SW.
«mam.

Ma .5» .o
; .meW‘

1. N15 ’
.(rfli \
. .
3.x LS .31.

f3. ”,3. psi"

54a»...

(5» ii («4:5
.. .Ilblilur ‘11..
V I i0 A z

1..
.l

vtu.

 

 

 

 

 

 

2 LIBRARY
2 078 Michigan State
Unwersrty J1
This is to certify that the
dissertation entitled

SINGLE-REFERENCE COUPLED-CLUSTER METHODS
EMPLOYING MULTl-REFERENCE PERTURBATION
THEORY

presented by
MARICRIS D. LODRIGUITO

has been accepted towards fulfillment
of the requirements for the

PhD. degree in Chemistry

le EQUNQA

Méfbr Professbr's Signature
Gag/07

Date

MSUlsanaflirrnefive-acfion.oqual-onpoflwfilyenployer

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/07 p:/CIRC/DaleDue.indd-p.1

 

 

SIN GLE—REFEREN CE COUPLED-CLUST ER METHODS
EMPLOYING MULTI-REFERENCE PERTURBATION THEORY

By

Maricris D. Lodriguito

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Chemistry

2007

ABSTRACT

SINGLE-REFEREN CE COUPLED-CLUSTER METHODS
EMPLOYING MULTIREFERENCE PERTURBATION THEORY

By
Maricris D. Lodriguito

A new class of noniterative coupled cluster (CC) methods, which improve the
results of standard CC and equation-of—motion (EOM) CC calculations for ground
and excited-state potential energy surfaces along bond breaking coordinates and for
excited states dominated by two-electron transitions, is explored. The proposed ap-
proaches combine the method of moments of coupled-cluster equations (MMCC), in
which the a posteriori corrections due to higher—order correlations are added to stan-
dard CC / EOMCC energies, with the multi-reference many-body perturbation the-
ory (MRMBPT), which provides information about the most essential nondynamical
and dynamical correlation effects that are relevant to electronic quasi-degeneracies.
The newly developed variant of the MMCC theory, termed MRMBPT—corrected
MMCC (MMCC / PT), is formulated using diagrammatic methods. The performance
of the basic MMCC / PT approximations, in which inexpensive corrections due to
triples (MMCC(2,3) / PT) or triples and quadruples (MMCC(2,4)/PT) are added to
ground- and excited-state energies obtained with the CC / EOMCC singles and doubles
(CCSD/EOMCCSD) approach, is illustrated by the results of benchmark calcula-
tions including bond breaking in HF, H20, and F2, and excited states of CH+. Test
calculations reveal that the MMCC(2,3)/PT and MMCC(2,4) / PT approaches pro-
vide an accurate description of bond breaking and excited states dominated by dou-

bles, eliminating the failures of the conventional CC / EOMCC methods at larger

internuclear distances and for excited states dominated by two-electron transitions
without invoking expensive steps of high-order CC / EOMCC methods. The efficient,
general-purpose, highly vectorized implementations of the triply and quadruply ex-
cited moments of the CCSD/EOMCCSD equations, relevant to the MMCC(2,3) / PT
and MMCC(2,4)/ PT approaches, are discussed and the details of the algorithm are

also presented.

ACKNOWLEDGMENTS

First and foremost, I owe my deepest gratitude to my Ph.D. supervisor, Prof.
Piotr Piecuch, for his advice, endless enthusiam, and guidance. It is a great privilege
to work with an exceptional scientist and teacher. He was always generous with his
time and knowledge and I definitely learned a lot from him. He knew when to give
me a little push when I needed it. I could not ask for a better advisor to guide my
graduate research.

I would also like to thank the rest of my thesis committee, including Prof. Katharine
C. Hunt, Prof. James K. McCusker, and Prof. Robert Cukier, for their advice, sup-
port, and willingness to review my work on a short notice.

I am indebted to many members of Piecuch group, both past and present. When
I first joined the group, Dr. Karol Kowalski helped me in the initial stages of my
research. After Karol’s departure, I had the great fortune to work with Dr. Marta
WIoch. I would like to thank them for setting me up and patiently guiding me. It has
been a real pleasant experience to collaborate with Dr. Peng-dong Fan and Jeffrey
Cour. The following people deserves special mention: Dr. Ian Pimienta, Dr. Michael
Mcguire, Dr. Dorothy Gearhart, Dr. Ruth Lafuente, Dr. Armagan Kinal, Jesse Lutz,
Anirban Mandal, and Afra Panahi, for providing a stimulating and fun environment
in which to grow and learn. They were always available to help anytime.

I gratefully acknowledge the research support from the the Chemical Sciences,
Geosciences, and Biosciences Division, Office of Basic Energy Sciences, Office of Sci-
ence, US. Department of Energy through the generosity of my advisor. I would also
like to thank James L. Dye Endowed Fellowship and the Michigan State University

(MSU) Dissertation Completion Fellowship for additional support.

iv

I am thankful to Janet Haun; she helped me out many times during pre-presentation
panics and made my stay here at MSU much easier.

I would like to acknowledge my MSU Filipino friends for the fun times and warm
camaraderie. Special mention to Maan, my dissertation buddy, who made navigating
the dissertation process much tolerable.

I would like to thank my family for their continuous love and guidance. They have
always encouraged me to pursue my dreams even if it means leaving home. I know
that they are always there for me wherever I will be.

Lastly and surely not the least, my most heartfelt gratitude to Drew for his love,
quiet patience, and profound understanding. He believed in me and kept me sane in
my journey through graduate school. I will never be able to thank him enough for

always sticking with me without complaints.

Contents

List of Tables ....................................................... viii
List of Figures ...................................................... x
1 Introduction ..................................................... 1
2 Project Objectives ............................................... 26
3 Theory .......................................................... 27
3.1 An Overview of Coupled-Cluster (CC) Theory ............. 27
3.1.1 Ground-State Formalism ..................... 27
3.1.2 Extension to Excited States via the Equation-of—Motion For-
malism (EOMCC) ........................ 31
3.2 Method of Moments of CC Equations (MMCC) ............ 34
3.2.1 An Overview of the Exact MMCC Formalism ......... 34
3.2.2 An Overview of the Approximate MMCC Approaches:
The MMCC(mA,m3) 'Ituncation ................ 40
3.3 The Existing MMCC(mA, m3) Approximations ............ 44
3.3.1 The CI-corrected MMCC(2,3) and MMCC(2,4) Methods . . . 44
3.3.2 The Renormalized and Completely Renormalized CC Methods
for Ground and Excited States .................. 49
3.4 The Basic Elements of Multi-Reference Many-Body Perturbation The-
ory (MRMBPT) .............................. 54
4 The MMCC Methods Employing Multi-Reference Many-Body Per-
turbation Theory (MMCC/ PT) .................................. 59
4.1 The Multi-Reference Many-Body Perturbation Theory Wavefunctions
Used in the MMCC / PT Approaches .................. 59
4.2 The MRMBPT—corrected MMCC(2,3) and MMCC(2,4) Approaches:
MMCC(2,3) / PT and MMCC(2,4) / PT ................. 65
4.3 Diagrammatic Formulation and Factorization of the Triply and Quadru-
ply Moments of the CCSD Equations .................. 71

4.4 Final Equations for err?" (2 ) and 93"“ (“1(2) and the Remaining De-
tails of the Implementation of the MMCC(2, 3) / PT and MMCC(2, 4) / PT

Approaches ................................ 87
Numerical Examples ............................................. 103
5.1 Bond Breaking in HF ........................... 103
5.2 Bond Breaking in F2 ........................... 107
5.3 Double Dissociation in H20 ....................... 110

5.3.1 The DZ basis set ......................... 110

5.3.2 The cc-pVDZ basis set ...................... 113
5.4 Excited States of CHJr .......................... 114
Summary, Concluding Remarks, and Future Perspectives ........ 123

vi

Appendix A. An Introduction to the Diagrammatic Methods of Many-

Body Theory .................................................... 125
Appendix B. The Structure of the Similarity-transformed Hamiltonian

of CCSD Equation ............................................... 133
Appendix C. Factorization of Coupled-Cluster Diagrams ............ 137
Bibliography ....................................................... 164

vii

List of Tables

1 The quality of the overall description of the dynamical and nondy-
namical electron correlation effects by selected wavefunction methods
of quantum chemistry. .......................... 19

2 Explicit algebraic expressions for one- and two—body matrix elements
elements of H CCSD (designated by h) and other intermediates (desig-
nated by I or 19), shown in Figures 12-14, used to construct the triply
and quadruply excited moments of the CCSD/EOMCCSD equations,
9313:,”(2) and 93132,“,(2), respectively. ................. 102

3 A comparison of the CC and MMCC ground-state energies with the
corresponding full CI results obtained for a few geometries of the HF
molecule with a DZ basis set.The full CI total energies are given in
hartree. The remaining energies are reported in millihartree relative
to the corresponding full CI energy values. The non-parallelity errors
(NPE), in millihartree, relative to the full CI results are given as well. 105

4 A comparison of various CC ground-state energies obtained for a few
geometries of the F2 molecule with a cc-pVDZ basis set. In all post-
RHF calculations, the lowest two orbitals were kept frozen. The Carte-
sian components of d orbitals were used. The CCSDT total energies are
given in hartree. The remaining energies are reported in millihartree
relative to the corresponding CCSDT energies. The non-parallelity
errors (NPE), in millihartree, relative to the full CCSDT results are
given as well. ............................... 108

5 A comparison of the CC and MMCC ground-state energies with the
corresponding full CI results obtained for the equilibrium and two dis-
placed geometries of the H20 molecule with the DZ basis set. The
full CI total energies are given in hartree. The remaining energies are
reported in millihartree relative to the corresponding full CI energy
values. .................................. 111

6 A comparison of various CC ground-state energies obtained for the
H20 molecule, as described by the cc—pVDZ basis set, at the equi-
librium geometry and several non-equilibrium geometries obtained by
stretching the 0—H bonds, while keeping the H-O—H angle fixed. The
spherical components of the d orbitals were used. In all post-RHF cal-
culations, all electrons were correlated. The full CI total energies are
given in hartree. The remaining energies are reported in millihartree
relative to the corresponding full CI energies. ............. 113

viii

10

11

A comparison of various CC and MMCC vertical excitation energies
of the CH+ ion, as described by the [5s3p1d/3slp] basis set, at the
equilibrium geometry R,: and two stretched geometries, 1.5126 and 2133,
with the corresponding full CI results. The full CI values are the
excitation energies. All other values are the deviations from the full
CI results. The n 1Y energy is the vertical excitation energy from the
ground state (X 12* E 1 12+) to the n—th singlet state of symmetry
Y. All energies are in eV. The equilibrium bond length R. in CH+ is
2.13713 bohr. ............................... 116

Explicit algebraic expressions for the one— and two-body body matrix
elements elements of H (CCSD) (designated by h) shown in Figures 20-21. 151

Explicit algebraic expressions for the three— an_d four-body body matrix
elements elements of H (CCSD) (designated by h) shown in Figures 22-23. 152

Explicit algebraic expressions for im3’tbc(2) obtained by reading the
diagrams shown in Figure 24. ...................... 153
Explicit algebraic expressions for 9.73:3“,(2) obtained by reading the
diagrams shown in Figure 25. ...................... 154

ix

List of Figures

The orbital classification used in the active-space CI, MRMBPT, and
the CI- and MRMBPT-corrected MMCC methods, such as
MMCC(2,3)/CI, MMCC(2,4) / CI, MMCC(2,3) / PT, and
MMCC(2,4) / PT. Core, active, and virtual orbitals are represented by
solid, dashed, and dotted lines, respectively. Full and open circles rep—
resent core and active electrons of the reference determinant |<I>) (the
closed-shell reference determinant |<I>) is assumed) ............

Diagrammatic representation of (a) T1 as defined by Eq. (6); (b) T2 as
defined by Eq. (7); (c) RM as defined by Eq. (20); ((1) R33 as defined
by Eq. (21); (e) FN = ng[a"aq], and (f) VN = fivgNMPaqasar], where
FN and VN are the one- and two-body components of the Hamiltonian
in the normal-ordered form (H N) and N [ . .] stands for the normal
product of the operators. ........................

Diagrammatic representation of 933’,“ “(2) using the Brandow forms of
the relevant Hugenholtz diagrams. Vertices with wavy lines correspond
to many-body components of H (CCSD) [3 in the last term represents

T001). ...................................

Diagrammatic representation of 9313’kl (110.1(2) using the Brandow forms
of the relevant Hugenholtz diagrams. MAS in Figure 3, vertices with
wavy lines corresponds to many-body components of H (CCSD) and El
represents Tom) ...............................

Factorized forms of the three-body components of H (CCSD) required 1n
the derivation of M30342), expressed as products of two-body matrix

elements of H (CCSD) and T2 cluster amplitudes ..............

Diagrammatic representation of 9R3’k abc(2) obtained by substituting the

three-body components of H (CCSD) in Figure 3 entering diagrams I-V
by their factorized analogs shown in Figure 5 and by expressing the
four-body component of H (CCSD) that defines diagram VI in terms of
VN, T2, and RN. .............................

Process of diagram factorization for the nonlinear terms in 9.113” 036(2)
shown in Figure 6 (diagrams IA—VI). (a) and (b) The T2 vertex is
factored out. (c) Diagrams in parenthesis in (b) that have a similar
structure are grouped together to define two intermediate vertices that
are linear in RM and RN .........................

Final factorized form of 9.113”c (2 ) 2The two-body vertices marked with
“= ” are defined 1n Figure 7 (c ). .....................

45

73

78

79

81

84

85

10

11

12

13

14

15

16
17
18
19
20
21
22
23
24

25

26

27

28

29

Factorized forms of the four-body components of H (CCSD) required 1n
the derivation of m3f233cd( ), expressed as products of two-body matrix

elements of H (CCSD) and T2 cluster amplitudes .............. 88

Diagrammatic representation of 9313’“ MG) obtained by substituting

the three- and four-body components of H (CCSD) in Figure 4 by their
factorized analogs shown in Figures 5 and 9. .............. 89

Final factorized form of 9313’“ (2 ). ................... 91

One-body matrix elements of H (CCSD) and other one-body intermedi-
ates needed to construct 9113“ “0(2) and 933’: ”1(2) ............ 92

Two-body matrix elements of H (CCSD) and other two—body intermedi-
ates needed to construct 911:3"; bc(2) and 9313”“1 (2 ) ............ 93

Three-body intermediates needed to construct 911;?" (2 ) and 9213’” (2.)

Hugenholtz representation of (a) FN; (b) VN; (c) T1; ((1) T2; (e) R ,1;

and (f) R“; ................................. 128
Hugenholtz representation of FN. .................... 129
Hugenholtz representation of VN. .................... 130
Brandow representation of FN ....................... 132
Brandow representation of VN ....................... 133
One-body components of H (CCSD) ..................... 138
Two-body components of H (CCSD). ................... 139
Three-body components of H (CCSD) .................... 142
Four-body components of H (CCSD). ................... 145
Diagrammatic representation of 9313"c (2 ). ............... 148
Diagrammatic representation of 9313’s,!“ (2). .............. 150
Diagrammatic representation of 71:. ................... 156
Example of diagram factorization ..................... 156
Fully Factorized Diagrammatic Formulation of 713 ........... 158
Factorized one—body components of H(CCSD) ............... 159

30 Factorized two-body components of H {(30311) ............... 160

xii

1 Introduction

Not long after the discovery of Schréidinger’s equation1 in 1926, one of the pi-
oneers of quantum mechanics, Dirac, made a widely publicized statement that “the
underlying physical laws necessary for the mathematical treatment of a large part of
physics and the whole of chemistry are thus completely known, and the difficulty is
only that the exact application of these laws leads to equations much too complicated
to be soluble” .2 Because of this statement, chemists at that time were skeptical about
the prospects of quantum theory as a way to study molecular properties. However,
what has not been publicized as broadly, in the same paper Dirac also wrote that “it
therefore becomes desirable that the approximate practical methods should be devel-
oped, which can lead to an explanation of the main features of complex atomic system
without too much computation” .2 Inspired by this less known remark and despite the
early reluctance of the chemical community, the field of quantum chemistry has un-
dergone revolutionary advances brought about in part by the advent of computers and
in part by much improved understanding of many-electron systems, particularly in
recent four decades. The tremendous improvements in accuracy and predictive power
of electronic structure methods and significant advances in the fundamental under-
standing of many-electron wavefunctions, which continues to improve every year, have
paved the way to widespread applicability of quantum-chemical methods in solving
increasingly complex chemical problems surpassing what was originally expected in
late 1920’s by a wide margin. Furthermore, because of its rigorous and predictive na-

ture, quantum chemistry has become a powerful, important tool not only for theorists

but also for experimentalists, while inspiring similar developments in computational
nuclear physics, biochemistry, molecular biology, and materials science, among others.

Nowadays, highly accurate ab initio (meaning from “first principles”) quantum
mechanical calculations for small and medium size molecular systems, with up to
20-30 light atoms, a few transition metal atoms and about 100 explicitly correlated
electrons, are routine. Theoretical calculations of the energetics and other molecular
properties of smaller molecules can often rival those obtained experimentally.

The above successes of quantum chemistry do not mean that there are no open or
challenging problems in modern electronic structure theory. Indeed, it is well known
and as further elaborated on below, the key to a successful description of chemical
systems is an accurate assessment of many-electron correlation effects. Electrons in
molecules move in a complicated, highly correlated fashion and one cannot describe
molecular properties by using a simple mean-field description which ignores electron
correlation effects, particularly in cases involving bond breakng and excited electronic
states. The least expensive ways of accurately accounting for electron correlation ef-
fects have computational steps that typically scale as JV" — W7 with the system size
JV and this limits the applicability of correlated methods to systems with about 100
correlated electrons (defined as electrons outside the frozen core). Thus, one of the
challenges of modern quantum chemistry is to go to systems with hundreds or thou-
sands of correlated electrons and basis functions. Another challenge is an accurate
treatment of bond breaking (reaction pathways) and excited states, particularly those
dominated by many-electron transitions, i.e. problems where a traditional description

in which one starts from a single Slater determinant and builds the electronic wave

2

functions on top of a single determinant through particle-hole excitations, and which
works for non-degenerate states of closed-shell molecules, is no longer adequate. This
thesis work addresses the latter challenge.

Considering the nature of this thesis research, which focuses on the development of
accurate new methods to describe many-electron correlation effects in cases involving
bond breaking and excited states, in this chapter, we outline some of the most popular
electron correlation methods developed over the recent years. We begin by describing
the exact solution to the Schrbdinger equation in a basis set. Then, we present the
various wavefunction-based approximations to solving electronic Schriidinger equation
including correlation, presenting the advantages and disadvantages of each approach
in terms of accuracy of the predicted energies and the computational cost require-
ments.

The conceptually most straightforward method of obtaining the exact solution to
the electronic Schriidinger equation, within a given one-electron basis set, is the full
configuration interaction (full CI) approach, which solves the Hamiltonian eigenvalue
problem by including all possible Slater determinants (antisymmetrized products of
one-electron wavefunctions, i.e. spin-orbitals) of the appropriate symmetry which can
be formed from the given one-electron basis set. Examples of full CI calculations at
a few representative geometries along potential energy curves of small, few electron
molecules can be found in Refs. 3—7. Advances in algorithms and computer hardware
have made it feasible to obtain full CI potential energy curves"_27 for several few-
electron systems. There has been interest in implementing parallel full CI algorithms

on massive parallel architectures and clusters.28_3’ The largest full CI calculation

3

performed to date has been reported by Can and co-workers for the C2 molecule
using the aug-cc—pVTZ basis set without the augmented diffuse d and f functions,
which required about 65 billion Slater determinants.33 This impressive calculation
has only been made possible by employing highly efficient parallel and vector full
CI implementation as well as the dedicated availability of massive supercomputing
resources.

Indeed, there is a major problem with full CI. The factorial growth of the required
number of determinants that defines the dimension of the Hamiltonian eigenvalue
problem with the number of electrons and basis functions makes full CI calculations
feasible only for small molecules, with up to ~10 correlated electrons and with modest
basis sets. This critical drawback can be illustrated by the following example: a
small system with six correlated electrons and 100 orbitals in a basis requires ~109
configuration state functions (spin-adapted combinations of Slater determinants). An
increase of the system size to 12 correlated electrons employing 100 orbitals in a
basis set already would require ~10l7 configuration state functions. This is beyond
the capabilities of present-day computers, and is likely to remain so in many years
to come. Hence, there is a need for alternative, affordable, approximate and yet
accurate approaches in order to study the majority of chemical problems of interest.
Nevertheless, the full CI method, providing exact solutions in a basis set, is invaluable
in benchmarking approximate quantum chemical methods, helping the development
work, including the development work described in this thesis.

The simplest approach to solving the electronic Schrodinger equation is the Hartree—
Fock (H—F) approach which approximates the electronic wavefunction by a single

4

Slater determinant. The H-F method is a mean field theory;3“_37 motion of an elec-
tron in a. molecule is defined by the effective one-electron Hamiltonian (Fock operator)
which describes the average field (Coulomb and exchange terms) created by the re-
maining electrons in the system. The H—F theory can account for about 99% of the
total electronic energy. Unfortunately, it is the remaining 1% that is very important
in describing chemical phenomena, such as the breaking of chemical bonds, electronic
excitations in molecules, response and spectroscopic molecular properties, weak in-
termolecular interactions, etc. Such situations cannot be properly accounted for by
H—F theory because of the neglect of the many-electron correlation effects in the the-
ory. This necessitates going beyond the H—F theory and using more sophisticated
and computationally more demanding techniques, to reach a reasonable description
of molecular systems. In other words, an accurate description of molecular systems
requires a precise determination of many-electron correlation effects.

At this point, it is worth introducing the term “correlation energy” which is for-
mally defined as the difference between the exact energy, defined in this work as the
full CI energy, and the corresponding H—F energy, calculated in the same basis set.
The true correlation energy would require a calculation in the infinite basis set. The
infinite basis set limit can be approached by performing a sequence of calculations
with basis sets of increasing size. In the development work, one does not have to
study the infinite basis set limit to learn about the performance of new methods. The
performance of new methods in finite basis set calculations and comparisons with full
CI in the same basis sets are enough to make judgements about the usefulness of the

newly proposed approaches.

There are two types of electron correlation: dynamical and nondynamical (static).
Dynamical correlations can be regarded as short-range correlations due to electrons
instantaneously avoiding each other, particularly when they come close to each other.
On the other hand, nondynamical correlations can be treated as long-range corre—
lations which arise from the multiconfigurational character of the systems having
quasi(near)-degenerate states, as in the case of electrons constituting an electron
pair, which are separated during the bond breaking process, biradicals, and the ma-
jority of excited states. A well-balanced account of both dynamical and nondynamical
electron correlation is important to obtain a uniformly accurate description of reac-
tants, products, reaction intermediates, and transition states as well as the precise
description of electronic excitations in molecules.

There has been tremendous progress in the development of electron correlation
methods in the recent history. Here, we primarily focus on the wavefunction-based
approaches which can be divided into three main categories: (1) configuration inter-
action (CI) approaches;38_"1 (2) many-body perturbation theory (MBPT);“"46 and
(3) coupled-cluster (CC) methods.“"-51 The electron correlation methods can also be
classified as single-reference and multi—reference methods. Single-reference electron-
correlation methods, from the name itself, are defined by building a correlated wave-
function through electronic excitations out of a single Slater determinant (e.g., a H—F
determinant). Multi-reference (MR) methods, on the other hand, utilize the multi-
determinantal reference wavefunction to build the desired correlated electronic state
or states. The single-reference based methods are presented first.

Of all single-reference methods, the CI approach is conceptually (but not compu-

6

tationally) the simplest. The CI wavefunctions are represented as linear combinations
of Slater determinants obtained by promoting or exciting electrons from occupied to
unoccupied orbitals, and the corresponding expansion coefficients are obtained vari-
ationally by diagonalizing the Hamiltonian matrix. One of the advantages of CI
methods is the ease of access to excited states and molecular properties other than
energy, which can be calculated as expectation values or transition matrix elements
involving the relevant CI states, and the fact that Cl energies are always the upper
bounds to the corresponding exact energies. The disadvantages of CI methods include
enormous computer costs, if one is aiming at high accuracy, and lack of size extensiv-
ity of the truncated CI approaches, which basically means that the resulting energy
of a system does not scale correctly with the system size and a loss of accuracy occurs
as the system becomes bigger. The basic CI technique, designated as CISD, which
includes all singly- and doubly-excited configurations from a H—F reference, typically
recovers ~90% of the correlation energy for small, few electron problem, as long as the
H—F determinant dominates the wavefunction.52 By including all triple and quadruple
excitations from the reference (CISDTQ), it is possible to recover nearly all of the
remaining ~10%.52 Furthermore, while the CISD method is inaccurate for excited
states and for systems away from the equilibrium geometry, the CISDTQ wavefunc-
tion recovers almost all of the correlation energy, even in those challenging situations,
as long as the number of correlated electrons is not too large,” since CISDTQ, as
any other truncated CI approach, is not size extensive. The CISDTQ method is a
high accuracy approach for smaller molecular problems, but, unfortunately, the cost

associated with a CISDTQ computation is usually prohibitively expensive due to the

7

large numbers of triply and quadruply excited determinants. The CPU operational
count associated with a single iteration of the CISDTQ calculation is 713113, where
no(n,,) is the number of ocuppied (unoccupied) orbitals in a basis set that are in-
cluded in correlated calculations. This is more or less equivalent to the ./V ’0 scaling
of the CPU time with the system size J.

The second type of approach to the many-electron correlation problem, referred
to as MBPT, uses the Rayleigh-Schrodinger or some other perturbation theory to
describe correlation effects. The MBPT methods offer the following advantages: (i)
unlike the truncated CI approaches, the MBPT formulations based on Rayleigh-
Schriidinger perturbation theory provide size extensive results, i.e. no loss of accu-
racy occurs as the system size increases, (ii) low-order MBPT methods are inexpensive
while offering reliable information about the most essential correlation effects. How-
ever, MBPT methods are non—variational and the MBPT series do not always con-
verge, particularly when bond breaking is examined. Furthermore, standard MBPT
methods do not apply to excited states, although one can address this problem by
switching to multi-reference MBPT schemes as mentioned below.

The simplest and most economical MBPT method for including electron correla-
tion is the second-order Moller—Plesset (or many-body) perturbation theory (MP2)42
which typically accounts for ~80% of the correlation energy when the H—F reference
determinant dominates the wavefunction. Although, this will generally not take us
to chemical accuracy, several molecular properties, including geometries and vibra-
tional frequences, are reasonably well described by MP2. In general, MP2 energy

calculation requires noniterative steps that scale as ngnfi. Essentially, one can af-

8

ford such calculations for any system for which one can afford the H—F calculation
(systems with hundreds of correlated electrons). The extension of MP2 to the fourth-
order (MP4)54 improves the electron correlation description and accounts for about
95-98% of the correlation energy in non-degenerate closed-shell cases, while requiring
noniterative steps that scale as 113713. The higher-order MP5 and MP655—58 theories
have been pursued as well in spite of the vast increase in the number of terms in-
volved. It is reasonable to infer that going to higher orders of MBPT should give
more accurate results. However, in the work of Olsen et al.,59 the authors found
out that going to higher—order MBPT for many-electron systems in extended basis
sets did not guarantee convergence, even for cases of trivial wavefunctions dominated
by a single-reference determinant which are generally assumed to produce conver-
gent MBPT series.”61 The MBPT expansions are always strongly divergent in cases
involving bond breaking, as shown, for example, by Piecuch et at.”

The third type of wavefunction-based electron correlation methods is coupled—
cluster (CC) theory, which is based on the exponential ansatz for the ground-state
wavefunction, allowing CC approaches to describe high-order correlation effects at the
low level approximation by generating the higher-order excitations in the wavefunc-
tions as products of low-order excitations. Of all approaches to the many-electron
correlation problem, the single-reference CC“.51 methods, in which the H—F wave-
function is usually chosen as the reference state, are generally considered as the best
compromise between high accuracy and relatively low computer cost. Another no-
table characteristic of CC theory is its ability to preserve size extensivity at any level

of truncation. These and other attractive features of the CC theory have inspired

9

and continue to inspire significant research work on developing high accuracy meth-
ods methods which can be used as general-purpose tools by experts and non-experts
(see Refs. 21, 25, 26, 54, 63—70 for selected reviews) and this thesis research is a
contribution to this area.

In the above description, we have focused on single-reference theories, which im-
plicitly assume that a single Slater determinant (e.g., a H-F determinant) dominates
the electronic wavefunction so that one can recover the remaining part of the wave-
function through excitations of electrons to unoccupied spin-orbitals. Because of this
underlying assumption, single-reference methods have the applicability which is es-
sentially limited to non-degenerate ground states of closed- and simple open-shell
molecules near the equilibrium geometries. However, large part of the chemistry
research deals with chemical reactions, where bonds are stretched or broken, and
processes involving excited electronic states, which cannot be described by conven-
tional singlereference methods, since the corresponding wavefunctions have a sig-
nificant multideterminantal character. In particular, there have been a great deal
of interest in extending single-reference CC methods to quasi-degenerate situations,
such as molecular bond dissociation and excited states dominated by two— and other
many-electron transitions, which are very challenging because of the large nondy-
namical electron correlation effects that traditional single-reference theories cannot
capture.63’66167171_73 The basic ground state CC approach with singles and doubles
(CCSD)""‘—76 is accurate in describing closed-shell systems and dynamical correlation

2 4
onut

effects with relatively low computer costs that scale as 11 but fails in describing
bond breaking because it neglects the higher-than-pair (e. g., triply and quadruply

10

excited or T3 and T4) clusters. The CCSDT”:78 and CCSDTQm—82 methods, which
incorporate T3 or T3 and T4 clusters, respectively, are capable of giving an accurate
description for certain classes of systems with electronic quasi-degeneracies, in spite
of their single-reference character, since they describe correlation effects to very high
orders, often compensating for the inadequacies of single-reference description, but
the computer costs of the CCSDT and CCSDTQ calculations are extremely high,
limiting their applicability to small molecular problems with 2—3 light atoms. In
particular, the CCSDT and CCSDTQ methods require iterative steps that scale as
112113 and nfinfi, respectively, which make them applicable to systems with up to ~10
correlated electrons.

To considerably reduce the computational costs of the CCSDT, CCSDTQ, and
other high-order CC schemes, several CC approaches, in which the effects of higher-
than-doubly excited clusters is approximately included have been developed. One of
the most popular CC methods in this category is the CCSD(T)83 approach in which
the connected triply excited (T3) clusters are incorporated in a computationally ef-
ficient manner through noniterative corrections to CCSD energy derived using argu-
ments that originate from MBPT. The CCSD(T) method and its CCSD[T] analogs”5
are currently available in the majority of popular quantum chemistry software pack-
ages, enabling highly accurate ab initio calculations of useful molecular properties by
experts as well as non-experts. However, while the CCSD[T] and CCSD(T) methods
improve the description of molecular properties in the region of equilibrium geometry,
they completely fail when chemical bonds are stretched or broken.1“—271621674018"—100

The inclusion of noniterative quadruples (T4 clusters) using the arguments originating

11

from MBPT, which leads to CCSD(TQf) and CCSDT(Q,) methods101 and their var-
ious modifications, further improves the results in the equilibrium region but cannot
help when the configurational quasi-degeneracy sets in.18'19121123125126168170188—4‘m192'9"196

Similar remarks about the performance and failures of single-reference CC meth-
ods can be made when extending the discussion to excited states calculations. The
most natural extensions of the singlereference CC formalism to excited states are the

102—107 and the closely related equation-of—motion

linear-response CC theory
(EOM) CCmfi—112 and symmetry-adapted cluster configuration interaction
(SAC—CI) approaches.”3_117 The basic linear response CCSDmfi'107 and
EOMCCSD")?111 approximations, which are characterized by the manageable com-
putational steps that scale as nfinf, or JV“ with the system size and the analo-
gous SAC-CISD method provide reliable information about excited states domi-
nated by one-electron transitions. Unfortunately, the linear response CCSD and
EOMCCSD methods cannot describe excited states having significant double exci-
tation components and excited-state potential energy surfaces along bond breaking
coordinateslo'21‘26169’97193'118—128.

High-order EOMCC methods including higher—than—double excitations, such as
the recently implemented full EOMCCSDT (EOMCC singles, doubles,
and triples)2“'125'129 and EOMCCSDTQ (EOMCC singles, doubles, triples, and
quadruples) 1281130 approaches, provide an excellent description of excited states dom-
inated by <10ubles’41125”128 as well as excited-state potential energy surfaces,24 but
large costs of the EOMCCSDT and EOMCCSDTQ calculations, which are defined

by the iterative steps that scale as W8 and W10 with the system size, respectively,

12

 

limit their applicability to small molecules with 2—3 light atoms and relatively small
basis sets. For this reason, a number of approximate and less expensive ways of
incorporating triple or triple and quadruple excitations in the EOMCC and lin-
ear response CC formalisms have been developed in order to make these methods
applicable to a wider range of molecular sizes. Among those are the noniterative
EOMCCSDT-n approaches and their noniterative EOMCCSD(T), EOMCCSD(T),
and EOMCCSD(T’) counterparts,“81119 and the analogous linear-response CC meth-
ods such as CC3“’1121_123 and CCSDR(3),”123 which use elements of MBPT to es-
timate triples effects. All of these methods are characterized by the relatively inex-
pensive ./V7 steps of the 113713 type and all of them improve the EOMCCSD/ linear
response CCSD results for excited states dominated by two—electron transitions, but
there are many cases where the results of EOMCCSDT-n, EOMCCSD(T), CC3, and
similar calculations are far from satisfactory or poor. This can be illustrated by the
large 0.4—0.5 and 0.9 eV errors in the description of the lowest 1H9 and ’11,, states
of the C2 molecule, respectively, by the EOMCCSDT-1 and CC3 approaches10 or the
failure of the CC3 and CCSDR(3) methods to provide accurate information about
excited-state potential energy surfaces along bond breaking coordinatesl2 (see also
Ref. 22 for an additional analysis).

The above problems encountered in single-reference CC / EOMCC calculations in
various cases of electronic quasi—degeneracy clearly indicate that a traditional single-
reference description is not sufficient and that more flexible models are needed. The
conventional wisdom is to turn to multi—reference approaches, in which instead of

using a single-determinantal reference state, one selects a certain number of refer-

13

ence determinants to construct the appropriate zero-order wavefunction(s) adjusted
to the type of bond breaking or excited states of interest. The less intuitive and yet
potentially very useful is an idea of improved single-reference methods, which com-
pletely or largely rely on a single-determinantal reference state, while being capable
of describing at least some of the most frequent cases of electronic quasi—degeneracies.
We first overview the traditional multi-reference methods, which we classify as multi-
reference CI (MRCI), multi-reference MBPT (MRMBPT), and multi-reference CC
(MRCC) approaches.

In C1, the multi-reference formulation is typically accomplished by adopting a
more sophisticated zeroth-order reference in the form of a multiconfiguration self-
consistent field (MCSCF) wavefunction instead of using a single H—F determinant.
The basic MRCISD approach (abbreviated here as MRCI) involves single and dou-
ble excitations out of all reference determinants. The most popular MRCI schemes
using complete active space self-consistent field (CASSCF) reference wavefunctions
(CASSCF is a variant of MCSCF obtained by a distribution of a number of active elec-
trons among a number of active orbitals in all possible ways) provide potential energy
surfaces which are closely parallel to full CI surfaces, at least for small molecules. Be-
cause of this, the MRCI methods are among the most useful benchmark techniques of
quantum chemistrym'l’l'132 particularly in cases where full CI results are not available,
and among the most popular approaches to calculation of potential energy surfaces
of smaller molecular system. One of the biggest advantages of MRCI techniques, in
addition to high accuracy in calculations of potential energy surfaces, is their ability
to describe ground and excited states, near-degeneracy effects, and all kinds of open-

14

 

shell systems. The fundamental drawback of the MRCI methods, their lack of size
extensivity, can be addressed through either the a posteriori Davidson-type energy

correctionsmmm3

or the a pn'on' refinements through suitable Hamiltonian dressing
techniques, such as the MR—average coupled pair functional (MR-ACPF)139 and MR-
average quadratic coupled cluster (MR-AQCC).1‘“”141 Neither of these propositions
is ideal, but benchmark calculations show that at least the main problems related to
inextensivity of MRCI can be addressed in this manner (see, e.g., Ref. 142). Several
other approximate extensive modifications of MRCI have been developed.1“3"145

In addition to inextensivity, the main challenge for the MRCI methods is the fact
that the lengths of the corresponding CI wavefunction expansions and the related
computational effort rapidly increase with the system and basis set sizes. Although
several clever ideas, such as the use of the configuration selection thresholds, com-
bined with the extrapolation tecniques,138 and internal contractions of configuration
state functionsl‘w'147 have been developed to reduce the costs of MRCI calculations,
all MRCI methods are generally costly in all aspects of computing resources.”8 Even
with the substantial progress in terms of algorithms, efficient implementation, and
parallelization (cf. e.g., Refs. 146-154), practically all applications of MRCI remain
limited to relatively small molecular systems. In addition, the existing MRCI ap-
proaches are significantly more complex than other computational chemistry methods
due to several choices that the user have to make to run the MRCI calculations in
a prOper manner. This makes MRCI approaches popular among experts, but much
less popular among non-experts.

A cost effective alternative to MRCI is represented by methods based on the

15

 

multi—reference extension of MBPT. Historically, the general MRMBPT formalisms
for quasi-degenerate and open-shell states have been formulated in the late 1960’s155
and early 1970’s,156 but much of the development work geared toward practical com-
putational schemes has been done in the last two decades. Two general categories of
MRMBPT methods are distinguished in the literature: the “perturb then diagonalize”
and the “diagonalize then perturb” approaches.1571158 The “perturb then diagonalize”
approaches involve deriving the effective Hamiltonian in the multiconfigurational ref-
erence space, which in the case of the most popular second-order MRMBPT schemes
is truncated at first-order terms, and then obtaining the final energies as eigenvalues
of this operator. 155458—162 The most popular second-order “diagonalize then perturb”
methods use the first-order contributions to the Schriidinger equation to define the
perturbed wavefunctions, and then use the second-order equation to get the energy.
Various methods, particularly of the latter family, differ from one another in the
choice of the zeroth-order Hamiltonian, and other details of the algorithms used in
the computer implementation.“‘3_187 Among the most popular and widely used vari-
ants of MRMBPT are the complete active space second-order perturbation theory
(CASPT2),1"3*164 the multi-reference Moller—Plesset perturbation theory (MRMP),165
and the multi-configurational quasi-degenerate perturbation theory (MC-QDPT).174

A general difficulty of MRMBPT is the choice of the zeroth-order Hamiltonian,
which is much less straightforward than in the single-reference case. Another difficulty
concerns the choice of reference determinants. Usually, CASSCF wavefunctions are
chosen as reference functions for MRMBPT considerations. However, CASSCF often

generates too many configurations, and the size of the active space can easily outgrow

16

the capacity of the present technology, causing tremendous difficulty in obtaining
converged results, which strongly vary wth the number of active electrons and orbitals
and the number of roots included in the ca.lculations.1‘”7_189 In addition, at least
some MRMBPT methods suffer from lack of size extensivity190 and intruder state
problems leading to divergent behavior. 191-193 Among the most promising approaches
to eliminate intruders, while retaining manifest size extensivity, are state-specific
MRMBPT methods that start with a multideterminantal reference space but target
one state at a time. The state-specific MRMBPT method advocated by Mukherjee et
01.177 seems particularly promising, showing smooth performance in and around the
region of intruders and reasonable accuracy.

Overall, in spite of the aforementioned problems the MRMBPT methods have
been successfully applied to many chemical and spectroscopic problems and have es-
tablished themselves as efficient techniques for treating nondynamical and leading
dynamical correlationsm":185 Compared to MRCI and genuine MRCC methods dis-
cussed below, the second-order MRMBPT approaches are much more practical. The
low-order MRMBPT approaches have a drawback that they are not as accurate as
MRCI or CC methods in describing dynamical correlation, but this can be taken care
of by combining the low-order MRMBPT theory with the CC theory, as demonstrated
in this thesis research through the MRMBPT-inspired corrections to CC energies.

The last type of multi-reference methods are the multi-reference coupled-cluster
(MRCC) approaches. The existing MRCC approaches can be classified into three
basic categories: the Fock-space (FS) or the valence-universal approaches,“1721194"196

Hilbert-space (HS) or the state-universal methods,73'197-”°5 and the state-specific or

17

 

state—selective (SS) approaches, such as, for example, those described in Refs. 206 and

207, or SSMRCC methods based on the wavefimction ansatz described in Refs. 82 and

208, which we nowadays referred to as the active-space CC methods.""9911°°112311241209—211

The FSMRCC theories employ a single valence universal exponential wave operator
to generate ground and excited states of a given system and its ions obtained by
removing active electrons one by one. Thus, they are very good for valence sys-
tems around closed-shells and differential properties, such as ionization energies and
electron affinities. The HSMRCC theories are based on the ansatz of Jeziorski and
Monkhorst, which employs different cluster operators for different reference deter-
minants, are particularly well suited for electronic quasi-degeneracies due to several
interacting states. Both the FSMRCC and HSMRCC approaches are multiroot pro-

cedures. This should be contrasted by SSMRCC theories, which treat one electronic

state at a time. We will return to the SSMRCC methods when we discuss active-space

CC approaches. With an exception to active-space methods,2“'8”99'10042311241208"211 the
MRCC approaches are greatly limited so far and no general purpose codes have been
developed due to recurring issues such as computational complexity, intruder—state
problems,212 and difficulties with retaining size extensivity in incomplete model space
considerations (cf. e.g., Refs. 213—216 for recent progress in this area). This implies
that it may be more worthwhile to focus on methods that rely, at least in part, on
a single-reference formalism. In general, single-reference methods are much easier to
implement and use than genuine MRCC approaches. Thus, it is useful to develop
new classes of single-reference CC methods that eliminate the failures of standard

CC / EOMCC approximation in the bond breaking region and for excited states hav-

18

Table l: The quality of the overall description of the dynamical and nondynamical
electron correlation effects by selected wavefunction methods of quantum chemistry.

 

Type of Electron Correlations

 

 

Method Dynamical Nondynamical
MBPT(2), MP2 low-order poor
CCSD(T), CCSD(TQ), excellent poor
MRMBPT low-order excellent
MRCI excellent* excellent*
MRCC excellent excellent

 

* MRCI is not size extensive, so there is a loss of accuracy as the system
becomes larger

ing a manifestly multiconfigurational character, without involving the complexities of
the FSMRCC and HSMRCC considerations.

As shown in Table 1, MRCI and MRCC are the only methods that offer an excel-
lent treatment of both dynamical and nondynamical correlation effects. As described
earlier, these methods are computationally very expensive and very diffith to use
which greatly limits their applicability. Thus, we need alternative approaches which
are more practical than MRCI and MRCC and and which are capable of balancing
dynamical and nondynamical correlations, particularly in studies of bond breaking
and excited states having a significant configuration mixing.

One can think of two general ways of developing such approaches, at least within
the CC formalism. The first way is to improve the existing single-reference CC meth-
ods such that they can describe at least the selected classes of bond breaking and ex-

cited states in a purely “black-box” fashion, i.e. without using any elements of multi-

19

 

reference calculation. A few ideas of this type have been proposed in recent years,

including the non-iterative single-reference CC methods based on the partitioning of

95,217—222 223—225

the similarity-transformed Hamiltonian, spin-flip CC approaches, and
renormalized CC methods.18’19:2112’125’’"mfi-m:871961981226"228 The latter approaches are
particularly successful. They are based on the idea of correcting the CC (e.g., CCSD)
or EOMCC (e.g., EOMCCSD) energies for the effects of higher-order clusters (e.g.,
triples) using expressions for the leading terms toward full CI obtained using the for-
malism of the moments of CC equations.18'19:21'2’’25126168""):871961981126’127"""""‘_229 Renor-
malized CC methods are particularly useful in calculations of single bond breaking,
reaction pathways involving biradicals, and excited states dominated by two-electron
transitions.18,19,21,22,25,26,68—70,87,95,96,98,187,188,226—236

Renormalized CC methods are very successful, enabling accurate, inexpensive
“black-box” CC calculations for some of the most frequent multi-reference situations
but there are quasi-degeneracy problems that cannot be handled in this manner.
A good example is provided by the excited states in a highly degenerate B83 sys-
tem, where one cannot obtain all excited states due to difficulties with converging
the underlying CCSD and EOMCCSD equations in some cases.2371238 In such cases
and in more general cases of electronic quasi-degeneracies, where pure “black-box”
single-reference solutions do not work, it may be useful to look for alternative ap-
proaches that mix single- and multi-reference concepts with a single mathematical
theory. One of the best and most successful examples of such theory is provided by
the active-space CC and EOMCC methods,”41271821861881991"’0'12“'1""'2°8"21m3”241 which

can be viewed as the state—selective MRCC approaches exploiting a single-reference

20

CC formalism.2718218612(“31239_241 These methods use active orbitals, which are normally
exploited to define reference determinants in a multi-reference calculation, to select
the dominant three-body and other higher-than-two—body clusters and excitation am-
plitudes within a standard single-reference CC or EOMCC formalism. As a result,
they offer a tremendous amount of flexibility, since one can always improve the results
by employing the active space. At the same time, they have a well-defined relationship
with higher-order single-reference CC or EOMCC methods. For example, the active-
space CC/EOMCC approach with up to triple excitations (CCSDt/EOMCCSDt)
becomes equivalent to full CCSDT/EOMCCSDT theory when all orbitals are active.
On the other hand, as demonstrated in numerous calculations by the inventors of
this method (Piecuch, Adamowicz, and co-workers) , and others who adopted active
space approaches (Bartlett, Head-Gordon, Gauss, Olsen, Krylov, Sherrill, Hirata,
and others), it is sufficient to use small numbers of active orbitals in active space
CC / EOMCC calculations to obtain extremely accurate results for bond breaking and
multideterminantal excited states. In particular, the active space EOMCC methods
lead to the virtually perfect description of the excited states of the aforementioned
Be3 system, were other methods fail.237'238

Active space CC / EOMCC methods mix single- and multi-reference CC concepts,
but one can go even one step further and mix CC and non-CC concepts, so that
the advantages of different kinds of electronic structure theory are utilized to the
utmost. One of the most successful methods in this broad category is the idea of
the externally corrected single-reference CC methods”?250 in which the T3 and T4
cluster components, instead of being calculated by solving CCSDTQ or the expen-

21

sive CC equations, are obtained by a cluster analysis of the wavefunctions provided
by some external non-CC source such as the projected unrestricted Hartree-Fock
(PUHF),242'247 valence bond (VB),2“31244 MCSCF or CASSCF,“5—2471249 and MRCI250
wavefunctions.

The MRCI-corrected CC methods, which define the so-called reduced MRCC ap-
proaches, are particularly impressive, since the MRCI method itself is already very
accurate in applications involving potential energy surfaces along bond breaking co-
ordinates and by reading T3 and T4 clusters extracted from MRCI calculations and
solving the resulting T3 and T4—corrected CCSD equations one obtains the virtu-
ally perfect description of both dynamical and nondynamical correlations. The only
problem of reduced MRCC methods is the fact that they are currently limited to
lowest-energy electronic states of a given symmetry. Moreover, the MRCI calculation
needed to estimate T3 and T4 clusters for the subsequent CCSD calculations can be
quite expensive.

There is however, a major lesson on the success of the reduced MRCC approach
in that it is very useful to mix multi-reference theory, which can be used to provide
information about the nondynamical and leading dynamical correlation effects, with
single-reference CC theory, which gives the information about the remaining corre-
lations. The success of the reduced MRCC method has inspired the development
of the CI (MRCI)-corrected approaches expoiting the aforementioned formalism of
the method of moments of CC equations (MMCC)31125126137112611271251 (see, also, Refs.
252—254) As already mentioned, the MMCC formalism enables one to determine
the mathematical structure of terms which, when added to CC or EOMCC (e.g.,

22

CCSD or EOMCCD) energies, give, in the exact limit, the exact full CI energies.
One can use the MMCC theory to design the “black-box” renormalized CC approx-
imation, which we mentioned earlier, or to develop the externally corrected MMCC
methods, such as the (MR)CI-corrected MMCC schemes, in which one uses selected
components of the (MR) CI wavefunctions to design the MMCC corrections due to
higher-order correlation effects on top of CCSD or EOMCCSD. The major advantage
of the (MR)CI—corrected MMCC schemes is their applicability to ground and excited
states, not just to lowest-energy states of a given symmetry, and high accuracy that
in some cases is even better than the results of reduced MRCC calculations?”—254
The (MR) CI-corrected MMCC methods work extremely well for single, double, or
even triple bond breaking and all kinds of excited states,”125126187112“:1271252—254 but
they still require additional CI calculations to generate trial wavefunctions that enter
the MMCC corrections to CCSD or EOMCCSD energies. Although the main idea
of the CI-corrected MMCC methods is straightforward, the CI-corrected MMCC cal-
culations can be quite expensive if the CI calculations used in designing the MMCC
corrections use larger active orbital spaces. Undoubtedly, it would be most desirable
to examine if one could use another, less expensive, multi-reference method to gen-
erate the wavefunctions that enter the MMCC corrections, while retaining the high
accuracy of the CI—corrected MMCC calculations.

In this dissertation, we examine the possibility of replacing the relatively ex-
pensive (MR)CI-like wavefunctions in the CI-corrected (MMCC / CI) schemes by the
wavefunctions obtained in the MRMBPT calculations. As described earlier, the low-

order MRMBPT methods are known to provide a reasonable description of non-

23

 

dynamical and leading dynamical correlation effects in the presence of electronic
quasi-degeneracies (cf., e.g., Refs. 159, 162—165, 167, 171, 173, 174, 177, 183, 184,
187, 255—259). At the same time, the computer costs of the low-order MRMBPT
(e.g., second-order) calculations are very small compared to the analogous MRCI
calculations. By combining the wavefunctions obtained in the low-order MRMBPT
calculations, which provide a reasonable description of electronic quasi-degeneracies,
with the MMCC formalism, in which these MRMBPT wavefunctions are used to de-
sign the MMCC corrections to single-reference CC or EOMCC energies, we obtain a
new class of the MRMBPT-corrected MMCC methods, referred to here and elsewhere
in this thesis as the MMCC/ PT approaches.69’97'98 Just like the MMCC/CI methods,
which combine the CC and CI concepts, the MMCC / PT approaches proposed, de-
veloped, implemented, and benchmarked in this work can be viewed as the externally
corrected MMCC methods. All externally corrected MMCC methods are similar, in
the overall philosophy, to the externally corrected CC methods pioneered by Paldus
and collaborators, in which the CC and non-CC concepts are combined together to
improve the CC results in the presence of electronic quasi-degeneracies. We demon-
strate in this work that by combining the single-reference CC and EOMCC methods
with the low-order MRMBPT-like wavefunctions via noniterative MMCC / PT cor-
rections to CC or EOMCC energies, we can considerably improve the results of the
standard CC and EOMCC calculations in the bond breaking region and for excited
states dominated by two-electron transitions, while keeping the computer costs at the
low level. Thus, the MRMBPT-corrected MMCC methods provide an inexpensive al-
ternative to other CC/EOMCC and MMCC schemes, and reduced MRCC approach

24

of Li and Paldus, which in the long-term may facilitate highly accurate calculations
of reaction pathways and electronic excitations in larger molecular systems.

The rest of this document is organized as follows:

In Chapter 2, we summarize the main objectives of this work.

In Chapter 3, we describe the mathematical details relevant in the understanding
and carrying out this reserach study. We outline, in particular, the single-reference
CC theory for ground states and its extension to excited states via the EOMCC
formalism. We also introduce the basic elements of the MMCC theory, in which
our MMCC/ PT work is based. The existing MMCC approaches, such as the CI-
corrected methods and the selected renormalized and completely renormalized CC
methods, and basic concepts of the MRMBPT methods are described in Chapter 3
as well.

In Chapter 4 , we describe the details of the new MRMBPT-corrected MMCC
(MMCC/PT) methods. One of the key parts of the chapter is the diagrammatic
formulation and factorization of the key components of the MMCC / PT equations,
called method of moments of CC / EOMCC equations, which leads to the highly effi-
cient computer implementation of the MMCC / PT and other MMCC methods.

In Chapter 5, we present some of the benchmark calculations to illustrate the
performance of the MMCC / PT approximations developed in this work.

Lastly, in Chapter 6, we present a summary and future perspectives of this work.

25

2 Project Objectives

The main objective of this dissertation is to combine the low-order multi-reference
many-body perturbation theory (MRMBPT) with coupled-cluster (CC) and equation-

of-motion CC (EOMCC) methodologies via the method of moments of CC (MMCC)

equations. The specific objectives of this work are:

A. to formulate the low-order MRMBPT approximation that can be used in the

MRMBPT-corrected MMCC (MMCC/ PT) formalism.

B. to formulate the MMCC / PT approach using diagrammatic methods.

C. to efficiently implement the leading MMCC / PT approximations, termed

MMCC(2,3) / PT and MMCC(2,4) / PT, using the idea of diagram factorization

and recursively generated intermediates.

D. to benchmark the proposed MMCC/ PT approaches in calculations of ground
and excited states of selected molecular systems, for which the exact, full CI

data can be generated.

26

3 Theory

In this chapter, an overview of single-reference coupled-cluster (CC) theory and its
extension to excited states via the equation-of-motion (EOM) CC theory is presented.
The method of moments of CC (MMCC) equations is introduced and the existing
MMCC approximations, namely, the CI-corrected MMCC methods as well as the

selected renormalized- and completely-renormalized CC methods are discussed.

3.1 An Overview of Coupled-Cluster (CC) Theory
3.1.1 Ground-State Formalism

The single-reference CC theory is based on the exponential ansatz for the ground-
state wavefunction,

1%) = 111302) a eTI<I>>. (1)

where T is a particle-hole excitation operator referred to as the cluster operator and
|<I>) is the reference determinant (usually, the Hartree-Fock determinant). In the exact
CC theory, T is a sum of all many-body cluster components that one can write for a

given N -electron system,

T=ETM (2)

where the n—body cluster component T, is defined in a usual way as

1 2 . .
T,, = (H) tat"; a“1 . . . aanain . . .a,-1 , (3)

27

with tiil'jfgn representing the corresponding antisymmetrized cluster amplitudes,
2'1 . . .in(a1 . . . an) referring to the single-particle states (spin-orbitals) occupied (unoc—
cupied) in the reference determinant, and aID (ap) designating the standard creation
(annihilation) operators associated with the orthonormal spin-orbitals lp). Here and
elsewhere in this dissertation, the Einstein summation convention over repeated upper
and lower indices is employed, so that the summation symbols corresponding to the
unrestricted summations over occupied and / or unoccupied spin-orbitals are omitted.

In all standard CC approximations, the many-body expansion for the cluster op-
erator T is truncated at a given excitation level m A < N (usually mA << N). The
general form of the truncated cluster operator defining a standard CC approximation

A, characterized by the excitation level m A, is

m4
T“) = Z T... (4)
n=1

An example of the standard CC approximation is the CCSD method. In this case,

m A = 2 and the cluster operator T is approximated by

T‘CCSD) = T1 +213, (5)
where
T1 = t3a“a,~ (6)
and
T2 = itgaaabaja, (7)

28

are the singly and doubly excited cluster components, and tf, and t3’3 are the cor-
responding singly and doubly excited cluster amplitudes. In accordance with our
general notation, z“, j . . . (a, b. . .) are the occupied (unoccupied) spin-orbitals in the
reference determinant IQ).

In all conventional CC approximations, the cluster amplitudes
t3‘l'jjz‘n are determined by solving a coupled system of energy-independent non-linear

algebraic equations of the form:

(Q?;_;;;?IFI(A)|<I>) = 0, 2'1 < < in, a1 < < an, (8)
where n = l,...,mA,
HUI) = €_T(A)H€T(A) = (H8T(A))C (9)

is the similarity-transformed Hamiltonian of the CC / EOMCC theory, subscript C des-

al...an) E

ignates the connected part of the corresponding operator expression, and Wilma"

a“l -- - awn,“ - - . a5, IQ) are the n—tuply excited determinants relative to reference IQ).
In particular, the standard CCSD equations for the singly and doubly excited cluster

amplitudes tf, and t3’3, defining T1 and T2, respectively, can be written as

<¢?|H‘CCSD)I¢) = 0, (10)

(ogtmiccsmlo) = 0, 2' < j, a < b, (11)

29

where

fflCCSD) = e_T(Ccsn) HeT(ccsn) = (HeT(ccsn))C (12)

is the similarity-transformed Hamiltonian of the CCSD/EOMCCSD approach. The
explicit and computationally efficient form of H (CCSD) and other equations used in the
CC calculations, in terms of one- and two-body matrix elements of the Hamiltonian
in the normal-ordered form, fg E (pI f Iq) and 22;; E (quvIrs) — (quvIsr), respec-
tively, where f is the Fock operator and v is the operator representing the electron-
electron interaction energy, and cluster amplitudes @1111?" or, in the CCSD (m A = 2)
case, tf, and t3’3, can be derived by applying the powerful diagrammatic techniques of
many-body theory260 combined with technique of diagram factorization, which yields
highly vectorized computer codes.791911226 The algebraic and diagrammatic structure
of H CCSD is presented in detail in Appendix B and the factorized forms of H CCSD is
shown in Appendix C. The introduction to diagrammatic methods used in this work
is given in Appendix A.

Unlike the CI approaches, which are variational in nature, the CC energy is ob-
tained by projecting the connected cluster form of the Schriidinger equation on the
reference configuration IQ). In other words, once the system of equations, Eq. (8), is
solved for T“) or till???" (or in the CCSD case, Eqs. (10) and (11) are solved for T1
and T2 or t3 and t3’3), the CC energy corresponding to approximation A, characterized

by the excitation level m A, is calculated using the equation

E3" = <<I>IFI<A>I<I>> s<<1>IH§£ZsI<I>t <13)

30

where Hg; is a “closed” part of H (A), which is represented by those diagrams
contributing to H (A) that have no external (uncontracted) Fermion lines (as opposed
to the “open” part of H (A) which is represented by the diagrams having external or

uncontracted Fermion lines).

3.1.2 Extension to Excited States via the Equation-of-Motion Formalism

(EOMCC)

The ground-state CC theory has a natural extension to excited electronic states

IQ“) via the EOMCC formalism, in which we write
|\II,,) = MP“) 5 RfleTIQ), (14)

where T is obtained in the ground-state CC calculations and R” is a linear particle-
hole excitation operator, similar to T, obtained by diagonalizing the
similarity-transformed Hamiltonian H = e’THeT in the space of excited determi-
nants IQ311‘33fgn) that typically correspond to excitations included in T.

In the following, we use a notation where p = 0 refers to the ground states while
11 > 0 designates excited states. Thus, the excitation operator R3 is defined as a unit
operator for u = 0, that is, root = 0) = 1 and rillffg‘nw = 0) = 0 for n 2 1, where

r001) is a coefficient defining the zero-body of R” and rillj'gm) are the excitation
amplitudes defining the n—body components of R3 when n > 1 (see discussion below).

In this way, the EOMCC ansatz, Eq. (14), reduces to the ground—state CC ansatz,

Eq. (1), when u = 0.

31

In the exact EOMCC theory, the cluster operator T and the excitation operators
R), are sums of all relevant many-body components that can be written for a given N -
electron systems. As in the ground-state case, the standard EOMCC approximations
are obtained by truncating the many-body expansion for the operator R3 at a given
excitation level mA < N, which typically is the same as the excitation level used
to define the truncated form of T. In general, when T is approximated by T“),
Eq. (4), the corresponding excitation operator R), defining the EOMCC method A is

approximated by

_ (A) A
123") _ RM, + 1233,“, (15)
where
Run = TOOL) 1, (16)

and the “open” part of RLA) is defined as

m4
12},pr = 2 RM, (17)
11:1
with
1 2 . .
Ru.“ = (a) r::-::.:::.(u)a mama (18)

representing the n—body component of RLA). For instance, in the EOMCCSD theory,

which is a basic EOMCC approximation where m A is set at 2, the excitation operator

R3008!” is approximated by

RLCCSD) = 314.0 + R ,1 + R ,2, (19)

32

where RM is given by Eq. (16) and

Run = Tit“) aaai (20)

and

Rn =17: to») a a (21)

RLCCSD), with r301) and r3’3(,u) representing

are the one- and two-body components of
the corresponding excitation amplitudes (1 in Eq. (16) is a unit operator).

The excitation amplitudes #1132310 ) defining the excitation operator R33”, Eq.
(15), are obtained by solving the eigenvalue problem involving the

similarity-transformed Hamiltonian H (A) in the space spanned by the excited de-

terminants IQfl‘j 1:3") with n = 1,. . . ,mA, i.e.

((1)01 an|(H(A)R R(A) )0 l<p> = wf‘A),ri1...in (“)1

11.1” open 11,0pen0 a1 ...an

2'1<---<z',,, a1<~~<am (22)
where
— — - - A
1153),: H<A>-H.<;:;..=H<A>—Es >1 <23)

is the “open” part of H (A), represented by the diagrams of H (A) that have external
Fermion lines, and
of,“ = Eff) — Eff” (24)

is the vertical excitation energy obtained with the EOMCC method A. In particular,

33

the r301), and r3’3(u) amplitudes of the EOMCCSD theory and the corresponding
excitation energies wLCCSD) are obtained by diagonalizing the similarity-transformed
Hamiltonian H (CCSD), Eq. (12), in the space of singly and doubly excited determi-
nants, IQ?) and IQgI’), respectively. Equation (22) alone does not provide information
about the coefficient r001) at the reference determinant IQ) in the corresponding
EOMCC excited-state wavefunction RLAIeTW IQ). This coefficient is determined a
posteriori using the equation

ro(u) = (MU-1“” R“) )cl‘P)/wf.’”, (25)

Open woven

once the excitation amplitudes r3113?" ()1) defining R313)” are determined (Eq. (25) is

valid for 11 > 0, meaning excited states only; for )u = 0, mm = 0) = 1, as explained

above) .

3.2 Method of Moments of CC Equations (MMCC)
3.2.1 An Overview of the Exact MMCC Formalism

We are now equipped with the basic elements of the CC / EOMCC theory which
are necessary to explain the noniterative MMCC approaches to ground and excited
electronic states. In this section, we focus on the exact MMCC formalism.

As described in Section 3.1, the standard CC and EOMCC equations are obtained
by projecting H WIQ) and H (AIRLA)IQ) on the excited determinants IQflljjji?) with
n = 1, . . . , m A that correspond to the particle-hole excitations included in the cluster

operator T“) and linear excitation operator R514). The corresponding ground-state

34

CC energy is obtained by projecting HWIQ) on the reference determinant IQ). It is,
therefore, quite natural to expect that in order to correct the results of the standard
CC / EOMCC calculations employing the cluster and excitation operators truncated
at mA-body terms, the projections of H WIQ) and H (AIRII’IHQ) on the excited de-
terminants @3122?) with n > m A, which span the orthogonal complement to the
subspace of the N -electron Hilbert space spanned by the reference determinant IQ)
and the excited determinants IQfl‘jjji?) with n = 1,. . . ,mA, have to be considered.
These projections, which are the essence of the MMCC formalism, designated by
M‘fifiaJmA), define the generalized moments of the CC / EOMCC equations corre-
sponding to method A.

The original single-reference MMCC theory,18'19121'22:25'26'68_7°*37’98 which is partic-
ularly useful in this work, is based on a simple idea that the exact, full CI, energies

of the electronic states 11, E,“ can be recovered by adding the state-selective, non-

iterative energy corrections

(A) = _ (A)
6,, _ E, E,

N n
= Z 2 (111.10 44ml) Miami)I<I>>/<\II..IRLA>eT“’14>) (26)

n=mA+l k=mA+l

to the ground (11 = 0) and excited (p > 0) states energies Eff” obtained in the stan-

dard CC/EOMCC calculations, such as CCSD/EOMCCSD, etc., which we continue
to designate by letter A. The alternative formulations of the MMCC theory includ-

ing the generalized MMCC formalism which applies to non—standard CC methods,26194

the numerator-denominator—connected MMCC expansions,96 the multi-reference ex-

35

tensions of MMCC,73'229’261 and the most recent biorthogonal MMCC formalism which

(18,226—228

leads to rigorously size extensive renormalized CC metho are of no use for

the MRMBPT-corrected MMCC methods and are not discussed here. The RE." and
TM) operators entering Eq. (26) are the truncated linear excitation and cluster op-
erators used in the underlying CC and EOMCC calculations, respectively, and in the

exact theory |\II,,) are the full CI states. The

Gem.) = (eT"").-. (27)

quantity is the (n — k)-body component of the wave operator eTW operator, defining
the CC method A, which is trivial to determine. The zero-body term, Co(m,4),
equals 1; the one-body term, 01(m ,4), equals T1; the two-body term, 02(mA), equals
T2 + 3T1” if mA 2 2; the three-body term 03(mA) equals TIT; + 3T? if mA = 2 and
T3 + T1T2 + 3T? if mA _>_ 3, etc. The

1 2 s ...1 a a
M3,),(mA) = (Ir-l) MialfflAmA) a 1 ...a “01,, . ..a,1 (28)

operator in Eq. (26) is the particle-hole excitation operator defined through the
aforementioned generalized moments of the CC / EOMCC equations of method A,
magi-3.0,: (mA) = ((1)611ka (H(A)R3A))I(p), (29)

films]:

which represent the projections of the CC/EOMCC equations of method A on the

36

excited determinants |Q,-11_'"° ,k k) with k > m A that are normally disregarded in the
standard CC / EOMCC calculations. Consistent with our notation in which R“) 3-0 = 1,
Eq. (29) includes the ground-state (p = 0) case as well. In this case, moments
MEEI’chmA) reduce to the generalized moments of the ground-state CC equations,

911030303 (m A), defining approximation A, i.e.

mfifal’kak (mA) E ”311°:ka (mil) = ((1)1113. .iikIH(A)I(p> (30)

As demonstrated, for example, in Refs. 22, 25, 26, 126, 127, the generalized mo-
ments of the CC / EOMCC equations can be calculated using the following expression:

mag-I‘kak ("l/l): ($01": '0’: I(H(Aope)n 120‘) )C IQ)

11k ..ik p,open

+ :(@"91"ak(H(A)R(/l) )DCI‘I’)

i1" "k ch—p
P=mA+l

+7.00” ”0301?..03, (mA)1 (31)

where r001) is the coefficient at the reference determinant IQ) in the many-body
expansion of RLA)IQ), defined by Eq. (25), subscripts “open,” 0, and DC refer to open
(i.e. having external lines), connected, and disconnected parts of a given operator
expression, 0 represents the j- -body component of operator 0, and M3141? .0]; (m A)
are the generalized moments of the ground-state CC equations defined by Eq. (30).

In particular, if the goal is to recover the full CI energies E“ by correcting the

CCSD/EOMCCSD energies ELCCSD) (the m A = 2 case), then the following a posteri-

37

. . CCSD
on correctlons 6f. ),

N 71
65.003”) = Z Belem) M.,.<2)I<I>>/<\P.IR§.CCSD’eT‘°CS°’I<I>>. (32)

71:3 k=3

have to be added to Ef‘ccsn) energies. Here, the generalized moments W1""“ (2) of

11,111 ...an

the CCSD/EOMCCSD equations corresponding to projections of these equations on

triply, quadruply, etc., excited determinants, i.e.

911212.42) = <¢:::I(FI‘CCSD’R§FCSD>)I<I>>, (33)
Mitre) = («as (H‘CCSD’RICCSD>)I<I>>. (34)

etc., should be evaluated. Again, H (CCSD) is the similiary—transformed Hamiltonian
of the CCSD method defined by Eq. (12) and RIPCSD) is the EOMCCSD excitation
operator described in Eq. (19).

Equation (26) defines the exact MMCC formalism for ground and excited states.
This equation can formally be derived by considering the asymmetric energy expres-
sion,

AI‘I'I = (‘1'|(H - EL‘>)R§.A>eT‘"I<I>>/<\P|R£.A>eT“’14>). (35)

referred to as the MMCC functional, which was introduced for the first time in the
original MMCC papers by Kowalski and Piecuch in Ref. 18 for the ground-state

case and Ref. 126 for the excited-state generalization. This expression satisfies the

38

property

Al‘pul = Eu " Eff): (36)

where Eu is the exact, full CI, energy, if I‘ll”) is a full CI state. The details of the
derivation of Eq. (26) using functional A[\II], Eq. (35), can be found in Refs. 18 and
126 (see, also, Ref. 25).

To this point we have focused the discussion on the exact forms of the MMCC
energy expansions, Eqs. (26 and (32). However, in order to develop practical MMCC
methods based on these equations, the following two issues must be addressed. First,
the exact MMCC correCtions 63”), Eq. (26), or 630081)), Eq. (32), are represented
by complete many-body expansions including the N -body contributions, where N
is the number of electrons in a system, corresponding to all many-body compo-
nents of the wavefunctions I‘ll“) that enter Eqs. (26) and (32) (cf. the summa-
tions over 11. in Eqs. (26) and (32)). In order to use Eqs. (26) or (32) in practical

53A) or 630081)), must be truncated at

calculations, the many-body expansions for
some, preferably low, excitation level m 3 > m A. This leads to the MMCC(mA, m3)
schemes.18119121'2212512‘5'68"7018718‘3 The second issue that has to be addressed is the fact
that the waveftmctions I\II,,) entering the exact Eqs. (26) and (32) are the full CI
states, which are not normally available. To resolve this dilemma, wavefunctions
IQ") must be approximated in some way. Depending on the form of I\II,,), we can
distinguish between externally corrected MMCC methods where I\II,,) is obtained in

the non-CC (e.g., CI) calculations, and renormalized CC approaches, in which the

form of I‘IIfl) is determined using CC arguments. The CI-corrected MMCC(2,3) and

39

MMCC(2,4) approaches, which are the basic CI-corrected MMCC approximations,
and the CR-CCSD(T) /CR—EOMCCSD(T) methods, which are examples of renormal-
ized CC / EOMCC approaches, are described in Section 3.3. The theoretical details
of the new variant of the MMCC theory, the MRMBPT-corrected MMCC method,
abbreviated as MMCC / PT, which is based on the idea of approximating I\II,,) by the
relatively inexpensive, low-order, MRMBPT-like expansions, and which is explored in
this dissertation, is presented in Chapter 4 and its performance is tested in Chapter
5. First, however, we discuss formal considerations that lead to all MMCC(mA,mB)

truncations.

3.2.2 An Overview of the Approximate MMCC Approaches:

The MMCC(mA,mB) Truncation

The main goal of all approximate MMCC calculations, including the CI—corrected
MMCC approaches and the renormalized and completely-renormalized methods pre-
sented in this section as well as the MRMBPT-corrected MMCC methods, which are
developed for this dissertation and discussed in Chapters 4 and 5, is to approximate

the exact corrections (5),”), Eq. (26), such that the resulting energies, defined as

MMCC _ A A
E3 >_E,<,>+.s,<,>, (37)

are close to the corresponding full CI energies E“. A systematic hierarchy of ap-
proximations that allows us to achieve this goal and that are called MMCC(mA,m3)

schemes is described below.

40

All MMCC(mA,m3) schemes are obtained by assuming that the CI expansions of
the ground- and excited-state wavefunctions IQ”) entering Eq. (26) do not contain
higher—than—mB-tuply excited components relative to the reference IQ), where m A <
m 3 < N. This requirement reduces the summation over 11 in Eq. (26) to E . The

n=mA+l

EfiMMCC) (m A,m3), can be given the following

resulting MMCC(mA, m3) energies,
form:

ELMMCC)(mA, m8) = Eff) + 6p(mAamB)1 (38)

where E34) is the energy of the u—th electronic state, obtained with some standard

CC/EOMCC method A, and

6.(m...me) = Z 2 (11.10 —k(mA)M...k(mA)l¢>/<\I’thf."’eTW|<I>> <39)

n=mA+l n=mA+l

is the relevant MMCC correction to Eff).

We restrict our discussion to the MMCC(mA,mB) schemes with mA = 2, which
enables us to correct the results of the CCSD or EOMCCSD calculations. In this
category, two schemes are particularly useful, namely, MMCC(2,3) and MMCC(2,4).
These schemes can be used to correct the results of the CCSD/EOMCCSD calcula-
tions for the effects of triple (the MMCC(2,3) case) or triple and quadruple excitations

(the MMCC(2,4) case). The MMCC(2,3) and MMCC(2,4) energy expressions are as

follows:

ELMMCC>(2, 3) = E55303”) +”HA/[143(2)Iq’l/I‘I’uIRLCCSD)€T(CCSmI‘D), (40)

41

EfiMMCCJQA) = ELCCSDJ+(Wu|M».3(2)+[Mu.4(2)+T1Mu.a(2)ll¢>

/<‘I,”| R£CCSD)eT(CCSD) |¢> , (41)

where EfiCCSD) is the CCSD (u = 0) or EOMCCSD (p > 0) energy, T‘CCSD) is the

cluster operator obtained in the CCSD calculations (cf. Eq. (5)), 135.00an is the

corresponding EOMCCSD excitation operator (cf. Eq. (19); when u = 0, R(CCSDJ=
l), and My,3(2) and M#,4(2) are defined as
M“,3(2) = — DJI‘J" abe(2)a“aba°akaja,-, (42)
and
M#,4(2)= 5%,; 931”“ “(2) aaabacadagakajah (43)

with 911'th(2) and 931”” “(2) representing the triply and quadruply excited moments
of the CCSD/EOMCCSD equations, respectively.

The explicit expression for the triply excited moment 931:1;(2), entering the
MMCC(2,3) and MMCC(2,4) approximations discussed in Sections 3.3 and 4.2 and
corresponding to the projections of the CCSD/EOMCCSD equations on the triply ex-
cited determinants |<I>fb°jk), in terms of the many-body components of the
CCSD/EOMCCSD similarity-transformed Hamiltonian Hmcsn), Eq. (12), and oper-

ator R‘CCSDJJZq Eq. '(19), is

...<2>= @- “(HéCCSD)R. 2)c|<1>>+<<P.jk|[H§CCSDJ(Ru.1+Rn.2)]c|<1>>
+<<I>.-,-.(H H‘CCSD’R..1>CI<I>>Mommas... <2) (44)

42

where the ground-state moment 9213542), obtained by projecting the CCSD equa-

tions on triply excited determinants, is given by

mg": “(2) (q, gbcjanMTg + T1T2 + le+ lTl2T2 + §T1T22 + éTfT2)lcl‘I’>- (45)

9;“in

The analogous expression for the quadruply excited momentsim “0(2) entering the

MMCC(2,4) approximation, has the form

22221“ ....(2) = (<2 .:.JJ;JI(H .SCCSDJR..2>CI<I>>+(<I>:;J:;JI[H£CCSDJ(H.1+H ,2)]o|<I>>

+(<I>:;J£;‘I(H§CCSDJH..1)DCI<I>> + r0002)! W“ ......(2) (46)

where the ground-state moment an” JJ (2,) obtained by projecting the CCSD equa-

tions on quadruply excited determinants, is expressed as

an?“ ......(2) = (in-‘3"? |[V~(%T2'" + lTszJ + éTé‘ + inT§)]cl<I>>- (47)

The operators H (CCSD) in Eqs. (44) and (46) represent the p-body components
of limos”) and H N = H — (<I>|H|<I>) is the Hamiltonian in the normal-ordered
form. The diagrammatic techniques of many-body theory greatly facilitate the derivar
tion of the explicit expressions for DJIJJJJ (2 ) and 9J1JJJJJ MO) The diagrams and al-
gebraic expressions representing the triply and quadruply excited moments of the
CCSD/EOMCCSD equations, in terms of molecular intergrals fg and 11;; and clus-

z 3

ter and excitation amplitudes ti, t3, 7'9 and r3, are shown in Section 4.3 and

43

Appendix B. The fully factorized expressions for mtgfibgz) and 91:33,,(2), which
can be used in the efficient implementations of all MMCC(2,3) and MMCC(2,4)
approximations, including the CI-corrected MMCC methods and the renormalized
and completely-renormalized CC approaches discussed in Section 3.3 as well as the
MRMBPT-corrected MMCC methods developed for this dissertation, are described

in Section 4.4.

3.3 The Existing MMCC(mA,mB) Approximations

Depending on the form of |\II,,) in the MMCC energy equations, the existing
MMCC(mA, m 3) methods that have been developed prior to this thesis work fall into
one of the following two categories: (i) the CI-corrected MMCC(mA, m 3) schemes
and (ii) the renormalized and completely renormalized CC methods for the ground-
state problem, and their excited-state completely renormalized EOMCC extensions.

We begin with the CI-corrected MMCC approaches.

3.3.1 The CI-corrected MMCC(2,3) and MMCC(2,4) Methods

In the CI-corrected MMCC(2,3) and MMCC(2,4) calculations, the wavefunc-
tions |\II,,) in Eq. (40) and (41) are replaced with the wavefunctions obtained in
the active-space CISDtJlJJJJle‘JJJJJJ251 and CISthmJ251 calculations, respectively, as

shown below:

Ef‘MMCC/CJ)(2,3) = ELCCSD) +(‘I’E‘CISDJJIMM3(2)|‘D)
Raff‘s”)IRlCCSD’eT‘°°S°’I<I>>. (48)

44

 

 

virtual (31, 32, . . .)

 

unoccupied (a1, a2, . . .)

 

active (A1, A2, . . .)

____.O._.O____._
-___.O._.O_____
—H——
—H——
——o——o—
—o—o——

 

21.0th8 (11, 12, . . .)

occupied (i1,z°2,. . .)
core (11, i2, . . .)

Figure 1: The orbital classification used in the active-space CI, MRMBPT,
and the CI— and MRMBPT-corrected MMCC methods, such as MMCC(2,3)/CI,
MMCC(2,4)/CI, MMCC(2,3)/PT, and MMCC(2,4)/PT. Core, active, and virtual
orbitals are represented by solid, dashed, and dotted lines, respectively. Full and
open circles represent core and active electrons of the reference determinant IQ) (the
closed-shell reference determinant IQ) is assumed).

ELMMCC/CJJQA) = ELCCSDJ+(‘I’LCJSDJ‘JJIMmQH[Mu.4(2)+T1Mu.3(2)ll<P)

/(\IJ§.C‘SDJJJIH£FCS°JeT“’°S°’I<I>>. (49)

In order to define the relevant CISDt and CISth wavefunctions, |\II(CISD‘)) and
I‘II(CJSDJ‘1)), respectively, we follow the philosophy of multi-reference calculations, i.e.,
we first divide the available spin-orbitals into four groups (see Figure 1) of core spin-
orbitals (i1, i2, or i, j, ), active spin-orbitals occupied in reference IQ) (I1, 12,

. or I, J, .. . ), active spin-orbitals unoccupied in reference IQ) (A1, A2, or A,

B, ), and virtual spin-orbitals (a1, a2, .. . or a, b, . . . ). The choice of active spin-

45

orbitals (typically, a few highest—energy occupied spin-orbitals and a few lowest-energy
unoccupied spin-orbitals) is dictated by the dominant orbital excitations in the ground
and excited states that we would like to calculate. A few examples of reasonable
choices of active orbitals in calculations involving bond breaking and excited states
are discussed in Chapter 5. Once the active orbitals are selected, the CISDt and

CISth wavefunctions are defined as follows:

IHLC‘S°JJ> =(C...o+0.1+0,2+c..,3)l<1>>. (so)
I‘IJE‘CISDJ’JJJ) = (Cmo + CAI + C ,2 + Cma + C#,4)I(I>), (51)

where
01,0 = com) 1. (52)
0.1 = c:(u> = 2cm) (53)

and
012.2 = i 301) aaabajaé = Z 031W) aaabajai (54)
i<j,a<b

are the usual reference, singly, and doubly excited contributions, respectively, to

IQLCJSDJJ) and mommy), and

012.3 = Z CIZEM) aAaJacaKaJ-a,, (55)
i<j<K,A<b<c
CM = Z cjififdm) aAaBaca‘JaLaKaJ-cu. (56)

i<j<K<L,A<B<c<d

46

Thus, in the CISDt approach, used in the CI—corrected MMCC(2,3) calculations (of
Eq. (48)), the waveflmctions I\II,,) are constructed by including all singles and doubles
from IQ) and a relatively small set of internal and semi-internal triples containing at
least one active occupied and one active unoccupied spin-orbital indices defined by
Eq. (55). For the CISth approach, used for the CI—corrected MMCC(2,4) approach,
an additional relatively small set of quadruples containing at least two active occupied
and at least two active unoccupied spin—orbital indices is also required. The CI coef-
ficients defining the CISDt and CISth wavefunctions are determined variationally,
as in all CI calculations.

One of the main advantages of the CI-corrected MMCC schemes, including the
MMCC(2,3)/CI and MMCC(2,4)/CI methods, is a very good control of the qual-
ity of wavefunctions |\II,,) used to construct the noniterative corrections 6,,(m A, m 3),
which is accomplished through the judicious choice of active orbitals that can al-
ways be adjusted to the type of bond-breaking or excited excited states of interest.
Another advantage of the CI-corrected MMCC methods is their relatively low com-
puter cost compared to many other approaches, which is a consequence of the fact
that it is usually sufficient to use very small active orbital spaces to obtain excellent
results.JJJJJ‘JJ’JJ‘JJJJJJ”6'12"“251 If N, (N) is the number of active orbitals occupied (unoc-
cupied) in IQ), the most expensive steps of the CISDt and CISth methods scale as
NoNungn: and Nngngnfi, respectively, which are considerable savings in the com-
puter effort compared to the 713713 and ngnfi scalings of the parent CISDT (CI singles,

doubles, and triples) and CISDTQ (CI singles, doubles, triples, and quadruples),

respectively. Furthermore, the numbers of triples and quadruples considered in the

47

CISDt and CISth calculations are ~N0Nun3nfi and ~N3N3ngn?“ respectively, which
is significantly less than the numbers of all triples and quadruples if the number of ac-
tive orbitals is small. For instance, the number of triples used in the CISDt-corrected
MMCC(2,3) calculations is usually very small (no more than ~30% of all triples,
in many cases even less than that). Additionally, the CPU times required to con-
struct the relevant 6,,(2, 3) corrections are often on the order of the CPU time of a
single CCSD/EOMCCSD iteration. Similar remarks apply to the CISth-corrected
MMCC(2,4) calculations.

Thus, the CI-corrected MMCC methods can be regarded as useful approaches
to accurate calculations of ground— and excited-state potential energy surfaces. One
may contemplate, however, an idea of reducing the costs of the CISDt-corrected
MMCC (2,3) and CISth—corrected MMCC(2,4) calculations even further by replacing
the CISDt or CISth wavefunctions in the MMCC(2,3) and MMCC(2,4) expressions
by the wavefunctions obtained with low-order MRMBPT approaches, as presented in
Chapter 4.

In the next subsection, we describe an alternative to externally corrected MMCC
schemes using nothing else but the cluster and excitation operators obtained in the
CC / EOMCC calculations instead of non-CC wavefunctions to define the relevant

wavefunction |\II,,).

48

3.3.2 The Renormalized and Completely Renormalized CC Methods for

Ground and Excited States

An interesting alternative to the CI-corrected MMCC methods, discussed in
Section 3.3.1, is offered by the renormalized (R) and completely renormalized (CR)
CC / EOMCC methods which can be considered as purely single-reference, “black-box”
methods based on the MMCC(mA,mB) approximations.

Let us begin with the ground-state R—CC and CR-CC methods. The R—CCSD(T)
and CR-CCSD(T) methods are examples of the MMCC(2,3) schemes, whereas the
R—CCSD(TQ) and CR—CCSD(TQ) approaches are examples of the the MMCC(2,4)

approximations. The energy formula defining CR—CCSD(T) method is

E3CR-CCSD(T)) = Eéccsn) + (\IISCCSDJTJJI Mo,3(2)|‘1’)/ (\I’IICCSDJTJJICTJCCSDJIQ). (57)

Here, again, T(CCSD) refers to the cluster operators obtained in the CCSD calculations
and Mo,3 (2) is given by Eq. (42) using the generalized moments 91:32,,(2) defined by
Eq. (44). The IQSCCSDJTJJ) wavefunction, entering Eq. (57), is a simple MBPT-like

expression

lwgCCSD‘T”) = [1 + T1 + T2 + R33’(VNT2)C + RIJJVNTIIIQ) (58)

with R33) representing the three-body component of the MBPT reduced resolvent and
VN designating the two-body part of the Hamiltonian in the normal-ordered form.

The R—CCSD(T) approach is obtained by replacing the 9123542) moments by their

49

lowest-order estimates, i.e. (Qg‘ff (VNT2)CIQ). The R-CCSD(T) approach reduces
to standard CCSD(T) method when the (\II(CCSD(T))IeTJCCSDJIQ) denominator in the
R-CCSD(T) energy formula is replaced by 1.

The idea of renormalizing the CCSD(T) approach can also be extended to the
CCSD(TQ) method. Two variants of the CR-CCSD(TQ) approach, which are labeled
“a” and “b” , are mentioned here. The CR—CCSD(TQ) energy formulas can be given

in the following form213119J87v389°

ESCR—CCSD(TQ)JXJ = EéCCSD) + <w3CCSD(TQ),x)|[M0’3(2) +T1Mo,3(2)

+ Mo..(2>1Ida/(waccsn‘m’J’IeT‘°°S°’|<I>> (x = am), (59)

where

(wacCSD‘T‘J’J) = IwéccsD‘T”> + éTsz‘J’FIJ) (60)
and

(wgJJJJ<JJ>J>> =(wgJJJJ<J>>>+-;T.J(<I>>, (61>

with T2“) representing the first-order MBPT estimate of T2. The Mo,3(2) operator is
defined by Eq. (42) and Mo,4(2) is defined by Eq. (43). The R-CCSD(TQ) meth-
ods are obtained by dropping the T1M0,3(2) term in Eq. (59) and by replacing the
9313:,”(2) and M33042) moments that enter the definitions of Mo,3(2) and Mo,4(2), re-
spectively, by their lowest-order estimates. As in the case of the R—CCSD(T) method,
one can obtain the standard CCSD(TQ) approaches by replacing the overlap denom-

inator (QSCCSDJTQJ’JJJIeTJCCSDJIQ) in the R-CCSD(TQ) energy expression by 1.

50

The standard CCSD(T) and CCSD(TQ) approaches and their (C)R-CCSD(T)
and (C)R—CCSD(TQ) counterparts have nearly identical computer costs. For exam-
ple, the costs of the conventional CCSD(T) calculations are 713113 in the iterative
CCSD steps and ngnfi in the noniterative steps defining the triples correction while
the CR—CCSD(T) approach scales as nan: in the iterative CCSD steps and 2nfinfi
in the noniterative triples correction part. Similar remarks aply to other R—CC and
CR—CC methods. In particular, the CR—CCSD(TQ),x approaches have computa-
tional steps that scale as 71371: in the CCSD part, 2113113 in the triples correction
parts, and 2113113 in parts that deal with the quadruples corrections. This should be
compared to the costs of conventional CCSD(TQ) calculations which are very sim-
ilar, namely, 123111 in the CCSD steps, 713113, in the (T) part, and 123712 in the (Q)
parts. As already mentioned, the renormalized CC methods, such as CR-CCSD(T)
or its recently formulated size extensive extension (not discussed here), termed CR-
CC(2,3) 30321228 are particularly useful in studies of reaction pathways involving single
bond breaking and biradicals.18’20‘2“'68—70'95J‘JJJJJJJJJJSJJJJJ‘JJJJJJJJJJJJJJ_236 They remove failures
of conventional CCSD(T), CCSD(TQ), and similar approximations without making
the calculations more complex or considerably more expensive.

The idea of renormalizing conventional CC methods via the MMCC formalism can
be extended to excited states. For example, in the CR—EOMCCSD(T)
approachJJJJ‘J’JJ‘JJJ26 which is an example of the excited-state MMCC(2,3) schemes,

the energies of ground and excited states E” are calculated as follows:

ELCR-EOMCCSD(T)) = Ef‘ccsn) + <W£CR—EOMCCSD(T))IMMQNQV

51

- (CCSD)
(WIFE EOMCCSD(T))|R’(‘CCSD) eT I‘D). (62)

Depending on the specific form of the wavefunction IQIFR-EOMCCSDJT») in Eq. (62),
several variants of the CR-EOMCCSD(T) approach can be considered. Here, only
one variant called ID (the CR—EOMCCSD(T),ID method), which represents one of
the most complete versions of the CR—EOMCCSD(T) approach and which usually
provides the most accurate description of excited states compared to other variants,
is presented. The CR—EOMCCSD(T) wavefunction entering Eq. (62) is defined in

this case as

leca-EOMCCSD(T)> = P(Ru.0 + Run + R“,2 + Rp,3)eT(CCSDJIQ)
= {Run + (32.1 + Ru.oT1)
+[R,,,2 + HAITI + R,,0(T2 + §Tf)]
+[R,,,3 + R,.,2T1 + 1J2,,,,(T2 + $3)

+Ry,0(TlT2 + éTiJllll‘I’la (63)

where P is a projection operator on the subspace spanned by the reference IQ) and
all singly, doubly, and triply excited determinants. The triple excitation operator

~

RM, entering Eq. (63), is calculated as

Rug, = 313 f$(u) aaabacakajag, (64)

52

where

site) = Mitre/0:22.. (65)

are the approximate values of the triple excitation amplitudes rJJ JJ(;1) resulting from

the analysis of the full EOMCCSDT eigenvalue problem.22 As implied by Eq. (65),

the approximate amplitudes rim) are calculated using exactly the same set of

triply excited moments 911‘th(2) of the CCSD/EOMCCSD equations that enters
the MMCC(2,3) energy expression, Eq. (40). This greatly facilitates the computer
coding effort, since one can reuse the moments 9.113342), which are needed to con-
struct the triples correction of CR—EOMCCSD(T) anyway, to determine the rJJJJ(Iu)
amplitudes. The DJth quantities that enter Eq. (65) represent the perturbative
denominators for triple excitations, which are defined as follows:

We CCSD abc ’ CCSD abc
Diabc = E; J " (QijkIHJ )quijk

(CCSD
= (2.80080)— (nglHi ’I<I>.-,-°J:>

—<<I>JJJIH CCSD)I2..°J£>

ijk

—(<I>.,°J:IH§CCSDJI<I>..JJ:> (66)

ngccsn)

where , p = 1 — 3, are the one-, two-, and three-body components of the

CCSD/EOMCCSD similarity-transformed Hamiltonian H(CCSD), respectively, and
wJCCSDJ is the EOMCCSD vertical excitation energy.

One of the main advantages of the renormalized and completely renormalized

methods is the fact these methods are as easy to use as the standard “black-box”

53

approaches of the CCSD(T) type while considerably improving the description of the
bond breaking region and excited states without the need to define active orbitals
or other elements of multi-reference theory. However, as mentioned earlier, there are
cases, where the degree of quasi-degeneracy is so big that one has to use elements of
multi—reference theory within the MMCC formalism. The CI—corrected MMCC meth—
ods are one possible way of incorporating multi-reference concepts into the MMCC
considerations. The MRMBPT-corrected MMCC methods developed in this work
represent another way. Brief information about the basic elements of MRMBPT is

presented next.

3.4 The Basic Elements of Multi-Reference Many-Body Per-

turbation Theory (MRMBPT)

All genuine multireference theories involve two fundamental concepts, namely,
that of the model or reference space flu and that of the wave operator U. The model
space lo,

.40 = “amt... (67)

is spanned by M determinants or configuration state functions IQ?) (p = 1, . . . , M)

that provide a reasonable zero—order description of the target space

H = “an“ <68)

u=0 ’

54

spanned by M quasi-degenerate eigenstates I\II,,) (u = 0, 1, . . . , M — 1) of the electronic
Hamiltonian H,

HIQ”) = EflI‘I’I‘)' (69)

In order to define the reference configurations IQp), the employed molecular orbital
basis set is partitioned into core, active, and virtual orbitals in a similar manner as
in Figure 1. The core orbitals are occupied and the virtual ones are unoccupied in
all reference configurations. The references IQ?) differ in the occupancies of active
orbitals. All possible distributions of active electrons among the active orbitals result
in a complete model or active space (CAS). The use of CAS is essential to obtain
size extensive results in the MRMBPT calculations, if the “perturb then diagonalize”
MRMBPT method, summarized below, is considered.

The wave operator U: .14, —> J! is defined as a one-to-one mapping between .240

and I that satisfies the intermediate normalization condition,

PU=R am

and the relation determining its kernel, i.e.,

UQ=0. (71)

Here, P and Q are the projection operators onto, respectively, model space [(0 and

55

its orthogonal complement floi in the N -electron Hilbert space,
M
P = Z P‘JJ. P‘JJ = |<I>p><<1>pl. (72)
p=1

Q=1—P. an

Based on Eqs. (70)-(73), although, unlike P and Q, the wave operator U is not
Hermitian, U 75 U l.

The MRMBPT wavefunctions I‘ll”) are calculated using the formula,

M on M
a.)=zc..(¢.>+z(zc..v<n>n¢.>). (74)

where U (1), U (2), etc. represent perturbative corrections to the wave operator U (ex-
pressed in terms of excited configurations from .x/(oJ', i.e., other than IQP), p=1,...,M).
These corrections describe dynamical correlation efiects.

The coefficients cw, which describe the nondynamical correlation effects, define

the zero-order states belonging to lo,

M
PPS») = ZCPHIJIJP) , (75)

In the “perturb then diagonalize” MRMBPT methods that follow the ideas described
in Ref. 155, these coefficients and the corresponding energies E” of the ground and

excited states I‘ll”), p: 0, 1, ..., M — l, are obtained by diagonalizing the effective

56

Hamiltonian,

Heff = PHUP = PHP + ZPHU‘JJJP, (76)

11:1
in the model space .x/(o,

Hefl’lxpgm) = Eylwff”). (77)

In the practical implementations of such MRMBPT theories, the wave operator U is
truncated at some, preferably low, perturbation theory order n. When U is truncated
at the first-order term U (I), i.e., U = P + U (1), the second-order MRMBPT flmction
(MRMBPT(2)) is obtained; second-order since the resulting energies E” are correct
through second order).

One can considerably simplify the above considerations, which are based on the
generalized Bloch formalism,262 in which the wave operator U is determined by solving
the multiroot generalized Bloch equation262 H U = U H U (in the case of MRMBPT,
using perturbation theory), by formulating the “diagonalize then perturb” MRMBPT
methods. In those methods, one continues to use Eq. (75) to determine the zero-
order states, but the coefficients cm, of the zero-order states are obtained by diag—
onalizing the bare Hamiltonian H rather than the efl’ective Hamiltonian H “J. Per-
turbative corrections to the zero-order energies 13(0) are calculated a .posteriori in a
state-selective manner without using the multiroot generalized Bloch formalism. As
mentioned in the Introduction, several methods, including the popular CASPT2 and
MC—QDPT approaches, are in this category. The MRMBPT approach used to de-
fine the MRMBPT-corrected MMCC schemes discussed in detail in the next chapter

(Section 4.1) belongs to the category of the “diagonalize then perturb” methods as

57

well.

58

4 The MMCC Methods Employing Multi-Reference

Many-Body Perturbation Theory (MMCC / PT)

In this chapter, the new variant of the MMCC theory, refered to as the MRMBPT-
corrected MMCC or, for bevity, MMCC/ PT, in which the full CI wavefunctions
I‘ll“) in Eqs. (26) or (39) are approximated by the wavefunctions obtained from the
low-order MRMBPT calculations, is described. The approximate MRMBPT wave—
functions used in the MMCC(m A,m 3)/ PT approaches implemented in this work are
discussed. Moreover, the diagrammatic formulation of the resulting MMCC(2,3) / PT
and MMCC(2,4) / PT schemes are presented. The details of the algorithm that en-
ables one to achieve a high degree of code vectorization for the generalized moments

of CC equations defining the MMCC methods are described as well.

4.1 The Multi-Reference Many-Body Perturbation Theory

Wavefunctions Used in the MMCC / PT Approaches

As in all multi-reference considerations and in analogy to the CISDt and CISth
methods discussed in Section 3.3.1, in order to define the computationally simple form
of the low-order MRMBPT waveftmctions IQLMRMBPTJ) for the MRMBPT-corrected
MMCC calculations, we begin by partitioning the molecular orbital basis set into
four groups as shown in Figure 1: core spin-orbitals (i1, i2, or i, j, ), active
spin-orbitals occupied in reference IQ) (II, 12, or I, J, ), active spin-orbitals

unoccupied in reference IQ) (A1, A2, .. . or A, B, .. . ), and virtual spin-orbitals (a1,

59

a2, . . . or a, b, . . .). As in the CI—corrected MMCC schemes, reference IQ) is one of the
many reference determinants that we choose as a Fermi vacuum for the CC / EOMCC
and MMCC calculations. By distributing active electrons among active spin-orbitals
in all possible ways, we generate a certain number (designated here by M) of reference
determinants IQp) (including, of course, IQ)), which span the complete model space
or P—space .14). Then, we define the zeroth-order wavefunctions of the ground and

excited states I‘IJILPJ) of interest as linear combinations of the reference configurations
I‘l’p),

M
Wfl=2amx W)
p=l

where the coefficients Em, and the corresponding zeroth-order eigenvalues E, are ob-
tained by diagonalizing the Hamiltonian H in the model space 4%.

Once the model space #0 is defined, we introduce the Q-space which in the
MMCC / PT method pursued here is a subspace of the orthogonal complement 10*
spanned by all singly and doubly excited determinants with respect to each reference
IQ?) (p = 1, . . . , M), as is done, for example, in the MRCISD calculations. After elim-
inating the repetitions in the list of Q—space configurations, the resulting MRMBPT
wavefunctions IQLMRMBPTJ), which will eventually be used to design the MMCC/ PT
(e.g. MMCC(2,3) / PT and MMCC(2,4) / PT) energy corrections, are then defined as

the linear combinations of the P-space and Q-space contributions,

M R
I‘I’IIMRMBPTJI = ZCPHIJIJPI + Z cqyqu), (79)

11:1 q=M+l

60

where IQ?) (p = 1,. . . , M) are the reference determinants and I‘I’q) (q = M+l, . . . , R)
are the Q-space determinants, as expressed above. In principle, we could use any
of the existing low-order MRMBPT methods to determine the approximate values
of the coefficients cw and cm, that enter Eq. (79) and perform the correspond-
ing MMCC(mA,m3)/PT calculations. In the simplified MRMBPT model used in
the present implementation of the MMCC(2,3) / PT method69J97J98 as well as the
MMCC(2,4)/PT scheme, we evaluate the relevant coefficients cw (p = 1,. . .,M)
and Cg). (q = M + 1, . . . , R) in Eq. (79) by applying the Liiwdin-style partitioning
technique'J‘J‘JJ264 to the Hamiltonian matrix in the space spanned by the P—space and
Q-space determinants IQ?) and IQ), respectively. Thus, if C?“ and Cqu are the
column vectors of coefficients cm, with p = 1,. . . , M and cm, with q = M + 1,. . . , R,
respectively, and if pr, Hpq, qu, and H00 are the corresponding PP, PQ, QP,
and QQ blocks of the Hamiltonian, we can write the Hamiltonian eigenvalue problem
for the wavefunctions I‘IIIPJRMBPTJ), Eq. (79), in the following manner:

HPP HP CP CP
Q ’J = E, ’J . (80)

HQP Hoe Co» Co»

In other words, we obtain

HPPCPp + HPQCQp = EpCPpa (81)

HQPCPI‘ + HQQCQH = EuCQu- (82)

61

If we approximate the QQ block of the Hamiltonian matrix entering Eq. (80), qu,

by the diagonal matrix elements (QQIHIQq), we can write
CQ)‘ z (Eul -- Dq)_l HQPCP)‘, (83)
or, more explicitly,

M
c,,, 8 2(3), — (<1>,|H|<1>,))-J (<1>,|H|<I>,,) cw, (q = M + 1, . . . , R), (84)
p=l

where DQ in Eq. (83) is the diagonal part of qu. In practice, we obtain the working
equation for the approximate values of the coefficients cg), (q = M + 1,. . . , R) by
replacing the energies E), and coefficients Cw (p = 1,. . . , M) in Eq. (84) by the zero—
order energies E” and coefficients 5p“, respectively, resulting from the diagonalization
of the Hamiltonian in the model space flu (diagonalization of pr; cf. Eq. (78)).

We use the resulting approximate values of the coefficients qu

M
54m = 2“?” -(ququJq))-1<JququJp)Epm (q = M + 1: - , , a R), (85)
p=l

along with the coefficients Em, obtained by diagonalizing H p p, to define the perturbed

states of the MRMBPT theory exploited in this thesis work, which are defined as

M R
l‘i’LMRMBPT» = Zépfiiqfio) + Z 5qu¢¢1>~ (86)

p=1 q=M+l
The wavefunctions I‘IILMRMBPTJ), Eq. (86), are used instead of the exact I‘ll”) states

62

in the MMCC(mA, m3) energy expressions, Eqs. (38) and (39), to define the family
of the MMCC(mA, m3) / PT approximations.

The above perturbative procedure based on the partitioning of the Hamiltonian
into the P-space and Q-space contributions equivalent to the MRCISD problem has an
advantage that we only have to consider the QP matrix elements of the Hamiltonian
(the (QqIHIQp) matrix elements) and the diagonal (QqIHIQq) matrix elements in the
process of defining I‘ll”). Another advantage of this procedure is the fact that the
zero-order states are obtained by simply diagonalizing the Hamiltonian in .210 to
obtain the relevant coefficients Ew-

We have developed computer codes for the MRMBPT model defined by Eqs.
(78), (85), and (86). In the actual construction of the MRMBPT wavefunctions
IQ“), we use a complete model space obtained by all possible distributions of ac-
tive electrons among active orbitals allowed by the spin and spatial symmetries. As
discussed earlier, we limit ourselves to the Q spaces corresponding to the MRCISD
problem. Because of the fact that we are mainly interested in the MMCC(2,3) and
MMCC(2,4) approximations and to further simplify our MRMBPT calculations, we
decided to consider only those Q-space configurations that are at most quadruply
excited with respect to the reference determinant IQ) used in the CCSD/EOMCCSD
and subsequent MMCC(2,3) and MMCC(2,4) calculations. Furthermore, the allowed
singly excited Q-space configurations from £0 to floi are:

a. core —-> active
b. core ——> virtual

c. active ——-> virtual

63

and for all cases the allowed excitations are of the a —> a and 6 —> fl types, since
we are interested in the S; = 0 states (in practice, singlet states) obtained out of
the restricted Hartree-Fock (RHF) reference. Similarly, the doubly excited Q—space
configurations from .110 to JIOJ can be divided into the following eight categories:

a. core, core ——> active, active

b. core, core —-> active, virtual

c. core, core —-> virtual, virtual

(1. core, active ——+ active, active

e. core, active ——2 active, virtual

f. core, active ——> virtual, virtual

g. active, active —-> active, virtual

h. active, active ———> virtual, virtual
Again, due to the conservation of spin, the allowed Q-space double excitations are
aa —+ aa, fifi —> 66, afi —2 afi, a3 —> Ho, 60 —> Ba, and 6a —> afl, if the RHF
determinant is used as reference IQ).

We have implemented the simplified MRMBPT scheme described above and tested

its usefulness in the MMCC calculations by examining its performance in several
benchmark calculations discussed in Chapter 5. The relevant MRMBPT-corrected

MMCC(2,3) and MMCC(2,4) approaches are discussed next.

64

4.2 The MRMBPT-corrected MMCC(2,3) and MMCC(2,4)

Approaches: MMCC(2,3)/PT and MMCC(2,4)/PT

As already mentioned, in this dissertation we focus on the basic MMCC(2,3) / PT

and MMCC(2,4) / PT approaches in which the CCSD/EOMCCSD energies are cor-

rected for the leading triples or triples and quadruples effects via corrections 6,,(2, 3)

and 6,,(2, 4), respectively, calculated with the MRMBPT wavefunctions IQLMRMBPTJ),

Eq. (86). In order to use the multi-reference wavefunctions I‘IILMRMBPTJ) in the

single-reference MMCC formalism, first we have to rewrite the IQLMRMBPTJ) states

in the form of CI expansions relative to the reference determinant IQ) used in the

CC/EOMCC and MMCC calculations, as shown below:
I‘I'JLMRMBPT) = (07.0 + 0...: + 02.2 + 02.3 + 02.4 + - - -)I<I>>.

where

(714,0 : 5001‘) 11

= c (u) = Z 2:.(2)
i,a

_ _ 1 _‘j
0.2 - anb(#) a a = 2: c.1(u)a a
£<j, a<b
' 1 4'1: b -t k b c
CM = 55 CancU‘) a‘Ja acakaja, = E J achJ) (1% a akajag,
i<j<k,a<b<c

65

(87)

(88)

(89)

(90)

(91)

and

C 4: 5—76 —EJanJ;Jd(p) aaabacadalakaja,- — Z éJJJJJ M01.) a‘JaJ’aca‘JagakaJ-a, (92)

i<j<k<La<b<c<d

are the corresponding particle-hole excitation operators relative to IQ) defining the

reference, singly, doubly, triply, and quadruply excited contributions to IQLMRMBPTJ)

’

respectively. In the specific case of the MMCC(2,3) / PT and MMCC(2,4) / PT approx-

imations explored in this work, we further simplify the wavefunctions IQLMRMBPTJ), in

Eq. (87), by truncating the CI expansions for the IQLMRMBPTJ)

at the triply excited
determinants (the CMIQ) term) in the MMCC(2,3) / PT case and at the quadruply
excited determinants (the (714.4(1)) term) in the MMCC(2,4) / PT case. The final en-
ergy expressions for the MMCC(2,3) / PT and MMCC(2,4) / PT energies, obtained by
replacing I\II,,) in Eqs. (40) and (41) by IQLMRMBPTJ), Eq. (87), truncated at the
triply (the MMCC(2,3) case) or triply and quadruply (the MMCC(2,4) case) excited

determinants, respectively, relative to IQ), are

ELMMCC/J’TJm) =HLCCJDJ+ Z (are). 221:3," ...(2>/D.. (93)

i<j<k,a<b<c

and

ELMMCC/J’T’m) = 3.0082 + 2 (22.3.» 22:12.42)

i<j<k, a<b<c

Z IJJ'J‘(u)1(2H;'.J;J....(2)

i<j<k<La<b<c<d

+ Ache/493322.42) 40/17.. , (94)

66

respectively, where the triply excited moments mtjffgbcm) are defined by Eqs. (44)
and (45) and the quadruply excited moments 931230,,(2) are defined by Eqs. (46) and

(47). The Adm/d partial antisymmetrizer is defined as

Aabc/d = 1 _ (ad) _ (bd) - (Cd), (95)

where (pq) is a usual transposition of indices p and q, and

XP'
| |

(@(‘MRMBP'H l RLCCSD)6T(CCSD) I‘D)

= App + Ami + Ape + 5.4.3 + Am + - -~ (96)

is the overlap denominator (\IIuIRLCCSD)eT(CCSD) IQ) entering Eq. (40), written for the
wavefunction IQ”) = IQfiMRMBPTb, Eq. (87), truncated at triples for MMCC(2,3) / PT
and at quadruples for MMCC(2,4)/PT. The Ago. A“, 574.2, [3.3, and AM contri-

butions to the denominator D”, Eq. (96), are calculated as

Ann = [5001)]. T001), (97)

A... = Elihu)? WM). (98)

5.4.2 = Z [3500]" [33m (99)
i<j,a<b

5.. = 2 [6.3le" fliitm), (100)

i<j<k,a<b<c

67

and

A..4= Z IJ..J...‘J‘(u)1JB 62.1..“(m (101)

i<j<k<l,a<b<c<d

where 500;), c My) ¢3,,01) E’aickM), andc 4’“ Mm) are the CI coefficients obtained by

rewriting the MRMBPT wavefunction, Eq. (86), in the single-reference

CI form of Eq. (87), and T001),

33.01) = (‘1’?|(Ru,1 + R.,0T1)|<I>), (102)
53,01) = <¢g§llRm2 + RMITI + Ru.0(T2 ‘l‘ éTfHI‘I’). (103)
fllbc’km ) = (‘I’ajkllRmTl + Ru.1(T2 + éTf) + Ru.o(T1T2 + JéTfHI‘I’). (104)

and

3;...J’J‘m) = <<I>:J.‘-J£IJI[R..2<T2+%Tf>+R».1<TIT2+%T3 >

+R..(%T§ + $1"an + 53mm» (105)

are the coefficients at the reference determinant IQ) and singly, doubly, triply, and
quadruply excited determinants, |Q?), IQ?” ), |be°jk), and |Qijabfl“), respectively, in the
CI expansion of the CCSD/EOMCCSD wavefunction RLCCSD)eT(CCSD) |Q), which can
be easily determined using the CCSD/EOMCCSD cluster and excitation amplitudes
t' ‘53., T001) 7‘ ...(u) and r ...(u)-

Although the summations over i < j < k, a < b < c in Eqs. (93) and (100)

and over i < j < k < l, a < b < c < din Eqs. (94) and (101) have the form of

68

the complete summations over triples and and quadruples, respectively, in reality the
wavefunctions IQLMRMBPT», Eq. (86), contain only small subsets of all triples and
quadruples, once we rewrite each IQLMRMBPT” in the form of the singlereference CI
expansion, Eq. (87). This is a consequence of using active orbitals in designing the
MRMBPT wavefunctions IQLMRMBPTU, which limit the triple and quadruple excita-
tions relative to the reference IQ) to a small class of triple and quadruple excitations
that carry a certain number of active spin-orbital indices. Although the actual number
of triples and quadruples in the IQLMRMBPT)) wavefunctions depends on the dimen—
sion of the active space used in the MRMBPT calculations, one usually needs a small
portion of all triples and quadriples in the MMCC(2,3)/PT and MMCC(2,4)/PT
considerations. As a result of using active orbitals in the MMCC(2,3) / PT and
MMCC(2,4) / PT methods, we only need a small subset of all triexcited coefficients
633p), Eq. (104), and similarly a small subset of triply excited moments M3342),
Eqs. (44) and (45), which match the nonzero coefficients 53:01), to calculate the
MMCC(2,3) / PT energi, Eq. (93). Similarly, for the MMCC(2,4)/PT approach, one
only needs a small subset of all quadruply excited coefficients 6213“”), Eq. (105), and
a similarly small subset of all quadruply excited moments 911330119), Eqs. (46) and
(47), which match the nonzero coefficients Eflm), in the MMCC(2,4) / PT energy
expression, Eq. (94).

One of the main advantages of the MRMBPT-corrected MMCC schemes, such
as MMCC(2,3)/ PT and MMCC(2,4) / PT, is their low computer cost, compared to
the already relatively inexpensive CI—corrected MMCC methods described in Section

3.3.1. As in the case of the CI—corrected MMCC approaches, such as MMCC(2,3)/CI

69

and MMCC(2,4)/CI, in the MMCC(2,3) / PT and MMCC(2,4) / PT methods we have a
very good control of accuracy through active orbitals defining model space 1.3, which
can always be adjusted to the excited states or the bond breaking problem of interest,
but unlike in the MMCC / CI schemes, we do not have to solve the iterative CISDt
and CISth equations to generate the wavefunctions IQLMRMBPT)) that enter the cor-
rections 6,.(mA, m3) of the MRMBPT-corrected MMCC theories. We calculate the

relevant CI-like coefficients, such as 500;), (1(a), 3&(p), é‘gfim), and Efim), by sim-

ply converting the expressions that define the MRMBPT wavefunctions IQLMRMBPF)),
Eq. (86), into the single-reference CI form defined by Eq. (87). Thus, the main com-
puter effort for MMCC(2,3) / PT approach goes into the calculations of a small subset
of triexcited moments mime), leading to the significant reduction of the ngnfi steps
that are normally needed to calculate all moments 9313:,”(2). Likewise, the main
computer effort for MMCC(2,4) / PT approach goes into the calculations of a small

subset of quadruply excited moments 93121220,,(2). Very little effort is needed to deter-

(MRMBPT)
u )

mine the wavefunctions IQ and the corresponding coefficients 5001.), 612,04),

cilia), alto), and amp).

At this point, the remaining issue that we have to address before calculating the
MMCC(2,3) / PT and MMCC(2,4) / PT energies is how to obtain the explicit algebraic
expressions for the triply and quadruply excited moments 9.1113342) and 931;: 220,1(2)
It turns out that the best way to handle these quantities is through the use of the di-

agrammatic methods of the many-body theory which are described in the subsequent

section.

70

4.3 Diagrammatic Formulation and Factorization of the Triply

and Quadruply Moments of the CCSD Equations

In this section, the diagrammatic derivation of the explicit equations for the triply

and quadruply excited moments of the CCSD/EOMCCSD equations, 911:1:bc(2) and
William): respectively, is presented. The procedure of diagram factorization, which
is deemed necessary to obtain highly efficient computer code, is discussed as well.

Historically, the use of diagrams originated in quantum field theory using the time-
dependent formalism. However, the time-independent formulation is sufficient in the
development of quantum chemical and other many-body methods that rely on the
time-independent Schrbdinger equation. Diagrams are a graphical representation of
Wick’s theorem, which is a basic theorem for the algebraic manipulations involving
operators in the second-quantized form. They carry information about interesting
physics (e.g., connected vs. disconnected clusters), while providing powerful tool to
derive and organize numerous mathematical expressions that almost any accurate
many-body theory generates.

The entire discussion of diagrammatic methods described in this dissertation fo-
cusses on the time-independent formulation. It is important to note that the sequence

in which the operators act (i.e. right to left in the bracket notation) is relevant; this

is indicated in the diagram by means of a so-called formal time axis as shown below:

71

Thus if we want to represent the operator product VNTI diagrammatically, we begin
with a diagrammatic representation of T1 at the bottom, followed by a diagram rep-
resenting the operator VN drawn above the T1 diagram. This bottom-top convention
is the convention we will be using throughout this dissertation. Another common
convention, which was pioneered by CiZek and Paldus, is to place the formal time

axis for the operator ordering horizontally, from right to left,

l
+—

corresponding to the way we normally write a sequence of operators acting on a func-
tion. The difference between the two arrangement is the 90° rotation of the diagrams.

The standard procedure for deriving the explicit algebraic expressions for 9.112242)
and Mazda), Eqs. (44) and (46), in terms of the individual cluster and linear ex-
citation amplitudes as well as one— and two—body integrals defining the Hamiltonian,
is to represent the formulas, as given by Eqs. (44) and (46), in a diagrammatic form,
obtain the so-called resulting diagrams by contractions of fermion lines represent-
ing the relevant creation and annihilation operators that enter the second-quantized
forms of the operators, and apply the diagrammatic rules to convert the resulting
diagrams back into algebraic language. Figure 2 shows the the basic diagrams used
to derive the equations for the generalized moments of the CCSD/EOMCCSD equa-
tions. An overview the diagrammatic methods of many-body theory, describing the
basic elements of the diagrammatic language used here, is presented in Appendix A.
The diagrammatic and algebraic structure of the one—, two-, three-, and four-body

components of the CCSD similarity-transformed Hamiltonian H‘CCSD), which are the

72

 

V VmV V V

E)— (d)

A A
(e) (T)

Figure 2: Diagrammatic representation of (a) T1 as defined by Eq. (6); (b) T2 as
defined by Eq. (7); (c) R,“ as defined by Eq. (20); (d) RM as defined by Eq. (21);
(e) FN = f;N[aPaq], and (f) VN = fivgNhPaqamr], where FN and VN are the one-
and two-body components of the Hamiltonian in the normal-ordered form (H N) and
N [ . .] stands for the normal product of the operators.

key elements for all CCSD—based methods, are shown in Appendix B.

We use the Hugenholtz (and Brandow) diagrams to derive the explicit many-body
expressions for all terms that correspond to the triply and quadruply moments of the
CCSD/EOMCCSD equations, 9362542) and 9113342), respectively. Other represen-
tations, such as that of Goldstone, could be used as well, but Hugenholtz diagrams
are best whenever we rely on the second-quantized operators using antisymmetrized
matrix elements, as is the case here. There are several methods of obtaining all re-
sulting diagrams. The usual approach, which is followed here, is to first draw all
elementary and then resulting Hugenholtz skeletons. The arrows are subsequently
added in all distinct ways to produce all the distinct resulting diagrams. In the
case of expressions for the generalized moments of CCSD equations, where we have

T(CCSD)

)CIQ) on the excited determinants lel‘.‘j_',-:"), we do not draw

to project (HNe

73

the diagram representing the bra state (Qfl‘finl. Instead, we draw all permissible

T(CCSD))C|Q) with n incoming and n outgoing external

resulting diagrams for (H Ne
fermion lines labeled by fixed indices 2'1, . . . ,2}, and (11,. . . ,an, corresponding to the
determinant Iq’?,l.'.'.i:"> on which we project. Similar applies to moments of EOMCCSD
equations. This greatly facilitates the process of drawing the resulting diagrams and

makes the resulting diagrams less complicated.

To interpret the diagram algebraically, we conform to the following rules:

a. Each upgoing external line is labeled with a “particle” (unoccupied) spin-orbital
label a, b, c, d, and each downgoing external line with a “hole” (i.e. occu-
pied) spin-orbital label 1', j, k, l, . . . . External lines should always be labeled in
canonical sequence as a,z'; b, j; c,k; etc. The internal hole lines are labeled with

m, n, . . ., whereas the internal particle lines are labeled with e, f, .. .

b. The one-body vertex representing the one-body component FN = f: N [apaq] of
H N carries the numerical value of the Fock matrix element (pI f Iq) = ,3, where

p is an outgoing line and q is an incoming line. For instance,

carries a value of matrix element f2.

0. The two—body vertex representing the two-body component VN = i—vgNIaPaqasar]

of H N carries the numerical value of the antisymmetrized interaction matrix e1-

74

ement 12;; = (quvIrs) — (quvIsr) where p and q are the outgoing lines and r

and s are the incoming lines. For example,

a b
c d
carries a value of 2123' = (ablecd) — (avaIdc). In general, 1);; = —v;"q = —v;; =

er
v“.

. The one- and two—body vertices representing the T1 and T2 cluster operators

and the linear excitation vertices representing RM and RM, i.e.,

My aw, ail}! we...

carry the numerical values of the t3, t3, r34”), and ram) amplitudes, re-
spectively. The two-body amplitudes are antisymmetric so that, for example,

t3 = —t2, = 43', = t’; (similar applies to r3,(p)).

. All the spin-orbitals label are summed over “internal” lines, which are obtained

by contracting external lines of FN, VN, T1, T2, RM, and RM.

. The sign of the diagram is obtained from (——1)‘+" where l is the number of loops

and h is the number of internal hole lines.

75

g. The combinatorial weight factor of the connected diagram is specified by 0%)”,
where m is the number of pairs of “equivalen ” lines. A pair of equivalent
lines is defined as being two lines originating at the same vertex and ending at
another, but also same vertex, and going in the same direction. This weight
rule is specific to a Hugenholtz representaton and lines that carry fixed labels
(such as those defining the (‘1’?1‘,IZ}?I bra states on which we project) are always

regarded as non-equivalent.

h. To maintain the full antisymmetry of a final expression for a quantity, such as
moment mega?” (m A), which is antisymmetric with respect to permutation of
indices 2'1, . . . ,in and a1, . . . , an, the algebraic expression for each diagram should

be preceded by a suitable complete or partial antisymmetrization operator,

permuting the external lines in all distinct ways.

The procedure outlined above (of, also, Appendix A) can be greatly simplified
if we realize that the generalized moments of CCSD/EOMCCSD equations, such
as 9321;42) and mil-304(2): are defined in terms of H‘CCSD). As shown in Ap-
pendix B, various many-body components of H‘CCSD) contain several diagrams of the
(FNeT‘+T’)C and (VNeT1+T’)C types, which are part of more complex diagrams repre-
senting ”$342) and armada). By treating many-body component of H‘CCSD) as
more basic diagrams, which we represent by effective vertices with wavy lines, we can
express moments M2242) and wtjffigeda) in terms of matrix elements 72:, fig, 71$,
etc. that define one-body, two-body, three-body, etc., components of the H‘CCSD). In

this way, instead of drawing large numbers of resulting diagrams corresponding to the

76

original definitions of 9113342 ) and 911"” ehed(2) in terms of FN, VN, T1, T2, Rm], and
RM, Eqs. (44)-(47), we can draw the relatively small number of diagrams in terms
of the precomputed matrix elements of 1:1(CCSD), which serve as natural intermediates
(cf. Appendix B). All of the diagrams representing 911:1;(2), obtained in this way,
are shown in Figure 3. Note that all of the diagrams in Figure 3 have six external lines

corresponding to the projections of triply excited determinant IQ?“ ). The diagram

ijk

that shows up as the last term in Figure 3 is the ground-state moment 9113’:he(2)
defined in Eq. (45). The fact that we can represent the entire ground-state moment
9113,’:he( 2() in this compact way is a consequence of the fact that 9113’,“ eh42) can be re-
garded as one of the three-body components of H‘CCSD); the complete set of diagrams
corresponding to 9113’ehe (2) is shown in Figure 24 in Appendix B. The squared dot,

EJa,t 9113’:he (2) in Figure 3 represents the constant mm). For the quadruply excited
moments 9113’ke3ed (2), Eq. (46), all of the resulting diagrams should have eight external
lines extending to the top and corresponding to the projection onto the quadruply
excited determinant Ingfh"). We can see this in Figure 4, where all diagrams repre-

senting 19123342) are presented. Again, in analogy to 911"eh42), the diagram

W

that shows up as the last term in Figure 4 corresponds to the ground-state moment

77

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IV

VI

1:

Figure 3: Diagrammatic representation of 9113442) using the Brandow forms of the
relevant Hugenholtz diagrams. Vertices with wavy lines correspond to many-body
components of H (CCSD). El in the last term represents r001).

 

 

 

 

 

 

 

 

 

XIV XV

 

 

 

 

XVI

Figure 4: Diagrammatic representation of 91133340) using the Brandow forms of the
relevant Hugenholtz diagrams. As in Figure 3, vertices with wavy lines corresponds

to many-body components of HwCSD) and El represents r0 (p).

79

W13’H ahe42) defined by in Eq. (47), which can also be regarded as one of the four-body
components of H (CCSD) the complete set of diagrams corresponding to W1" kl ahed(2)
is shown in Figure 25 in Appendix B. Again, the squared dot, El, represents the
coefficient r001). All of the diagramms that represent the two-, three-, and four-body
components of H (CCSD) which enter W1" th42) and W1”H ahe42), including the ground-
state triply and quadruply moments of the CCSD equations described above, are
given in in Appendix B (see Figures 20—25). The diagrams representing the one-body
components of H (CCSD) do not show up in Figures 3 and 4 directly, but do show up
indirectly as intermediates when the diagram factorization discussed in Section 4.4,
which leads to efficient computer codes, is fully carried out (see, also, Appendix C).

Diagrams shown in Figures 3 and 4, with two-, three-, and four-body components
of H (CCSD) represented diagrammatically in Figures 2025, provide correct algebraic
expressions for moments W1”the(2) and W1" t3e42), but these expressions, if coded term
by term, do not yield efficient computer programs. In order to reduce the number of
CPU operations that are requested to calculate W1jfthe(2) andW1 W1"H ahed(2) in the most
eflicient manner, one has to factorize the 911”" ahe(2) and W1""’ ahe42) diagrams shown in
Figures 3 and 4, that relateW1”the(2) and W121“ ahed(2) to the expensive three- and four—
body components of H (CCSD) , shown in Figures 20—25. The main ideas behind diagram
factorization that lead to considerable reduction in the number of CPU operations
characterizing the resulting many-body expressions are explained in Appendix C. We
illustrate the key steps that lead to the factorized, computationally efficient form of
W1zfth42) in Figures 5—8.

In the initial step, shown in Figure 5, we rewrite the expensive three-body matrix

80

(a)

M

(b)

m

(C)

m

(d)

M+M
+111

3:2 :E E E

Figure 5: Factorized forms of the three-body components of H (CCSD) required in the
derivation of W1;fthe(2), expressed as products of two-body matrix elements of H (CCSD)
and T2 cluster amplitudes.

W m m
W= W M
Ply NW

elements of H (CCSD) that enter the W132he(2) diagrams shown in Figure 3 (see diagrams

where

(f)

Figure 5: continued.

I-V in Figure 3) as tensor products of two-body matrix elements of H (CCSD) and T2
cluster amplitudes. We also rewrite diagram VI of Figure 3, which uses a four-
body component or H (CCSD), in a more explicit form in terms of VN, T2, and RN.
As results of these operations, the original diagrams shown in Figure 3 acquire a
new form shown in Figure 6. The first two diagrams in Figure 6 are in their final,
computationally efficient, form. However, diagrams IA—VI are no longer linear in

T or Rh, and represent multiple tensor products that need to be factorized further

by bringing them to a linearized (vectorized) form and reusing one— and two-body
matrix elements of H (CCSD), which are easy to generate, as much as possible. This is
accomplished in Figure 7 in three steps shown in Figure 7 (a), (b), and (c) where we
first factor out T2 vertices (Figure 7 (a) and (b)) and then group terms that have a
similar overall structure to define the final set of intermediates linear in RM and Rm
and shown in Figure 7 (c). As a result of all these operation, all of the diagrams in
Figure 3 or 6 acquire a compact, computationally, efficient form shown in Figure 8.

A similar procedure can be performed for the original diagrams representing
W13t3ed(2) shown in Figure 4. Again, by replacing the three-body matrix elements
of H (CCSD) that enter diagrams in Figure 4 (diagrams VIII and IX) by their fac-
torized analogs shown in Figure 5 and by replacing the four-body H‘CCSD) vertices
in the diagrams in Figure 4 (diagrams X—XVI) by their factorized analogs shown
in Figure 9, we obtain the set of diagram representing W133hed(2) shown in Figure
10. The factorization of non-linear diagrams in Figure 10 by factoring out common
terms, grouping diagram that have a similar struture, and reusing recursively gen-
erated one— and two-body matrix elements of H (CCSD) and other intermediates gives
the final, computationally efficient and compact form of W13:3ed(2) shown in Figure
11.

Figures 8 and 11 show the most compact, fully factorized, diagrammatic forms
of W13:he(2) and W13:3ed(2), that yield the highly efficient computer codes that are
5qu

characterized by the 113713 steps in the W133he(2) and 113113 steps in the W1”,ehee(2) case.

Figures 12-14 show the diagrams representing all recursively generated intermediates

83

 

 

 

 

 

 

 

VI IA VIIB

Figure 6: Diagrammatic representation of W13 the(2) obtained by substituting the
three-body components of H (CCSD) in Figure 3 entering diagrams I—V by their fac-
torized analogs shown in Figure 5 and by expressing the four-body component of
H (can) that defines diagram VI in terms of V”, T2, and RM.

84

 

 

 

 

us m N

 

 

 

l A
II
I

L:

 

 

 

 

 

ll
\lexlSVQW“ 1‘

Ill VA VB

 

 

(b)

Figure 7: Process of diagram factorization for the nonlinear terms in W13" ehe(2) shown
in Figure 6 (diagrams IA-VI). (a) and (b) The T2 vertex is factored out. (0) Diagrams
in parenthesis in (b) that have a similar structure are grouped together to define two
intermediate vertices that are linear in RM and Rm.

85

Ly

 

ll

" 7
+

i7

 

 

 

I" VA

 

 

W
T = Q7)

(C)
Figure 7: continued.
that are needed to calculate W13the(2) and W13"l ehee(2) using diagrams shown in Figure
8 and 11. The explicit algebraic expressions for the intermediates shown in Figures
12-14 are given in Table 2. The final algebraic expressions for W13:hJ2) and W1333ed(2)
used in this work and the remaining details of the computer implementation of the

MMCC(2,3)/PT and MMCC(2,4) / PT methods are discussed next.

86

bia !kc bia

(IA. IIB, NLVA)

J k

c b i a i k c
I ab + E k 0' {3333333 'l" I ab H

“B. IIA IV, VB. VI)

Figure 8: Final factorized form of W13the(2) The two-body vertices marked with “=”
are defined in Figure 7 (c).

4.4 Final Equations for W13;';he(2) and W1333ed(2) and the Re-
maining Details of the Implementation of the MMCC(2,3) / PT

and MMCC(2,4) / PT Approaches

The final, fully factorized expression for the triply excited moments of the
CCSD/EOMCCSD equations 9313342), in terms of the amplitudes defining the
CCSD/EOMCCSD cluster and excitation operators, T1, T2, Rap, RM, and R33,
and molecular integrals f3 and 1%, obtained fi'om the diagrams shown in Figure 8,
which can be used in the highly efficient, vectorized, computer implementations of
all MMCC(2,3) approximations, including the externally corrected MMCC(2,3) and
MMCC(2,4) schemes, such as the MMCC(2,3) / PT and MMCC(2,4)/PT approaches

pursued to this work, and their CI—corrected MMCC and renormalized CC/EOMCC

87

=,,\M ,, +
(a)
=,):h e, +
(b)
)

MN

Map

(c

Figure 9: Factorized forms of the four-body components of H (CCSD) required in
the derivation of 931323,,(2), expressed as products of two-body matrix elements of

H (CCSD) and T2 cluster amplitudes.

 

+ . ab cl d +
9 m
VIIIB
I R J c
I 8b cl (1 + ab Kl d +
m n m 9
IXA IXB

 

 

ll

X

(a) ..
Figure 10: Diagrammatic representation of 93222040) obtained by substituting the
three- and four-body components of H (CCSD) in Figure 4 by their factorized analogs
shown in Figures 5 and 9.

89

 

 

 

 

 

 

 

 

 

ab cl d

+

XVIC

(b)

Figure 10: continued.

90

CIDia kjbia
Idk + ldc
f n

(VIIIA IXB. XA. XII) (vms, IXA. XB, XIA.X| B, xm. XIV) (XVA XVB)
c I ' a l b l
+ E] ' d k + I d c
f n
(XVIA. XVI B) (XVIC)

Figure 11: Final factorized form of 93133,“,(2 )

analogs, can be given the following compact form:

Mitch AMT}: (2 ) (106)

where

ab ec mc Tab

Title) = AJ/J’JH eh“ J’J— W J" —-;-I,J;i:t::+1.izti5)

mc ab

+gro<u><t th: — I'J’J “)1 (107)

For simplicity, we dropped the symbol u in the amplitudes 735m), rz'g‘M), and mat)
entering Eq. (107). The final, fully factorized and computationally highly efficient

expression for the quadruply excited moments of the CCSD/EOMCCSD equations

91

 

 

0’ (D
II
0 m
I
I
>k
+
O‘ (D
I
ol
3
+

(c) l3"

0

 

 

(d) E?

Figure 12: One-body matrix elementsof F1 (CCSD) and other one-body intermediates
needed to construct mtgfibcm) and mma).

92

(d) Bf;

Figure 13: Two-body matrix elements‘of Elmo's” and other two-body intermediates
needed to construct 9.7212542) and Mfikm).

93

 

 

 

 

O'

 

 

 

b a
+ _ __
ayi e m c bVi 60m 0
(

g) 713%

Figure 13: continued.

94

 

 

 

 

 

 

(j) 13;"

Figure 13: continued.

95

 

 

 

 

(n) :3

Figure 13: continued.

96

 

 

C I
b'
= + 'a
d d m
kibia " '
+VV= b|a
l+ l m
(Mil?
b Ia Ikc
+Iab
m
c l
+ bia
d m

 

(d) 153:

Figure 14: Three-body intermediates needed to construct 9312342) and Dfljftgdfl).

97

WV: ..

 

 

 

 

(e) 125."

Figure 14: continued.

93"” “5“,,(2) obtained from the diagrams shown in Figure 11, can be written as

Wilda) = mzzttdm), (108)

where

T3“ 0504(2) = AW“ 1"thkl_ Aijk/l Ilijktnl+ +éA""/‘9fi2{§bc(2)rfiz

+%ro(u) (A‘J/‘J‘ Iiiitit —A‘J”°/' 13:11:15) . (109)

with 933’33,c (2) representing the ground-state triexcited moments of CCSD (see Table

2). The antisymmetrizers Am, AW, qu/r, qur/sa Jim/rs, qu/r/s, AW” etc. which

98

enter Eqs. (106)-(109) directly or through the matrix elements of H‘CCSD), and other
intermediates that are needed to construct Eqs. (107) and (109) and that are listed

in Table 2, are defined in a usual way,

Am 5 Am =1pq _ (pq), (110)
AW 5 AW = lpqr — (W) — (PT) — (47‘) + (P47) + (PM): (111)
AP/qr ‘3 Ap/qr E Avr/p 5 AW”, = lpqr - (P9) - (1”), (112)

Amr/a 5 AW” 5 As/pqr 5 AW" = 1mm - (p3) - (<18) - (7‘3), (113)

Am/rs 5 AW” = 1pm - (pr) - (p8) - (qr) - ((18) + (mos), (114)

Am/r/s E Af/m/s __—: Ar/pq/s E Ar/s/pq E Ar/s/m

Am/r/s
= 1m -- (pr) - (p8) - (qr) - ((18) + (PTXQS) - (T8)
+ (W3) + (1987') + (m) + ((18?) - (qu8)

= qu/rs Ann (115)

AW. :- AW“ = 1W. - (pa) - (pr) - (p8) - (qr) - (<18) - (T8) + (m)
+ (487") + (W) + (1948) + (pm) + (P73) + (1284)
+ (1087‘) + (pq)(r8) + (pr)((18) + (ps)(qr) - (para)

- (pqsr) - (PTSQ) - (WIS) - (psrq) - (1984?), (116)

99

with (pq), (pqr) and (pqrs) representing the cyclic permutations of two, three, and four
spin-orbital indices, respectively. As mentioned in the previous section, the explicit
spin-orbital expressions for one- and two-body matrix elements of HCCSD, 71g and 5;},
respectively, and other recursively generated intermediates entering Eqs. (107) and
(109), in terms of cluster amplitudes t; and ti’h, excitation amplitudes r; E rim) and
ri’h E raw), and molecular integrals f: and 1);, are given in Table 2.

By using the idea of recursively generated intermediates and by reusing, as much as
possible, the one and two-body matrix elements of H(CCSD) , which are generated in the
CCSD/EOMCCSD calculations that precede the calculations of moments 93:3:he(2)
and MEMO), we achieve a very high degree of code vectorization, while avoiding the
explicit construction and storing of the most expensive three- and four-body matrix
elements of FNCCSD). Since all of the expressions defining 21.113542) and 932:3“,(2)
and the corresponding recursively generated intermediates are binary tensor (matrix)
products, one can very effectively calculate 933542) and 93:3:3ed(2) and the required
intermediates with fast matrix multiplication routines available in the BLAS library.

Once the CCSD/EOMCCSD equations are solved for th, tjfh, r001), r301) and
73(11): and the relevant moments M3342) and mama) and the corresponding
coefficients my). arm), 62:0»). ditto), mu), am), my), and mm) are de
termined using Eqs. (106)-(109), MRMBPT wavefunctions, IWLMRMBP'T’), and Eqs.
(102)-(105), we construct the overlap denominators D”, as in Eqs. (96)-(101), and,
finally, the energy corrections due to triples or triples and quadruples defining the
MMCC(2,3) / PT and MMCC(2,4) / PT methods, using Eqs. (93) and (94). Our

MMCC(2,3)/PT and MMCC(2,4)/PT computer programs are interfaced with the

100

RHF and integral transformation and sorting routines available in the GAMESS

package.265

101

m mil-.7555” ""

Table 2: Fixplicit algebraic expressions for one- and two-body matrix elements ele-
ments of H CCSD (designated by h) and other intermediates (designated by I or 19),
shown in Figures 12—14, used to construct the triply and quadruply excited moments

of the CCSD/EOMCCSD equations, 933342) and 933:3ed(2), respectively.

 

 

 

 

 

Ezpmssz'on“ Figure
71? ff + tgnvfif, 12 (a)
I? r2113; 12 (b)
7&2 f2 + t2”v2‘m - %t3"vf£n 4271?. 12 (C)
712 ff + $223,; + 5:;ng +1272: 12 (d)
7135 ”35 - trvfs. 13 (a)
7139 v3“ + tfvg’ 13 (b)
713 12% + 3thnh"v,°,'fn — $712?" + thnvgfn 13 (c)
71?} 12:“; + étfi‘fvfjf + tfehff — tfvfi 13 (d)
:3, ”3": + tiny; 13 (e)
317, 19;; + my; - triage” 13 (f)
“is, v35, — tjgg’; + tgvgg + twig“ — tgnl‘zjgh + §t3hmfiffm + 11;ng - 3‘2;ng 13 (g)
Eff: vi: + 13,571: + tgvff + §t§§v§j + $1933 — trfifi + mafia; 13 (h)
131‘ Aikizfirf — mm + tail: + $531.5} — AJ'Jfisgw-ghm 13 (i)
13" Eff — 1:17:71: 13 (j)
1;; hazy; - ijhrr + ififinrg" — Bgsnrggn 13 (k)
133 1037.73" - 71$"? 13 (1)
I3 ivzh’ri’) + 3.4" 71251;]; 13 (m)
1,: —v;J,:,.r;;,JJ + Egrg — my? 13 (n)
13‘: 3mg: - 3A‘jtffhf‘l-1fie 14 (a)
13: §A‘/j"l_z{,'f,t;’h‘ 14 (b)
9313342) “MAJ/“(723112: — 13:33) 14 (c)
1,11“ hfigrg — #0333;ng + #:331th 14 (d)
133* AJ/J*(§B;§r;;r - $43,513?) + AJJ/ngrrg + mtg) 14 (e)
" Summation over repeated upper and lower indices is assumed; f: = (pl f Iq) and

1);; = (pqlvlrs) — (pqlvlsr) are the one- and two-electron integrals in a molecular spin-
orbital basis {p} corresponding to the Fock operator (f) and the two-body part of the

Hamiltonian (1)).

102

 

 

5 Numerical Examples

We illustrate the performance of the MRMBPT-corrected MMCC approaches
developed in this work by discussing the results of the benchmark MMCC(2,3) / PT
and MMCC(2,4) / PT calculations for the single bond breaking in the HF and F2
molecules, the simultaneous dissociation of both O—H bonds in the H20 molecule,
and the valence excited states of the CH‘r ion. We focus on a comparison of the
MMCC(2,3) / PT and MMCC(2,4) / PT results with a few other ways of incorporating
the triple and quadruple excitations in the CC and EOMCC formalisms, and the

exact, full CI data also obtained with GAMESS.

5.1 Bond Breaking in HF

In order to test the ability of the MRMBPT-corrected MMCC approaches to
improve the poor description of bond breaking by the standard CCSD and CCSD(T)
methods, we applied the MMCC(2,3) / PT approach to the potential energy curve of
HF. We used a double zeta (DZ) basis set,266 for which the exact, full CI energies27
and several other useful results, including, for example, the full CCSDT energies27
and their standard and completely renormalized CCSD(T) analogs,18 are available.
We also compare the MMCC(2,3) / PT results with the results of the CI-corrected
MMCC(2,3) (MMCC(2,3)/CI) calculations,251 which provide yet another way of cor-
recting the CCSD energies for the most essential effects due to triple excitations. We
focus on the triples methods because generally triply excited clusters along with the

singly and doubly excited clusters are sufficient to obtain a virtually exact description

103

of single bond breaking. In all calculations reported in this work, we used the ground-
state RHF determinant as a reference |<I>). The active space employed in the CI- and
MRMBPT-corrected MMCC methods consisted of three highest-energy occupied or-
bitals, 30, hr, and 211, and the lowest-energy unoccupied orbital 40 that correlate
with valence shells of the H and F atoms. This is a natural choce of active space for
the description of bond breaking in HF, since for larger internuclear separations R1“:

the full CI wavefunction of HF is dominated by the ground-state RHF configuration,

I‘D) = l(10)2(20)2(17r)2(27r)2(30)2|, (117)

the doubly excited configuration,

|<1>') = l(10)2(20)2(11r)2(211)2(40)2I, (118)

corresponding to the (30)2 -+ (40)2 excitation, and the (30) —> (40) singly excited
configuration.

The results of our MMCC(2,3) / PT calculations for the potential energy curve of
HF are shown in Table 3. In this particular case, there is a 1.634 millihartree difference
between the CCSD and full CI energies at the equilibrium geometry, RH-F = Re,
which increases to 12.291 millihartree at R1”: = SR: (for most practical purposes,
R1“: = 5Re can be regarded as a dissociation limit). As in other cases of single bond
breaking, the large differences between the CCSD and full CI energies at larger values

of R34: are primarily caused by the absence of the connected T3 clusters in the CCSD

104

 

 

Table 3: A comparison of the CC and MMCC ground-state energies with the cor-
responding full CI results obtained for a few geometries of the HF molecule with a
DZ basis set.The full CI total energies are given in hartree. The remaining energies
are reported in millihartree relative to the corresponding full CI energy values. The
non-parallelity errors (NPE), in millihartree, relative to the full CI results are given
as well.

 

 

Method R; 211., 317. 517. NPE
r1111 CIb —100.160300 —100.021733 -99.985281 —99.983293

CCSD 1.634 6.047 11.596 12.291 10.657
CCSDTb 0.173 0.855 0.957 0.431 0.784
CCSD(T)° 0.325 0.038 —24.480 —53.183 53.508
CR—CCSD(T)° 0.500 2.031 2.100 1.650 1.600
MMCC(2,3)/CI“ 1.195 2.708 3.669 3.255 2.474
MMCC(2,3)/PT“ 1.544 1.116 0.025 —0.889 2.433

 

“R, = 1.7328 bohr is the equilibrium value of the internuclear H—F distance.
b Fi‘om Ref. 27.

° Horn Ref. 18.

d From Ref. 251.

° The active space consisted of the 30, 111, 217, and 4a orbitals.

wavefunction. Indeed, the full CCSDT method, which includes T3 clusters, reduces
large errors in the CCSD results, relative to full CI, to as little as 0.173 millihartree
at RH.F = Re and 0.431 millihartree at R1“: = 5R6.

The full CCSDT approach works, but, CCSDT is a rather impractical method.
Unfortunately, the much more practical CCSD(T) approach completely fails at large
internuclear separations RH-p. Indeed, the small, 0.325 millihartree, error in the
results of the CCSD(T) calculations at R34: = Re increases (in absolute value) to
24.480 millihartree at R1142 = 31‘?e and 53.183 millihartree at R114: = 512.. (cf. Table
3). As shown, for example, in Refs. 18, 21, 25—27, 87, the CCSD(T) potential energy

curve lies significantly below the full CI curve at larger internuclear separations and

105

.1-_._—...— Mi!
‘ I '\- I |

is characterized by an unphysical hump in the region of intermediate Rwy values.

The CR-CCSD(T) approach, which is one of the “black-box” variants of the
MMCC theory, considerably improves the results of the CCSD(T) calculations, elim-
inating the unphysical hump on the CCSD(T) curve and reducing the 53.183 milli-
hartree error in the CCSD(T) results at RH-p = 5R... to 1.650 millihartree.18J25J26J87
The errors in the CR-CCSD(T) energies, relative to full CI, do not exceed 2.1 mil-
lihartree over the entire range of R1”: values. The CI-corrected MMCC(2,3) ap-
proach, employing the 30, 177, Zn, and 4a orbitals as active orbitals, provides similar
improvements""*""’J87‘251 (see Table 3).

As shown in Table 3, the MMCC(2,3) / PT calculations employing the same set
of 30, hr, 21r, and 40 active orbitals as used in the more expensive MMCC(2,3)/CI
calculations reported, for example, in Ref. 251, provide the results which are better
in the R1”: > Re region than the already very good results of the CR-CCSD(T) and
MMCC(2,3)/CI calculations. In particular, the MMCC(2,3)/PT approach reduces
the large errors in the CCSD(T) results at R34: = 3& and R1“: = 51?.2 to small errors
that do not exceed 1 millihartree. The signed errors in the MMCC(2,3)/ PT results
vary between 1.544 millihartree at RH-F = Re and —0.889 millihartree at RIM-.1 =
5R... The overall qualities of the MMCC(2,3)/PT and MMCC(2,3)/CI results, as
measured by the corresponding N PE values, are quite similar, Indeed the NPE values
characterizing the MMCC(2,3)/PT and MMCC(2,3)/CI calculations are 2.433 and
2.474 millihartree, respectively. An obvious issue that may need further attention is
the the quality of the MMCC(2,3) / PT energy at R1“: = Re, which is only slightly
better than that obtained with CCSD. This is related to the fact that we use the

106

ground-state RHF orbitals and a small active space which is designed to describe the
most essential nondynamical correlation effects in the region of larger H—F distances.
The suitable orbital optimization scheme would have to be developed to improve the
results of the MMCC(2,3) / PT calculations at 123.15 = Re. One possiblity could be to

use CASSCF orbitals.

5.2 Bond Breaking in F2

We now turn to the more challenging case of single bond breaking in F2. The
results for the F2 molecule, as described by the cc-pVDZ basis set,267 are shown in
Table 4. Again, the ground-state RHF determinant was used as a reference. In the
post-RHF calculations, the lowest two orbitals were kept frozen. The active space used
in the MRMBPT-corrected MMCC calculation consisted of the five highest-energy
occupied orbitals, 309, hr“, 277", 17rg, and 2119, as well as the lowest-energy unoccupied
orbital, 30“. No CI-corrected MMCC and full CI calculations were performed for this
system, so we have to rely on the full CCSDT energies, reported in Ref. 88, to assess
the performance of the MMCC(2,3) / PT approach.

In this case, the CCSDT energies can serve as the reference values, since it is well
known that the full CCSDT approach provides an excellent description of a single
a-bond breaking (the previously discussed case of the HF molecule was an illustration
of this statement). As one can see, the CCSD approach provides very poor results at
all values of the F—F distance RF_F, even at the equilibrium geometry Re, where the

difl’erence between the CCSD and CCSDT energies is already 9.485 millihartree.

107

£358 :3 v8 aka .3; 5km 55 8.6m 05 «89:2: 88% 2,58 23.. 5
.ww .3 83m 5
.88...va “Tm 8205588 B... mo 28> 823233 85 fl Eon Swoofi N am a

 

 

 

88 was 88.... 83. 3.3. 88 83 82.. 5%.” 985.3822

33 88 84.5 98.8 38 we; 83. as; 88 2388.5

89% 88.81 SET @881 ”2.21 :81 831 was 8:. 53800

3.2. 28.2. 885 88.3. 83. 5.8 :2: 88 33. 800

88.87 88.521 888.87 «838T 888.87 88.87 888%: 388.821 2 .880

82 .5. .8 .8 55.5.3 £3 .52 um 553 8852
483 8

83» 88 83mm: EQmUO :3 on... 3 95.33 688558 E .357: much» 5:38.898: BB @3928 BQmOO magnoamoboo
one o... @388.“ 8.58458 3 89893 08 83.88 33888 23. .858: 5 83m 88 83.88 8.8... BQmUO 23. $8: 8.8?
233.8 m mo mummaoaaoo 86080 SE. .888“ ES— 883 Edufiuo 95 «830— wfi £83830? mmmémoa =8 5 .88 28A
NQ>98 8 8:3 38308 «m 2: mo momuumaomw Bow 8 8m 3830 83.88 088653» 00 £535 mo a8m8aaoo < ”v @588

108

The CCSD results become even worse for larger values of Rpup. In fact, even the
relatively small stretches of the F—F bond, such as R1221: = 1.5Rc, lead to very large,
> 30 millihartree errors in the CCSD energies relative to full CCSDT. The failure of
the CCSD approach illustrates the important role played by the triply excited clusters
in describing the F2 molecule. The CCSD(T) method is very successful at describing
the effects due to triply excited clusters at the equilibrium geometry, reducing the
large error in the CCSD result relative to full CCSDT to 0.248 millihartree but,
unfortunately, the CCSD(T) approach completely fails at larger F—F distances, where
the unsigned errors in the CCSD(T) energies become as large as 39.348 millihartree
at REL]? = 5R...

The MRMBPT-corrected MMCC theory, explored in this dissertation, dramati-
cally improve the CCSD and CCSD(T) results. As shown in Table 4, the
MMCC(2,3)/ PT method employing the small active space described above reduces
the 9.485 millihartree error in the CCSD energy relative to CCSDT at Rp.p = Re
to 3.725 millihartree. In contrast to CCSD(T), the MMCC(2,3) / PT method remain
accurate in the bond breaking region. It is capable of producing the results of nearly
CCSDT quality along the entire potential energy curve of F2 from RIP-F = 0.75126 to
RIP-F = 511;. The largest error relative to CCSDT characterizing the MMCC(2,3) / PT
calculations along the whole curve is 5.618 millihartree. Even more striking is the fact
that the NPE value characterizing the MMCC(2,3) / PT energies (calculated relative
to the full CCSDT results) is only 2.698 millihartree, demonstrating the ability of the
relatively inexpensive MMCC(2,3) / PT approach to produce potential energy curves
that accurately mimic the full CCSDT potential curve. The MMCC (2,3) / PT methods

109

tumult. f A. .A- .'..

seem to perform better in this regard than the CR—CCSD(T) which work reasonably
well at larger F-F separations, but are characterized by somewhat larger NPE values.
Only the recently developed size extensive modification of CR-CCSD(T), termed
CRCC(2,3), shows the NPE error similar to that observed in the MMCC(2,3)/PT

calculations.98

5.3 Double Dissociation in H20
5.3.1 The DZ basis set

Similar improvements in the relatively poor CCSD and CCSD(T) results are
observed when the MMCC(2,3) / PT approach is applied to the simultaneous dissoci-
ation of both O—H bonds in H20. As explained in Ref. 86 (cf., also, Ref. 251), in this
case, a reasonable choice of active orbitals, which is needed to obtain a fairly uniform
description of the equilibrium and bond breaking regions, is provided by the lbl, 3a1,
lbg, 401, 2b1, and 2b2 orbitals. We used these orbitals to determine the MRMBPT-like
wavefunctions that enter the MMCC / PT corrections to CCSD energies. Since double
bond dissociations are characterized by significant quadruple effects, in addition to
large effects due to triples, we also performed the MMCC(2,4) / PT calculations to ex-
plore the effects of quadruples. The simultaneous stretching or breaking of both O—H
bonds in water provides us with an example of a situation, where both T3 and their
T4 counterparts are sizable and difficult to describe with the approximate CCSDT
and CCSDTQ approaches.

As shown in Table 5, the MMCC(2,3) / PT method employing a small set of the

110

Table 5: A comparison of the CC and MMCC ground-state energies with the corre-
sponding full CI results obtained for the equilibrium and two displaced geometries of
the H20 molecule with the DZ basis set. The'full CI total energiw are given in hartree.
The remaining energies are reported in millihartree relative to the corresponding full
CI energy values.

 

 

Method R,a 1.512,.b 2R.b
51111 CI —76.157866“ —76.014521" 45.905247b
CCSD 1.790 5.590 9.333
CCSDTc 0.434 1.473 —2.211
CCSD(T)d 0.574 1.465 -—7.699
CR-CCSD(T)d 0.738 2.534 1.830
MMCC(2,3)/CI“ 0.811 2.407 1.631
MMCC(2,3)/PTf 1.265 2.174 0.335
CCSDTQK 0.015 0.141 0.108
CCSD(TQ)“ 0.166 0.094 —5.914
CR-CCSD(TQ),ad’h 0.195 0.905 1.461
CR—CCSD(TQ),b‘ 0.195 0.836 2.853
MMCC(2,4)/CI“ 0.501 0.942 2.416
MMCC(2,4)/PTf 1.069 0.380 —1.815

 

“ The equilibrium geometry and full CI result from Ref. 268.

b The geometry and full CI result from Ref. 53.

c From Ref. 77.

d H'om Ref. 18.

° From Ref. 251.

f The active space consisted of the 1b1, 301, lb;, 401, 2b1, and 252 orbitals.

3 From Ref. 80.

h The “a” variant of the completely renormalized CCSD(TQ) method. The results
are from Ref. 25.

i The “b” variant of the completely renormalized CCSD(TQ) method. The results
are from Ref. 25.

111

lbl, 3a], 152, 401, flu, and 2b: active orbitals reduces the 9.333 and 7.699 milli-
hartree unsigned errors in the CCSD and CCSD(T) results at R041 = 2R, (R03
is the O—H distance and Re is the equilibrium value of 30-11) to 0.335 millihartree.
The overall description of the simultaneous stretching of both O—H bonds in H20
by the MMCC(2,3)/ PT approach, which produces the relatively small errors that
do not exceed 2.2 millihartree in the entire R0}; = Re — 2R9 region, is very good.
The CR—CCSD(T) and MMCC(2,3)/CI methods (the active orbital space used in
the MMCC(2,3)/CI calculations, which were originally reported in Ref. 251, was the
same as that used in the present MMCC( 2,3) / PT calculations) provide similar results.
We also examined the effect of quadruples on the MMCC / PT results by considering
higher-order MMCC corrections to CCSD energies employing the selected triples and
quadruples contributions relative to the RHF determinant |<I>) that originate from the
low-order MRMBPT wavefunctions |\II).MRMBPT)), Eq. (86). The MMCC(2,4)/PT
method reduces the error of MMCC(2,3)/PT at R0.“ = 1.5R,a further from 2.174
to 0.380 millihartree. However, the MMCC(2,4) / PT overestimates the ground-state
energies of the H20 molecule at R0}; = 2R, by 1.815 millihartree. We believe this is
due to the simplified version of the MRMBPT theory used in this work. Clearly, it is
encouraging to observe that the inexpensive MMCC calculations, in which the simple
MRMBPT-like wavefunctions IQLMRMBPTB, Eq. (86), truncated at triple excitations
relative to the RHF determinant |<I>), are inserted into the MMCC(2,3) energy cor-
rections, provide a much better overall description of bond breaking in H20 than the

standard CCSD and CCSD(T) approaches.

112

Table 6: A comparison of various CC ground-state energies obtained for the H20
molecule, as described by the cc-pVDZ basis set, at the equilibrium geometry and
several non-equilibrium geometries obtained by stretching the O—H bonds, while keep-
ing the H-O—H angle fixed. The spherical components of the d orbitals were used.
In all post-RHF calculations, all electrons were correlated. The full CI total energies
are given in hartree. The remaining energies are reported in millihartree relative to
the corresponding full CI energies.

 

 

Method Re“ 1.517.e 2R... 2.5R,z 3Re
1:111chb -76.241860 —76.072348 -75.951665 -75.917991 —75.911946
CCSD 3.744 10.043 22.032 20.307 10.849
CCSDTb 0.493 1.423 —1.405 —24.752 —40.126
CCSD(T)b 0.658 1.631 —3.820 -42.564 —90.512
CR-CCSD(T)° 1.025 3.355 7.252 -2.270 —15.040
MMCC(2,3)/PTd 2.780 3.329 4.251 —7.607 —21.456

 

‘ The equilibrium value of the O—H distance Re equals 1.84345 bohr and the H-
O-H bond angle is fixed at 110.6°. For further details of the equilibrium and non-
equilibrium geometries used in this work, see Ref. 9.

b Horn Ref. 9.

c From Ref. 95.

d The active space consisted of the lbl, 102, 3m, 401, 2b2, 5m, and 3b2 orbitals.

5.3.2 The cc—pVDZ basis set

To explore the effect of basis set as well as to examine the performance of
MRMBPT-corrected MMCC theory in a somewhat more complete scan of the poten-
tial energy surface of the doubly dissociating H20 molecule, we tested our methods
on the H20 system as described by the cc-pVDZ basis set.

Though the CCSDT method provides an excellent description of the equilibrium
region, producing an error relative to full CI of only 0.493 millihartree at R041 = Be,
it completely fails at larger O—H separations (R041 > 217.), where the errors in the

CCSDT energies grow up to 40.126 millihartree at R0.“ = 3R... These results Show

113

that even the full inclusion of T3 clusters is not sufficient to guarantee the proper
description of the double dissociation of water if we go to very large stretches of both
O—H bonds. As both O—H bonds in H20 are simultaneously stretched, the effects of
triples as well as quadruples (T4 clusters) become very important due to a significant
increase of a multi-reference character of the ground-state electronic wavefunction.
The MMCC(2,3) /PT approach provides a very good description of the O—H stretches
up to R041 = 217,, reducing the 10.043 and 22.032 millihartree errors in the CCSD
results at R041 = 1.5122 and R0.“ = 2R,” respectively, to 3.329 and 4.251 millihartree.
The errors in the MMCC(2,3) / PT results increase as we enter the Ron > 2Re region,
where quadruples (neglected in the MMCC(2,3)/ PT calculations) become important,
but it is more it is also interesting to compare the errors relative to full CI produced by
the MMCC(2,3)/ PT approach with the errors produced by the full CCSDT method.
As one can see in Table 6, the use of the MRMBPT-like wavefunction in determining
the ground-state triples correction 60(2, 3) of the MMCC(2,3) / PT approach seems to

reduce the degree of the failure of the CCSDT method in the R0.“ > 217.. region.

5.4 Excited States of CH+

One of the main advantages of the MMCC / PT formalism is that we can study
ground as well as excited states. In principle, for a given M dimensional model space
91%, we can calculate up to M different MRMBPT wavefunctions I‘ll-194mm“), Eqs.
(86) or (87), which represent the approximate forms of ground and excited states III")

that enter the MMCC (e.g., MMCC(2,3)) corrections to CC and EOMCC (e.g., CCSD

114

and EOMCCSD) energies. If we have an a priori knowledge about the dominant or-
bital excitations that define the excited states of interest (and the leading EOMCCSD
amplitudes r; and r3, may help us in this regard), we can use the corresponding or-
bitals as active orbitals for the MRMBPT and subsequent MMCC / PT calculations.
There is, of course, an issue of matching the MRMBPT wavefunctions IQLMRMBPT»,
Eq. (86) or (87 ), with the corresponding EOMCC states Ewe TM) |<I>) (in the case of
the MMCC(2,3) / PT calculations, the EOMCCSD states RLCCSD)6T(CCSD)I<I>)), so that
we read the correct wavefunctions I‘IILMRMBPT)) into the MMCC corrections 6“), but
this issue can be resolved by constructing, for example, the overlaps of all zero-order
states |Q),”), Eq. (78), obtained by diagonalizing the Hamiltonian in the model
space lo, with all EOMCC states R(A)e T‘ )|<I>) of interest. In most cases, only
one specific zero-order state |Q),”) gives a large overlap with a given EOMCC state
RM) eTM)|<I>). If there are two or more states |\II).P)) that form similar overlaps with
a given EOMCC state Rune T‘ )|<I>), we can calculate the more complete MRMBPT
wavefunctions ITLMRMBPTU, Eq. (86) or (87), that correspond to these zero-order
states |\I').P)) and search for the I‘IILMRMBPT)) state that gives the maximum overlap
with a given EOMCC state Ewe TM)|<I>). The resulting overlap enters the MMCC
correction 6):” anyway (see, e.g., the overlap denominator (\IILMRMBPT)|R£A)6T(A)|<I>)
in Eq. (26)), and we can see now that the same denominator serves as a very im-
portant diagnostic for matching the EOMCC states RWe T”) |<I>) and the MRMBPT

wavefunctions IQLMRMBPT» for the purpose of determining the corresponding energy

corrections 6,94). In the specific case of the MMCC(2,3) / PT approach tested in this

115

88538 3. E8 .5 m an; .5 m 55 6m 23 mo wEEmnoo 88% 83.88 25. .

 

 

 

116

 

 

.RH 88 see .

.8 8: 85H .888 35802838 58 8 .88.. B 2:. 5

.8 8m .88 .

.NNH 8m 88 2

.2 .88 58 88 583 .28. 88 £2 .8 £88 8 8.. 2H... H58 8“ 88 58 88 83 .m 8 8:85 8 H8 58. .
EST I 82. 8.8 2.3 881 82. EHH 83. S H
83. 8:7 58.? EST 22. 8.8 82 88.... H: N
28 83 H88 8.3 H88 ~86 82 82. E H
88.? H85: 5:; 8o... 83 88 ~88 H88 +8 8
E; 83: ”Hod :3. ”8... 83.: 83 83.. +8 m

.8 n 85m
83. .85 82. 83 33 88 :2 8.... 9 H
8%: SET 88.8.. 83 8H; 83 82 88 H: m
8H; 83 was 83. H85 H88 83 25 E H
58...: 83: 83 83 HHS. 83 83 Hand +8 ...
8.8: 83 8H; 83 88 82. 88 8.8 +9 8
..me u 85m
83. 83 H86 H88 E; 2.3. 82. 83 8% S H
:21 83: 83. BS 23 82. 22 82 81: H: m
:2. £3 28 83. SH; 83: 2.3 H88 83 H: H
H83 H86 H85: 83 :3 H88 22. 83 4.8.2 +9 m
88 8:. 82H 83 :3 E; 82 88 88.8 +2 8
5m .1. 80.:
585.3 5858 885.3 .8283 53882088 958028 .80 nmoozom .8 :8 88m
0022

 

.an mgmfid 2 +m0 5 ox £32 «:82 533238 25. .>8 E
8.8 .8888 :< ...m @8884? we 288 $298 fié 2: 3 A +NH m +NH NV 888 959% 2: Sod 43.88 non—388 8282, 23
2 9.88 \r 2 23. .338.“ HO :3 2: Bob 82.83% 23 88 83.8, .850 :< .8888 285388 25 8.8 828, HO :3 2S.
.338“ HO :3 wfivqommotoo 23 :23 8mm 98 5mm; .88888m 8888a 95 «28 um D8898 82.5538 2: 8 .8m 225
382388 58 .3 8888 8 as :8 58 8 8888 8:888 888 0022 H8 00 89.8, 8 588558 .4 g 538.

work, we matched the MRMBPT wavefunctions NILMRMBPT’) with the EOMCCSD

(CCSD)eT(CCSD)

states Ru it”),

|<I>) by analyzing first the overlaps of the zero-order states |\II
Eq. (78), with the EOMCCSD wavefunctions of interest. If this was not sufficient
for determining the matching pairs of the MRMBPT and EOMCCSD states, we
calculated the complete overlap denominators (\IILMRMBPT) IRLCCSD)6T(CCSD) |Q), which
we need to determine the MMCC(2,3) / PT corrections to EOMCCSD energies anyway
(see the denominators D" in Eqs. (93) and (96)).

We illustrate the performance of the MRMBPT-corrected MMCC theory in excited-
state calculations by analyzing the results of benchmark MMCC(2,3) / PT and
MMCC(2,4) / PT calculations for the valence excited states of the CH+ ion (see Table
7). We compare the MMCC(2,3)/PT and MMCC(2,4)/PT results for a few low—
lying excited states of CH+ of the 153“, 1II, and 1A symmetries, obtained with the
[533p1d/ 331p] basis set described in Ref. 269 and the ground-state RHF orbitals,
with the results of the corresponding full CI calculations reported in Refs. 13, 269.
Along with the MMCC / PT and full CI data, we show the EOMCCSD and full EOM-
CCSDT results (the latter ones obtained in Ref. 24) and the results obtained with
the perturbative triples response CC3 model.122 In addition to the equilibrium ge-
ometry RC-“ = Re (RC4; is the C—H separation and R.i is the equilibrium value of
Ron), we consider two stretched geometries of CH+, so that we can see how good
the MMCC / PT theory can be in calculations of excited—state potential energy curves
along bond breaking coordinates. We also compare the MMCC / PT results with the
results of the MMCC/CI calculations reported in Refs. 126, 127. In calculating the
MRMBPT wavefunctions IQLMRMBW)), Eq. (86), that enter the MMCC(2,3) / PT and

117

 

MMCC(2,4)/PT energy formulas, Eqs. (93) and (94), we used the same set of ac-
tive orbitals as used in the previously reported MMCC(2,3) / CI and MMCC(2,4)/CI
calculationsm’127 Thus, the active space employed in the calculations for CH+ con-
sisted of the highest-energ' occupied orbital, 3a, and the three lowest-energy unoc-
cupied orbitals, 171', 5 hr, lvry E 211’, and 40. This choice of active space reflects
the nature of orbital excitations defining the valence excited states of CH+ shown
in Table 7 and the nature of the bond breaking in this system. (see Refs. 124—126
for details). Finally, we compare the results of the MMCC(2,3) / PT calculations
with the CR—EOMCCSD(T) results reported in Ref. 22. Let us recall that the CR-
EOMCCSD(T) method is a “black-box” variant of the MMCC(2,3) approximation,
in which we do not have to select active orbitals to determine the triples corrections
to CCSD/EOMCCSD energies derived from the general MMCC formalism.

We begin our discussion with the vertical excitation energies at the equilibrium
geometry. In this case, the doubly excited nature of the first-excited 12+ (2 1E“)
state and the lowest-energy 1A (1 1A) state, and the partially biexcited character of
the second 1II (2 1II) state (cf. Refs. 111, 124—127, 269) cause significant problems for
the EOMCCSD approach. The errors in the EOMCCSD excitation energies for these
three states, relative to the corresponding full CI values, are 0.560, 0.924, and 0.327
eV, respectively. The conventional linear response CC approach to triple excitations

19121-123 reduces these large errors

via the CC3 method of Jorgensen and co-workers
to 0219—0318 eV,122 which is a significant improvement, but if we want to obtain
errors which are much than 0.1 eV with the standard EOMCC methodology, we must
use the full EOMCCSDT approach (or its active—space EOMCCSDt variant125'24).

118

F

 

The full EOMCCSDT approach reduces the relatively large unsigned errors in the
EOMCCSD results for the 2 12+, 1 1A, and 2 1II states to 0.074, 0.040, and 0.060
eV, respectively.

As shown in Table 7, the CR—EOMCCSD(T), MMCC(2,3)/CI, and
MMCC(2,3) / PT methods, which represent three different flavors of the
MMCC(2,3) theory and which are all much less expensive than the iterative CC3 and
EOMCCSDT approaches, are capable of providing the results of near—EOMCCSDT
quality. Indeed, the errors in the vertical excitation energies calculated at Ron =11:3
for the 2 123+, 1 1A, and 2 1II states of CH+, which have significant double excitation
components, obtained with the noniterative CR—EOMCCSD(T), MMCC(2,3)/CI,
MMCC(2,3)/PT approximations, are 0.084—0.117 eV for the 2 12"” state, 0.027—
0.090 eV for the 1 123 state, and 0105—0176 eV for the 2 1II state. This should
be compared to the 0.560, 0.924, and 0.327 eV errors, respectively, in the EOM-
CCSD results and the 0.230, 0.318, and 0.219 eV errors, respectively, obtained with
the CC3 method. For the remaining two states shown in Table 7 (the third 12+
state and the lowest-energy ll'I state), which at Ron = R, are dominated by sin-
gle excitations,“hurl?“269 the errors in the CR—EOMCCSD(T), MMCC(2,3)/CI,
and MMCC(2,3)/PT results are 0.000—0.051 eV and 0007—0015 eV, respectively.
In this case, the CR—EOMCCSD(T), MMCC(2,3)/CI, and MMCC(2,3)/PT meth-
ods provide the results of the CC3 or near-EOMCCSDT quality. The standard
EOMCC methods, including the basic EOMCCSD approximation, have no trou-
bles with describing excited states dominated by one-electron transitions, but it is

encouraging to observe that even in this case all three MMCC(2,3) methods, includ-

119

ing the MMCC(2,3) / PT approximation developed in this work, improve the EOM-
CCSD results. The MMCC(2,4)/ PT approach provides small improvements in the
MMCC(2,3) / PT results for the 212+ and 1A states, while keeping the high accuracy
of the MMCC(2,3) / PT calculations for the remaining states.

The very good performance of the MMCC(2,3) / PT and other MMCC(2,3) meth-
ods is not limited to vertical excitation energies at the equilibrium geometry. As
shown in Table 7, the CR—EOMCCSD(T), MMCC(2,3)/CI, and MMCC(2,3)/PT
approaches are capable of providing a highly accurate description of excited-state po-
tentials of CH+ at larger values of Ron, where all excited states listed in Table 7 gain
a considerable multi-reference characterm—m. The very large (even ~ 1 eV) errors
in the EOMCCSD results for the excited-state potential energy curves of CH+, rel-
ative to the corresponding full CI potentials, are reduced in the CR—EOMCCSD(T),
MMCC(2,3)/CI, and MMCC(2,3)/PT calculations to 0.1 eV or less. Indeed, the
errors in the EOMCCSD excitation energies, relative to full CI, for the two lowest
excited states of the 123+ symmetry, the two lowest 111 states, and the lowest 1A state
are 0.668, 0.124, 0.109, 0.564, and 1.114, respectively, at RC4; = 1.5R¢, and 0.299,
0.532, 0.234, 0.467, and 1.178 eV, respectively, at RC-“ = 2B,. The MMCC(2,3)/ PT
method proposed in this work reduces these large unsigned errors to 0.102, 0.053,
0.083, 0.022, and 0.085 eV, respectively, at R041 = 1.5R¢, and 0.079, 0.021, 0.133,
0.123, and 0.005 eV, respectively, at RC_H = 2R... As in the RC4; = R. case, the only
standard EOMCC approach that can provide the results of similar quality is the very
expensive full EOMCCSDT method (cf. Table 7). The MMCC(2,4) / PT approach

provides further improvements in a few cases or does not change the already very

120

good MMCC(2,3)/PT energies.

The results in Table 7 show that all three MMCC(2,3) approximations, including
CR-EOMCCSD(T), MMCC(2,3) / CI, and MMCC(2,3) / PT, provide similar improve-
ments in the EOMCCSD energies. The improvements are particularly substantial
for excited states dominated by doubles and for excited-state potentials at stretched
internuclear geometries, where all excited states of CH+ gain a significant multi-
reference character. It is interesting to learn that the basic and relatively simple
MMCC( 2,3) approximation is capable of providing considerable improvements in the
EOMCCSD results, independent of the form of the wavefunction |\II,,) used in the
MMCC(2,3) correction formula. The CR—EOMCCSD(T) approach uses the pertur-
bative, EOMCCSDT-like, wavefunctions I‘ll"), the MMCC(2,3)/CI method uses the
MRCI-like wave functions I‘ll“), and the MMCC(2,3)/ PT scheme proposed in this
work uses the MRMBPT-like wavefunctions |\II,,) defined by Eqs. (86) or (87), and
yet the resulting MMCC(2,3) excitation energies for CH+ are very similar. This
demonstrates the robustness of the MMCC formalism, which is capable of improving
the results of conventional CC and EOMCC calculations independent of the specific
form of |\II,,) used to calculate the energy corrections 6,94) or JLCCSD). Rom the point
of view of this work, it is important that we can considerably improve the CCSD and
EOMCCSD results using low-order MRMBPT-like wavefunctions |‘Il,,) defined by
Eq. (86). As explained earlier, the MMCC(2,3) / PT is much less expensive than the
MMCC(2,3) /CI approach, since we do not have to solve iterative MRCI-like equations
to obtain the wavefunctions I‘ll”) that enter the MMCC(2,3) / PT energy expression.

Also, if the active space is small, the MMCC(2,3) / PT method is less expensive than

121

the CR-EOMCCSD(T) approach, since the summation over i < j < k,a < b < c in
Eq. (93) includes only the selected types of triple excitations relative to |<I>) that are
included in the MRMBPT wavefrmctions IQLMRMBPTU, Eqs. (86) or (87). Finally,
it is worth noticing that there is no apparent need to reoptimize orbitals to obtain
an accurate description of excited states of CH+ in the MMCC(2,3) / PT calcula-
tions (the ordinary RHF orbitals seem to suffice), although it would be interesting to
examine the role of orbital optimization in MRMBPT calculations that precede the
MMCC(2,3) / PT calculations. The MMCC(2,4) / PT method improves the description
of the CH’r system somewhat, but not to a degree that would favor this approach

over the less expensive MMCC(2,3) / PT approximation.

122

6 Summary, Concluding Remarks, and Future Per-

spectives

In this dissertation, a new electronic structure theory, termed MMCC/ PT, which
combines the CC / EOMCC method with low-order multi-reference perturbation the-
ory, has been developed. The MMCC / PT approaches have been formulated using
diagrammatic methods. The MMCC(2,3)/PT and MMCC(2,4)/PT methods have
been efficiently implemented using the idea of diagram factorization. The perfor-
mance of the basic MMCC / PT approximations has been illustrated by the results of
test calculations for the bond breaking in HF, F2, and H20, and the excited states of
CH+. The test calculations show that the MMCC(2,3) / PT and MMCC(2,4) / PT ap-
proaches provide a good description of bond breaking and excited states dominated by
doubles, eliminating the failures of the conventional CC / EOMCC methods at larger
internuclear distances and for excited states dominated by two—electron transitions
without invoking expensive steps of high-order CC / EOMCC methods

Clearly, several issues need further study. The role of different choices of active
orbitals should be examined. The majority of multi-reference calculations are per-
formed with the orbitals optimized at the CASSCF level. In this work, we have only
used the ground-state RHF orbitals. It would be also interesting to examine different
types of the MRMBPT wavefunctions that enter the MMCC / PT corrections. In this
work, we have tested a simplified MRMBPT-like scheme based on the partitioning of
the Hamiltonian into the model-space (P-space) and Q-space components (the latter

component originates from single and double excitations from a multi-dimensional

123

. r433: _I-

 

model space). The majority of contemporary MRMBPT calculations are performed
with schemes, such as CASPT2, MC-QDPT, or MRMP2, mentioned in the Introduc-
tion. Several other low-order MRMBPT have been proposed in literature as discussed
in the Chapter 1. It would be useful to test how the conclusions of this work depend
on the particular form of the MRMBPT theory used to provide wavefunctions |\II,,)

for MMCC calculations.

124

 

 

 

Appendices

125

Appendix A. An Introduction to the Diagrammatic Methods of
Many-Body Theory

To introduce the diagrammatic methods of many-body theory, we begin our con-
siderations with selecting a Fermi vacuum state |<I>), which is a single-determinantal
state that is typically chosen to provide a reasonable approximation to the ground
electronic state of a given many-fermion system of interest. The Fermi vacuum state
(reference state) is diagrammatically represented by an empty space. All other Slater
determinants are represented with oriented lines, pointing either upward for parti-
cle states (spin-orbitals unoccupied in the reference |<I>)) or downward for hole states
(spin-orbitals occupied in the reference |<I>)), with labels associated with the corre-
sponding spin-orbitals excitations relative to Iq)), and outgoing or incoming into a
simple vertex.

We represent the second—quantized operators entering a given operator product
with basic diagrams. In the design of these diagrams, we use vertices with incoming
lines representing annihilation operators and outgoing lines representing the creation
operators. Each basic vertex contains information about matrix elements in a spin—
orbital basis defining the operator. For instance, the Slater determinants |<I>;') =

[a“a,]|0) are represented by

and (@fl = (0|(aaa,)l = (0|a‘aa are represented by

126

W. l . 1"“: ifI-fim m.”

 

OJ

We assign difierent vertices to represent different operators. Since the one-body
operator brings a pair of creation and annihilation operators, its diagrammatic rep—
resentation contains only two oriented lines. Likewise, the diagram representing the
two-body operator, which brings two pairs of creation and annihilation operators,
contains four oriented lines, and so on and so forth.

There are several diagrammatic representations; the most popular are the Hugen-
holtz and the Goldstone representations. The Hugenholtz representation employs
the antisymmetrized matrix elements in the second-quantized definitions of operators
(e.g., 1);; = (pqlvlrs) — (pq|v|sr) for the two-body operator VN) while the Goldstone
representation is based on the second-quantized form of operators that uses non-
symmetric matrix elements (e.g. (pq|v|rs) in the case of VN). In our analysis, we use
the Hugenholtz representation, since this approach produces fewer distinct resulting
diagrams than the Goldstone representation. The Goldstone representation is useful
in developing spin-adapted formalisms for spin-free Hamiltonians.

The basic diagrams (in the Hugenholtz representation) for the one- and two—body

parts of the electronic Hamiltonian in the normal-ordered form, FN and VN,

FN = 2&1than E f:N[a”aq] (119)

P41

127

p p r
q q s
(a) (b)
a t b i .
\J V
(C) (d)
a I b l .
V 8‘)?
(e) (0

Figure 15: Hugenholtz representation of (a) By; (b) VN; (c) T1; (d) T2; (e) R”; and
(f) Rye.

and

VN = % Z v;N[aPaqa,a,.] E ivgthaqamr] , (120)

p,q,r.s

respectively, and the cluster and excitation operators T1, T2, R“, 1, and Rm are shown
in Figure 15.

The spin-orbitals that are attached to the oriented lines are referred to as free
if they represent summation indices and as fixed otherwise. As shown in Figure 15,
the cluster operators and the linear excitation operators have the same diagrammatic

form, the only difference is with their corresponding vertices. In particular, T1 and

128

Figure 16: Hugenholtz representation of F N.

T2 use unfilled oval vertices, while RM and Rm» use circled dots. A diagram stripped
of the free labels is called a skeleton or, better, the oriented skeleton. The oriented
skeleton determines the weight of the diagram (a combinatorial coefficient that enters
the algebraic expression). The spin—orbital indices p, q, r, s can either be occupied
or unoccupied in the Fermi vacuum. According to standard convention, the indices
i, j, . .. label occupied spin-orbitals, while a, b, . .. refer to unoccupied spin-orbitals in
the Fermi vacuum |<I>). For example, the one-body component of the Hamiltonian

H N can be expressed as

FN = Z f:N[a°ab] + Z f{N[a‘a,] + foNWaa] + Zf;N[a“a,]. (121)

0,6

The corresponding diagrams are shown in Figure 16. The two-body component of
H N, VN, can be partitioned in a similar way, as shown in Figure 17.
After assigning skeletons (or diagrams) to the basic operators, the diagrammatic

calculation proceeds as follows:

1. Represent the operators by appropriate skeletons following the vertical time axis

(as described in Section 4.3), that is, placing the operators from the bottom to

129

 

 

   

b i .
k a a l
+ + + V
i i b c
a i

 

Figure 17: Hugenholtz representation of VN.

the top corresponding to the left to right order of the operators, when they

act on the functions (vectors) in the Fock space. For instance, the operator

%(VNT3) is represented as

 

2. Form all permissible resulting skeletons by connecting the lines in all possible

ways. Then, form all nonequivalent resulting diagrams, in which the oriented

130

 

lines are labeled with their corresponding spin-orbital indices, representing only
the nonvanishing contraction schemes that result when the Wick’s theorem is

applied, that is,

t
i

(a) (b)

l—l m
where (a) a? aq = 6WH(q) and (b) ap a9 = 6m[1—H(q)].

The H (q) step function is equal to 1 for q being a hole and 0 for q representing

a particle and 5m is the usual Kronecker delta.

3. Lastly, assign the algebraic expressions to all nonequivalent permissible result-
ing diagrams. The final expression for the operator product of interest is a sum
of the algebraic expressions corresponding to all nonequivalent resulting dia-
grams allowed by a given many-body formalism. In general, the corresponding
algebraic expression of a given diagram is a product of (a) the weight factor, (b)
the sign factor, (c) the scalar factor, and (for example, in the wavefunction ex-
pressions) (d) the operator part, accompanied by a summation over all relevant

hole and particle indices, if such summations exist in the operators. For the

131

——x

———x

Figure 18: Brandow representation of FN.

connected Hugenholtz diagrams, the weight factor is specified by G)” where
m is the number of pairs of “equivalent” lines. A pair of equivalent lines is
defined as being two lines originating at the same vertex and ending at another,

but same vertex, and going in the same direction. We must remember that the

“m (II. I finickflllm
."
I
i.

identically oriented lines carrying fixed (i.e. not summed) spin-orbital indices
can never be regarded as equivalent lines. The scalar factor is a product of
matrix elements associated with the individual vertices entering the resulting
diagram. The operator part (if it appears in the expression) is a product of the

creation and annihilation operators associated with the uncontracted external

lines. 1

The drawback of the Hugenholtz representation is that Hugenholtz diagrams do

 

not specify the overall sign of the contribution of the diagram. This is due to the fact j
that the basic Hugenholtz diagrams, such as VN or T2, use antisymmetrized matrix E3
elements 2);, ti, etc. In order to determine the sign, it is necessary to draw one
Goldstone representative, called the Brandow diagram, for each Hugenholtz diagram.
This can be done by “expanding” the Hugenholtz vertices. In other words, we replace

all basic Hugenholtz vertices by Brandow vertices (the Brandow representations of

132

Figure 19: Brandow representation of VN.

FN and VN are shown in Figures 18 and 19), while keeping the directions of the lines
and the connectivity intact. Usually more than one possibility exists, since one can
usually draw several Goldstone diagrams for each Hugenholtz diagram,’ but this is
not a problem here: we choose any Goldstone representative of a given Hugenholtz
diagram as a Brandow diagram. The final results do not depend on the choice. Once

the Brandow diagram is drawn, we determine the sign factor for it by counting the

 

number of loops (1) and the number of internal hole lines (h) and by using the sign

formula (—1)'+".

133

Appendix B. The Structure of the Similarity-transformed Hamiltonian
of the CCSD Theory

The H = e‘THeT operator is a quantity of interest for several reasons. Due to
the Campbell-Hausdorff-Baker formula, H can be expanded in terms of commutators
only, which is equivalent to its expansion in terms of the connected diagrams, as in
Eqs. (9) and (12). This means that H is represented by a finite set of diagrams. The
other reason H is worth considering is the fact that it is a quantity that occurs in
various computational schemes based upon the CC wavefunction ansatz.

In order to construct the H(CCSD) diagrams, we follow the diagrammatic rules
discussed in Appendix A. Thus, we contract the Hamiltonian in the normal ordered

(CCSD) .
expansron.

form, H N, with a number of T‘CCSD) operators appearing in the eT
Since Hamiltonians used in chemistry contain at most two-body interactions, H N can
be contracted with at most four cluster operators to produce connected diagrams of
- (CCSD)

N .

The one- and two-body components of the H(CCSD) have similar diagrammatic
forms as the one- and two-body component of the Hamiltonian, H N, the difference
is only with the interaction line, the former uses the wavy interaction line, while the
latter uses the dashed lines, as in Figures 18 and 19.

To illustrate how we use the diagrammatic methods in deriving the explicit alge-

braic expressions for H(CCSD),

H(CCSD) = (H NeT1+T2)c = figapaq + fifigapaqaflr + . . . , (122)

134

w...- _h tr- L
I

 

we show an example on how we obtain the contributing diagrams to 1.2.5:,

which is one of the one-body components of H(CCSD). In this case, our goal is to

generate diagrams of this form from the connected product of H N and eflccsm. The

 

“fig...” 1. .-. .h
d .
I
I

resulting diagrams must contain an oriented line above and below the vertex; both
lines should be directed upward as particle lines a and b. We can expand the H(CCSD)

operator in the following manner:

T(CCSD)

380“” = (Hts )0
= [(FN(1+T1+T2+%T12+...)+

VN(1 + T1+ T2 + %T12 + . . ..)]c (123)

Let us identify which terms in Eq. (123) contributes to It: by forming first the

nonoriented Hugenholtz skeletons. We begin with the first term, FN, which is repre-

 

sented diagrammatically in Figure 16. As shown in Figure 16, the leftmost diagram in
that figure satisfies the above criteria and hence contributes to fig. Next, we consider
the (FNT1)c terms, which produces one resulting nonoriented Hugenholtz skeleton
that contains one line above the vertex as well as one line below the vertex as shown

below:

135

When we analyze other terms in Eq. (123) in the same fashion, we obtain all of the

resulting nonoriented Hugenholtz skeletons of B3, which are

<+W6+a+m

The corresponding oriented Hugenholtz diagrams for it: with their respective weights

-
I
1
4
as
'-
Ii
4
o

are as follows:

a a
+ m b + b + b + b
b a 9 m 9 ha a n 9m
w=1 w=1 w=1 w=U2 “(=1

The corresponding Brandow diagrams for It: with their corresponding sign factors are

shown below

a ----x a ______
-.- , "" """ +
X b + m b + e m b + 6 mb n 9 mb n
a a a .
s=+1 s=-1 s=+l s=-1 s=-1

 

136

These diagrams yield the following algebraic expression for I13:

72: = f: — at? + 22"" t'" — lif” tm" — v‘b tmt" (124)

mac 27717380 mac 0'

All of the resulting diagrams for one-, two-, three, and four—body components of

H(CCSD) are obtained in a similar way. They are shown in Figures 20-23. Figures 24

and 25 show all of the resulting diagrams corresponding to ground-state triply and P

i.
'3!
1
[a
w
v
.I
i
I
I
i ..

quadruply excited moments of the CCSD equations, 9.113;,(2) and 5111330,,(2), respec-
tively, which can also be regarded as three- and four-body components of H(CCSD).
Tables 8—11 summarize all the algebraic expressions of the one-, two-, three, and four-

body components of H(CCSD), including 9123:“(2) and ”31:11:19), which are obtained

by reading the diagrams shown in Figures 20—25.

 

137

(a) 13‘:

a a . _b_
= —-—-x + —— + n
e m
b b e m a
_b_ b __.)(

e m n m

a a

(b) 713

—i- i --_-9<
+ 6 m I + e
l i
(

c) 7115

Figure 20: One-body components of F1 (CCSD).

138

= . 7)“ + in

(a) 71'”.

a:

= , "it + {£318

0)) if?

_ a . . "'3‘ .
— -—— + m n + -d-—
0 d a b c m
b

c .___
d

+ m n
a b

(c) 712%

k I i —-.- k
= "‘ + e I f + -.-
l

i -..-
+ e ' f
k I
((1)712?

Figure 21: Two—body components of H(CCSD).

139

 

 

+ sit/"x"; + at}
b i a b
a I

+ Mg“: + “ °

b b a

(f) 713%

Figure 21: continued.

140

l k a , -‘-'x l kVa k l
= I | + __ + ....-
i a k e i m i
a

   

.. a
+ —— i + -- i
. SE ° 8 i
a j k l
a k
—_ “'- I
+ f‘/‘ Vi i§lm + 9Q'“. E 5%
l k e a l
a k __ __
l f l k a

(g) 53:

Figure 21: continued.

 

141

 

 

 

(a)fi”‘

abc

Figure 22: Three-body components of H(CCSD).

142

 

 

wmmAJ. flwfim

 

 

 

 

 

(b) 87.3.:

Figure 22: continued.

 

143

 

(e) 58;:

 

Figure 22: continued.

144

 

 

 

-u: “i
(a) hmite

Figure 23: Four-body components of H(CCSD).

145

 

 

...a,"
r
A

 

 

 

 

 

(b) 72:22;

Figure 23: continued.

146

 

 

 

 

(c) 72525.:

I «_‘vllw4
"I

 

 

 

"dk .t.
(e) hlcbja

Figure 23: continued.

 

147

 

' k b a
l ab c k C] l
+ -- + -- +
m n f e
(30) (3d)

 

b i l a
k C j a k C b i
_ .. + _ _ +
(4c) (4d)

(a)

Figure 24: Diagrammatic representation of 936]:me-
148

W.“ A'i‘dlcrm
3"? _-L.'n-.‘V.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)

Figure 24: continued.

149

 

Vaughn . l -
‘. 5"

 

 

 

 

 

 

 

(12b) (120)
Figure 25: Diagrammatic representation of 931330119).

150

man—.‘..
-‘. '7
" l

Table 8: Explicit algebraic expressions for _the one- and two-body body matrix ele-
ments elements of H (CCSD) (designated by h) shown in Figures 20-21.

 

 

 

Expressiona Figure
11: f,0 + 21mg" 20 (a)
72:; f2+v3‘mt2”- teats—vats:- it." 20 (b)
133 ff + vfgtg' + é-vgtgj + vfifitg‘t’} + fit: 20 (c)
53‘: v2? - vii-t2." 21 (a)
71:39 1;ij + offtf; 21 (b)
712‘}, 123 + %v,°,‘,‘,,t;"b“ — Aobvfjfntf," + vfintftg 21 (c)
hf} 22f} + §vfjftflf + Ak’vfftf, + vfjftft} 21 (d)
’ 31;; 2);; + eggtf, - eggjtgn + vfijtg‘ — eggtgtgn 21 (e)
’25, —f::.t:§.r + v8 + 223th — Aevrmtr + Amati? 21 (f)

+88%" + vfzntrtz: - Aabvzfntiti,” + entree

ecmin cc imn lecmni
—v ttab—Aabv t tb—Evmntab te

mn e mn ac
— 'k . 'k 'k . . . .
hi. ftt’: + v’... - vimtz‘ + A’kvfft: + A’kztztzz" 21 (g)
+8113: 1’}: - Ajkvfttzti + 21,-fat: - eggtgntgt}
+v,’,f,t;"t": + Ajkvfgtgnakfi — fivfitjttg‘

 

° Summation over repeated upper and lower indices is assumed; f: = (pl f Iq) and
v; = (pqlers) — (pqlvlsr) are the one- and two—electron integrals in a molecular
spin-orbital basis {p} corresponding to the Fock operator (f) and the two-body part

of the Hamiltonian (2)).

151

V

H... 1.1:;

. ___

Table 9: Explicit algebraic expressions for the three- and four—body body matrix
elements elements of H (CCSD) (designated by h) shown in Figures 22-23.

 

Expression" Figure

 

’43: Ao/bcvnmté‘tm: + Aat/cA‘WS‘mtztitm‘ 22 (a)
_Aa/bcA”Wm ”2385' + tAa/tcvfmtgmtti
+A°b/°A"”¥?mt3tm’ Aobcvifttm: - Aa/tA‘ivtfititE‘
+Au/bcveb ”g; - Aab/cAijvgtg‘

575: -AatA"'/"v{,f,t’;t;"tfi _ Ai/jkvfetlfctjtmt 22 (b)
+AotA‘/j*v,{f,t{’;tgg _ l -A'/""v{,,,t, tat...-
«222222 + ...a/agree;
_AobAtk/jzfiltg‘tg + AabAij/kvfietz; __ Ai/jkvlkitznai

Egg; —A.b,.v;$,tg‘ + A..,.v;$,,tgtm‘ 22 (0)
71:3, A‘j/kvfifitfli + A‘j/kvfitfitg; 22 (d)
72:5: was; — «45122.82: 22 (e)

—A‘itfl,,f,,t,";‘ — A‘jvzfntitm‘

 

5:13,; —Aab/CA‘/j*/‘vfif,nt'gtfl + A..,.A2'/“ef° t3; t'; 23 (a)
—A../.A‘/J"°/'v,f,f,,t‘,t§gt3; — AM/tA‘j/k‘vgfntgtntfi
hm -Aob/c MA” #11:; t3tm‘ + Aw/dAik/ngntgtgg 23 (b)

+Aab/c/dA‘/jkvf,{,,t3t:} 3‘ + Ab/dAik/jvzatgfitfli

h5.5: Aw/evst.t5.'"thf 23 (c)
hf“: A‘k/j'vgfnflgttj, 23 (d)
hfi: —.Aob/.A‘/j’°vf‘;,t§§tg‘ 23 (e)

 

“ Summation over repeated upper and lower indices is assumed; f: = (pl f Iq) and
1);; = (pqlvlrs) - (pqlvlsr) are the one- and two—electron integrals in a molecular
spin-orbital basis {p} corresponding to the Fock operator (f) and the two-body part

of the Hamiltonian (2)).

152

 

Table 10: Explicit algebraic expressions for 931321542) obtained by reading the dia-

grams shown in Figure 24.

 

 

Term Expression“

1 — Ana/.AW" f3. 1:232:
2a — AW. AW" 222,1; t3};

2b Aab/c AW" vii ti:

3a — AWN/1"“ vfgbtg‘ til:

3b — A0,... A‘i" v33“ t; tag"

30 AM. AW" ”5;?“ ti? t2

3d chA‘h'kvfii gt”;

4a AW. 21‘0"“ 22:3,, tg‘ t: 23;;
4b - Aw. 24‘0"“ 12;; t3: tg' t';
4c — Auk/""225; tittitz"
4d Au/bcA‘jkvi‘intitWL‘c"
5a — Aa/bc A‘j" v53" t2: t3:

5b AubCA‘mvsztflf 21";

50 % Ash/c AW" 033.. t3," til:
5d "i-Aob/cA‘H’WSLti’l‘ 15'}
6a 24.2.2122" v2. t5" t5 t: t":
6b AW. Ai/J'" 12:1,, tfg‘ tg‘ t'; t:
7 -— AW. AW“ 223;“ t2}; t3; tgn
88‘ i Aab/c AW" 2:5. til? 1‘33} 2
8b 5 Aw. AW v3“ tgf t3" t}
8c — A... 21”“ v35, t3;1 ti; 2
8d — AW. A‘j" 2251,, t} t2; tg"

 

“ Summation over repeated upper and lower indices is assumed; f; = (pl f Iq) and
v" = (pqlvlrs) - (pq|v|sr) are the one- and two-electron integrals in a molecular

P9

spin-orbital basis {p} corresponding to the Fock operator (f) and the two-body part

of the Hamiltonian (1)).

153

Table 11: Explicit algebraic expressions for 911332;“,(2) obtained by reading the dia-
grams shown in Figure 25.

 

 

Term Ezpressz'on“

9a AW... Ajk/i‘ 223;; t3: :3;

9b AW. A‘j/k‘ v3 t3; t’;f,

9‘3 — Aa/b/od 44'7”” v5.5. t2. t2?

10a 24./W A‘i/kfl v31" t3; t3* t3
10b — AW... A‘i/kfl vgfb tff, t3" t1
10c Aw... A‘k/J'” v3“ t3; t3" tf,
10d — AW.“ AW *1 v3; tff', t5; t3"
11a — Am.” A‘i/W 1,3,5; t3, tg" t3
11b iAob/od AW 035. ti? $152.5
llc -;-A../,,. AW“ v55, ta ta" t’;‘d
123 AM... A‘W" 125,5; tfg‘ tg' t’; t3;
12b AW.” 21:/1*” v35, t3: tiff t1, t3
12c And/b. A‘j/k‘ v35; t3” tg" t3 t’;f,

 

° Summation over repeated upper and lower indices is assumed; f: = (p| f |q) and
2);; = (pqlvlrs) - (pqlvlsr) are the one- and two-electron integrals in a molecular
spin-orbital basis {19} corresponding to the Fock operator (f) and the two-body part
of the Hamiltonian (1)).

154

Appendix C. Factorization of Coupled-Cluster Diagrams

The diagrammatic technique is a very powerful tool, however, if we program the
diagrams one by one, this will give an inefficient and hence impractical computer code,
even in a situation where each of the diagrams is computed at a modest cost. The
purpose of this section is to reformulate the one- and two-body components of H(CCSD)
in such a way that all the common pieces could be factored out, computed only once,
and then substituted in different places. This procedure is possible due to the fact
that coupled-cluster diagrams are, in fact “denominator-less”, which means that the
MBPT denominators are implicitly included in the cluster operator vertices, which
avoids assigning the denominators to each diagram that are normally considered in
MBPT diagrams. This enables us to sum over all internal lines belonging to one T
vertex independent of the other T vertex, which helps the factorization.

In this section, we rederive the expressions for the one- and two-body components
of H(CCSD), FIECCSD) and HéCCSD), respectively, in such a way that only linear terms
with redefined vertices are retained in each equation. Thus, in each nonlinear term,
we retain one t amplitude, usually (but not always) corresponding to the highest
level of excitation, while the remaining amplitudes are absorbed to an efiective vertex.

{0081” and ELECCSD) are used to define the higher-

Ultimately, the factorized forms of H

rank components 0f H(CCSD), such as the three- and four-body components 10030)

and HECCSD), as well as other intermediates entering the CC expressions of interest,

such as 9332,42) and 931:1?“(2L which are naturally expressed via -§CCSD) and

13110030) .

155

u

i
r<:>'

.

Figure 26: Diagrammatic representation of 55;.

-b- -b- n
eQmfi —> .QmA —* A“. —> b "
8 nVa nVa a
A B

C D

Figure 27: Example of diagram factorization.

To illustrate the diagrammatic factorization technique and computational benefits
that it offers, we analyze an example of fig, which is expressed in a diagrammatic form
in Figure 26.

Clearly, diagram A in Figure 26 is a nonlinear contribution. Hence, diagram A is
a good candidate for demonstrating the idea and benefits of factoriZation. The cost
of evaluating this diagram scales as W5 with the system size ./V or 12312:.

Figure 27 presents the factorization of diagram A and its decomposition, for com-

putational efficiency, into independently calculated parts. After factorization, the

original computational cost associated with diagram A, 123,123 is reduced to nonfi (the

u,

156

 

‘ .'.." f '..
1’ ..r .
1

 

cost associated with diagram D) plus ngnfi (the cost associated with the intermediate
C which is defined as diagram B). Furthermore, if we realize that the intermediate
C actually takes one of the forms of chcsn), which may have been calculated al-
ready, we can take advantage of this fact by extending the definition of the above
intermediate C as one of the one-body components of H(CCSD), in this case the 77.3,,

which is defined in part (a) of Figure 20. As a result, diagram D, which has initially

represented a single diagram A in Figure 27, becomes now equivalent to
— — — -X
”I?” = Q l. + 1

D A E
corresponding to diagrams A and E in the original, not factorized, diagrammatic
formulation of 53, as shown in Figure 26. Thus, the fully factorized version of 113, as
shown in Figure 28, uses four diagrams, which are considerably cheaper to calculate
than the original five diagrams shown in Figure 26.

The same factorization and cost reduction procedure can be applied to other com-
ponents of H(CCSD). The computational savings offered by the factorization procedure
in the case of the two-body and other many-body components of H(CCSD) are even
more substantial than in the above example. Figures 29-30 shows the factorized dia-

grams of the one and two-body components of H(CCSD), which can be generated in

a recursive, fully vectorized (linearized) manner.

157

Figure 28: Fully Factorized Diagrammatic Formulation of 72'},

158

u
>:
k
>.
IC:>'

--—x + "O +

(b) 715’.

 

(c) E:

Figure 29: Factorized one-body components of H(CCSD).

159

= 7\ — + $7)

(a) hb‘

at

= 7) + 527)

(b) 715‘,"

5::

+ 7i/
(d) 7153‘

Figure 30: F actorized two-body components of HfCCSD).

160

References

163

" * ‘ I t ' - . .
" r

References

...a

10

ll

l2

l3

14

15

16

17

18

19

E. SCHRODINCER, Ann. Phys. 384, 361, 489, 734 (1926), ibid. 385, 439 (1926).
P. A. M. DIRAC, Proc. R. Soc. London, Ser. A 123, 714 (1929).

C. W. BAUSCHLICHER JR, S. R. LANGHOFF, P. R. TAYLOR, N. C. HANDY,
and P. J . KNOWLES, J. Chem. Phys. 85, 1469 (1986).

C. W. BAUSCHLICHER JR. and P. R. TAYLOR, J. Chem. Phys. 86, 5600 (1987).
C. W. BAUSCHLICHER JR. and P. R. TAYLOR, J. Chem. Phys. 86, 1420 (1987).

C. W. BAUSCHLICHER JR. and S. R. LANGHOFF, Chem. Phys. Lett. 135, 67
(1987).

C. W. BAUSCHLICHER JR. and P. R. TAYLOR, J. Chem. Phys. 85, 2779 (1986).
T. VAN MOURIK and J. H. VAN LENTHE, J. Chem. Phys. 102, 7479 (1995).

J. OLSEN, P. JORCENSEN, H. KOCH, A. BALKOVA, and R. J. BARTLETT, J.
Chem. Phys. 104, 8007 (1996).

O. CHRISTIANSEN, H. KOCH, P. JORCENSEN, and J. OLSEN, Chem. Phys. Lett.
256, 185 (1996).

A. I. KRYLov, C. D. SHERRJLL, E. F. C. BYRD, and M. HEAD-GORDON, J.
Chem. Phys. 109, 10669 (1998).

H. LARSEN, J. OLSEN, P. JORCENSEN, and O. CHRISTIANSEN, J. Chem. Phys.
113, 6677 (2000), ibid. 114, 10985 (2001) [Erratum].

A. I. KRYLov, C. D. SHERRILL, and M. HEAD-GORDON, J. Chem. Phys. 113,
6509 (2000).

A. DUTTA and C. D. SHERRILL, J. Chem. Phys. 118, 1610 (2003).

J. M. JUNQUERA-HERNANDEZ, J. SANCHEZ-MARI’N, G. L. BENDAZZOLI, and
S. EVANGELISTI, J. Chem. Phys. 120, 8405 (2004).

J. M. JUNQUERA-HERNANDEZ, J. SANCHEZ-MARI’N, G. L. BENDAZZOLI, and
S. EVANCELISTI, J. Chem. Phys. 121, 7103 (2004).

M. L. ABRAMS and C. D. SHERRILL, J. Chem. Phys. 121, 9211 (2004).
K. KOWALSKI and P. PIECUCH, J. Chem. Phys. 113, 18 (2000).
K. KOWALSKI and P. PIECUCH, J. Chem. Phys. 113, 5644 (2000).

M. J. MCGUIRE, K. KOWALSKI, and P. PIECUCH, J. Chem. Phys. 117, 3617
(2002).

164

 

\'..I

21

22

23

24

26

27

28

31

32

35

36

37

38

P. PIECUCH, K. KOWALSKI, I. S. O. PIMIENTA, and S. A. KUCHARSKI, in
Low-Lying Potential Energy Surfaces, edited by M. R. HOFFMANN and K. G.
DYALL, volume 828, pp. 31—64, American Chemical Society, Washington, DC,
2002.

K. KOWALSKI and P. PIECUCH, J. Chem. Phys. 120, 1715 (2004).
C. D. SHERRILL and P. PIECUCH, J. Chem. Phys. 122, 124104 (2005).
K. KOWALSKI and P. PIECUCH, Chem. Phys. Lett. 347, 237 (2001).

P. PIECUCH, K. KOWALSKI, I. S. O. PIMIENTA, and M. J. MCGUIRE, Int.
Rev. Phys. Chem. 21, 527 (2002).

P. PIECUCH, K. KOWALSKI, I. S. O. PIMIENTA, P.-D. FAN, M. LODRIGUITO,
M. J. MCGUIRE, S. A. KUCHARSKI, T. KUS, and M. MUSIAL, Theor. Chem.
Ace. 112, 349 (2004).

K. B. GHOSE, P. PIECUCH, and L. ADAMOWICz, J. Chem. Phys. 103, 9331
(1995).

S. EVANCELISTI, G. L. BENDAZZOLI, R. ANSALONI, F. DURI, and E. ROSSI,
Chem. Phys. Lett. 252, 437 (1996).

E. ROSSI, G. L. BENDAZZOLI, and S. EVANGELISTI, J. Comput. Chem. 19, 658
(1998).

R. ANSALONI, G. L. BENDAZZOLI, S. EVANGELISTI, and E. ROSSI, Comput.
Phys. Commun. 128, 496 (2000).

M. KLENE, M. A. ROBB, M. J. FRISCH, and P. CELANI, J. Chem. Phys. 113,
5653 (2000).

Z. GAN, Y. ALEXEEV, M. S. GORDON, and R. A. KENDALL, J. Chem. Phys.
119, 47 (2003).

Z. GAN and R. J. HARRISON, Calibrating quantum chemistry: A multi-terafiop,

parallel-vector, full-configuration interaction program for the Cray-X1, in SC ’05:

Proceedings of the 2005 ACM/IEEE conference on Supercomputing, p. 22,1EEE
Computer Society, Washington, D. C, 2005.

D. R. HAR’I‘REE, Proc. Cambridge Philos. Soc. 24, 89 (1928), ibid. 24, 111
(1928), ibid. 24,426 (1928).

D. R. HARTREE, W. HARTREE, and B. SWIRLES, Philos. Trans. R. Soc. London,
Ser. A 238, 229 (1939).

V. FOCK, Z. Physik 61, 126 (1930).
J. C. SLATER, Phys. Rev. 35, 210 (1930).
S. F. BOYS, Proc. R. Soc. London, Ser. A 201, 125 (1950).

165

39

40

41

42

43

44

45

46

47

48

49

51

52

55

57

58

59

60

61

P.-O. LOWDIN, Phys. Rev. 97, 1474 (1955).

P.-O. LOWDIN, Phys. Rev. 97, 1490 (1955).

P.-O. LOWDIN, Phys. Rev. 97, 1509 (1955).

C. MOLLER and M. S. PLESSET, Phys. Rev. 46, 618 (1934).
K. A. BRUECKNER, Phys. Rev. 97, 1353 (1955).

J. GOLDSTONE, Proc. R. Soc. London, Ser. A 239, 267 (1957).
J. HUBBARD, Proc. R. Soc. London, Ser. A 240, 539 (1957).
N. M. HUCENHOLTz, Physica 23, 481 (1957).

J. Chin, J. Chem. Phys. 45, 4256 (1966).

J. CiZEK, Adv. Chem. Phys. 14, 35 (1969).

J. CfZEK and J. PALDUS, Int. J. Quantum Chem. 5, 359 (1971).
F. COESTER, Nucl. Phys. 7, 421 (1958).

F. COESTER and H. KiiMMEL, Nucl. Phys. 17, 477 (1960).

I. SHAVITT, in Methods of Electronic Structure, edited by H. F. SCHAEFER III,
volume 3 Of Modern Theoretical Chemistry, pp. 189-275, Plenum, New York, 1977.

R. J. HARRISON and N. C. HANDY, Chem. Phys. Lett. 95, 386 (1983).
R. J. BARTLETT, Ann. Rev. Phys. Chem. 32, 359 (1981).
D. CREMER and Z. HE, J. Phys. Chem. 100, 6173 (1996).

S. A. KUCHARSKI, J. NOCA, and R. J. BARTLETT, J. Chem. Phys. 90, 7282
(1989).

S. A. KUCHARSKI and R. J. BARTLETT, Chem. Phys. Lett. 237, 264 (1995).

K. RACHAVACHARI, J. A. POPLE, E. S. REPLOGLE, and M. HEAD-GORDON,
J. Phys. Chem. A 94, 5579 (1990).

J. OLSEN, O. CHRISTIANSEN, H. KOCH, and P. JORCENSEN, J. Chem. Phys.
105, 5082 (1996).

P. J. KNOWLES, K. SOMASUNDRAM, N. C. HANDY, and K. HIRAO, Chem.
Phys. Lett. 113, 8 (1985).

W. D. LAIDIC, G. FITZGERALD, and R. J. BARTLETT, Chem. Phys. Lett. 113,
151 (1985).

166

 

 

 

62 P. PIECUCH, A. E. KONDO, V. SPIRKO, and J. PALDUS, J. Chem. Phys. 104,
4716 (1996).

63 J. PALDUS, in Methods In Computational Molecular Physics, edited by S. WIL-
SON and G. H. F. DIERCKSEN, volume 293 of NATO Advanced Study Institute,
Series B: Physics, pp. 99—194, Plenum, New York, 1992.

64 J. GAUSS, in Encyclopedia of Computational Chemistry, edited by P. V. R.
SCHLEYER, N. L. ALLINCER, T. CLARK, J. GASTEICER, P. A. KOLLMAN,
H. F. SCHAEFER III, and P. R. SCHREINER, volume 1, p. 615, Wiley, Chichester,
U.K., 1998.

65 T. D. CRAWFORD and H. F. SCHAEFER 111, Rev. Comp. Chem. 14,33 (2000).

66 J. PALDUS, in Handbook of Molecular Physics and Quantum Chemistry, edited
by S. WILSON, volume 2, pp. 272—313, Wiley, Chichester, 2003.

67 J. PALDUS and X. LI, Adv. Chem. Phys. 110, 1 (1999).

68 P. PIECUCH, K. KOWALSKI, P.-D. FAN, and I. S. O. PIMIENTA, in Advanced
Topics in Theoretical Chemical Physics, edited by J. MARUANI, R. LEFEBVRE,
and E. BRANDAS, volume 12 of Progress in Theoretical Chemistry and Physics,
pp. 119—206, Kluwer, Dordrecht, 2003.

 

6" P. PIECUCH, M. WLOCH, M. D. LODRICUITO, and J. R. GOUR, in Recent Ad-
vances in the Theory of Chemical and Physical Sciences, edited by J.-P. J ULIEN,
J. MARUANI, D. MAYOU, S. WILSON, and G. DELCADO-BARRIO, volume 15
of Progress in Theoretical Chemistry and Physics, p. 45, Springer, Berlin, 2006.

70 P. PIECUCH, M. WLOCH, and A. J. C. VARANDAS, in Progress in Theoretical
Chemistry and Physics, edited by S. LAHMAR, J. MARUANI, S. WILSON, and
G. DELGADO-BARRJO, volume 16, p. 65, Springer, Berlin, 2007.

7‘ I. LINDCREN and D. MUKHERIEE, Phys. Rep. 151, 93 (1987).

72 D. MUKHERJEE and S. PAL, Adv. Quantum. Chem. 20, 291 (1989).

73 P. PIECUCH and K. KOWALSKI, Int. J. Mol. Sci 3, 676 (2002).

74 G. D. PURVIS III and R. J. BARTLETT, J. Chem. Phys. 76, 1910 (1982).

75 G. E. SCUSERIA, A. C. SCHEINER, T. J. LEE, J. E. RICE, and H. F. SHAE—
FER III, J. Chem. Phys. 86, 2881 (1987).

76 P. PIECUCH and J. PALDUS, Int. J. Quantum Chem. 36, 429 (1989).

77 J. NOGA and R. J. BARTLETT, J. Chem. Phys. 86, 7041 (1987), ibid. 89, 3401
(1988) [Erratum].

78 G. E. SCUSERIA and H. F. SCHAEFER III, Chem. Phys. Lett. 152, 382 (1988).
79 S. A. KUCHARSKI and R. J. BARTLETT, Theor. Chim. Acta 80, 387 (1991).

167

 

80

81

82

83

87

89

91

92

93

94

95

97

S. A. KUCHARSKI and R. J. BARTLETT, J. Chem. Phys. 97,4282 (1992).

N. OLIPHANT and L. ADAMOWICZ, J. Chem. Phys. 94, 1229 (1991), ibid. 96,
3739 (1992), Int. Rev. Phys. Chem. 12, 339 (1993).

P. PIECUCH and L. ADAMOWICZ, J. Chem. Phys. 100, 5792 (1994).

K. RACHAVACHARI, G. W. TRUCKS, J. A. POPLE, and M. HEAD—GORDON,
Chem. Phys. Lett. 157, 479 (1989).

M. URBAN, J. NOGA, S. J. COLE, and R. J. BARTLETT, J. Chem. Phys. 83,
4041 (1985).

P. PIECUCH and J. PALDUS, Theor. Chim. Acta 78, 65 (1990).
P. PIECUCH and L. ADAMOWICZ, J. Chem. Phys. 102, 898 (1995).

P. PIECUCH and K. KOWALSKI, in Computational Chemistry: Reviews of Cur-
rent Trends, edited by J. LESZCZYNSKI, volume 5, pp. 1—104, World Scientific,
Singapore, 2000.

K. KOWALSKI and P. PIECUCH, Chem. Phys. Lett. 344, 165 (2001).

P. PIECUCH, S. A. KUCHARSKI, and K. KOWALSKI, Chem. Phys. Lett. 344,
176 (2001).

P. PIECUCH, S. A. KUCHARSKI, V. SPIRKO, and K. KOWALSKI, J. Chem.
Phys. 115, 5796 (2001).

P. PIECUCH, S. A. KUCHARSKI, K. KOWALSKI, and M. MUSIAL, Comp. Phys.
Commun. 149, 71 (2003).

I. S. O. PIMIENTA, K. KOWALSKI, and P. PIECUCH, J. Chem. Phys. 119, 2951
(2003).

M. J. MCGUIRE, P. PIECUCH, K. KOWALSKI, S. A. KUCHARSKI, and M. MU-
SIAL, J. Phys. Chem. A 108, 8878 (2004).

P.-D. FAN, K. KOWALSKI, and P. PIECUCH, Mal. Phys. 103, 2191 (2005).

S. HIRATA, P.-D. FAN, A. A. AUER, M. NOOIJEN, and P. PIECUCH, J. Chem.
Phys. 121, 12197 (2004).

K. KOWALSKI and P. PIECUCH, J. Chem. Phys. 122, 074107 (2005).

M. D. LODRICUITO, K. KOWALSKI, M. WLOCH, and P. PIECUCH, J. Mal.
Struct.: THEOCHEM 771, 89 (2006).

M. WLOCH, M. D. LODRICUITO, P. PIECUCH, and J. R. GOUR, Mal. Phys.
104, 2149 (2006).

168

 

 

99 P. PIECUCH, S. A. KUCHARSKI, and R. J. BARTLETT, J. Chem. Phys. 110,
6103 (1999).

10° P. PIECUCH, S. A. KUCHARSKI, and V. SPIRKO, J. Chem. Phys. 111, 6679
(1999).

"’1 S. A. KUCHARSKI and R. J. BARTLETT, J. Chem. Phys. 108, 9221 (1998).
102 H. MONKHORST, Int. J. Quantum Chem. Symp. 811, 421 (1977).

103 H. SEKINO and R. J. BARTLETT, Int. J. Quantum Chem. Symp. $18, 255
(1984).

10“ E. DALCAARD and H. J. MONKHORST, Phys. Rev. A 28, 1217 (1983).
105 M. TAKAHASHI and J. PALDUS, J. Chem. Phys. 85, 1486 (1986). l

1°“ H. KOCH, H. J. A. JENSEN, P. JERGENSEN, and T. HELGAKER, J. Chem.
Phys. 93, 3345 (1990).

107 H. KOCH and P. JORCENSEN, J. Chem. Phys. 93, 3333 (1990). " i

 

 

108 K. EMRJCH, Nucl. Phys. A 351, 379 (1981). ’7

‘09 J. GEERTSEN, C. M. L. RITTBY, and R. J. BARTLETT, Chem. Phys. Lett. 164,
57 (1989).

11" D. C. COMEAU and R. J. BARTLETT, Chem. Phys. Lett. 207, 414 (1993).
1“ J. F. STANTON and R. J. BARTLETT, J. Chem. Phys. 98, 7029 (1993).
“2 P. PIECUCH and R. J. BARTLETT, Adv. Quantum Chem. 34, 295 (1999).
“3 H. NAKATSUJI and K. HIRAO, Chem. Phys. Lett. 47, 569 (1977).

1“ H. NAKATSUJI and K. HIRAO, J. Chem. Phys. 68, 2053 (1978), ibid. 68, 4279 .I
(1978).

“5 H. NAKATSUJI, Chem. Phys. Lett. 59, 362 (1978).
“6 H. NAKATSUJI, Chem. Phys. Lett. 67, 329 (1979), ibid. 67, 334 (1979). '

 

“7 H. N AKATSUJI, in Computational Chemistry: Reviews of Current Trends, edited
by J. LESZCZYNSKI, volume 2, pp. 62—124, World Scientific, Singapore, 1997.

“8 J. D. WATTS and R. J. BARTLETT, Chem. Phys. Lett. 233, 81 (1995).
“9 J. D. WATTS and R. J. BARTLETT, Chem. Phys. Lett. 258, 581 (1996).
12° J. D. WATTS and R. J. BARTLETT, J. Chem. Phys. 101, 3073 (1994).

121 H. KOCH, O. CHRISTIANSEN, P. JERGENSEN, and J. OLSEN, Chem. Phys. Lett.
244, 75 (1995).

169

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

O. CHRISTIANSEN, H. KOCH, and P. JCRGENSEN, J. Chem. Phys. 103, 7429
(1995).

O. CHRISTIANSEN, H. KOCH, and P. JORCENSEN, J. Chem. Phys. 105, 1451
(1996).

K. KOWALSKI and P. PIECUCH, J. Chem. Phys. 113, 8490 (2000).
K. KOWALSKI and P. PIECUCH, J. Chem. Phys. 115,643 (2001).
K. KOWALSKI and P. PIECUCH, J. Chem. Phys. 115, 2966 (2001).
K. KOWALSKI and P. PIECUCH, J. Chem. Phys. 116, 7411 (2002).
S. HIRATA, J. Chem. Phys. 121, 51 (2004).

S. A. KUCHARSKI, M. WLOCH, M. MUSIAL, and R. J. BARTLETT, J. Chem.
Phys. 115, 8263 (2001).

M. KALLAY and J. GAUSS, J. Chem. Phys. 121, 9257 (2004).

R. MCWEENY, in Molecular Quantum Mechanics, Academic, New York, 2nd
edition, 1989.

R. SHEPPARD, I. SHAVITT, R. M. PITZER, D. C. COMEAU, M. PEPPER,
H. LISCHKA, P. G. SZALAY, R. AHLRICHS, R. B. BROWN, and J. ZHAO,
Int. J. Quantum Chem. Symp. 822, 149 (1988).

E. R. DAVIDSON, in The World of Quantum Chemistry, edited by R. DAUDEL
and B. PULLMAN, pp. 17—30, Rcidel, Dordrecht, Boston, 1974.

S. R. LANGHOFF and E. R. DAVIDSON, Int. J. Quantum Chem. 8, 61 (1974).

W. BUTSCHER, S.-K. SHIH, R. J. BUENKER, and S. D. PEYERIMHOFF, Chem.
Phys. Lett. 52, 457 (1977).

K. JANKOWSKI, L. MEISSNER, and J. WASILEWSKI, Int. J. Quantum Chem.
28, 931 (1985).

J. PALDUS, in New Horizons of Quantum Chemistry, edited by P.-O. LOWDIN
and B. PULLMAN, p. 31, Reidel, Dordrecht, 1983.

P. J. BRUNA and S. D. PEYERIMHOFF, Adv. Chem. Phys. 67, 1 (1987).
R. J. GDANITz and R. AHLRICHS, Chem. Phys. Lett. 143, 413 (1988).

P. G. SZALAY and R. J. BARTLETT, J. Chem. Phys. 103, 3600 (1995).

P. G. SZALAY and R. J. BARTLETT, Chem. Phys. Lett. 214, 481 (1993).
L. FfiSTI-MOLNAR and P. G. SZALAY, Chem. Phys. Lett. 258, 400 (1996).

170

 

u .5,
' ‘1':

i__ .-

 

‘43 J.-P. DAUDEY, J.-L. HEULLY, and J.-P. MALRIEU, J. Chem. Phys. 99, 1240
(1993).

1‘“ J.-P. MALRIEU and J.-P. DAUDEY, J. Chem. Phys. 101, 8908 (1994).

145 P. J. A. RUTTINK, J. H. VAN LENTHE, R. ZWAANS, and G. GROENENBOOM,
J. Chem. Phys. 94, 7212 (1991).

‘46 H.-J. WERNER and P. J. KNOWLES, J. Chem. Phys. 89, 5803 (1988).
14" P. J. KNOWLES and H.-J. WERNER, Chem. Phys. Lett. 145, 514 (1988).

”8 A. J. DOBBYN, P. J. KNOWLES, and R. J. HARRISON, J. Comput. Chem. 19,
1215 (1998).

”9 P. STAMPFUSS, W. WENZEL, and H. KEITER, J. Comput. Chem. 20, 1559
(1999).

15° M. SCHI’iLER, T. KONVAR, H. LISHKA, R. SHEPARD, and R. J. HARRISON,
Theor. Chim. Acta 84, 489 (1993).

('5
i

‘51 H. DACHSEL, H. LISCHKA, R. SHEPARD, J. NIEPLOCHA, and R. J. HARRISON,
J. Comput. Chem. 18, 430 (1997).

‘5’ H. DACHSEL, R. J. HARRISON, and D. A. DIXON, J. Phys. Chem. 103, 152
(1999).

153 F. STEPHAN and W. WENZEL, J. Chem. Phys. 108, 1015 (1998).

‘54 B. SUO, G. ZHAI, Y. WANG, Z. WEN, X. H0, and L. LI, J. Comput. Chem.
26,88 (2004).

155 B. H. BRANDOW, Rev. Mod. Phys. 39, 771 (1967).
156 I. LINDGREN, J. Phys. B: Atom. Molec. Phys. 7, 2441 (1974).
15" P. DURAND and J.-P. MALRIEU, Adv. Chem. Phys. 67, 321 (1987).

158 J. J. W. MCDOUALL, K. PEASLEY, and M. A. ROBB, Chem. Phys. Lett. 148,
183 (1988).

159 P. M. KOZLOWSKI and E. R. DAVIDSON, J. Chem. Phys. 100, 3672 (1994),
Chem. Phys. Lett. 222, 615 (1994).

 

‘50 J. P. FINLEY, J. Chem. Phys. 108, 1081 (1998).
161 J. P. FINLEY and H. A. WITEK, J. Chem. Phys. 112, 3958 (2000).

162 K. F. FREED, in Lecture Notes in Chemistry, volume 52, p. 1, Springer, Berlin,
1989, and references therein.

‘63 K. ANDERSSON, P.-A. MALMQVIST, B. 0. R003, A. J. SADLEJ, and
K. WOLINSKI, J. Phys. Chem. 94,5483 (1990).

171

164

K. ANDERSSON, P.-A. MALMQVIST, and B. 0. Boos, J. Chem. Phys. 96, 1218
(1992).

165 K. Hume, Chem. Phys. Lett. 190, 374 (1992), ibid. 196, 397 (1992), ibid. 201,

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

59 (1993).
P. CELANI and H.-J. WERNER, J. Chem. Phys. 112, 5546 (2000).

K. HIRAO, Int. J. Quantum Chem. Symp. $26, 517 (1992), and references
therein.

R. B. MURPHY and R. P. MESSMER, Chem. Phys. Lett. 183, 443 (1991).
R. B. MURPHY and R. P. MESSMER, J. Chem. Phys. 97, 4170 (1992).
M. SEJPAL and R. P. MESSMER, J. Chem. Phys. 114, 4796 (2001).

K. WOLINSKI and P. PULAY, J. Chem. Phys. 90, 3647 (1989).

R. B. MURPHY, W. T. POLLARD, and R. A. FRIESNER, J. Chem. Phys. 106,
5073 (1997).

M. R. HOFFMANN, Chem. Phys. Lett. 195, 127 (1992).
H. NAKANO, J. Chem. Phys. 99, 7983 (1993).

C. ANGELI, R. CIMIRAGLIA, S. EVANGELISTI, T. LEININGER, and J .-P. MAL-
RIEU, J. Chem. Phys. 114, 10252 (2001).

C. ANCELI, R. CIMIRAGLIA, and J.-P. MALRIEU, J. Chem. Phys. 117, 9138
(2002).

U. S. MAHAPATRA, B. DATTA, and D. MUKHERJEE, Chem. Phys. Lett. 299,
42 (1999), J. Phys. Chem. A 103, 1822 (1999).

P. GHOSH, S. CHATTOPADHYAY, D. JANA, and D. MUKHERJEE, Int. J. Mal.
Sci. 3, 733 (2002).

Z. ROLIK, A. SZABADOS, and P. R. SURJAN, J. Chem. Phys. 119, 1922 (2003).

P. R. SURJAN and A. SZABADOS, in Fundamental World of Quantum Chemistry,
A Tribute to the Memory of Per-010v Lo'wdin, edited by E. J. BRANDAS and
E. S. KRYACHKO, volume 111, pp. 129—185, Kluwer, Dordrecht, 2004.

G. HOSE, Theor. Chim. Acta 72, 303 (1987).

J.-P. MALRIEU, P. DURAND, and J. P. DAUDEY, J. Phys. A: Math. Gen. 18,
809 (1985).

H. NAKANO and H. HIRAO, Bull. Korean Chem. Soc. 24, 812 (2003).

172

 

 

13“ K. WOLINSKI, H. L. SELLERS, and P. PULAY, Chem. Phys. Lett. 140, 225
(1987)

‘35 B. 0. R008, K. ANDERSSON, M. P. FiiLSCHER, P.-A. MALMQVIST,
L. SERRANO—ANDRES, K. PIERLOOT, and M. MERCHAN, Adv. Chem. Phys.
93, 219 (1996), and references therein.

‘86 C. MURRAY and E. R. DAVIDSON, Chem. Phys. Lett. 187, 451 (1991).

187 C. J. CRAMER, M. WLOCH, P. PIECUCH, and L. GACLIARDI, J. Phys. Chem.
A 110, 1991 (2006).

‘38 C. J. CRAMER, A. KINAL, M. WLOCH, P. PIECUCH, and L. GAGLIARDI, J.
Phys. Chem. A 110, 11557 (2006).

139 M. RODE and H.-J. WERNER, Theor. Chem. Ace. 114, 309 (2005).

19° J. RINTELMAN, I. ADAMOVIC, S. VARCANOV, and M. S. GORDON, J. Chem.
Phys. 122, 044105 (2005).

191 T. H. SCHUCAN and H. A. WEIDENMiiLLER, Ann. Phys. 73, 108 (1972).
192 G. HOSE and U. KALDOR, J. Phys. B: Atom. Molec. Phys. 12, 3827 (1979).

‘93 J. P. FINLEY, R. K. CHAUDHURI, and K. FREED, J. Chem. Phys. 103, 4990
(1995).

194 I. LINDGREN, Int. J. Quantum Chem. Symp. 12, 33 (1978).

195 D. MUKHERJEE, R. K. MOITRA, and A. MUKHOPADHYAY, Mal. Phys. 30, 1861
(1975), ibid. 33, 955 (1977).

196 B. JEZIORSKI and J. PALDUS, J. Chem. Phys. 90, 2714 (1988).
197 B. JEZIORSKI and H. J. MONKHORST, Phys. Rev. A 24, 1668 (1981).
193 B. JEZIORSKI and J. PALDUS, J. Chem. Phys. 88, 5673 (1988).

‘99 A. BALKOVA, S. A. KUCHARSKI, L. MEISSNER, and R. J. BARTLETT, Theor.
Chim. Acta 80, 335 (1991).

20° J. PALDUS, L. PYLYPOW, and B. JEZIORSKI, in Many-Body Methods in Quan-
tum Chemistry, edited by U. KALDOR, volume 52 of Lecture Notes in Chemistry,
p. 151, Springer, Berlin, 1989.

201 P. PIECUCH and J. PALDUS, Theor. Chim. Acta 83,69 (1992).
202 P. PIECUCH and J. PALDUS, J. Chem. Phys. 101, 5875 (1994).

203 J. PALDUS, B. JEZIORSKI, and L. PYLYPOW, in Recent Progress in Many-Body
Theories, edited by T. L. AINSWORTH, C. E. CAMPBELL, B. E. CLEMENS, and
E. KROTSCHEK, volume 3, pp. 287—303, Plenum, New York, 1992.

173

 

 

20“ P. PIECUCH and J. PALDUS, Phys. Rev. A 49, 3479 (1993).
205 P. PIECUCH, R. TOBOLA, and J. PALDUS, Chem. Phys. Lett. 210, 243 (1993).

2°“ U. S. MAHAPATRA, B. DATTA, B. BANDYOPADHYAY, and D. MUKHERIEE,
Adv. Quantum Chem. 30, 169 (1998).

207 U. S. MAHAPATRA, B. DATTA, andD. MUKHERJEE, J. Chem. Phys. 110, 6171
(1999).

208 P. PIECUCH, N. OLIPHANT, and L. ADAMOWICz, J. Chem. Phys. 99, 1875
(1993).

209 J. R. GOUR, P. PIECUCH, and M. WLOCH, J. Chem. Phys. 123, 134113 (2005). 7‘4

I .
21° J. R. GOUR, .P. PIECUCH, and M. WLOCH, Int. J. Quantum Chem. 106, 2854 l
(2006).

2“ J. R. GOUR and P. PIECUCH, J. Chem. Phys. 125, 234107 (2006).
212 K. KOWALSKI and P. PIECUCH, Phys. Rev. A 61, 052506 (2000). a“
213 . LI and J. PALDUS, J. Chem. Phys. 119, 5320 (2003).

. LI and J. PALDUS, J. Chem. Phys. 119, 5334 (2003).

. LI and J. PALDUS, J. Chem. Phys. 119, 5346 (2003).

. LI and J. PALDUS, J. Chem. Phys. 119, 5890 (2003).

R. GWALTNEY and M. HEAD-GORDON, Chem. Phys. Lett. 323, 21 (2000).

 

214
215
216
217

218 R. GWALTNEY, C. D. SHERRILL, M. HEAD-GORDON, and A. I. KRYLOV,

Chem. Phys. 113, 3548 (2000).

219 R. GWALTNEY and M. HEAD-GORDON, J. Chem. Phys. 115, 2014 (2001).

@STI’JXXXX

22° S. R. GWALTNEY, E. F. C. BYRD, T. VAN VOORHIS, andM. HEAD-GORDON,
Chem. Phys. Lett. 353,359 (2002).

”1 M. HEAD-GORDON, T. VAN VOORHIS, S. R. GWALTNEY, and E. F. C. BYRD,
in Low-Lying Potential Energy Surfaces, edited by M. R. HOFFMANN and K. G.
DYALL, volume 828, p. 93, American Chemical Society, Washington, DC, 2002.

 

2” R. J. BARTLETT, I. GRABOWSKI, S. HIRATA, and S. IVANOV, J. Chem. Phys.
122, 034104 (2005).

223 A. I. KRYLOV, Chem. Phys. Lett. 338, 375 (2001).
22“ A. I. KRYLOV and C. D. SHERRILL, J. Chem. Phys. 116, 3194 (2002).
”5 L. V. SLIPCHENKO and A. I. KRYLOV, J. Chem. Phys. 117, 4694 (2002).

174

 

226

227

228

230

231

232

239

240

241

242

243

244

245

247

M. WLOCH, J. R. GOUR, K. KOWALSKI, and P. PIECUCH, J. Chem. Phys.
122, 214107 (2005).

P. PIECUCH and M. WLOCH, J. Chem. Phys. 123, 224105 (2005).

P. PIECUCH, M. WLOCH, J. R. GOUR, and A. KINAL, Chem. Phys. Lett. 418,
467 (2006).

K. KOWALSKI and P. PIECUCH, J. Mal. Struc.: THEOCHEM 547, 191 (2001).
I. OZKAN, A. KINAL, and M. BALCI, J. Phys. Chem. A 108, 507 (2004).
A. KINAL and P. PIECUCH, J. Phys. Chem. A 110, 367 (2006).

S. COUSSAN, Y. FERRO, A. TRIVELLA, M. RAJZMANN, P. ROUBIN, R. WIEC-
ZOREK, C. MANCA, P. PIECUCH, K. KOWALSKI, M. WLOCH, S. A.
KUCHARSKI, and M. MUSIAL, J. Phys. Chem. A 110, 3920 (2006).

M. MCGUIRE and P. PIECUCH, J. Am. Chem. Soc. 127, 2608 (2005).

A. KINAL and P. PIECUCH, J. Phys. Chem. A 111, 734 (2007).

M. WLOCH, J. R. GOUR, and P. PIECUCH, J. Phys. Chem. A , in press.
Y. GE, M. S. GORDON, and P. PIECUCH, J. Chem. Phys. , in press.

K. KOWALSKI, H. HIRATA, M. WLOCH, P. PIECUCH, and T. L. WINDUS, J.
Chem. Phys. 123, 074319 (2005).

P. PIECUCH, S. HIRATA, K. KOWALSKI, P.-D. FAN, and T. L. WINDUS, Int.
J. Quantum Chem 106, 79 (2006).

P. PIECUCH and L. ADAMOWICZ, Chem. Phys. Lett. 221, 121 (1994).

K. B. GHOSE and L. ADAMOWICZ, J. Chem. Phys. 103, 9324 (1995).

L. ADAMOWICZ, P. PIECUCH, , and K. B. GHOSE, Mal. Phys. 94, 225 (1998).
J. PALDUS, J. CIZEK, and M. TAKAHASHI, Phys. Rev. A 30, 2193 (1984).

J. PALDUS and J. PLANELLES, Theor. Chim. Acta 89, 13 (1994).

J. PLANELLES, J. PALDUS, and X. LI, Theor. Chim. Acta 89, 33 (1994), ibid.
89, 59 (1994).

L. Z. STOLARCZYK, Chem. Phys. Lett. 217, 1 (1994).

X. LI, G. PERIS, J. PLANELLES, F. RAJADALL, and J. PALDUS, J. Chem.
Phys. 107, 90 (1997).

P. PIECUCH, R. TOBOLA, and J. PALDUS, Phys. Rev. A 54, 1210 (1996).

175

248

249

250

251

252

254

257

259

261

262

266

267

X. LI, 1. GRABOWSKI, K. JANLOWKSI, and J. PALDUS, Adv. Quantum Chem.
36, 231 (2000).

G. PERIS, J. PLANELLES, and J. PALDUS, Int. J. Quantum Chem. 62, 137
(1997).

x. LI and J. PALDUS, J. Chem. Phys. 107, 6257 (1997), ibid. 108, 637 (1998),
ibid. 110, 2844 (1999), Chem. Phys. Lett. 286, 145 (1998), Mol. Phys. 98, 1185
(2000), J. Chem. Phys. 113, 9966 (2000).

P. PIECUCH, K. KOWALSKI, and I. S. O. PIMIENTA, Int. J. Mal. Sci. 3, 475
(2002).

X. LI and J. PALDUS, J. Chem. Phys. 115, 5759 (2001).
X. LI and J. PALDUS, J. Chem. Phys. 115, 5774 (2001).
X. LI and J. PALDUS, J. Chem. Phys. 117, 1941 (2002).

B. O. ROOS, P. LINSE, P. E. M. SIEGBAHN, and M. R. A. BLOMBERG, Chem.
Phys. 66, 197 (1982).

P. M. KOZLOWSKI, M. DUPUIS, and E. R. DAVIDSON, J. Am. Chem. Soc. 117,
774 (1995).

K. WOLINSKI, Theor. Chim. Acta 82, 459 (1992).
M. W. SCHMIDT and M. S. GORDON, Annu. Rev. Phys. Chem. 49, 233 (1998).
H. NAKANO, Chem. Phys. Lett. 207, 372 (1993).

J. PALDUS, in Diagrammatic Methods for Many-Fermion Systems, University of
Nijmegen.

K. KOWALSKI and P. PIECUCH, Mal. Phys. 104, 2425 (2004).
C. BLOCH, Nucl. Phys. 6, 329 (1958).
P.-O. LOWDIN, J. Math. Phys. 3, 969 (1962).

P.-O. LOWDIN, in Perturbation Theory and Its Applications in Quantum Me-
chanics, edited by C. H. WILCOX, p. 255, Wiley, New York, 1966.

M. W. SCHMIDT, K. K. BALDRIDGE, J. A. BOATz, S. T. ELBERT, M. S.
GORDON, J. H. A. JENSEN, S. KOSEKI, N. MATSUNAGA, K. A. NGUYEN,
S. J. SU, T. L. WINDUS, M. DUPUIS, and J. A. MONTGOMERY, J. Comput.
Chem. 14, 1347 (1993).

T. H. DUNNING JR., J. Chem. Phys. 53, 2823 (1970).
T. H. DUNNING, 1989 90, 1007 (1989).

176

 

 

 

263 P. SAXE, H. F. SCHAEFER III, and N. C. HANDY, Chem. Phys. Lett. 79, 202
(1981).

269 J. OLSEN, A. M. SANCHEZ DE MERAS, H. J. A. JENSEN, and P. JERGENSEN,
Chem. Phys. Lett. 154, 380 (1989).

 

 

 

 

 

177

          
  

 

u(It)ritually)

1293 0 5

515(5))

3867