Studies in multireference many-electron theories

Studies in multireference

many-electron theories

Peter Jeszenszki

Ph.D. Thesis

Supervisor: Prof. Dr. Peter Surjan

Consultant: Dr. Agnes Szabados

Doctoral School of Chemistry

Head of the doctoral school: Prof. Dr. Gyorgy Inzelt

Theoretical and Physical Chemsitry,

Structural Chemistry Doctoral Programme

Head of the doctoral programme: Prof. Dr. Peter Surjan

Laboratory of Theoretical Chemistry

Institute of Chemistry

Eotvos Lorand University, Budapest

2014

Contents

Abbreviations iii

1 Introduction 1

2 Theoretical Background 3

2.1 Possible reference functions for multireference calculations . . . . . 3

2.1.1 Introduction to Multiconfigurational Self-Consistent Fieldtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Antisymmetrized Product of Strongly Orthogonal Geminals 10

2.1.3 Restricted-Unrestricted Singlet-type Strongly orthogonal Gem-inals and its spin projected form . . . . . . . . . . . . . . . . 16

2.2 Perturbative corrections for multireference functions . . . . . . . . . 21

2.2.1 Introduction to multireference perturbation theory . . . . . 22

2.2.2 State-Specific Multireference Perturbation Theory . . . . . . 30

3 Redundancy in Spin-Adapted SSMRPT 36

3.1 Spin-adaptation in SSMRPT . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Redundancy in the amplitude equations . . . . . . . . . . . . . . . 39

3.3 Removal of redundancies in T µ . . . . . . . . . . . . . . . . . . . . 42

3.4 Redundancy-free amplitude equations . . . . . . . . . . . . . . . . . 49

3.5 Construction of the effective Hamiltonian . . . . . . . . . . . . . . . 56

3.6 Sensitivity analysis in SA-SSMRPT . . . . . . . . . . . . . . . . . . 57

3.7 Demonstrative examples . . . . . . . . . . . . . . . . . . . . . . . . 61

3.7.1 Kinks due to small coefficients . . . . . . . . . . . . . . . . . 63

3.7.2 Kinks due to redundancy . . . . . . . . . . . . . . . . . . . . 65

3.7.3 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . 66

3.7.4 Determinantal versus spin-adapted formulation and alterna-tive redundancy treatments . . . . . . . . . . . . . . . . . . 66

4 Role of local spin in geminal-type theories 70

4.1 Size consistency in strongly orthogonal geminal type theories . . . . 70

4.1.1 Single bond dissociation . . . . . . . . . . . . . . . . . . . . 71

4.1.2 Multiple bond dissociation . . . . . . . . . . . . . . . . . . . 73

4.2 Size consistency of spin purified geminal type methods . . . . . . . 77

4.3 Local spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.4 Assessment of local spin by strongly orthogonal geminals . . . . . . 84

i

Contents ii

4.4.1 Water symmetric dissociation . . . . . . . . . . . . . . . . . 85

4.4.2 Nitrogen molecule dissociation . . . . . . . . . . . . . . . . . 87

4.4.3 The H4 system . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.5 MR-LCC corrections for geminal based reference functions . . . . . 91

4.6 Implementation of MR-LCC corrected geminal theories and somedemonstrative examples . . . . . . . . . . . . . . . . . . . . . . . . 97

4.6.1 Water symmetric dissociation . . . . . . . . . . . . . . . . . 98

4.6.2 The H4 system . . . . . . . . . . . . . . . . . . . . . . . . . 101

5 Summary 103

A Exponential form of the APSG wave function 105

B Spin-unrestricted and restricted forms of RUSSG 107

C Possible structures of geminals in water symmetric dissociation 114

Bibliography 118

Acknowledgements 118

Abstract 134

Tudomanyos osszefoglalo 135

Abbreviations

AGP Antisymmetric Geminal Product

APSG Antisymmetric Product of Strongly Orthogonal Geminals

CAS-SCF Complete Active Space Self-Consistent Field

EHF Extended Hartree Fock

FCI Full Configuration Interaction

MC-SCF Multiconfigurational Self-Consistent Field

MRPT Multireference Perturbation Theory

RHF Restricted Hartree Fock

RSSG Restricted Strongly Orthogonal Singlet-type Geminals

RUSSG Restricted-Unrestricted Strongly Orthogonal Singlet-type Geminals

UHF Unrestricted Hartree Fock

USSG Unrestricted Strongly Orthogonal Singlet-type Geminals

SA-SSMRPT Spin-Adapted State-Selective Multireference Pertubation Theory

SCF Self-Consistent Field

SSMRPT State-Selective Multireference Pertubation Theory

SP-RUSSG Spin Projected Restricted-Unrestricted Strongly Orthogonal Singlet-type

Geminals

iii

Chapter 1

Introduction

Existing quantum chemical methods generally describe chemical compounds at

equilibrium geometries properly. However, several chemically interesting phenom-

ena (e.g.: covalent bond dissociation and electronic structure of radical compounds

or transition metals) still induce methodological challenges. One possible way to

treat these systems is based on partitioning the electron correlation into dynamical

and static parts. Dynamical correlation corresponds to the relative movement of

electrons, which is not treated by mean-field (Hartree-Fock) techniques. In order

to gain a deeper insight into this phenomenon, we should expand the exact wave

function in the basis of all possible Slater determinants (Full Configuration Inter-

action, FCI expansion), which are constructed from the Hartree-Fock one-electron

orbitals. If there is only one dominant coefficient in this expansion, then the corre-

sponding determinant is the Hartree-Fock determinant, while the remaining small

coefficients represent dynamical correlation. Static correlation emerges, when the

expansion contains more than one dominant coefficient. The part of the wave

function described by these coefficients provides the static correlation.

If there is only dynamical correlation in the system, then it can be determined

by single reference perturbation methods. Although static correlation can also be

handled appropriately by Multiconfigurational Self-Consistent Field (MC-SCF)

methods, there is no generally accepted recipe for the treatment of the remaining

dynamical correlation. In this thesis two related methods are investigated. The

first is the State-Specific Multireference Perturbation Theory (SSMRPT), which

1

Chapter 1. Introduction 2

has some advantageous features (size-extensivity and intruder independence), but

its spin-adapted form (SA-SSMRPT) might produce unphysical kinks on the po-

tential energy surfaces. One of the main objectives of this work was to explore the

origin of the unphysical kinks and find a way to eliminate these from the potential

energy surfaces.

The second method is related to the treatments of the static correlation, where

two-electron functions (geminals) are used. In that case, interactions between

the two electrons are taken into account explicitly, while interactions among pairs

are considered only at the mean-field level. This method can properly describe

single bond dissociation processes, but it may produce spurious results in case of

multiple bond dissociation, which are caused by the improper description of the

(local) spin states of the fragments. Another important goal of my work is to

examine the connection between the improper description of spin states and the

spurious energy profiles of dissociation processes in the context of geminal wave

functions. I also investigated, how this wave function can be applied as a reference

function in multireference calculations.

The structure of the thesis can be divided according to the two methods men-

tioned above. The discussions are based on Refs. [1] and [2] respectively.

In the next chapter theoretical background for MC-SCF methods, particularly

geminal-type methods, are devised. Multireference Perturbation Theory (MRPT)

is also outlined with special focus on SSMRPT and geminal-based perturbation

theories. Subsequently, these methods are presented in some details and their

properties are analyzed through demonstrative examples.

Chapter 2

Theoretical Background

2.1 Possible reference functions for multirefer-

ence calculations

In this section those methods are presented that give an appropriate descrip-

tion of the static correlation and provide a proper reference wave function for

multireference perturbation calculations as well. First, the Multiconfigurational

Self-Consistent Field (MC-SCF) method is introduced, which has a detailed de-

scription in fundamental quantum chemical books [3–5] and reviews [6–9]. Here,

we would only like to outline the fundamental concepts of MC-SCF theory and

to introduce the basic notions. Thereafter geminal based methods are presented

from the aspect of composite particles [10–16]. The connections with standard

quantum chemical (multiconfigurational) methods are also discussed. Afterwards,

the unrestricted form of these geminals [17, 18] are outlined and compared to the

Unrestricted Hartree-Fock (UHF) method.

3

Chapter 2. Theoretical Background 4

2.1.1 Introduction to Multiconfigurational Self-Consistent

Field theory

The main objective of MC-SCF calculations is to find the multiconfigurational

wave function corresponding to the lowest energy with a given number of deter-

minants:

|ΨMC〉 =M∑

µ

cµ|Φµ〉 , (2.1)

where |ΨMC〉 is the multiconfigurational wave function, |Φµ〉 is the µ-th deter-

minant, cµ is the linear variational parameter and M is the dimension of the

multiconfigurational space. Determinants |Φµ〉 are constituted by orthogonalised

one-electron functions ϕiσ, where i is the index of the spatial orbitals, while σ is

the spin index. In order to obtain appropriate multiconfigurational wave function,

two types of parameters should be optimized parallelly: the coefficients cµ and

the one-electron orbitals ϕiσ. This optimization can also be described by unitary

operators, which transform the initial wave function (|ΨMC0 〉) into the desired wave

function:

|ΨMC〉 = eReS|ΨMC

0 〉 ,

where R and S are antihermitian operators as a condition for the unitary of oper-

ators eR and eS. Operator S is responsible for the optimization in configurational

space with the following definition:

S =∑

K 6=0

SK0

|ΨMC

K 〉〈ΨMC

0 | − |ΨMC

0 〉〈ΨMC

K |︸︷︷︸

XK0

, (2.2)

where SK0 is the parameter of rotation in the multiconfigurational space and |ΨMCK 〉

is the multiconfigurational wave function, which is perpendicular to the initial

state. The coefficients cµ (Eq.(2.1)) can be expressed with the help of SK0. The


operator R is used to optimize the one-electron orbitals:

R =∑

i>j

Rij

(

Eij − Eji

)

︸︷︷︸

E−ij

,

where Rij is the parameter of the orbital optimization and Eij is the generator

of the unitary group [4, 19]. Eij can be expressed in standard second quantized

notation as:

Eij =∑

σ

ϕ+iσϕ

−jσ , (2.3)

where σ is the spin index (σ = α, β). In order to obtain the appropriate parameters

Rij, let us calculate the energy of the multiconfiguration wave function:

E = 〈ΨMC

0 |eR†

HeR|ΨMC

0 〉 , (2.4)

where for the sake of simplicity the coefficient cµ is fixed. Using the fact that

operator R is antisymmetric (R† = −R) the Baker-Campbell-Hausdorff (BCH)

formula [4] can be applied for Eq.(2.4):

E = 〈ΨMC

0 |H|ΨMC

0 〉 + 〈ΨMC

0 |[

H, R]

|ΨMC

0 〉 + (2.5)

1

2〈ΨMC

0 |[[

H, R]

, R]

|ΨMC

0 〉 + . . . ,

Eq.(2.5) can be easily rewritten in the usual Taylor-series form:

E = E0 +∑

ij

fijRij +1

2

∑

ijkl

GijklRijRkl + . . . , (2.6)

where E0, fij and Gijkl are the zeroth-, first- and the second-order coefficients at

point R = 0:

E0 = 〈ΨMC

0 |H|ΨMC

0 〉 , (2.7)

fij =∂E

∂Rij

∣∣∣∣∣R=0

= 〈ΨMC

0 |[

H, E−ij

]

|ΨMC

0 〉 , (2.8)


Gijkl =∂2E

∂Rij∂Rkl

∣∣∣∣∣R=0

= (2.9)

=1

2〈ΨMC

0 |([[

H, E−ij

]

, E−kl

]

+[[

H, E−kl

]

, E−ij

])

|ΨMC

0 〉.

Due to definitions in Eqs.(2.8) and (2.9) f and G are usually referred to as the

gradient and the Hessian respectively. In order to find the energy minimum the

following equations should be satisfied for all i and j:

∂E

∂Rij

= 0 . (2.10)

Assuming that the wave function is already in the optimum with respect to all the

orbital rotations (|ΨMC0 〉 = |ΨMC

Ropt〉), the following condition is satisfied:

fRopt

ij = 〈ΨMC

Ropt|[

H, E−ij

]

|ΨMC

Ropt〉 = 0 , (2.11)

which is obtained by the substitution of Eq.(2.8) into Eq.(2.10) . The last equal-

ity in Eq.(2.11) is called the generalized Brillouin condition, which simplifies to

the well-known Brillouin condition with a Hartree-Fock determinant (ΨHF) in the

expectation value instead of ΨMCRopt:

〈ΨHF|[

H, Eij

]

|ΨHF〉 = 0 .

The other resemblance with the Hartree-Fock method is the Fock operator like

quantity, which has a following relation to the gradient f [3, 4]:

fij = 2Fij − 2Fji , (2.12)

where matrix F is the generalized Fock matrix. F can be expressed with the

spin-free one- and two-particle density matrices (P and Γ):

Fij =∑

k

hikPkj +∑

klm

[im|kl] Γkljm , (2.13)


where hik and [im|kl] are the one- and two-electron integrals in [12|12] convention.

The density matrices can be evaluated in the following way:

Pkj = 〈ΨMC

0 |Ekj|ΨMC

0 〉 , (2.14)

Γkljm = 〈ΨMC

0 |EkjElm − δljEkm|ΨMC

0 〉 . (2.15)

The relation between the generalized Fock matrix and the traditional Fock matrix

becomes visible when applying the expression of the two-particle density matrix

(ΓHF) by one-particle density matrices (PHF) in Hartree-Fock theory:

ΓHF

kljm = P HF

kj PHF

lm − 1

2P HF

kmPHF

lj , (2.16)

and substituting this into Eq.(2.13):

F HF

ij =∑

k

(

hik +∑

lm

(

[im|kl] − 1

2[im|lk]

)

P HF

lm

)

︸︷︷︸

fHFik

P HF

kj . (2.17)

It is important to note that the generalized Fock matrix is not hermitian (Eq.(2.13))

unless the gradient is zero (Eq.(2.12)). Naturally by substituting in the condition

Eq.(2.16) the hermiticity of the traditional Fock-matrix is achieved (Eq.(2.13)).

One of the most popular optimization methods in MC-SCF calculations are the

Newton-Raphson type methods. There the energy in Eq.(2.6) is approximated up

to the second order to obtain the following form:

0 =∂E

∂Rij

= fij +∑

k>l

GijklRkl . (2.18)

If hyperindex I (J) is introduced instead of indices i and j (k and l), one obtains

a more compact form of Eq.(2.18):

0 = fI +∑

J

GIJRJ ,


from which parameters RJ can be determined by inverting the Hessian:

RJ = −∑

I

G−1JI fI . (2.19)

Due to the existence of redundant rotations, to which the energy is invariant, the

Hessian matrix contains zero blocks leading to singularities in matrix G−1. There-

fore these redundant rotations should be omitted. Similar redundant rotations

also appear in Hartree-Fock calculations at occupied-occupied or virtual-virtual

orbital rotations.

In Eq.(2.19) it is assumed that the coefficients cµ are fixed. However, they

also are variational parameters, the effects of which should be included. Simi-

larly to Eq.(2.18) the Newton-Raphson equations can be written for both type of

parameters as:

0 =∂E

∂Rij

= fij +∑

k>l

Gijkl Rkl +∑

P

LijP SP0 , (2.20)

0 =∂E

∂SP0

= eP +∑

i>j

L†Kij Rij +

∑

Q

KPQ SQ0 , (2.21)

where the unknown quantities are the following:

eP =∂E

∂SP

∣∣∣∣∣R=0

= 〈ΨMC

0 |[

H, XP0

]

|ΨMC

0 〉 ,

LijP =∂2E

∂Rij∂SP

∣∣∣∣∣R=0

= 〈ΨMC

0 |[[

H, E−ij

]

, XP0

]

|ΨMC

0 〉 ,

KPQ =∂2E

∂SP∂SQ

∣∣∣∣∣R=0

= 〈ΨMC

0 |[[

H, XP0

]

, XQ0

]

|ΨMC

0 〉 .

These optimization processes are typically implemented in two cycles: an outer

and an inner cycle. In the outer cycle the eigenvalue problem of the Hamiltonian is

solved to obtain the coefficients cµ. Meanwhile in the inner cycle the one-electron

orbitals are optimized with the approximated form of Eqs.(2.20) and (2.21). One

favorable approximation is to assume optimal coefficients cµ in the inner cycle


(eP = 0). Substituting this back into Eqs.(2.20) and (2.21), Rij can be obtained:

Rij =∑

k>l

B−1ijklfkl ,

where Bijkl has the following form:

Bijkl = −Gijkl +∑

PQ

LijPK−1PQL

†Qkl .

However, due to possible poor convergence properties of Newton-Raphson method

and high cost of the inversion of Hessian matrices, numerous modifications and

alternative methods were also introduced [7, 20–24].

MC-SCF methods are typically distinguished by the choice of multireference

space. One of the most popular MC-SCF methods is the Complete Active Space

Self-Consistent Field (CAS-SCF) method [22, 23, 25], where the selection of the

determinants is based on the partitioning of the orbital space. Three types of

orbitals are distinguished: core, active and virtual orbitals. In the core part all

the spatial orbitals are occupied by two electrons, while the remaining (active)

electrons are in the active part (no electrons in the virtual part). In the active

space all the configurations are generated, with which the core part constitute

the multiconfigurational space. The main advantage of CAS-SCF method is the

straightforward exclusion of redundant orbital rotations (rotations within the sub-

spaces). Therefore in the optimization only those rotations are included, which

incorporate different subspaces.

One of the main difficulties in CAS is the factorial scaling of configurational

space. Assuming that the number of α and β active electrons are equal, the

dimension of the multiconfigurational space (M) is:

M =

(n

ne

)2

, (2.22)

where n is the number of active orbitals and ne is the number of active α (or β)

electrons. Although by using spin symmetry this dimension can be reduced [3], the


scaling property is maintained making systems with large active spaces (e.g.: tran-

sition metal complexes) unmanageable. One possible solution is to improve our

algorithms to be able to treat such large matrices and their diagonalization pro-

cedures. To this, developers of Density Matrix Renormalisation Group (DMRG)

theory have recently given a great contribution [26, 27] and also produced some

promising results [28, 29].

Another alternative is to restrict the original CAS Ansatz by reducing the

dimension of the configurational space. Usually these methods are based on the

partition of the active space [30–36] with different orbital occupancy restrictions

in each partition. In the so-called Constrained Complete Active Space (CCAS)

method [3, 31] the active orbitals are separated into mutually orthogonal subspaces

with a given number of electrons. In that case, the dimension of the configurational

space is a product of the dimensions of these small distinct subspaces:

MCCAS =∏

i

(ninei

)2

,

where i runs over these subspaces, ni is the number of spatial orbitals and nei is

the number of active electrons in the i-th subspace. If nei is equal to one for all

i, which means two electrons in every subset (with α and β spin), one obtains

the Antisymmetrized Product of Strongly Orthogonal Geminals (APSG) wave

function Ansatz, which is introduced in more details in the following subsection.

2.1.2 Antisymmetrized Product of Strongly Orthogonal Gem-

inals

Let us start our discussion with Hartree-Fock theory, where the electronic repul-

sion is approximated by an effective mean-field interaction providing one-electron

orbitals. The antisymmetric products of these orbitals constitute the Hartree-Fock

determinant, which does not contain any explicit electron-electron (Coulomb) cor-

relation. It is a natural extension of this to construct the antisymmetric product

from two-electron functions (geminals) [37, 38]. In this way the explicit two-

electron interaction is included, which plausibly provides the main part of the


electronic correlations. Although these geminals are typically constructed from

one-electron functions [10, 39–41], the actual two-electron function is also some-

times used to describe the electronic cusp by the distance of electrons as a variable

[42–46].

Geminal-type models also have a special role in the description of low-temperature

superconductivity. In Bardeen-Cooper-Schrieffer (BCS) model [47] the Cooper-

pairs are described by geminal functions, the antisymmetric products of which

constitute the BCS wave function. When this is projected onto a subspace with

a certain number of particles, it is called the Antisymmetric Geminal Power

(AGP) [48, 49], which has already been applied to describe electronic structures

of molecules [50–53]. However, the calculation of necessary matrix elements is

demanding, therefore strong orthogonality is introduced to avoid these difficulties

[38]. In that case, the integral of the product of geminals is zero even though the

integral is evaluated only over one of the electrons’ coordinates:

∫

ψi (x1,x2)ψj (x1,x2) dx1 = 0 , (2.23)

where for i 6= j, ψi and ψj are geminals, x1 and x2 are coordinates of the electrons.

The antisymmetric product of these geminals is called Antisymmetric Product of

Strongly Orthogonal Geminals (APSG) or Separated Electron Pair wave func-

tion [10, 38, 54]. The strong orthogonality condition can easily be satisfied with

Arai theorem [55], which states that two geminals are strongly orthogonal to each

other if and only if they can be expanded in mutually orthogonal subspaces (Arai

subspaces).

The APSG model has a close relation to the Generalized Valence Bond method

[30]. In this method the perfect pairing approximation is considered, where the

orbitals are ordered in pairs and they are expanded in the mutually orthogonal

subspaces, which ensures the strong orthogonality between them. These pairs can

be defined with the following geminals:

ψI(xi,xj) = A[ϕI1(ri) ϕ

I2(rj)Θ(σi, σj)

], (2.24)


where A is the Antisymmetrization operator, ri and σi refer to the spatial and

spin coordinate of electron i, Θ is the singlet coupled spin function (Θ(σi, σj) =

α(σi)β(σj)− β(σi)α(σj)), ϕI1 and ϕI2 are the one-electron orbitals. The product of

these geminals provide the GVB wave function,

ΨGVB(x1,x2, . . . ,xn) = A [ψ1(x1,x2) ψ2(x3,x4) . . . ψn(x2n−1,x2n)] ,

where the spatial functions are optimized to reach the lowest energy. The ϕI1 and

ϕI2 in Eq.(2.24) are linearly independent, but are not necessary orthogonal to each

other. This overlap provides a multiconfigurational character of the geminal ψI ,

which becomes visible, if one orthogonalizes the orbitals:

ϕIa =1

Na

(ϕI1 + ϕI2

), (2.25)

ϕIb =1

Nb

(ϕI1 − ϕI2

), (2.26)

where Na and Nb are the corresponding normalisation factors. In that case,

Eq.(2.24) has the following form [30, 56]:

ψI(xi,xj) =N 2a

2A[ϕIa(ri) ϕ

Ia(rj)α(σi) β(σj)

]− N 2

b

2A[ϕIb(ri) ϕ

Ib(rj)α(σi) β(σj)

],

which can be given in the second quantized notations as well:

ψ+I =

N 2a

2ϕIaα

+ϕIaβ

+ − N 2b

2ϕIbα

+ϕIbβ

+. (2.27)

It should be emphasized here that the relative sign of the terms in Eq.(2.27) are

fixed due to conditions Na > 0 and Nb > 0. Nowadays this restriction is usually

resolved and the GVB geminals are defined in the following generalized expression:

ψ+I =

2∑

ij

CIij ϕ

I+iα ϕI+jβ , (2.28)


where geminals are normalized to one, therefore elements of CI satisfies the fol-

lowing relation:

2∑

ij

CIij

2= 1 . (2.29)

In APSG the subspaces can be expanded in more than two basis functions [10, 30]:

ψ+I =

nI∑

ij

CIij ϕ

I+iα ϕI+jβ , (2.30)

where nI is the dimension of the I-th subspace and the coefficient matrix CI satisfy

the similar normalization condition as in Eq.(2.29):

nI∑

ij

CIij

2= 1 . (2.31)

The APSG wave function can be constructed as a product of geminal creation

operators:

|ΨAPSG〉 =

N/2∏

I

ψ+I |vac〉 , (2.32)

where N is the number of the electrons and |vac〉 is the physical vacuum. The

geminal expression in Eq.(2.30) is usually given by a natural orbital basis expansion

[57], which diagonalizes CI . To prove this, let us evaluate the elements of the

density matrix:

Pij = 〈ΨAPSG|Eij|ΨAPSG〉 ,

where |ΨAPSG〉 is the APSG wave function. If indices i and j of the one-electron

orbitals belonging to different geminals, then operator Eij (Eq.(2.3)) annihilates

an electron in one two-electron subspace thus providing a one-electron state, and

creates an electron in another two-electron state while producing a three-electron

state. The excited wave function obtained in this way is orthogonal to the original


APSG wave function, because the numbers of electrons are different in the corre-

sponding subspaces. Therefore Pij is equal to zero and the density matrix can be

given in a block diagonal form:

P Iij = 〈ΨAPSG|EI

ij|ΨAPSG〉 , (2.33)

where index I refers to the indices i and j belong to the same subspace I. Op-

erator EIij affects geminal ψI only, therefore Eq.(2.33) simplifies to the following

expression:

P Iij = 〈ψI |EI

ij|ψI〉 . (2.34)

Substituting Eq.(2.30) into Eq.(2.34), the density matrix element can be evaluated

as

P Iij = 2

(∣∣CI∣∣2)

ij,

hence PI is diagonal only if∣∣CI∣∣2

is diagonal.

As mentioned above, the APSG wave function is a special case of CCAS wave

function, where the number of active electrons is restricted to two in every sub-

space. In CCAS the orbitals are partitioned to three parts: core, active and virtual

parts. The active subspaces can be related to Arai-subspaces and the core part

of the wave function can be prepared in APSG as well, if these subspaces con-

tain only one spatial orbital. The virtual orbitals are unoccupied in both cases.

As the Ansatz is the same in both methods and their parameters are optimized

variationally, they are expected to provide the same results.

In APSG and also in CCAS the same Newton-Raphson type expression can

be used to determine the optimal rotational parameters as in CAS (Eq.(2.19)).

On the other hand in APSG the process of coefficient optimization is a little bit

different than in (C)CAS. While in CCAS the partitioning of the active orbitals

is used to restrict the configurational space for matrix diagonalization, in APSG


the process of diagonalization is usually reduced to the iterative solution of inter-

acting two-particle problems. The difference between these methods is also visible

in the parametrization. In CCAS the parameters are the coefficients of the de-

terminants (Eq.(2.1)) or the corresponding rotation parameters (Eq.(2.2)), while

in APSG those geminal coefficients are considered (Eq.(2.30)), which belong to

two-particle states. The direct product of these states produces N -particle de-

terminants and similarly, the products of the corresponding geminal coefficients

provide the determinant coefficients.

APSG optimization procedure can be regarded as the generalization of the

Hartree-Fock method. In this method the one-particle Fock operator is derived to

obtain one-particle orbitals, in APSG the effective two-electron Hamiltonian can

be similarly constructed [10], which provides the geminals:

HI |ψI〉 = EI |ψI〉 , (2.35)

HI =

nI∑

ij

∑

σ

h′ijϕI+iσ ϕI−jσ +

1

2

nI∑

ijkl

∑

σσ′

[ij|lk]ϕI+iσ ϕI+jσ′ ϕI−kσ′ ϕ

I−lσ , (2.36)

where HI is the effective Hamiltonian of the I-th subspace, EI is the energy of

geminal ψI and h′ij is the following:

h′ij = hij +∑

K 6=I

nK∑

k,l

PKkl

(

[ik|jl] − 1

2[ik|lj]

)

. (2.37)

As it can be seen in Eq.(2.36), one should construct an effective Hamiltonian for

every geminal ψI . However, in Hartree-Fock method one needs only one Fock

operator providing all the one-electron orbitals. In APSG one solves N/2 pieces

of eigenvalue equations to obtain all the geminals.

Substituting Eq.(2.30) into Eq.(2.35) and then projecting onto the two-electron

states, the geminal coefficients can be obtained:

nI∑

kl

HIij,klC

Ikl = EICI

ij , (2.38)

HIij,kl = 〈vac|ϕI−iβ ϕI−jα HI ϕI+kα ϕ

I+lβ |vac〉 . (2.39)


We should emphasize here that the geminal energy obtained in Eq.(2.38), does

not have a real physical meaning. However, by using the optimal coefficients CIij

the APSG energy can be determined as:

EAPSG =

N/2∑

I

nI∑

ij

h′ijPIij +

1

2

N/2∑

I

nI∑

ijkl

[ij|kl]CIijC

Ikl −

− 1

2

N/2∑

I 6=K

nI∑

ij

nK∑

kl

(

[ik|jl] − 1

2[ik|lj]

)

P IijP

Kkl .

There are other (variational) parameters in the APSG optimization; namely

the dimensions of the subspaces. Similar problem appears in CAS calculation

when selecting the active orbitals, which is usually based on chemical intuition

[58]. Although the bonding and the antibonding orbital pairs, obtained by Boys

localization procedure [59], can usually be assigned to geminal functions [10], it is

hard to determine how large the optimal dimensions of the subspaces should be.

Rassolov introduced a process, in which the dimensions of subspaces are also

optimized [17] parallelly with the orbitals and the geminal coefficients. However,

this optimization process may produce steps on the potential energy surfaces, when

the dimensions of the subspaces vary at adjacent geometric points. Anyway, this

does not influence the results in practice [17].

The other important property of APSG is size extensivity, which is ensured by

exponential parametrization [56, 60–63] (c.f. Appendix A). This feature is applied

to construct the simplified Coupled Cluster like equations, however, these methods

loose their variational character [61, 64, 65].

2.1.3 Restricted-Unrestricted Singlet-type Strongly orthog-

onal Geminals and its spin projected form

The Unrestricted Singlet-type Strongly orthogonal Geminals (USSG) [17] and

Restricted-Unrestricted Singlet-type Strongly orthogonal Geminals (RUSSG) [18]

are spin-unrestricted forms of APSG. The geminals in USSG and in RUSSG can


be defined similarly to Eq.(2.30):

U/RUψ+I =

nI∑

ij

CuIij φI+iα χI+jβ ,

where U/RUψ+I is a geminal in USSG and in RUSSG, φIiα is the α spin orbital and

χI+jβ is β spin orbital, while matrix CuI remains hermitian. In USSG, similarly to

the Unrestricted Hartree-Fock (UHF) method, the α and β orbitals are optimized

independently. This extra variational freedom provides energy decreasing, how-

ever, in this case the spin-contamination appears as well. In RUSSG the α and β

orbitals have to expand the same Arai-subspaces, therefore the spin-contamination

can be assigned to the geminals. Due to this restriction in RUSSG the geminals

can be transformed to the restricted one-electron basis, where the spatial parts of

the α and β orbitals are equivalent (ϕIiα and ϕIjβ, c.f. Appendix B):

RUψ+I =

nI∑

ij

CIij ϕ

I+iα ϕI+jβ . (2.40)

In this case, the matrix CrI is no longer symmetric, it can be partitioned to

symmetric (sCrI) and antisymmetric (tCrI) parts, which regard the singlet and

the triplet parts of the geminal:

CrIij = sCrI

ij + tCrIij .

In the further discussion the geminals are usually expressed in the restricted basis,

therefore we abandon the index r to obtain a simpler expressions.

Let us continue by analysing the UHF wave function, which has a strong relation

to strongly orthogonal geminal type wave functions. Making use of the Lowdin’s

pairing theorem [66–68] the occupied spin orbitals can be transformed to the so-

called corresponding orbital pairs. The corresponding α and β spin functions

overlap only each other, therefore every pair is expanded in different mutually

orthogonal subspaces. Due to the Arai-theorem[55] these pairs can be regarded as


strongly orthogonal geminals (Eq.(2.23)) [69]:

|ΨUHF〉 =∏

I

φI+oα χI+oβ︸︷︷︸

UHFψ+I

|vac〉 , (2.41)

where φIoα and χIoβ is the corresponding orbital pair and UHFψI is the UHF geminal.

Applying Karadakov’s extended pairing theorem [68, 70, 71] we can relate an

occupied-virtual orbital pair of α (φIoα, φIvα) and β spin (χIoβ, χIvβ) via unitary

transformation. Let us express now the φIoα, φIvα and χIoβ, χIvβ pair using another

pair of orthonormal functions (ϕIo, ϕIv):

φI+oα

φI+vα

=

cosαI sinαI

sinαI − cosαI

ϕI+oα

ϕI+vα

, (2.42)

χI+oβ

χI+vβ

=

cos βI − sin βI

sin βI cos βI

ϕI+oβ

ϕI+vβ

. (2.43)

Hence φI+vα (χI+vβ ) is a corresponding orbital from the virtual space of alpha (beta)

functions and the spatial parts of ϕI+oα and ϕI+oβ (ϕI+vα and ϕI+vβ ) are the same. The

rotation matrices in Eqs.(2.42) and (2.43) are chosen to satisfy the equations in

Ref. [68, 71].

In the following part of this subsection an alternative form of geminal UHFψ+I is

given. For the sake of simplicity index I is abandoned, however, we should notice

that these derivations can be done for all geminals. First, let us express UHFψ+

with ϕ+!

UHFψ+ = φ+oα χ

+oβ =

(cosα ϕ+

oα + sinα ϕ+vα

) (cos β ϕ+

oβ − sin β ϕ+vβ

).

After rearrangement we get the expression of geminal UHFψ with restricted orbitals:

UHFψ+ = cosα cos β ϕ+oα ϕ

+oβ − cosα sin β ϕ+

oα ϕ+vβ +

+ sinα cos β ϕ+vα ϕ

+oβ − sinα sin β ϕ+

vα ϕ+vβ .


The conventional compact form of UHFψ can be written as:

UHFψ+ =∑

µ,ν∈o,vCUHF

µν ϕ+µα ϕ

+νβ , (2.44)

CUHF =

cosα cos β − cosα sin β

sinα cos β − sinα sin β

. (2.45)

Let us separate CUHF to singlet (symmetric) and triplet (antisymmetric) parts!

CUHF = sCUHF + tCUHF

,

sCUHF =

cosα cos β 1

2sin(α− β)

12

sin(α− β) − sinα sin β

,

tCUHF

=

0 −1

2sin(α + β)

12

sin(α + β) 0

.

The condition sin(α − β) = 0 restricts the above forms by transforming the coef-

ficients sC into diagonal form:

sCUHF

nat =

cos2 α 0

0 − sin2 α

, (2.46)

tCUHF

nat =1√2

sin(2α)

0 − 1√

2

1√2

0

. (2.47)

The spin-free density matrix [10] is also expressed by using parameter α:

PUHF = CUHF

nat CUHF

nat† =

2 cos2 α 0

0 2 sin2 α

, (2.48)

which is also diagonal, therefore this basis is a natural basis. The diagonal elements

in Eq.(2.48) represent the occupation of natural orbitals, where we can observe the

well-known theorem that the sum of the occupation number of the corresponding


occupied and virtual pairs equals two [72]. We can observe that while φo, φvare α-natural orbitals (i.e. eigenvectors of PUHF,α) and similarly χo, χv are β-

natural orbitals, the natural orbitals (i.e. eigenvectors of PUHF = PUHF,α +PUHF,β)

are given by ϕo, ϕv with the condition α = β+kπ (k ∈ N). Therefore, Eqs.(2.42)

and (2.43) provide the relation between eigenvectors (natural orbitals) of PUHF,α,

PUHF,β and PUHF assuming sin(α− β) = 0.

Based on Eqs.(2.46) and (2.47) the connection of UHF to other geminal methods

can be established by comparison of coefficient matrices. For example if Eq.(2.47)

is omitted then the geminal remains singlet and one retrieves the original APSG

geminal Ansatz (Eq.(2.30)), where the dimension of the subspaces is restricted to

two. Although, introducing an extra parameter one obtains the RUSSG geminal

Ansatz (Eq.(2.40)) with the same restriction of two dimensional subspaces:

sCRUSSG

nat = cos δ

cos γ 0

0 − sin γ

, (2.49)

tCRUSSG

nat = sin δ

0 − 1√

2

1√2

0

. (2.50)

The main difference between UHF and RUSSG is that the latter contains two

parameters in each geminal function (i.e. γ and δ), while the former contains

only one (i.e. α). Therefore the UHF occupancies of natural orbitals ϕo and ϕv

can not be changed independently from the relative weight of singlet and triplet

components. The difference between UHF and RUSSG usually does not appear

at equilibrium geometries because the equilibrium structure is generally well de-

scribed by a single determinant (α = 0 in UHF, γ = 0 and δ = 0 in RUSSG).

The energy difference also fades out in the dissociation limit because the singlet-

triplet solutions become degenerate. For this reason there are no extra penalties

of the energy from the triplet component in UHF when the occupancies of natural

orbitals are optimized. However, this is not true at other regions of the potential

energy surface. As RUSSG involves an extra variational parameter compared to

UHF, it provides a better energy (or at least not worse).


However, similarly to UHF, RUSSG also suffers from spin-contamination, which

has to be eliminated for a proper description. A possible way to obtain the

spin-purified wave function is to spin project the original spin-contaminated wave

function (projection after the variation methods) [69, 73–79]. However, it may

cause discontinuity on the potential energy surface at the point where the spin-

contamination appears. A better, but usually more demanding possibility is the

variation after projection [69, 79, 80], where the parameters are optimized to min-

imize the energy of the spin-projected wave function.

Rassolov and Xu introduced a spin-projection method to obtain spin projected

form of RUSSG wave function (SP-RUSSG) [81], which is based on the itera-

tive diagonalization of matrix S2 to decrease the calculation demand. However,

it belongs to the former group producing unphysical discontinuities [2]. Head-

Gordon et al. also developed the approximate spin-purification scheme for their

appropriate Unrestricted in Active Pairs (UAP) function [82], though their results

seem more inaccurate due to the approximation. Despite of the arising prob-

lem SP-RUSSG can describe those systems, where APSG fails [2], which will be

particularly outlined in Chapter 4.

2.2 Perturbative corrections for multireference

functions

In the previous section those methods were presented, which are usually applied

to describe static correlations. Although in these cases the potential energy surface

usually can be described qualitatively well, there are quantitative failures due to

the remaining dynamical correlations. These correlations can be incorporated

with the multireference perturbation theories (MRPT), which can be more or less

classified into two groups: ”perturb-then-diagonalize” [83–87] and ”diagonalize-

then-perturb” [88–96] methods. In the first subsection general remarks on these

methods and the basic notions of MRPT are outlined. Subsequently, State-Specific

Multireference Perturbation Theory (SSMRPT) [97–100] is discussed in details.


2.2.1 Introduction to multireference perturbation theory

In Rayleigh-Schrodinger perturbation theory [4, 68, 101] the Hamilton operator

is partitioned to two parts:

H = H0 + λV , (2.51)

where H0 is the zeroth-order Hamilton operator, V is the perturbation opera-

tor and λ is a dimensionless parameter. Let us substitute Eq.(2.51) into the

Schrodinger equation and expand the energy and the wave function in power se-

ries of λ:

(

H0 + λV)∑

i=0

λi|Ψ(i)〉 =∑

j=0

λjE(j)∑

i=0

λi|Ψ(i)〉 . (2.52)

Parameter λ can be chosen arbitrarily, therefore Eq.(2.52) has to be satisfied order

by order:

λ0 : H0|Ψ(0)〉 = E(0)|Ψ(0)〉 ,λ1 : H0|Ψ(1)〉 + V |Ψ(0)〉 = E(0)|Ψ(1)〉 + E(1)|Ψ(0)〉 ,λ2 : H0|Ψ(2)〉 + V |Ψ(1)〉 = E(0)|Ψ(2)〉 + E(1)|Ψ(1)〉 + E(2)|Ψ(0)〉 ,

......

λn : H0|Ψ(n)〉 + V |Ψ(n−1)〉 =n∑

i=0

E(i)|Ψ(n−i)〉 .

(2.53)

Projecting Eq.(2.53) onto the zeroth-order wave function 〈Ψ(0)|, the energy terms

can be evaluated:

λ0 : E(0) = 〈Ψ(0)|H0|Ψ(0)〉 ,λ1 : E(1) = 〈Ψ(0)|V |Ψ(0)〉 ,λ2 : E(2) = 〈Ψ(0)|V |Ψ(1)〉 ,

......

λn : E(n) = 〈Ψ(0)|V |Ψ(n−1)〉 ,

(2.54)

where the orthogonality between the zeroth- and higher-order wave functions is

applied, and the wave function is normalized by an intermediate normalization


(〈Ψ(0)|Ψ〉 = 1). In order to obtain higher order corrections of the wave function,

let us expand them in basis |Φµ〉:

|Ψ(i)〉 =∑

µ 6=0

c(i)µ |Φµ〉 , (2.55)

where the restriction µ 6= 0 excludes the zeroth-order wave function (|Ψ(0)〉 = |Φ0〉)and c

(i)µ is the linear expansion coefficient. Projecting Eq.(2.53) onto 〈Φk|, c(i)µ can

be determined:

λ1 : c(1)µ =

∑

ν

R0µνVν0 ,

λ2 : c(2)µ =

∑

ν 6=0

∑

κ 6=0

R0µν

(Vνκ − E(1)δνκ

)c(1)κ ,

......

λn : c(n)µ =

n∑

i=1

∑

ν 6=0

∑

κ 6=0

R0µν

(Vνκδi,n−1 − E(n−i)δνκ

)c(i)κ ,

(2.56)

where R0 is a so-called reduced resolvent [68, 102, 103], which projects matrix(

E(0) − H0

)

onto subspace Q and inverts it there. R0 can be written in the

following form:

R0 = Q(

E(0) − QH0Q)−1

Q =Q

E(0) − H0

, (2.57)

where operator Q is a projector onto subspace Q. Due to the expansion in

Eq.(2.55), subspace Q contains all the |Φi〉 basis functions except for the zeroth-

order wave function:

Q = I − |Φ0〉〈Φ0| ,

where I is the unity operator.

A crucial point of the perturbation theory is how to partition in Eq.(2.51).

In case of single reference perturbation theory, the use of the Fock operator as

the zeroth-order operator would be a reasonable choice, because it provides the


lowest energy for a one-determinant wave function (H0 = F , Møller-Plesset par-

tition [104]). However, when general dissociation processes are considered, a one-

determinant wave function may not be described qualitatively well at the disso-

ciation limit. In that case, the excited states of H0 may approach E(0) so close

that the reduced resolvent may become singular (Eq.(2.57)). This type of excited

states is called intruder state [4].

In order to avoid these singularities, the quasi-degenerate perturbation theory

was introduced [83, 84]. In this theory one constructs a model space, where all the

low lying states are included. The functions of the model space can be defined by

the eigenvalue equation of the zeroth-order Hamiltonian:

H0|Φ0µ〉 = E(0)µ |Φ0µ〉 . (2.58)

Let us denote the desired I-th eigenfunction of operator H with |ΨI〉 and its

projected form in the model space with |Ψ0I〉:

|Ψ0I〉 = O|ΨI〉 , (2.59)

where O is a projector onto the model space and function |ΨI〉 can be expressed

with the eigenfunctions of H(0) (|Φ0µ〉) as:

|Ψ0I〉 =∑

I

c0µI |Φ0µ〉 .

Let us introduce the wave operator (Ω) [103, 105, 106], which transforms the

zeroth-order functions into the corresponding eigenfunctions of H as:

Ω =∑

I

|ΨI〉〈Ψ0I | . (2.60)

Hence the Schrodinger equation can be written in the following form:

HΩ|Ψ0I〉 = EIΩ|Ψ0I〉 . (2.61)


Operating Eq.(2.61) from the left with O, we obtain the following effective Hamil-

tonian equation:

OHΩ︸︷︷︸

Heff

|Ψ0I〉 = EI |Ψ0I〉 , (2.62)

where Heff is the effective Hamiltonian. Projecting Eq.(2.62) onto 〈Φ0ν |, we obtain:

∑

µ

Heff

νµcµI = EIcνI , (2.63)

where

Heff

νµ = 〈Φ0ν |Heff|Φ0µ〉 = 〈Φ0ν |HΩ|Φ0µ〉 . (2.64)

Due to the definition of Heff (Eq.(2.62)), Eq.(2.63) becomes a non-hermitian eigen-

value equation, the dimension of which is equal to the dimension of the model

space. Substituting Eq.(2.51) into the definition of Heff and assuming λ = 1, we

obtain:

Heff = OH0Ω + OV Ω . (2.65)

Using Eq.(2.59), we can easily see that:

OΩ =∑

I

O|ΨI〉〈Ψ0I | =∑

I

|Ψ0I〉〈Ψ0I | = O . (2.66)

This allows to rewrite Eq.(2.65) as:

Heff = H0O + OV Ω , (2.67)

since H0 and O commute with each other (Eq.(2.58)).

In order to evaluate the eigenvalue equation in Eq.(2.63) the explicit form of

Ω is required. First let us partition Ω according to the model (O) and the outer


space (Q = I − O):

Ω = OΩO + OΩQ+ QΩO + QΩQ . (2.68)

Applying the definition of Ω (Eq.(2.60)) and O (=∑

I |Ψ0I〉〈Ψ0I |), the following

relations can be derived:

ΩO =∑

IJ

|ΨI〉〈Ψ0I |Ψ0J〉〈Ψ0J | =∑

I

|ΨI〉〈Ψ0I | = Ω , (2.69)

ΩQ = Ω(

I − O)

= Ω − ΩO = 0 . (2.70)

Substituting Eqs.(2.66), (2.69) and (2.70) into Eq.(2.68), we obtain:

Ω = O + QΩO . (2.71)

The explicit expression of the operator QΩO can be given in the form of perturba-

tive series. These perturbative terms can be derived order by order, by following

the generalized Bloch equation [86, 87, 103, 106]:

Ω(0) = O ,

QΩ(1)O =∑

I

RI V |Ψ0I〉〈Ψ0I | ,...

QΩ(n)O =∑

I

RI

(

V QΩ(n−1) −n−1∑

i=1

Ω(i)OV QΩ(n−i−1)

)

|Ψ0I〉〈Ψ0I | ,

(2.72)

where

RI =Q

E(0)I − H0

.

Substituting Eq.(2.71) into Eq.(2.67), we obtain the cumulative perturbative series

of the effective Hamiltonian, where the n-th cumulative order contains all the lower


order terms as well:

Heff[0]

= H0O ,

Heff[1]

= H0O + OV Ω(0) = H0O + OV O ,

Heff[2]

= H0O + OV Ω(1) + OV Ω(2)

= H0O + OV O +∑

I

OV RI V |Ψ0I〉〈Ψ0I | ,...

Heff[n]

= H0O +n−1∑

i

OV Ω(n−1) .

(2.73)

Substituting Eqs.(2.73) into Eq.(2.63), the cumulative n-th order energy can be

obtained as:

∑

µ

Heff

νµ[n]c

[n]µI = E

[n]I c

[n]νI , (2.74)

where c[n]µI is a right eigenvector of the n-th cumulative order effective Hamiltonian.

This type of perturbation method, when the functions are perturbed first to build

the effective Hamiltonian and diagonalization is performed afterwards, is called

”perturb-then-diagonalize” method [83–87].

However, the above mentioned intruder problem is not completely solved with

this approach. Eq.(2.74) provides as many functions as the dimension of the

model space. Among these we can also obtain such high energy solutions that

may become nearly degenerate with the virtual states. Solutions for this problem

exist in quantum chemical literature [84, 85, 107].

There is another type of perturbation methods, where a multiconfigurational

zeroth-order function is considered. In that case, the expressions of the energies

and coefficients are analogous with the standard Rayleigh-Schrodinger perturba-

tion expressions (Eqs.(2.54) and (2.56)). The zeroth-order multiconfigurational

function is usually obtained by diagonalization processes, therefore this type of

methods is referred to as ”diagonalize-then-perturb” methods [88–96]. There is

only one state obtained, which avoids the singularities of the intruder problem.

However, these methods can be size inconsistent [4, 96, 108]. The above men-

tioned ”perturb-then-diagonalize” method can be constructed in size extensive


form, which is size consistent in localized orbitals, with the help of diagrammatic

formulation [85, 100]. Meanwhile in the ”diagonalize-then-perturb” methods the

size consistency depends on the appropriate partition of Hamiltonian [4, 96, 108].

For example, in Complete Active Space Perturbation Theory (CASPT) [3, 4, 91]

the so-called generalized Fockian is applied as a zeroth-order Hamiltonian:

F gen = O F CAS O + QSD FCAS QSD + QTQ F

CAS QTQ + . . . , (2.75)

where O projects onto one of the CAS states and QSD (QTQ) projects onto the

space of single- and double (triple- and quadruple) excited functions. Operator

F CAS is a one-electron operator such that

F CAS =∑

ij

fCAS

ij Eij ,

where matrix element fCASij has the same form as the Hartree-Fock counterpart

(Eq.(2.17)). However, in this case the density matrix of the CAS wave function is

used to build fCASij as

fCAS

ij = hij +∑

lm

(

[im|jl] − 1

2[im|lj]

)

P CAS

lm .

Projector O is introduced in Eq.(2.75) so that the CAS wave function becomes an

eigenfunction of the zeroth order Hamiltonian (Eq.(2.75)):

F gen|ΨCAS〉 = O F CAS O|ΨCAS〉 = E(0)CAS|ΨCAS〉 .

While the projectors QSD and QTQ are applied to obtain a simple structure for the

zeroth-order Hamiltonian. However, due to this block-diagonal form, the energy

corrections are not separable among the independent subsystems (size inconsis-

tency) [4, 108].

Dyall introduced another zeroth-order operator, where the two-electron terms

are also included so that the multireference function can be an eigenfunction of

the zeroth-order Hamiltonian without any projection to subspaces [108]. This


generalization is able to ensure the size consistency as well. A similar zeroth-order

Hamiltonian is used in N-Electron Valence State Perturbation Theory (NEVPT)

[96]. However, in this case the evaluation of the zeroth-order Hamiltonian becomes

more laborious due to the complicated structure of the zeroth-order operator.

In APSG the zeroth-order operator can be naturally considered as the effective

Hamiltonian (Eq.(2.36)) [109, 110]. In addition to this, similarly to Dyall’s zeroth-

order operator [108] it contains two-electron operators as well, which ensures that

the APSG wave functions are the eigenfunctions of this operator Eq.(2.35). There-

fore, the size consistency of this method is guaranteed. Despite of the two-electron

terms, the evaluation of matrix elements is easier than in case of Dyall’s pertur-

bation theories due to the strong orthogonality approximation. The first efficient

implementation of this perturbation theory was performed by Rosta and Surjan

[111], which has been recently reformulated in Density Matrix Functional Theory

[112] and in Block Correlated Perturbation Theory [113] as well.

Alternative perturbation methods for APSG are also known. Rassolov et al.

introduced two simplified perturbation theories to examine the error caused by

the strong orthogonality approximation [114, 115]. These corrections do not give

any significant contributions to the energy in the small molecule examples used

by them.

Another alternative philosophy is followed by Head-Gordon et al., where the

previously mentioned Coupled Cluster like structure of the APSG wave function

(c.f. Appendix A) is used. In that case, the APSG cluster operator expansion

is completed by additional excitations, which are derived from the Valence Bond

Theory and solved by standard single-reference Coupled Cluster techniques [116–

119].

APSG is also examined with those general multi-reference perturbation meth-

ods, which do not use the geminal structure of the wave function explicitly [93, 120–

122]. Detailed descriptions of these methods and the previously mentioned APSG

based techniques can be found in Refs. [63, 123].

In order to obtain better dynamical correlation treatment, alternative APSG


based methods are investigated such as Multi-Reference Configuration Interac-

tion (MRCI) [124, 125], Multi-Reference Coupled Cluster (MRCC) [126], Multi-

Reference Linearized Coupled Cluster (MR-LCC) [127] and Extended Random

Phase Approximation methods [128].

2.2.2 State-Specific Multireference Perturbation Theory

The State-Specific Multireference Perturbation Theory (SSMRPT) [97–100] is

usually classified as a ”perturb-then-diagonalize” method, because the perturba-

tive corrections are obtained by the diagonalization of the effective Hamiltonian.

However, in that case only one physically relevant solution is achievable (state-

specificity). The SSMRPT has some advantageous features such as size extensivity

and intruder independence [98], therefore it was successfully applied to describe

bond dissociation [129, 130], excited and ionized sates [131] and radical structures

[132]. Compared to the other multireference perturbation methods it has simi-

lar characteristics with respect to the accuracy of calculations [133, 134]. The

SSMRPT is obtained by the quasi-linearization of State-Specific Multireference

Coupled Cluster Theory (SSMRCC) [135, 136] thoroughly examined in the last

few years [137–142]. The main problem with SSMRCC is the weak coupling among

virtual functions [139, 142], which can cause serious problems in case of property

calculations [141].

The wave operator, Ω in SSMRCC is defined by the Jeziorski-Monkhorst Ansatz

[143]:

ΩSSMRCC =∑

µ

eTµ |Φ0µ〉〈Φ0µ| , (2.76)

where µ runs over the model space and T µ is the cluster operator generating the

outer space (Q-space) functions from the model space function |Φ0µ〉. T µ can be

partitioned according to excitation ranks as:

T µ =N∑

n=1

T µn ,


where operator T µn is an n-rank excitation operator. In the following only the first

and second order excitation ranks are considered, which can be given in second

quantized form as:

T µ1 =∑

ai

∑

σ

taσiσ (µ) ϕ+aσϕ

−iσ , (2.77)

T µ2 =1

4

∑

abij

∑

σσ′

(2 − δσσ′) taσbσ′

iσjσ′(µ) ϕ+

aσϕ+bσ′ϕ

−jσ′ϕ

−iσ , (2.78)

where taσiσ (µ) and taσbσ′

iσjσ′(µ) are the so-called amplitudes, i, j (a, b) stand for the

occupied (virtual) indices of |Φ0µ〉. In Eqs.(2.77) and (2.78) only those excitations

are included, which generate Q-space functions from function |Φ0µ〉 and factor

2−δσσ′

4in Eq.(2.78) is introduced to eliminate redundancies.

The wave operator of SSMRPT can be obtained by linearizing the cluster op-

erator in Eq.(2.76) as:

ΩSSMRPT =∑

µ

(

1 + T µ)

|Φ0µ〉〈Φ0µ| = O +∑

µ

T µ|Φ0µ〉〈Φ0µ| , (2.79)

where the definition of operator O is used. Substituting Eq.(2.79) into Eq.(2.64)

the following expression can be obtained for the effective Hamiltonian:

Heff

νµ = 〈Φ0ν |H|Φ0µ〉 + 〈Φ0ν |HT µ|Φ0µ〉 . (2.80)

In order to determine the amplitudes in operator T λ, let us substitute Eq.(2.76)

into Eq.(2.61):

∑

µ

HeTµ |Φ0µ〉 cµI = EI

∑

µ

eTµ |Φ0µ〉 cµI . (2.81)

Inserting I = eTµ(

O + Q)

e−Tµ

in front of H in Eq.(2.81):

∑

µk

eTµ |Υk〉Hµ

kµ cµI +∑

µν

eTµ |Φ0ν〉Hµ

νµ cµI = EI∑

µ

eTµ |Φ0µ〉 cµI , (2.82)


while considering the function |Υk〉 from the Q-space (Q =∑

k |Υk〉〈Υk|) with the

following notations:

Hµ

= e−Tµ

H eTµ

,

Hµ

kµ = 〈Υk|Hµ|Φ0µ〉 ,

Hµ

νµ = 〈Φ0ν |Hµ|Φ0µ〉 .

Interchanging indices µ and ν in the second term of Eq.(2.82), we obtain:

∑

µ

(∑

k

eTµ |Υk〉Hµ

kµ cµI +∑

ν

eTν |Φ0µ〉Hν

µν cνI − EI eTµ |Φ0µ〉 cµI

)

︸︷︷︸

Zµ

= 0 . (2.83)

In SSMRCC Eq.(2.83) is satisfied by setting Zµ = 0 for every µ, which means the

following:

∑

k

eTµ |Υk〉Hµ

kµ cµI +∑

ν

eTν |Φ0µ〉Hν

µν cνI − EI eTµ |Φ0µ〉 cµI = 0 . (2.84)

Eq.(2.84) is also referred to as sufficiency condition, because it produces sufficient

number of equations to determine the amplitudes. However, this sufficiency con-

dition is arbitrary and other types of sufficiency conditions are known to exist

[144–146]. Projecting Eq.(2.84) onto 〈Υl|e−Tµ

, we obtain the SSMRCC amplitude

equation:

Hµ

lµ cµI +∑

ν

〈Υl|e−Tµ

eTν |Φ0µ〉Hν

µν cνI = 0 . (2.85)

After linearizing the exponential cluster operators in Eq.(2.85) and omitting all

the non-linear terms in T µ, we get the following expression:

HlµcµI + 〈Υl|[

H, T µ]

|Φ0µ〉cµI +∑

ν

(tνlµ − tµlµ

)Hµν cνI = 0 , (2.86)


where

tνlµ = 〈Υl|T ν |Φ0µ〉 , (2.87)

tµlµ = 〈Υl|T µ|Φ0µ〉 . (2.88)

The matrix elements tνlµ (tµlµ) can be assigned to an amplitude taσiσ (µ) or taσb′σiσj′σ

(µ)

due to which excitation makes |Υl〉 from |Φ0µ〉. Using partition H = H0 + V into

Eq.(2.90), where H0 has the following diagonal form:

H0 =∑

µ

E(0)µ |Φ0µ〉〈Φ0µ| +

∑

k

E(0)k |Υk〉〈Υk| , (2.89)

we obtain

VlµcµI +(

E(0)l − E(0)

µ

)

tµlµcµI +∑

k

(Vlkt

µkµ − tµlkVkµ

)cµI −

−∑

ν

tµlνVνµcµI +∑

ν

(tνlµ − tµlµ

)Vµν cνI = 0 . (2.90)

Substituting the zeroth-order coefficients into Eq.(2.90) and omitting second or

higher-order terms, we obtain the following expression:

(

E(0)l − E(0)

µ

)

tµ(1)lµ c

(0)µI = −Vlµc(0)µI , (2.91)

where tµ(1)lµ can be determined by dividing to

(

E(0)l − E

(0)µ

)

c(0)µI as:

tµ(1)lµ = − 1

(

E(0)l − E

(0)µ

)

c(0)µI

Vlµc(0)µI . (2.92)

However, Eq.(2.92) suffers from the intruder problem due to the zero-division at

E(0)l = E

(0)µ . In order to avoid this singularity a second order term is included

as well [99], which modifies the original first-order expression (Eq.(2.91)) in the

following way:

(

E(0)l − E(0)

µ

)

tµ(1)lµ c

(0)µI +

∑

ν

(

tν(1)lµ − t

µ(1)lµ

)

Vµν c(0)νI = −Vlµc(0)µI . (2.93)


Eq.(2.93) is called Rayleigh-Schrodinger type SSMRPT amplitude equation, which

Brillouin-Wigner type formulation is also known [99].

The second-order energy of SSMRPT is usually calculated in two different man-

ners. One possibility is to determine the amplitudes from Eq.(2.93) and substi-

tute them into Eq.(2.80), where the non-hermitian eigenvalue equation provides

the second order energy with ”unrelaxed coefficients” [98, 147]. Another possibil-

ity is to solve Eq.(2.93) and Eq.(2.80) iteratively, where the amplitudes and the

coefficients are optimized. The obtained energy is called second-order SSMRPT

energy with ”relaxed coefficients” [98, 99]. In both cases the energy is obtained

by diagonalizing the effective Hamiltonian, which provides as many eigenvalues

as the dimension of the model space. However, from these energies only one has

physical relevance, which is determined by the zeroth-order coefficients (c(0)νI ).

In addition to the first-order amplitude equation contains second-order term

as well (Eq.(2.93)), the another discrepancy between SSMRPT and the standard

perturbation theory is that the zeroth-order coefficients are obtained from a CAS

calculation. However, the zeroth-order Hamiltonian is typically not a CAS Hamil-

tonian [99, 147]. Evangelista et al. developed an alternative formulation of SSM-

RPT, where a more rigorous treatment of the perturbation is introduced, i.e. it is

calculated order-by-order [148].

Spin-adapted SSMRPT (SA-SSMRPT) is also introduced to reduce the dimen-

sion of the model space and the interacting virtual space [149]. Recently, Mao

et al. implemented an effective SA-SSMRPT [147], which was also extended with

the explicit electron correlation [150].

One of the substantial difficulty with SSMRPT is that it can produce unphysical

kinks on the potential energy surfaces [134, 147, 151]. Some of these kinks are

mainly related to the small CAS coefficients, which analogously to the intruder

problems can decrease the value of the denominator in Eq.(2.92) to zero providing

singularities. Similar irregular properties appear in SSMRCC theory [138], where

the kinks can be eliminated by using Tikhonov regularization [152] as

1

c(0)µ

→ c(0)µ

c(0)µ

2+ ω2

, (2.94)


where ω is the damping parameter. This regularization can eliminate most kinks

in SSMRPT as well [147, 151], however, some kinks still remain when spin-adapted

theory is applied. These remaining kinks can be related to the possible redundan-

cies in the amplitude equation, which is outlined in the following Chapter.

Chapter 3

Redundancy in Spin-Adapted

SSMRPT

In this Chapter the previously mentioned redundancy problems of Spin Adapted

SSMRPT (SA-SSMRPT) are discussed in details. First, the formalism of SA-

SSMRPT is introduced, then the way it leads to redundancies in amplitude equa-

tions is shown. These redundancies are eliminated by canonical orthogonaliza-

tion. The differences from the original SA-SSMRPT are also presented with some

demonstrative examples. The following discussion is based on Ref. [1].

3.1 Spin-adaptation in SSMRPT

Usually in quantum chemical calculations the Hamilton operator is independent

of spin, therefore the following commutation relation holds:

[

H, S2]

= 0 , (3.1)

where S2 is the total spin-squared operator. Due to relation Eq.(3.1), the wave

function is eigenfunction of operator S2 as well. Therefore it seems reasonable to

choose spin-adapted function as basis in order to eliminate the unnecessary com-

ponents belonging to different spin eigenfunctions. However, the determinants are

generally not spin eigenfunctions, therefore Configuration State Functions (CSF)

are defined [19], where these spin-adapted basis functions are obtained as a linear

36

Chapter 3. Redundancy in Spin-Adapted SSMRPT 37

combination of determinants:

|ΦCSF

0µ 〉 =∑

ν

dνµ|Φ0ν〉 ,

where |ΦCSF0µ 〉 is the CSF and dνµ is the linear expansion coefficient.

Let us consider the Complete Active Space for the model space, where the

determinants are unitary transformed to the spin-adapted (CSF) basis. Therefore

the cost of the calculation can be reduced by considering only those functions,

which belong to the same spin-states.

In order to spin-adapt the Q-space function as well, we express T µ with the

operators preserving the total spin [149, 153, 154]. The unitary group generators

(Eq.(2.3)) commute with operator S2 [19] by producing a spin eigenfunction:

S2Eqp |ΦCSF

0µ 〉 = EqpS

2|ΦCSF

0µ 〉 = S(S + 1)Eqp |ΦCSF

0µ 〉 .

where S is the spin quantum number. However, this is only one possibility for

spin-adaptation and other techniques are also known in the quantum chemical

literature [155, 156].

In the determinant based theory (Eqs.(2.77) and (2.78)) the spin orbitals are

classified in T µ into two groups according to they are occupied or unoccupied in

|Φ0µ〉. Using the CSF as a reference function, this occupied-virtual classification of

spin orbitals is ambiguous if the CSF is constituted of more than one determinant.

However, this classification can be done for the spatial orbitals, the notions of

which are introduced in the following:

p, q, · · · general indices,

i, j, · · · doubly occupied indices,

a, b, · · · unoccupied indices,

ud, vd, · · · active double occupied indices,

us, vs, · · · active single occupied indices,

uz, vz, · · · active unoccupied indices.


In order to gain a deeper insight into this classification problem, let us examine

the open-shell singlet case of two active electrons (|ΦCSF0µ 〉) as:

|ΦCSF

0µ 〉 =1√2

ϕ+usαϕ

+vsβ

|Φc〉︸︷︷︸

|Φ10µ〉

+ ϕ+vsαϕ

+usβ

|Φc〉︸︷︷︸

|Φ20µ〉

, (3.2)

where us 6= vs and |Φc〉 contains double occupied orbitals (core part) only. It is

apparent that in Eq.(3.2) one of spin the functions associated with ϕus and ϕvs is

occupied in |Φ10µ〉, and the other spin function is unoccupied in |Φ2

0µ〉. Therefore,

to generate all interacting Q-space functions, single occupied orbitals should be

included in both creation and annihilation part of T µ. However, it can be visible

by simple substitution of operator Eqp (Eq.(2.3)) that in that case the excitation

operators do not commutate with each other:

[

Eaus , E

usi

]

= Eai .

It leads to difficulties in parent coupled cluster theory [154], but fortunately these

problems do not appear in the perturbation counterparts [149] due to the linear

parametrization.

Using the spin-symmetry relation:

tqαpα(µ) = tqβpβ(µ) = tqp(µ) , (3.3)

Eq.(2.77) can be written in the following form:

T µ1 =∑

i

∑

a

tai (µ)

Eai

c+∑

i

∑

us

tusi (µ)

Eusi

c+∑

us

∑

a

taus(µ)

Eaus

c,(3.4)

where c denotes the normal ordering to the common core part of the reference

function |ΦCSF0µ 〉 [147]. In Eq.(3.4) similarly to Eq.(2.77) the excitations are ex-

cluded, when both indices belong to the active part, to avoid the model space

function generation. Because the single occupied orbitals are in the active part,

we also have to exclude those excitations, where two single occupied orbitals are


incorporated Eq.(3.4).

In double excitations we can use the similar spin-symmetry consideration as in

Eq.(3.3), which provides the following relations to tqspr as:

tqαsαpαrα(µ) = tqαsβpαrβ(µ) = t

qβsαpβrα(µ) = t

qβsβpβrβ(µ) = tqspr(µ) .

Therefore T µ2 can be given as:

T µ2 =1

2

∑

ij

∑

ab

(1 + δijδab)tabij (µ)

Eabij

c+

+∑

ij

∑

aus

tausij (µ)

Eausij

c+∑

ius

∑

ab

tabius(µ)

Eabius

c+

+1

2

∑

ij

∑

usvs

tusvsij (µ)

Eusvsij

c+

1

2

∑

usvs

∑

ab

tabusvs(µ)

Eabusvs

c+

+∑

ius

∑

avs

tavsius(µ)

Eavsius

c+∑

ius

∑

avs

tvsaius (µ)

Evsaius

c+

+∑

ius

∑

vsws

tvsws

ius(µ)

Evsws

ius

c+∑

usvs

∑

aws

tawsusvs(µ)

Eawsusvs

c, (3.5)

where

Eqspr

c=

EqpE

sr

c.

The factor 12(1 + δijδab) is introduced to eliminate a redundancy from

Eabij

c=

Ebaji

c. However, the factor 1

2is enough at excitations

Eusvsij

cand

Eabusvs

c

since this excitation provides zero when us = vs.

3.2 Redundancy in the amplitude equations

Let us start our discussion with deriving the simplified form of the determinant

based amplitude equation. As it can be seen, the last term in Eq.(2.93) is zero

when ν = µ due to the factor tν(1)lµ − t

µ(1)lµ . Therefore, by adding the zero term to


Eq.(2.93) and abandoning the state index I,

∑

ν

(

tν(1)lµ − t

µ(1)lµ

)

c(0)ν E(0)µ δµν = 0 ,

we recognize the matrix element of the Hamiltonian:

(

E(0)l − E(0)

µ

)

tµ(1)lµ c(0)µ +

∑

ν

(

tν(1)lµ − t

µ(1)lµ

) (Vµν + E(0)

µ δµν)

︸︷︷︸

Hµν

c(0)ν = −Vlµc(0)µ . (3.6)

Using the CAS Hamiltonian eigenvalue equation:

−tµ(1)lµ

∑

ν

(Vµν + E(0)µ δµν)

︸︷︷︸

Hµν

c(0)ν = −tµ(1)lµ ECASc(0)µ ,

Eq.(2.93) can be written in the simplified form:

∑

ν

(

Hµν + (E(0)l − E(0)

µ − ECAS)δµν

)

︸︷︷︸

Alµν

c(0)ν tν(1)lµ = −Vlµc(0)µ . (3.7)

As mentioned earlier in Chapter 2, when the cluster operators are defined by

Eqs.(2.77) and (2.78), tν(1)lµ can be assigned to amplitude t

A(1)I (ν), where I (A)

stands for the annihilation (creation) indices of the excitation. Using indices Iand A instead of index l, Eq.(3.7) can be written in the following form:

∑′

ν

AIAµν c(0)ν t

A(1)I (ν) = − VIAµ,µ c

(0)µ , (3.8)

where the prime stands for that the summation index do not run for the whole

active space at every excitation. When index I (A) contains an active index, it

can be unoccupied (occupied) in some reference functions, therefore this excitation

is not included in the definition of T µ. Multiplying Eq.(3.8) from the left with(AIA)−1

λµand summing up to µ, t

A(1)I (λ) can be expressed as:

tA(1)I (λ) = − 1

c(0)λ

∑′

µ

(AIA)−1

λµVIAµ,µ c

(0)µ . (3.9)


In a spin-adapted theory the above derivation cannot be applied due to the pos-

sibility to assign more than one amplitude to tν(1)lµ . In order to gain a deeper

insight into this, let us examine the effects of Eavsius

c and Evsaius

c on the following

open-shell singlet state |ΦCSF0µ 〉 (Eq.(3.2)) as:

|µΥavsius

CSF〉 =1

2Eavs

iusc|ΦCSF

0µ 〉 =1√2

(ϕ+aαϕ

+iβ + ϕ+

iαϕ+aβ

)ϕ+vsαϕ

+vsβ

|Φic〉 , (3.10)

|µΥvsaius

CSF〉 = Evsaius

c|ΦCSF

0µ 〉 = − 1√2

(ϕ+aαϕ

+iβ + ϕ+

iαϕ+aβ

)ϕ+vsαϕ

+vsβ

|Φic〉 , (3.11)

where the factor 12

is introduced for normalization and

|Φic〉 = ϕ−

iβϕ−iα|Φc〉 .

As it can be seen in Eqs.(3.10) and (3.11), functions |µΥavsius

CSF〉 and |µΥvsaius

CSF〉 have

only a sign factor difference as:

|µΥavsius

CSF〉 = −|µΥvsaius

CSF〉 ,

therefore by calculating the matrix element tµlµ (|ΥCSFl 〉 = |µΥavs

iusCSF〉), we obtain

both amplitudes as:

tµ(1)lµ = 〈ΥCSF

l |T µ|ΦCSF

0µ 〉 = tavsius(1)(µ) − tvsaius

(1)(µ) . (3.12)

This type of redundancy has already known in internally contracted multirefer-

ence perturbation theory [4, 90, 94] and it is typically solved by canonical orthog-

onalization [157]. Mukherjee et al. [147, 149, 158] introduced extra redundancy

conditions, where the direct couplings (Eq.(3.12)) between the amplitudes are

eliminated and similarly to the determinantal case every amplitude is determined

by Eq.(3.9). The only exception is the handling of excitation

Equspus

c, which pro-

vides the same excited functions as the corresponding single excitation (

Eai

c),

when they act on their own reference function as:

|µΥausius

〉 =

Eausius

c|ΦCSF

0µ 〉 = Eai E

usus |ΦCSF

0µ 〉︸︷︷︸

|ΦCSF0µ 〉

=

Eai

c|Φ0µ〉 = |µΥa

i 〉 .


This type of excitation, when two indices are equivalent at the same spin part

(Equspus ), is called direct spectator. However, these two excitations are not com-

pletely equivalent. As it can be seen in the amplitude equation (Eq.(2.93)), the

effect of operator T µ is not restricted to function |ΦCSF0µ 〉, it can effects on all model

space functions as well. When the ϕusσ is an unoccupied orbital in function |ΦCSF0ν 〉

then the operator

Equspus

cerases the function |ΦCSF

0ν 〉 as:

|νΥausius

CSF〉 =

Eausius

c|ΦCSF

0ν 〉 = Eai E

usus |ΦCSF

0ν 〉︸︷︷︸

0

= 0 .

In order to eliminate redundancy and maximize the coupling between the ampli-

tudes the direct spectators are simply omitted from the definition of the cluster

operator [147, 149]. We also consider the above assumption for the direct spec-

tators to follow the original theory, but the remaining redundancies are removed

by canonical orthogonalization. This method will be outlined in detail in the next

section.

In the following part of this Chapter we work only with CSF’s. For simplifying

the notations the index CSF in |ΦCSF0µ 〉 and |νΥaus

iusCSF〉 is abandoned. We also

consider only the first order amplitudes, therefore the label (1) is also abandoned

in what follows.

3.3 Removal of redundancies in T µ

In this section we modify the original definition of T µ by introducing an operator

ˆT µ, where the excitations are linearly independent and orthogonal to each other.

The elimination of redundancies is based on the canonical orthogonalization [157]

of virtual functions, where we consider only the case of two active electrons.

Let us start our examination with the single excitations in T µ (T µ1 ), where the

excitations are distinguished by the occupations of the spatial orbitals, which are

incorporated into the excitations. When our reference function has a closed-shell

structure, there is only one type of excitation, which annihilates one electron from

doubly occupied orbital and creates an electron on an unoccupied orbital (first

term in Eq.(3.4)). In that case, the CSF remains a one-determinant function,


therefore no redundancy arises and we do not need to modify the original cluster

operator definition. However, if the reference function is an open-shell CSF, we

have the all types of excitations (Eq.(3.4)). In the open-shell case of two active

electron, these three excitations are distinguished by the following notations:

core → empty , from double occupied orbital to unoccupied orbital ,

core → active(1) , from double occupied orbital to single occupied orbital ,

active(1) → empty , from single occupied orbital to unoccupied orbital .

As mentioned in the previous section, the single excitations and the direct spec-

tator excitations generate the same Q-space functions form the reference function,

therefore the direct spectators are omitted from the definition of Eq.(3.5). How-

ever, this subspace remains linearly dependent due to the so-called exchange spec-

tators (Eusqpus ). First, let us examine the core→ empty excitation, where the three

linearly dependent excited functions have the following forms:

|µΥai 〉 =

Eai

c√2

|Φ0µ〉 =1

2

(ϕ+aαϕ

+iβ + ϕ+

iαϕ+aβ

) (ϕ+uαϕ

+vβ + ϕ+

vαϕ+uβ

)|Φi

c〉, (3.13)

|µΥusai us

〉 =

Eusaius

c√2

|Φ0µ〉 =1

2

(ϕ+uαϕ

+iβ + ϕ+

iαϕ+uβ

) (ϕ+aαϕ

+vβ + ϕ+

vαϕ+aβ

)|Φi

c〉, (3.14)

|µΥvsai vs

〉 =

Evsaivs

c√2

|Φ0µ〉 =1

2

(ϕ+vαϕ

+iβ + ϕ+

iαϕ+vβ

) (ϕ+uαϕ

+aβ + ϕ+

aαϕ+uβ

)|Φi

c〉, (3.15)

where in the derivation the expression in Eq.(3.2) is used. In canonical orthogo-

nalization [157] the linearly independent orthogonal vectors are obtained by diag-

onalization of the overlap matrix. Using Eqs.(3.13)-(3.15), the overlap matrix can

evaluated, for example the matrix element S12(= 〈µΥai |µΥusa

i us〉) can be obtained


by applying Eqs.(3.13) and (3.14) as:

S12 =1

4

(

〈Φic|ϕ−

vβϕ−uαϕ

−iβϕ

−aαϕ

+uαϕ

+iβϕ

+aαϕ

+vβ|Φi

c〉︸︷︷︸

−1

+ (3.16)

+ 〈Φic|ϕ−

uβϕ−vαϕ

−aβϕ

−iαϕ

+iαϕ

+uβϕ

+vαϕ

+aβ|Φi

c〉︸︷︷︸

−1

)

,

S12 = −1

2.

The remaining matrix element of the overlap matrix can be determined in a similar

manner:

S =

1 −12

−12

−12

1 −12

−12

−12

1

. (3.17)

The overlap matrix Eq.(3.17) has two non-zero degenerate eigenvalues, the eigen-

vectors of which construct a two dimensional subspace. Due to this degeneracy,

we have a unitary freedom in selecting the eigenvectors. We choose the following

two orthogonal excited functions:

|µ1Υai 〉 =

2

3

(

− |µΥai 〉 +

1

2

(|µΥusa

i us〉 + |µΥvsa

i vs〉))

,

|µ2Υai 〉 =

1√3

(|µΥusa

i us〉 − |µΥvsa

i vs

), u < v ,

where the restriction u < v is introduced for the unequivocally of |µ2Υai 〉, which

changes sign when indices us and vs are interchanged. The original and linearly

independent excited functions are collected in Table 3.1 along with those belonging

to the remaining single excitations (core→ active(1) and active(1) → empty ).

Applying the obtained linearly independent excitations, the new cluster opera-

tor can be constructed as:

ˆT µ1 (us < vs; i→ a) = 11tai (µ)

[

−

Eai

c+

1

2

(

Eusaius

c+

Evsaivs

c

)]

+

+ 22tai (µ)

(

Eusaius

c−

Evsaivs

c

)

. (3.18)


Table 3.1: Overlapping sets of normalized excited functions and their or-thonormalized counterparts for single excitations. Model function |Φ0µ〉 is atwo-determinantal open-shell, as given by Eq.(3.2). See text for labeling con-

vention.

overlapping functions orthonormal functions

core→empty

|µΥai 〉 = 1√

2

Eai

c|Φ0µ〉

|µΥusai us

〉 = 1√2

Eusaius

c|Φ0µ〉

|µΥvsai vs

〉 = 1√2

Evsaivs

c|Φ0µ〉

|µ1Υai 〉 = 2

3

(− |µΥa

i 〉 + 12

(|µΥusa

i us〉 + |µΥvsa

i vs〉))

|µ2Υai 〉 = 1√

3

(|µΥusa

i us〉 − |µΥvsa

i vs

), u < v

core→active(1)

|µΥusi 〉 =

Eusi

c|Φ0µ〉

|µΥvsusivs

〉 = 12

Evsusivs

c|Φ0µ〉

|µΥusi 〉 = 1

2

(|µΥus

i 〉 − |µΥvsusivs

〉)

active(1)→empty

|µΥaus〉 =

Eaus

c|Φ0µ〉

|µΥvsausvs〉 =

Evsausvs

c|Φ0µ〉

|µΥaus〉 = 1

2

(|µΥa

us〉 + |µΥvsausvs〉

)

The connection between the original cluster operator expansion becomes apparent,

when the expression in Eq.(3.18) is reordered by the excitation operators as:

ˆT µ1 (us < vs; i→ a) =−11tai (µ)

︸︷︷︸

tai (µ)

Eai

c+

(1

211tai (µ) + 2

2tai (µ)

)

︸︷︷︸

tusai us(µ)

Eusaius

c,

+

(1

211tai (µ) − 2

2tai (µ)

)

︸︷︷︸

tvsai vs(µ)

Evsaivs

c, (3.19)

where the connections between amplitudes t and the new amplitudes t are collected

in Table 3.2 as well.

Similarly, the remaining linearly independent single excitations can be derived

with the overlap matrices diagonalization (Table 3.1). The cluster operators of

the core → active(1) and active(1) → empty excitation parts have the following


Table 3.2: Relation between cluster amplitudes in the redundant parametriza-

tion Eq.(3.4) of Tµ and the orthogonal parametrization of Eqs.(3.18), (3.20) and(3.21). Case of single excitations. Model function |Φ0µ〉 is a two-determinantal

open-shell, as given by Eq.(3.2). See text for labeling convention.

core → empty

tai (µ) = − 11tai (µ)

tusai us(µ) = 1

211tai (µ) + 2

2tai (µ)

tvsai vs(µ) = 1211tai (µ) − 2

2tai (µ)

core → active(1 )

tusi (µ) = tusi (µ)

tvsusivs(µ) = −1

2tusi (µ)

active(1 ) → empty

taus(µ) = taus(µ)

tavsvsus(µ) = taus(µ)

form:

ˆT µ1 (i→ us) = tusi (µ)

(

Eusi

c− 1

2

Evsusivs

c

)

(3.20)

ˆT µ1 (us → a) = taus(µ)(

Eaus

c+

Evsaus vs

c

)

. (3.21)

Additional linearly dependency can be found in double excitation parts, where

the 2 cores→2 active(1)’s, 2 active(1)’s→2 empties and core, active(1)→active(1),

empty excitations suffer from this problem. The linear dependent and indpendent

virtual functions can be found in Table 3.3, with which the new cluster operators

can be defined:

ˆT µ2 (ij → usvs) = tusvsij (µ)(

Eusvsij

c+

Evsusij

c

)

, i < j ,

ˆT µ2 (usvs → ab) = tabusvs(µ)(

Eabusvs

c+

Ebausvs

c

)

, i < j ,

ˆT µ2 (ius → vsa) = tavsius(µ) (−1)p(vsa)

(

Evsaius

c− 1

2

Eavsius

c

)

,

where p(vsa) stands for the parity of the permutation ordering of pair (ua). The

connection to the original amplitudes can be found in Table 3.4.

Although the remaining excitations generate linearly independent functions,

they can overlap each other. These overlapping functions can be applied by mod-

ifying the projector Q expression as:

Q =∑

kl

|Υk〉(S)−1kl 〈Υl|,


Table 3.4: Relation between cluster amplitudes in the redundant parametriza-

tion of Tµ and the orthogonal parametrization of ˆTµ. Case of double excitations.Model function |Φ0µ〉 is a two-determinantal open-shell, as given by Eq.(3.2).See text for labeling convention. Index ordering i < j, a < b and u < v isassumed. Notation p(ua) stands for the parity of the permutation ordering the

pair (ua).

2 cores→2 empties

tabij (µ) = 11tabij (µ) + 2

2tabij (µ)

tbaij (µ) = 11tabij (µ) − 2

2tabij (µ)

2 cores→active(1), empty

tusaij (µ) = 11tusaij (µ) + (−1)p(usa) 22t

usaij (µ)

tausij (µ) = 11tusaij (µ) − (−1)p(usa) 22t

usaij (µ)

core, active(1)→2 empties

tabius(µ) = 11tabius(µ) + 2

2tabius(µ)

tbaius(µ) = 11tabius(µ) − 2

2tabius(µ)

2 cores→2 active(1)’s

tusvsij (µ) = tusvsij (µ)tvsusij (µ) = tusvsij (µ)

2 active(1)’s→2 empties

tabusvs(µ) = tabusvs(µ)tbausvs(µ) = tabusvs(µ)


tavsius(µ) = (−1)p(vsa) 1

2tvsaius (µ)

tvsaius (µ) = (−1)p(vsa) tvsaius (µ)

and similarly the amplitude and effective Hamiltonian equations are changed as

well. Here the orthogonality of the Q-space functions are used to resolved this

simplified form of working equation. However, the handling of these excitations

can be very tedious, which prevent one to apply this method for systems with

large number of active electrons.

In Tables 3.3 can also be found those functions, which are linearly independent,

but they are not orthogonal to each other (2 cores→2 empties, 2 cores→active(1),

empty and core, active(1)→2 empties). Using the same canonical orthogonaliza-

tion procedure orthogonal excited functions are constructed (Table 3.3), which


provide the following cluster operators:

ˆT µ(ij → ab) = 11tabij (µ)

(

Eabij

c+

Ebaij

c

)

+ 22tabij (µ)

(

Eabij

c−

Ebaij

c

)

ˆT µ(ij → usa) = 11tusaij (µ)

(

Eusaij

c+

Eausij

c

)

+ 22tusaij (µ) (−1)p(usa)

(

Eusaij

c−

Eausij

c

)

,

ˆT µ(ius → ab) = 11tabius(µ)

(

Eabius

c+

Ebaius

c

)

+ 22tabius(µ)

(

Eabius

c−

Ebaius

c

)

.

It is important to note that the orthogonality problem is not restricted to the

open-shell case, as 2 cores→2 empties excitation can be found at the closed-shell

CSF as well.

3.4 Redundancy-free amplitude equations

Let us review the determinant based amplitude equation again (Eqs.(3.7) and

(3.8)) in the following form:

∑′

ν

Hµν c(0)ν C (µ, I → A; ν) tAI (ν) = − 〈 µΥA

I |V |Φ0µ〉 c(0)µ , (3.22)

where

C (µ, I → A; ν) = 1 +1

Hµµ

(XAI (µ) − ECAS)δµν , (3.23)

XAI (µ) = 〈µΥA

I |H(0)|µΥAI 〉 − 〈Φ0µ|H(0)|Φ0µ〉 . (3.24)

As it can be seen in Eq.(3.22), the amplitude equations are decoupled according

to excitation I → A. As we mentioned in the previous subsection, these equations

are coupled when the unitary group generators are used as excitation operators.

While in determinantal case the matrix element 〈 µΥAI |T ν |Φ0µ〉 can be assigned to

one amplitude tAI (ν), in the spin-adapted case we obtain the linear combination


of amplitudes (Eq.(3.12)) as

〈 µΥAI |T ν |Φ0µ〉 =

∑

J ,B〈 µΥA

I |

EBJ

c|Φ0µ〉

︸︷︷︸

D(µ,I→A;ν,J→B)

tBJ (ν) , (3.25)

where the definition of T ν is the following:

T ν =∑

J ,BtBJ (ν)

EBJ

c. (3.26)

Modifying Eq.(3.22) to include these remaining amplitudes in Eq.(3.25), we obtain

∑′

ν

∑

J ,BHµν c

(0)ν C ′ (µ, I → A; ν,J → B) tBJ (ν) = − 〈 µΥA

I |V |Φ0µ〉 c(0)µ , (3.27)

where

C ′ (µ, I → A; ν,J → B) = C (µ, I → A; ν) D (µ, I → A; ν,J → B) . (3.28)

As it was discussed in the previous section, the unitary group based linearly

independent excitation belonging to the same reference CSF are orthogonal to

each other. However, this orthogonality does not hold for the excitations belonging

to different CSF’s. Therefore in that case, the amplitude equations also remain

coupled as:

∑′

ν

∑

J ,B

∑

g

Hµνc(0)ν C (µ, I → A, f ; ν,J → B, g) g

g tBJ (ν) =

= −〈µf ΥAI |V |Φ0µ〉c(0)µ , (3.29)


Table 3.5: Elements C of Eq.(3.30), for core→virtual excitation, i → a. Ab-breviations: ‘cs’ – closed-shell, ‘os’ – open-shell. Indices f and g are left blank,when not applicable. Virtual function |µf Υa

i 〉 is assumed to be normalized. See

text for labeling convention and the definition of ffXai (µ).

µ f ν g C (µ, i→ a, f ; ν, i→ a, g)

cl cl√

2 +√

2 δµν (Xai (µ) − ECAS) /Hµµ

cl, xd os, us < vs 1 −√

2(1 + 1

2(δxu + δxv)

)

cl, xd os, us < vs 2 −√

2(δxu − δxv)

os, xs < ys 1 cl −√

2

os, xs < ys 1 os, us < vs 1

√2(1 + 1

4(δxu + δyu + δxv + δyv)

)+

+ 3√2δµν (11X

ai (µ) − ECAS) /Hµµ

os, xs < ys 1 os, us < vs 2 1√2

(δxu − δxv + δyu − δyv)

os, xs < ys 2 cl 0

os, xs < ys 2 os, us < vs 1√3

2√2(δxu + δxv − δyu − δyv)

os, xs < ys 2 os, us < vs 2

√3√2

(δxu − δxv − δyu + δyv) +

+√

6 δµν (22Xai (µ) − ECAS) /Hµµ

where indices f and g are introduced according to notations in Tables 3.1 - 3.4

and the remaining quantities are defined in the following way:

C (µ, I → A, f ; ν,J → B, g) =

=

(

1 +1

Hµµ

(ffXAI (µ) − ECAS)δµν

)

︸︷︷︸

C(µ,I→A,f ;ν)

〈 µf ΥAI |gg

ˆEBJ (ν)

c|Φ0µ〉

︸︷︷︸

D(µ,I→A,f ;ν,J→B,g)

, (3.30)

ffX

AI (µ) = 〈µfΥA

I |H(0)|µfΥAI 〉 − 〈Φ0µ|H(0)|Φ0µ〉 , (3.31)

and operatorgg

ˆEBJ (ν)

cgenerates virtual function |νgΥB

J 〉 from reference function

|Φν〉. Let us determine D (µ, I → A, f ; ν,J → B, g) for excitation

Eai

cwhen i

is a core and a is a virtual orbital and |Φ0µ〉 has a following closed-shell structure:

|Φ0µ〉 = ϕ+xαϕ

+xβ|Φc〉 .


Table 3.6: Elements C of Eq.(3.30), for core→active excitations, i → w.Abbreviations: ‘cs’ – closed-shell, ‘os’ – open-shell. Indices f and g are leftblank, when not applicable. Virtual function |µf Υw

i 〉 is assumed to be normalized.

See text for labeling convention and the definition of ffXwi (µ).

µ f ν g C (µ, i→ w, f ; ν, i→ w, g)

cl, wz cl, wz√

2 +√

2 δµν (Xwi (µ) − ECAS) /Hµµ

cl, xd, wz os, us < vs, wz 1 −√

2(1 + 1

2(δxu + δxv)

)

cl, wz os, us < vs, wz 2 −√

2(δxu − δxv)

cl, wz os, us, ws√

2(1 + 12δxu)

os, xs < ys, wz 1 cl, wz −√

2

os, xs < ys, wz 1 os, us < vs, wz 1

√2(1 + 1

4(δxu + δxv + δyu + δyv)

)+

+ 3√2δµν(

11X

wi (µ) − ECAS)/Hµµ

os, xs < ys, wz 1 os, us < vs, wz 2 1√2

(δxu − δxv + δyu − δyv)

os, xs < ys, wz 1 os, us, ws −√

2(1 + 1

4(δxu + δyu)

)

os, xs < ys, wz 2 cl, wz 0

os, xs < ys, wz 2 os, us < vs, wz 1√3

2√2

(δxu + δxv − δyu − δyv)

os, xs < ys, wz 2 os, us < vs, wz 2

√3√2

(δxu − δxv − δyu + δyv) +

+√

6 δµν (22Xwi (µ) − ECAS) /Hµµ

os, xs < ys, wz 2 os, us, ws√3

2√2

(−δxu + δyu)

os, xs, ws cl, wz 1

os, xs, ws os, us < vs, wz 1 − (1 + δxu + δxv)

os, xs, ws os, us < vs, wz 2 2 (−δxu + δxv)

os, xs, ws os, us, ws (1 + δxu) + 2 δµν (Xwi (µ) − ECAS) /Hµµ


Table 3.7: Elements C of Eq.(3.30), for active→virtual excitations, i.e. w → a.

Abbreviations: ‘cs’ – closed-shell, ‘os’ – open-shell. Virtual functions |µΥaw〉 are

assumed to be normalized. See text for for labeling convention and the definitionof Xa

w(µ).

µ ν C (µ,w → a; ν, w → a)

cl, wd cl, wd√

2 +√

2 δµν (Xaw(µ) − ECAS) /Hµµ

cl, wd os, us, ws√

2

os, xs, ws cl, wd 1

os, xs, ws os, us, ws (1 + δxu) + 2 δµν (Xaw(µ) − ECAS) /Hµµ

The corresponding excited function can be determined by operating

Eai

con

function |Φ0µ〉 as:

|µΥai 〉 =

Eai

c|Φ0µ〉 =

(ϕ+iαϕ

+aβ + ϕ+

aαϕ+iβ

)ϕ+xαϕ

+xβ|Φi

c〉 . (3.32)

The excitation

Eai

ccan be found in every closed-shell function, therefore D can

be evaluated in the following way:

D (µ, i→ a; ν, i→ a) = 〈µΥai |

Eai

c|Φ0µ〉 =

√2 , (3.33)

where indices f and g are abandoned in the absence of other types of excitations in

this subspace. In the open-shell case there are two linearly independent excitations,

which generate overlapping functions with |µΥai 〉 as:

D (µ, i→ a;λ, i→ a, 1) = 〈µΥai |[

−

Eai

c+

1

2

(

Eusaius

c+

Evsaivs

c

)]

|Φ0µ〉 ,

D (µ, i→ a;λ, i→ a, 2) = 〈µΥai |(

Eusaius

c−

Evsaivs

c

)

|Φ0µ〉 ,

where |Φ0λ〉 stands for the open-shell CSF with us and vs singly occupied or-

bital. Using second quantized relations, we can derive the following expression to


Table 3.8: Elements C of Eq.(3.30), for non coupled double excitations, i.e.

f = g. Values of C are collected for the possible combinations of I → A withreference functions, |Φ0µ〉. Admissible types for CSF ν agree with types listed forµ. Description of CSF µ is indicated in the rows, indices I,A and f are given incolumn headers. Abbreviations: ‘cs’ – closed-shell, ‘os’ – open shell, ‘n.a.’ – notapplicable. Shorthand Y stands for ffY

abij (µ) = 1 + δµν(

ffX

abij (µ)− ECAS)/Hµµ).

See text for the definition of ffXabij (µ). Virtual functions |µf Υab

ij 〉 are assumed tobe normalized.

excitation 2 cores→2 virtuals: (i, j) → (a, b)i < j i = j i < j i = ja < b a < b a = b a = b

type of µ f = 1 f = 2

any 2 11Y

abij 2

√3 2

2Yabij

√2 Y ab

ii

√2 Y aa

ij 2 Y aaii

excitation 2 cores→active, virtual: (i, j) → (u, a)i < j i = ju < a u < a


cs or os, uz 2 11Y

uaij 2

√3 2

2Yuaij

√2 Y ua

ii

os, us√

2 11Y

uaij

√6 2

2Yuaij Y ua

ii

excitation core, active→2 virtuals: (i, u) → (a, b)i < u i < ua < b a = b


cs or os, ud 2 11Y

abiu 2

√3 2

2Yabiu

√2 Y aa

iu

os, us√

2 11Y

abiu

√6 2

2Yabiu Y aa

iu

excitation 2 cores→2 actives: (i, j) → (u, v)i < j i = j i < j i = ju < v u < v u = v u = v


cs or os, uz, vz 2 11Y

uvij 2

√3 2

2Yuvij

√2 Y uv

ii

√2 Y uu

ij 2 Y uuii

os, us, vz or uz, vs√

2 11Y

uvij

√6 2

2Yuvij Y uv

ii n.a. n.a.

os, us, vs 2 11Y

uvij 0

√2 Y uv

ii n.a. n.a.


Table 3.8: Cont’d.

excitation 2 actives→2 virtuals: (u, v) → (a, b)u < v

type of µ a < b a = b

os, us, vs 2 Y abuv

√2 Y aa

uv

excitation 2 actives→active, virtual: (u, v) → (w, a)u < v

type of µ w < aos, us, vs, wz 2 Y wa

uv

Table 3.9: Matrix C of Eq.(3.30), for core, active→2 actives excitations,(i, w) → (u, v). Description of CSF µ is indicated in the rows, together with in-dex f , when applicable. Characterization of CSF ν is given in column headers,together with index g, when applicable. Abbreviations: ‘cs’ – closed-shell, ‘os’– open shell. Shorthand Y stands for µfY

uviw = 1 + δµν(

ffX

uviw (µ) − ECAS)/Hµµ).

See text for the definition of ffXuviw (µ). Virtual functions |µf Υuv

iw〉 are assumedto be normalized. The table applies for the core, active→active, virtual excita-tions also, with a substituted for vz. Column and row referring to a vs is not

applicable in this case.

f g = 1 g = 2cs os cs os os oswd ws wd ws ws ws

uz < vz uz < vz uz < vz uz < vz us < vz uz < vscs, wd, uz < vz 1 2 µ

fYuviw 2 0 0 1

2− 1

2

os, ws, uz < vz 1√

2√

2 µfY

uviw 0 0 1

2√2

− 12√2

cs, wd, uz < vz 2 0 0 2√

3 µfY

uwiw 2

√3 3

√3

23√3

2

os, ws, uz < vz 2 0 0√

6√

6 µfY

uviw

3√3

2√2

3√3

2√2

os, ws, us < vz − 1 − 1 3 3 2 µY uviw

52

os, ws, uz < vs 1 1 3 3 52

2 µY uviw


D (µ, i→ a;λ, i→ a, 1) and D (µ, i→ a;λ, i→ a, 2):

D (µ, i→ a;λ, i→ a, 1) = −√

2

(

1 +1

2(δxu + δxv)

)

, (3.34)

D (µ, i→ a;λ, i→ a, 2) = −√

2(δxu − δxv) . (3.35)

Substituting Eqs.(3.33)-(3.35) into Eq.(3.30), C’s can be obtained as:

C (µ, i→ a; ν, i→ a) =

(

1 +1

Hµµ

(ffXAI (µ) − ECAS)δµν

)√2 , (3.36)

C (µ, i→ a;λ, i→ a, 1) = −√

2

(

1 +1

2(δxu + δxv)

)

, (3.37)

C (µ, i→ a;λ, i→ a, 2) = −√

2(δxu − δxv) . (3.38)

The above expressions are collected in Table 3.5 with other core→virtual excita-

tions. The values of C for the remaining single excitations can be found in Tables

3.6-3.7. Double excitations are uncoupled in index f in most of the cases. The

corresponding values for C are collected in Table 3.8. Couplings among double

excitations occur for two types: core, active→2 actives and core, active→active,

virtual. The values of C for these excitations is shown in Table 3.9. Type core,

active→active, virtual is not tabulated, as it can be derived from matrix C of the

core, active→2 actives case, by substituting index a for vz, and omitting the last

row and column.

3.5 Construction of the effective Hamiltonian

In order to obtain the second order SA-SSMPRT energy, we construct the effec-

tive Hamiltonian matrix (Eq.(2.80)) with the linearly independent parametrization

as:

Heff

νµ[2] = 〈Φ0ν |H|Φ0µ〉 + 〈Φ0ν |H ˆT µ|Φ0µ〉. (3.39)


Using definition of ˆT µ (Eq.(3.26)), the action of ˆT µ on |Φ0µ〉 can be evaluated:

ˆT µ|Φ0µ〉 =∑

IA

∑

f

ff t

AI (µ)|µf ΥA

I 〉 NAI , (3.40)

where NAI is the normalization factor of function |µf ΥA

I 〉. Substituting Eq.(3.41)

into Eq.(3.39), we can express the effective Hamiltonian with amplitude tAI (µ):

Heff

νµ[2] = 〈Φ0ν |H|Φ0µ〉 +

∑

IA

∑

f

〈Φ0ν |H|µf ΥAI 〉 f

f tAI (µ) NA

I . (3.41)

However, the matrix element of 〈Φ0ν |H|µf ΥAI 〉 can be difficult to evaluate due to

the complex structure of function |µf ΥAI 〉. In order to simplify this matrix element

let us use the original definition of cluster operator as:

ˆT µ|Φ0µ〉 = T µ|Φ0µ〉 =∑

IAtAI (µ)|µΥA

I 〉 NAI , (3.42)

which provides the following expression of the effective Hamiltonian:

Heff

νµ[2] = 〈Φ0ν |H|Φ0µ〉 +

∑

IA〈Φ0ν |H|µΥA

I 〉 tAI (µ) NAI . (3.43)

In that case, the amplitude tAI (µ) is determined from the amplitudes tAI (µ) by

using the relations in Tables 3.2 and 3.4. The expression in Eq.(3.43) is equivalent

to the original formulation of effective Hamiltonian [147, 149], and its explicit

expression can be used without any modification in our formulations as well.

3.6 Sensitivity analysis in SA-SSMRPT

Sensitivity analysis is a mathematical tool, where we monitor how the exam-

ined quantities change with the variation of the input parameters. A detailed

description and also several applications of this method in the context of reac-

tion kinetics can be found in Ref. [159]. In SSMRPT the sensitivity analysis is

used as a diagnostic tool in order to find the origin of the unphysical kinks on

the potential energy surface [151]. In that case, the input parameters are the


CAS coefficients (c(0)µ ), which determine the second-order energy (E [2]) and coeffi-

cients (c[2]µ ) with the help of the amplitude equations and the effective Hamiltonian

eigenvalue equation. The main assumption behind this analysis is that the energy

and the coefficients should not change much while varying the CAS coefficients.

Therefore the outstanding response values may indicate the problematic points on

the potential energy surface.

In order to quantify sensitivity, let us examine the relative error of the second-

order coefficient

ec =∑

µ

c[2]µ

(

c(0)0 + ∆c(0)

)

− c[2]µ

(

c(0)0

)

c[2]µ

(

c(0)0

)

2

, (3.44)

where c(0)0 is the vector of CAS coefficients and ∆c(0) is a little disturbance of

the coefficients. Expanding c[2]µ as a Taylor-series in c

(0)0 up to the first order, we

obtain the following expression:

c[2]µ

(

c(0)0 + ∆c(0)

)

= c[2]µ

(

c(0)0

)

+∑

ν

∂c[2]µ

∂c(0)ν

∣∣∣∣∣c(0)0

∆c(0)ν + O(2) . (3.45)

Substituting Eq.(3.45) back into Eq.(3.44), we obtain the following approximate

expression of the relative error

ec ≈∑

µ

∑

ν

∂c[2]µ

∂c(0)ν

∣∣∣∣∣c(0)0

c(0)0ν

c[2]µ (c

(0)0 )

︸︷︷︸

Sc

µν

∆c(0)ν

c(0)0ν

︸︷︷︸

dcν

2

. (3.46)

where Sc is the sensitivity matrix and dc provides the relative changes in the

parameters. Using matrix Sc and vector dc, Eq.(3.46) can be written in the

following alternative form:

ec ≈ dc† Sc† Sc dc . (3.47)


The sensitivity matrix is not-hermitian, but it can be given in a diagonal form by

singular value decomposition (SVD) [160] as

Sc = WσcU , (3.48)

where W and U are unitary matrices and σc is the diagonal matrix, which contains

the singular values σcµ. Measures of these singular values indicate how sensitive

the second-order coefficients are for the variation of CAS coefficients. Substituting

Eq.(3.48) into Eq.(3.47), we obtain the following expression for the relative error:

ec ≈ dc† U†σ

c† W†W︸︷︷︸

I

σc U dc =

∑

µ

(σcµ

)2

(∑

ν

Uµνdcν

)2

︸︷︷︸

(δcµ)2

,

where δcµ stands for the input parameter corresponding to singular value σµ.

In order to simplify the expression for the derivative∂c

[2]µ

∂c(0)ν

in matrix Scµν , only the

unrelaxed theory is considered, where the amplitudes and the effective Hamiltonian

are solved only once to obtain the second-order energy. In that case, the derivative

∂c[2]µ

∂c(0)ν

can be expressed in the following form:

∂c[2]µ

∂c(0)ν

=∑

IA

∑

λ

∂c[2]µ

∂tAI (λ)

∂tAI (λ)

∂c(0)ν

, (3.49)

where the chain rule is applied.

In Ref. [151] the sensitivity analysis is applied to examine the original SA-

SSMRPT method [149], where the amplitude equations are decoupled as we men-

tioned previously (Sec. 3.2). In that case, the derivative∂tAI (λ)

∂c(0)ν

can be derived

from the decoupled amplitude equations (Eq.(3.9)) as

∂tAI (λ)

∂c(0)ν

= − 1

c(0)λ

(

δλνtAI (λ) +

(AIA)−1

λµ〈 µΥA

I |V |Φ0µ〉)

. (3.50)


While the derivative∂c

[2]µ

∂tAI (λ)can be determined from the effective Hamiltonian equa-

tion (Eq.(3.43)) as

∂c[2]µ

∂tAI (λ)= −

∑

νρτ

Lµν GνρKρτ 〈Φ0τ |H|µΥAI 〉 c[2]µ , (3.51)

where the corresponding quantities are:

Lµν = δµν − c[2]µ c[2]∗ν ,

Kρτ = δρτ − c[2]ρ c[2]∗τ ,

G =I − |c[2]〉〈c[2]|Heff[2] − E [2]

.

Similarly to the sensitivity of coefficients Sc the sensitivity of energy can be

defined as

SEν =∂E [2]

∂c(0)ν

c(0)ν

E [2]. (3.52)

As it can be seen above SE is only a vector as the energy is a one component

quantity. Therefore, there is no need to perform SVD and the singular value

corresponds to the norm of vector SE as

σE =

√∑

ν

SEν2 . (3.53)

The derivative in Eq.(3.52) can be expressed similarly with the help of the chain

rule as

∂E [2]

∂c(0)ν

=∑

I,A

∑

λ

∂E [2]

∂tAI (λ)

∂tAI (λ)

∂c(0)ν

, (3.54)

where∂tAI (λ)

∂c(0)ν

is determined by Eq.(3.50) and ∂E[2]

∂tAI (λ)can be derived from the effective

Hamiltonian equations [151]:

∂E [2]

∂tAI (λ)=

∑

ν

c[2]∗ν 〈Φ0ν |H|λΥAI 〉 c[2]λ . (3.55)


The expressions above can be applied in the non-redundant parametrized ap-

proach with a small modification. In that case, the amplitudes are determined from

different amplitude equations Eq.(3.29), which modifies Eqs.(3.49) and (3.54) in

the following way:

∂c[2]µ

∂c(0)ν

=∑

I,A

∑

λ

∑

J ,B

∑

g

∂c[2]µ

∂tAI (λ)

∂tAI (λ)

∂gg tBJ (λ)

∂gg tBJ (λ)

∂c(0)ν

,

∂E [2]

∂c(0)ν

=∑

I,A

∑

λ

∑

J ,B

∑

g

∂E [2]

∂tAI (λ)

∂tAI (λ)

∂gg tBJ (λ)

∂gg tBJ (λ)

∂c(0)ν

,

where∂c

[2]µ

∂c(0)ν

and ∂E[2]

∂c(0)ν

are evaluated by Eqs.(3.51) and Eq.(3.55). The derivative

∂tAI (λ)

∂gg tBJ (λ)

can be easily determined from the relations in Tables 3.2 and 3.4, while

∂gg tBJ (λ)

∂c(0)ν

can be derived from the amplitude equation Eq.(3.29) as

∂ gg t

AI (λ)

∂c(0)ν

=

= − 1

c(0)λ

(

δλνgg t

AI (λ) +

∑

f

Hλν C (λ, I → A, g; ν, I → A, f) 〈 νf ΥAI |V |Φν〉

)

.

3.7 Demonstrative examples

In order to show the main characteristics of this method, single bond dissocia-

tion processes are considered in two systems: the HF and the LiH molecules. The

HF molecule is computed in Dunning’s polarized double zeta correlation consistent

(cc-pVDZ) basis set[161]. The CAS reference function is generated by distribut-

ing two active electrons on two active orbitals, with symmetry labels 3a1 and 4a1,

classified according to C2v. The bond dissociation curve of the LiH molecule is

computed in Dunning’s double zeta plus polarization (DZP) basis set[162]. The

CAS wave function is constructed with the use of two active electrons and five

active orbitals of symmetry 2a1, 3a1, 4a1, 1b1, 1b2 in the C2v molecular point

group. As the basis sets are relatively small, the results corresponding to the

full configuration interaction (FCI) are feasible, and enable the determination of

methodological errors.


Møller-Plesset (MP) and Epstein-Nesbet (EN) partitionings are applied within

SSMRPT. In the MP case the partitioning of the Hamiltonian depends on the

reference CSF, in analogy with the so-called ”multipartitioning” applied in many-

body PT schemes[163]. Fockian matrix elements are constructed as

f−pq(µ) = f 0

pq(µ) +∑

us∈|Φ0µ〉([pus|qus] − δqus [pus|usus])

f+pq(µ) = f 0

pq(µ) +∑

us∈|Φ0µ〉[pus|qus]

where

f 0pq(µ) = hpq +

∑

i∈|Φ0µ〉(2[pi|qi] − [pq|ii])

standing for matrix elements of the Fockian corresponding to the CSF |Φ0µ〉. Note,

that f−pq(µ) and f+

pq(µ) differ only if index q is singly occupied.

Quantity ggX

AI (µ), defined in Eq.(3.31), is considered in a similar manner as in

Ref. [147, 149, 151]. In these references the amplitude equations are solved in a

decoupled form (Eq.(3.9)), and in case of MP partitioning XAI (µ) is defined in the

following way:

Xpq (µ) = f+

pp(µ) − f−qq(µ) (3.56)

for single excitations, and for double excitations as

Xpqrs (µ) = f+

pp(µ) + f+qq(µ) − f−

rr(µ) − f−ss(µ) . (3.57)

It is easy to see that quantity Xpqrs (µ) is symmetric to the interchange of indices p

and q as well as r and s. Moreover only those doubly excited functions are com-

bined in canonical orthogonalization, which have the same spatial indices (Table

3.3). Therefore, ggXpqrs (µ) can be related to Xpq

rs (µ), which makes the expression of

ggX

pqrs independent of index g in the following way:

ggX

pqrs (µ) = f+

pp(µ) + f+qq(µ) − f−

rr(µ) − f−ss(µ) . (3.58)


However, the single excitations are linearly dependent with the exchange specta-

tors, which provides the following extra term [usus|usus] for Xpq :

Xuspqus (µ) = Xp

q + [usus|usus].

Here for the sake of simplicity in quantity ggX

pq (µ) we only consider Xp

q (µ), which

is also independent of index g:

ggX

pq (µ) = f+

pp(µ) − f−qq(µ) . (3.59)

In EN partitioning the zeroth-order operator is the diagonal part of the Hamil-

tonian, accordingly H can be substituted in H(0) in Eq.(3.31). In this case, many-

body expressions for ggX

AI (µ) are given both in Ref. [147] and Ref. [151], however,

they are not in a complete agreement. To eliminate every ambiguity, a numeri-

cal code, based on Wick’s theorem, was used for constructing the EN excitation

energies in the newly developed, redundancy-filtered formulation. In calculations,

relying on the redundant parametrization of T µ, expressions of Ref. [151] are used

for the EN excitation energies.

One-electron orbitals used to generate the PT results are either pseudo-canonical

or natural. In the former case, the active block of the generalized Fockian of the

target root is diagonal, where the Fockian is built with the same expression as it

is used in CASPT calculations (Eq.(2.75)). The natural orbitals are obtained by

diagonalization of the density matrix in the active block.

In the following we present the results of the various aspects of SA-SSMRPT

with unrelaxed coefficients. Let us first examine the kinks in more details, which

can be easily recognized on the order of 1-10 mEh by plotting the difference be-

tween the SA-SSMRPT second-order energy and FCI [164].

3.7.1 Kinks due to small coefficients

As mentioned earlier in Subsection 2.2.2, kinks can appear on the potential

energy surface, when the value of the zeroth-order coefficient is close to zero.

Such an example was reported by Mao et al.[129] in case of the HF molecule by


−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

∆E /

Eh

bond distance / Å

MP, no threshEN, no thresh

MP,thresh=1E−8EN,thresh=1E−8

Figure 3.1: Errors of second-order SSMRPT energies for the ground state ofthe HF system in cc-pVDZ basis[161]. Reference function is CAS(2,2), activeorbitals are naturals. Full CI values are subtracted from the total SSMRPTenergies. Partitioning is either MP or EN. Label ’thresh=1E-8’ correspondsto omitting model space CSF’s with coefficients smaller than 10−8 in absolute

value. No such treatment is applied for label ’no thresh’.

using natural orbitals. To alleviate the problem, Mao and coworkers applied the

previously mentioned Tikhonov regularization[152] (Eq.(2.94)).

The kinks observed by Mao are also apparent in Figure 3.1. As it can be seen

there, the effect is considerably larger than a few mEh and appears both in MP

and EN partitioning at the same bond distance, at around 2 A . Both curves

are smoothened, when setting a numerical threshold of 10−8 for the model space

coefficients. This means the dropping those CSF’s from the reference function,

which have coefficients with an absolute value smaller than 10−8. Omitting small

coefficients – either in the form of Tikhonov damping or by a numerical threshold

– is necessary for SSMRPT since a division by c(0)µ has to be carried out at some

point to obtain amplitudes.

The kinks disappearing due to this treatment are not interesting from our

present point of view. In the following, we focus on only those effects, which


−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

∆E /

Eh

bond distance / Å

MP, TEN, T

MP, T, no dir specEN, T, no dir spec

MP, T~

EN, T~

(a)

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

1 2 3 4 5 6 7 8

∆E /

Eh

bond distance / Å

MP, TEN, T

MP, T, no dir specEN, T, no dir spec

MP, T~

EN, T~

(b)

Figure 3.2: Errors of second-order SSMRPT energies for the ground state ofthe HF system (panel a) and LiH system (panel b). Active orbitals are pseudo-canonicals. Full CI values are subtracted from the total SSMRPT energies.Basis set, reference function and partitionings agree with those in Figure 3.1for the HF system. Dunning’s DZP basis[162] and CAS(2,5) reference func-tion are used for the LiH system. Key legends: ’T ’ refers to the redundantparametrization of Tµ with direct spectators included, ’T , no dir spec’ appliesthe parametrization of Eqs.(3.4) and (3.5) without direct spectators, ’T ’ uses

the non-redundant parametrization of ˆTµ.

appear even if an appropriate numerical threshold is set for c(0)µ ’s. The results re-

ported below are calculated only in pseudo-canonical basis, on which a numerical

threshold does not have any significant effect, hence it is not applied.

3.7.2 Kinks due to redundancy

Examples for kinks that show up even when setting proper numerical threshold

for small c(0)µ ’s are given by the curves labeled ’MP, T ’ and ’EN, T ’ in Fig. 3.2.

These calculations were carried out with the redundant parametrization of T µ,

including direct spectator excitations and using the decoupled form of the ampli-

tude equations, cf. Eqs. (3.9). Omitting direct spectators but keeping all other

features unchanged, we obtain the curves labeled ’T , no dir spec’. As Fig. 3.2

demonstrates, the error curves are smoothened by the exclusion of direct spec-

tators. Redundancy in T µ, however, is not eliminated completely by omitting

direct spectators. If working with the non-redundant parametrization of ˆT µ, am-

plitude equations Eq.(3.29) and the effective Hamiltonian of Eq.(3.43), the curves


labeled ’MP, T ’ and ’EN, T ’ are obtained. Apparently, in MP partitioning the

non-redundant parametrization has only a minor numerical effect compared to the

redundant parametrization of T µ without direct spectators. In EN partitioning

the curve is shifted even by cca. 10 mEh for the HF molecule. The larger effect

is attributed to the fact that besides orthogonalization of virtual functions, the

quantity XAI (µ) also differs in EN partitioning.

3.7.3 Sensitivity analysis

The largest singular value of sensitivity matrices, cf. Eqs.(3.46) and (3.52),

are presented for the HF molecule in Figure 3.3 and for the LiH molecule in

Figure 3.4. The unphysical kinks appear on sensitivity curves of both the second-

order energy and the coefficients as well, usually at the same bond length as on the

energy curves. These curves show a smoothening when instead of the redundant

parametrization of T µ including direct spectators (’MP or EN, T ’) we apply the

method where direct spectators are excluded (’T , no dir spec’). The values of

singularities are further diminished when applying the orthogonal parametrization

of ˆT µ in most of the cases.

3.7.4 Determinantal versus spin-adapted formulation and

alternative redundancy treatments

Spin-adapted results are compared with the determinantal formulation [? ] in

the MP partitioning in Fig. 3.5. Curves with the redundant parametrization of

T µ without direct spectators as well as those in the redundancy filtered method

show good correspondence with the determinantal formulation, for both molecules.

Among the two molecules, HF shows the worse picture, but even here the difference

does not exceed 1 mEh.

It is important to emphasize that the SA-SSMRPT energy is not invariant to the

type of orthogonalization used for redundancy filtering in T µ. Results of canonical

orthogonalization are compared with those obtained by alternative procedures in

Figure 3.5. An alternative orthogonalization scheme omits direct spectators and


0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

sin

g. v

alu

e o

f c(0

) sen

sisi

tivi

ty o

f E

[2]

bond distance / Å

MP, TMP, T, no dir spec

MP, T~

0

0.005

0.01

0.015

0.02

0.025

0.03

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

sin

g. v

alu

e o

f c(0

) sen

sisi

tivi

ty o

f E

[2]

bond distance / Å

EN, TEN, T, no dir spec

SS−MRPT,EN, T~

0.01

0.1

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

sin

g. v

alu

e o

f c(0

) sen

sisi

tivi

ty o

f c

bond distance / Å


MP, T~

0.01

1

100

10000

1e+06

1e+08

1e+10

1e+12

1e+14

1e+16

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

sin

g. v

alu

e o

f c(0

) sen

sisi

tivi

ty o

f c

bond distance / Å


SS−MRPT,EN, T~

Figure 3.3: The largest singular value of the sensitivity matrix of SA-SSMRPTenergy (Eq.(3.52)) and coefficients (Eq.(3.46)) corresponding to the ground stateof the HF molecule. Basis set, reference function and partitioning agree with

those in Figure 3.2. Key legends are also given at Figure 3.2.

follows a different scheme for constructing functions |µΥl〉. Formulae of this or-

thogonalization are summarized in Table 3.10. Results produced by this approach

are labeled ’T , alternative ort’ in Figure 3.5. As it can be seen there, the nonpar-

allelism error of the curve ’T , alternative ort’ is closer to that of the determinantal

curve, compared to the error produced by canonical orthogonalization. The ’T ,

alternative ort’ runs along the curve ’T , canonical ort’ for the LiH molecule on the

scale of the figure, therefore a common label ’T ’ is used for them.

In order to examine the effect of couplings among the amplitudes belonging to

different reference functions, only exchange spectators are used in the cluster op-

erator in Eqs.(3.18)-(3.21). This decreases these couplings, because the exchange

spectator can only affect those reference functions, which have a common single

occupied orbital index with the excitation operator. The results of this method

is shown by the curve labelled ’T , drop out’ in Figure 3.5. Apparently, it suffers

from the largest nonparallelism error, indicating that coupling among amplitudes

within T µ might be important.


1e−08

1e−07

1e−06

1e−05

0.0001

0.001

0.01

0.1

1

1 2 3 4 5 6 7 8

sin

g. v

alu

e o

f c(0

) sen

sisi

tivi

ty o

f E

[2]

bond distance / Å


MP, T~

1e−08

1e−07

1e−06

1e−05

0.0001

0.001

0.01

1 2 3 4 5 6 7 8

sin

g. v

alu

e o

f c(0

) sen

sisi

tivi

ty o

f E

[2]

bond distance / Å


EN, T~

0.001

0.01

0.1

1

10

100

1000

1 2 3 4 5 6 7 8

sin

g. v

alu

e o

f c(0

) sen

sisi

tivi

ty o

f c

bond distance / Å


MP, T~

0.0001

0.01

1

100

10000

1e+06

1e+08

1e+10

1e+12

1e+14

1e+16

1 2 3 4 5 6 7 8

sin

g. v

alu

e o

f c(0

) sen

sisi

tivi

ty o

f c

bond distance / Å


EN, T~

Figure 3.4: The largest singular value of the sensitivity matrix of SA-SSMRPT energy (Eq.(3.52)) and coefficients (Eq.(3.46)) corresponding theground state of the LiH molecule. Basis set, reference function and partitioning

agree with those in Figure 3.2. Key legends are given at Figure 3.2.

Table 3.10: Virtual spin functions of an alternative orthogonalization treat-ment, differing from those generated by canonical orthogonalization. Differenceswith the functions shown in Tables 3.1 and 3.3 affect elements of matrices C of

Tables 3.5-3.9. Derivation of changes to C is left to the reader.


core→active(1)

|µΥusi 〉 =

Eusi

c|Φ0µ〉

|µΥvsusivs

〉 = 12

Evsusivs

c|Φ0µ〉

|µΥusi 〉 =

(|µΥus

i 〉 + |µΥvsusivs

〉)


|µΥvsaius

〉 =

Evsaius

c|Φ0µ〉

|µΥavsius

〉 = 12

Eavsius

c|Φ0µ〉

|µΥvsaius

〉 = (−1)p(vsa)(

|µΥvsaius

〉 + |µΥavsius

〉)


0

0.002

0.004

0.006

0.008

0.01

0.012

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

∆E /

Eh

bond distance / Å

MP, T, no dir specMP, T

~, canonical ort

MP, T~

, alternative ortMP, T, drop out

MP, det

(a)

0.0012

0.0013

0.0014

0.0015

0.0016

0.0017

0.0018

0.0019

0.002

0.0021

1 2 3 4 5 6 7 8

∆E /

Eh

bond distance / Å

MP, T, no dir specMP, T

~

MP, T, drop outMP, det

(b)

Figure 3.5: Errors of second-order SA-SSMRPT energies in MP partitioningcorresponding to the ground state of the HF system (on panel a) and LiHsystem (on panel b). Full CI values are subtracted from the total SA-SSMRPTenergies. The basis sets and reference functions agree with those in Figure 3.2.Key legends: ’T , no dir spec’ applies the parametrization of Eqs.(3.4) and (3.5)without direct spectators. Label ’T ’ uses the non-redundant parametrization ofTµ, where ’canonical ort’ refers to the virtual functions of Tables 3.1 and 3.3,while ’alternative ort’ refers to the virtual functions of Table 3.10. Label ’T ,drop out’ refers to the method where single excitations and the correspondingamplitudes are omitted in Tµ of Eqs.(3.4) and (3.5) to reduce the couplingbetween the amplitudes. Label ’det’ refers to the determinantal approach [? ].

Chapter 4

Role of local spin in geminal-type

theories

In this Chapter we would like to examine the geminal based techniques in

covalent bond dissociations. First, the single bond dissociation is investigated,

which can be described appropriately by these methods. Afterwards, we present

the possible failures of multiple bond dissociations, which can be related to the

spurious description of spin states of fragments. Then the Linearized Coupled

Cluster (LCC) corrected form is examined with APSG and SP-RUSSG references.

The presented results partially can be found in Ref. [2].

4.1 Size consistency in strongly orthogonal gem-

inal type theories

In the description of chemical bond dissociation the same accuracy is required

at every geometric point on the potential energy surface. For example RHF usually

provides appropriate energies at equilibrium geometries, but it may produce delu-

sive results at the dissociation limit, especially if the molecule dissociates to open

shell fragments. To examine whether the method under investigation provides a

qualitatively correct description, we should calculate the exact (FCI) solution and

compare them. However, exact calculations can only be done for small molecular

systems. Therefore it is more expedient to define related quantities, which can be

70

Chapter 4. Role of local spin in geminal-type theories 71

checked analytically or by a less demanding numerical calculation and which may

shed light on the problematic parts of approximate methods. For these reasons

the notion size consistency [165] was introduced with the following definition: a

method is size consistent if the energy of a system composed of two (infinitely

separated) non-interacting fragments is equal to the sum of the energies of the

fragments calculated independently.

4.1.1 Single bond dissociation

In order to investigate size consistency we have to take a look at the expressions

for the energy. If the wave function is product separable over non-interacting

subsystems A and B, size consistency is fulfilled due to:

(

HA + HB

)

|ΨAΨB〉 = HA|ΨA〉︸︷︷︸

EA|ΨA〉

|ΨB〉 + HB|ΨB〉︸︷︷︸

EB |ΨB〉

|ΨA〉 = (EA + EB) |ΨAΨB〉 ,

where ΨA (ΨB) is the wave function of the subsystem A (B) and EA (EB) is the

corresponding energy. Therefore size consistency for the APSG wave functions

(Eq.(2.32)) automatically fulfils, if every geminal localizes on only one fragment:

|ΨAPSG〉 =A∏

I

ψ+I

︸︷︷︸

Ψ+A

B∏

J

ψ+J

︸︷︷︸

Ψ+B

|vac〉 ,

where index I runs over the geminals of fragment A, and similarly J runs over the

geminals of fragment B.

However, wave function separability is a sufficient, but not necessary condition

for size consistency. For example in case of H2 molecule in a minimal basis, the

system contains two electrons, which can be described by only one geminal. As

this model considers correlation among the two electrons explicitly, it provides FCI

solution in that simple case. In order to gain a deeper insight into this example,

let us write the wave function in the dissociation limit, when the electrons are


localized on different H atoms in the following way:

|ΨAPSG

H2〉 =

1√2

(ϕ+Aαϕ

+Bβ + ϕ+

Bαϕ+Aβ

)|vac〉 , (4.1)

where |ΨAPSGH2

〉 represents the singlet state of dissociated H2 molecule, ϕAα (ϕBβ)

is the one electron orbital, which localizes on H atom A (B). As it can be seen in

Eq.(4.1), the geminal is delocalized on the whole system and it cannot be described

by the product of the fragment states. Despite of this, using that the function ϕAσ

(σ = α, β) is the eigenfunction of operator HA,

HA|ϕAσ〉 = EA|ϕAσ〉 , (4.2)

and similarly for subsystem B, the following expression can be derived:

〈ΨAPSG

H2|HA + HB|ΨAPSG

H2〉 = EA + EB . (4.3)

Thus the total energy is additive over the subsystems.

It is important to note here is that though Eq.(4.3) shows additivity, the frag-

ment problem of Eq.(4.2) is not a geminal problem, since the eigenfunction is a

one-electron (instead of two-electron) function. Therefore size consistency in case

of single-bond dissociation cannot be investigated with APSG, as the method does

not apply to the odd-electron fragments produced in the dissociation limit.

However, it can be examined by extending the APSG model according to the

Restricted Singlet-type Strongly Orthogonal Geminals (RSSG) recipe [17]. This

wave function includes extra one-particle functions in addition to geminal func-

tions:

|ΨRSSG〉 =

(nα∏

i

ϕ+iα

)(N∏

I

ψ+I

)

|vac〉, (4.4)

where ϕiα is a one-particle function with spin α, nα = 1 and N = 0 in the H-atom

case, c.f. Eq.(4.2). The wave function of the dissociated H2 (Eq.(4.1)) is not a


product of the fragment RSSG functions, which have the following forms:

|ΨRSSG

A 〉 = |ϕAα〉 ,

|ΨRSSG

B 〉 = |ϕBα〉 .

However, the energy still satisfies Eq.(4.3), as the spin of the unpaired electron

does not affect the energy in the absence of external magnetic field. Thus RSSG

is size consistent in that case. The considerations above can be extended to the

general single-bond dissociation by using the assumption that the remaining part

of the system does not need to be considered in the process.

4.1.2 Multiple bond dissociation

In case of multiple bond dissociation, the correlation between the different

bonds may have relevance in a proper description. These intergeminal correlations

are generally completely omitted in the RSSG model.

This can lead to size consistency problems, for example in case of the symmetric

dissociation of water. Here, there are five geminals: at equilibrium geometries two

geminals correspond to the OH bonds, while the remaining three geminals compose

the core part of the oxygen atom. In the dissociation limit the two OH geminals

are delocalized on the two fragments and each one-electron orbital is localized on

one non-interacting fragment (derivation can be found in Appendix C). The two

OH geminals in a minimal basis have the following expressions:

ψ+A = CA

12 ϕO+1α ϕH+

2β + CA21 ϕ

H+2α ϕO+

1β (4.5)

ψ+B = CB

34 ϕO+3α ϕH+

4β + CB43 ϕ

H+4α ϕO+

3β , (4.6)

where for the sake of the generality the matrixces CA and CB do not need to be

symmetric, but condition Eq.(2.31) is applied to normalize the geminals to one.

In order to examine size consistency let us recalculate the energy of the frag-

ments. In case of hydrogen atom, only one orbital constructs the system, which

provides the energy of the hydrogen atom in a similar manner as in H2 molecule

dissociation. Meanwhile the oxygen atom has a more complicated structure as it


has more electrons. It can be partitioned to two parts, the first part is the core and

the second is the valence part, which is constructed of two oxygen orbitals from

the OH geminals. In order to check size consistency, let us calculate the energy of

the oxygen atom (EO) as

EO = 〈θ|ψ−B ψ

−A HO ψ

+A ψ

+B |θ〉 , (4.7)

where HO is the Hamilton operator of the oxygen atom and |θ〉 represents the

core part. Substituting Eq.(4.5) and Eq.(4.6) into Eq.(4.7) and assuming that

the operator HO does not affect the hydrogen functions, we obtain the following

expression for the energy:

EO =∣∣CA

12

∣∣2 ∣∣CB

34

∣∣2HααO +

∣∣CA

12

∣∣2 ∣∣CB

43

∣∣2HαβO +

+∣∣CA

21

∣∣2 ∣∣CB

34

∣∣2HβαO +

∣∣CA

21

∣∣2 ∣∣CB

43

∣∣2HββO , (4.8)

where the notation Hσσ′

O (σ, σ′ = α, β) is introduced as

Hσσ′

O = 〈θ| ϕO−3σ′ ϕ

O−1σ HO ϕO+

1σ ϕO+3σ′ |θ〉 .

To examine the energy of the oxygen atom, let us partition it according to the spin

states. The quantities HααO and Hββ

O correspond to the high spin triplet states,

while HαβO and Hβα

O are the mixture of the singlet and triplet spin-states. Hence,

the following two-electron quantities are introduced:

S+O =

1√2

(ϕO+1α ϕ

O+3β + ϕO+

3α ϕO+1β

),

T +O =

1√2

(ϕO+1α ϕ

O+3β − ϕO+

3α ϕO+1β

),

where S+O (T +

O ) creates the singlet (triplet) two-electron state. The combinations

of operators S+O and T +

O provide the mixed spin-states as

ϕO+1α ϕO+

3β =1√2

(S+O + T +

O

), (4.9)

ϕO+3α ϕO+

1β =1√2

(S+O − T +

O

). (4.10)


Substituting Eq.(4.9) into HαβO , and assuming relation 〈θ|S−

O HOT +O |θ〉 = 0 as the

Hamiltonian preserves the spin quantum number of the wave function, we obtain

the following spin-pure quantities:

HαβO =

1

2

〈θ|S−

O HOS+O |θ〉

︸︷︷︸

HSO

+ 〈θ|T −O HOT +

O |θ〉︸︷︷︸

HTO

. (4.11)

The same expression can be derived for HβαO in the same way as

HβαO =

1

2

(HSO +HT

O

). (4.12)

Finally, substituting Eq.(4.11) and Eq.(4.12) into Eq.(4.8), we obtain the following

expression for the energy of oxygen atom:

EO =∣∣CA

12

∣∣2 ∣∣CB

34

∣∣2HααO +

∣∣CA

21

∣∣2 ∣∣CB

43

∣∣2HββO

+

∣∣CA

21

∣∣2 ∣∣CB

34

∣∣2

+∣∣CA

12

∣∣2 ∣∣CB

43

∣∣2

2

(HSO +HT

O

). (4.13)

In RSSG matrices CA and CB are symmetric, therefore by using the normalization

condition Eq.(2.31), the matrix elements of CA and CB can be given as

CA12 = CA

21 = CB34 = CB

43 =1√2. (4.14)

Substituting this back into Eq.(4.13), the RSSG energy of oxygen atom can be

obtained as a mixture of singlet and triplet states:

EO =1

4

(

HααO + Hββ

O + HSO + HT

O

)

.

To examine the problem from another point of view, let us determine the atomic

spin of the oxygen in the dissociation limit. In that case, it can be determined by


the so-called local squared spin operator of the oxygen atom (S2O) as

S2O =

3

4

∑

i

(ϕO+iα ϕ

O−iα + ϕO+

iβ ϕO−iβ

)+

1

4

∑

ij

(ϕO+iα ϕ

O+jα ϕ

O−jα ϕ

O−iα + ϕO+

iβ ϕO+jβ ϕ

O−jβ ϕ

O−iβ

)+

+1

2

∑

ij

(2 ϕO+

iβ ϕO+jα ϕ

O+jβ ϕ

O+iα − ϕO+

iα ϕO+jβ ϕ

O+jβ ϕ

O+iα

).

This form of the second quantized expression is equivalent to the total squared spin

operator (S2) [19], only the summations are restricted to the orbitals of the oxygen

atom. The expectation value of the local spin of the oxygen can be evaluated,

similarly to the energy of the oxygen in Eq.(4.13):

SO(SO + 1) =∣∣CA

12

∣∣2 ∣∣CB

34

∣∣2 〈θ|ϕO−

2α ϕO−1α S

2Oϕ

O+1α ϕ

O+2α |θ〉

︸︷︷︸

2

+ (4.15)

+∣∣CA

21

∣∣2 ∣∣CB

43

∣∣2 〈θ|ϕO−

2β ϕO−1β S

2Oϕ

O+1β ϕ

O+2β |θ〉

︸︷︷︸

2

+

+

∣∣CA

21

∣∣2 ∣∣CB

34

∣∣2

+∣∣CA

12

∣∣2 ∣∣CB

43

∣∣2

2

〈θ|S−O S

2OS+

O |θ〉︸︷︷︸

0

+ 〈θ|T −O S

2OT +

O |θ〉︸︷︷︸

2

,

where SO is the local spin of oxygen. Substituting the RSSG coefficients in

Eq.(4.14) into Eq.(4.15), it gives 32

for SO(SO + 1). This spin state is a mixture of

a singlet and a triplet, which does not provide a proper description.

As it is well-known, the ground state of the oxygen atom is a triplet state, which

can be reproduced by RSSG, when calculating the oxygen atom independently. In

this case, two one-electron functions with spin α describe the valence part of the

oxygen atom (Eq.(4.4)). Therefore it is evident that RSSG is not size consistent in

this case, because when calculating the whole system the RSSG provides incorrect

spin-state for the oxygen fragment contrary to the calculation on independent

fragments.

If spin symmetry is broken in the geminals (RUSSG model), then symmetry of

matrices CA and CB is not preserved. This allows for the elimination of the last

term in Eq.(4.13), which causes the mixing of singlet and triplet states. Hence,


the following two solutions can be obtained:

CA12 = CB

34 = 1 CA21 = CB

43 = 0 |ΨaH2O

〉 = ϕO+1α ϕH+

2β ϕO+3α ϕH+

4β |θ〉 ,

CA12 = CB

34 = 0 CA21 = CB

43 = 1 |ΨbH2O

〉 = ϕO+1β ϕH+

2α ϕO+3β ϕH+

4α |θ〉 .

The above cases describe the high spin (MS = 1) triplet oxygen, where both

electrons on the oxygen atom have the same spin. In these cases the RUSSG wave

function reduces to a single determinant UHF solution. This UHF solution usually

appears, when the molecule dissociates to open-shell fragments. Consequently,

due to the advantageous properties of the UHF wave function, RUSSG is size

consistent [17], if the unpaired electrons produced upon fragmentation correspond

to single-electron functions ϕiα and they are not paired to form geminals ψI .

4.2 Size consistency of spin purified geminal type

methods

In Sec. 4.1 we found that RSSG may have problematic characteristics in case

of multiple dissociation processes, which can be solved by introducing spin con-

taminated geminals. By doing so, the total wave function also suffers from spin

contamination, which has to be eliminated for a proper description. The variation

after spin projection method [69, 79, 80], which was previously mentioned in Sec.

2.1.3, is performed to avoid unphysical steps on potential energy surfaces. In that

case, the parameters are optimized to minimize the energy corresponding to the

spin-projected wave function. If the initial spin contaminated wave function is the

UHF determinant, variation after projection method is called Extended Hartree-

Fock (EHF) method [80]. This method produces smooth potential energy surface,

but it violates size-extensivity [69, 79, 166, 167] and size-consistency [69, 79].

Neuscamman has recently suggested a method [168], where new non-linear vari-

ational parameters are introduced in order to eliminate size consistency and size

extensivity errors in the AGP wave function. Afterwards, Henderson and Scuseria

used a linearized version of this theory in the EHF wave function [169]. The price


to pay is the abandonment of the one-determinant picture, which increases the

computational scaling of this method.

Maybe the simplest system, where size-inconsistency of EHF can be examined

are two infinitely separated H2 molecules in minimal basis. In order to examine

this, let us take the UHF wave function with the parametrization Eqs.(2.46) and

(2.47) as

|ΨUHF

(H2)2〉 =

S+1 (α1) +

sin (2α1)√2

0T +1

︸︷︷︸

ψ+1 (α1)

·

S+2 (α2) +

sin (2α2)√2

0T +2

︸︷︷︸

ψ+2 (α2)

|vac〉 ,

where αI is the parameter of the I-th geminal, S+I (αI) is the singlet part of ψ+

I (αI),

while 0T +I is the triplet part. Index 0 in 0T +

I refers to MS = 0 eigenvalue of Sz:

S+I (αI) = cos2 (αI) ϕI+1α ϕI+1β − sin2 (αI) ϕI+2α ϕI+2β ,

0T +I =

1√2

(ϕI+1α ϕI+2β − ϕI+2αϕ

I+1β

).

Assuming that these geminals are localized on the H2 molecules, the singlet spin

projected UHF wave function of which can be easily derived as

P s|ΨUHF

H2〉 = P s

(

S+1 (α1) +

sin (2α1)√2

0T +1

)

|vac〉 = S+1 (α1)|vac〉 , (4.16)

where P s is the projection operator to the singlet space [19]. In minimal basis the

spin-projected wave function in Eq.(4.16) is equivalent to the FCI wave function of

the H2 molecule. As the H2 molecules are independent, the product of these wave

functions provides the exact solution of the H4 system. However, if we spin-project

the wave function of the dimer, we obtain:

P s|ΨUHF

(H2)2〉 =

(

S+1 (α1)S+

2 (α2) +sin (2α1) sin (2α2)

2√

3Π+

12

)

|vac〉 , (4.17)

where Π+12 creates a four-electron singlet state:

Π+12 =

√

1

3

(+1T +

1−1T +

2 + −1T +1

+1T +2 − 0T +

10T +

2

). (4.18)


Operator +1T +I ( −1T +

I ) above creates a two-electron triplet state with the ap-

propriate eigenvalue MS = 1 (MS = −1) of Sz. These triplet creation operators

can be defined with the spin raising and lowering operators as

+1T +I =

1√2S+

0T +I = ϕI+1α ϕI+2α , (4.19)

−1T +I =

1√2S−

0T +I = ϕI+1β ϕI+2β . (4.20)

As it is visible in Eq.(4.17), the triplet component appears in the spin-projected

wave function of the dimer, which can not be eliminated without changing the

singlet state as S+I depends on αI . This may lead to spin contamination in the

fragments. Consequently, the energy of the spin-projected wave function contains

an additional term, which may cause a size consistency error.

The EHF solution can be obtained with the wave function in Eq.(4.17), if the

parameters and basis functions are optimized to minimize the energy. The op-

timization of parameters αi (i = 1, 2) does not change the Ansatz of the wave

function, therefore the size consistency problem remains. In case of basis opti-

mization, the one-electron functions can be delocalized, which may improve the

local spin values. However, in most of these structures ionic terms (H+2 and H−

2 )

may appear producing high energy contributions. In addition to localized geminals

only one structure exists, which does not contain ionic terms. In that case, the

geminal is delocalized and it is constituted of two H-atom orbitals, which belong

to two different fragments (the proof can be done in the same manner as we did

in Appendix C). However, the explicit interaction between the hydrogen atoms in

the H2 molecule is neglected, which also leads to higher energy result. Moreover

in this structure local spin values still belong to the mixed spin states, similarly

to the local spin of the oxygen atom in water in Sec. 4.1.2.

It is important to note that this error appears only, when both fragments have a

multiconfigurational character (α1 6= 0 and α2 6= 0). Therefore at equilibrium bond

length the system is well-described by the singlet closed shell determinant, where

the triplet part in Eq.(4.17) disappears providing the spin-pure singlet states.

The size consistency error vanishes in the H2 molecule dissociation limit as well.


In that case, the singlet and triplet states become degenerate, hence the spin

contamination does not have any energy contribution. Consequently, the size

consistency error only arises, when both bond lengths are elongated, but they still

interact each other.

Let us examine the RUSSG wave function and its spin-projected form in the

same situation. These wave functions can be written in the following form by

using Eqs.(2.49) and (2.50):

|ΨRUSSG

(H2)2〉 =

cos (δ1)S+

1 (γ1) + sin (δ1)0T +

1︸︷︷︸

ψ+1 (γ1,δ1)

· (4.21)

·

cos (δ2)S+

2 (γ2) + sin (δ2)0T +

2︸︷︷︸

ψ+2 (γ2,δ2)

|vac〉 ,

P s|ΨRUSSG

(H2)2〉 =

[

cos (δ1) cos (δ2)S+1 (γ1)S+

2 (γ2) +sin (δ1) sin (δ2)√

3Π+

12

]

|vac〉, (4.22)

where

S+I (γI) = cos (γI) ϕI+1α ϕI+1β − sin (γI) ϕI+2α ϕI+2β .

If δ1 and δ2 are chosen to be zero, both of the wave functions above provide the

exact FCI Ansatz in minimal basis, which ensures size consistency.

However, the parametrization of RUSSG is not general enough to describe all

systems in a size consistent manner. In case of three geminals the size consistency

can be hold with RUSSG, but in four geminal cases this error may arise. For

example, let us consider four H2 molecules in minimal basis, where pairs of H2

molecules interact with each other, but there is no interaction between the two H4

clusters. The RUSSG wave function can be written in the following form for this

system:

|ΨRUSSG

(H4)2〉 =

4∏

I=1

ψ+I (γI , δI) |vac〉 , (4.23)


where the geminals with I = 1, 2 represent two interacting H2 bonding geminals

and I = 3, 4 correspond to the other pair. After spin-projection, we obtain:

P s|ΨRUSSG

(H4)2〉 =

[4∏

I=1

cos (δI)S+I (γI) + (4.24)

+

(4∏

J=1

sin (δJ)

)(1

3Π+

12Π+34 +

2

3√

5Ω+

1234

)

+

+4∑

K,K,L,L=1K<L,K<LK 6=K,L 6=L

cos (δK) cos (δL) sin (δK) sin (δL)√3

S+K (γK)S+

L (γL)Π+KL

]

|vac〉,

where Ω+1234 creates the following singlet state:

Ω+1234 =

√

1

5

(

+2Q+12

−2Q+34 − +1Q+

12−1Q+

34 + 0Q+12

0Q+34 −

− −1Q+12

+1Q+34 + −2Q+

12+2Q+

34

)

,

with

0Q+IJ =

√

1

6+1T +

I−1T +

J +

√

2

30T +

I0T +

J +

√

1

6−1T +

I+1T +

J .

Then, similarly to Eqs.(4.19) and (4.20) the MS 6= 0 quintet creation operators

(mQ+, m = ±1,±2) can be defined with operators S+ and S−. If the four H2

molecules do not interact with each other then only the first term in Eq.(4.24) has

a physical relevance. The other terms can be easily eliminated with the following

choice:

δI = 0 , ∀I .

However, if H4 clusters appear due to interaction within pairs then the triplet

component is needed for a proper description. To gain a more accurate picture

about these systems let us take a look at the spin-projected wave function of the


fragments:

P s|1ΨRUSSG

H4〉 =

[


2 (γ2) +sin (δ1) sin (δ2)√

3Π+

12

]

|vac〉, (4.25)

P s|2ΨRUSSG

H4〉 =

[


4 (γ4) +sin (δ3) sin (δ4)√

3Π+

34

]

|vac〉. (4.26)

As the two H4 clusters do not interact with each other, the wave function of the

whole system can be obtained in a product form. However, it is not equivalent to

the wave function obtained by spin-projection to the whole system:

P s|ΨRUSSG

(H4)2〉 6=

(

P s|1ΨRUSSG

H4〉)(

P s|2ΨRUSSG

H4〉)

.

The difference comes from the extra term Ω+1234 in Eq.(4.24), which leads to size

consistency error.

Finally, let us compare the obtained RUSSG function for the H4 system (Eq.(4.25))

with the APSG Ansatz:

|ΨAPSG

H4〉 = S+

1 (γ1)S+2 (γ2) .

As it can be seen in Eq.(4.25), we have an additional singlet component Π12,

which provides an extra variational freedom for the calculations. However, this

wave function still does not describe the H4 system at the FCI level due to the

absence of ionic terms. As we discussed earlier in Eqs.(4.21) and (4.22), this

parametrization can be described well at the dissociation limit. Moreover it was

also shown that this Ansatz is enough to describe the H4 system at symmetric

square geometry arrangement [81] by using completely delocalized orbitals instead

of the localized ones. Unfortunately, no computer codes have yet been written

to optimize the basis for the spin-projected RUSSG wave function. However,

the optimized RUSSG basis is also able to follow this localization-delocalization

transition, the obtained basis is used to examine this dissociation process and

other small examples in Sec. 4.4.


4.3 Local spin

In the following section we would like to demonstrate the local spin properties

of APSG, RUSSG and SP-RUSSG methods in some small molecules. In order

to follow this property on the whole dissociation curve the previously used local

spin quantity has to be generalized to the interacting subsystems as well. A

straightforward approach to determine such a quantity is based on the partitioning

of operator S2. In that case, the local spin can be calculated as an expectation

value of the appropriate term. Clark and Davidson defined this partitioning with

the help of the population analysis [170, 171], where atomic and diatomic local

spin contributions are distinguished. The diatomic part is proportional to the bond

order, meaning that singlet coupled electron pairs in covalent bonds have non-zero

contribution to the local spin. This contradicts the physical concept that magnetic

properties are determined by open-shell or ”actually free” electrons[172, 173]. For

this reason Mayer introduced an alternative definition for single determinant wave

function[174], based on the decomposition of the expectation value of S2, instead

of the operator itself. The advantage of this definition is that every term depends

on the spin density matrix (Ps = Pα −Pβ), and becomes zero for singlet coupled

electrons. Since the first formulation, the way of partitioning has evolved and

generalization for multi-determinant wave function has been developed[175–179].

Final version of the theory was settled by Ramos-Cordoba et al. [180], who also

studied basis set dependence and compared the benefits of decomposing in Hilbert-

space or in 3D-space[181].

In the present study atomic and diatomic terms of the local spin are computed

according to Ref. [180]:

〈S2〉A =3

4

∑

µ∈A

[2PS− (PS)2

]

µµ+

1

4(psA)2 − 1

4

∑

µ,ν∈A(PsS)µν (PsS)νµ

+1

2

∑

µ,ν∈A

∑

τ,ρ

[Λµνρτ − Λµντρ]SρµSτν (4.27)

〈S2〉AB =1

4psAp

sB − 1

4

∑

µ∈A

∑

ν∈B(PsS)µν (PsS)νµ

+1

2

∑

µ∈A

∑

ν∈B

∑

τ,ρ

[Λµνρτ − Λµντρ]SρµSτν , (4.28)


where A,B refer to atoms, S is the overlap matrix, P is the spin-less density

matrix (Eq.(2.14)), and psA is the gross spin population:

psA =∑

µ∈A(PsS)µµ .

The spin-less cumulant, Λ can be expressed as:

Λµνρτ = Γµνρτ − PµρPντ +∑

σ

P σµτP

σνρ,

where Γ(=∑

σ,σ′ Γσσ′

) is the spin-less two-particle density matrix and σ, σ′ label

spin indices. In the above, µ and ν refer to atomic orbitals, which implies a

Mulliken-like partitioning[182] of 〈S2〉.Due to the geminal structure the one-particle density matrix has a block diag-

onal form for APSG and RUSSG, P σmn being equal to zero if m and n belong to

different geminals. The two-particle Γσσmnls is zero too if l and s or m and n belong

to the same geminal. Otherwise:

Γσσmnls = P σmlP

σns − P σ

msPσnl.

The different spin term Γσσ′

(σ 6= σ′) can be written in the following form:

Γσσ′

mnls = P σmlP

σ′

ns + Λσσ′

mnls

where Λσσ′

mnls is non-zero only if all of its indices belong to the same geminal.

4.4 Assessment of local spin by strongly orthog-

onal geminals

In this section we present the local spin properties of APSG (RSSG), RUSSG

and SP-RUSSG in a few simple examples. All spatial orbitals are part one of occu-

pied geminal subsets, which are optimized by the previously mentioned Rassolov

algorithm in Ref. [17]. The method ”SP-RUSSG, opt” is also introduced, where


the geminal coefficient matrices are optimized to provide a lower energy. However,

it is only a partial variation after projection scheme as the basis is not optimized

in that case. To compute the local spin of Eq.(4.27) one needs overlap matrices,

the one-particle density matrix and the cumulant (i.e. two-particle density ma-

trix). Density matrices with APSG, RUSSG and SP-RUSSG were generated by

the modified version of Q-Chem [183]. Density matrices with SP-RUSSG were

obtained by a direct Full Configuration Interaction (FCI) code [164]. Due to the

large memory and computational time requirement of FCI, only small test systems

were affordable. All reported FCI energies assume the cores being frozen.

4.4.1 Water symmetric dissociation

While single bond dissociation is accurately described by established geminal

methods, dissociation of multiple bonds may be problematic. To describe e.g.

the symmetric dissociation of water four active electrons are needed. Geminal

type methods can be unsatisfactory because of artificial separation of the four

electrons into two pairs. The potential curve in Fig. 4.1, computed in 6-31G**

basis [184, 185], does not show any qualitative failure for APSG, RUSSG or SP-

RUSSG. When looking at error curves computed with FCI (c.f. panel (b) of

Fig. 4.1), one sees that geminal methods produce an error on the scale of a few

tens of millihartree in the 1-3 A bond length interval. Curves of APSG, RUSSG

and SP-RUSSG run together until about 2 A , just where the spin contamination

appears in the RUSSG wave function. From this point the energy of RUSSG and

SP-RUSSG gets markedly deeper than APSG causing larger nonparallelism error.

Local spin curves cast a different light on the case. When forming a molecule

from atoms, the high multiplicity of the free atom typically drops to a low value

at molecular equilibrium. This is apparent on the FCI curves in Figs. 4.2(a)

and 4.3(a). Local spin of hydrogen and oxygen behave in a completely different

manner: 〈S2〉H with geminal methods estimate the FCI result well, which does

not hold for 〈S2〉O. As mentioned in Sec. 4.1.2, APSG provides 32

instead of 2 in

the dissociation limit. This qualitative error is eliminated both by RUSSG and


−76.05

−76

−75.95

−75.9

−75.85

−75.8

−75.75

1 2 3 4 5 6 7

E [

Eh]

bond distances [Å]

APSGRUSSG

SP−RUSSGFCI

(a)

0.085

0.09

0.095

0.1

0.105

0.11

0.115

0.12

0.125

0.13

1 2 3 4 5 6 7

E−

EF

CI [

Eh]

bond distances [Å]

APSGRUSSG

SP−RUSSG

(b)

Figure 4.1: Total energy (a) and energy difference with respect to FCI (b) forH2O symmetric dissociation, in 6-31G** basis set [184, 185].

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

1 2 3 4 5 6 7

<S

2 >O

bond distances [Å]

APSGRUSSG

SP−RUSSGFCI

(a)

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

1 2 3 4 5 6 7

<S

2 >O

− [

<S

2 >O

] FC

I

bond distances [Å]

APSGRUSSG

SP−RUSSG

(b)

Figure 4.2: Local spin of oxygen (a) and local spin difference with respect toFCI (b) for H2O symmetric dissociation, in 6-31G** basis [184, 185].

SP-RUSSG. More details are revealed by the difference curves in panels (b). The

largest errors can be found in the 1.5-3 A bond distance range. RUSSG goes

parallelly with the APSG curve until the spin contamination appears at about 2

A , after that the (signed) error of RUSSG starts to increase forming a positive

peak about 2.5 A . A similar positive peak is apparent on the hydrogen local spin

curve (Fig. 4.3). In the dissociation limit local spin by RUSSG tends to the correct

value. Spin purification diminishes the error of RUSSG by shaving off the positive


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 2 3 4 5 6 7

<S

2 >H

bond distances [Å]

APSGRUSSG

SP−RUSSGFCI

(a)

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

1 2 3 4 5 6 7

<S

2 >H

− [

<S

2 >H

] FC

I

bond distances [Å]

APSGRUSSG

SP−RUSSG

(b)

Figure 4.3: Local spin of hydrogen (a) and local spin difference with respectto FCI (b) for H2O symmetric dissociation, in 6-31G** basis. [184, 185]

peaks from the difference curves, though the negative peak still remains.

4.4.2 Nitrogen molecule dissociation

In the complete active space approach nitrogen dissociation has to include six

active electrons. Geminal methods assign the six electrons to three bonds, which

may prevent correct description of the quartet state of free nitrogen atoms. The

total energy curve, computed in 6-31G basis [186], moves together for APSG,

RUSSG and SP-RUSSG (see Fig. 4.4) until spin contamination appears around

2 A . Beyond 2 A RUSSG and SP-RUSSG produce significantly deeper energies

increasing the nonparallelism error.

Similarly to the water example, near the dissociation limit APSG gives a mixed

local spin state again, c.f. Fig. 4.5, which can be determined from the symmetry

restrictions of geminal coefficients (Eq.(4.15)). Meanwhile RUSSG and SP-RUSSG

provide qualitatively correct local spin, where the nitrogen atom, analogously to

oxygen atom in water, is described by the one-determinant high-spin function in

the dissociation limit. There is a small peak on the local spin difference curve

around 2 A , which is diminished by spin purification. The small (about 10 mil-

lihartree) hump appears in the same distance range on the total energy curve of

RUSSG, which is flattened by SP-RUSSG.


−109.05

−109

−108.95

−108.9

−108.85

−108.8

−108.75

−108.7

−108.65

1 1.5 2 2.5 3 3.5 4 4.5 5

E [

Eh]

bond distances [Å]

APSGRUSSG

SP−RUSSGFCI

(a)

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

1 1.5 2 2.5 3 3.5 4 4.5 5

E−

EF

CI [

Eh]

bond distances [Å]

APSGRUSSG

SP−RUSSG

(b)

Figure 4.4: Total energy (a) and energy difference with respect to FCI (b) forN2 dissociation, in 6-31G basis set [186].

0

0.5

1

1.5

2

2.5

3

3.5

4

1 1.5 2 2.5 3 3.5 4 4.5 5

<S

2 >N

bond distances [Å]

APSGRUSSG

SP−RUSSGFCI

(a)

−2

−1.5

−1

−0.5

0

0.5

1

1 1.5 2 2.5 3 3.5 4 4.5 5

<S

2 >N

− [

<S

2 >N

] FC

I

bond distances [Å]

APSGRUSSG

SP−RUSSG

(b)

Figure 4.5: Local spin of nitrogen (a) and local spin difference with respectto FCI (b) for N2 dissociation, in 6-31G basis set [186].

4.4.3 The H4 system

The last example is the H4 system, computed with 6-31G** basis [184]. The

four hydrogen atoms are confined to a circle and the initially drawn rectangle is

gradually distorted to a square. Change of geometry is characterized by the (H-

X-H) angle where X refers to the centre of mass. The challenge of this system is

the simultaneous breaking and formation of covalent bonds.

As mentioned in Sec. 4.2, RUSSG and SP-RUSSG can describe the H4 system


−2.14

−2.12

−2.1

−2.08

−2.06

−2.04

−2.02

−2

85 86 87 88 89 90

E [

Eh]

angle (H−X−H) [degree]

APSG locAPSG deloc

RUSSGSP−RUSSG

SP−RUSSG, optFCI

(a)

0.02

0.03

0.04

0.05

0.06

0.07

0.08

85 86 87 88 89 90

E−

EF

CI [E

h]


APSG locAPSG deloc

RUSSGSP−RUSSG

SP−RUSSG,opt

0.028

0.029

0.03

0.031

0.032

0.033

0.034

0.035

87.8 87.9 88 88.1 88.2

(b)

Figure 4.6: Total energy (a) and energy difference with respect to FCI (b) forH4, in 6-31G** basis set [184]. The four hydrogen atoms are confined to circlewith a radius of

√2 bohr. The angle of two neighbouring hydrogen atoms (H)

and the centre of mass (X) is labelled angle(H-X-H).

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

85 86 87 88 89 90

<S

2>

H


APSG locAPSG deloc

RUSSGSP−RUSSG

SP−RUSSG,optFCI

(a)

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

85 86 87 88 89 90

<S

2>

H −

[<

S2>

H] F

CI


APSG locAPSG deloc

RUSSGSP−RUSSG

SP−RUSSG,opt

(b)

Figure 4.7: Local spin of hydrogen (a) and local spin difference with respectto FCI (b) for H4 rectangular to square distortion, in 6-31G** basis set [184].

For geometry see Fig. 4.6.


qualitatively well at the H2-H2 dissociation limit, and a similar behavior can be

derived for APSG as well. However, the energy curve of APSG shows a charac-

teristic cusp at the square geometry, formed by the crossing of two distinct solu-

tions: one with orbitals localized on horizontally aligned hydrogen molecules, the

other corresponding to vertically aligned H2 systems. The two resonance struc-

tures are degenerate at exactly 90o. As apparent in Fig. 4.6, no cusp appears

on the RUSSG and SP-RUSSG curves which can mainly be attributed to orbital

delocalization[81]. The curve labelled ”APSG deloc” underlines this statement,

showing a stationary solution of the APSG equations, giving delocalized orbitals

and higher energy than APSG, but no cusp at 90o. Triplet component of the

geminals appear somewhat below 88o, causing a step on the SP-RUSSG curve

but not on RUSSG, see the inset in panel (b) of Fig. 4.6. Discontinuity on the

SP-RUSSG curve is not surprising, bearing in mind that the underlying procedure

is essentially projection after variation. Optimizing the geminal coefficients, the

discontinuity can be removed, c.f. curve ’SP-RUSSG, opt’.

The dissociation like process is manifested by an increase in the local spin of

hydrogen as can be seen in Fig. 4.7. While APSG with localized orbitals can not

reflect this behavior, RUSSG repairs local spin abruptly once spin-contamination

appears. Appearance of triplet components of geminals is accompanied by orbital

delocalization, occurring just below 88o for RUSSG. Spin-purification has a de-

creasing effect on local spin, setting the error larger than RUSSG near to square

geometry. Coefficient optimization of SP-RUSSG improves at square geometry

and removes the step near to 88o. There occurs however a switch between local-

ized and delocalized solutions for ’SP-RUSSG, opt’ also. This is responsible for

the smaller step on the local spin curve just above 86o. Interestingly, when taking

the solution of APSG with delocalized orbitals, one gets a local spin curve parallel

with FCI and the best values at smaller angles.

Overlap values with the FCI vector, displayed in Fig. 4.8, give another look on

the quality of the wave function and largely lead to conclusion similar to the above.

Overlap of APSG with localized orbitals decreases approaching the square geom-

etry. Allowing spin-contamination of geminals gives worse results, c.f. RUSSG.


0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

85 86 87 88 89 90

Overl

ap


APSG locAPSG deloc

RUSSGSP−RUSSG

SP−RUSSG,opt

Figure 4.8: Overlap with the FCI vector for H4 rectangular to square distor-tion, in 6-31G** basis set [184]. For geometry see Fig. 4.6.

It is spin projection which improves the overlap at larger angles, as apparent on

the SP-RUSSG curve. Coefficient optimization, when in effect, pushes the overlap

very close to 1. The overlap of APSG with delocalized orbitals here also gives

a flat curve impressively close to 1. Notably, APSG with localized orbitals has

smaller overlap with FCI, than the higher energy, delocalized solution.

4.5 MR-LCC corrections for geminal based ref-

erence functions

As shown in the previous section, APSG may fail to describe the local spin prop-

erly, especially near the dissociation limit. In this section the recently introduced

APSG based Multireference Linearized Coupled Cluster (MR-LCC) method1 [127]

1The general considerations of LCC and the related Coupled Electron Pair Approximationare not discussed here in details, the particular description can be found in Refs. [187] and [9].


is discussed, which is partially based on Tamas Zoboki’s upcoming Ph.D. work

[188]. In the second part we generalize this formalism to SP-RUSSG based LCC

calculations.

Assuming the following form of the cluster operator:

T =∑

k

tkXk ,

where excitation operator Xk creates the k-th excited function (|ΨAPSGk 〉 = Xk|ΨAPSG

0 〉),the LCC equation can be derived [127, 189] as

ELCC = 〈ΨAPSG

0 |H|ΨAPSG

0 〉︸︷︷︸

EAPSG

+∑

k

〈ΨAPSG

0 |H|ΨAPSG

k 〉tk︸︷︷︸

Ecorr

, (4.29)

〈ΨAPSG

j |H|ΨAPSG

0 〉︸︷︷︸

bj

=∑

k

(

EAPSG δjk − 〈ΨAPSG

j |H|ΨAPSG

k 〉)

︸︷︷︸

Ajk

tk . (4.30)

As it can be seen, the LCC energy is determined by Eq.(4.29), while the amplitudes

are calculated by solving Eq.(4.30) as

tj =∑

k

(A−1

)

jkbk .

The excitations Xk are usually classified according to whether the electrons remain

on the same geminal or not. When one of the electrons is moved out to another

geminal, it is called charge transfer excitation, otherwise it is referred to as in-

trageminal excitation. The single and double intrageminal excitations are defined

by the geminal creation and annihilation operators as

T IN

1 =∑

I

∑

µ

tIIµ0 ψ+Iµψ

−I0 , (4.31)

T IN

2 =1

4

∑

IJ

∑

µν

tIJIJµν00 ψ+Iµψ

+Jνψ

−J0ψ

−I0 , (4.32)


where indices I and J refers to the geminals, while the Greek letter indices run

over the geminal states. The charge transfer excitations have the following form:

TCT1 =∑

I 6=A

∑

ia

tAIai∑

σ

ϕA+aσ ϕI−iσ , (4.33)

TCT2 =∑

IJAB

∑

abij

tABIJabij

∑

σσ′

Dσσ′

ABIJ

2 − δσσ′

4ϕA+aσ ϕ

B+bσ′ ϕ

J−jσ′ϕ

I−iσ , (4.34)

where factor (2− δσσ′)/4 is applied to eliminate redundancies and factor Dσσ′

ABIJ is

introduced to avoid the generation of intrageminal excitations as

Dσσ′

ABIJ = (1 − δAIδBJ)(1 − δAJδBIδσσ′) .

The geminal states in Eqs.(4.31) and (4.32) are obtained by solving the local

Schrodinger equation (Eq.(2.35)). These two-electron states can be assigned to two

types of spin states: singlet (S+Iµ) or triplet states with quantum number MS = 0

(0T +Iµ). As discussed in Sec. 2.1.2, the APSG wave function is the direct product

of ground state singlet geminals providing that the APSG is a singlet state as

well. Using the single intrageminal excitations (Eq.(4.31)), the triplet states can

be generated by exciting one of the singlet geminals to triplet geminal. Separating

the singlet (|sΨAPSGk 〉) and the triplet (|tΨAPSG

k 〉) excited geminals in Eq.(4.29), we

obtain the following expression:

ELCC = 〈sΨAPSG

0 |H|sΨAPSG

0 〉 + (4.35)

+∑

k

〈sΨAPSG

0 |H|sΨAPSG

k 〉stk +∑

l

〈sΨAPSG

0 |H|tΨAPSG

l 〉︸︷︷︸

0

ttl ,

where the matrix element 〈sΨAPSG0 |H|sΨAPSG

k 〉 is zero since the Hamiltonian pre-

serves the spin quantum number. As it can be seen in Eq.(4.35) the LCC energy


does not depend on the triplet amplitudes. Moreover, the singlet and triplet am-

plitudes decouple in the amplitude equations (Eq.(4.30)) as

〈sΨAPSG

j |H|sΨAPSG

0 〉 =∑

k

(

EAPSG δjk − 〈sΨAPSG

j |H|sΨAPSG

k 〉)stk +

−∑

l

〈sΨAPSG

j |H|tΨAPSG

l 〉︸︷︷︸

0

ttk ,

〈tΨAPSG

j |H|sΨAPSG

0 〉︸︷︷︸

0

= −∑

k

〈tΨAPSG

j |H|sΨAPSG

k 〉︸︷︷︸

0

stk +

+∑

l

(

EAPSG δjl − 〈tΨAPSG

j |H|tΨAPSG

l 〉)ttl .

Therefore, the triplet states do not have any effects on the LCC energy. However,

the double intrageminal excitations (or dispersive excitations) can create a product

of triplet geminals, which is a mixture of singlet and quadruplet states as

|ΨAPSG

k 〉 = csk |sΨAPSG

k 〉 + cqk |qΨAPSG

k 〉 .

Although the LCC energy depends explicitly only on the singlet contributions as

ELCC = 〈sΨAPSG

0 |H|sΨAPSG

0 〉 +

+∑

k

csk 〈sΨAPSG

0 |H|sΨAPSG

k 〉 + cqk 〈sΨAPSG

0 |H|qΨAPSG

k 〉︸︷︷︸

0

tk ,

the amplitude equations do not decouple, since

csj∗〈sΨAPSG

j |H|sΨAPSG

0 〉 =∑

k

(

EAPSG δjk −

−csj∗csk〈sΨAPSG

j |H|sΨAPSG

k 〉 − (4.36)

− cqj∗cqk〈qΨAPSG

j |H|qΨAPSG

k 〉)

tk .

This may cause spurious spin dependence of the LCC energy. In order to eliminate

this type of error, let us exchange the triplet-triplet product with the following


singlet state:

Π IJ+λκ =

√

1

3

(+1T +

Iλ−1T +

Jκ + −1T +Iλ

+1T +Jκ − 0T +

Iλ0T +

Jκ

), (4.37)

where +1T +Iλ and −1T +

Jκ can be defined similarly to Eqs.(4.19) and (4.20) with

operators S+ and S−. In that case, the intrageminal cluster operator in Eq.(4.32)

is modified in the following way:

sT IN

2 =1

4

∑

IJ

(∑

µν

StIJIJµν00 S+Iµ S+

Jν S−J0 S−

I0 +∑

λκ

T tIJIJλκ00 ΠIJ+λκ S−

J0 S−I0

)

. (4.38)

An additional problem occurs in charge transfer excitations as they can also gen-

erate mixed spin state functions. In this case, we can redefine the original charge

transfer excitations (Eqs.(4.33) and (4.34)) with spin-free excitation operators (Eqp

and Eqspr ). However, it leads to similar redundancy problems as in SA-SSMRPT

(c.f. Chapter 3). Therefore, an alternative procedure is implemented for the elim-

ination of fake couplings, which is described in Sec. 4.6.

A simple generalization of this formalism to SP-RUSSG based LCC metod is

based on considering a four-electron system only. Assuming two spatial orbitals

for every geminals (GVB), the SP-RUSSG can be constructed in a similar way as

in Eq.(4.22), such that

|ΨSP-RUSSG

g 〉 =(cSg S+

10 S+20 + cTg Π+

12

)|vac〉 . (4.39)

where indices 1 and 2 are related to the geminals and index g refers to the ground

state. In that case, only one Π IJ+λκ state is used, therefore to simplify the notation,

it is substituted by Π+12. The coefficients cSg and cTg are chosen so that |ΨSP-RUSSG

g 〉remains normalized to one as

∣∣cSg∣∣2

+∣∣cTg∣∣2

= 1 .

The SP-RUSSG based charge transfer excitations can be defined similarly as APSG


based excitations (Eqs.(4.33) and (4.34)). Meanwhile, the definition of intragem-

inal excitations becomes problematic due to the four-electron Π+12 state, thus we

have to leave the two-electron formalism.

In case of APSG the wave function is a product of two singlet geminals (S+10 S+

20),

while the intrageminal excited states can be collected into two groups (Eqs.(4.31)

and (4.38)): the products of singlet geminals (S+1µ S+

2ν , S+1µ S+

20 and S+10 S+

2ν , where

µ, ν > 0) and the triplet geminals projected to the singlet spin state (Π+12). In

SP-RUSSG the wave function is a linear combination of states S+10 S+

20 and Π+12.

Therefore, to obtain the same space as in the APSG case, we have to generate the

products of singlet geminals, which is excited in the APSG case as well (S+1µ S+

2ν ,

S+1µ S+

20 and S+10 S+

2ν , where µ, ν > 0). Moreover another state is needed to be able

to describe the two dimensional plane of S+10 S+

20 and Π+12. This excited state can

be given in the following form:

|ΨSP-RUSSG

e 〉 =(cSe S+

10 S+20 + cTe Π+

12

)|vac〉 , (4.40)

where the coefficients cSe and cTe are chosen so that |ΨSP-RUSSGe 〉 is orthogonal to

|ΨSP-RUSSGg 〉 as

〈ΨSP-RUSSG

e |ΨSP-RUSSG

g 〉 = cSe∗cSg + cTe

∗cTg = 0 .

Using Eqs.(4.39) and (4.40), the intrageminal excitations (Eqs.(4.31) and (4.32))

can be modified in the following way:

SPT IN

1 =1

2

2∑

I=1I 6=I

∑

µ

SPtIIµ0 S+IµS+

I0ΨSP-RUSSG−g , (4.41)

SPT IN

2 =1

2

2∑

I=1I 6=I

∑

µν

SPtIIµν S+IµS+

IνΨSP-RUSSG−g + SPte ΨSP-RUSSG+

e ΨSP-RUSSG−g . (4.42)

With the help of these cluster operators, the LCC equations (Eqs.(4.29) and (4.30))

can be solved in the same way as in case of APSG based LCC.


4.6 Implementation of MR-LCC corrected gem-

inal theories and some demonstrative exam-

ples

Two steps of LCC energy calculation can be separated. First, the amplitudes

are determined by solving the system of linear equations in Eq.(4.30). Substitut-

ing these amplitudes into Eq.(4.29), the LCC energy can be obtained. For the

expressions in Eqs.(4.29) and (4.30), matrix elements of the Hamiltonian have to

be evaluated. These matrix elements are determined by the general matrix el-

ement evaluation code, where the wave function and Hamiltonian are used in a

second quantized form. This code was originally implemented by Zoltan Rolik and

Agnes Szabados and further developed by Tamas Zoboki and Agnes Szabados for

this APSG based LCC method.

The excited functions are generated in two different ways. One part of the

excited functions are obtained by operating with the intrageminal excitations

(Eqs.(4.31) and (4.32)) on the APSG wave function. In order to determine the

charge transfer excited functions as well, the APSG wave function is expressed in

determinant basis. Afterwards, all single and double excited functions are gener-

ated by using a reference space, where each APSG determinant serves as a single

reference state. To avoid potential redundancies in the virtual space, we consider

every excited determinant only once, and omit those terms, which belong to APSG

and intrageminal excited functions. The remaining determinants correspond to the

charge transfer excited states.

As mentioned in the previous Section, the singlet LCC energy may have a spu-

rious dependence on higher spin states. However, the energy is invariant to the

unitary transformation of the excited functions.2 Therefore, the amplitude equa-

tions decouple according to spin, if the excited functions can be transformed into

spin eigenfunctions. For example, if only intrageminal excitations are considered,

excitations in Eqs.(4.31) and (4.32) couple the different spin eigenfunctions in the

2This is easy to show by substituting |ΨAPSGk 〉 =

∑

j

Ukj |ΨAPSGj

′〉 and tk =∑

j

U†kjt

′j into

Eqs.(4.29) and (4.30).


amplitude equation (Eq.(4.36)). However, if the spin polarized terms (+1T +Iλ

−1T +Jκ

and −1T +Iλ

+1T +Jκ) are also included, then the singlet spin state Π IJ+

λκ can be con-

structed (4.37), thus eliminating the spurious dependence.

Supplementing the intrageminal excitations with the charge transfer excitations

(Eqs.(4.33) and (4.34)), the missing triplet spin polarized terms are automatically

generated. However, the charge transfer excitations still bring additional excited

functions with mixed spin states. The missing excited functions are obtained by

generating all spin-flip configurations of these excited determinants.

In Ref. [127] this method is applied for intrageminal excitations, but the charge

transfer excitations were not handled appropriately. Here, the charge transfer

excitations are treated correctly by including the missing excited states discussed

above.

To reduce the number of the excited functions only two spatial orbitals are

considered for each occupied geminal (GVB model), while the remaining functions

are moved into the virtual geminal subspace. In Sec. 4.4 the modified Q-Chem

[183] is applied for the calculation of SP-RUSSG, however, it cannot be used here

as it can only treat the occupied geminal space. Therefore, the SP-RUSSG is

simulated here by diagonalizing the Hamiltonian in ASPG (|S10S20〉) and in |Π12〉basis. As the coefficients cSg and cTg in this simulated SP-RUSSG are obtained by

diagonalization (variational) process, it provides better results than the original

SP-RUSSG, where these coefficients are obtained by the spin-projection scheme.

Therefore, this method is more similar to the ’SP-RUSSG, opt’ scheme, which has

been already applied in Sec. 4.4 to optimize the geminal coefficients. Another

difference is that the APSG one-electron orbitals are used here instead of RUSSG

orbitals. However, it does not lead to any qualitative failures in our examples.

4.6.1 Water symmetric dissociation

Water symmetric dissociation is revisited here in 6-31G basis set [186]. The first

innermost shell of the oxygen atom is kept frozen in the geminal based LCC cal-

culations to reduce the computational demands. The two OH bonds are described

by two geminals, each of them contains two spatial orbitals. The remaining four


−76.4

−76.2

−76

−75.8

−75.6

−75.4

−75.2

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

E [

Eh]

bond distances [Å]

APSGSP−RUSSGAPSG LCC

SP−RUSSG LCCFCI

(a)

−0.05

0

0.05

0.1

0.15

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

E−

EF

CI [

Eh]

bond distances [Å]


SP−RUSSG LCC

(b)

Figure 4.9: Total energy (a) and energy difference with respect to FCI (b) forH2O symmetric dissociation, in 6-31G basis set [186], where the innermost shell

of the oxygen atom is kept frozen.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

<S

2 >O

bond distances [Å]


SP−RUSSG LCCFCI

(a)

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

<S

2 >O

− [

<S

2 >O

] FC

I

bond distances [Å]


SP−RUSSG LCC

(b)

Figure 4.10: Local spin of oxygen (a) and local spin difference with respectto FCI (b) for H2O symmetric dissociation, in 6-31G basis set [186], where the

innermost shell of the oxygen atom is kept frozen.

electrons of the oxygen atom are described by two geminals, each of which pos-

sesses one spatial orbital. In that case, the geminal contains only one closed shell

singlet state. The Eqs.(4.41) and (4.42) include the two-geminal contributions

only, for the four-geminal case additional spin-couplings are needed. However,

those geminals, which possess just one spatial orbital, do not bring extra triplet

components. Therefore we can use the original two-geminal expressions.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

<S

2 >H

bond distances [Å]


SP−RUSSG LCCFCI

(a)

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

<S

2 >H

− [

<S

2 >H

] FC

I

bond distances [Å]


SP−RUSSG LCC

(b)

Figure 4.11: Local spin of hydrogen (a) and local spin difference with respectto FCI (b) for H2O symmetric dissociation, in 6-31G basis set [186], where the

innermost shell of the oxygen atom is kept frozen.

As it can be seen in Figs. 4.9 - 4.11, the GVB restriction does not change

the characteristics of the potential energy and local spin curves of APSG and

SP-RUSSG methods significantly (Figs. 4.1 - 4.3). Above the equilibrium bond

length the SP-RUSSG energies go below the APSG energy, thus providing larger

nonparallelism error. In case of the local spin of the oxygen atom, SP-RUSSG

provides an appropriate description at all bond distances, meanwhile the APSG

gives false results near the dissociation limit. Despite of these, the local spin of

hydrogen is described qualitatively well by both methods, only the domain of 1-2.5

A bond length exhibits a significant deviation from the FCI value.

At short bond distances the energies of APSG based LCC method are equivalent

to the FCI results to millihatree precision (Figure 4.9). However, a singularity

appears at about 2 A bond distance, after this the accuracy cannot be restored

to millihatree scale. The local spin of the oxygen atom offers a deeper insight

into this process. After the local spin of APSG and FCI are separated, the kink

emerges on the LCC curve. At even larger distances APSG based LCC provides

lower local spin value than APSG. The spurious values can be originated from the

fact that APSG incorporates the triplet and singlet states of the oxygen atom near

the dissociation limit. Therefore the APSG wave function is not an appropriate

reference to describe the triplet oxygen atom. This problem with the local spin


−2.2

−2.15

−2.1

−2.05

−2

−1.95

−1.9

−1.85

−1.8

−1.75

85 86 87 88 89 90

E [

Eh]

H−X−H angle

APSG, locSP−RUSSG, locAPSG LCC, loc

SP−RUSSG LCC, locAPSG, deloc

SP−RUSSG, delocAPSG LCC, deloc

SP−RUSSG LCC, delocFCI

(a)

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

85 86 87 88 89 90

E−

EF

CI [E

h]

H−X−H angle

APSG, locSP−RUSSG, loc

APSG, delocSP−RUSSG, delocAPSG LCC, deloc

SP−RUSSG LCC, deloc

(b)

Figure 4.12: Total energy (a) and energy difference with respect to FCI (b)for H4, in 6-31G** basis set [184]. The four hydrogen atoms are confined to acircle with radius

√2 bohr. The angle of the two neighbouring hydrogen atoms

(H) and the centre of mass (X) is labelled by angle(H-X-H).

has been already discussed by Li et al. in the context of Pair-Correlated Coupled

Cluster (PCCC) method [126], where the coupled cluster correction is calculated

for APSG. In that case, the energy results near the dissociation limit also have

significant errors, which were caused by the inappropriate reference state of APSG.

Using the SP-RUSSG, the energy and the local spin curves improve, which is

originated from the proper description of triplet oxygen atom at the dissociation

limit.

4.6.2 The H4 system

The rearrangement of H4 is calculated in 6-31G** basis set [184] with the same

geometric parameters as in Sec. 4.4.3. Similarly to the previous example, the

two geminals contain two spatial orbitals. During the optimization process two

possible one-electron orbital bases can be obtained.

In one of the solutions one-electron orbitals are localized onto the H2 molecules.

In that case, the cusp emerges at the square geometry due to the previously

described rearrangement of one-electron orbitals (Figure 4.12). The APSG and

SP-RUSSG solutions are equivalent in this scale. The local spin of the hydrogen

atoms stays very close to the singlet state, it does not change significantly in the


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

85 86 87 88 89 90

<S

2>

H

H−X−H angle

APSG, locSP−RUSSG, locAPSG LCC, loc

SP−RUSSG LCC, locAPSG, deloc

SP−RUSSG, delocAPSG LCC, deloc

SP−RUSSG LCC, delocFCI

(a)

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

85 86 87 88 89 90

<S

2>

H −

[<

S2>

H] F

CI

H−X−H angle

APSG, delocSP−RUSSG, delocAPSG LCC, deloc

SP−RUSSG LCC, deloc

(b)

Figure 4.13: Local spin of hydrogen (a) and local spin difference with respectto FCI (b) for H4 rectangular to square distortion, in 6-31G** basis set [184].

For geometry see Fig. 4.12.

rearrangement process (Figure 4.13). The LCC energy has a singularity on the

energy curves in APSG and SP-RUSSG based treatments. The values of local spin

do not correlate with the FCI results.

Using delocalized basis, we obtain similar results as in Sec. 4.4.3. Although this

produces larger energies in APSG, the cusp disappears and energies of SP-RUSSG

also decrease. Moreover it provides appropriate local spin values in both cases.

Using the LCC correction, the energy and local spin curves improve. The SP-

RUSSG results show better agreement with the FCI values, though they mainly

depend on the choice of one-electron basis rather than on the type of the method

applied.

Chapter 5

Summary

This thesis consists of two main parts. The first part focuses on SS-MRPT,

a ”diagonalize-then-perturb” type multireference perturbation theory, the spin-

adapted form of which may produce unphysical kinks on potential energy sur-

faces. These originate from the redundancy emerging among virtual functions.

The elimination of this redundancy by canonical orthogonalization is examined on

two examples: LiH and HF molecules. Due to this method, the unphysical kinks

disappear from the potential energy curves, which can be seen on the sensitiv-

ity curves as well. However, this elimination can also be done by excluding the

so-called direct spectator excitations. Although after these exclusions additional

redundancies still remain, they do not affect the potential energy curves signifi-

cantly. Similar results are obtained by alternative orthogonalization methods and

determinant based approaches.

In the second part of the thesis strongly orthogonal geminal-type theories

(APSG, RUSSG and SP-RUSSG) are investigated, which can provide an alterna-

tive reference function in multireference calculations. One of the main drawback

of APSG is that it may not describe the spin-states of fragments in multiple bond

dissociation correctly, which leads to size consistency problems. Applying the un-

restricted version of this theory (RUSSG), the size consistency can be restored in

most of the cases. However, this method suffers from spin contamination. After

spin-purification (SP-RUSSG), additional higher spin-components may also con-

tribute to the fragment wave functions, which again induces problems with size

103

Chapter 5. Summary 104

consistency.

In the numerical examples only two- and three-bond dissociation processes are

examined, where RUSSG and SP-RUSSG can describe the local spin properties

properly. Near the dissociation limit APSG provides spurious local spin values.

Despite of this, the potential energy curves show a smaller nonparalellism error

than the RUSSG and SP-RUSSG curves do. Although the MR-LCC type correc-

tions do not improve the local spin values of APSG and it even produce singu-

larities on the potential energy curves, the SP-RUSSG based LCC can properly

reproduce the FCI results on millihatree scale.

Bond rearrangement is examined as an example, where APSG produces an

unphysical cusp on the potential energy curves besides the wrong local spin values.

Nevertheless, RUSSG and SP-RUSSG provide appropriate values in that case.

Using delocalized basis, the APSG energy increases, but the cusp disappears and

the local spin values also improve. Application of the LCC type corrections results

in proper potential energy curves. As it can be expected, using SP-RUSSG as a

reference function, the energy and local spin values improve further.

Appendix A

Exponential form of the APSG

wave function

Exponential parametrization of APSG has been discussed by many authors

[56, 60–63]. Here we would like to derive this exponential form according to the

reference [63]. Let us consider the APSG function in the following form:

|ΨAPSG〉 = N∏

I

(1 + TI)|0〉, (A.1)

where N is a normalization factor, |0〉 is the Fermi vacuum, I is the index, which

runs over strongly orthogonal subspaces, and TI is the excitation operator, which

affects the I-th subspace. The Fermi vacuum and operator TI have the following

form in natural orbital basis:

|0〉 =∏

I

( ϕI+oα ϕI+oβ )|vac〉

TI =1

4

∑

µ 6=otIµ ϕ

I+µα ϕ

I+µβ ϕ

I−oβ ϕI−oα ,

where |vac〉 is the physical vacuum and tIµ is the so-called amplitude. From the

above equations above it can be seen that

T nI |0〉 = 0 n ≥ 2.

105

Appendix A. Exponential form of the APSG wave function 106

Therefore Eq.(A.1) can be written in the following form:

|ΨAPSG〉 = N∏

I

(1 + TI +1

2!T 2I +

1

3!T 3I + . . . )|0〉 = N

∏

I

eTI |0〉.

Because of strong orthogonality operators TI commutate with each other and the

APSG wave function can be written in an exponential form:

|ΨAPSG〉 = N e∑

I TI |0〉.

Appendix B

Spin-unrestricted and restricted

forms of RUSSG

Written with spin unrestricted orbitals (φµα and χνβ), RUSSG wave function

has the following form:

|ΨRUSSG〉 =∏

I

(∑

µ,ν

CuIµν φ

I+µα χ

I+νβ

)

︸︷︷︸

RUψ+I

|vac〉 , (B.1)

where |ΨRUSSG〉 is RUSSG wave function and index u refers to the spin-unrestricted

geminals in CuIµν . The spatial part of α and β orbitals of RUψI (Eq.(2.40)) span

the same space, the so-called Arai-subspace [55] of RUψI . Abandoning index I and

using restricted (orthonormal) orbitals (ϕµ) to express φµ and χν , one can write

the down following transformations:

φ1α

φ2α

...

φNα

= Uα

ϕ1α

ϕ2α

...

ϕNα

, (B.2)

107

Appendix B. Spin-unrestricted and restricted forms of RUSSG 108

χ1β

χ2β

...

χNβ

= Uβ

ϕ1β

ϕ2β

...

ϕNβ

. (B.3)

The normalization condition of RUSSG reads:

∑

µ,ν

∣∣Cu

νµ

∣∣2

= 1 . (B.4)

Let us relate now the spin-unrestricted form Eq.(B.1) of RUSSG to the expression

using restricted natural orbitals ϕµ. Diagonalizing the density matrices

Pαµν = 〈ΨRUSSG|φ+

µαφ−να|ΨRUSSG〉 =

(CC†)

µν,

P βµν = 〈ΨRUSSG|χ+

µβχ−νβ|ΨRUSSG〉 =

(C†C

)

µν,

one obtains α-natural and β-natural orbitals (pseudo-natural orbitals). Let us

assume that Cuνµ is symmetric. Without any restriction one can suppose then that

Cu is diagonal, too. Consider first the 2-dimensional Arai-subspaces case, which

leading to the form put forward by Head-Gordon under the acronym Unrestricted

in Active Pairs (UAP) [190, 191]:

RUψ+ = Cuoo φ

+oα χ

+oβ + Cu

vv φ+vα χ

+vβ . (B.5)

The normalization condition of function RUψ requires condition Eq.(B.4) to sim-

plify as:

Cuoo

2 + Cuvv

2 = 1 .

Therefore these coefficients are chosen as trigonometric functions, according to:

Cuoo = cos ǫ , (B.6)

Cuvv = sin ǫ . (B.7)


Substituting Eqs.(B.6) and (B.7) into Eq.(B.5) and using of the two dimensional

form of Eqs.(B.2) and (B.3):

φI+oα

φI+vα

=

cosαI sinαI

sinαI − cosαI

ϕI+oα

ϕI+vα

,

χI+oβ

χI+vβ

=

cos βI − sin βI

sin βI cos βI

ϕI+oβ

ϕI+vβ

,

we obtain:

RUψ+ = cos ǫ(cos(α) ϕ+

oα + sin(α) ϕ+vα

) (cos(β) ϕ+

oβ − sin(β) ϕ+vβ

)+

+ sin ǫ(sin(α) ϕ+

oα − cos(α) ϕ+vα

) (sin(β) ϕ+

oβ + cos(β) ϕ+vβ

).

From the above expression of matrix Cr in

RUψ+ =∑

µ,ν∈o,vCrνµ ϕ

µα+ ϕνβ

+

can be given as:

Cr =

cos ǫ cosα cos β + sin ǫ sinα sin β − cos ǫ cosα sin β + sin ǫ sinα cos β

cos ǫ sinα cos β − sin ǫ cosα sin β − cos ǫ sinα sin β − sin ǫ cosα cos β

.

Matrix Cr can be separated to singlet and triplet parts by constructing the sym-

metric and antisymmetric matrices:

Cr = sCr + tCr ,

sCr =

cos ǫ cosα cos β + sin ǫ sinα sin β cos ǫ+sin ǫ

2sin (α− β)

cos ǫ+sin ǫ2

sin (α− β) − cos ǫ sinα sin β − sin ǫ cosα cos β

,

tCr =

0 sin ǫ−cos ǫ

2sin (α + β)

cos ǫ−sin ǫ2

sin (α + β) 0

.


Applying condition sin (α− β) = 0, the coefficient matrices transform into natural

orbital basis (i.e. P gets diagonal):

sCrnat =

sin ǫ sin2 α + cos ǫ cos2 α 0

0 − cos ǫ sin2 α− sin ǫ cos2 α

, (B.8)

tCrnat =

0 sin ǫ−cos ǫ

2sin (2α)

cos ǫ−sin ǫ2

sin (2α) 0

. (B.9)

Comparing Eq.(2.50) with Eq.(B.9) sin γ can be read:

sin γ =cos ǫ− sin ǫ√

2sin(2α) , (B.10)

from which cos γ can be derived using the well-known trigonometric relation:

cos γ = ±√

1 − sin2 γ = ±√

sin4 α + cos4 α + 2 sin2 α cos2 α sin (2α) . (B.11)

Let us substitute into Eq.(B.8):

sCrnat = cos γ

± sin ǫ sin2 α+cos ǫ cos2 α√sin4 α+cos4 α+2 sin2 α cos2 α sin(2α)

0

0 ± − cos ǫ sin2 α−sin ǫ cos2 α√sin4 α+cos4 α+2 sin2 α cos2 α sin(2α)

,

where the diagonal elements have a same relation as cos δ and sin δ in Eq.(2.49).

This connection can be seen by taking equal sin δ to the second diagonal element

in Eq.(B.9):

sin δ = ± − cos ǫ sin2 α− sin ǫ cos2 α√

sin4 α + cos4 α + 2 sin2 α cos2 α sin (2α),

and using the trigonometric relation:

cos δ = ±√

1 − sin2 δ

cos δ = ±√

1 − cos2 ǫ sin4 α + sin2 ǫ cos4 α− 2 cos ǫ sin ǫ cos2 α sin2 α

sin4 α + cos4 α + 2 sin2 α cos2 α sin (2α)

cos δ = ± sin ǫ sin2 α + cos ǫ cos2 α√

sin4 α + cos4 α + 2 sin2 α cos2 α sin (2α),


which equals to the first diagonal element in Eq.(B.9).

The similarity of UHF Ansatz can also be seen in expression of Eq.(B.1), when

Cuoo equals one (and correspondingly Cu

vv equals zero ). In that case RUSSG wave

function falls back to the UHF parametrization.

Let us examine now the case of an N -dimensional Arai subspace. First we

should notice that for N > 2, the natural orbital basis is not equivalent to the

basis where the matrix sCr is diagonal. In order to show that these bases are

different, let us evaluate the spin-free density matrix:

Pµν = 〈ΨRUSSG|ϕ+µα ϕ

−να + ϕ+

µβ ϕ−νβ|ΨRUSSG〉 , (B.12)

where µ and ν are in the Arai subspace of geminal RUψ, which simplifies expression

Eq.(B.12) to:

Pµν = 〈RUψ|ϕ+µα ϕ

−να + ϕ+

µβ ϕ−νβ|RUψ〉 . (B.13)

Substituting expression of RUψ into Eq.(B.13) and assuming matrix sC is diagonal,

RUψ+ =∑

µν

(sCr

µνδµν + tCrµν

)ϕ+µα ϕ

+νβ , (B.14)

Pµν takes the form:

Pµν = 2

[(sCr2

)

µνδµν +

(tCr2

)

µν

]

.

As apparent form above, the density matrix is diagonal if tCr2 is diagonal as

well. In the two dimensional case tCr2 is diagonal, but this is not true for higher

dimensions. Therefore, the equivalence of the natural basis and the sCr-diagonal

basis holds only for two dimensional Arai subspaces.

In order to decide the equivalence of the restricted and unrestricted forms, we

count the number of free parameters in them. In restricted case sCr (Eq.(B.14))

contains N free parameters supposing it is diagonal, while tCr has N(N − 1)/2

free parameters due to being antihermitian. If the norm of the geminal is chosen


to be one the coefficients satisfy the condition as:

∑

µ,ν

(sCr

µν + tCrµν

)2= 1 ,

therefore this restricted Ansatz provides N(N+1)2

− 1 (= N + N(N−1)2

− 1) free

parameters.

The unrestricted Ansatz form of Eq.(B.1), where the spatial part of orbitals

alpha and beta can be different. Assuming a one electron basis where Cu is

diagonal (i.e. pseudo-natural orbitals), there are N − 1 free parameters provided

(the number of the diagonal elements minus one parameter from the norm of

the geminal). This Ansatz contains two unitary transformations for the φµα and

χµβ orbitals c.f. Eqs.(B.2) and (B.3), which providea N(N−1)2

free parameters.

Therefore the total number of parameters is N(N+1)2

− 1, which equals to the

number of free parameters in the restricted Ansatz. We should notice that Cu

can be transformed to the restricted basis in a similar manner as done in the two

dimensional case. This allows to find the correspondence between the parameters

introduced in Cu an in sCr, tCr.

Let us finally study a possible generalization of the original RUSSG Ansatz in

Eq.(B.1) with breaking the symmetry of matrix Cu to obtain Cu,gen. To compare

this Ansatz with the symmetric Ansatz let us transform χνβ to φνβ basis and

separate the matrix to symmetric and antisymmetric part:

RUψ+I =

∑

µ,ν

Cu,genµν φ+

µα χ+νβ =

∑

µ,ν

(sCr

µν + tCrµν

)φ+µα φ

+νβ . (B.15)

Now let us transform orbitals φµα and χνβ to ϕµα and ϕνβ, making sCr diagonal.

Hence RUψ+I takes the form:

RUψ+I =

∑

µ

sCr,dµµϕ

+µα ϕ

+µβ +

∑

µν

tCr,dµν ϕ

+µα ϕ

+νβ , (B.16)


where we should notice that the antihermitian matrix remains antihermitian upon

unitary transformation:

A† = −A ,

A′† =(U†AU

)†= U†A†U = −U†AU = −A′ .

The expression in Eq.(B.16) has the same form as a symmetric Ansatz (Eq.(B.14)).

Therefore the number of the variational parameters should be equal. This means,

that there are an energy invariant transformation, which takes from the non-

symmetric unrestricted Ansatz (Eq.(B.15)) to the symmetric unrestricted Ansatz

(Eq.(B.14)). This can be found by diagonalizing Cu,gen by singular value decom-

position:

Cu = U†Cu,genW = U†Cu,gen WU†︸︷︷︸

V

U , (B.17)

where U, W and V are unitary matrices. Unitary matrix V controls the difference

between the α and the β functions, which does not provide any extra variational

parameters:

RUψ+I =

∑

µ,ν

Cu,sµν φ+

µα χ+λβ =

∑

µ,ν

Cu,genµν φ+

µα

∑

λ

Vνλ χ+λβ

︸︷︷︸

χ′+νβ

.

Matrix U in Eq.(B.17) fixes the same freedom as the requirement for a diagonal

sCr in the restricted case. One can conclude, that a non-symmetric coefficient

matrix in the unrestricted form of RUSSG does not represent any extension of the

Ansatz, it simply offers an alternative form.

Appendix C

Possible structures of geminals in

water symmetric dissociation

Consider the process of the symmetric dissociation of water, where the hydrogen

atoms are simultaneously removed from the oxygen atom to infinite distance, while

the HOH angle is kept constant. For the sake of simplicity, minimal basis is

considered, which means an s-type function on each hydrogen atom, 2 s- and 3

p-type functions on the oxygen atom. At equilibrium geometries the two bonding

geminals are constituted of the hydrogen orbitals and two hybrid orbitals (mixture

of s- and p-type orbitals) of the oxygen atom. It is enough to consider only

these four orbitals in the description of the dissociation process, therefore the

remaining orbitals of oxygen atom are not involved in the analysis. To see which

geminal structures appear in the dissociation limit, let us introduce the one-particle

function ξiσ, which is a linear combination of atomic orbitals:

ξ+iσ =2∑

p=1

DHpi ϕ

H+pσ +

2∑

q=1

DOqi ϕ

O+qσ , (C.1)

where ϕH+pσ (ϕO+

qσ ) is the orbital of hydrogen (oxygen) atom, and DHpi (DO

pi) is the

corresponding coefficient. The strongly orthogonal geminals can be constructed

114

Appendix C. Possible structures of geminals at water symmetric dissociation 115

by ξ+iσ as

ψ+A =

2∑

i,j=1

CAij ξ

+iα ξ

+jβ , (C.2)

ψ+B =

4∑

k,l=3

CBkl ξ

+kα ξ

+lβ , (C.3)

where for the sake of generality the coefficient matrices CA and CB do not need

to be symmetric (RUSSG model). The complete wave function of the system can

be obtained as the product of these geminals, such that:

Ψ+AB = ψ+

Aψ+B =

2∑

i,j=1

4∑

k,l=3

CAij C

Bkl ξ

+iα ξ

+jβ ξ

+kα ξ

+lβ . (C.4)

Let us substitute Eq.(C.1) into Eq.(C.4):

Ψ+AB =

2∑

i,j=1

4∑

k,l=3

CAijC

Bkl

2∑

p,q,r,s=1

(

DHpiD

HqjD

HrkD

Hsl ϕ

H+pα ϕ

H+qβ ϕ

H+rα ϕH+

sβ + (C.5)

+ DOpiD

OqjD

OrkD

Osl ϕ

O+pα ϕ

O+qβ ϕ

O+rα ϕO+

sβ +

+ POH(DHpiD

HqjD

HrkD

Osl ϕ

H+pα ϕ

H+qβ ϕ

H+rα ϕO+

sβ

)+

+ POH(DOpiD

OqjD

OrkD

Hsl ϕ

O+pα ϕ

O+qβ ϕ

O+rα ϕH+

sβ

)+

+ POH(DHpiD

HqjD

OrkD

Osl ϕ

H+pα ϕ

H+qβ ϕ

O+rα ϕO+

sβ

)

)

,

where operator POH generates all combinations of the determinants by interchang-

ing H and O indices. For example the expansion of the last term is given by the

following determinants:

POH(DHpiD

HqjD

OrkD

Osl ϕ

H+pα ϕ

H+qβ ϕ

O+rα ϕO+

sβ

)=

(

DHpiD

HqjD

OrkD

Osl ϕ

H+pα ϕ

H+qβ ϕ

O+rα ϕO+

sβ + DHpiD

OqjD

HrkD

Osl ϕ

H+pα ϕ

O+qβ ϕ

H+rα ϕO+

sβ +

+ DHpiD

OqjD

OrkD

Hsl ϕ

H+pα ϕ

O+qβ ϕ

O+rα ϕH+

sβ + DOpiD

HqjD

HrkD

Osl ϕ

O+pα ϕ

H+qβ ϕ

H+rα ϕO+

sβ +

+ DOpiD

HqjD

OrkD

Hsl ϕ

O+pα ϕ

H+qβ ϕ

O+rα ϕH+

sβ + DOpiD

OqjD

HrkD

Hsl ϕ

O+pα ϕ

O+qβ ϕ

H+rα ϕH+

sβ

)

.


In the dissociation limit three neutral atoms are expected, however, in Eq.(C.5)

only the last term belongs to this structure. The remaining ionic terms should be

eliminated by the manipulations of matrices CA, CB, DH and DO. For example in

the context of the hydrogen atom 1, we should exclude those terms, which contain

ϕH+1α ϕ

H+1β . In order to perform this, the spatial function ϕ1σ belongs to only one

ξiσ such that

DH1i = 1 DH

2i = 0 DO1i = 0 DO

2i = 0 .

Therefore in the product ξ+iαξ+jβ (i 6= j) this unwanted expression does not appear.

However, the product ξ+iαξ+iβ can still generate such terms, therefore only the off-

diagonal parts of CA and CB are considered. Therefore the geminals in Eqs.(C.2)

and (C.3) have the following expression:

ψ+A = CA

12 ξ+1α ξ

+2β + CA

21 ξ+2α ξ

+1β , (C.6)

ψ+B = CB

34 ξ+3α ξ

+4β + CB

43 ξ+4α ξ

+3β . (C.7)

Hence, only two type of geminal structures exist. The first type is when one of

the geminals is localized on fragment H2 and the other one is localized on fragment

O as

ξ+1σ = ϕH+1σ ξ+2σ = ϕH+

2σ ξ+3σ = ϕO+1σ ξ+4σ = ϕO+

2σ .

It can only describe the singlet oxygen in RSSG, because it is a product of singlet

geminals. However, the ground state of oxygen atom is a triplet state, which cannot

be described by this geminal structure. In RUSSG the geminals are unrestricted,

therefore they can describe the triplet geminal as well. In the second type the

geminals are constituted of a hydrogen function and an oxygen function as

ξ+1σ = ϕH+1σ ξ+2σ = ϕO+

1σ ξ+3σ = ϕH+2σ ξ+4σ = ϕO+

2σ . (C.8)

Let us substitute Eq.(C.8) into Eqs.(C.6) and (C.7) and introduce a new indexing

scheme for localized functions to be able to assign them to the indices of matrices


CA and CB. In this way we obtain the following expression for the geminals:

ψ+A = CA

12 ϕO+1α ϕH+

2β + CA21 ϕ

H+2α ϕO+

1β ,

ψ+B = CB

34 ϕO+3α ϕH+

4β + CB43 ϕ

H+4α ϕO+

3β ,

As mentioned above, by using the first geminal structure, the RSSG model provides

qualitatively false results due to the incorrect local spin value of the oxygen atom.

In the second geminal structure the local spin is not as easy to determine as in

the first case. A detailed examination can be found in Sec. 4.1.2.

Acknowledgements

First I would like to express my gratitude to my supervisor, Peter Surjan, who

despite of his academic obligations always found some time to contribute to this

work with his research experience. Without his guidance this dissertation would

not have been possible.

It is also important for me to show my gratitude to my consultant, Agnes

Szabados, who was always ready for fruitful discussions about my research. She

is the other person without who I would not be able to complete this thesis.

I also would like to express my thanks to the other members of the group,

Peter Nagy, Tamas Zoboki and Zsuzsanna Toth, for the discussions about several

different topics and the pleasant working environment they provided during these

years.

I am thankful to Vitaly Rassolov for the efficient cooperation in the local spin

topic and I would like to express my gratitude to Debasish Mukherjee and Rahul

Maitra for the valuable discussions about the topic of SSMRPT.

Finally, I would like to express my thanks to my girlfriend, Niki, for her help-

ful advices about the wording of my thesis and the supportive background she

provided together with my family.

118

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Abstract

Studies in multireference many-electron

theories

Peter Jeszenszki

Theoretical and Physical Chemsitry,

Structural Chemistry Doctoral Programme

Doctoral School of Chemistry

Eotvos Lorand University

Existing quantum classical methods generally give an appropriate description

of chemical compounds at equilibrium geometries. However, several chemically

interesting phenomena (e.g.: covalent bond dissociation and electronic structure

of radical compounds or transition metals) still induce methodological challenges.

The multireference technique has a potential in the treatment of these systems,

but there is no trivial way to do this properly.

In this thesis two multireference methods are investigated. The first is the

State-Specific Multireference Perturbation Theory (SSMRPT), which has some

advantageous features (size-extensivity and intruder independence), but its spin-

adapted form (SA-SSMRPT) might produce unphysical kinks on the potential

energy surfaces. In this work I show that these kinks are related to the emerging

redundancy in SA-SSMRPT equations. By the elimination of these redundancies

the kinks disappear from the potential energy surfaces.

In the second method geminals are utilized in the examinations. This method

can properly describe single bond dissociation processes, but it may produce spuri-

ous results in case of multiple bond dissociation, which are caused by the improper

description of the spin states of the fragments. In this work I addressed these is-

sues with several geminal functions. I also investigated, how these wave functions

can be applied as reference functions in multireference calculations.

Tudomanyos osszefoglalo

Tanulmany a multireferencia sok-elektron

elmeletekrol

Jeszenszki Peter

Elmeleti es Fizikai Kemia,

Anyagszerkezetkutatas Doktori Program

Kemiai Doktori Iskola

Eotvos Lorand Tudomanyegyetem

Standard kvantumkemiai modszerekkel megfeleloen nagy pontossag erheto el

egyszerubb vegyuletek targyalasakor egyensulyi geometrianal, azonban nehany

kemiai szempontbol erdekes pelda (kovalens kotes disszociacio, szabad gyokok es

atmeneti femek elektronszerkezete) meg mindig kihıvast jelent. A multireferencia

technikak lehetseges megoldast nyujtanak ezeknek a rendszereknek a kezelesere,

de ezen modszerek megfelelo hasznalata meg nem kiforrott.

Ez az ertekezes ket multireferencia alapu modszerrel foglalkozik. Az elso az

ugynevezett Allapot-Specifikus Multireferencia Perturbacios Elmelet (State-Spe-

cific Multireference Perturbation Theory, SSMRPT), ami ugyan szamos elonnyel

rendelkezik (meretkonzisztencia es intruder fuggetlenseg), a potencialis energia-

feluleten azonban gyakran fizikailag nem indokolt csucsokat produkal. Ebben a

doktori munkaban bemutatom a nem-fizikai csucsok es az SA-SSMRPT egyen-

letekben megjeleno redundancia kapcsolatat. A redundancia megszuntetesevel az

energia feluletek is kisımulnak.

A dolgozatban targyalt masodik modszer a ket-elektron fuggvenyek (geminalok)

hasznalatan alapul. Habar ez a modszer jo kozelıtessel le tudja ırni az egysz-

eres kotes disszociaciot, tobbszoros kotes disszociacio eseten hamis eredmenyek

adodnak. A hiba elsosorban annak tulajdonıthato, hogy a modszer rosszul ırja

le a fragmensek spin allapotait. Ebben a doktori munkaban a fenti problemat

analizalom nehany geminal fuggveny tıpusra. Ezen felul fontos kerdeskent vetodott

fel a geminalok referenciafuggvenykent valo alkalmazhatosaga multireferencias sza-

molasok soran.

Studies in multireference many-electron theories

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