Application and development ofquantum chemical methods. Densityfunctional theory and valence bond

theory

Fuming Ying

Theoretical ChemistrySchool of Biotechnology

Royal Institute of TechnologyStockholm 2010

Copyright c©2010 by Fuming YingTRITA-BIO Report 2010:13ISSN 1654-2312ISBN 978-91-7415-633-1

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Printed by US-AB, Stockholm 2010

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AbstractThis thesis deals with two subdiciplines of quantum chemistry. One is the

most used electronic structure method today, density functional theory (DFT),and the other one of the least used electronic structure methods, valence bondtheory (VB). The work on DFT is based on previous developments in the groupin density functional response theory and involves studies of hyperfine cou-pling constants which are measured in electron paramagnetic resonance exper-iments. The method employed is a combination of a restricted-unrestricted ap-proaches which allows for adequate description of spin polarization withoutspin contamination, and spin-orbit corrections to account for heavy atom ef-fects using degenerate perturbation theory. The work on valence bond theory isa new theoretical approach to higher-order derivatives. The orbital derivativesare complicated by the fact that the wave functions are constructed from de-terminants of non-orthogonal orbitals. An approach based on non-orthogonalsecond-quantization in bi-orthogonal basis sets leads to straightforward deriva-tions without explicit references to overlap matrices. These formulas are rele-vant for future applications in time-dependent valence bond theory.

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List of publications

Paper I. Restricted-unrestricted density functional theory for hyperfine coupling con-stants: vanadium complexes Xing Chen, Fuming Ying, Zilvinas Rinkevicius, OlavVahtras, Hans Agren, Wei Wu and Zexing Cao. submitted

Paper II. An analytical valence bondHessian, by means of second quantization in biorthonor-mal basis sets Fuming Ying, Olav Vahtras, Zilvinas Rinkevicius and Wei Wu.Manuscript

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My contribution

• I carried out calculations in paper I and participated in analysis and au-thoring

• I derived the Hessian formulas in paper II and participated in authoring

• I participated in the preparations of the manuscripts for all papers

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Acknowledgments

I would thank Prof. Yi Luo, Hans Agren and Prof Wei Wu in Xiamen Uni-versity who gave me the opportunity to study in KTH.

I appreciate my supervisor Olav Vahtras andmy tutor Zilvinas Rinkeviciousfor their guidance, advice in each project. I still remember the situation whenZilvinas and I discussed the details of DALTON program, just as happened yes-terday.

I should also give my gratitude to my colaborator Xing Chen for her helpand discussion during the work.

I may also say ”Thank you” to Prof. Magareta Bromberg, Aato Laaksonenand Alexander Lyubartsev in SU for the courses and Prof. Poul Jøgensen, JeppeOlsen and Trygve helgaker for their teaching Quantum Chemistry in the sum-mer school.

Thanks to my friends in Department of Theoretical Chemistry, Xiaofei Li,Keyan Lian, Hao Ren, et al. for their kind help in my life in Stockholm.

Contents

1 Introduction 1

2 Density functional theory 32.1 Density optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Variational perturbation theory . . . . . . . . . . . . . . . . . . . . 4

2.2.1 The restricted-unrestricted method . . . . . . . . . . . . . . 5

3 Valence bond theory 73.1 VB Wave Function and Hamiltonian . . . . . . . . . . . . . . . . . 8

3.1.1 VB Wave Function . . . . . . . . . . . . . . . . . . . . . . . 83.1.2 Constructing VB Hamiltonian with Determinant Expansion 9

3.2 Different types of VB Orbitals . . . . . . . . . . . . . . . . . . . . . 93.2.1 Localized VB Orbitals . . . . . . . . . . . . . . . . . . . . . 93.2.2 Semi-Localized VB Orbitals . . . . . . . . . . . . . . . . . . 10

3.3 VB Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3.1 Generalized VB . . . . . . . . . . . . . . . . . . . . . . . . . 113.3.2 CASVB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.3 VB Self-Consistent Field . . . . . . . . . . . . . . . . . . . . 123.3.4 Breathing Orbital VB . . . . . . . . . . . . . . . . . . . . . . 133.3.5 VB Configuration-Interaction . . . . . . . . . . . . . . . . . 143.3.6 VBPCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Various VB Programs . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.1 TURTLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.2 VB2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4.3 XMVB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Summary of Included Papers 174.1 Restricted-unrestricted density functional theory . . . . . . . . . . 174.2 The Valence bond Hessian . . . . . . . . . . . . . . . . . . . . . . . 18

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Chapter 1

Introduction

The work of this thesis deals with two methods of electronic structure theory.The first, density functional theory (DFT), is by far the most important methodtoday with its competitive combination of speed and accuracy. The numberof articles based on DFT increase with an exponential rate making a completeoverview if not impossible but certainly beyond the scope of this thesis. Thesecond, valence bond (VB) theory, was the most important theory in the infancyof this science.

In chapter 2 we provide the theoretical background of DFT for the appli-cations used in paper I. It is customary to introduce an expose of DFT to twopioneering articles, the first by Hohenberg and Kohn[1] who established themathematical foundation of a ground state theory based on the electron densityas a fundamental variable, and the other by Kohn and Sham[2] who introduceda practical single-determinant theory for practical calculations. It was becauseof this work, which started a tremendous development in computational chem-istry, that Walther Kohn was awarded the Nobel Prize in Chemistry in 1998.

In chapter 3 we give a background of valence bond theory. It dates back toG. N. Lewis[3] who introduced concepts such as electron pair bonding and theoctet rule, which has influenced the way of thinking for generations of chemists.The theorywhich gradually lost importancewhen the rivalingmolecular orbital(MO) theory grew in popularity. The latter had its origin in spectroscopy andallowed for efficient computer implementations with predictive power. Recentdecades have witnessed a revival of valence bond theory, with new algorithmsand new computer implementation. Paper II. is a contribution to this develop-ment.

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2 CHAPTER 1. INTRODUCTION

Chapter 2

Density functional theory

The implications of the fundamental theorems of density functional theory areenormous[1]. The ground state energy of a many-electron system is a uniquefunctional of the electron density and that the part of the functional which in-dependent of the external field, the Hohenberg-Kohn functional, is a universalfunctional independent of the number of electrons. This implies that any many-electron problem can be reduced to an equation in three variables, the coordi-nates of a point in space. The problem is that the equation, or the functional, isnot known which has prompted a huge development of approximate function-als which have been parameterized to reproduce a certain class of experimentalresults. The success of DFT has been due to basis set expansions and the formu-lation of DFT as a single determinant theory[2]. In practice DFT can be viewedas a Hartree-Fock theory with a semi-empirical element which includes electroncorrelation.

In this chapter we review the aspects of DFT that are relevant for paper I;the optimization procedure, the restricted-unrestricted method and perturba-tion theory with application to hyperfine coupling tensors.

2.1 Density optimization

The density is optimized for a given energy functional E[ρ].

δE[ρ] = 0 (2.1)

In the most general case the optimization is carried out with the auxiliary con-dition that the total number of particles is constant

∫

dV ρ(~r) = N (2.2)

which leads to a Lagrangian multiplier which is interpreted as the chemicalpotential[1]. If the density is constructed from a determinantal wave function

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4 CHAPTER 2. DENSITY FUNCTIONAL THEORY

the condition 2.2 is automatically fulfilled. The variation of a functional (func-tion of a function) gives

δE[ρ(~r)] =

∫

dVδE

δρ(~r)δρ(~r) (2.3)

In practical calculations the density is parameterized for some finite set of pa-rameters λ = (λ1, λ2 . . .) such that

δρλ(~r) =∂ρλ(~r)

∂λδλ (2.4)

Salek et al.[4] formulated the optimization in second quantization as follows;the density is the expectation value of a density operator

ρλ(~r) = 〈λ|ρ(~r)|λ〉 (2.5)

where the density operator projected on a basis set φp is

ρ(~r) =∑

pq

φp(~r)∗φq(~r)a

†paq (2.6)

When the parameterization is of exponential unitary form,

|λ〉 = e−λpqa†paq |0〉 (2.7)

i.e. the with parameters describing a change from a reference set of orbitals, thetransformation is expanded around λ = 0 and the density variation a becomesa commutator

∂ρ

∂λpq= 〈λ|[a†paq, ρ(~r)|λ〉 (2.8)

Defining a Fock operator

F =

∫

dV ρ(~r)δE

δρ(~r)(2.9)

the energy optimization reduces to

〈0|[a†paq, F ]|0〉 = 0 (2.10)

which are equivalent to the Kohn-Sham equations[2].

2.2 Variational perturbation theory

If a perturbation V is introduced in the Hamiltonian with a field strength x,

H = H0 + xV (2.11)

2.2. VARIATIONAL PERTURBATION THEORY 5

the optimized parameters will be a function of the field strength λ(x). The en-ergy change due to this perturbation can be identified from the power expan-sion

E[ρλ(x); x] = E[ρλ(0); 0] + xdE

dx

∣

∣

∣

∣

x=0λ=λ(0)

(2.12)

and in a strictly variational theory we have

dE

dx

∣

∣

∣

∣

x=0λ=λ(0)

= 〈0|V |0〉 (2.13)

2.2.1 The restricted-unrestricted method

The restricted-unrestricted method were developed by Fernandez et al.[5] andlater generalized for DFT by Rinkevicius et al.[6] for describing spin polariza-tion in spin-restricted methods. In spin-restricted methods, the alpha- and betaspin-orbitals always have the same spatial parts. For the calculation of a prop-erty such as the hyperfine coupling constant whose main component, the Fermicontact contribution, is a measure of the spin density at the nucleus this is prob-lematic. An approximate wave function models such as MCSCF does not havesufficient flexibility at the nucleus because the core orbitals are normally inac-tive, and to allow for correlation in the core orbitals is computationally veryexpensive. On the other hand an unrestricted which does have this flexibilityhave other problems such as spin contamination in the wave function

The restricted-unrestricted theorywas developed to avoid the problemswithspin contamination by optimizing the wave function in a restricted framework(singlet optimization), but by including extra parameters (triplet optimization)in the perturbation calculation. What this means is that the gradient with re-spect to some of the parameters is non-zero

∂E

∂λ= τ 6= 0 (2.14)

but we introduce Lagrangian multipliers so that the same relation holds in thepresence of the perturbation, i.e. for all x. A direct way of obtaining the finalrelations is to differentiate Eq. (2.14) we have

d

dx

∂E

∂λ=

∂2E

∂λ∂x+∂2E

∂λ2∂λ

∂x= 0 (2.15)

which defined a linear system of equations for the wave function response. Theenergy change then consists of two terms; the first as an expectation value ofthe perturbation and the second from the response in the wave function

dE

dx=∂E

∂x+∂E

∂λ

∂λ

∂x(2.16)

6 CHAPTER 2. DENSITY FUNCTIONAL THEORY

The combination of Eqs.(2.15,2.16) gives

dE

dx=∂E

∂x− ∂E

∂λ

(

∂2E

∂λ2

)−1∂2E

∂λ∂x(2.17)

Chapter 3

Valence bond theory

Valence bond(VB) theory is an ancient as well as modern theory in quantumchemistry. It was first introduced by Heitler and London[10] in 1927. Lateron, Slater[11] and Pauling[12] developed and added several concepts into thattreatment in order to generalize it for polyatomic molecules. The theory thatwas developed has the name ”valence bond theory”.

The concept of electron-pairing by Lewis was used in dealing with the H2

molecule which made VB very easy to understand by chemists. To explainthe geometry (bonding character) in polyatomic systems, another concept ”hy-bridization” and the corresponding ”hybrid orbital” were introduced by Paul-ing. A hybrid orbital is a kind of atomic orbital (HAO) which can be obtained bycombining original atomic orbitals to get the maximum bonding ability. Thusthe original atomic orbitals can be treated as a special type of HAOs. In thefollowing, we will use HAO to denote all kinds of atomic orbitals. Perhaps themost important concept in VBwas resonance. It means that the ”real” electronicstructure of a molecule can be explained as the linear combination between a setof ”extreme” structures. For the sake of resonance, VB is a multi-configurationalmethod. Some molecular properties like conjugation can be easily explained byresonance. A derivative of resonance is the resonance energy. It is defined asthe difference between the energies of ”real” structure and themost stable struc-ture(s) and can be a measure of aromaticity.

A severe technical disadvantagewas discovered soon. The use of non-orthogonallocal bonding orbitals made quantitative calculations impossible at that time sothat a lot of simplification had to be made. Some of them may lead to wrongconclusions. On the other hand, molecular orbital(MO) theorymade implemen-tation and calculation much easier due to its delocalized orthogonal orbitals.From the late of 1960s, there was a log of development and implementations inMO theory, while the role of VB diminished.

Despite of such a disadvantage, VB was always preferred by chemists inqualitative analyses. Furthermore, the increasing of computer power made it

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8 CHAPTER 3. VALENCE BOND THEORY

possible to perform ab initio VB calculations. Thus, from the late of 1970s, VBwas reborn from the ashes.

VB developments, i.e. generalized Slater-Condon rules[13], spin-free VBmethod[14, 15] and the paired-permanent-determinant[16, 17] approach, havebeen published to reduce computational costs so that pure ab initio VB may beperformed faster and be a practical method. Because of these developments,the cost of ab initio VB calculation now can be reduced to O(N4) where N is thenumber of electrons. Furthermore, a lot of VB methods have been developed tohandle different situations, including electron correlation, but which maintainsthe main features and analytical power of VB theory. [18, 19, 20, 21, 22, 23, 24,25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36].

To display concept of resonance, tools like VB wave function, VB orbitals,and VB computational methods are essential. In the following sections, suchtools will be revealed.

3.1 VB Wave Function and Hamiltonian

3.1.1 VB Wave Function

A VB wave function with Rumer spin function stands for a VB structure. It iscalled HLSP(Heitler-London-Slater-Pauling) wave function. It can be expressedas

ΦK = AΩKΘK (3.1)

where A is the antisymmetric operator and ΩK and ΘK are the spacial and spinfunction respectively. Spacial function has the form of the multiplication of aset of orbitals

ΩK = φ1(1)φ2(2) · · ·φN(N) (3.2)

and spin function has the form

ΘK =∏

i,j

1√2[α(i)β(j)− α(j)β(i)]

∏

l

α(l) (3.3)

where i, j run over all paired electrons and l runs over all unpaired electrons.Real wave function can be expressed as the linear combination of a set of

HLSP wave function

ΨV B =∑

K

CKΦK (3.4)

and weight of ΦK can be obtained by Coulson-Chirgwin’s formula[37]

WK =∑

L

CKCLMKL (3.5)

3.2. DIFFERENT TYPES OF VB ORBITALS 9

where CK is the combination coefficient of HLSP wave function ΦK andMKL isthe overlap between ΦK and ΦL. The linear independent HLSP that are used tobuild real wave function can be obtained with Rumer rules or standard Youngtableau.

3.1.2 Constructing VB Hamiltonian with Determinant Expan-

sion

A VB structure in the form of Eqn (3.1) can be expanded into a set of determi-nants

ΦK =2m∑

i=1

CiKDiK (3.6)

where m is the number of covalent bonds in the VB structure ,DiK is the deter-minant belonging to the structure and CiK is the corresponding coefficient. Forexample, normalized wave function of covalent bond A-B can be expressed asthe determinant expansion

Φcov =1√2

(

|φaφb| − |φaφb|)

(3.7)

The Hamiltonian of VB wave function is constructed in terms of VB struc-tures. Thus, Hamiltonian matrix elements 〈ΦK |H|ΦL〉 can be expanded into a

set of Hamiltonian elements 〈DiK |H|DjL〉

〈ΦK |H|ΦL〉 =2m∑

i=1

2n∑

j=1

CiKCjL〈DiK |H|DjL〉 (3.8)

and overlap can be generated in the same way.

3.2 Different types of VB Orbitals

VB orbitals are essential to construct Hamiltonian and describe VB wave func-tions. Although all ab initio VB methods deal with nonorthogonal orbitals, thedegree of localization various from on the another. There are 2 kinds of VB or-bitals in modern VB theory, named localized and semi-localized orbitals. Bothare used for different purposes.

3.2.1 Localized VB Orbitals

A localized VB orbital is simply an HAOs. ”Pure” VB methods are based onsuch kind of VB orbital. In a system with one single bond A-B, the VB wave

10 CHAPTER 3. VALENCE BOND THEORY

function for this bond can be expressed as

Ψ = c1(

|φaφb| − |φaφb|)

+ c2|φaφa|+ c3|φbφb| (3.9)

where ci is the coefficient of ith VB structure and φa and φb are HAOs located on

atom A and B respectively. The first term corresponds to a covalent structure ofA-B bond, and the last two terms correspond to ionic terms. The advantage oflocalized VB orbital is that a clear distinction between covalent and ionic struc-ture of a bond can be provided.

A bond in a polyatomic molecule, i.e. one C-H bond in methane moleculeCH4, may be dealt in the following way: the localized orbital on H keeps thetype of HAO and the orbital on C becomes fragment orbital(FO) which meansthat orbital on C bonding with H may have a slight delocalization to the otherH atoms of the remaining CH3 group.

3.2.2 Semi-Localized VB Orbitals

When a many bonds need to be considered, the number of VB structures interms of local VB orbitals increases exponentially and becomes quite hard to re-solve. Hence, some simplification is needed to reduce the number of VB struc-tures. Two types of semi-localized orbitals are used in VB calculations. Thegeneral form was suggested by Coulson and Fischer[38] as

ψa = φa + λφb (3.10)

andψb = φb + λ′φa (3.11)

to represent the A-B bond where λ and λ′ are coefficients and can be a measure-ment of delocalization. In this case, wave function can be expressed as

Ψ = |ψaψb| − |ψaψb|= (1 + λλ′)

(

|φaφb| − |φaφb|)

+ λ′|φaphia|+ λ|φbφb|(3.12)

Thus, a VB wave function with 3 VB structures can be reduced to a wave func-tion in covalent form containing covalent and ionic structures.

A VB orbital that is allowed to delocalize freely to all other parts of themolecule if no other restriction is added onto it, is called overlap-enhanced or-bital(OEO). The wave function can be constructed with several VB structures(inform of (3.12), known as perfect-pairing).

If some restriction is added to the orbital that a VB orbital can be only de-localized onto the atom with which the orbital is bonded in the VB structurein consideration, OEO becomes bond-distorted orbital(BDO). For instance, A

3.3. VB METHODS 11

semi-localized orbital for C-C bond in H3C-CH3 with OEO can be an orbital onone CH3 group and delocalizing to whole part of the other CH3 group, while aBDO is an orbital that can only delocalize to the other C atom.

The advantage of semi-localized VB orbitals is that it may reduce a largenumber of VB structures and thenmake the wave functionmuch simpler. Whenall orbitals become delocalized to the whole system, the wave function will be-come an MO wave function. Semi-localized orbital make a link between VBand MO theory. The disadvantage of semi-localized orbitals, as we have shownabove, is that the distinction of covalent and ionic pictures is impossible.

3.3 VB Methods

3.3.1 Generalized VB

GVB[18, 19, 20] was introduced byWilliam A. Goddard III and was the first im-portant case of Coulson-Fischer’s formula. The restricted form of GVB methodis referred as GVB-SOPP, which is often used by GVB. GVB-SOPP has two sim-plifications. The first is so called ”perfect pairing”(PP) approximation that aGVBwave function can be expressed as a single determinant with a set of ”gem-inal” two-electron functions each corresponding to a chemical bond or a lonepair

ΨGV B = | (|ϕ1aϕ1b| − |ϕ1aϕ1b|) (|ϕ2aϕ2b| − |ϕ2aϕ2b|) · · · (|ϕNaϕNb| − |ϕNaϕNb|)(3.13)

where ϕia and ϕib are OEOs for ith chemical bond or lone pair.

The second one in GVB is the strong orthogonality for computational con-venience. In a strong orthogonality framework, the orbitals are nonorthogonalif in the same geminal function

〈ϕia|ϕib〉 6= 0 (3.14)

and orthogonal if they are in different geminal functions

〈ϕi|ϕj〉 = 0 (3.15)

GVB can also be represented in another form with natural orbitals as

|ϕiaϕib| − |ϕiaϕib| = Ci|φiφi|+ C∗i |φ∗

iφ∗i | (3.16)

where φi and φ∗i are natural orbitals respect to bonding and antibonding MOs

and Ci and C∗i are corresponding coefficients. The transformation between

forms shown in Eqn gvb) and 3.16 is

ϕia =φi + λφ∗

i√1 + λ2

ϕib =φi − λφ∗

i√1 + λ2

λ2 = −C∗i

Ci

(3.17)

12 CHAPTER 3. VALENCE BOND THEORY

3.3.2 CASVB

The underlying idea of CASVB is quite simple. A spin-coupled VB wave func-tion should be close to CASSCF wave function with the same number of elec-trons and the same active space. Thus, a transformation from CASSCF to VBwave function is reasonable. Since CASSCF is easily obtained, this idea pro-vides a way to generate a VB wave function.

Thorsteinsson and D.L. Cooper[21, 22, 23] suggested a way to generate theVB wave function from CASSCF. After the transformation, the final VB wavefunction is restrictly equivalent to starting CASSCF. The way can be expressedas following formula

ΨCAS = SΨCASV B + (1− S2)ΨCASV B⊥ (3.18)

whereΨCAS andΨCASV B are original CASSCF and final CASVB wave functionsrespectively,ΨCASV B

⊥ is the perpendicular complement of CASVBwave functionand

S = 〈ΨCAS|ΨCASV B〉 (3.19)

is the overlap between CASSCF and CASVB wave function. One way to getCASVB wave function is to maximize the overlap S and another way is to mini-mize the energy of ΨCASV B . It’s a simple way to implement but the orbitals willmore or less keep the MO type, similar to localized molecular orbitals(LMOs).

Different fromThorsteinsson’smethod, Hirao[24, 25] developed anotherwayto get CASVBwave function with nonorthogonal VB orbitals. Since LMOs haveobvious delocalization tails and may enlarge the ionic weights and mix cova-lent and ionic pictures, nonorthogonal orbitals will give us more reasonable VBwave function and clearer resonance picture.

3.3.3 VB Self-Consistent Field

VB self-consistent Field(VBSCF) was first introduced by van Lenthe[26, 27, 28]and has been one of the fundamental computational methods of VB theory.Hamiltonian and overlap matrix are generated in terms of VB structures

HKL = 〈ϕK |H|ϕL〉 (3.20)

MKL = 〈ϕK |ϕL〉 (3.21)

After getting hamiltonian and overlap,

HC = EMC (3.22)

3.3. VB METHODS 13

we will get the coefficient CK of VB wave function ϕK and the energy E. Min-imize the energy with optimizing orbital and CK simultaneously in a SCF pro-cedure and we will get the ground state of the system.

VBSCF accepts VB orbitals in various types such as OEOs, BDOs and HAOs.There is no constraint of orthogonality for orbitals and thus the clear picture ofbonding may be kept. Particularly, when a set of fully delocalized orthogonalorbital is used with a wave function in single determinant, VBSCF will go tothe form of Hartree-Fock (HF) and have the same energy. The wave functionmay differ from the HFwave function only by a unitary transformation withoutchanging the energy.

3.3.4 Breathing Orbital VB

Breathing orbital VB(BOVB) theory was developed by Sason Shaik and WeiWu[29, 30, 31, 32, 33]. In VBSCF, different VB structures share the same setof VB orbitals. However, same orbital in different VB structures is treated as”different” orbitals in BOVB and optimized separately. Thus, BOVB will gen-erate more orbitals than VBSCF if there are more than one VB structure for thesystem. If there is only one VB structure, nothing will be changed and BOVB isequivalent to VBSCF.

F F F F F F+ C2C1 C3+

F FC1 + C2 + C3F F FF

Figure 3.1: Pictures of VBSCF(above) and BOVB(below) results

Fig 3.1 shows results of VBSCF and BOVB of F2 molecule with localized VBorbitals. As we can see, different VB structures share the same set of VB orbitalsin VBSCF, no matter whether it is covalent or ionic structure. In BOVB, sameorbital is treated as different. orbitals in cation atom become more contractedand orbitals in anion atom become more diffused. Thus, part of dynamical cor-

14 CHAPTER 3. VALENCE BOND THEORY

relation is included in BOVB result.

3.3.5 VB Configuration-Interaction

BOVB improves computational results because it provides some dynamical cor-relation. The VBConfiguration-Interaction(VBCI)method[34, 35] is a VBmethodbased on CI model aiming to provide better results and keep the clear picture ofVBSCF. In a VBCI calculation, the orbitals from aVBSCF are used but not furhteroptimized. Virtual orbitals and structures will be constructed to describe exci-tations.

The VBCI procedure contains several steps. First, a VBSCF wave functionshould be obtained by a VBSCF calculation. Although VBSCF orbitals can bestrictly localized, a unitary transformation is still allowed without changing theenergy. Thus, an extra Boys localization[39] is needed to gain a unique VBSCFwave function for a zeroth-order wave function

Ψ0 =∑

K

C0KΦ

0K (3.23)

Second, VB virtual orbitals and VB virtual structures will be constructed.Virtual orbitals ϕi

a are restricted to the same block of the corresponding occu-pied orbital ϕa

ϕia =

∑

µ

χµciaµ (3.24)

where ciaµ is the LCAO coefficient of basis function χµ in orbital ϕia. To fulfill the

condition that N basis functions can only construct N linearly independent or-bitals, occupied orbitals are restricted in the different blocks, which means thatno basis function is shared in more than 1 VB orbital. Orbitals are orthogonal inthe same block, but not in the different blocks

〈ϕa|ϕia〉 = 0 (3.25)

〈ϕia|ϕj

a〉 = 0 (3.26)

〈ϕib|ϕj

a〉 6= 0 (3.27)

VB virtual structures can be obtained by replacing some occupied orbitalwith a virtual one, meaning the excitation. Excitation shall be done accordingto the rule that one occupied orbital ϕa can be replaced only by the virtual onesϕi

a that in the same block. If only one electron is excited to the virtual orbitalin a structure, a VBCIS calculation will be obtained. If at most two electrons canbe excited, a VBCISD calculation will be obtained. A VBCI wave function thenis generated containing all VBSCF and excited structures.

ΨV BCI =∑

K

CV BCIK ΦV BCI

K (3.28)

3.4. VARIOUS VB PROGRAMS 15

where CV BCIK is coefficient of ΦV BCI

K that should be solved later.

Finally, a secular equation is solved to get the coefficientsCV BCIK . Theweights

of the VBCI structures are obtained in the same way as VBSCF.

3.3.6 VBPCM

All the above VB methods are applicable only in the gas phase and cannot beused to solve chemical problems in solution, i.e. SN2 reactions. In order to havea VB theory for problems in solution a combination with the Polarizable Con-tinuumModel (PCM)[40] has been developed.

VBPCM[36] is the combined method of VB theory and the PCM model. Inthe PCMmodel, the solvent is treated as a homogeneous medium characterizedby a dielectric constant and can be polarized by the charge of the solute. Theinteraction between solute and solvent is taken as an effective potential in theHamiltonian, and optimized in a self-consistent reaction field procedure.

VBPCM is amethod based onVBSCF. Thus, the representation of the VBPCMwave function is the same as VBSCF. However, the Hamiltonian now is nolonger the same as the VBSCFHamiltonian and added by a effective one-electronHamiltonian to describe the interaction between solvent and solute

H = H0 +Vs (3.29)

where H0 is the Hamiltonian in gas phase for a bare molecule and Vs is the

extra potential in solution.

3.4 Various VB Programs

3.4.1 TURTLE

TURTLE(refs needed) is a VB software developed by van Lenthe et al. focusingon ab initio multi-structure VB calculation. Currently, it is available for non-orthogonal CI calculation and non-orthogonal MCSCF calculation with simul-taneous optimization of VB coefficients and orbitals. Different types of VB or-bitals, such as OEOs, BDOs and HAOs are allowed to construct VB structuresfor VBSCF, BOVB, SCVB, BLW, etc. Optimization of wave function and geome-try are both available with analytical gradients. The message passing interface(MPI) has been used for the parallelization.

16 CHAPTER 3. VALENCE BOND THEORY

3.4.2 VB2000

VB2000 is a software based on ”Algebrant Algorithm”. Group function theoryis also implemented into VB2000 so that a large molecule can be described asphysically identifiable electron groups. Non-orthogonal CI, multi-structure VB,GVB, SCVB and CASVB developed by Hirao.

3.4.3 XMVB

XMVB is developed by Prof. Wei Wu etc. based on spin-free VB theory andconventional Slater determinant expansion methods. This program is enabledfor different kinds of VB theories such as VBSCF, BOVB, VBCI, VBPCM, GVB,SCVB, BLW, etc. with OEOs, BDOs, or HAOs. VB orbitals and VB coefficientscan be optimized simultaneously. XMVB program can be obtained either as astand-alone program or a module in GAMESS-US. Parallel version of XMVBwith MPI is also available and can be obtained only as a stand-alone program.

Chapter 4

Summary of Included Papers

4.1 Restricted-unrestricted density functional theory

In this study we apply restricted-unrestricted density functional theory devel-oped in our group[6] for the calculation of hyperfine coupling constants of vana-dium complexes. These parametersmeasure the spin density at a given nucleus.A spin restricted method has the advantage that it is free from spin contamina-tion, but the disadvantage that it is not sufficiently flexible at the core to describethe spin polarization that occurs in an electron paramagnetic resonance exper-iment with magnetic nuclei. On the other hand an unrestricted method hasopposite features, it is flexible enough at the nucleus but spin contamination isknown to give poor results in special cases. The philosophy of the combinationof these methods, the restricted-unrestricted model, is to merge the best of bothworlds. The wave function is optimized with spin restricted method while thespin-polarization is introduced as a perturbation. The price to pay is that the cal-culations are more complicated as we need to solve response equations of oneorder higher than would have been the case in a purely unrestricted scheme.This idea was first developed and applied in multi-configuration self-consistentfield theory by Fernandez et al. [5]

Furthermore, spin-orbit effects were included using a degenerate perturba-tion theory, developed in ref[9]. This gives a one-to-one relationship betweenexperimental parameters and calculated quantities at a level which is goes be-yond the textbook formulations.

The results obtained indicate in general a good agreement with experimentsfor vanadocenes(IV) complexes while for vanadium oxide complexes the devia-tions are slightly higher. The accuracy of DFT calculations depend in general onthe choice of functional, here the BHLYP give the best results while the popularB3LYP fail in the prediction of hyperfine coupling constants. This is believed tobe due to insufficient Hartree-Fock exchange.

17

18 CHAPTER 4. SUMMARY OF INCLUDED PAPERS

4.2 The Valence bond Hessian

The orbital derivatives of valence bond theory are complicated by the fact thatthe wave function is constructed from determinants of non-orthogonal orbitals.To date there is no computer implementation of an analytical valence bondHes-sian which is the key quantity necessary for applications in time-dependentvalence bond theory. In this paper outline a theoretical framework based onsecond quantization in non-orthogonal orbitals and a theory of non-orthogonaldeterminants that we credit to P-O Lowdin[41]

The main elements are the dual sets of orbitals bi-orthogonal to each otherwhich are connected by a transformation of the overlap matrix

a†p = Spqaq† (4.1)

where the notation with subscripts for one set and superscripts for the dual setintroduces tremendous simplifications. The algebraic rules of differentiationand commutation allows one to identify the derivative of a determinant withrespect to a molecular orbital coefficient in terms of excitation operator

∂

∂Cµm

|Ψ〉 = a†µam|Ψ〉 (4.2)

This gives that the derivative of an inner product between two determinantsis an element of a transition density matrix, and all energy derivatives can beexpressed in terms of integrals and transition density matrices. It is shown thatgiven a Hamiltonian in an arbitrary basis

H = hpqap†aq +

1

2gpqrsa

p†ar†asaq (4.3)

(summation implied by repeated subscript-superscript pair) the derivative of aHamiltonian matrix element is given by

∂

∂Cµm

〈K|H|L〉 = 2〈K|L〉(

D(KL)mρF (KL)ξρ∆(KL)ξµ +H(KL)D(KL)mµ

)

(4.4)

where D(KL) is the intermediately normalized transition density matrix,

D(KL)pq =〈K|ap†aq|L〉

〈K|L〉 (4.5)

∆ given by∆(KL)pm = δpq −D(KL)pq (4.6)

and F the Fock operator

F (KL)pq = hpq + (gpqrs − gpsrq)D(KL)rs (4.7)

4.2. THE VALENCE BOND HESSIAN 19

The relations thus obtained are transparent and without explicit reference tooverlapmatrices which appear in great number using standardmatrix notation.For the gradient it can also be shown that the derivative reproduces the resultspublished previously by Song et al[42].

This approach is now easily generalized to Hessian evaluations, from iden-tification of the equivalence between a second derivative and a two-particle ex-citation

∂2

∂CµmCν

n

|Ψ〉 = a†µa†νa

nam|Ψ〉 (4.8)

the second derivative of a Hamiltonian matrix element is

∂2

∂CµmCν

n

〈K|H|L〉 = 〈K|H|L〉D(KL)mµ(D(KL) +D(LK))νn

+ [D(KL)F (KL)∆(KL))]mµ (D(KL) +D(LK))nµ

+D(KL)mµ (D(KL)F (KL)∆(KL))nν

+D(KL)mµ (∆(KL)F (KL)D(KL))nν

+ gm nµ ν − gm n

ν µ + g mnµ ν + gnmµν

(4.9)

The non-Fock contributions are effectively integral transformations with transi-tion density matrices, e.g.

gm nµ ν = D(KL)mρD(KL)nτgξρστ∆(KL)ξµ∆(KL)σν (4.10)

This enables us to implement second-order methods, but most importantly theHessian is the key to time-dependent valence bond theory, which until now hasbeen a completely unexplored field of research.

20 CHAPTER 4. SUMMARY OF INCLUDED PAPERS

Bibliography

[1] P. Hohenberg and W. Kohn. Phys. Rev., 136:B864, 1964.

[2] W. Kohn and L. J. Sham. Phys. Rev., 140:A1133, 1965.

[3] G. N. Lewis. The atom and the molecule. J. Am. Chem. Soc., 38:762, 1916.

[4] Pawel Salek, Olav Vahtras, Trygve Helgaker, and Hans Agren. Density-functional theory of linear and nonlinear time-dependent molecular prop-erties . J. Chem. Phys., 117:9630, 2002.

[5] Berta Fernandez, Poul Jørgensen, Jørgen Byberg, Jeppe Olsen, Trygve Hel-gaker, and Hans Jørgen Aa. Jensen. J. Chem. Phys., 97:3412, 1992.

[6] Zilvinas Rinkevicius, Lyudmyla Telyatnyk, Olav Vahtras, andHans Agren.Density functional theory for hyperfine coupling constants with therestricted-unrestricted approach . J. Chem. Phys., 121:7614, 2004.

[7] P.-O. Lowdin and O. Goscinski. Int. J. Quant. Chem., 5:685, 1971.

[8] H. Bolvin. Chem. Phys. Chem., 7:1575, 2006.

[9] Z. Rinkevicius, K.J. de Almeida, C.I. Oprea, O. Vahtras, K. Ruud, andH. Agren. Degenerate perturbation theory for electronic g tensors:leading-order relativistic effects . J. Chem. Theor. Comp., 4:1810, 2008.

[10] W. Heitler and F. London. Z. Phys., 44:455, 1927.

[11] J. C. Slater. Phys. Rev., 35:509, 1930.

[12] L. Pauling. The Nature of the Chemical Bond, 3rd ed. 1960.

[13] Jacob Verbeek and Joop H. Van Lenthe. The generalized Slater-Condonrules. Int. J. Quant. Chem., 40(2):201–210, 1991.

[14] Roy McWeeny. A spin-free form of valence bond theory. Int. J. Quant.Chem., 34(1):25–36, 1988.

21

22 BIBLIOGRAPHY

[15] Xiangzhu Li andQianer Zhang. Bonded tableau unitary group approach tothe many-electron correlation problem. Int. J. Quant. Chem., 36(5):599–632,1989.

[16] Wei Wu, Anan Wu, Yirong Mo, and Qianer Zhang. Paired-permanent ap-proach to valence bond theory. Sci. Chin. B., 39(5):456, 1996.

[17] Wei Wu, Anan Wu, Yirong Mo, Menghai Lin, and Qianer Zhang. Efficientalgorithm for the spin-free valence bond theory. i. new strategy and pri-mary expressions. Int. J. Quant. Chem., 67(5):287–297, 1998.

[18] W. A. Gorddard III. Phys. Rev., 157:81, 1967.

[19] R. C. Ladner and W. A. Goddard III. J. Chem. Phys., 51:1073, 1969.

[20] W. A. Goddard III., T. H. Dunning Jr., W. J. Hunt, and P. J. Hay. Acc. Chem.Res., 6:368, 1973.

[21] Thorstein Thorsteinsson, David L. Cooper, Joseph Gerratt, Peter B.Karadakov, and Mario Raimondi. Modern valence bond representationsof CASSCF wavefunctions. Theor. Chem. Acc., 93(6):343–366, 1996.

[22] David L. Cooper, Thorstein Thorsteinsson, and Joseph Gerratt. Modernvb representations of casscf wave functions and the fully-variational op-timization of modern vb wave functions using the casvb strategy. Adv.Quant. Chem., 32:51 – 67, 1998.

[23] Thorstein Thorsteinsson and David L. Cooper. Exact transformations of CIspaces, VB representations of CASSCF wavefunctions and the optimiza-tion of VB wavefunctions. Theor. Chem. Acc., 94(4):233–245, 1996.

[24] Kimihiko Hirao, Haruyuki Nakano, Kenichi Nakayama, and MichelDupuis. A complete active space valence bond (casvb) method. J. Chem.Phys., 105(20):9227–9239, 1996.

[25] Kenichi Nakayama, Haruyuki Nakano, and Kimihiko Hirao. Int. J. Quant.Chem., 66(2):157–175, 1998.

[26] J. H. Van Lenthe and G. G. Balint-Kurti. The valence-bond SCF (VB SCF)method. : Synopsis of theory and test calculation of oh potential energycurve. Chem. Phys. Lett., 76(1):138, 1980.

[27] J. H. van Lenthe and G. G. Balint-Kurti. The valence-bond self-consistentfield method (VB–SCF): theory and test calculations. J. Chem. Phys.,78(9):5699, 1983.

[28] Jacob Verbeek and Joop H. Van Lenthe. On the evaluation of non-orthogonal matrix elements. J. Mol. Struct.(THEOCHEM), 229:115, 1991.

BIBLIOGRAPHY 23

[29] P. C. Hiberty, J. P. Flament, and E. Noizet. Compact and accurate valencebond functions with different orbitals for different configurations: applica-tion to the two-configuration description of f2. Chem. Phys. Lett., 189(3):259– 265, 1992.

[30] P. C. Hiberty, S. humbel, J.H. van Lanthe, and C.P. Byrman. J. Chem. Phys.,101:5969, 1994.

[31] Philippe C. Hiberty and Sason Shaik. Breathing-orbital valence bondmethod - a modern valence bond method that includes dynamic correla-tion. Theor. Chem. Acc., 108(5):255–272, 2002.

[32] Lingchun Song, Wei Wu, Philippe C. Hiberty, David Danovich, and SasonShaik. An accurate barrier for the hydrogen exchange reaction from va-lence bond theory: Is this theory coming of age? Chem. Eur. J., 9(18):4540–4547, 2003.

[33] Philippe C. Hiberty, Stephane Humbel, and P. Archirel. Nature of thedifferential electron correlation in Three-Electron bond dissociations. effi-ciency of a simple Two-Configuration valence bondmethodwith breathingorbitals. J. Phys. Chem., 98(45):11697–11704, 1994.

[34] Wei Wu, Lingchun Song, Zexing Cao, Qianer Zhang, and Sason Shaik.Valence bond configuration interaction: A practical ab initio valencebond method that incorporates dynamic correlation. J. Phys. Chem. A.,106(11):2721–2726, 2002.

[35] Lingchun Song, Wei Wu, Qianer Zhang, and Sason Shaik. A practical va-lence bond method: A configuration interaction method approach withperturbation theoretic facility. J. Comp. Chem., 25(4):472–478, 2004.

[36] Lingchun Song, Wei Wu, Qianer Zhang, and Sason Shaik. Vbpcm: A va-lence bond method that incorporates a polarizable continuum model. J.Phys. Chem. A., 108(28):6017–6024, 2004.

[37] H.B. Chirgwin and C.A. Coulson. The electronic structure of conjugatedsystems. vi. Proc. R. Soc. London Ser A, 201:196, 1950.

[38] C. A Coulson and I. Fischer. Notes on the molecular orbital treatment ofthe hydrogen molecule. Phil. Mag. Ser 7, 40(303):386 – 393, 1949.

[39] S. F. Boys. Construction of some molecular orbitals to be approximatelyinvariant for changes from one molecule to another. Rev. Mod. Phys.,32(2):296–299, 1960.

[40] S. Miertus, E. Scrocco, and J. Tomasi. Chem. Phys., 55:117, 1981.

[41] P.-O. Lowdin. Phys. Rev., 97:1474, 1955.

24 BIBLIOGRAPHY

[42] Lingchun Song, Jinshuai Song, Yirong Mo, and Wei Wu. An efficient algo-rithm for energy gradients and orbital optimization in valence bond theory.J. Comp. Chem, 30:399, 2008.