Application and development of quantum chemical methods. Density functional and bond theory

Fuming Ying

Theoretical School of Biotechnology Royal Institute of Technology Stockholm 2010 Copyright c 2010 by Fuming Ying TRITA-BIO Report 2010:13 ISSN 1654-2312 ISBN 978-91-7415-633-1 3

Printed by US-AB, Stockholm 2010 i Abstract This thesis deals with two subdiciplines of . One is the most used method today, density functional theory (DFT), and the other one of the least used electronic structure methods, valence bond theory (VB). The work on DFT is based on previous developments in the group in density functional response theory and involves studies of hyperfine cou- pling constants which are measured in paramagnetic exper- iments. The method employed is a combination of a restricted-unrestricted ap- proaches which allows for adequate description of spin polarization without spin contamination, and spin-orbit corrections to account for heavy atom ef- fects using degenerate perturbation theory. The work on valence bond theory is a new theoretical approach to higher-order derivatives. The orbital derivatives are complicated by the fact that the wave functions are constructed from de- terminants of non-orthogonal orbitals. An approach based on non-orthogonal second-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. ii List of publications

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

Paper II. An analytical valence bond Hessian, by means of second quantization in biorthonor- mal basis sets Fuming Ying, Olav Vahtras, Zilvinas Rinkevicius and Wei Wu. Manuscript iii 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 • iv 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 and my tutor Zilvinas Rinkevicious for their guidance, advice in each project. I still remember the situation when Zilvinas 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 help and discussion during the work.

I may also say ”Thank you” to Prof. Magareta Bromberg, Aato Laaksonen and Alexander Lyubartsev in SU for the courses and Prof. Poul Jøgensen, Jeppe Olsen 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 3 2.1 Densityoptimization ...... 3 2.2 Variational perturbation theory ...... 4 2.2.1 The restricted-unrestricted method ...... 5

3 Valence bond theory 7 3.1 VBWaveFunctionandHamiltonian ...... 8 3.1.1 VBWaveFunction ...... 8 3.1.2 Constructing VB Hamiltonian with Determinant Expansion 9 3.2 DifferenttypesofVBOrbitals ...... 9 3.2.1 LocalizedVBOrbitals ...... 9 3.2.2 Semi-Localized VB Orbitals ...... 10 3.3 VBMethods ...... 11 3.3.1 GeneralizedVB ...... 11 3.3.2 CASVB...... 12 3.3.3 VB Self-Consistent Field ...... 12 3.3.4 BreathingOrbitalVB...... 13 3.3.5 VB Configuration-Interaction ...... 14 3.3.6 VBPCM ...... 15 3.4 VariousVBPrograms...... 15 3.4.1 TURTLE ...... 15 3.4.2 VB2000...... 16 3.4.3 XMVB ...... 16

4 Summary of Included Papers 17 4.1 Restricted-unrestricted density functional theory ...... 17 4.2 The Valence bond Hessian ...... 18

v

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 method today with its competitive combination of speed and accuracy. The number of articles based on DFT increase with an exponential rate making a complete overview if not impossible but certainly beyond the scope of this thesis. The second, valence bond (VB) theory, was the most important theory in the infancy of 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 two pioneering articles, the first by Hohenberg and Kohn[1] who established the mathematical foundation of a theory based on the as a fundamental variable, and the other by Kohn and Sham[2] who introduced a practical single-determinant theory for practical calculations. It was because of 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 to G. N. Lewis[3] who introduced concepts such as electron pair bonding and the , which has influenced the way of thinking for generations of chemists. The theory which gradually lost importance when the rivaling (MO) theory grew in popularity. The latter had its origin in spectroscopy and allowed for efficient computer implementations with predictive power. Recent decades have witnessed a revival of valence bond theory, with new algorithms and new computer implementation. Paper II. is a contribution to this develop- ment.

1 2 CHAPTER 1. INTRODUCTION Chapter 2

Density functional theory

The implications of the fundamental theorems of density functional theory are enormous[1]. The ground state energy of a many-electron system is a unique functional of the electron density and that the part of the functional which in- dependent of the external field, the Hohenberg-Kohn functional, is a universal functional independent of the number of . 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, is not known which has prompted a huge development of approximate function- als which have been parameterized to reproduce a certain class of experimental results. The success of DFT has been due to expansions and the formu- lation of DFT as a single determinant theory[2]. In practice DFT can be viewed as a Hartree-Fock theory with a semi-empirical element which includes electron correlation. 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) Z which leads to a Lagrangian multiplier which is interpreted as the chemical potential[1]. If the density is constructed from a determinantal

3 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 δE[ρ(r)] = dV δρ(r) (2.3) Z δρ(r)

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

∂ρ (r) δρ (r)= λ δλ (2.4) λ ∂λ 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 ρˆ(r) λ (2.5) | | where the density operator projected on a basis set φp is { } ∗ † ρˆ(r)= φp(r) φq(r)apaq (2.6) Xpq

When the parameterization is of exponential unitary form,

† λ = e−λpqapaq 0 (2.7) | | i.e. the with parameters describing a change from a reference set of orbitals, the transformation is expanded around λ = 0 and the density variation a becomes a commutator ∂ρ † = λ [apaq, ρˆ(r) λ (2.8) ∂λpq | | Defining a Fock operator δE Fˆ = dV ρˆ(r) (2.9) Z δρ(r) the energy optimization reduces to

† 0 [a aq, Fˆ] 0 =0 (2.10) | p | 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 dE E[ρλ(x); x]= E[ρλ(0);0]+ x (2.12) dx x=0 λ=λ(0) and in a strictly variational theory we have

dE = 0 V 0 (2.13) dx x=0 | | λ=λ(0)

2.2.1 The restricted-unrestricted method The restricted-unrestricted method were developed by Fernandez et al.[5] and later generalized for DFT by Rinkevicius et al.[6] for describing spin polariza- tion in spin-restricted methods. In spin-restricted methods, the alpha- and beta spin-orbitals always have the same spatial parts. For the calculation of a prop- erty such as the hyperfine coupling constant whose main component, the Fermi contact 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 have sufficient flexibility at the nucleus because the core orbitals are normally inac- tive, and to allow for correlation in the core orbitals is computationally very expensive. On the other hand an unrestricted which does have this flexibility have other problems such as spin contamination in the wave function The restricted-unrestricted theory was developed to avoid the problems with spin 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 = τ =0 (2.14) ∂λ but we introduce Lagrangian multipliers so that the same relation holds in the presence of the perturbation, i.e. for all x. A direct way of obtaining the final relations is to differentiate Eq. (2.14) we have

d ∂E ∂2E ∂2E ∂λ = + =0 (2.15) dx ∂λ ∂λ∂x ∂λ2 ∂x which defined a linear system of equations for the wave function response. The energy change then consists of two terms; the first as an expectation value of the perturbation and the second from the response in the wave function

dE ∂E ∂E ∂λ = + (2.16) dx ∂x ∂λ ∂x 6 CHAPTER 2. DENSITY FUNCTIONAL THEORY

The combination of Eqs.(2.15,2.16) gives

dE ∂E ∂E ∂2E −1 ∂2E = (2.17) dx ∂x − ∂λ  ∂λ2  ∂λ∂x Chapter 3

Valence bond theory

Valence bond(VB) theory is an ancient as well as modern theory in quantum chemistry. It was first introduced by Heitler and London[10] in 1927. Later on, Slater[11] and Pauling[12] developed and added several concepts into that treatment in order to generalize it for polyatomic molecules. The theory that was 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 explain the 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 (HAO) which can be obtained by combining original atomic orbitals to get the maximum bonding ability. Thus the original atomic orbitals can be treated as a special type of HAOs. In the following, we will use HAO to denote all kinds of atomic orbitals. Perhaps the most important concept in VB was resonance. It means that the ”real” electronic structure of a molecule can be explained as the linear combination between a set of ”extreme” structures. For the sake of resonance, VB is a multi-configurational method. Some molecular properties like conjugation can be easily explained by resonance. A derivative of resonance is the resonance energy. It is defined as the difference between the energies of ”real” structure and the most stable struc- ture(s) and can be a measure of .

A severe technical disadvantage was discovered soon. The use of non-orthogonal local bonding orbitals made quantitative calculations impossible at that time so that a lot of simplification had to be made. Some of them may lead to wrong conclusions. On the other hand, molecular orbital(MO) theory made 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 in MO theory, while the role of VB diminished.

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

7 8 CHAPTER 3. VALENCE BOND THEORY possible to perform ab initio VB calculations. Thus, from the late of 1970s, VB was reborn from the ashes. VB developments, i.e. generalized Slater-Condon rules[13], spin-free VB method[14, 15] and the paired-permanent-determinant[16, 17] approach, have been published to reduce computational costs so that pure ab initio VB may be performed 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 the number of electrons. Furthermore, a lot of VB methods have been developed to handle different situations, including electron correlation, but which maintains the 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, such tools 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 is called HLSP(Heitler-London-Slater-Pauling) wave function. It can be expressed as ΦK = AˆΩK ΘK (3.1) where Aˆ is the antisymmetric operator and ΩK and ΘK are the spacial and spin function respectively. Spacial function has the form of the multiplication of a set of orbitals ΩK = φ (1)φ (2) φN (N) (3.2) 1 2 and spin function has the form

1 ΘK = [α(i)β(j) α(j)β(i)] α(l) (3.3) √ − Yi,j 2 Yl 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 VB Ψ = CK ΦK (3.4) XK and weight of ΦK can be obtained by Coulson-Chirgwin’s formula[37]

WK = CK CLMKL (3.5) XL 3.2. DIFFERENT TYPES OF VB ORBITALS 9 where CK is the combination coefficient of HLSP wave function ΦK and MKL is the overlap between ΦK and ΦL. The linear independent HLSP that are used to build real wave function can be obtained with Rumer rules or standard Young tableau.

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 2m

ΦK = CiK DiK (3.6) Xi=1 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. For example, normalized wave function of A-B can be expressed as the determinant expansion

cov 1 Φ = φaφb φaφb (3.7) √2 | |−| |  The Hamiltonian of VB wave function is constructed in terms of VB struc- tures. Thus, Hamiltonian matrix elements ΦK Hˆ ΦL can be expanded into a | | set of Hamiltonian elements DiK Hˆ DjL | | 2m 2n

ΦK Hˆ ΦL = CiK CjL DiK Hˆ DjL (3.8) | | | | Xi=1 Xj=1 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, the degree 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. Both are used for different purposes.

3.2.1 Localized VB Orbitals A localized VB orbital is simply an HAOs. ”Pure” VB methods are based on such kind of VB orbital. In a system with one A-B, the VB wave 10 CHAPTER 3. VALENCE BOND THEORY function for this bond can be expressed as

Ψ= c φaφb φaφb + c φaφa + c φbφb (3.9) 1 | |−| | 2| | 3| |  th where ci is the coefficient of i VB structure and φa and φb are HAOs located on atom A and B respectively. The first term corresponds to a covalent structure of A-B bond, and the last two terms correspond to ionic terms. The advantage of localized 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 molecule CH4, may be dealt in the following way: the localized orbital on H keeps the type of HAO and the orbital on C becomes fragment orbital(FO) which means that orbital on C bonding with H may have a slight delocalization to the other H 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 in terms 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. The general form was suggested by Coulson and Fischer[38] as

ψa = φa + λφb (3.10) and ′ ψb = 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 | |−| | ′ ′ (3.12) = (1+ λλ ) φaφb φaφb + λ φaphia + λ φbφb | |−| | | | | |  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 the molecule 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(in form 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 structure in 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 on one CH3 group and delocalizing to whole part of the other CH3 group, while a BDO 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 large number of VB structures and then make the wave function much simpler. When all orbitals become delocalized to the whole system, the wave function will be- come an MO wave function. Semi-localized orbital make a link between VB and MO theory. The disadvantage of semi-localized orbitals, as we have shown above, 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 by William A. Goddard III and was the first im- portant case of Coulson-Fischer’s formula. The restricted form of GVB method is 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 a GVB wave function can be expressed as a single determinant with a set of ”gem- inal” two-electron functions each corresponding to a or a GV B Ψ = ( ϕ aϕ b ϕ aϕ b )( ϕ aϕ b ϕ aϕ b ) ( ϕNaϕNb ϕNaϕNb ) | | 1 1 |−| 1 1 | | 2 2 |−| 2 2 | | |−| | (3.13) th where ϕia and ϕib are OEOs for i 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 nonorthogonal if in the same geminal function

ϕia ϕib =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 φ φ (3.16) | |−| | | | i | i i | ∗ where φi and φi are natural orbitals respect to bonding and antibonding MOs ∗ and Ci and Ci are corresponding coefficients. The transformation between forms shown in Eqn gvb) and 3.16 is ∗ ∗ ∗ φi + λφi φi λφi 2 Ci ϕia = ϕib = − λ = (3.17) √1+ λ2 √1+ λ2 − Ci 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 VB wave 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 the VB wave function from CASSCF. After the transformation, the final VB wave function is restrictly equivalent to starting CASSCF. The way can be expressed as following formula

ΨCAS = SΨCASV B + (1 S2)ΨCASV B (3.18) − ⊥ where ΨCAS and ΨCASV B are original CASSCF and final CASVB wave functions CASV B respectively, Ψ⊥ is the perpendicular complement of CASVB wave function and S = ΨCAS ΨCASV B (3.19) | is the overlap between CASSCF and CASVB wave function. One way to get CASVB 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 will more or less keep the MO type, similar to localized molecular orbitals(LMOs).

Different from Thorsteinsson’s method, Hirao[24, 25] developed another way to get CASVB wave function with nonorthogonal VB orbitals. Since LMOs have obvious delocalization tails and may enlarge the ionic weights and mix cova- lent and ionic pictures, nonorthogonal orbitals will give us more reasonable VB wave 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 Hˆ 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 of bonding may be kept. Particularly, when a set of fully delocalized orthogonal orbital is used with a wave function in single determinant, VBSCF will go to the form of Hartree-Fock (HF) and have the same energy. The wave function may differ from the HF wave function only by a unitary transformation without changing the energy.

3.3.4 Breathing Orbital VB Breathing orbital VB(BOVB) theory was developed by Sason Shaik and Wei Wu[29, 30, 31, 32, 33]. In VBSCF, different VB structures share the same set of 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 the system. If there is only one VB structure, nothing will be changed and BOVB is equivalent to VBSCF.

C1 F F + C2 F F + C3 F F

C1 F F + C2 F F + C3 F F

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 VB orbitals. As we can see, different VB structures share the same set of VB orbitals in VBSCF, no matter whether it is covalent or ionic structure. In BOVB, same orbital is treated as different. orbitals in cation atom become more contracted and 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 VB Configuration-Interaction(VBCI) method[34, 35] is a VB method based on CI model aiming to provide better results and keep the clear picture of VBSCF. In a VBCI calculation, the orbitals from a VBSCF are used but not furhter optimized. Virtual orbitals and structures will be constructed to describe exci- tations.

The VBCI procedure contains several steps. First, a VBSCF wave function should be obtained by a VBSCF calculation. Although VBSCF orbitals can be strictly localized, a unitary transformation is still allowed without changing the energy. Thus, an extra Boys localization[39] is needed to gain a unique VBSCF wave function for a zeroth-order wave function 0 0 0 Ψ = CK ΦK (3.23) XK Second, VB virtual orbitals and VB virtual structures will be constructed. Virtual orbitals ϕi are restricted to the same block of the corresponding occu- { a} pied orbital ϕa i i ϕa = χµcaµ (3.24) Xµ i i where caµ is the LCAO coefficient of basis function χµ in orbital ϕa. 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 that no basis function is shared in more than 1 VB orbital. Orbitals are orthogonal in the same block, but not in the different blocks i ϕa ϕ =0 (3.25) | a ϕi ϕj =0 (3.26) a| a ϕi ϕj =0 (3.27) b| a VB virtual structures can be obtained by replacing some occupied orbital with a virtual one, meaning the excitation. Excitation shall be done according to the rule that one occupied orbital ϕa can be replaced only by the virtual ones ϕi that in the same block. If only one electron is excited to the virtual orbital { a} in a structure, a VBCIS calculation will be obtained. If at most two electrons can be excited, a VBCISD calculation will be obtained. A VBCI wave function then is generated containing all VBSCF and excited structures.

V BCI V BCI V BCI Ψ = CK ΦK (3.28) XK 3.4. VARIOUS VB PROGRAMS 15

V BCI V BCI where CK is coefficient of ΦK that should be solved later.

V BCI Finally, a secular equation is solved to get the coefficients CK . The weights 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 be used to solve chemical problems in solution, i.e. SN2 reactions. In order to have a VB theory for problems in solution a combination with the Polarizable Con- tinuum Model (PCM)[40] has been developed.

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

VBPCM is a method based on VBSCF. Thus, the representation of the VBPCM wave function is the same as VBSCF. However, the Hamiltonian now is no longer the same as the VBSCF Hamiltonian and added by a effective one-electron Hamiltonian to describe the interaction between solvent and solute

s H = H0 + V (3.29)

s where H0 is the Hamiltonian in gas phase for a bare molecule and V 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. focusing on 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 structures for 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 theory is also implemented into VB2000 so that a large molecule can be described as physically 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 and conventional Slater determinant expansion methods. This program is enabled for different kinds of VB such as VBSCF, BOVB, VBCI, VBPCM, GVB, SCVB, BLW, etc. with OEOs, BDOs, or HAOs. VB orbitals and VB coefficients can be optimized simultaneously. XMVB program can be obtained either as a stand-alone program or a module in GAMESS-US. Parallel version of XMVB with 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 parameters measure 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 describe the spin polarization that occurs in an electron paramagnetic resonance exper- iment with magnetic nuclei. On the other hand an unrestricted method has opposite features, it is flexible enough at the nucleus but spin contamination is known to give poor results in special cases. The philosophy of the combination of these methods, the restricted-unrestricted model, is to merge the best of both worlds. The wave function is optimized with spin restricted method while the spin-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 one order higher than would have been the case in a purely unrestricted scheme. This idea was first developed and applied in multi-configuration self-consistent field 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 between experimental 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 experiments for vanadocenes(IV) complexes while for vanadium oxide complexes the devia- tions are slightly higher. The accuracy of DFT calculations depend in general on the choice of functional, here the BHLYP give the best results while the popular B3LYP fail in the prediction of hyperfine coupling constants. This is believed to be 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 that the wave function is constructed from determinants of non-orthogonal orbitals. To date there is no computer implementation of an analytical valence bond Hes- sian which is the key quantity necessary for applications in time-dependent valence bond theory. In this paper outline a theoretical framework based on second quantization in non-orthogonal orbitals and a theory of non-orthogonal determinants that we credit to P-O Lowdin[41]¨ The main elements are the dual sets of orbitals bi-orthogonal to each other which are connected by a transformation of the overlap matrix

† q† ap = Spqa (4.1) where the notation with subscripts for one set and superscripts for the dual set introduces tremendous simplifications. The algebraic rules of differentiation and commutation allows one to identify the derivative of a determinant with respect to a molecular orbital coefficient in terms of excitation operator

∂ † m µ Ψ = aµa Ψ (4.2) ∂Cm | | This gives that the derivative of an inner product between two determinants is an element of a transition density matrix, and all energy derivatives can be expressed in terms of integrals and transition density matrices. It is shown that given a Hamiltonian in an arbitrary basis 1 H = h ap†aq + g ap†ar†asaq (4.3) pq 2 pqrs (summation implied by repeated subscript-superscript pair) the derivative of a Hamiltonian matrix element is given by

∂ mρ ξ m µ K H L =2 K L D(KL) F (KL)ξρ∆(KL) µ + H(KL)D(KL) µ ∂Cm | | |   (4.4) where D(KL) is the intermediately normalized transition density matrix,

K ap†aq L D(KL)pq = | | (4.5) K L | ∆ given by ∆(KL)p = δp D(KL)p (4.6) m q − q and F the Fock operator

rs F (KL)pq = hpq +(gpqrs gpsrq)D(KL) (4.7) − 4.2. THE VALENCE BOND HESSIAN 19

The relations thus obtained are transparent and without explicit reference to overlap matrices which appear in great number using standard matrix notation. For the gradient it can also be shown that the derivative reproduces the results published 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 ∂ † † n m µ ν Ψ = aµaνa a Ψ (4.8) ∂CmCn | | the second derivative of a Hamiltonian matrix element is

2 ∂ m ν µ ν K H L = K H L D(KL) µ(D(KL)+ D(LK))n ∂CmCn | | | | m n +[D(KL)F (KL)∆(KL))]µ (D(KL)+ D(LK))µ m n (4.9) + D(KL)µ (D(KL)F (KL)∆(KL))ν m n + D(KL)µ (∆(KL)F (KL)D(KL))ν + gm n gm n + g mn + gnm µ ν − ν µ µ ν µν The non-Fock contributions are effectively integral transformations with transi- tion density matrices, e.g.

m n mρ nτ ξ σ g µ ν = D(KL) D(KL) gξρστ ∆(KL)µ∆(KL)ν (4.10)

This enables us to implement second-order methods, but most importantly the Hessian is the key to time-dependent valence bond theory, which until now has been a completely unexplored field of research. 20 CHAPTER 4. SUMMARY OF INCLUDED PAPERS Bibliography

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