studies in multireference many-electron theories
TRANSCRIPT
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
Chapter 2. Theoretical Background 5
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)
Chapter 2. Theoretical Background 6
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)
Chapter 2. Theoretical Background 7
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 ,
Chapter 2. Theoretical Background 8
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
Chapter 2. Theoretical Background 9
(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
Chapter 2. Theoretical Background 10
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
Chapter 2. Theoretical Background 11
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)
Chapter 2. Theoretical Background 12
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)
Chapter 2. Theoretical Background 13
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
Chapter 2. Theoretical Background 14
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
Chapter 2. Theoretical Background 15
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)
Chapter 2. Theoretical Background 16
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
Chapter 2. Theoretical Background 17
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
Chapter 2. Theoretical Background 18
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 .
Chapter 2. Theoretical Background 19
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
Chapter 2. Theoretical Background 20
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).
Chapter 2. Theoretical Background 21
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.
Chapter 2. Theoretical Background 22
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
Chapter 2. Theoretical Background 23
(〈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
Chapter 2. Theoretical Background 24
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)
Chapter 2. Theoretical Background 25
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
Chapter 2. Theoretical Background 26
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
Chapter 2. Theoretical Background 27
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
Chapter 2. Theoretical Background 28
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
Chapter 2. Theoretical Background 29
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
Chapter 2. Theoretical Background 30
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 ,
Chapter 2. Theoretical Background 31
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)
Chapter 2. Theoretical Background 32
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)
Chapter 2. Theoretical Background 33
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)
Chapter 2. Theoretical Background 34
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)
Chapter 2. Theoretical Background 35
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.
Chapter 3. Redundancy in Spin-Adapted SSMRPT 38
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
Chapter 3. Redundancy in Spin-Adapted SSMRPT 39
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
Chapter 3. Redundancy in Spin-Adapted SSMRPT 40
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)
Chapter 3. Redundancy in Spin-Adapted SSMRPT 41
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 〉 .
Chapter 3. Redundancy in Spin-Adapted SSMRPT 42
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,
Chapter 3. Redundancy in Spin-Adapted SSMRPT 43
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
Chapter 3. Redundancy in Spin-Adapted SSMRPT 44
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)
Chapter 3. Redundancy in Spin-Adapted SSMRPT 45
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
Chapter 3. Redundancy in Spin-Adapted SSMRPT 46
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|,
Chapter 3. Redundancy in Spin-Adapted SSMRPT 47
Table 3.3: Overlapping sets of normalized excited functions and their or-thonormalized counterparts for double excitations. Model function |Φ0µ〉 is atwo-determinantal open-shell, according to Eq.(3.2). Index ordering i < j, a < band u < v is assumed. See text for labeling convention. Notation p(ua) stands
for the parity of the permutation ordering the pair (ua).
overlapping functions orthonormal functions
2 cores→2 empties
|µΥabij 〉 = 1
2
Eabij
c|Φ0µ〉
|µΥbaij 〉 = 1
2
Ebaij
c|Φ0µ〉
|µ1Υabij 〉 = |µΥab
ij 〉 + |µΥbaij 〉
|µ2Υabij 〉 = 1√
3
(|µΥab
ij 〉 − |µΥbaij 〉)
2 cores→active(1), empty
|µΥusaij 〉 = 1√
2
Eusaij
c|Φ0µ〉
|µΥausij 〉 = 1√
2
Eausij
c|Φ0µ〉
|µ1Υusaij 〉 = |µΥusa
ij 〉 + |µΥausij 〉
|µ2Υusaij 〉 = (−1)p(usa) 1√
3
(|µΥusa
ij 〉 − |µΥausij 〉)
core, active(1)→2 empties
|µΥabius〉 = 1√
2
Eabius
c|Φ0µ〉
|µΥbaius = 1√
2
Ebaius
c|Φ0µ〉
|µ1Υabius〉 = |µΥab
ius〉 + |µΥbaius〉
|µ2Υabius〉 = 1√
3
(|µΥab
ius〉 − |µΥbaius〉)
2 cores→2 active(1)’s
|µΥusvsij 〉 =
Eusvsij
c|Φ0µ〉
|µΥvsusij =
Evsusij
c|Φ0µ〉
|µΥusvsij 〉 = 1
2
(|µΥusvs
ij 〉 + |µΥvsusij 〉
)
2 active(1)’s→2 empties
|µΥabusvs〉 =
Eabusvs
c|Φ0µ〉
|µΥbausvs〉 =
Ebausvs
c|Φ0µ〉
|µΥabusvs〉 = 1
2
(|µΥab
usvs〉 + |µΥbausvs〉
)
core, active(1)→active(1), empty
|µΥvsaius
〉 =
Evsaius
c|Φ0µ〉
|µΥavsius
〉 = 12
Eavsius
c|Φ0µ〉
|µΥvsaius
〉 = (−1)p(vsa) 12
(|µΥvsa
ius〉 − |µΥavs
ius〉)
Chapter 3. Redundancy in Spin-Adapted SSMRPT 48
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)
core, active(1)→active(1), empty
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
Chapter 3. Redundancy in Spin-Adapted SSMRPT 49
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
Chapter 3. Redundancy in Spin-Adapted SSMRPT 50
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)
Chapter 3. Redundancy in Spin-Adapted SSMRPT 51
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〉 .
Chapter 3. Redundancy in Spin-Adapted SSMRPT 52
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
Chapter 3. Redundancy in Spin-Adapted SSMRPT 53
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
Chapter 3. Redundancy in Spin-Adapted SSMRPT 54
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
type of µ f = 1 f = 2
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
type of µ f = 1 f = 2
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
type of µ f = 1 f = 2
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.
Chapter 3. Redundancy in Spin-Adapted SSMRPT 55
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
Chapter 3. Redundancy in Spin-Adapted SSMRPT 56
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)
Chapter 3. Redundancy in Spin-Adapted SSMRPT 57
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
Chapter 3. Redundancy in Spin-Adapted SSMRPT 58
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)
Chapter 3. Redundancy in Spin-Adapted SSMRPT 59
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)
Chapter 3. Redundancy in Spin-Adapted SSMRPT 60
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)
Chapter 3. Redundancy in Spin-Adapted SSMRPT 61
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.
Chapter 3. Redundancy in Spin-Adapted SSMRPT 62
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)
Chapter 3. Redundancy in Spin-Adapted SSMRPT 63
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
Chapter 3. Redundancy in Spin-Adapted SSMRPT 64
−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
Chapter 3. Redundancy in Spin-Adapted SSMRPT 65
−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
Chapter 3. Redundancy in Spin-Adapted SSMRPT 66
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
Chapter 3. Redundancy in Spin-Adapted SSMRPT 67
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, TMP, T, no dir spec
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
EN, TEN, T, no dir spec
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.
Chapter 3. Redundancy in Spin-Adapted SSMRPT 68
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, TMP, T, no dir spec
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, TEN, T, no dir spec
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, TMP, T, no dir spec
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, TEN, T, no dir spec
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.
overlapping functions orthonormal functions
core→active(1)
|µΥusi 〉 =
Eusi
c|Φ0µ〉
|µΥvsusivs
〉 = 12
Evsusivs
c|Φ0µ〉
|µΥusi 〉 =
(|µΥus
i 〉 + |µΥvsusivs
〉)
core, active(1)→active(1), empty
|µΥvsaius
〉 =
Evsaius
c|Φ0µ〉
|µΥavsius
〉 = 12
Eavsius
c|Φ0µ〉
|µΥvsaius
〉 = (−1)p(vsa)(
|µΥvsaius
〉 + |µΥavsius
〉)
Chapter 3. Redundancy in Spin-Adapted SSMRPT 69
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
Chapter 4. Role of local spin in geminal-type theories 72
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
Chapter 4. Role of local spin in geminal-type theories 73
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
Chapter 4. Role of local spin in geminal-type theories 74
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)
Chapter 4. Role of local spin in geminal-type theories 75
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
Chapter 4. Role of local spin in geminal-type theories 76
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,
Chapter 4. Role of local spin in geminal-type theories 77
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
Chapter 4. Role of local spin in geminal-type theories 78
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)
Chapter 4. Role of local spin in geminal-type theories 79
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.
Chapter 4. Role of local spin in geminal-type theories 80
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)
Chapter 4. Role of local spin in geminal-type theories 81
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
Chapter 4. Role of local spin in geminal-type theories 82
fragments:
P s|1ΨRUSSG
H4〉 =
[
cos (δ1) cos (δ2)S+1 (γ1)S+
2 (γ2) +sin (δ1) sin (δ2)√
3Π+
12
]
|vac〉, (4.25)
P s|2ΨRUSSG
H4〉 =
[
cos (δ3) cos (δ4)S+3 (γ3)S+
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.
Chapter 4. Role of local spin in geminal-type theories 83
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)
Chapter 4. Role of local spin in geminal-type theories 84
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
Chapter 4. Role of local spin in geminal-type theories 85
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
Chapter 4. Role of local spin in geminal-type theories 86
−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
Chapter 4. Role of local spin in geminal-type theories 87
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.
Chapter 4. Role of local spin in geminal-type theories 88
−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
Chapter 4. Role of local spin in geminal-type theories 89
−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]
angle (H−X−H) [degree]
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
angle (H−X−H) [degree]
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
angle (H−X−H) [degree]
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.
Chapter 4. Role of local spin in geminal-type theories 90
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.
Chapter 4. Role of local spin in geminal-type theories 91
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
angle (H−X−H) [degree]
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].
Chapter 4. Role of local spin in geminal-type theories 92
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)
Chapter 4. Role of local spin in geminal-type theories 93
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
Chapter 4. Role of local spin in geminal-type theories 94
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
Chapter 4. Role of local spin in geminal-type theories 95
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
Chapter 4. Role of local spin in geminal-type theories 96
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.
Chapter 4. Role of local spin in geminal-type theories 97
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).
Chapter 4. Role of local spin in geminal-type theories 98
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
Chapter 4. Role of local spin in geminal-type theories 99
−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]
APSGSP−RUSSGAPSG LCC
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]
APSGSP−RUSSGAPSG LCC
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]
APSGSP−RUSSGAPSG LCC
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.
Chapter 4. Role of local spin in geminal-type theories 100
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]
APSGSP−RUSSGAPSG LCC
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]
APSGSP−RUSSGAPSG LCC
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
Chapter 4. Role of local spin in geminal-type theories 101
−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
Chapter 4. Role of local spin in geminal-type theories 102
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)
Appendix B. Spin-unrestricted and restricted forms of RUSSG 109
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
.
Appendix B. Spin-unrestricted and restricted forms of RUSSG 110
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),
Appendix B. Spin-unrestricted and restricted forms of RUSSG 111
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
Appendix B. Spin-unrestricted and restricted forms of RUSSG 112
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)
Appendix B. Spin-unrestricted and restricted forms of RUSSG 113
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
)
.
Appendix C. Possible structures of geminals at water symmetric dissociation 116
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
Appendix C. Possible structures of geminals at water symmetric dissociation 117
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
Bibliography
[1] Peter Jeszenszki, Peter R. Surjan, and Agnes Szabados. The Journal of
Chemical Physics, 138(12):124110, 2013.
[2] Peter Jeszenszki, Vitaly Rassolov, Peter R. Surjan, and Agnes Szabados.
Molecular Physics, 113(3-4):249–259, 2015.
[3] Bjorn O. Roos and Per-Olof Widmark. European Summer School in Quan-
tum Chemistry, Book II. Chemical Centre Printshop, Lund, 2000.
[4] Trygve Helgaker, Poul Jørgensen, and Jeppe Olsen. Molecular Electronic
Structure Theory. John Wiley & Sons, LTD, Chichester, 2000.
[5] Roy McWeeny and Brian T. Sutcliffe. Methods of molecular quantum me-
chanics. Academic Press London, New York, 1996.
[6] Hans-Joachim Werner. Advances in Chemical Physics, pages 1–62, 1987.
[7] Ron Shepard. Advances in Chemical Physics, 69:63–200, 1987.
[8] M. W. Schmidt and M. S. Gordon. Annu Rev Phys Chem, 49:233–66, 1998.
[9] Peter G. Szalay, Thomas Muller, Gergely Gidofalvi, Hans Lischka, and Ron
Shepard. Chemical Reviews, 112(1):108–181, 2012.
[10] Peter R. Surjan. Topics in current chemistry, 203:63–88, 1999.
[11] Marvin D. Girardeau. J. Math. Phys, 4:1096, 1963.
[12] Vladimir Kvasnicka. Czech. J. Phys., B32:947, 1982.
[13] Carmela Valdemoro. Phys. Rev. A, 31:2114, 1985.
119
Bibliography 120
[14] Peter R. Surjan. Phys. Rev. A, 30:43–50, 1984.
[15] Peter R. Surjan, Istvan Mayer, and Istvan Lukovits. Phys. Rev. A, 32:748,
1985.
[16] Tamas Zoboki, Peter Jeszenszki, and Peter R. Surjan. International Journal
of Quantum Chemistry, 113(3):185–189, 2013.
[17] Vitaly A. Rassolov. J. Chem. Phys., 117:5978, 2002.
[18] Vitaly A. Rassolov and Feng Xu. J. Chem. Phys., 126:234112, 2007.
[19] Ruben Pauncz. Spin Eigenfunctions. Plenum Press, New York, 1979.
[20] Hans-Joachim Werner and Wilfried Meyer. The Journal of Chemical Physics,
73(5):2342–2356, 1980.
[21] Hans-Joachim Werner and Peter J. Knowles. The Journal of Chemical
Physics, 82(11):5053–5063, 1985.
[22] Bjorn O. Roos. Advances in Chemical Physics, 69:399–445, 1987.
[23] Bjorn O. Roos, Peter R. Taylor, and Per E. M. Siegbahn. Chemical Physics,
48(2):157 – 173, 1980.
[24] Byron H. Lengsfield and Bowen Liu. The Journal of Chemical Physics, 75
(1):478–480, 1981.
[25] Jeppe Olsen. International Journal of Quantum Chemistry, 111(13):3267–
3272, 2011.
[26] Dominika Zgid and Marcel Nooijen. The Journal of Chemical Physics, 128
(14):144116, 2008.
[27] Debashree Ghosh, Johannes Hachmann, Takeshi Yanai, and Garnet Kin-Lic
Chan. The Journal of Chemical Physics, 128(14):144117, 2008.
[28] Yuki Kurashige, Garnet Kin-Lic Chan, and Takeshi Yanai. Nature Chem-
istry, 5:660–666, 2013.
Bibliography 121
[29] Sandeep Sharma, Kantharuban Sivalingam, Frank Neese, and Garnet Kin-
Lic Chan. Nature Chemistry, 6:927–933, 2014.
[30] F. W. Bobrowicz and W. A. Goddard-III. The Self-Consistent Field Equa-
tions for Generalized Valence Bond an Open-Shell Hartree-Fock Wave Func-
tions. In H. F. Schaefer-III, editor, Methods of Electronic Structure Theory,
page 79. Plenum, New York, 1977.
[31] Stephen P. Walch, Charles W. Bauschlicher Jr., Bjorn O. Roos, and Con-
stance J. Nelin. Chemical Physics Letters, 103(3):175 – 179, 1983.
[32] Haruyuki Nakano and Kimihiko Hirao. Chemical Physics Letters, 317(1–2):
90 – 96, 2000.
[33] Joseph Ivanic. The Journal of Chemical Physics, 119(18):9364–9376, 2003.
[34] A. I. Panin and O. V. Sizova. Journal of Computational Chemistry, 17(2):
178–184, 1996.
[35] Jeppe Olsen, Bjorn O. Roos, Poul Jørgensen, and Hans Jørgen Aa. Jensen.
The Journal of Chemical Physics, 89(4):2185–2192, 1988.
[36] Dongxia Ma, Giovanni Li Manni, and Laura Gagliardi. The Journal of
Chemical Physics, 135(4):044128, 2011.
[37] Vladimir A. Fock. Dokl.Akad.Nauk.USSR, 73:735, 1950.
[38] Andrew C. Hurley, John Lennard-Jones, and John A. Pople. Proc. R. Soc.
Lond. A, 220:446, 1953.
[39] Paul A. Johnson, Paul W. Ayers, Peter A. Limacher, Stijn De Baerdemacker,
Dimitri Van Neck, and Patrick Bultinck. Computational and Theoretical
Chemistry, 1003:101 – 113, 2013.
[40] Peter A. Limacher, Paul W. Ayers, Paul A. Johnson, Stijn De Baerdemacker,
Dimitri Van Neck, and Patrick Bultinck. Journal of Chemical Theory and
Computation, 9:1394–1401, 2013.
Bibliography 122
[41] Gustavo E. Scuseria, Carlos A. Jimenez-Hoyos, Thomas M. Henderson,
Kousik Samanta, and Jason K. Ellis. The Journal of Chemical Physics,
135(12):124108, 2011.
[42] Samuel F. Boys. Proceedings of the Royal Society of London. Series A.
Mathematical and Physical Sciences, 258(1294):402–411, 1960.
[43] K. Szalewitz, B. Jeziorski, H. J. Monkhorst, and J. G. Zabolitzki. J. Chem.
Phys., 78:1420, 1983.
[44] K. Szalewitz, B. Jeziorski, H.J. Monkhorst, and J.G. Zabolitzki. J. Chem.
Phys., 79:5543, 1983.
[45] Andrew Komornicki and Harry F. King. The Journal of Chemical Physics,
134(24):244115, 2011.
[46] Seiichiro Ten-no and Jozef Noga. Wiley Interdisciplinary Reviews: Compu-
tational Molecular Science, 2(1):114–125, 2012.
[47] J. Bardeen, L. N. Cooper, and J. R. Schrieffer. Phys. Rev., 108:1175–1204,
1957.
[48] J. M. Blatt. Prog.Theor.Phys, 23:447, 1960.
[49] A. J. Coleman. J.Math.Phys, 6:1425, 1965.
[50] S. Bratoz and Ph. Durand. The Journal of Chemical Physics, 43(8):2670–
2679, 1965.
[51] Brian Weiner, Hans-Jørgen Jensen, and Yngve Ohrn. The Journal of Chem-
ical Physics, 80(5):2009–2021, 1984.
[52] J. V. Ortiz, B. Weiner, and Y. Ohrn. Int. J. Quantum Chem., S15:113, 1981.
[53] E. Sangfelt, O. Goscinski, N. Elander, and H. Kurtz. Int. J. Quantum Chem.,
S15:133, 1981.
[54] J. M. Parks and R. G. Parr. J. Chem. Phys., 28:335, 1957.
Bibliography 123
[55] T. Arai. J. Chem. Phys., 33:95, 1960.
[56] David W. Small and Martin Head-Gordon. J. Chem. Phys., 137:114103,
2012.
[57] Werner Kutzelnigg. J. Chem. Phys., 40:3640, 1964.
[58] Valera Veryazov, Per Ake Malmqvist, and Bjorn O. Roos. International
Journal of Quantum Chemistry, 111(13):3329–3338, 2011.
[59] J. M. Foster and S. F. Boys. Rev. Mod. Phys., 32:300–302, 1960.
[60] I. I. Ukrainskii. Theoret. Math. Phys., 32:816, 1978.
[61] John Cullen. Chem. Phys., 202:217–229, 1996.
[62] Werner Kutzelnigg. Chemical Physics, 401:119 – 124, 2012.
[63] Peter R. Surjan, Agnes Szabados, Peter Jeszenszki, and Tamas Zoboki. Jour-
nal of Mathematical Chemistry, 50(3):534–551, 2012.
[64] Troy Van Voorhis and Martin Head-Gordon. Chem. Phys. Letters, 317:575–
580, 2000.
[65] Troy Van Voorhis and Martin Head-Gordon. J. Chem. Phys., 115(17):7814–
7821, 2001.
[66] Per-Olov Lowdin. Journal of Applied Physics, 33(1):251–280, 1962. Suppl.
[67] A. T. Amos and G. G. Hall. Proc. Roy. Soc. (London), 263:483, 1961.
[68] I. Mayer. Simple Theorems, Proofs, and Derivations in Quantum Chemistry.
Kluwer Academic/Plenum Publisher, 2003.
[69] Jason K. Ellis, Richard L. Martin, and Gustavo E. Scuseria. Journal of
Chemical Theory and Computation, 9(7):2857–2869, 2013.
[70] Peter Karadakov. International Journal of Quantum Chemistry, 27(6):699–
707, 1985.
Bibliography 124
[71] Istvan Mayer. International Journal of Quantum Chemistry, 29(1):31–34,
1986.
[72] John E. Harriman. The Journal of Chemical Physics, 40(10):2827–2839,
1964.
[73] Per-Olov Lowdin. Phys. Rev., 97:1509–1520, 1955.
[74] Per-Olov Lowdin. Rev. Mod. Phys., 36:966–976, 1964.
[75] Alexander A. Ovchinnikov and Jan K. Labanowski. Phys. Rev. A, 53:3946–
3952, 1996.
[76] K. Yamaguchi, Y. Takahara, T. Fueno, and K. N. Houk. Theoretica Chimica
Acta, 73(5-6):337–364, 1988.
[77] Yasutaka Kitagawa, Toru Saito, Masahide Ito, Mitsuo Shoji, Kenichi
Koizumi, Shusuke Yamanaka, Takashi Kawakami, Mitsutaka Okumura, and
Kizashi Yamaguchi. Chemical Physics Letters, 442(4–6):445 – 450, 2007.
[78] Yves G. Smeyers and Gerardo Delgado-Barrio. International Journal of
Quantum Chemistry, 8(5):733–743, 1974.
[79] Carlos A. Jimenez-Hoyos, Thomas M. Henderson, Takashi Tsuchimochi, and
Gustavo E. Scuseria. J. Chem. Phys., 136(16):164109, 2012.
[80] Istvan Mayer. The Spin-Projected Extended Hartree-Fock Method. vol-
ume 12 of Advances in Quantum Chemistry, pages 189 – 262. Academic
Press, 1980.
[81] Vitaly A. Rassolov and Feng Xu. J. Chem. Phys., 127:044104, 2007.
[82] A. M. Mak, K. V. Lawler, and M. Head-Gordon. Chemical Physics Letters,
515:173 – 178, 2011.
[83] Baird H. Brandow. Rev. Mod. Phys., 39:771–828, 1967.
[84] G. Hose and U. Kaldor. Journal of Physics B: Atomic and Molecular Physics,
12(23):3827, 1979.
Bibliography 125
[85] Leszek Meissner and Rodney J. Bartlett. The Journal of Chemical Physics,
91(8):4800–4808, 1989.
[86] I. Lindgren. Journal of Physics B: Atomic and Molecular Physics, 7(18):
2441, 1974.
[87] Vladimir Kvasnicka. Czechoslovak Journal of Physics B, 24(6):605–615,
1974.
[88] Krzysztof Wolinski and Peter Pulay. The Journal of Chemical Physics, 90
(7):3647–3659, 1989.
[89] Peter Pulay. International Journal of Quantum Chemistry, 111(13):3273–
3279, 2011.
[90] Kerstin. Andersson, Per-Ake. Malmqvist, Bjoern O. Roos, Andrzej J. Sadlej,
and Krzysztof. Wolinski. The Journal of Physical Chemistry, 94(14):5483–
5488, 1990.
[91] Kerstin Andersson, Per-Ake Malmqvist, and Bjorn O. Roos. The Journal of
Chemical Physics, 96(2):1218–1226, 1992.
[92] Bjorn O Roos, Kerstin Andersson, Markus P Fulscher, Per-Ake Malmqvist,
Luis Serrano-Andres, Kristin Pierloot, and Manuela Merchan. Advances in
Chemical Physics, 93:219–331, 1996.
[93] Zoltan Rolik, Agnes Szabados, and Peter R. Surjan. The Journal of Chemical
Physics, 119(4):1922–1928, 2003.
[94] Paolo Celani and Hans–Joachim Werner. The Journal of Chemical Physics,
112(13):5546–5557, 2000.
[95] Peter R. Nagy and Agnes Szabados. International Journal of Quantum
Chemistry, 113(3):230–238, 2013.
[96] C. Angeli, R. Cimiraglia, S. Evangelisti, T. Leininger, and J.-P. Malrieu.
The Journal of Chemical Physics, 114(23):10252–10264, 2001.
Bibliography 126
[97] Uttam Sinha Mahapatra, Barnali Datta, and Debashis Mukherjee. The
Journal of Physical Chemistry A, 103(12):1822–1830, 1999.
[98] Uttam Sinha Mahapatra, Barnali Datta, and Debashis Mukherjee. Chemical
Physics Letters, 299(1):42 – 50, 1999.
[99] Pradipta Ghosh, Sudip Chattopadhyay, Debasis Jana, and Debashis
Mukherjee. Int. J. Mol. Sci., 3(6):733 – 754, 2002.
[100] Dola Pahari, Sudip Chattopadhyay, Sanghamitra Das, and Debashis
Mukherjee. Chemical Physics Letters, 381(1–2):223 – 229, 2003.
[101] Erwin Schrodinger. Annalen der Physik, 385(13):437–490, 1926.
[102] PerOlov Lowdin. International Journal of Quantum Chemistry, 2(6):867–
931, 1968.
[103] Isaiah Shavitt and Rodney J. Bartlett. Many-Body Methods in Chemistry
and Physics. Cambridge University Press, 2009.
[104] Chr. Møller and M. S. Plesset. Phys. Rev., 46:618–622, Oct 1934.
[105] Per-Olov Lowdin. Journal of Mathematical Physics, 3(5):969–982, 1962.
[106] I. Lindgren and J. Morrison. Atomic many body theory. Springer, 1982.
[107] J. P. Malrieu, P. Durand, and J. P. Daudey. Journal of Physics A: Mathe-
matical and General, 18(5):809, 1985.
[108] Kenneth G. Dyall. The Journal of Chemical Physics, 102(12):4909–4918,
1995.
[109] Ede Kapuy. Theoretica chimica acta, 6(4):281–291, 1966.
[110] Ede Kapuy. Theoretica chimica acta, 12(5):397–404, 1968.
[111] Edina Rosta and Peter R. Surjan. J. Chem. Phys., 116:878–890, 2002.
[112] Mario Piris. The Journal of Chemical Physics, 139(6):064111, 2013.
Bibliography 127
[113] Enhua Xu and Shuhua Li. The Journal of Chemical Physics, 139(17):174111,
2013.
[114] Vitaly A. Rassolov. J. Chem. Phys., 120:10385, 2004.
[115] Brett A. Cagg and Vitaly A. Rassolov. The Journal of Chemical Physics,
141(16):164112, 2014.
[116] Gregory J. O. Beran, Martin Head-Gordon, and Steven R. Gwaltney. The
Journal of Chemical Physics, 124(11):114107, 2006.
[117] John A. Parkhill and Martin Head-Gordon. The Journal of Chemical
Physics, 133(12):124102, 2010.
[118] David W. Small and Martin Head-Gordon. The Journal of Chemical Physics,
137(11):114103, 2012.
[119] David W. Small, Keith V. Lawler, and Martin Head-Gordon. Journal of
Chemical Theory and Computation, 10(5):2027–2040, 2014.
[120] Peter R. Surjan and Agnes Szabados. International Journal of Quantum
Chemistry, 90(1):20–26, 2002.
[121] Agnes Szabados, Zoltan Rolik, Gabor Toth, and Peter R. Surjan. The Jour-
nal of Chemical Physics, 122(11):114104, 2005.
[122] Masato Kobayashi, Agnes Szabados, Hiromi Nakai, and Peter R. Surjan.
Journal of Chemical Theory and Computation, 6(7):2024–2033, 2010.
[123] Peter Jeszenszki, Peter R. Nagy, Tamas Zoboki, Agnes Szabados, and
Peter R. Surjan. International Journal of Quantum Chemistry, 114(16):
1048–1052, 2014.
[124] Mihaly Kallay and Peter R. Surjan. Chemical Physics Letters, 312(2–4):221
– 228, 1999.
[125] Jing Ma, Shuhua Li, and Wei Li. Journal of Computational Chemistry, 27
(1):39–47, 2006.
Bibliography 128
[126] Shuhua Li, Jing Ma, and Yuansheng Jiang. The Journal of Chemical Physics,
118(13):5736–5745, 2003.
[127] Tamas Zoboki, Agnes Szabados, and Peter R. Surjan. Journal of Chemical
Theory and Computation, 9:2602–2608, 2013.
[128] Katarzyna Pernal. Journal of Chemical Theory and Computation, 10(10):
4332–4341, 2014.
[129] Shuneng Mao, Lan Cheng, Wenjian Liu, and Debashis Mukherjee. The
Journal of Chemical Physics, 136(2):024106, 2012.
[130] Sudip Chattopadhyay, Uttam Sinha Mahapatra, and Rajat K. Chaudhuri.
Theoretical Chemistry Accounts, 131(4):1213, 2012.
[131] Sudip Chattopadhyay, Uttam Sinha Mahapatra, and Rajat K. Chaudhuri.
Chemical Physics Letters, 488(4–6):229 – 234, 2010.
[132] Sudip Chattopadhyay, Uttam Sinha Mahapatra, and Rajat K. Chaudhuri.
Chemical Physics, 401:15 – 26, 2012.
[133] Rajat K. Chaudhuri, Karl F. Freed, Gabriel Hose, Piotr Piecuch, Karol
Kowalski, Marta W loch, Sudip Chattopadhyay, Debashis Mukherjee, Zoltan
Rolik, Agnes Szabados, Gabor Toth, and Peter R. Surjan. The Journal of
Chemical Physics, 122(13):134105, 2005.
[134] Mark R. Hoffmann, Dipayan Datta, Sanghamitra Das, Debashis Mukherjee,
Agnes Szabados, Zoltan Rolik, and Peter R. Surjan. The Journal of Chemical
Physics, 131(20):204104, 2009.
[135] Uttam Sinha Mahapatra, Barnali Datta, Barun Bandyopadhyay, and De-
bashis Mukherjee. State–Specific Multi-Reference Coupled Cluster Formu-
lations: Two Paradigms. volume 30 of Advances in Quantum Chemistry,
pages 163 – 193. Academic Press, 1998.
[136] Uttam Sinha Mahapatra, Barnali Datta, and Debashis Mukherjee. The
Journal of Chemical Physics, 110(13):6171–6188, 1999.
Bibliography 129
[137] Francesco A. Evangelista and Jurgen Gauss. The Journal of Chemical
Physics, 133(4):044101, 2010.
[138] Sanghamitra Das, Mihaly Kallay, and Debashis Mukherjee. The Journal of
Chemical Physics, 133(23):234110, 2010.
[139] Sanghamitra Das, Mihaly Kallay, and Debashis Mukherjee. Chemical
Physics, 392(1):83 – 89, 2012.
[140] Dipayan Datta, Liguo Kong, and Marcel Nooijen. The Journal of Chemical
Physics, 134(21):214116, 2011.
[141] Thomas-C. Jagau and Jurgen Gauss. The Journal of Chemical Physics, 137
(4):044115, 2012.
[142] Leonie Anna Muck and Jurgen Gauss. The Journal of Chemical Physics,
136(11):111103, 2012.
[143] Bogumil Jeziorski and Hendrik J. Monkhorst. Phys. Rev. A, 24:1668–1681,
1981.
[144] Jozef Masik and Ivan Hubac. Multireference Brillouin–Wigner Coupled-
Cluster Theory. Single-root approach. volume 31 of Advances in Quantum
Chemistry, pages 75 – 104. Academic Press, 1998.
[145] Ivan Hubac, Jirı Pittner, and Petr Carsky. The Journal of Chemical Physics,
112(20):8779–8784, 2000.
[146] Jirı Pittner. The Journal of Chemical Physics, 118(24):10876–10889, 2003.
[147] Shuneng Mao, Lan Cheng, Wenjian Liu, and Debashis Mukherjee. The
Journal of Chemical Physics, 136(2):024105, 2012.
[148] Francesco A. Evangelista, Andrew C. Simmonett, Henry F. Schaefer III,
Debashis Mukherjee, and Wesley D. Allen. Phys. Chem. Chem. Phys., 11:
4728–4741, 2009.
Bibliography 130
[149] Dola Pahari, Sudip Chattopadhyay, Sanghamitra Das, Debashis Mukherjee,
and Uttam Sinha Mahapatra. Chapter 22 – size–consistent state–specific
multi–reference methods: A survey of some recent developments. In Clif-
ford E. Dykstra, Gernot Frenking, Kwang S. Kim, and Gustavo E. Scuseria,
editors, Theory and Applications of Computational Chemistry, pages 581 –
633. Elsevier, 2005.
[150] Robin Haunschild, Shuneng Mao, Debashis Mukherjee, and Wim Klopper.
Chemical Physics Letters, 531:247 – 251, 2012.
[151] Agnes Szabados. The Journal of Chemical Physics, 134(17):174113, 2011.
[152] Heinz Werner Engl, Martin Hanke, and Gunther Neubauer. Regularization
of Inverse Problems. Springer, 1996.
[153] Xiangzhu Li and Josef Paldus. J. Chem. Phys., 101(10):8812–8826, 1994.
[154] Dipayan Datta and Debashis Mukherjee. J. Chem. Phys., 134(5):054122,
2011.
[155] Miriam Heckert, Oliver Heun, Jurgen Gauss, and Peter G. Szalay. J. Chem.
Phys., 124(12):124105, 2006.
[156] Pavel Neogrady and Miroslav Urban. International Journal of Quantum
Chemistry, 55(2):187–203, 1995.
[157] Per-Olov Lowdin. Advances in Physics, 5(17):1–171, 1956.
[158] Rahul Maitra, Debalina Sinha, and Debashis Mukherjee. The Journal of
Chemical Physics, 137(2):024105, 2012.
[159] Tamas Turanyi. Journal of Mathematical Chemistry, 5(3):203–248, 1990.
[160] William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P.
Flannery. Numerical Recipes 3rd Edition: The Art of Scientific Computing.
Cambridge University Press, New York, NY, USA, 3 edition, 2007.
[161] T. H. Dunning, Jr. J. Chem. Phys., 90:1007, 1989.
Bibliography 131
[162] Thom H. Dunning. The Journal of Chemical Physics, 53(7):2823–2833, 1970.
[163] Andrei Zaitsevskii and Jean-Paul Malrieu. Chemical Physics Letters, 233:
597 – 604, 1995.
[164] The FCI code is based on sparse-FCI algorithm by Olsen [35], it is imple-
mented by Zoltan Rolik previously in our laboratory, 2007.
[165] John A. Pople, J. Stephen Binkley, and Rolf Seeger. International Journal
of Quantum Chemistry, 10(S10):1–19, 1976.
[166] Istvan Mayer and Miklos Kertesz. International Journal of Quantum Chem-
istry, 10(6):961–966, 1976.
[167] Obis Castano and Peter Karadakov. Chemical Physics Letters, 130(1–2):123
– 126, 1986.
[168] Eric Neuscamman. Phys. Rev. Lett., 109:203001, Nov 2012.
[169] Thomas M. Henderson and Gustavo E. Scuseria. The Journal of Chemical
Physics, 139(23):234113, 2013.
[170] A. E. Clark and E. R. Davidson. J. Chem. Phys., 115:7382, 2001.
[171] E. R. Davidson and A. E. Clark. Mol. Phys., 100:373, 2002.
[172] I. Mayer. Faraday Discuss., 135:146, 2007.
[173] C. Herrmann, M. Reiher, and B. Hess. J. Chem. Phys., 122:034104, 2005.
[174] I. Mayer. Chem. Phys. Letters, 440:357–359, 2007.
[175] D. Alcoba, A. Torre, L. Lain, and R. Bochicchio. Chem. Phys. Letters, 470:
136–139, 2009.
[176] I. Mayer. Chem. Phys. Letters, 478:323–356, 2009.
[177] D. Alcoba, A. Torre, L. Lain, and R. Bochicchio. Chem. Phys. Letters, 504:
236–240, 2011.
Bibliography 132
[178] D. Alcoba, A. Torre, L. Lain, and R. Bochicchio. J. Chem. Theory. Comput.,
7:3560–3566, 2011.
[179] I. Mayer. Chem. Phys. Letters, 539:172–174, 2012.
[180] Eloy Ramos-Cordoba, Eduard Matito, Istvan Mayer, and Pedro Salvador.
J. Chem. Theory. Comput., 8:1270–1279, 2012.
[181] E. Ramos-Cordoba, E. Matito, I. Mayer, and P. Salvador. Phys. Chem.
Chem. Phys., 14:15291–15298, 2012.
[182] R. S. Mulliken. The Journal of Chemical Physics, 23:1833, 1955.
[183] Y. Shao, L. F. Molnar, Y. Jung, J. Kussmann, C. Ochsenfeld, S. T. Brown,
A. T. B. Gilbert, L. V. Slipchenko, S. V. Levchenko, D. P. O’Neill, R. A. DiS-
tasio, R. C. Lochan, T. Wang, G. J. O. Beran, N. A. Besley, J. M. Herbert,
C. Y. Lin, T. Van Voorhis, S. H. Chien, A. Sodt, R. P. Steele, V. A. Ras-
solov, P. E. Maslen, P. P. Korambath, R. D. Adamson, B. Austin, J. Baker,
E. F. C. Byrd, H. Dachsel, R. J. Doerksen, A. Dreuw ans B. D. Dunietz,
A. D. Dutoi, T. R. Furlani, S. R. Gwaltney, A. Heyden, S. Hirata, C. P.
Hsu, G. Kedziora, R. Z. Khalliulin, P. Klunzinger, A. M. Lee, M. S. Lee,
W. Liang, I. Lotan, N. Nair, B. Peters, E. I. Proynov, P. A. Pieniazek, Y. M.
Rhee, J. Ritchie, E. Rosta, C. D. Sherrill, A. C. Simmonett, J. E. Subotnik,
H. L. Woodcock, W. Zhang, A. T. Bell, A. K. Chakraborty, D. M. Chip-
man, F. J. Keil, A. Warshel, W. J. Hehre, H. F. Schaefer, J. Kong, A. I.
Krylov, P. M. W. Gill, and M. Head-Gordon. Phys. Chem. Chem. Phys., 8:
3172–3191, 2006.
[184] P. C. Hariharan and J. A. Pople. Theoretica chimica acta, 28(3):213–222,
1973.
[185] Michelle M. Francl, William J. Pietro, Warren J. Hehre, J. Stephen Binkley,
Mark S. Gordon, Douglas J. DeFrees, and John A. Pople. The Journal of
Chemical Physics, 77(7):3654–3665, 1982.
Bibliography 133
[186] W. J. Hehre, R. Ditchfield, and J. A. Pople. The Journal of Chemical
Physics, 56(5):2257–2261, 1972.
[187] Peter G. Szalay. Towards State-Specific Formulation Of Multireference
Coupled-Cluster Theory: Coupled Electron Pair Approximation (CEPA)
Leading To Multirefernce Configurational Interaction (MR-CI) Type Equa-
tions. In Rodney J. Bartlett, editor, Recent Advances in Coupled-Cluster
Methods, volume 3 of Recent Advances in Computational Chemistry, pages
81–124. World Scientific, 1990.
[188] Tamas Zoboki. Methodological Developments in the Theory of Geminals.
PhD thesis, Doctoral School of Chemistry, Eotvos Lorand University, 2015.
[189] Attila Szabo and Neil S. Ostlund. Modern Quantum Chemistry: Introduction
to Advanced Electronic Structure Theory (Dover Books on Chemistry). Dover
Publications, 1996.
[190] Gregory J. O. Beran, Brian Austin, Alex Sodt, and Martin Head-Gordon.
The Journal of Physical Chemistry A, 109:9183–9192, 2005.
[191] K. V. Lawler, D. W. Small, and M. Head-Gordon. The Journal of Physical
Chemistry A, 114:2930–2938, 2010.
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.