function methods, clearly show that DFTconsistently overestimates the associative reaction energy and underestimates thedissociative reaction energy. For the pure functionals without HF-exchange the un-derestimation is in average 21 and 16 kJ mol"1 for the ground- and luminescent staterespectively. The two hybrid functionals indicate that the increase of HF-exchangeimproves the result although the approach to just increase the HF-exchange in anexisting functional is not recommended. Rotzinger150 noted that DFT tends to over-estimate the energy of transition metal complexes with a low coordination number,thus favoring dissociative reactions over associative reactions (cf. Table 8.2), in agree-ment with the result presented in this chapter. This sentence was wrongly formulatedin Paper 1, see the erratum in 13.1.

The functionals are thus not suited to describe energetics in actinides for thistype of reactions and it is plausible that this is due to the valence 5f orbitals. Sinceall functionals where developed using much lighter atoms in comparison, the resultmight not be so surprising.

Chapter 9

Chemical modelsExperimental investigations of ligand exchange reactions can provide informationabout possible stoichiometric mechanisms for the reaction or activation enthalpy.However, it is not possible to extract information of the details concerning the mech-anism, as e.g. the molecular behavior along the reaction coordinate. For reactionsin solutions, information of this type can only be obtained, in principal, by compar-ing experimental activation parameters with calculated values obtained through acombination of chemical and quantum chemical models.

In quantum chemistry it is often advisable to use the smallest possible chemicalmodel that can describe the reaction of interest. The minimal six-water model thathas been used in previous water exchange reactions111,112,117 consists of structureswhere the outer sphere water molecule(s) bind to the inner sphere with two hydrogenbonds each. As mentioned in section 8.2, Bühl and Kabrede140 used a chemical modelthat included a D-intermediate structure where the two outer sphere water moleculesbond to the inner sphere water molecules with single hydrogen bonds, see Figure9.2. The reactant structure also includes an outer sphere water molecule and the aimwas to investigate which of the configurations that was favored for the reactant andD-intermediate structures in gas phase. This issue was addressed in the first part ofPaper 2.

9.1 The chemical modelReaction energies for the small six water model were calculated at the MP2 levelusing reactants and D-intermediate structures with both single hydrogen bonds andtwo hydrogen (double) bonds between the inner and outer coordination sphere. Allgeometries were optimized at the B3LYP level. The structure with single hydrogenbonds between the inner and outer sphere water molecules was found to be morestable than the original configuration for the D-intermediate structure, while themost stable configuration for the reactant was with double hydrogen bonds betweenthe inner and outer coordination sphere. The geometry for the single hydrogen bondmodel caused some doubts due to the rather large di!erence (0.1 Å) in bond lengthsfor the inner sphere water molecules. Figure 9.1 and 9.2 show the two possiblegeometries for the reactant and the D-intermediate respectively. This opens for thequestion if a chemical model that consists of one uranyl and six water molecules issu"cient to describe the mechanism in the water exchange reaction or whether alarger model is required for a correct interpretation of the chemistry.

45

46 9 Chemical models

Chemical models for hydrated uranyl with more than six water molecules can befound in the literature. One example is the quantum chemical study by Gutowskiand Dixon.148 They used chemical models with twelve and fifteen water molecules,among others, in an investigation concerning energies and free energy of solvationfor uranyl ions in aqueous solution. The present investigation started initially witha model including eighteen water molecules. However, due to the large number oflocal energy minima it was not possible to find a global or at least a reasonableminimum for the reactant. The smallest extension of the six-water model whichallows a symmetric arrangement of the second shell water molecules is obtained byadding four more water molecules. This results in a partially saturated reactant withfive water molecules in the first coordination shell and five in the second.

In accordance with the small model, all geometries/structures for the ten-watermodel were optimized at the B3LYP level and (in the final step) without symmetryconstraints, and the reaction energies were obtained at the MP2 level of theory. Theten-water model provides information about whether a small six-water model withsingle or double hydrogen bonds are favored between the inner and outer spherewater molecule(s). Reactions (9.1) and (9.2) describe the water exchange for the six-and ten-water model respectively.

[UO2((OH2)(OH2)4]

2+(H2O) ! [UO2(OH2)5]2+(H(

2O) (9.1)

[UO2((OH2)(OH2)4]

2+(H2O)5 ! [UO2(OH2)5]2+(H(

2O)(H2O)4 (9.2)

9.1.1 The six-water modelThe A-intermediate structure has no water molecules in the outer sphere in thismodel and is not a!ected by the question concerning the bonding between the innerand outer sphere water molecules.

The reactantThe reactant with double hydrogen bonds, depicted in Figure 9.1a, was found to be9 kJ mol"1 more stable than the geometry with a single hydrogen bond illustratedin Figure 9.1b. The energy relative to the structure with a single hydrogen bond ispresented in Table 9.1.

(a) (b)

Figure 9.1: The two geometrical options for the reactant in the six-water model. Theconfiguration with double hydrogen bonds is presented in (a) and the configurationwith a single hydrogen bond in (b).

9.1 The chemical model 47

The D-intermediateThe opposite situation was found for the two D-intermediate structure, illustrated inFigure 9.2, where the geometry with single hydrogen bonds was 20 kJ mol"1 morestable than the geometry with double hydrogen bonds, see Table 9.1 for energiesrelative to the structures with single hydrogen bond(s).

(a) (b)

Figure 9.2: The two geometrical options for the D-intermediate in the six-watermodel. The configuration with double hydrogen bonds is presented in (a) and theconfiguration with single hydrogen bonds in (b).

As shown in the table with geometrical information (9.3), the D-intermediatestructure with single hydrogen bonds between the inner and outer sphere has largedi!erences in the U-OH2 bond length; it is a much shorter bond length for the twoaccepting inner sphere water molecules compared to the two without bonds to theouter sphere, the di!erence is close to 0.1 Å. It should be noted that these geometricaldi!erences really points to a key question: is the six-water model reasonable?

Table 9.1: Energies in kJ mol"1 relative to the structure with single hydrogen bond(s)(1H) obtained at the MP2 level for the reactant and D-intermediate structure in thesix-water model.

UO2+2 (H2O)6

1H 2H

Reactant 0 -9D-intermediate 0 20

9.1.2 A comparison of the six- and ten-water modelsTo investigate the applicability of the minimal six-water model a comparison is madewith a larger ten-water model. This model includes a partially saturated reactantwith five water molecules in the inner and outer sphere respectively. The reactionenergies (in gas phase) for the six- and ten-water models are presented in Table 9.2.It should be noted that the additional water molecules in the ten-water model willrecover part of the solvation e!ect. The reactants and intermediate structures forboth models are illustrated in Figures 9.3 – 9.2.


Table 9.2: Reaction energies in kJ mol"1 at the MP2 level for the six- and ten-watermodels in gas phase.

UO2+2 (H2O)6 UO2+

2 (H2O)10

A-int D-int 1H D-int 2H A-int D-int

Gas phase 37 16 35 36 47

The associative reaction energies for both the small six-water model and theten-water model di!er with 1 kJ mol"1 in gas phase. This may indicate that theassociative reaction path is reasonable well described with a six-water model wherethe reactant has a double hydrogen bond to the outer coordination shell. The ge-ometrical comparison in Figure 9.3 confirms this; all outer sphere water moleculesin the larger model bind with double hydrogen bonds to the inner sphere and theU-OH2 distances are in agreement between the two models, as can be seen in Table9.3. In similarity with the reactant, the A-intermediate structure in the ten-watermodel only includes double hydrogen bonds between the two coordination spheresas can be seen in Figure 9.4.

(a) (b)

Figure 9.3: The reactant geometry in the (a) six-water model and (b) ten-watermodel.

(a) (b)

Figure 9.4: The A-intermediate geometry in the (a) six-water model and (b) ten-water model.

For the dissociative reaction energies, the six-water model with double hydrogenbonds is closer in energy to the ten-water model than the model with single hydrogenbonds. It should be kept in mind that the outer sphere water molecules in the ten-water model will contribute to the solvation e!ect, something that could explainpart of the large di!erence in the reaction energies for the six-water model with

9.2 Conclusions 49

single hydrogen bonds and the ten-water model. The solvation e!ect will, however,be addressed in the following chapter.

For the six-water model the D-intermediate with single hydrogen bonds was foundto be 20 kJ mol"1 more stable than the structure with double hydrogen bonds. Onlysingle hydrogen bonds were also found for ten-water model as can be seen in Figure9.5, and for the large D-intermediate with eighteen water molecules.

Adding the geometrical information about the bonds between inner and outersphere water molecules together with the more stable dissociative reaction energyfor the D-intermediate with single hydrogen bonds, the results indicate that thedissociative reaction in a six-water model is best described with an intermediatewith single hydrogen bonds.

(a) (b)

Figure 9.5: The D-intermediate geometry in the (a) six–water model and (b) ten-water model.

Information about the geometries for both the six- and ten-water model arepresented in Table 9.3

Table 9.3: Geometry information for the uranyl(VI) in a six- and ten-water modelobtained at the B3LYP level. All distances in Ångström and n is the number ofcoordinated water molecules in the sphere.



UO2+2 (H2O)6 Reac. 1.75 5 2.50 1 4.26

A-Int 1.76 6 4)2.52, 2)2.65 -D-inta 1.75 4 2)2.37, 2)2.46 2 4.46D-intb 1.75 4 4)2.42 2 3.93

UO2+2 (H2O)10 Reac. 1.76 5 2.47 5 3)4.26, 2)4.37

A-Int 1.77 6 4)2.49, 2)2.65 4 2)4.49, 2)4.24D-int 1.76 4 2)2.35, 2)2.41 6 4)4.44, 2)4.58

aSingle hydrogen bonds (1H)bDouble hydrogen bonds (2H)

9.2 ConclusionsThe results presented in this chapter can be formulated in condensed form as: Aminimal six-water model for the water exchange reaction includes a reactant andassociative intermediate as in the original six-water model, but the dissociative in-termediate reaction is better described with two water molecules in the second hy-


dration shell that bind to the first with single hydrogen bonds. The two hydrogenbond model might be useful but is certainly less accurate than a chemical modelwith single hydrogen bonds between the second sphere water and a water moleculein the first coordination sphere.

Chapter 10

Solvation modelsIn this chapter the solvation e!ect for the water exchange reaction of uranyl(VI)is discussed and how this can be taken into account by applying conductor-likepolarizable continuum models (CPCM, see Chapter 6). A limited chemical modelwith explicit water molecules, as the six- and ten-water models discussed in Chapter9, cannot account for all interaction between uranyl and its aqueous surrounding.Part of the e!ect can be obtained by applying a continuum model to the chemicalmodel, as described in section 6.1, to mimic the solvent water. The solvent in acontinuum model is represented as a continuous dielectric, only described by thedielectric constant without any internal structure. One of the limitations of thecontinuum model is that it only simulates the macroscopic properties of the solventand thus it does not reproduce detailed solute-solvent interactions.

In the first part of this chapter the investigation concerning the choice of solvationmodel, published in Paper 2, is discussed. The model used for the water exchangemechanism is a!ected both by the description of the second coordination sphere, asdiscussed in Chapter 9, and by the choice of solvation model. In this study boththe small six-water model and the ten-water model have been used to probe whichsolvation model to use for actinide systems. The solvation e!ects are obtained with aconductor-like polarizable continuum model using the two most common topologicalmodelsa: the united atom (UA)b 104 and the individual cavity (IC)c 104 models. TheUA model can be regarded as an approximation of the IC and some di!erences areto be expected. However, as discussed in Paper 2, the UA and IC continuum modelsresult in significantly di!erent solvation e!ects when applied on both the six- andten-water models. In an attempt to understand the reason behind the inconsistencyin the calculated solvation e!ect, the question of what influence the level of theoryhas on the solvation e!ect is addressed in the first part of this chapter.

In the second part of this chapter an unpublished investigation, concerning pos-sible reasons for the di!erence in solvation e!ects obtained with UA and IC, ispresented. This unpublished part discusses to what extent the sphere radii, usedto build up the cavity, could be related to the found di!erence in solvation e!ectstogether with the parameters of electronic structure and charge on the central ion.In connection to the investigation concerning the electronic structure of the centralion, a reinvestigation of the water exchange reaction for uranyl(V), using the same

aAs implemented in the Gaussian 03 package.bThe implemented default setting is the UA0 model.cThe atomic radii for the implemented IC default setting is based on the universal force field (UFF).

51

52 10 Solvation models

ECP and basis sets that have been used for the all uranyl(VI) complexes in Chapter8-11 in this thesis, has been included.

Since this chapter is mainly about solvation e!ects from di!erent aspects, thetables will in most cases (but not all) include solvation e!ects and not informationconcerning reaction energies or enthalpies.

10.1 Solvation models from Paper 2During the initial work with the project concerning the water exchange reactionfor the first excited state of uranyl(VI) (Paper 3), it was found that the UA andIC solvation models resulted in reaction energies that deviate more from each otherthan expected. The calculated solvation e!ect obtained for associative reactions usingboth the six-water and ten-water model di!ers with close to 20 kJ mol"1 when theresult obtained with UA and IC are compared. For the dissociative reactions evenlarger di!erences are found, as can be seen in Table 10.1.

10.1.1 Wave function vs. density functional theory

The solvation e!ect can be calculated at di!erent levels of theory. Table 10.1 containsthe solvation e!ects obtained for the water exchange reaction for uranyl(VI) in theelectronic ground state for the six- and ten-water model at the MP2 and B3LYPlevel. Although there are significant di!erences between UA and IC, the solvatione!ects for the di!erent reaction paths for a given topological model are in goodagreement regardless of whether this is calculated at the B3LYP or MP2 level oftheory. For the six-water model, the di!erence is at most 2 kJ mol"1 and for theten-water model the corresponding value is 4 kJ mol"1. This result shows that it ispossible to calculate the solvation e!ects at the B3LYP level and then add it to asingle point gas phase energy obtained with a wave function based method and viceversa.

Table 10.1: Solvation e!ects in kJ mol"1 at the MP2 and B3LYP level of theoryobtained with UA and IC models for the water exchange reaction for uranyl(VI) inthe electronic ground state, using the six- and ten-water models.

UO2+2 (H2O)6 UO2+

2 (H2O)10

Cavity A-int D-int 1H D-int 2H A-int D-int

MP2 UA -19 21 27 -24 -8IC -3 19 2 -5 23

B3LYP UA -18 22 29 -21 -4IC -2 21 3 -2 27

10.1 Solvation models from Paper 2 53

10.1.2 Reaction enthalpiesOne quantity that can give further information about how to calculate the solvatione!ects is the enthalpy. The reaction enthalpyd, H, involves the change of internalenergy and the e!ects on the surrounding bulk water. Activation enthalpies forchemical reactions can be obtained experimentally and therefore information aboutenthalpy corrections to the reaction energies have been included, as can be seenin Table 10.2. The enthalpy correction #H has been calculated at B3LYP level oftheory at a temperature of 298.15 K and pressure of 1 atm.

Vallet et al.112 showed that the activation barriers surrounding the intermediatesin the exchange reaction are low, therefore the experimental activation enthalpyshould be a good estimate of the reaction enthalpy between the reactant and theintermediates. The experimental activation enthalpy was found by Farkas et al.111to be 26 ± 1 kJ mol"1 for the uranyl(VI) water exchange reaction path.

Table 10.2: Reaction enthalpies in kJ mol"1 computed at the MP2 level for the six-and ten-water models in gas phase and in solvation using CPCM with both UA andIC topological models.

UO2+2 (H2O)6 UO2+

2 (H2O)10

A-int D-int 1H D-int 2H A-int D-int

#Ha298 0.3 -4 0 1 -8

Gas phase 37 12 35 36 39UA 17 33 63 12 31IC 34 31 37 31 62

a!H298 has been calculated in gas phase at B3LYP level of theory.

The associative reactionIt is a consistent agreement between the reaction enthalpies in the ten- and six-watermodels for the associative reaction paths. UA stabilizes the associative reaction by20 and 24 kJ mol"1 relative to the gas phase for the six- and ten-water modelrespectively, while the IC only lowers the reaction enthalpies marginally.

The dissociative reactionThe dissociative reaction enthalpies obtained with the six-water model with singlehydrogen bonds di!er with only 2 kJ mol"1 between the IC and UA models. For thedissociative reaction paths with double hydrogen bonds, the two topological modelsgenerate significantly di!erent enthalpies. In this case UA gives rise to the largeste!ect, as can be seen in Table 10.2. A large di!erence was also found between the twosolvation models for the ten-water model; UA stabilize the reaction with a moderate8 kJ mol"1 whereas the IC model gives rise to a larger destabilization of 23 kJ mol"1.

The calculated UA and IC solvation e!ects have in a later step been comparedwith the experimental activation enthalpy for the water exchange reactions in anattempt to determine which of the UA and IC models that are more suited todescribe the solvation e!ects in the uranyl(VI) water exchange reactions.

dThe enthalpy is defined as the heat content of a chemical system, H = U + pV , where U is theinternal energy, p is the pressure associated with a fixed volume V. The enthalpy change !H is theamount of heat released or absorbed during a chemical reaction.


If only the D-intermediate with single hydrogen bonds is to be considered (basedon the result from Chapter 9), the UA model gives a reasonable agreement betweenthe six- and ten-water model.Using gas phase geometries in a continuum modelTo validate the use of a gas phase geometry in a solvent model, a re-optimization ofboth the six- and ten-water model, using CPCM with the UA topology models, wereperformed. No significant di!erences were found. For the six-water model the U-OH2

bond decreased in average 0.05 Å and for the lager ten-water model the change waseven smaller. The distance from the uranium to the second shell water decreasedwith 0.1 Å for the larger model.

When the reaction energies at the B3LYP level were compared, the di!erencebetween using gas phase optimized geometries in a solvent model and performing amore time consuming solvent optimization were in most cases less than 1 kJ mol"1.The largest di!erence was 2 kJ mol"1 for the six-water model and 7 kJ mol"1 for thelarger ten-water model. The conclusion is therefore that the compromise of placinggas phase geometries in a solvent model gives accurate results for reaction energies. Itmust be noted however that the enthalpy correction and other thermodynamic func-tions should not be calculated within a continuum model, there are non-cancellingdi!erences in the thermal contributions related to entropic changes associated withstructural reorganization of the solvent that are indirectly incorporated into the cal-culated solvation e!ect through a parameterization, as for example discussed by Hoet al.151

As Table 10.3 shows, the incorrect enthalpy correction, derived with the gasphase geometries in an UA or IC model are di!erent from the correction obtainedin gas phase. The correction for the associative reaction in gas phase was close to 0whereas the two solvation results di!er with 18 kJ mol"1, the UA enthalpy correctionstabilized the reaction with 6 kJ mol"1 and the IC correction destabilized the samewith 12 kJ mol"1. For the other two reaction paths, the di!erence is smaller, at most5 kJ mol"1.

Table 10.3: Enthalpy correction, #H298, in kJ mol"1, using gas phase geometries ina continuum model with UA and IC.

UO2+2 (H2O)6

Reaction Gas Phase UA IC

A-int 0 -6 12D-int 1H -4 -6 -9D-int 2H 0 8 3

10.2 Conclusion from Paper 2The almost identical calculated solvation e!ects that are obtained at di!erent levelsof theory show that it is possible to use DFT to obtain solvation e!ects that can beadded to a more accurate wave function based energy result.

The question is then, which solvation model should be used? The rather scatteredresults for solvation e!ects that are obtained with the UA and IC models concerning

10.3 Factors that can contribute to the obtained di!erence 55

the water exchange mechanism as can be seen in Table 10.1, make it di"cult todetermine which of the solvation models to prefer.

The experimental activation enthalpy for uranyl(VI) water exchange is 26±1 kJmol"1 111 and can be compared with the calculated reaction enthalpies in Table10.2. The calculated reaction path enthalpies are equally plausible if only the D-intermediate structure with single hydrogen bonds is considered for the small six-water model. Also the results for the larger model make it di"cult to draw anyconclusion concerning which solvation model to chose. The UA model gave moreconsistent solvation e!ects for both the associative reaction and the dissociativereaction (1H) for the six-water model, 17 kJ mol"1 and 33 kJ mol"1 respectively,compared to the solvation e!ect obtained for the larger ten-water model, 12 kJ mol"1

and 31 kJ mol"1. The corresponding result obtained with the IC model are 34 kJmol"1 and 31 kJ mol"1 compared to 31 kJ mol"1 and 62 kJ mol"1 respectively forthe larger model. It therefore seems as if the experimental activation enthalpy is inbetween the calculated reaction enthalpies obtained with UA and IC for both thesmall six-water model and the larger ten-water model and thus cannot be used inthe decision concerning solvation models.

M. Bühl and H. Kabrede140 found, using a constrained Car-Parrinello molec-ular dynamics152 (CPMD) simulationse and thermodynamic integration including59 water molecules, that the simulated free energy of activation for an associa-tive/interchange process was 28 kJ mol"1. This result is in agreement with theexperimental activation enthalpy of 26 kJ mol"1 found by Farkas et al.111 and sig-nificantly lower than the one they obtained for a purely dissociative mechanism, 45kJ mol"1.

There is at most a qualitative agreement between the results in this investigationand the results presented by Bühl and Kabrede, but a comparison of results does notallow any further conclusions on the relative merits of the two topological models sofar.

The conclusion concerning the use of an UA or IC solvation model was reachedfrom a consistency point of view. The UA model resulted in more consistent solvatione!ects for associative reaction and for the dissociative reaction for both the six- andten-water models. This together with the fact that choosing the same solvation modelas in preceding studies112,117 might be the best option until the di!erence is fullyunderstood and since the combined data from both the six- and ten-water modelsinfer that the exchange reaction follows an associative path.

10.3 Factors that can contribute to the obtained di!erenceIn the following part, the di!erence between the two topological models is furtherinvestigated in terms of sphere radii, electronic structure and charge on the centralion. As shown in section 10.1.1, the level of theory does not influence the calculatedsolvation e!ect, and in the following only solvation results at the B3LYP level oftheory are presented.

eThe CPMD simulation was performed using the BLYP functional.


10.3.1 The topological model — sphere radiiAs the numbers in Table 10.1 show, there are large di!erences between the solva-tion e!ect obtained with the UA and IC topological models. One factor that cancontribute with a small di!erence is that the implementations of the CPCM modeldi!er between the available computational chemistry packages. To avoid confusiondue to di!erent implementation, calculated solvation e!ects obtained with both theMolcas and Turbomole packages have been included in addition to Gaussian 03153

results.To explore the e!ect of the sphere radii this parameter test has been carried

out by varying the sphere radii in the two groups of topological models using pre-defined cavities that are available in the Gaussian 03 package. All other parameters,regardless of which quantum chemical package, have been set to be the same as in thedefault settings in Gaussian 03. The radii information for the set of UA topologicalmodels are shown in Table 10.4, and the corresponding information for the group ofIC topological models are shown in Table 10.6. In this part of the investigation thesix-water model for the uranyl(VI) water exchange reaction in the electronic groundstate has been used to test the sphere radii e!ect.

UA topological modelsThe topological models, using di!erent sphere radii to build the cavities in this in-vestigation, are UA0, UAHF154 and UAKS153 from Gaussian 03 and the default UAtopological model in the Molcas package.92 The sphere radii for each model are pre-sented in Table 10.4. As shown in Table 10.5, it is only a small variation in solvation

Table 10.4: The radii parameters in Å for the UA topological models.

Cavity Radii U Radii O Radii H2O

UA0 1.698 1.75 1.95UAHF 1.698 1.59 1.68UAKS 1.698 1.50 1.68MOLCAS 1.860 1.50 1.68

e!ects among the di!erent UA models. The result for the UA model implemented inMolcas was in good agreement with the other UA models for the associative and dis-sociative reaction with single hydrogen bonds. The dissociative reaction with doublehydrogen bonds gave a smaller solvation e!ect than that obtained with Gaussian 03.

Table 10.5: Solvation e!ects calculated at B3LYP level of theory in kJ mol"1 for thewater exchange reaction of aqueous uranyl(VI) with di!erent UA models.

UO2+2 (H2O)6

Cavity A-int D-int 1H D-int 2H

UA0 -18 22 29UAHF -15 21 25UAKS -15 22 24MOLCAS -13 22 16


IC topological modelsThe set of cavity radii for IC models that has been used are presented in Table 10.6.The most common IC model (the default setting for the IC model) in Gaussian 03 isthe UFF, whereas the Pauling155 and Bondi156 are two other IC models available inGaussian 03 that have been investigated. Information of solvation e!ects obtainedwith the Molcas and Turbomole package are included in the same table. Only small

Table 10.6: The radii parameters for the IC topological models, in Å

Cavity Radii U Radii O Radii H

UFF 1.698 1.75 1.44COSMO 1.698 1.72 1.30Pauling 1.860 1.40 1.20Bondi 1.860 1.52 1.20

variations in the solvation e!ects were found for the associative reaction. The twodissociative reactions however, show a larger di!erence. The UFF and COSMO mod-els results are consistent, while the Pauling and Bondi models resulted in solvatione!ects that are di!erent, as can be seen in Table 10.7. The Pauling model, and tolesser extent the Bondi model, give much larger solvation e!ects, with 16 kJ mol"1

and 9 kJ mol"1 respectively compared to UFF. The dissociative reaction with dou-ble hydrogen bonds Pauling and Bondi di!ers with 33 kJ mol"1 and 14 kJ mol"1

relative to the UFF respectively.

Table 10.7: The solvation e!ects in kJ mol"1 calculated at B3LYP level of theoryusing di!erent IC models for the water exchange reaction of aqueous uranyl(VI) inthe electronic ground state.

UO2+2 (H2O)6

Cavity A-int D-int 1H D-int 2H

UFF -2 21 3COSMO -2 19 3COSMOa -2 22 1Pauling -3 5 -30Bondi -3 12 -11MOLCASa -3 19 3aCavities are set to be the same as for UFFimplemented in Gaussian 03

SummaryThe Molcas and UAKS models only di!er in the uranium radii, resulting in 8 kJmol"1 smaller solvation e!ect for the dissociative reaction with double hydrogenbonds. As can be seen in Table 10.5 the sensitivity to the uranium radii in UAmodels is smaller for the associative reaction and the reaction energy is apparentlynot sensitive to the uranium cavity-radius. The sensitivity to the water radius is alsosmall. The e!ect of the oxygen radii is much larger as can be seen for the two ICmodels Pauling and Bondi. The in comparison small oxygen has a large e!ect onthe dissociative reaction energy as can be seen in Table 10.7. However, the result


from this part of the investigation does not make it possible to draw any definitiveconclusions about the choice of topological model from the investigations so far.

10.3.2 The water exchange reaction mechanism for uranyl(V)The section below will discuss the e!ect on the solvation e!ect due to the elec-tronic structure of the central ion. Di!erent uranium complexes in six-water models(optimized for the individual system) were used in this part of the investigationand the water exchange mechanism for uranyl(V) was one of the systems used toprobe the possible e!ect. The uranyl(V) reaction mechanism has previously beeninvestigated by Vallet et al.117 and in that study only the D-intermediate structurefor uranyl(V) was found. The geometries were optimized at the SCF level and theA-intermediate structure disintegrated at this level of theory. The water exchangereaction for uranyl(V) was therefore re-calculated in the present investigation, us-ing the same ECP and basis sets that have been used for all uranyl(VI) systems inChapter 8 – 11 in this thesis (see section 7.1 for details). The optimizations werenow performed at the U-B3LYP level in the solvent using the UA solvent model inGaussian 03. The reaction energies were calculated at the minimal CASPT2 leveland are presented in Table 10.9.

The dissociative reaction with single hydrogen bonds was found to be 19 kJmol"1 more stable than with double hydrogen bonds for uranyl(V) (in gas phase)to be compared with 20 kJ mol"1 for uranyl(VI) in Table 9.1. The dissociative re-actions with double and single hydrogen bonds were 8 kJ mol"1 and 13 kJ mol"1

less stable compared to the associative reaction respectively. It is an overall geo-metrical similarity between the complexes in the uranyl(VI) and uranyl(V) systemsalthough in the latter, all bond distances are as expected larger. Information aboutthe uranyl(V) geometry is included in Table 10.8.

Table 10.8: Geometry information for the uranyl(V) in the electronic ground state.Optimized at the U-B3LYP level using an UA solvation model. All distances inÅngström and n is number of coordinated water molecules in the sphere.



UO+2 (H2O)6 Reac. 1.84 5 2.54 1 3.72

A-Int 1.84 6 4)2.57, 2)2.78 –D-int 1.84 4 2)2.45, 2)2.47 2 3.97

10.3.3 The central ionThe significant di!erence that was found between the solvation e!ect obtained withthe UA and IC topological models in Table 10.1 is related to the choice of usingexplicit cavities for the hydrogens or not. In this section the di!erence is furtherinvestigated by applying the UA and IC solvation models to a set of actinide modelcomplexes with di!erent electronic structures on the central ion.

The e!ect of the electronic structure was investigated by replacing the centraluranyl(VI) ground state ion with uranyl(VI) in the first excited state, uranyl(V) and


Table 10.9: Reaction energies in kJ mol"1 for uranyl(V) in a six-water model at theCASPT2 level. All structures were optimized at the U-B3LYP level using the UAsolvation model in Gaussian 03.

UO+2 (H2O)6

A-int D-int 1H D-int 2H

Gas phase 61 13 32UA 16 29 23IC 32 20 12

UN2. All model complexes were optimized in a six-water model for each system. Theground state of uranyl(VI) and UN2 are closed shells systems with di!erent charges,2+ and neutral respectively. The other two have di!erent spin states, a triplet anddoublet, with charge 2+ and 1+ respectively. Table 10.10 shows the solvation e!ectfor the uranium model complexes.

Table 10.10: Solvation e!ects in kJ mol"1 for the water exchange reaction for ura-nium complexes obtained at the B3LYP (and U-B3LYP) level using UA and ICmodels. The complexes are UO2+

2 (H2O)6, the first excited state U(O2+2 (H2O)6,

UO+2 (H2O)6 and UN2(H2O)6.

UO2+2 (H2O)6 U#O2+

2 (H2O)6

Cavity A-int D-int 1H D-int 2H A-int D-int 1H D-int 2H

UA -18 22 29 -19 24 30IC -2 21 3 -1 24 5

UO+2 (H2O)6 UN2(H2O)6


UA -45 16 -9 61 14 23IC -28 7 -20 68 9 13

The same large di!erences in the solvation e!ect between UA and IC that wasfound in the ground state for the associative intermediate and the dissociative in-termediate with double hydrogen bonds was also found for the luminescent state.The solvation e!ect regardless of model is stabilizing for the associative reactionand destabilizing for the two dissociative reactions. The di!erence in solvation e!ectthus seems not to be related to electronic configuration. For uranyl(V), the di!er-ence between solvation e!ects obtained with UA and IC shows the same trend as theone found for the ground and luminescent states of uranyl(VI). The actual numbersdi!er, presumably because of the di!erent charges. The di!erence in solvation e!ectobtained with UA and IC for the neutral UN2 system is smaller, but still noticeable.

10.3.4 The charge on the central ionIn the investigation of the electronic structures of the central ion and the possibleimpact on the solvation e!ect, a trend in the di!erence in solvation e!ects that might


be related to the charge of the central ion was noted and this possibility is furtherexplored in this part of the solvation study.

First uranyl(VI) in the six-water model was used to probe to what extent thecharge on the central ion influences the calculated solvation e!ect. In this investiga-tion the central uranyl(VI) ion in the surrounding water cage was replaced with alighter ion. Both the H+ and Be2+ ions were used as point-like charges placed in thesame water cage as in the original uranyl(VI) system, keeping the geometry frozen.The results are presented in Table 10.11.

Table 10.11: Solvation e!ects in kJ mol"1 at the B3LYP level for uranyl(VI) in thesix-water model, the neutral water cage and the systems where the uranyl(VI) ionis exchanged with H+ and Be2+.

UO2+2 (H2O)6 Water Cage (H2O)6


UA -18 22 29 45 -24 -9IC -2 21 3 48 -20 -12

Water Cage with H+ Water Cage with Be+2


UA 6 -9 -2 65 8 4IC 13 -7 -10 65 6 -12

The solvation e!ects that are obtained for H+ and Be2+ ions inside the watercage indicate that the charge on the central ion is not the sole reason for the di!er-ence between the UA and IC model. The di!erence is significantly smaller but stillnoticeable compared to uranyl(VI).

In the next step the larger ten-water model was used both for uranyl(VI) and theTh4+ ion as an example of a model complex with higher charge, in order to comparesolvation e!ects obtained with UA and IC. All CPCM calculations for the two ten-water systems were also performed with the Molcas package and the Turbomolepackages (COSMO), no significant di!erences between the di!erent implementationswere found and the discussion will be restricted to results obtained with Gaussian03. As Table 10.12 shows, uranyl(VI) system in the ten-water model shows the samelarge di!erences in the solvation e!ect between UA and IC that was found for thesix-water model. For the Th4+ system, the solvation e!ect is stabilizing for theassociative reaction and destabilizing for the dissociative reactions but the absolutedi!erence between UA and IC is similar to the one found for the uranyl(VI) complexregardless of whether applied to the six- or ten-water model.

The same exchange of the central actinide ion that was performed for the six-water model has been performed for the two ten-water model systems as well. Thetwo actinide ions were replaced by point-like charges in terms of the H+ and Be2+ions for the uranyl(VI) system and C4+ for Th4+.

When the H+ and Be2+ atoms are put in the position of the uranyl(VI) iona reasonable consistency between the solvation e!ects obtained with UA and IC isfound. The largest di!erence between topological models is found for the dissociativereaction with Be2+, 9 kJ mol"1, where other di!erences are neglectable. The charge

10.4 Conclusions 61

does thus not appear to be primarily responsible for the di!erence found betweenUA and IC in uranyl(VI).

Table 10.12: B3LYP solvation e!ects in kJ mol"1 for uranyl and Th4+ in the ten-water model, the plain water cage and water cage with H+ and Be2+ in the positionof the uranyl and C4+ in the position of Th4+.

UO2+2 (H2O)10 Water Cage Water Cage Water Cage

(H2O)10 with H+ with Be+2

Cavity A-int D-int A-int D-int A-int D-int A-int D-int

UA -21 -5 112 20 31 -5 31 -29IC -2 27 110 31 31 -6 28 -20

Th4+(H2O)10 Water Cage Water Cage(H2O)10 with C4+

Cavity A-int D-int A-int D-int A-int D-int

UA -60 66 42 -59 -66 69IC -47 41 40 -65 -47 42

The di!erence in solvation e!ects obtained with UA and IC models for whenthe Th4+ ion is replaced with C4+ is similar to the results obtained for the originalTh4+(H2O)10 system. For this case, the rather large di!erence between solvatione!ects obtained with UA and IC can be attributed to the large charge on the centralion.

10.4 ConclusionsAs mentioned in the summary of section 10.3.1, the parameter of the atomic radiiused to build up the cavities does not explain the di!erence in solvation e!ect.Although the small oxygen radii (1.40 Å and 1.52 Å) used in the Pauling and Bondimodels has a large e!ect on the dissociative reaction energy, as can be seen in Table10.7, this does not explain the di!erence between the UA and IC solvation modeldiscussed in section 10.1.

The electronic structure of the central ion does not seem to significantly a!ectthe discrepancy between UA and IC, since the singlet ground state, the luminescenttriplet state and the doublet uranyl(V) all show the same trend for the solvatione!ects. Even though the neutral UN2 system results in a smaller di!erence, theproblem seems not to be connected with uranyl(VI), it is rather a more generalproblem.

The charge was a promising explanation since a smaller di!erence in solvatione!ect was obtained with UA and IC for the neutral UN2 system, but when the resultsobtained with the point-like H+ and Be2+ ions were compared there was a muchsmaller discrepancy between the two topological models. This leads to the conclusionthat the charge itself is not the cause for the large discrepancies between the UAand IC models for uranyl(VI). Instead, it seems as if the problems encounteredare connected to the uranyl and the water molecules in the first hydration shell,presumably due to a di!erent response for UA and IC to the charge distribution in


this region. However, in the Th4+ system it is clear that the di!erence between UAand IC is due to the high charge on the thorium ion.

It can be noted that a more consistent solvation e!ect was obtained with UA forboth the associative reaction and the dissociative reaction using a D-intermediatestructure with single hydrogen bonds than the one obtained with the IC model. TheCar-Parrinello results from the M. Bühl and H. Kabrede140 investigation concerningthe activation for an associative/interchange process was found to be 28 kJ mol"1f.Combining this information with the fact that the UA solvation model has been usedin preceding ligand exchange investigations, the conclusion is that it is reasonableto use the UA solvation model in the investigation of the water exchange reactionfor the luminescent state of uranyl(VI).

fClose to the experimental activation enthalpy of 26 ± 1 kJ mol!1 found by Farkas et al.111

Chapter 11

Water exchange mechanism inthe luminescent state ofuranyl(VI)The investigation of the water exchange reaction in the luminescent state of uranyl(VI)is a continuation of the series of ligand exchange studies by Wahlgren, Grenthe andcoworkers111–117 mentioned before. In Paper 3, the water exchange reaction in theluminescent state of uranyl(VI) is compared with the corresponding ground statereaction. In this study the outcome of the investigations of the performance of DFTfor actinide reaction mechanisms (discussed in Chapter 8) has been used togetherwith the results from the model investigation concerning the chemical model andhow the aqueous surrounding should be modeled, discussed in Chapter 9 and 10.The investigation was done with the six-water model mechanism, as illustrated inFigure 11.1, to approximate the exchange reaction mechanism for the luminescentstate of uranyl(VI).

11.1 The exchange mechanism

Figure 11.1: The reaction mechanisms used to describe the water exchange reactionin hydrated uranyl.

63

64 11 Water exchange mechanism in the luminescent state of uranyl(VI)

Although the main subject of this investigation was to compare the ground stateand the luminescent state, it is also of some interest to make a comparison with thecorresponding reactions for uranyl(V). The water exchange mechanism for uranyl(V)was used in Chapter 10 to explore if the electronic structure could contribute to thedi!erence in the calculated solvation e!ects obtained with the UA and IC solvationmodels within CPCM. Included in this chapter is an unpublished comparison be-tween the ground- and first excited states of uranyl(VI) with the ground state ofuranyl(V). The comparison is based on reaction enthalpies and geometries. Basedon the results from the model investigation described in Chapter 9, a structure withsingle hydrogen bonds between the inner and outer sphere water molecules has beenused for the D-intermediate.

11.2 GeometriesThe geometries for the luminescent state were in an initial step obtained at the TD-DFT level using the B3LYP functional, in accordance with the method investigationdescribed in Chapter 8. It was for technical reasonsa not possible to obtain a dis-sociative intermediate geometry at the TD-B3LYP level, where the inner and outersphere water molecules bind with single hydrogen bonds (see Chapter 9) with a con-firmed energy minimum. All structures for the luminescent state in this investigationare therefore optimized with unrestricted B3LYP formalism. As discussed in Chap-ter 8, a transition state for the associative reaction was not identified. However, ina preceding investigation concerning the water exchange mechanism for the groundstate of uranyl(VI)112 it was found that the di!erences in enthalpies and geometrybetween the transition states and the intermediate structures were so small that itwas assumed that the intermediate structures could be used as approximations tothe transition states. The adiabatic energies that are presented in Table 8.4 wereused as an argument indicating that the reaction energy for the ground state andthe first excited state are parallel and that the assumption of using the intermediatestructures is acceptable.

The geometries, shown in Table 11.1 are quite similar for the intermediates inall systems. The main di!erence is the position of the second shell water molecules,shown for the reactants in Figure 11.2, and an increasing U-Oyl bond distance.

(a) (b) (c)

Figure 11.2: The reactants for the uranyl(VI) in the ground state in (a), the lumi-nescent state in (b) and uranyl(V) in (c)

aIt can be mentioned that it was not possible to obtain optimized geometries at the U-B3LYP levelwith the implementation in Gaussian 03 whereas it was possible with the Turbomole package.

11.3 Reaction energies 65

The position of the second sphere water can be correlated with the negativecharge on the Oyl which decreases for the luminescent state (-0.30) and increases foruranyl(V) (-0.49) compared to the ground state of uranyl(VI) (-0.31).

Table 11.1: Geometry information for the uranyl(VI) in the ground and luminescentstate and uranyl(V) obtained at the B3LYP and U-B3LYP level. All distances inÅngström and n is number of coordinated water molecules in the sphere.



UO2+2 (H2O)6 Reac. 1.75 5 2.50 1 4.26

A-Int 1.76 6 4)2.52, 2)2.65 -D-int 1.75 4 2)2.36, 2)2.45 2 4.36

U#O2+2 (H2O)6 Reac. 1.80 5 2.50 1 4.37

A-Int 1.81 6 4) 2.52, 2)2.62 -D-int 1.81 4 2)2.37, 2)2.46 2 4.43

UO+2 (H2O)6 a Reac. 1.84 5 2.54 1 3.72

A-Int 1.84 6 4)2.57, 2)2.78 -D-int 1.84 4 2)2.45, 2)2.47 2 3.97

a Optimized in the UA solvation model.

11.3 Reaction energiesSpin-free energies were obtained with minimal and near-minimal CASPT2 usingU-B3LYP geometries in accordance with the results described in Chapter 8. Thereaction enthalpies were estimated by adding corrections obtained at the U-B3LYPgas phase level,151 as described in Chapter 10. The solvation e!ect is included withCPCM using the UA topological modelb. The choice of solvent model is motivatedin Chapter 10.

11.3.1 Spin-orbit correctionsThe electron correlation and spin-orbit interactions have been treated simultaneouslyusing the two-step SO-CI method, using the EPCISO87 and the MOLCAS packages.In the first step, the spin-free electron correlation was obtained at the near-minimalCASPT2 level. This was then followed by the spin-orbit interaction, calculated byusing an e!ective Hamiltonian scheme where electron correlation is included usingthe level shift technique outlined in section 4.4. The reference space used for the spin-orbit calculations in the luminescent state included five orbitals, the .u, the two +uand the two #u, while the remaining electrons were kept uncorrelated. All singleexcitations from the active space were included in the spin-orbit CI, ensuring thatboth electronic orbital relaxation and di!erential spinor relaxation will be accountedfor in the final wave function.87,88

bAs implemented in the Gaussian 03 package.

66 11 Water exchange mechanism in the luminescent state of uranyl(VI)

11.3.2 ResultsThe reaction enthalpies for uranyl(VI) in the ground and luminescent states in solva-tion, both the spin-free and spin-orbit levels, are shown in Table 11.2 together withthe spin-free reaction enthalpies for the uranyl(V) mechanism. In the associativereaction, the enthalpies are essentially identical for all three systems. In the disso-ciative reaction, the ground state and the luminescent state of uranyl(VI) are verysimilar at the spin-free level, although the di!erence increases from 2 to 12 kJ mol"1

when the spin-orbit e!ect is taken to account. The reaction enthalpy for uranyl(V) issomewhat lower than for the uranyl(VI) complexes, by 4 and 6 kJ mol"1 respectivelyat the spin free level. No spin-orbit calculations were performed for the uranyl(V)system.

Table 11.2: Reaction enthalpy in kJ mol"1 at the MP2/CASPT2 spin-free (SF)and spin-orbit (SO) level, for the water exchange mechanism for UO2+

2 (H2O)6, theluminescent state U(O2+

2 (H2O)6 and UO+2 (H2O)6. The solvation e!ects obtained

with the UA model

UO2+2 (H2O)6 U#O2+

2 (H2O)6 UO+2 (H2O)6

A-int D-int 1H A-int D-int 1H A-int D-int 1H

SF 18 35 16 37 19 31SO 16 32 18 44 - -

11.4 ConclusionsAt the spin free level the three systems are quite similar, and it is not meaningful tomake any statements about the relative similarity between the luminescent state andthe ground states of uranyl(VI) and uranyl(V). If the ground state is compared withthe luminescent state, it is clear that the mechanism and the activation parametersfor the water exchange reaction are hardly a!ected by the excitation to the lumines-cent state. A reasonable conclusion is that the same is true for other ligand exchangereactions involving hard inorganic ligands, and that no new chemistry appears in theluminescent state for this type of reactions.

Of course photochemical and charge transfer reactions are di!erent for uranyl(VI)in the ground and luminescent state, for example, photo-excited uranyl(VI) is apowerful oxidant.157,158

Chapter 12

Multi-reference short-rangeDFTIn this chapter the methodological test of the multi-reference srDFT methoda sug-gested by Fromager et al.125 is discussed. It is a short-range/long-range separationof the two-electron Coulomb interaction in this method that enable a multi-referencedescription of the wave function. The range separation is obtained by using an errorfunction together with a free parameter µ within the half-closed interval [0,+([.The optimal µ (µopt) value is defined as the largest value of µb for which the wavefunction is a single determinant in systems with no significant static correlation anddispersion interaction e!ects.

This project was a continuation of the investigation by Fromager et al.,125 wherethe original test set of elements with no significant static correlation and dispersioninteraction e!ects, was replaced by a set of neutral and charged isoelectronic f0actinide complexes. The aim was to see if the suggested µopt = 0.4125 is applicableon actinide complexes, that often need a multi-reference approach due to strongstatic correlation e!ects. A prerequisite for the method is that the same µ value canbe used for all elements in the periodic table.

The results are published in Paper 4, but it should be noted that in this thesis,the interpretation of the results concerning the static correlation is modified.

12.1 Chemical compounds and active spaceThe test set included the closed shell compounds ThO2, PaO+

2 , UO2+2 , UN2 and

CUO as examples of complexes assumed to have limited static correlation, theNpO3+

2 molecule was included as an example where the static correlation shouldbe more important, as discussed by Straka et al.159 However, static correlation isnon-negligible also in for example the uranyl complex. The coe"cient for the dom-inating configuration obtained from a CASSCF calculation has the coe"cient 0.91for NpO3+

2 and 0.94 UO2+2 and the next coe"cients, origin from excitation within

the --space, are about 0.1 and 0.07 for the two actinyls respectively. It is clear thatthe di!erence in static correlation is not very large between NpO3+

2 and UO2+2 as

the µ-dependent result will show.

aDescribed in section 3.3.1bµ is given in atomic units (a.u.)

67

68 12 Multi-reference short-range DFT

The MCSCF-srDFT calculations were performed with the same srLDA andsrPBE functionals as used in reference.125 The energies were calculated using dif-ferent active spaces, where the choice of active space is based on the bonding andanti-bonding orbitals in f0 actinyls. There are six doubly occupied bonding orbitals.g, -u, .u and -g and six anti-bonding orbitals .(

g , -(u, .(

u and -(g . For the actinyl

group, four di!erent sizes of the active space were used. The expression Ne/Nao isused to denote the active space where Ne stands for the number of electrons andNao is the number of active orbitals resulting in the 6/6, 10/10, 12/12 and 12/15active spaces. A 0/0 active space corresponds to a Hartree-Fock calculation and thelargest 12/15 active space includes three of the non-bonding +u and #u f -orbitals.

12.2 The optimal µ for ThO2, PaO+2 , UO2+

2 , UN2 and CUOThe energies as a function of the µ parameter are plotted in Figure 12.1 for ThO2,PaO+

2 and UO2+2 . The criterion for a wave function to be considered as a single deter-

minant is the same as that described in section 3.3.1: the energy di!erence betweena HF-srDFT and a multi-reference-srDFT energy should be less than 10"3 a.u. As

(a) (b)

(c)

Figure 12.1: The energies with respect to the µ parameter for ThO2 (a), PaO+2 (b)

and UO2+2 (c) at the ECP level. The uppermost curve is HF-srDFT whereas the

lower curves are MCSCF-srDFT obtained with the srLDA and srPBE functionalsusing di!erent active spaces.

12.2 The optimal µ for ThO2, PaO+2 , UO2+

2 , UN2 and CUO 69

Figure 12.1 illustrates, the MCSCF-srDFT µ-dependent energy is close to the singlereference HF-srDFT up to a µ value at about 0.3, where the splitting for all curves isaround 10"3 a.u. The uppermost curve in the figure is the HF-srDFT energy whereasthe lower curves are MCSCF-srDFT energies obtained with the srLDA and srPBEfunctionals using di!erent active spaces. The µ-dependent curves obtained with thevalence active space (12/12) and the active space including three additional +u and#u f -orbitals (12/15) are almost identical within the plotted µ-range, showing thatthe valence active space is satisfactory for determining a value for µopt.

The MCSCF-srDFT calculation for UN2 and CUO were only performed withthe valence active space (12/12). The results are shown in Figure 12.2 together withthe corresponding HF-srDFT energies, and it is clear that the splitting takes placearound µ = 0.3 for those two actinide compounds as well. Since the static correlation

(a) (b)

Figure 12.2: The energies with respect to the µ parameter for (a) UN2 and (b)CUO at the ECP level. The uppermost curve is HF-srDFT whereas the lower curveis MCSCF-srDFT energy obtained with the srLDA and srPBE functionals for the12/12 active space.

is more pronounced in the actinide complexes than in the test set used by Fromageret al.,125 it is not surprising that the µopt is smaller, 0.3 rather than 0.4. The samee!ect was observed in ref.125 for the Be, N2 and Mg complexes where the staticcorrelation is significant. However, the static correlation is more important in theactinide systems than in the original test set with compounds without significantstatic correlation and dispersion interaction e!ects. The splitting obtained at µ =0.4 has increased with at factor of 2 relative to the splitting at µopt = 0.3 for thetest set. The µopt = 0.4 as suggested in ref.125 seems thus to be within an acceptablerange for this part of the test set.

12.2.1 ECP versus AETo investigate the reliability of the relativistic ECPs, calculations at the all electron(AE) level were carried out with the LDA and PBE sr-functionals for all complexesin the actinide test set. The AE results were in good agreement with the ECP forthe separation of the static and dynamic correlations as illustrated in Figure 12.3aand 12.3b, where the results for the UN2 complex are presented as an example. Thedi!erence in energy between HF-srDFT and the 12/12 MCSCF-srDFT obtained withECP and AE are 0.8%10"3 a.u. and 0.7%10"3 a.u. for srLDA and srPBE respectivelyat µ = 0.3. Similar agreement between AE and ECP concerning the splitting werefound for all complexes.


(a) (b)

Figure 12.3: The total energy with respect to the µ parameter, obtained with thesrLDA and srPBE functionals at both ECP (a) and AE (b) level. The HF-srDFTenergy is the uppermost curve and the 12/12 MCSCF-srDFT result is the lowercurve. The results for the UN2 complex is presented as an example, the same trendwas obtained for all other complexes as well.

Somewhat surprisingly there are some striking di!erences between the AE andECP results. The energy di!erence between the two functionals at the ECP and AElevel is very large as indicated in Figure 12.3b (see text in figure); the AE energycurves for LDA are shifted by 15.6 a.u. The shape of the curves are also di!erent,in particular for LDA. The e!ect could be due to the fact that both the core andthe valence densities are used at the all-electron level, and the densities are notadditive in the DFT functionals. For large µ the curves converge to the same energyas expected.

12.2.2 Natural orbital occupationAnother option to explore the single determinant character of the wave function isthe natural orbital occupation. The µ dependency of the natural orbital occupationobtained for the valence active space with the MCSCF-srLDA was therefore inves-tigated. The orbital occupation versus µ for all actinyls are plotted in Figure 12.4.

(a) (b)

Figure 12.4: The µ dependency of the natural orbital occupation. (a) shows the -u

orbital and (b) the -g orbital, obtained for the valence active space with MCSCF-srLDA for all actinyls.

12.3 The optimal µ for NpO3+2 71

As can be seen in the figure, no natural orbital occupation outside the occupiedspace was larger than 10"3 at µ = 0.3 for ThO2, PaO+

2 and UO2+2 . For a µ = 0.4

the corresponding result is 10"2 a.u. The natural orbital occupations in Figure 12.4verify that the static correlation is similar for the Th – U actinyls. The occupationis somewhat smaller and thus the static correlation should be somewhat larger forNpO3+

2 in consistency with the CASSCF coe"cients discussed in section 12.1.

12.3 The optimal µ for NpO3+2

The static correlation in the NpO3+2 ion is the largest in the sequence of complexes

previously investigated by Straka et al.,159 and include in this test set, as discussedin the introduction.

The results from the MCSCF-srDFT calculations with the same active spacesas for the other actinyls, is presented in Figure 12.5a. For NpO3+

2 , the HF-srDFTand MCSCF-srDFT energies di!er with 2%10"3 a.u. for µ = 0.3, a somewhat largerenergy di!erence than around 10"3 a.u. that was found for the other actinide com-plexes. In Paper 4, it was wrongly stated that the energy di!erence was 10 timeslarger for NpO3+

2 than for the other actinide complexes, see Erratum 13.2.The rather small increase in the splitting is consistent with the limited di!er-

ences in static correlation; recall the values 0.91 and 0.94 for the coe"cients of thedominating configuration obtained from CASSCF for NpO3+

2 and UO2+2 mentioned

above.

(a) (b)

Figure 12.5: Total ground state energies, (a), for NpO3+2 calculated at the ECP level,

HF-srDFT is the uppermost curve and MCSCF-srDFT results are the lower curves.The natural MCSCF-srLDA orbitals occupancies, (b), of NpO3+

2 bonding orbitalsusing the 12/12 active space are plotted with respect to the µ parameter.

12.3.1 Natural orbital occupationAt µ = 0.3, the static correlation is dominated by double excitations from the -g

orbitals as can be seen in Figure 12.5b. It is clear from calculations that the con-figurations corresponding to double excitations from the -g and .g orbitals to the-(g and .(

g orbitals are dominant in the MCSCF-srDFT wave functions. In the samefigure, it can be seen that the smallest occupancy is found for the -g natural orbital,


whereas in a CASSCF wave function it is found that the -u is less occupied (obtainedwith the same active space). An occupancy result similar as the one obtained withCASSCF, is obtained with MCSCF-srDF for a µ = larger than 0.6 (the crossing ofthe red and green curve, see the figure caption in Figure 12.5b).

12.4 Equilibrium geometriesA test of the µ-dependence of the geometries was done by comparing results fromHF-srDFT and 12/12 MCSCF-srDFT. The equilibrium geometry obtained with thesrPBE functionals for µ equals 0.3 and 0.4 are shown together with geometriesoptimized at the CCSD(T) and the hybrid B3LYP level in Table 12.1. In this inves-tigation, the geometries obtained with CCSD(T) are regarded as the most accurateones.

Table 12.1: Equilibrium geometries for the ground state of ThO2, PaO+2 , UO2+

2 ,UN2 and NpO3+

2 at the 12/12 MCSCF-srPBE level together with results at theCCSD(T) and B3LYP levels. For the bent structures, the angle (in degrees) is givenin parentheses.

Method ThO2 PaO+2 UO2+

2 UN2 CUO NpO3+2

MCSCF-srPBE µ = 0.3 1.89(123) 1.76 1.68 1.71 1.79/1.74 1.67(155)MCSCF-srPBE µ = 0.4 1.89(123) 1.76 1.69 1.71 1.79/1.74 1.66(160)B3LYP 1.90(119) 1.77 1.69 1.72 1.79/1.74 1.67(165)

CCSD(T) 1.91(116) 1.77 1.70 1.72 1.79/1.75 1.68

The bond distances obtained with DFT methods are underestimated with atmost 0.02 Å relative to the CCSD(T) geometry, which is a reasonable result. ForNpO3+

2 , all density functional methods result in an incorrect bent structure. TheMCSCF-srPBE with µ = 0.3 gives a geometry that deviate with 25 degrees froma linear ion, and for µ = 0.4 the corresponding deviation is 20 degrees. Also theB3LYP functional fails to obtain a linear geometry for neptunyl(VII).

Jackson et al.160 report variations in the bending frequency by up to 30 cm"1

in DFT calculations. Ramakrishnan et al.161 found that PBE resulted in a bendingfrequency for UO2+

2 of 92 cm"1, to be compared with the CCSD(T) result of 178cm"1 for the same ion in ref.160

In NpO3+2 , the e!ect of a too weak bending frequency is, as shown in Table

12.1, a bending of the ion. This e!ect is illustrated in Figure 12.6a where srLDAis compared with CASSCF and standard LDA. Furthermore, it follows from Figure12.6b that, srLDA gives a low bending frequency for UO2+

2 compared to CASSCFin agreement with the results in ref.160,161

The bending angle for ThO2 obtained with DFT methods are somewhat larger,3–7 degrees, than that obtained with CCSD(T). For all actinide (An) compounds inthis investigation except ThO2, the An-Oc bond is dominated by the An5f orbitalsoverlap with the O2p orbitals, leading to linear compounds, whereas the longer Th-O bond is dominated by the Th6d overlap with the O2p orbitals, which favours abent structure.162

cAnd the corresponding U-N and U-C bonds.

12.5 Conclusions 73

(a) (b)

Figure 12.6: The energy, in a.u. for the NpO3+2 (a) and UO2+

2 (b) compounds relativeto the An-O distance.

12.5 ConclusionsThe optimal µ for the investigated actinide compounds was found to be 0.3 ratherthan 0.4 due to the larger amount of static correlation compared to the original testset.125 Considering this, and the fact that the splitting at µ = 0.4 still is reasonblysmall, 0.4 seems to be an acceptable choice also for the actinides.

The bond distances agree well for µ 0.3 and 0.4 but there are significant dif-ferences between CCSD(T) and srDFT concerning the bending angles. DFT canunderestimate the bending vibration frequencies and for the NpO3+

2 ion all densityfunctional methods resulted in an incorrect bent structure. It should be noted thatfor a µ = 0.6 a linear NpO3+

2 ion is recovered, Table III in Paper 4.An universal µ seems to be possible for all elements in the periodic table, which is,

as mentioned in the introduction, a prerequisite for the suggested MC-DFT methodand the results are in that sense promising.

The method is under development. In a recent study by E. Fromager, R. Cimi-raglia and H. -J. A Jensen,163 the multi-reference van der Waals systems Be2, Mg,and Ca2 were investigated by using the sc-NEVPT2164-srDFT method for model-ing van der Waals systems which also have significant static correlation e!ects. Theresult for sc-NEVPT2-srPBE shows that the method can describe the London dis-persion forces in the complexes in the test set. As also noticed for the actinide testset, the short-range density functionals need to be improved and work is in progressin this direction.

Chapter 13

Concluding remarksAs mentioned in the introduction, theoretical studies of actinide compounds areessential for an increased understanding of their behavior and possible impact on theenvironment. In this context there are some areas that need to be further improved.

The density functional based methods that have become popular in computa-tional actinide chemistry are useful but, as pointed out in this thesis, better func-tionals, especially for the heavier elements, are much needed for accurate results.One approach to develop better functionals would be to expand the test set usedfor benchmarking DFT functionals to include also transition metal compounds andlanthanide and actinides complexes. Development of new functionals are in progressbut there is still more work to be done.

It is clear that the approach of using a continuum model to explore the conse-quences of the solubility of actinide complexes in water is associated with significantuncertainties. The solvation e!ect is a!ected by the choice of cavity model. It ismandatory to describe the e!ect of the solute-solvent interaction on the reactions,and it is for several reasons not possible to include enough water molecules in aquantum chemical model to describe solvation properly. In the short term the mostreasonable way forward is to use methods such as QM/MM which combines quan-tum mechanics with molecular mechanics. One variant of this, the Car-Parinellomethod where the solute is described at the DFT level has been successfully appliedto ligand exchange reactions in uranyl(VI)).140 However, since DFT can be quiteunreliable when applied to actinides new functionals should be developed for thispurpose. Alternatively, wave-function based methods could be used. A complicationis that the solute must include a number of solvent molecules in addition to thecentral ion which can make the QM/MM approach complicated.

As mentioned earlier, the density functional theory has become increasingly pop-ular in computational actinide chemistry since wave function based calculations arevery time- and memory consuming. The drawback is that the available function-als cannot take dispersion and the full e!ect of electron exchange and correlationinto account. One attractive possibility is to combine DFT and wave function basedmethods as, for example, in the short-range/long-range separation approach dis-cussed in Chapter 12, to obtain a time-e"cient method with acceptable accuracy. Anumber of methods are suggested but most of them are still under development.

75

76 13 Concluding remarks

13.1 Erratum Paper 1

Page 573 ”Rotzinger noted that DFT tends to overestimate the energy of tran-sition metal complexes with a high coordination number, thus favor-ing associative reactions over dissociative reactions (cf. Table 2).”

Should be ”Rotzinger noted that DFT tends to overestimate the energy of tran-sition metal complexes with a low coordination number, thus favor-ing dissociative reactions over associative reactions (cf. Table 2).”

13.2 Erratum paper 4

Page 054107-4 ”As shown in Fig. 1 and Fig. B in the supplementary materials,46 theHF-srDFT and CAS-srDFT energies at both RECP and all electronlevel di!er for µ= 0.3 a.u. by 10"2 a.u., which is ten times largerthan what was found for the static correlation-free systems studiedin Sec. IV A 1.”

Should be ”As shown in Fig. 1 and Fig. B in the supplementary materials,46 theHF-srDFT and CAS-srDFT energies at both RECP and all electronlevel di!er for µ= 0.3 a.u. by 2%10"3 a.u., which is two times largerthan what was found for the static correlation-free systems studiedin Sec. IV A 1.”

List of Figures

6.1 The two groups of topological models, the individual cavity topolog-ical model (IC) in (a) and the united atom topological model, (UA)in (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7.1 The reaction mechanisms for the ligand exchange in hydrated uranyl(VI)used in preceding investigations. . . . . . . . . . . . . . . . . . . . . 33

8.1 The ground state structures of the aqueous uranyl(VI) used to probethe DFT accuracy. The reactant (and product) is depicted in (a), theassociative intermediate in (b) and (c) shows the dissociative inter-mediate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8.2 The spin-free reaction energy for the ground state relative to CCSD(T)in kJ mol"1. The GGA functionals are plotted in red, the hybrids inblue (DFT-MRCI dashed blue) and the MP2 result is plotted dashedgreen. The CCSD(T) energy corresponds to the black line. . . . . . 41

8.3 The spin-free reaction energy for the luminescent state relative tonear-minimal CASPT2 in kJ mol"1. The TD-DFT results for theGGA functionals are plotted in red and the hybrids in blue (DFT-MRCI dashed blue). The near-minimal CASPT2 energy (35.6 and 36.6kJ mol"1 for the A- and D-reaction paths respectively) correspondsto the black line. The B3LYP and BHLYP TD-DFT and unrestrictedresults are almost identical in dissociative reaction and the B3LYP*and BHLYP* refer to both unrestricted and time-dependent results. 43

9.1 The two geometrical options for the reactant in the six-water model.The configuration with double hydrogen bonds is presented in (a) andthe configuration with a single hydrogen bond in (b). . . . . . . . . . 46

9.2 The two geometrical options for the D-intermediate in the six-watermodel. The configuration with double hydrogen bonds is presented in(a) and the configuration with single hydrogen bonds in (b). . . . . . 47

9.3 The reactant geometry in the (a) six-water model and (b) ten-watermodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

9.4 The A-intermediate geometry in the (a) six-water model and (b) ten-water model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

9.5 The D-intermediate geometry in the (a) six–water model and (b) ten-water model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

77

78 LIST OF FIGURES

11.1 The reaction mechanisms used to describe the water exchange reactionin hydrated uranyl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

11.2 The reactants for the uranyl(VI) in the ground state in (a), the lumi-nescent state in (b) and uranyl(V) in (c) . . . . . . . . . . . . . . . . 64

12.1 The energies with respect to the µ parameter for ThO2 (a), PaO+2

(b) and UO2+2 (c) at the ECP level. The uppermost curve is HF-

srDFT whereas the lower curves are MCSCF-srDFT obtained withthe srLDA and srPBE functionals using di!erent active spaces. . . . 68

12.2 The energies with respect to the µ parameter for (a) UN2 and (b)CUO at the ECP level. The uppermost curve is HF-srDFT whereasthe lower curve is MCSCF-srDFT energy obtained with the srLDAand srPBE functionals for the 12/12 active space. . . . . . . . . . . . 69

12.3 The total energy with respect to the µ parameter, obtained with thesrLDA and srPBE functionals at both ECP (a) and AE (b) level. TheHF-srDFT energy is the uppermost curve and the 12/12 MCSCF-srDFT result is the lower curve. The results for the UN2 complex ispresented as an example, the same trend was obtained for all othercomplexes as well. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

12.4 The µ dependency of the natural orbital occupation. (a) shows the -u

orbital and (b) the -g orbital, obtained for the valence active spacewith MCSCF-srLDA for all actinyls. . . . . . . . . . . . . . . . . . . 70

12.5 Total ground state energies, (a), for NpO3+2 calculated at the ECP

level, HF-srDFT is the uppermost curve and MCSCF-srDFT resultsare the lower curves. The natural MCSCF-srLDA orbitals occupan-cies, (b), of NpO3+

2 bonding orbitals using the 12/12 active space areplotted with respect to the µ parameter. . . . . . . . . . . . . . . . . 71

12.6 The energy, in a.u. for the NpO3+2 (a) and UO2+

2 (b) compoundsrelative to the An-O distance. . . . . . . . . . . . . . . . . . . . . . . 73

List of Tables

8.1 Geometry information for the uranyl(VI) in the ground and lumi-nescent states of uranyl(V) obtained at the B3LYP and TD-B3LYPlevel. All distances in Ångström and n is number of coordinated watermolecules in the sphere. . . . . . . . . . . . . . . . . . . . . . . . . . 39

8.2 Reaction energies in kJ mol"1 for the associative and dissociativereactions for uranyl(VI) in the electronic ground state. . . . . . . . . 40

8.3 Reaction Energies in kJ mol"1 for the associative reaction and disso-ciative reaction computed for uranyl(VI) in the electronic first excitedstate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8.4 The adiabatic transitions, in cm"1 computed for uranyl(VI). For allfunctionals, except the U-B3LYP, the TD-DFT results have been usedin this table for the excited state. . . . . . . . . . . . . . . . . . . . 43

9.1 Energies in kJ mol"1 relative to the structure with single hydro-gen bond(s) (1H) obtained at the MP2 level for the reactant andD-intermediate structure in the six-water model. . . . . . . . . . . . 47

9.2 Reaction energies in kJ mol"1 at the MP2 level for the six- and ten-water models in gas phase. . . . . . . . . . . . . . . . . . . . . . . . . 48

9.3 Geometry information for the uranyl(VI) in a six- and ten-watermodel obtained at the B3LYP level. All distances in Ångström and nis the number of coordinated water molecules in the sphere. . . . . . 49

10.1 Solvation e!ects in kJ mol"1 at the MP2 and B3LYP level of theoryobtained with UA and IC models for the water exchange reaction foruranyl(VI) in the electronic ground state, using the six- and ten-watermodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

10.2 Reaction enthalpies in kJ mol"1 computed at the MP2 level for thesix- and ten-water models in gas phase and in solvation using CPCMwith both UA and IC topological models. . . . . . . . . . . . . . . . 53

10.3 Enthalpy correction, #H298, in kJ mol"1, using gas phase geometriesin a continuum model with UA and IC. . . . . . . . . . . . . . . . . 54

10.4 The radii parameters in Å for the UA topological models. . . . . . . 5610.5 Solvation e!ects calculated at B3LYP level of theory in kJ mol"1 for

the water exchange reaction of aqueous uranyl(VI) with di!erent UAmodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

10.6 The radii parameters for the IC topological models, in Å . . . . . . 57

79

80 LIST OF TABLES

10.7 The solvation e!ects in kJ mol"1 calculated at B3LYP level of theoryusing di!erent IC models for the water exchange reaction of aqueousuranyl(VI) in the electronic ground state. . . . . . . . . . . . . . . . 57

10.8 Geometry information for the uranyl(V) in the electronic groundstate. Optimized at the U-B3LYP level using an UA solvation model.All distances in Ångström and n is number of coordinated watermolecules in the sphere. . . . . . . . . . . . . . . . . . . . . . . . . . 58

10.9 Reaction energies in kJ mol"1 for uranyl(V) in a six-water model atthe CASPT2 level. All structures were optimized at the U-B3LYPlevel using the UA solvation model in Gaussian 03. . . . . . . . . . . 59

10.10Solvation e!ects in kJ mol"1 for the water exchange reaction for ura-nium complexes obtained at the B3LYP (and U-B3LYP) level usingUA and IC models. The complexes are UO2+

2 (H2O)6, the first excitedstate U(O2+

2 (H2O)6, UO+2 (H2O)6 and UN2(H2O)6. . . . . . . . . . 59

10.11Solvation e!ects in kJ mol"1 at the B3LYP level for uranyl(VI) in thesix-water model, the neutral water cage and the systems where theuranyl(VI) ion is exchanged with H+ and Be2+. . . . . . . . . . . . 60

10.12B3LYP solvation e!ects in kJ mol"1 for uranyl and Th4+ in the ten-water model, the plain water cage and water cage with H+ and Be2+in the position of the uranyl and C4+ in the position of Th4+. . . . . 61

11.1 Geometry information for the uranyl(VI) in the ground and lumi-nescent state and uranyl(V) obtained at the B3LYP and U-B3LYPlevel. All distances in Ångström and n is number of coordinated watermolecules in the sphere. . . . . . . . . . . . . . . . . . . . . . . . . . 65

11.2 Reaction enthalpy in kJ mol"1 at the MP2/CASPT2 spin-free (SF)and spin-orbit (SO) level, for the water exchange mechanism for UO2+

2 (H2O)6,the luminescent state U(O2+

2 (H2O)6 and UO+2 (H2O)6. The solvation

e!ects obtained with the UA model . . . . . . . . . . . . . . . . . . . 66

12.1 Equilibrium geometries for the ground state of ThO2, PaO+2 , UO2+

2 ,UN2 and NpO3+

2 at the 12/12 MCSCF-srPBE level together withresults at the CCSD(T) and B3LYP levels. For the bent structures,the angle (in degrees) is given in parentheses. . . . . . . . . . . . . . 72

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Acknowledgments

First and foremost I would like to thank my advisor Professor Ulf Wahlgren, hisguidance, help and awesome knowledge of quantum chemistry has been invaluableduring these years. Ulf has always been available for questions and good adviceregardless of subject, something that I really appreciate. It is also a pleasure to thankDr Valérie Vallet and Dr Bernd Schimmelpfennig that both clearly have contributedto my increased understanding of the theoretical actinide chemistry, their adviceand willingness to answer questions has been of great importance during my timeas a Ph.D. student. I have also had the great privilege to take part of ProfessorIngmar Grenthe vast experience in experimental actinide chemistry. I would alsolike to thank Dr Jean-Pierre Flament for taking his time to answer my questions.

I am indebted to Dr Fernando Ruipérez, Dr Michael Leetmaa, Thor Wikfeldt,Dr Emmanuel Fromager and Dr Mathias Ljungberg, who have taken the time tocomment on parts of this thesis. Also a special thanks to Ann-Britt Hellmark whoproofread all the text, all remaining errors are my own.

I would like to mention all my colleagues in the quantum chemistry group at thephysics department that I have encountered over the years. I have had the privilegeto first share o"ce building with Professor Per Siegbahn and Professor MargaretaBlomberg’s research group and later with Professor Lars Pettersson’s group. Theinteresting discussions and pleasant atmosphere have made me appreciated theseyears, not only from a research perspective but also socially.

Finally, I would like thanking all the friends I met here, no one mentioned andnone forgotten.

Documents

Theoretical Actinide Chemistry - DiVA portal398617/FULLTEXT02.pdf · CASSI Complete Active Space State Interaction CC Coupled Cluster CCSD Coupled Cluster Singles and Doubles CCSD(T)