DEVELOPMENT AND APPLICATIONS OF COUPLED-CLUSTER METHODS AND
POTENTIAL ENERGY SURFACE EXTRAPOLATION SCHEMES
By
Jesse J. Lutz

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Chemistry
2011

ABSTRACT
DEVELOPMENT AND APPLICATIONS OF COUPLED-CLUSTER
METHODS AND POTENTIAL ENERGY SURFACE EXTRAPOLATION
SCHEMES
By
Jesse J. Lutz

The generation of highly accurate potential energy surfaces (PESs) for reactive processes
represents a difficult challenge for modern electronic structure theory. Since chemical reactions often involve breaking and forming bonds or intermediate and transition state species,
one must employ a methodology that provides a balanced and highly accurate description
of varying levels of electronic degeneracy, but that is also practical enough to be applied
to a wide range of chemical problems. Using small to medium sized systems, we examine
the performance of two classes of coupled-cluster (CC) methods which are capable of accounting for the diverse electron correlation effects encountered in the majority of groundand excited-state PES considerations. The first class of methods are the size-extensive completely renormalized (CR) CC approaches for ground-states and their equation of motion
(EOM) CC extensions for excited-states, in which noniterative corrections due to higherorder excitations are added to the energies obtained with the standard CC and EOMCC
approximations, such as CCSD (CC with singles and doubles) or EOMCCSD (EOMCC with
singles and doubles), respectively. In particular, we focus on the left-eigenstate CR-CC(2,3)
and CR-EOMCC(2,3) methods, in which a noniterative correction due to triple excitations is
added to the CCSD or EOMCCSD energy, respectively, and, when necessary, a noniterative
correction for quadruple excitations is also included via the CR-CC(2,3)+Q approach. A new
variant of the CR-EOMCC(2,3) method, abbreviated as δ-CR-EOMCC(2,3), that can pro-

vide a size-intensive treatment of excitation energies, is discussed as well. The second class
of methods considered here is the active-space variants of the electron-attached (EA) and
ionized (IP) EOMCC theories, which utilize the idea of applying a linear electron-attaching
or ionizing operator to the correlated, ground-state CC wave function of an N -electron
closed-shell system in order to generate the ground and excited states of the related (N ± 1)electron radical species of interest. These approaches use a physically motivated set of active
orbitals to a priori select the dominant higher-order correlation effects to be included in the
calculation, which significantly reduces the costs of the high-level EA- and IP-EOMCC approximations needed for obtaining accurate results for open-shell species without sacrificing
accuracy. We have also developed a general extrapolation strategy for reducing the cost of
generating PESs with correlated electronic structure methods using the concept of correlation energy scaling. Benchmark studies were performed to demonstrate typical accuracies
for two types of PES extrapolation schemes, namely, the single-level PES extrapolation
schemes, in which the essential quantity, the correlation energy scaling factor, is generated
using only the quantum chemistry method of interest, and the dual-level PES extrapolation
schemes, where lower-order approaches are used to estimate the correlation energy scaling
factor corresponding to the method of interest. Unifying features of these PES extrapolation
techniques are discussed, including the role of pivot geometries and base wave functions, and
PES extrapolation to the complete basis set limit is examined as well. Finally, the most
essential details of the new open-shell EOMCCSD and EA- and IP-EOMCC computer codes
for the GAMESS software package, developed as part of this thesis research, are described.

Copyright by
JESSE J. LUTZ
2011

This dissertation is dedicated to my loving and ever supportive parents who taught me the
most important things I could ever learn.

v

ACKNOWLEDGMENT

I owe a great deal of gratitude to my Ph.D. advisor, Professor Piotr Piecuch for countless
opportunities to contribute important research which never would have been possible otherwise. He and his group were nurturing and supportive in many ways, particularly regarding
research, but never limited to it. I would also like to thank the rest of my guidance committee, namely my second reader, Professor Katharine L. C. Hunt, as well as Professor Robert
I. Cukier and Professor James E. Jackson for lending their time and support, and for all
their comments and advice regarding research and life in general.
It is also important to mention my great appreciation for the efforts of Professor Marta
Wloch and Dr. Jeffery R. Gour towards my professional development. As a result of their
able guidance, I progressed much more quickly then I would have without them, and I am
deeply grateful for their help and collaboration in many of the projects described in this
dissertation. I would also like to thank other members of the Piecuch group, both past and
present, for the help and companionship they provided, including Dr. Maricris Lodriguito,
Dr. Wei Li, Dr. Jun Shen, and Mr. Jared Hansen.
Finally, I would like to acknowledge Michigan State University and the Department of
Chemistry for several fellowships that supported part of my research. I would also like to
acknowledge the MSU High Performance Computing Center for providing resources which
were utilized for part of this research. Most of the research discussed in this thesis was
supported by the U. S. Department of Energy, Office of Science, Office of Basic Energy
Sciences, through grants awarded to my advisor and research fellowships to me, and I am
appreciative for this support as well.

vi

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Figures

ix

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

1 Introduction

1

2 Project Objectives

17

3 Applications of Coupled-Cluster and Equation-of-Motion Coupled-Cluster
Methods
19
3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 Single-Reference Coupled-Cluster Theory for Ground States . . . . . 20
3.1.2 Equation-of-Motion Coupled-Cluster Theory for Electronically Excited,
Electron-Attached, and Ionized States . . . . . . . . . . . . . . . . . 25
3.1.2.1 Excited States . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.2.2 Electron-Attached and Ionized States . . . . . . . . . . . . . 29
3.1.3 The Method of Moments of Coupled-Cluster Equations . . . . . . . . 32
3.1.3.1 Completely Renormalized Coupled-Cluster and Equation-ofMotion Coupled-Cluster Approaches . . . . . . . . . . . . . 35
3.1.3.2 Biorthogonal Formulation of the MMCC Equations and the
CR-CC(2,3) and CR-EOMCC(2,3) Methods . . . . . . . . . 43
3.1.3.3 A Size-Intensive Variant of CR-EOMCC(2,3): The δ-CREOMCC(2,3) Method . . . . . . . . . . . . . . . . . . . . . 49
3.1.4 The Active-Space Coupled-Cluster and Equation-of-Motion CoupledCluster Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.1 The DBH24 Benchmark Database for Thermochemical Kinetics . . . 57
3.2.2 Addition Reactions of Ethylene and Acetylene to Ozone . . . . . . . 67
3.2.3 Mechanism of the Isomerization of Bicyclobutane to Butadiene . . . . 75
3.2.4 Excited-State Potential Energy Surfaces for the Dissociation of Water 81
3.2.5 Excitation Energies and Hydrogen-Bonding-Induced Spectral Shifts in
Complexes of cis-7-Hydroxyquinoline . . . . . . . . . . . . . . . . . . 91
3.2.6 Geometries and Adiabatic Excitation Energies of CNC, C2 N, N3 , and
NCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4 Potential Energy Surface Extrapolation Schemes
129
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
vii

4.2
4.3

4.4

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Single-Level Potential Energy Surface Extrapolation Schemes . . . . . . . . . 137
4.3.1 Potential Energy Surface Extrapolation to Larger Basis Sets . . . . . 137
4.3.2 Potential Energy Surface Extrapolation to the Complete Basis Set Limit138
4.3.3 The Role of Pivot Geometries and Base Wave Functions in Single-Level
PES Extrapolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.3.4 Application to the Isomerization of Bicyclobutane to Butadiene . . . 141
Dual-Level Potential Energy Surface Extrapolation Schemes . . . . . . . . . 153
4.4.1 Using Lower-Order Methods to Produce Correlation Energy Scaling
Factors in High-Level Calculations . . . . . . . . . . . . . . . . . . . 154
4.4.2 Application to Single Bond-Breaking Potential Energy Curves . . . . 156
4.4.3 Application to the Isomerization of Bicyclobutane to Butadiene . . . 163
4.4.4 The Role of Pivot Geometries and Base Wave Functions in Dual-Level
PES Extrapolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5 Development of Computer Codes for the GAMESS software Package
5.1 Key Details of Efficient Computer Implementations . . . . . . . . . . . . . .
5.1.1 Standard Equation-of-Motion Coupled-Cluster Theory with Singles
and Doubles for Open-Shell Systems . . . . . . . . . . . . . . . . . .
5.1.2 Electron-Attached and Ionized Equation-of-Motion Coupled-Cluster
Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

174
175

6 Summary and Concluding Remarks

187

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191

viii

176
184

LIST OF TABLES

3.1

Representative barrier heights database DBH24 taken from Ref. [204].

. . .

59

3.2

Mean signed error (MSE) and mean unsigned error (MUE) of coupled cluster
methods calculated with all electrons correlated compared against the DBH24
benchmark database (in kcal/mol). . . . . . . . . . . . . . . . . . . . . . . .

61

Mean signed error (MSE) and mean unsigned error (MUE) of coupled cluster methods calculated with frozen core approximation compared against the
DBH24 benchmark database (in kcal/mol). . . . . . . . . . . . . . . . . . .

62

Errors resulting from CR-CC(2,3) calculations of forward and reverse barrier
=
=
=
=
heights, Vf and Vr respectively, reported as ǫ(Vf ) / ǫ(Vr )) at varying basis
set levels compared against DBH24 benchmark values (in kcal/mol). . . . .

66

Mean signed errors (MSE) and mean unsigned errors (MUE) resulting from
CR-CC(2,3) calculations of all DBH24 forward and reverse barrier heights,
=
=
=
=
Vf and Vr respectively, reported as ǫ(Vf ) / ǫ(Vr )) at varying basis set
levels (in kcal/mol). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

3.6

Benchmark values for the C2 H2 and C2 H4 ozonolysis reaction pathways. . .

70

3.7

Energetics of stationary points relative to reactants, in kcal/mol, produced
at various levels of coupled-cluster theory for the ozonolysis of ethylene and
acetylene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

Relative enthalpies, in kcal/mol, with respect to reactant as calculated at
various levels of theory for the conrotatory and disrotatory bicbut → tbut
isomerization pathways. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

Ground-state X 1 A′ energies for the asymmetric single-bond breaking of H2 O
at a series of the O-H bond lengths, R, in atomic units. The last two rows give
the mean unsigned errors (MUE) and non-parallelity errors (NPE) relative to
the full CI PES obtained in Ref. [271]. . . . . . . . . . . . . . . . . . . . . .

83

3.3

3.4

3.5

3.8

3.9

3.10 Same as Table (3.9) for the three lowest-lying 1 A′ states.
ix

. . . . . . . . . .

86

3.11 Same as Table (3.9) for the three lowest-lying 3 A′ states.

. . . . . . . . . .

87

3.12 Same as Table (3.9) for the two lowest-lying 1 A′′ states.

. . . . . . . . . . .

88

3.13 Same as Table (3.9) for the two lowest-lying 3 A′′ states.

. . . . . . . . . . .

89

3.14 The basis-set dependence of the vertical excitation energies ωπ→π ∗ and the
environment-induced shifts ∆ωπ→π ∗ (in cm−1 ) obtained with the EOMCCSD
approach corresponding to the lowest π → π ∗ transition in the cis-7HQ chromophore and its complexes with the water and ammonia molecules. . . . .

95

3.15 The vertical excitation energies ωπ→π ∗ and the environment-induced shifts
∆ωπ→π ∗ (in cm−1 ) obtained with the EOMCCSD/6-31+G(d), EOMCCSD/6311+G(d), δ-CR-EOMCC(2, 3), A/6-31+G(d), and δ- CR-EOMCC(2, 3), D/631+G(d) approaches, and their composite EOMCC corresponding to the lowest π → π ∗ transition in the cis-7HQ chromophore and its various complexes.

97

3.16 Total and adiabatic excitation energies for the ground and low-lying excited
states of CNC, as obtained with the different EA-EOMCC approaches using
the DZP [4s2p1d] and cc-pVXZ (X = D, T, Q) basis sets and extrapolating
to the CBS limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.17 Total and adiabatic excitation energies for the ground and low-lying excited
states of C2 N, as obtained with the different EA-EOMCC approaches using
the DZP [4s2p1d] and cc-pVXZ (X = D, T, Q) basis sets and extrapolating
to the CBS limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.18 Comparison of the optimized equilibrium geometries for the low-lying states
of CNC and C2 N, as obtained with the EA-EOMCC and SAC-CI-SDT-R/PS
approaches using the DZP[4s2p1d] and cc-pVXZ (X = D, T, Q) basis sets. . 125
3.19 Total and adiabatic excitation energies for the ground and low-lying excited
states of NCO, as obtained with the different IP EOMCC approaches using
the DZP [4s2p1d] and cc-pVXZ (X = D, T, Q) basis sets and extrapolating
to the CBS limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.20 Total and adiabatic excitation energies for the ground and low-lying excited
states of N3 , as obtained with the different IP EOMCC approaches using the
DZP [4s2p1d] and cc-pVXZ (X = D, T, Q) basis sets and extrapolating to
the CBS limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
x

3.21 Comparison of the optimized equilibrium geometries for the low-lying states
of N3 and NCO, as obtained with the IP EOMCC and SAC-CI-SDT-R/PS
approaches using the DZP [4s2p1d] and cc-pVXZ (X = D, T, Q) basis sets.

128

4.1

Computer costs of typical ab initio wave function calculations at the aug-ccpVTZ basis set level, taken from Ref. [204] . . . . . . . . . . . . . . . . . . 133

4.2

The calculated CR-CC(2,3)/cc-pVDZ and CR-CC(2,3)/cc-pVTZ energies and
the calculated and extrapolated CR-CC(2,3)/cc-pVQZ energies at the stationary points defining the conrotatory and disrotatory pathways characterizing
the bicbut→t-but isomerization. The bicbut reactant defines the pivot geometry for the PES extrapolations. . . . . . . . . . . . . . . . . . . . . . . . 143

4.3

The calculated CR-CC(2,3)/cc-pVDZ and CR-CC(2,3)/cc-pVTZ energies and
the calculated and extrapolated CR-CC(2,3)/cc-pVQZ energies at the stationary points defining the conrotatory and disrotatory pathways characterizing
the bicbut→t-but isomerization. The con TS transition state defines the
pivot geometry for the PES extrapolations. . . . . . . . . . . . . . . . . . . 144

4.4

The calculated CR-CC(2,3)/cc-pVDZ and CR-CC(2,3)/cc-pVTZ energies and
the calculated and extrapolated CR-CC(2,3)/cc-pVQZ energies at the stationary points defining the conrotatory and disrotatory pathways characterizing
the bicbut→t-but isomerization. The dis TS transition state defines the
pivot geometry for the PES extrapolations. . . . . . . . . . . . . . . . . . . 145

4.5

The calculated CR-CC(2,3)/cc-pVDZ and CR-CC(2,3)/cc-pVTZ energies and
the calculated and extrapolated CR-CC(2,3)/cc-pVQZ energies at the stationary points defining the conrotatory and disrotatory pathways characterizing
the bicbut→t-but isomerization. The g-but intermediate defines the pivot
geometry for the PES extrapolations. . . . . . . . . . . . . . . . . . . . . . 146

4.6

The calculated CR-CC(2,3)/cc-pVDZ and CR-CC(2,3)/cc-pVTZ energies and
the calculated and extrapolated CR-CC(2,3)/cc-pVQZ energies at the stationary points defining the conrotatory and disrotatory pathways characterizing
the bicbut→t-but isomerization. The gt TS transition state defines the
pivot geometry for the PES extrapolations. . . . . . . . . . . . . . . . . . . 147

4.7

The calculated CR-CC(2,3)/cc-pVDZ and CR-CC(2,3)/cc-pVTZ energies and
the calculated and extrapolated CR-CC(2,3)/cc-pVQZ energies at the stationary points defining the conrotatory and disrotatory pathways characterizing
the bicbut→t-but isomerization. The t-but product defines the pivot geometry for the PES extrapolations. . . . . . . . . . . . . . . . . . . . . . . . 148
xi

4.8

The calculated CR-CC(2,3)/cc-pVDZ, CR-CC(2,3)/cc-pVTZ, and CR-CC(2,3)/ccpVQZ energies and the CBS values of the CR-CC(2,3) energies obtained using
the point-wise extrapolations exploiting Eq. (3.98) and the CBS extrapolation
procedure combining Eqs. (4.1)–(4.3) with Eq. (3.98) discussed in the text
at the stationary points defining the conrotatory and disrotatory pathways
characterizing the bicbut→t-but isomerization. The bicbut reactant was
used to define the pivot geometry for the PES extrapolations based on Eqs.
(4.1)–(4.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

4.9

A comparison of calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies of the H2 O molecule in which one of the two O-H bonds (R) is stretched,
while keeping the other O-H bond at the equilibrium length and the H-O-H
angle fixed at 104.5 ◦ . The equilibrium geometry defines the pivot geometry
Re and RHF defines the base wave function for the PES extrapolations. In
all post-RHF calculations, the lowest orbital, correlating with the 1s shell of
the oxygen atom was kept frozen. . . . . . . . . . . . . . . . . . . . . . . . 160

4.10 A comparison of calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies for several internuclear separations RH-Cl of the HCl molecule. The
equilibrium geometry defines the pivot geometry Re and RHF defines the
base wave function for the PES extrapolations. In all post-RHF calculations,
the lowest five orbitals, correlating with the 1s, 2s, and 2p shells of Cl, were
kept frozen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

4.11 A comparison of calculated and extrapolated CR-CC(2,3)/aug-ccpVQZ energies for several internuclear separations RF-F of the F2 mole- cule. The
equilibrium geometry defines the pivot geometry Re and RHF defines the
base wave function for the PES extrapolations. In all post-RHF calculations,
the lowest two orbitals, correlating with the 1s shells of the F atoms, were
kept frozen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

4.12 A comparison of calculated and extrapolated CR-CC(2,3)/cc-pVQZ energies
at the stationary points defining the conrotatory and disrotatory pathways
characterizing the bicbut→t-but isomerization. The bicbut reactant defines the pivot geometry Re and RHF defines the base wave function for the
PES extrapolations. In all post-RHF calculations, the lowest four orbitals,
correlating with the 1s shells of the carbon atoms, were kept frozen. . . . . 165
xii

4.13 A comparison of calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies of the H2 O molecule, in which one of the two O-H bonds R is stretched,
while keeping the other O-H bond at the equilibrium length and the H-O-H
angle fixed at 104.5 ◦ . Each geometry serves as its own pivot geometry and
RHF defines the base wave function for the PES extrapolations. In all postRHF calculations, the lowest orbital, correlating with the 1s orbital of the
oxygen atom, was kept frozen. . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.14 A comparison of calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies for several internuclear separations R of the HCl molecule. Each geometry
serves as its own pivot geometry and RHF defines the base wave function for
the PES extrapolations. In all post-RHF calculations, the lowest five orbitals,
correlating with the 1s, 2s, and 2p shells of Cl, were kept frozen. . . . . . . 170
4.15 A comparison of calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies for several internuclear separations R of the F2 molecule. Each geometry
serves as its own pivot geometry and RHF defines the base wave function for
the PES extrapolations. In all post-RHF calculations, the lowest two orbitals,
correlating with the 1s orbitals on the fluorine atoms, were kept frozen. . . 171
4.16 A comparison of calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies at the stationary points defining the conrotatory and disrotatory pathways characterizing the bicbut→t-but isomerization. Each geometry serves
as its own pivot geometry and RHF defines the base wave function for the
PES extrapolations. In all post-RHF calculations, the lowest four orbitals,
correlating with the 1s orbitals of the carbon atoms, were kept frozen. . . . 172
4.17 A summary of the necessary calculations and the corresponding computer
resources required to utilize different tiers of the PES extrapolation scheme
based on Eqs. (4.1) – (4.3) to scale the bicyclobutane isomerization pathway
from the CR-CC(2,3)/cc-pVTZ level of theory to the CR-CC(2,3)/cc-pVQZ
level, along with the corresponding extrapolation errors. The bicbut structure is used to provide the pivot geometry. . . . . . . . . . . . . . . . . . . 173
5.1

Explicit algebraic expressions for the one- and two-body matrix elements of
¯ (CCSD)
HN,open . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

xiii

LIST OF FIGURES

3.1

3.2

Stationary points along the C2 H2 (top row) and C2 H4 (bottom row) ozonolysis
reaction pathways. In each row, the structures from left to right represent the
van der Waals minimum (vdW), transition state (TS), and the cycloadduct,
respectively. The oxygen, carbon, and hydrogen atoms are represented by the
red, yellow, and grey spheres, respectively. For interpretation of the references
to color in this and all other figures, the reader is referred to the electronic
version of this dissertation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

The conrotatory and disrotatory pathways describing the isomerization of
bicyclo[1.1.0]butane to trans-buta-1,3-diene, along with the enthalpy values
relative to the reactant at all stationary points obtained in the explicit CRCC(2,3)/cc-pVQZ//CASSCF(10,10)/cc-pVDZ calculations (numbers in roman) and the enthalpy values obtained with the PES extrapolation procedure
using the CR-CC(2,3)/cc-pVDZ, CR-CC(2,3)/cc-pVTZ, and RHF/cc-pVQZ
energies at all stationary points resulting from the CASSCF(10,10)/cc-pVDZ
optimizations, and the CR-CC(2,3)/cc-pVQZ correlation energy at the reactant geometry (numbers in bold italics). The available experimental enthalpy
values are in parentheses. All enthalpies are in kcal/mol. . . . . . . . . . . .

77

3.3

Cuts of the PESs for a few ground- and excited-states of a single-bond stretching model of H2 O as obtained with (a) the CCSD/EOMCCSD and (b) the
CR-CC(2,3)/CR-EOMCC(2,3) approaches, and the TZ basis set. Lines are
used to represent the CC/EOMCC data, while the corresponding full CI values are represented by points (which are identically replicated in (a) and (b)). 92

3.4

The eight hydrogen-bonded complexes of the cis-7HQ molecule. . . . . . . .

xiv

94

Chapter 1
Introduction
Potential energy surfaces (PESs) play a central role in the theoretical description of molecular
structures, properties, and reactivities, making them of great utility in many areas of chemical research, including spectroscopy and kinetics, investigations of reaction mechanisms, and
the design of force fields for biological and materials science applications. Unfortunately,
obtaining a molecular PES is typically very difficult due to the enormous mathematical
complexity one faces when trying to solve the electronic Schr¨dinger equation. Development
o
efforts in quantum chemistry have focused on the design of computationally manageable, yet
reliable, approximation methods for the generation of energies and properties that can be applicable to a wide range of molecular systems. In the interest of reducing mathematical complexity some simplifying approximations are introduced from the outset, including the neglect
of relativistic effects, which are small in systems with light atoms [1], and the decoupling
of the nuclear and electronic wave functions, accomplished under the Born-Oppenheimer
approximation [2], which allows the electronic energy of a molecule to be expressed as a
function of its geometry, thus providing the conceptual basis for a PES. Despite these and
1

other commonly employed simplifications, the electronic Schr¨dinger equation remains too
o
formidable to be solved exactly for any system with two or more electrons [3]. One of the
principal challenges in the conventional determination of a PES, in which one is required
to solve the electronic Schr¨dinger equation point-by-point for each fixed nuclear geometry
o
of interest, is the development, implementation, and benchmarking of new approximate ab
initio electronic structure methods that can provide suitably accurate results for the wide
variety of chemical species commonly encountered when studying reactive PESs, while requiring only modest computational resources. Thus, the first goal of this dissertation is
to make a significant contribution toward the development and benchmarking of leading
modern methods of electronic structure theory.

It is difficult at the outset to determine what characteristics are important when developing new ab initio methods of electronic structure theory. One of the ultimate goals
of quantum chemistry is to provide a predictive tool which can guide experimental efforts.
In order to be considered a quantitatively predictive model, an ab initio method must be
able to reliably produce numerical values for reaction energies which are accurate to within
≃ 1 kcal/mol of widely acknowledged benchmark data, a threshold often called “chemical
accuracy”. It should also be clear what cases a method is and is not appropriate for, and,
when necessary, it should be clear what to do to improve a poor result. A shortcoming of
many ab initio methods is that they do not offer a balanced treatment of the short-range
or “dynamical” and long-range or “non-dynamical” electronic interactions and therefore are
unreliable for applications involving chemically reactive processes, since the relative importance of these competing effects can vary rapidly moving from one region to another on the
corresponding PES. The goal is to develop a method which can give a balanced treatment
2

of both types of electronic interactions, while also having a straightforward and logical way
of improving the result in the case that the predictions are shown to be inadequate.

The starting point for single-reference (SR) electronic structure methods is the independent particle model (IPM), usually Hartree-Fock (HF), approximation [4–7], and the more
practical discretized algebraic form of the resulting equations based on the linear combination of atomic orbitals (LCAO) self-consistent field (SCF) formalism, which is the origin of
the basis set approximation defining ab initio models [8]. The HF method produces the best
single determinant description of the electronic wave function of interest and although it is
by now well established that the HF approximation yields over 99% of the total energy, its
ability to provide a reliable description of chemical phenomena is very limited due to the
need to describe relatively small energy differences in characterizing chemical processes. The
inadequacies of HF have been shown time and again, but as an outstanding early example,
Wahl’s study of the F2 molecule demonstrated that an SCF description predicts that F2
is unbound [9]. Despite the completely unphysical relative energetics often produced by
the HF approximation, the LCAO SCF formalism is still a valuable tool as it provides the
HF model, a critical element to our conceptual understanding of electronic structure and
chemical reactivity. To retain the HF picture and be able to make predictive calculations,
methods which improve upon the basic molecular orbital approximation must be developed.
The focus of these so-called ’post-HF methods’ is to accurately recover the small amount of
energy neglected at the HF level, a quantity which is known as the correlation energy. The
most popular ab initio post-HF approaches can be grouped into the variational and perturbative classes. Variational approaches provide an upper bound to the exact energy, as in
the configuration interaction (CI) theory [10–14], while perturbative approaches, such as the
3

many-body perturbation theory (MBPT) [15–22] and coupled cluster (CC) methods [23–27],
have other advantages which will be discussed shortly. To understand how these approaches
may be utilized to produce powerful new ab initio electronic structure approaches, their
individual strengths and weaknesses must first be understood.

The CI method is the most straightforward way of accounting for electronic correlation.
The CI wave function is a linear combination of the reference (usually HF) determinant and
Slater determinants obtained by exciting electrons from occupied to unoccupied orbitals. If
this determinantal expansion is complete (full CI), the correlation energy and total energy of
the molecular system will be exact within whatever basis set approximation was used used
in the calculation. Due to the fact that the number of Slater determinants grows factorially
with the size of the system, full CI calculations are impractical for systems of more than a few
electrons. Approximations are commonly made in order to reduce this expense, for example,
by truncating the CI expansion to include only the simplest classes of excitations, singles
and doubles (SD), which is called the CISD method. A particularly appealing feature of
the CI methods is that, by systematically adding classes of excitations, e.g., CI with singles,
doubles, and triples, CI with singles, doubles, triples, and quadruples, and so on, one may
obtain a series of increasingly accurate energies approaching the full CI value. Without this
systematically improvable nature, it is difficult to claim a particular level of convergence in
electronic structure calculations unless there is benchmark data available for comparison.
Unfortunately, truncated CI methods lack size-extensivity, i.e., the correct linear scaling
with the number of electrons, and the hierarchy converges slowly with the rank of excitation
included. While the lack of size-extensivity can be approximately accounted for at the CISD
level by adding Davidson corrections [28, 29], the slow convergence with level of excitation
4

to the full CI limit makes the CI method comparatively inefficient in the context of other
modern electronic structure methods.

Perturbation theory offers another systematically improvable hierarchy of methods for
determining the correlation energy. Although these methods are not variational, they are
strictly size-extensive at every level. In the SR-MBPT approach, the electronic Hamiltonian
is divided into an unperturbed part, corresponding to a single-determinantal reference description, usually the HF determinant, and a perturbation part, which usually describes the
electronic correlation. In this form of MBPT, introduced by Møller and Plesset (MP) [15],
the perturbation corrections to the reference wave function and energy are then calculated
using the Rayleigh-Schr¨dinger perturbation theory. Systematic improvements in the eleco
tronic energy may be made by considering perturbative corrections of increasing orders, i.e.,
second-order MP (MP2), third-order MP (MP3), and so on. Additionally, comparisons of
each order of perturbation with various CI ranks have made possible further division within
levels of perturbation, allowing for the selection of certain classes of excitations. As an
example, one may choose to calculate only certain components within the fourth-order of
MP theory, e.g., MP4 with only contribution from doubles (MP4D), doubles and quadruples (MP4DQ), singles, doubles, and quadruples (MP4SDQ), or singles, doubles, triples, and
quadruples (full MP4). This allows for certain computationally inexpensive higher-order
corrections to be included without significantly increasing time requirements. In general,
if the unperturbed HF determinant describing the molecular system is close to the exact
wave function for that system, the convergence of the MBPT series is usually very rapid;
however, when chemical bonds are stretched, the MBPT series becomes divergent. Thus,
MBPT produces very good energies near the equilibrium geometries of molecules; however,
5

unlike CI, in regions of PESs where bonds are broken or formed MBPT energies become
unphysical, making them inadequate for describing reactive PESs.

The conventional SR coupled-cluster (CC) theory is currently considered the preeminent
ab initio method for ground-state calculations of modest-sized molecular systems. Contrary
to the CI approach, which is characterized by a linear expansion of configuration state
functions, the CC approach uses an exponential ansatz for the wave function, inherently
assures that truncated forms of CC theory remain size-extensive, and produces a much
faster convergence to the full CI wave function. It can be shown using a perturbation theory
analysis that the improved convergence of CC as compared with the same level of truncation
in CI theory is due to higher-order excitations being folded in as products of lower-order
excitations by the exponential form of the CC ansatz. At the same time, thanks to the use of
diagram factorization techniques commonly employed in efficient computer implementations
of CC methods, the computer costs of CC calculations are similar to those characterizing
the CI approaches truncated at the same excitation levels. This is why CC methods can
offer higher accuracy at relatively lower costs as compared with CI or MBPT methods and
even though the energies produced are not variational, they are typically considerably more
accurate than those produced by CI at the same level of truncation. Excited states may also
be accessed in CC theory through the equation-of-motion (EOM) CC formalism [30–34] or
its symmetry-adapted-cluster configuration-interaction (SAC-CI) [35–39] or linear-response
CC analogs [40–44]. The most appealing electronic structure methods come as a result of
combining the different abovementioned approaches. EOMCC, for example, is just a CIlike expansion starting from an approximate ground-state wave function obtained using CC
theory. Approaches based on the CC and EOMCC formalisms which utilize perturbation
6

theory to obtain inexpensive corrections due to higher-order correlation effects also exist and
will be discussed in more detail shortly.

All of these appealing features of CC theory make it the most promising electronic structure approach of those considered so far, but there are still open problems yet to be completely addressed in CC theory. For instance, the most widely used and computationally
practical CC approximation, the CC with singles and doubles (CCSD) method [45–48], fails
completely when applied to describe a PES involving bond breaking. The CCSD method,
which is based on an iterative procedure with central processing unit (CPU) steps scaling
as n2 n4 , where no and nu are the numbers of occupied and unoccupied orbitals in the refero u
ence model, respectively, or as N 6 , where N is the size of the system expressed as the sum
of the exponents of the basis functions, neglects the important triply excited, quadruply
excited, and other higher-order clusters needed to describe bond breaking. Unfortunately,
the CC method with singles, doubles, and triples (CCSDT) [49, 50] and the CC approach
with singles, doubles, triples, and quadruples (CCSDTQ) [51–54] which include these clusters are prohibitively expensive for anything but small molecules, as they require iterative
steps that scale as n3 n5 (N 8 ), and n4 n6 (N 10 ), respectively. A parallel problem exists
o u
o u
in the case of the EOMCC or response CC methods where the basic singles and doubles
approximation (EOMCCSD) [31–33], which is characterized by iterative n2 n4 (N 6 ) scaling
o u
steps, fails to describe excited states dominated by two-electron and other many-electron
transitions out of the ground state. Again, the EOMCC theory with singles, doubles, and
triples (EOMCCSDT) [55–58] and the EOMCC approach with singles, doubles, triples, and
quadruples (EOMCCSDTQ) [59], which can describe such states, do not offer a suitable alternative in the majority of applications because of their prohibitively expensive iterative N 8
7

and N 10 scaling steps, respectively. The massive success of the SRCC theories in molecular applications [60–66], particularly for nondegenerate ground-states of molecules near their
equilibrium geometry and electronically excited states dominated by one-electron transitions,
in addition to substantial progress in recent years in code parallelization [67–77], and in the
development of various local correlation CC techniques (see, e.g., Refs. [78–83]), including,
for example the cluster-in-molecule CC method developed by the Piecuch group [84–86], continues to stimulate parallel effort toward the extension of CC theory to handle quasidegerate
states characterizing bond breaking and many-electron excitations with the same or close to
the same level of computational effort as that required by CCSD.

The most natural way to handle quasi-degenerate electronic states is to either turn to
one of the variants of multi-reference (MR) perturbation theory, such as the popular MCQDPT2 [87,88] and CASPT2 [89] methods, which are designed to handle large nondynamical
correlation effects and low-order dynamical correlation effects, the MRCI approaches, such
as the popular MRCI(Q) approximation [90, 91], or the genuine MRCC methods, which, in
analogy to the previously discussed SR analogs, offer a better treatment of the dynamical
correlation effects as compared with the MRMBPT or MRCI methods. The genuine MRCC
methods can be categorized into two types. The first is the hierarchy of the Fock-space or
valence-universal methods [92, 93], in which a single valence-universal wave operator operates on the system of interest and its ions which are obtained by removing one, two, etc.
active electrons from active orbitals. This is a convenient formalism when one is interested
in ionization potentials and electron affinities, however, these methods are unfavorable when
a wide range of geometries must be considered, as is the case in the generation of reactive
PESs. The other category of genuine MRCC methods, the Hilbert-space or state-universal
8

(SU) approaches [94], employ the Jeziorski-Monkhorst wave function ansatz in conjunction
with the multi-root Bloch wave-operator formalism. The SU-MRCC approaches have been
shown to produce very accurate results for the ground and excited states of systems undergoing severe geometrical transformations (see, e.g., Refs. [95–100]), making them a very
attractive choice within the context of PES generation. However, their routine use for such
applications is complicated by many factors. First, a model space of reference states must be
pre-defined by the user, which often contains many states irrelevent to a given problem, and
then a truncation scheme must be chosen which is compatible with the model space. This
requires expert-level decisions on a case-by-case basis for each molecular problem. Then,
even when appropriate choices are made, intruder states may appear [95–97], which can
severely complicate interpretation of results. The aforementioned ambiguous parameters
become even more undesirable when considered in the context of practical implementation.
The need to accomodate such general choices in combination with the massive number of
cluster amplitudes which must be computed make writing general SU-MRCC computer programs excessively difficult. While widespread routine use of genuine MRCC methods seems
unlikely in the near future, these methods have provided great insight into ways to improve
existing SRCC formalisms and inspired activity toward MRCC approaches dealing with a
single quantum state. We refer the reader to Ref. [101], and references therein for further
detailed discussion.

Due to the difficulties with implementing practical, user-friendly MRCC methods and the
prohibitive computational expenses characterizing the SR CCSDT and CCSDTQ approaches
and their EOM analogs, substantial research effort has been directed toward developing approximate approaches for including higher-order correlation effects within a SRCC formal9

ism. This led to both approximate iterative approaches, such as CCSDT-n [102–104], and
CCSDTQ-1 [105], as well as the more popular non-iterative approaches, such as CCSD[T]
[104, 106] and CCSD(T) [107], where triply excited T3 clusters are approximated using perturbative arguments which take the form of a relatively inexpensive non-iterative n3 n4 (N 7 )
o u
scaling step in addition to the cost of the underlying CCSD calculation. Methods were also
derived for cases when quadruple excitations should not be disregarded, where perturbative
noniterative corrections for both triples and quadruples are added, e.g., the CCSD(TQf )
method [108], characterized by a n2 n5 (N 7 ) scaling step. These and other noniterative
o u
perturbative CC approaches have become extremely popular because they efficiently account for most of the important connected triple or triple and quadruple excitations in a
user-friendly (“black-box”) fashion, while avoiding the steep iterative N 8 or N 10 scaling
steps required by full CCSDT or CCSDTQ. Unfortunately, their applicability is limited to
molecules near the equilibrium geometries, since the perturbative arguments used to derive the noniterative corrections of CCSD(T) and similar approaches fail when bonds are
streched or broken. A parallel challenge is found within the development of the standard
response CC and EOMCC approximations for excited states, in which the effects of triply
or triply and quadruply excited configurations are estimated using arguments originating
from MBPT (see, e.g., Refs. [109–114]). The perturbatively corrected EOMCC methods,
˜
such as, for example, EOMCCSD(T) [110], where the basic EOMCCSD approximation is
corrected for triples via a noniterative n3 n4 or N 7 scaling correction, break down for excited
o u
states having larger contributions due to doubly excited configurations, which is particularly
common for excited-state PESs along bond breaking coordinates (see, e.g., Ref. [115]).

Quite a few approaches have been suggested in recent years which attempt to overcome
10

the failures of the conventional perturbative SR CC and EOMCC methods at larger internuclear separations and for excited states dominated by many-electron transitions, while
avoiding the complexity of the genuine MRCC approaches. Examples which will be considered here include the externally corrected CC methods, such as the reduced MRCC (RMRCC) approaches [116–122], the completely renormalized (CR) CC methods [65,115,123–132],
and the active-space CC theories [54–57, 133–149] (see Ref. [101] for a recent review). All
of these methods are related in that they are concerned with improving the description of
bond-breaking processes and other cases involving electronic quasidegeneracies, while relying
on a SR-like formulation. Each of these methods will now be discussed, paying particular attention to the strengths of each approach and making mention of selected recent applications
in the literature.

The externally corrected SRCC methods represent an alternative to the perturbativelyderived corrections in which the CCSD equations corrected for terms containing the triply
(T3 ) and quadruply (T4 ) excited clusters are solved after replacing T3 and T4 amplitudes
by their values obtained in the cluster analysis of some non-CC wave function which exhibits good behavior at large internuclear separations, such as the projected unrestricted
HF [150, 151], valence bond [152–154], multiconfigurational SCF or complete active-space
SCF (CASSCF) [155–157], or MRCI wave functions [116–121]. Although all of these methods help in bond breaking situations, the latter, MRCI-corrected CCSD approach, referred
to as RMR-CCSD, and its RMR-CCSD(T) extension [121, 122], have shown the most substantial improvements in the CCSD results. Unfortunately, the generation of the MRCISD
wave function is very expensive when compared to the CCSD approach and MRCI is not sizeextensive, which limits the applicability of the RMR-CC approaches to smaller systems. One
11

can partly address these deficiencies by turning to the so-called (N ,M )-CCSD methods [158].
This notation implies that an M -reference general-model-space (GMS) SU-MRCCSD calculation [99,100] is corrected for higher-order clusters by drawing the relevent information from
M pertinent wave functions [159] produced by an N -reference MRCISD calculation. Extensive benchmarking and testing have shown the RMR-CCSD method and its aforementioned
extension with a perturbative correction for triples, RMR-CCSD(T) [121, 122, 160–172], as
well as the (N ,M )-CCSD methods [158, 173–181] to be quite accurate in practice, although
all of these approaches are very complex and require a lot of expertise.

The ground-state CR-CC [65, 123–131] and excited-state CR-EOMCC [65, 115, 123, 126,
129, 132] methods represent another class of approaches which were developed with the intention of removing the pervasive failings of the conventional CC/EOMCC perturbative
methods. These approaches are based on the more general formalism of the method of
moments of CC equations (MMCC) [65, 123–125, 127–129, 131, 182, 183]. In analogy to the
conventional perturbative methods, the CR-CC and CR-EOMCC methods allow one to calculate noniterative state-specific energy corrections corresponding to selected higher-order
excitations which are added to the energies obtained from conventional CC/EOMCC calculations, such as CCSD or EOMCCSD. These CR-CC and CR-EOMCC corrections are
based on the asymmetric energy expressions and resulting moment expansions which form
the underlying framework for all MMCC methods. The original CR-CC approaches, such as
CR-CCSD(T) [124,125] suffered from small errors due to a lack of strict size-extensivity [65],
but these issues were addressed in the more recent left-eigenstate CR-CC approaches, including CR-CC(2,3), which are based on yet another form of the of the moment expansion
of the full CI energy that defines the biorthogonal MMCC formalism [127–132]. The main
12

advantage of the CR-CC, CR-EOMCC, and other MMCC approaches is that the resulting
ground- and excited-state energies are not dependent on any choice of active orbitals or
other subjective parameters that one has to choose in MR calculations. Another advantage is the computational expense of the CR-CC and CR-EOMCC methods, which is on
the order of the conventional perturbative SRCC methods. For example, the noniterative
triples correction in the CR-CC(2,3) approach scales as n3 n4 or N 7 , in analogy to the scalo u
ing of CCSD(T). The CR-CC(2,3) method has already been proven to be very accurate and
robust, particularly in applications involving single bond breaking [127–130,184–188], mechanistic studies involving biradicals [127, 128, 130, 131, 189–193], and singlet-triplet gaps in
biradical/magnetic systems [130, 131, 192]. Similar successes have been reported for the CREOMCC(2,3) method [129, 132] and its CR-EOMCCSD(T) predecessor [65, 115, 123, 126],
where noniterative, n3 n4 or N 7 scaling corrections for triples are added to the underlyo u
ing EOMCCSD energies. For example, the CR-EOMCC(2,3) method has recently been
shown to accurately reproduce adiabatic excitation energies for various closed- and openshell molecules which are believed to be dominated by two-electron transitions out of the
ground-state [132].

The third and final effort toward constructing a SR formalism capable of handling
stronger non-dynamical correlations which will be considered here are the active-space CC
and EOMCC methods [54, 133–149] (see Ref. [101] for a recent review). By specifically
targeting the higher-order cluster and excitation amplitudes which become large in such
situations by assigning a small subset of active orbitals defining these excitations, highly
accurate results may be obtained while avoiding much of the expense of the higher-order
parent methods. For example, the CCSDt and CCSDtq methods are based on the idea of
13

selecting T3 and T4 clusters within the CCSDT and CCSDTQ systems of equations using
active orbitals [54, 133–146]. When properly implemented, the most expensive CPU steps of
2 2
the full CCSDt and CCSDtq approaches scale as No Nu n2 n4 and No Nu n2 n4 , respectively,
o u
o u

where No and Nu are the numbers of active occupied and unoccupied orbitals, respectively.
All active-space approaches have a few distinct advantages over competing methods. As
an example, take again the CCSDt and CCSDtq approaches. These methods recover the
exact results of their parent CCSDT and CCSDTQ approaches, respectively, in the limit
that all orbitals are assigned as active. They are also systematically improvable, approaching the CCSDT and CCSDTQ limits as the number of active orbitals is increased. Given
an appropriate selection of a usually small number of active orbitals, the active-space CC
results are typically virtually perfect when compared to the values produced by the parent
methods. Another advantage is the fact that all active-space CC methods are characterized
by relatively low computational scalings, which are small prefactors times the n2 n4 steps
o u
of the CCSD type. Naturally, following the development of the ground-state CCSDt and
CCSDtq active-space approaches, the EOMCCSDt and EOMCCSDtq methods were developed, with the first implementation of EOMCCSDt and the proposal for all such EOMCCbased methods occurring in the Piecuch group [55–58]. In addition to the EOMCCSDt
approach, another class of the active-space EOMCC methods was also developed by the
Piecuch group, namely the active-space electron attached (EA) and ionized (IP) EOMCC
approaches [147–149], which may be used to generate ground and excited states of valence
open-shell systems out of a related closed-shell ground-state, as in radical species. Generally,
the EA- and IP-EOMCC methods (see Refs. [66,147–149] and references therein for information) have the distinct advantage over the traditional open-shell EOMCC approaches based

14

on restricted open-shell HF (ROHF) or unrestricted HF (UHF) references in the fact that
they automatically generate orthogonally spin-adapted wave functions (a difficult thing to
accomplish in CC theory). For comparison, traditional ROHF- or UHF-based open-shell
CC/EOMCC methods introduce spin-contamination to the resulting wave functions. Spincontamination can have non-negligible effects on the energy of the states generated, but,
more importantly, it prohibits the designation of the spin-symmetry of a given state, causing
the identification of a particular state to become most inconvenient. This is an important
issue, particularly when the excited states of open-shell systems are examined, which will be
returned to periodically throughout this dissertation.

While promising electronic structure approaches have been and continue to be developed,
many of which provide a highly accurate and balanced description of chemical species typically encountered while scanning molecular PESs, reliable methods are still prohibitively
expensive when one faces the calculation of the hundreds or thousands of points which are
typically needed to sample these PESs. Electronic structure methods could certainly benefit
from an auxiliary approach for predicting points on the PES based on inexpensive calculations with either less expensive quantum chemistry approaches or smaller basis sets, which
would ameliorate the staggering computational expense of generating hundreds or thousands
of points using high levels of electronic structure theory. To address this problem, an ab initio
extrapolation scheme has been proposed that predicts the PES corresponding to expensive
high-level calculations from the results of a series of comparatively inexpensive lower-level
calculations using the concept of correlation energy scaling. The PES extrapolation scheme
of this type was originally suggested in Ref. [184] and was designed such that it can be
used in conjunction with any typical post-HF or post-CASSCF electronic structure method.
15

Thus, the second principal goal of this dissertation is to develop and perform benchmark
studies on the PES extrapolation schemes based on correlation energy scaling ideas, which
allow one to generate chemically reactive molecular PESs with very little or almost no information from high-level calculations while retaining the desired accuracy that high-level ab
initio electronic structure methods provide.
Finally, after new methods are developed and shown to be useful in benchmark studies,
general-purpose computer programs should be written and distributed based on the successful theories to give scientists and engineers around the world a tool for interpreting, or even
predicting, experimental results. The third and final goal in this dissertation is to outline
the key details for the computer implementations of the open-shell EOMCCSD and EAand IP-EOMCC methods which which were written for the GAMESS electronic structure
software package [194], a freely available suite of computer codes with tens of thousands
of registered users. The three goals of this dissertation, summarized in Chapter 2, are addressed in this thesis, with a chapter being devoted to each. Thus, Chapter 3 is devoted to
the development and application of new ground and excited-state CC/EOMCC methods for
chemically reactive systems including the CR-CC/EOMCC methods and the active-space
EA- and IP-EOMCC approaches. Chapter 4 begins with a discussion motivating the need
for auxiliary methods to aid in the generation of molecular PESs and moves toward a presentation of the theory and various applications which help demonstrate various ways the PES
extrapolation schemes based on correlation energy scaling may be used. Chapter 5 covers
the development of computer codes for the GAMESS software package in detail, presenting
programmable equations for a few of the theories implemented in this work as well as a brief
discussion of how they are solved in practice.

16

Chapter 2
Project Objectives
The main objectives of this work are:

A. Performing benchmark calculations using the new generations of CR-CC approaches,
including barrier heights of hydrogen transfer, heavy-atom transfer, nucleophilic substitution, and unimolecular and association reactions, and PESs for addition and isomerization reactions involving species with varying degrees of electronic degeneracy in
order to demonstrate what levels of theory are appropriate in different situations.
B. Developing and performing benchmark applications for the new generations of CREOMCC approaches, including the calculations of vertical excitation energies and
environment-induced spectral shifts of organic chromophores and the calculation of
excited-state PESs along bond-breaking channels.
C. Performing benchmark applications for the EA- and IP-EOMCC methods including
geometry optimizations and the calculation of adiabatic excitation energies of small
open-shell molecules.
17

D. Developing the PES extrapolation schemes based on correlation energy scaling and
performing benchmark applications to demonstrate the full range of capabilities offered
and typical accuracies which should be expected in practice.
E. Outlining the key details of computer implementations of the ROHF-based EOMCCSD and RHF- or ROHF-based EA- and IP-EOMCC programs recently developed
for GAMESS.

18

Chapter 3
Applications of Coupled-Cluster and
Equation-of-Motion Coupled-Cluster
Methods

3.1

Theory

As explained in the Introduction, the SR CC and EOMCC methods are the preeminent
methods for the determination of electronic energies and properties in chemistry. The majority of this dissertation is concerned with the new generations of the CC and EOMCC
methods which are useful in situations where the conventional CC and EOMCC approximations fail. We begin by reviewing the conventional CC and EOMCC theories in Sects. (3.1.1)
and (3.1.2), respectively. The CR-CC and CR-EOMCC approaches and the active-space CC
and EOMCC methods, with special emphasis on their EA- and IP-EOMCC extensions are
discussed afterwards, in Sects. (3.1.3) and (3.1.4), respectively.
19

3.1.1

Single-Reference Coupled-Cluster Theory for Ground States

In the SRCC theory, the ground-state wave function |Ψ0 of an N -electron system is expressed using the exponential ansatz,

|Ψ0 = eT |Φ ,

(3.1)

where T is the cluster operator and |Φ is an IPM reference configuration, e.g., the HF
determinant (throughout this thesis, the RHF or ROHF determinant). Typically, we truncate
the many–body expansion of T at a conveniently chosen excitation level mT , to obtain an
approximate T , i.e., T ≃ T (A) , hoping that one can reach the desired accuracies with
mT << N . The truncated cluster operator T (A) defining the approximate CC method A is
given by
T (A) =

mT

Tn ,

(3.2)

n=1

with
i ...i

Tn =
i1 <···<in ,a1 <···<an

n
ta1 ...an aa1 · · · aan ain · · · ai1 ,
1

(3.3)

i ...i

n
where Tn is the n-body component of T (A) , ta1 ...an are the cluster amplitudes, i1 , i2 , i3 ,. . . or
1

i, j, k,. . . (a1 , a2 , a3 ,. . . or a, b, c,. . . ) are the spin-orbitals occupied (unoccupied) in the reference determinant |Φ , and ap (ap ) are the creation (annihilation) operators associated with
the orthonormal spin-orbital basis set {|p }. In Eq. (3.2), mT defines the maximum many–
body component included in the truncated cluster operator T (A) , returning the exact, full
CI ground-state wave function when mT = N (recall that N is the number of electrons in
the system of interest). When the cluster operator is truncated such that mT < N , Eq.
20

(3.1) leads to the well-known hierarchy of standard CC approximations: CCSD when T is
truncated at doubly excited clusters (T ≃ T (CCSD) = T1 + T2 , mT = 2); CCSDT when T is
truncated at triply excited clusters (T ≃ T (CCSDT) = T1 +T2 +T3 , mT = 3), CCSDTQ when
T is truncated at quadruply excited clusters (T ≃ T (CCSDTQ) = T1 + T2 + T3 + T4 , mT = 4),
etc.
The SRCC equations are formally obtained by inserting the CC wave function |Ψ0 , Eq.
(3.1), into the electronic Schr¨dinger equation,
o

H|Ψ0 = E0 |Ψ0 ,

(3.4)

(A)
to obtain the connected cluster
premultiplying both sides of Eq. (3.4) on the left by e−T

form of the Schr¨dinger equation [25–27],
o

¯
H (A) |Φ = E0 |Φ ,

(3.5)

(A)
(A)
(A)
¯
H (A) = e−T HeT
= (HeT )C ,

(3.6)

with

a ...a

and projecting Eq. (3.5) onto the excited determinants |Φi 1...inn = aa1 . . . aan ai1 . . . ain |Φ
1
corresponding to the many-body compoenents Tn included in T (A) . Here, the subscript
C indicates the connected part of a given operator expression. Eq. (3.6) is known as the
similarity-transformed Hamiltonian of the CC theory. The resulting system of equations for
i ...in
cluster amplitudes ta1 ...an defining T (A) has the following general form:
1

a ...a

¯
Φi 1...inn |H (A) |Φ = 0, i1 < · · · < in , a1 < · · · < an ,
1
21

(3.7)

a ...a

where n = 1, . . . , mT and |Φi 1...inn are the n-tuply excited determinants relative to the
1
reference determinant |Φ . As an example, the CCSD amplitude equations are obtained by
projecting Eq. (3.5), where T (A) = T (CCSD) = T1 + T2 , onto all singly and doubly excited
determinants, |Φa and |Φab , respectively, such that
i
ij

¯
Φa |H (CCSD) |Φ = 0,
i
¯
Φab |H (CCSD) |Φ = 0,
ij

(3.8)

where
¯
H (CCSD) = e−(T1 +T2 ) He(T1 +T2 ) = (HN eT1 +T2 )C

(3.9)

is the similarity-transformed Hamiltonian of the CCSD approach. These equations are solved
ij

for the one- and two-body cluster amplitudes, ti and tab , respectively, which appear in the
a
definitions of the one- and two-body cluster operators,

ti aa ai
a

(3.10)

ij

T1 =

(3.11)

i,a

and
tab aa ab aj ai .

T2 =
i<j,a<b

As explained in the Introduction, the most expensive CPU steps of the CCSD calculations,
based on Eq. (3.8) scale as n2 n4 or N 6 , where N is a measure of the system size.
o u

Once the general system of nonlinear, energy-independent equations for cluster ampli(A)

tudes, Eq. (3.7), is solved for T (A) , the energy E0
22

corresponding to the standard SRCC

method A is calculated by projecting Eq. (3.5) onto the reference determinant |Φ such that

(A)

E0

= Φ|(HeT

(A)

)C |Φ .

(3.12)

By introducing the normal product form of the Hamiltonian, HN = H − Φ|H|Φ , this
equation can be rewritten as

(A)

∆E0
(A)

where ∆E0
(A)

E0

= Φ|(HN eT

(A)

)C |Φ ,

(3.13)

(A)

is the total energy of the system relative to reference energy, i.e., ∆E0

=

− Φ|H|Φ , which is equivalent to the correlation energy when |Φ is the HF state.

For CCSD and all higher-order CC methods it is interesting to note that, at least for the
quantum-chemistry Hamiltonians of interest in this dissertation, which include only twobody electron-electron interactions, the SRCC energy expression is

(A)

E0

1 2
= Φ|H|Φ + Φ|[HN (T1 + T2 + 2 T1 )]C |Φ

(3.14)

or
(A)

∆E0

1 2
= Φ|[HN (T1 + T2 + 2 T1 )]C |Φ .
(A)

From Eq. (3.15) it can be seen that ∆E0

(3.15)

depends only on T1 and T2 clusters, independent

of the excitation level mT defining SRCC method A as long as mT ≥ 2. It follows then
that the T1 and T2 clusters obtained at the level of the basic CCSD approximation are
already sufficient to calculate the CC energy in a complete manner. This does not make
the CCSD approach the exact theory. Adding higher-than-doubly excited clusters in the

23

cluster expansion can significantly improve the quality of the CC energy through coupling
of higher-order clusters with the T1 and T2 amplitudes in the SRCC equations, Eq. (3.7).
This suggests, however, an approximate treatment of higher-than-doubly excited cluster
amplitudes may be sufficient to improve the quality of the CC energy in a satisfactory way.

As a final note about the ground-state CC theory, it should be mentioned that spincontamination can become an issue depending on the choice of method used to generate
the reference determinant. For example, if one is interested in describing a closed-shell
molecule, the RHF reference |Φ may be employed, which is itself a spin eigenfunction,
ˆ
ˆ
S 2 |Φ = S(S + 1)¯2 |Φ , where S is the total spin operator, as is any approximate CC wave
function produced from it through Eq. (3.1). In this case, spin-contamination is not a
problem in the ground-state wave function or for any wave functions derived from it, since T
ˆ
and eT commute with S 2 . However, if instead one wishes to describe an open-shell system,
i.e., a radical or a system with even number of electrons of non-singlet multiplicity, the
restricted open-shell HF (ROHF) or unrestricted HF (UHF) references may be employed,
but one must then be careful when doing so. While the ROHF wave function is spin-adapted
(UHF wave functions are not), in contrast to the RHF closed-shell case, a CC wavefunction
constructed from an ROHF reference determinant |Φ will not be automatically spin-adapted
due to the nonlinear nature of CC theory. The spin-contamination can introduce small
errors in energies and present other difficulties when UHF-based and ROHF-based CC wave
functions are employed. This issue will be further discussed in the next section of this
dissertation which addresses the generation of electronically excited, electron-attached, and
ionized states out of ground-state CC wave functions.
24

3.1.2

Equation-of-Motion Coupled-Cluster Theory for Electronically Excited, Electron-Attached, and Ionized States

The ground-state CC theory, described in Sect. (3.1.1), can be extended to excited, electronattached, and ionized states by application of a linear excitation operator Rµ to the CC
ground state, |Ψ0 . This leads to the EOMCC formalisms. Many different EOMCC methods may be formulated by modifying the basic definition of Rµ . In this dissertation, the
particle-conserving excitation energy (EE) EOMCC as well as the particle-nonconserving
electron-attatched (EA) EOMCC and ionized (IP) EOMCC theories are considered. While
discussions of the EA- and IP-EOMCC models are constrained to schemes where only one
electron is added or removed, other approaches in these catagories can easily be imagined
with more than one electron added or removed. In the remainder of this section, an overview
is presented introducing important concepts common to all EOMCC theories, while the specific details defining the EE-EOMCC and the EA- and IP-EOMCC theories are outlined in
Sections (3.1.2.1) and (3.1.2.2), respectively.
In general, an ansatz may be written expressing the exact excited-state, electron-attached,
or ionized wave function |Ψµ corresponding to state µ of interest as a linear excitation
operator Rµ applied to the ground-state SRCC wave function, i.e.,

|Ψµ = Rµ |Ψ0 ,

(3.16)

where |Ψ0 is defined by Eq. (3.1). In the exact EOMCC theory, the cluster operator T
and the exciting, electron-attaching, or ionizing operator Rµ are sums of all relevant manybody components needed to generate the system of interest. To obtain Rµ one must solve
25

the eigenvalue problem resulting from the substitution of Eq. (3.16) into the Schr¨dinger
o
equation, H|Ψµ = Eµ |Ψµ , to obtain

¯
(H N,open Rµ,open )C |Φ = ωµ Rµ |Φ ,

(3.17)

in the subspace spanned by all determinants corresponding to the many-body components
¯
included in Rµ . Here, H N,open = (HN eT )C,open = e−T HN eT − (HN eT )C,closed is the
similarity-transformed Hamiltonian of the CC theory in the normal-ordered form relative
to the Fermi vacuum |Φ , where the subscripts “open” and “closed” refer to the open (i.e.,
having external lines) and closed (i.e., having no external lines) parts of a given operator expression, obtained by solving the corresponding ground-state CC equations for T as described
in the previous section, and ωµ = Eµ − E0 is the vertical excitation (or electron-attachment
or ionization) energy.

3.1.2.1

Excited States

In the particle conserving EE-EOMCC theory, excited-state energies and wave functions are
obtained for an N -electron system by the application of a linear excitation operator Rµ of
the form
(A)
(A)
(A)
Rµ = Rµ,0 + Rµ,open ≡ rµ,0 1 +

mR

Rµ,n ,

(3.18)

n=1

where mR ≤ N and

Rµ,n =
i1 <···<in ,a1 <···<an

i1 ...in
rµ,a1 ...an aa1 · · · aan ain · · · ai1 ,

26

(3.19)

(A)

onto the N -electron ground-state SRCC wave function |Ψ0

. The EE-EOMCC method is

(A)

characterized by the linear excitation operator, Rµ , in Eq. (3.18) having exactly the same
number of creation and annihilation operators in each many-body component Rµ,n . Just
as in the cluster expansion of the ground-state wave function, in practice the many-body
(A)

expansion of Rµ

in Eq. (3.18) is truncated at some excitation level mR < N (usually

the same level of excitation as in the cluster operator used in the preceding ground-state
i ...i

1 n
CC calculation, i.e., mR = mT ). To determine the amplitudes rµ,a1 ...an with n > 1, we

solve the eigenvalue problem given by Eq. (3.18) by diagonalizing the similarity-transformed
a ...a
¯
Hamiltonian H (A) , Eq. (3.6), in a space spanned by the excited determinants |Φi 1...inn ,
1
(A)

with n = 1, ..., mR , corresponding to the many-body excitation operators included in Rµ .
In general, in order for Eq. (3.18) to hold and to obtain a size-intensive description [44, 196]
of vertical excitation energies, mR should not exceed mT [34], but, as already mentioned,
one typically chooses mR = mT .
For example, in the EOMCCSD method (note that the EE-EOMCC methods are often
abbreviated as simply EOMCC), where Rµ is approximated as

(CCSD)

Rµ

= Rµ,0 + Rµ,1 + Rµ,2 ,

(3.20)

with
i
rµ,a aa ai

(3.21)

ij

Rµ,1 =

(3.22)

i,a

and
rµ,ab aa ab aj ai ,

Rµ,2 =
i<j,a<b

ij

i
we obtain the singly and doubly excited rµ,a and rµ,ab amplitudes and EOMCSD vertical

27

excitation energies
(CCSD)

ωµ

(CCSD)

= Eµ

(CCSD)

− E0

(3.23)

by solving the system

(CCSD) i
¯ (CCSD) (CCSD)
Φa |(H N,open Rµ,open )C |Φ = ωµ
rµ,a
i
(CCSD)

(CCSD)

(CCSD) ij
rµ,ab .

¯
Φab |(H N,open Rµ,open )C |Φ = ωµ
ij

(3.24)
(3.25)

In other words, the EOMCCSD amplitudes and energies are determined by diagonalizing
the matrix representing the similarity-transformed Hamiltonian of CCSD, Eq. (3.9), in a
space of all singly and doubly excited determinants,

¯ SS H SD
¯

 H
¯
H CCSD = 
.
¯
¯ DS H DD
H


(3.26)

In analogy to the ground-state CCSD calculations, the most expensive CPU steps needed to
do this scale as n2 n4 or N 6 .
o u
While the EE-EOMCC methods can generate very accurate excited-state energetics and
properties, spin-contamination of the ground-state when open-shell systems are treated with
ROHF and UHF references can be sometimes problematic, introducing small errors into the
calculations and preventing the identification of multiplicities of excited states, which is a
useful guide when attempting to sort out states, particularly in situations involving neardegeneracies. One way to address this issue is to employ particle-nonconserving EOMCC
approaches such as the EA- and IP-EOMCC methods, which build open-shell ground and
excited states out of a related closed-shell CC wave function generated with a RHF reference,
28

automatically assuring that the resulting ground- and excited-state wave functions are spinadapted. A brief description of the EA- and IP-EOMCC methods is given in the following
section.

3.1.2.2

Electron-Attached and Ionized States

In the particle-nonconserving EA- and IP-EOMCC approaches, ground- and excited-state
wave functions are generated by solving the eigenvalue problem given by

(N ±1)

¯
(HN,open Rµ

(N ±1)

where Rµ

(N ±1) (N ±1)
Rµ
|Φ ,

)C |Φ = ωµ

(3.27)

are particle-nonconserving operators, generating electronic states of (N ± 1)-

electron systems, given by
(N +1)
Rµ
=

mR

Rµ,(n+1)p-nh

(3.28)

Rµ,(n+1)h-np ,

(3.29)

n=0

and
(N −1)
Rµ
=

mR
n=0

where
i ...i

Rµ,(n+1)p-nh =
i1 <···<in ,a<a1 <···<an

n
raa1 ...an aa aa1 . . . aan ain . . . ai1
1

(3.30)

ii1 ...in
r a1 ...an aa1 . . . aan ain . . . ai1 ai ,

(3.31)

and
Rµ,(n+1)h-np =
i1 <···<in <i,a1 <···<an

with mR = N in the exact case and mR < N in the approximate schemes. In the above
equations, |Φ is an N -electron reference determinant (e.g., the RHF reference) and T , used to
¯
define HN in Eq. (3.27) and |Ψ0 in Eq. (3.16), is the cluster operator of the SRCC theory, as
29

applied to the N -electron reference system. If a closed-shell reference system is used to define
(N ±1)

the wave functions |Ψµ through Eq. (3.16), with Rµ being one of the Rµ

operators,

the similarity transformed Hamiltonian defining the EA-EOMCC and IP-EOMCC methods
commutes with the S 2 and Sz operators. The result is that these methods produce openshell eigenstates which are orthogonally spin-adapted, which means that spin-contamination
issues which plague the standard spin-orbital-based open-shell EOMCC implementations
that utilize ROHF or UHF references are avoided entirely.
In general, and in analogy to the EE-EOMCC case, when constructing approximate EAEOMCC and IP-EOMCC schemes, the connected form of the eigenvalue problem displayed
in Eq. (3.27) and the size intensivity of the resulting electron-attachment or ionization
energies
(N ±1)

ωµ

(N ±1)

= Eµ

(N )

− E0

(3.32)

are retained when mR ≤ mT [34]. The common approaches to designing the EA- and
IP-EOMCC approximations are built upon approximate N -electron CCSD reference wave
(N ±1)

functions with Rµ

truncated such that mR = mT − 1 or mR = mT . In the basic

EA-EOMCCSD(2p-1h) and IP-EOMCCSD(2h-1p) methods, we use mR = 1 and mT = 2.
Thus, after solving the CCSD equations for an N -electron reference system, we diagonalize
¯
the similarity transformed Hamiltonian H (CCSD) in the (N + 1)-electron subspace of the
|Φa = aa |Φ and |Φab = aa ab aj |Φ determinants in the EA-EOMCCSD(2p-1h) case and
j
b
the (N − 1)-electron subspace of the |Φi = ai |Φ and |Φij = ab aj ai |Φ determinants in

the IP-EOMCCSD(2h-1p) case, obtaining the 1p amplitudes ra and the 2p-1h amplitudes
(N +1)

j

rab , along with the corresponding energies ωµ

, in the former case, or the 1h ampli(N −1)

ij

tudes ri , the 2h-1p amplitudes r b , and the corresponding energies ωµ
30

in the latter

case. While these methods provide an initial approximation, the EA-EOMCCSD(3p-2h)
and IP-EOMCCSD(3h-2p) approaches, in which mR = mT = 2, provide a much better
description by incorporating the 3p-2h and 3h-2p components in the electron-attaching and
(N +1)

ionizing operators, Rµ

(N −1)

and Rµ

, respectively. Unfortunately, the full inclusion of

the Rµ,3p-2h and Rµ,3h-2p terms in the EA- and IP-EOMCC calculations comes at a rather
high price, increasing the N 5 -like no n4 and n2 n3 operations defining the iterative diagou
o u
nalization steps of EA-EOMCCSD(2p-1h) and IP-EOMCCSD(2h-1p), respectively, to the
N 7 -like n2 n5 and n3 n4 steps. In the EA-EOMCCSD(3p-2h) and IP-EOMCCSD(3h-2p)
o u
o u
¯
schemes, the similarity-transformed Hamiltonian H (CCSD) , obtained in the CCSD calculations for the N -electron reference system, is diagonalized in the N + 1-electron subspace
spanned by the |Φa , |Φab , and |Φabc = aa ab ac ak aj |Φ determinants in the former case or
j
jk
a
ab
the N − 1-electron subspace spanned by the |Φi , |Φij , and |Φijk = ab ac ak aj ai |Φ deterj

minants in the latter case. From these diagonalizations, the ra , rab , and 3p-2h amplitudes
jk

ij

ijk

N
rabc , along with the corresponding energies ωµ +1 , or the ri , r b , and 3h-2p amplitudes r bc ,
N
along with the corresponding energies ωµ −1 are produced, which define the results of the

EA-EOMCCSD(3p-2h) and IP-EOMCCSD(3h-2p) calculations, respectively. In many cases,
(N ±1)

it appears that the 3p-2h and 3h-2p effects brought through the Rµ

operators play a

much more significant role than the triply excited components of the cluster operator T . In
fact, it is usually not necessary to include the T3 clusters in T until the 4p-3h and 4h-3p effects
become important. In Sect. (3.1.4) the active-space variants of the EA-EOMCCSD(3p-2h)
and IP-EOMCCSD(3h-2p) methods are presented, which is one way to retain the accuracy
of these methods while avoiding their steep computer cost increase with no and nu described
above.

31

3.1.3

The Method of Moments of Coupled-Cluster Equations

The conventional CC and EOMCC methods summarized in Sects. (3.1.1) and (3.1.2) are
useful, but one usually faces a challenge of having to correct the results of the low-level
CC/EOMCC calculations, such as CCSD or EOMCCSD, for the higher-order correlation
effects neglected by lower levels of CC/EOMCC theory without making the calculations
prohibitively expensive. One would also like to make sure that the corrections to the CCSD,
EOMCCSD, or other CC/EOMCC energies are robust in situations, such as bond breaking
or excited states dominated by two-electron transitions, where the traditional perturbative
corrections of the CCSD(T) type fail. The MMCC theory summarized below provides such
robust and computationally attractive corrections to the CCSD, EOMCCSD, and other
conventional CC/EOMCC energies.
The central focus of the MMCC theory is obtaining the non-iterative, state-specific,
energy corrections
(A)

δµ

(A)

≡ Eµ − Eµ ,

(3.33)

which recover the exact, full CI energies Eµ when added to the corresponding ground- and
(A)

excited-state energies, Eµ , obtained from the conventional CC/EOMCC approximation A.
The goal of any method based on MMCC theory is to estimate these corrections using the
underlying moment energy expansions, such that the resulting MMCC energies, defined as

(MMCC)

Eµ

(A)

= Eµ

(MMCC)

+ δµ

,

(3.34)

are good approximations to the corresponding exact energies Eµ .
(A)

All MMCC corrections δµ

are obtained via expansions of Eµ in terms of the general32

ized moments of CC/EOMCC equations defining a given CC/EOMCC approximation. The
ground-state moments are simply projections of the connected cluster form of the Schr¨dinger
o
a ...a

equation, Eq. (3.5), on the excited determinants, |Φi 1...inn with n > mA disregarded in the
1
conventional CC calculations

a ...a ¯
i1 ...in
M0,a ...an (mA ) = Φi 1...inn |H (A) |Φ ,
1
1

(3.35)

where mA = mT is the maximum level of excitation included in the CC calculation being
corrected. The excited-state moments needed to correct the EOMCC energies resulting from
truncating T and Rµ at the mA -body components (so that mT = mR = mA ) are projections
a ...a

of the EOMCC equations on the excited determinants |Φi 1...inn ,
1
i1 ...in
a ...a ¯ (A) (A)
Mµ,a1 ...an (mA ) = Φi 1...inn |Hopen Rµ,open |Φ .
1

(3.36)

These moments are central to the ground- and excited-state MMCC theory and will be
shown to be crucial quantities for evaluating the desired CR-CC and CR-EOMCC energy
corrections in the following sections.

(A)

Several ways of expressing the δµ

i ...i

1 n
corrections in terms of moments Mµ,a1 ...an (mA )

have been proposed to date [65,123–125,127–129,131,182,183]. The original and historically
oldest formula, obtained in [124] for the ground states and [182] for excited states has the
33

following form:

(A)

δµ

(A)

≡ Eµ − Eµ
N

n

Ψµ |Cn−k (mA ) Mµ,k (mA )|Φ /

=
n=mA +1 k=mA +1
(A)

Ψµ |Rµ eT

(A)

|Φ .

(3.37)

Here,
Cn−k (mA ) = (eT

(A)

)n−k

(3.38)

(A)
are the (n − k)-body components of the exponential wave operator eT , defining the CC

method A, |Ψµ is the full CI ground- (µ = 0) or excited- (µ > 0) state, and

i ...i

Mµ,k (mA ) =
i1 <···<ik ,a1 <···<ak

1 k
Mµ,a1 ...ak (mA ) aa1 · · · aan ain · · · ai1 (n ≥ 1)

(3.39)

i ...i

1 k
where moments Mµ,a1 ...ak (mA ) are defined by Eqs. (3.35) and (3.36). Thus, Eq. (3.37)
i ...i

1 k
states that one has to calculate quantities Cn−k (mA ), Eq. (3.38), and moments Mµ,a1 ...ak (mA ),
(A)

with k > mA , to determine the noniterative energy correction δµ , Eq. (3.33). The
Cn−k (mA ) terms are very easy to calculate. The zero-body term, C0 (mA ), equals 1; the
2
one-body term, C1 (mA ), equals T1 ; the two-body term, C2 (mA ), equals T2 + 1 T1 if mA ≥ 2;
2
3
3
the three-body term C3 (mA ) equals T1 T2 + 1 T1 if mA = 2 and T3 + T1 T2 + 1 T1 if mA ≥ 3,
6
6
i ...i

1 n
etc. The computation of moments Mµ,a1 ...an (mA ) for the most interesting cases of correct-

ing the CCSD or EOMCCSD energies (mA = 2) is straightforward too, particularly if we
limit ourselves to the corrections due to triples (k = 3) or quadruples (k = 4).
As an example, if one is interested in recovering the exact ground-state energy E0 through
34

(CCSD)

the addition of the full correction δ0

(CCSD)

to the CCSD energy Eµ

(where mA = 2),

i1 ...ik
one has to consider the generalized moments of the CCSD equations M0,a ...a (2) with
1

k

k > 2. After a quick diagrammatic analysis, we can show that this seemingly long expansion
i1 ...ik
contains moments M0,a ...a (2) with k ≤ 6 only, since the electronic Hamiltonian contains
1

k

(CCSD)

only up to two-body interactions. Thus, all terms which make up the correction δ0
approaching the exact energy contain relatively few moments, namely,

a ...a ¯
i1 ...ik
M0,a ...a (2) = Φi 1...i k |H (CCSD) |Φ ,
1 k
1 k

k = 3 − 6.

(3.40)

The projections of the CCSD equations on higher-than-hextuply excited configurations do
not have to be calculated, since for Hamiltonians containing up to two-body interactions
i1 ...ik
the generalized moments M0,a ...a (2) with k > 6 vanish. Similar simplifications occur
1

k

in the case of correcting the excited-state EOMCCSD energies, where the only moments
i1 ...ik
M0,a ...a (2) that matter are those with k = 3 − 8. Although this is a considerable reduction
1

k

of the computer effort, it is usually not computationally feasible to calculate up to six-body or
higher moments to obtain a given MMCC correction. The CR-CC and CR-EOMCC methods
discussed in Sects. (3.1.3.1)-(3.1.3.3) address this issue by focusing on the approximate
i ...i

1 k
corrections due to triples and quadruples which use moments Mµ,a1 ...ak (2) with k = 3 and

4 only and simplify Eq. (3.37) accordingly.

3.1.3.1

Completely Renormalized Coupled-Cluster and Equation-of-Motion CoupledCluster Approaches

In general, the CR-CC and CR-EOMCC approaches are obtained by approximating |Ψµ
in Eq. (3.37) by a quasi-perturbative form that brings information about the desired cor35

(A)

relation effects we want Eµ

to be corrected for. In particular, the CR-CCSD(T) [124],

CR-CCSD(TQ) [124, 125] and CR-EOMCCSD(T) [115] methods, which are of interest in
this dissertation, are obtained when the wave functions |Ψµ in Eq. (3.37) are approximated by low-order MBPT-like expressions. In the CR-CCSD(T) method which corrects
(CCSD)

the ground-state CCSD energy E0

for triply excited clusters, |Ψ0 in Eq. (3.37), where

mA = 2, is replaced by the following second-order-type, MBPT(2)[SDT]-like expression

CCSD(T)

|Ψ0

[2]

= (1 + T1 + T2 + T3 + Z3 )|Φ ,

(3.41)

where T1 and T2 are the singly and doubly excited clusters obtained in the CCSD calculations,
the
[2]

(3)

T3 |Φ = R0 (VN T2 )C |Φ

(3.42)

term is an approximation of the connected triples (T3 ) contribution, which is correct through
second order, and
(3)

Z3 |Φ = R0 VN T1 |Φ

(3.43)

is the disconnected triples correction, which distinguishes the CCSD(T) approach from its
(3)

CCSD[T] predecessor. Conventional MBPT notation is used here, in which R0

designates

the three-body component of the MBPT reduced resolvent and VN is the two-body part
of HN . In the CR-CCSD(TQ) method (note that here and elsewhere the so-called CRCCSD(TQ),b variant is implied and that a discussion of the other variants, which can be
found, for example, in [65, 123], will be omitted for the sake of brevity), which corrects the
ground-state CCSD energy for the combined effect of triples and quadruples, |Ψ0 in Eq.
(3.37), where mA = 2, is replaced by the following second-order-type, MBPT(2)[SDTQ]-like
36

expression
CCSD(TQ)

CCSD(T)

|Ψ0
CCSD(T)

where |Ψ0

= |Ψ0

1 2
+ 2 T2 |Φ ,

(3.44)

is given by Eq. (3.41).

Once these approximations have been established, the following compact formulas for the
CR-CCSD(T) and CR-CCSD(TQ) energies can be written:

(CR-CCSD(T))

E0

(CCSD)

= E0

+ N CR(T) /D(T)

(3.45)

and
(CR-CCSD(TQ))

E0

(CCSD)

= E0

+ N CR(TQ) /D(TQ) ,

(3.46)

where the N CR(T) and N CR(TQ) numerators are defined as

[2]

N CR(T) = Φ|(T3 )† M0,3 (2)|Φ + Φ|(Z3 )† M0,3 (2)|Φ

(3.47)

†
1
N CR(TQ) = N CR(T) + 2 Φ|(T2 )2 [T1 M0,3 (2) + M0,4 (2)]|Φ ,

(3.48)

and

CCSD(T)

and the D(T) and D(TQ) denominators, representing the overlaps between the |Ψ0
CCSD(TQ)

and |Ψ0

wave functions, Eqs. (3.41), and (3.44), respectively, with the CCSD
37

ground state, as in Eq. (3.37), are calculated as

CCSD(T) T1 +T2
|e
|Φ

D(T) ≡

Ψ0

†
†
1 2
= 1 + Φ|T1 T1 |Φ + Φ|T2 (T2 + 2 T1 )|Φ
[2]
1 3
+ Φ|(T3 )† (T1 T2 + 6 T1 )|Φ
†
1 3
+ Φ|Z3 (T1 T2 + 6 T1 )|Φ

(3.49)

and

D(TQ) ≡

CCSD(TQ) T1 +T2
|e
|Φ

Ψ0

†
1 2
1 2
1 4
= D(T) + 1 Φ|(T2 )2 ( 2 T2 + 2 T1 T2 + 24 T1 )|Φ .
2

(3.50)

The quantities M3 (2) and M4 (2) in Eqs. (3.47) and (3.48) are expressed in terms of the
triply and quadruply excited moments of the CCSD equations, easily calculated as in Eqs.
(3.39) and (3.40), with k set at 3 and 4, respectively. Specifically,

ijk

M0,abc (2) |Φabc ,
ijk

M0,3 (2)|Φ =

(3.51)

i<k<j,a<b<c

where

ijk

M0,abc (2) =

1 2
1 2
Φabc |[HN (T2 + T1 T2 + 2 T2 + 2 T1 T2
ijk
2
1
1 3
+ 2 T1 T2 + 6 T1 T2 )]C |Φ ,

38

(3.52)

and
ijkl

M0,abcd (2) |Φabcd ,
ijkl

M0,4 (2)|Φ =

(3.53)

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

where

ijkl

2
1
1 3
1 2
Φabcd |[HN ( 2 T2 + 2 T1 T2 + 6 T2
ijkl

M0,abcd (2) =

1 2 2
+ 4 T1 T2 )]C |Φ .

(3.54)

ijkl

ijk

It is interesting to note that if we replace moments M0,abc (2) and M0,abcd (2), entering
the above equations through the M3 (2)|Φ and M4 (2)|Φ quantities as shown above, by
their lowest-order estimates and the overlap denominators D(T ) and D(T Q) by 1, the CRCCSD(T) and CR-CCSD(TQ) approaches reduce to the standard CCSD(T) and CCSD(TQ)
methods.

ijk

For example,

if we replace M0,abc (2) by the lowest-order estimate

Φabc |(VN T2 )C |Φ and D(T ) by 1, the CR-CCSD(T) energy, Eq. (3.45), simplifies to
ijk
(CCSD(T))

E0

(CCSD)

= E0

[4]

[5]

+ ET + EST ,

(3.55)

where
[4]

[2]

ET = Φ|(T3 )† (VN T2 )C |Φ

(3.56)

and
[5]

EST = Φ|(Z3 )† (VN T2 )C |Φ ,

(3.57)

which is the well-known formula for the CCSD(T) energy. In other words, the CCSD(T) and
CCSD(TQ) methods can be derived through the MMCC theory and are shown to be natural
simplifications of the CR-CCSD(T) and CR-CCSD(TQ) methods, respectively. It is even
39

more important to note that the assumption that D(T) and D(TQ) can be approximated
by 1 is correct only for molecules near the equilibrium geometries. As bonds are streched
or broken, D(T) and D(TQ) increase, damping the unphysical values of the conventional
(T) and (TQ) correction. This is the key idea of “renormalization” of non-iterative energy
corrections behind the CR-CC and CR-EOMCC approaches, which assume excessive values
in conventional perturbative approaches of the CCSD(T) type when bonds are broken.

The CR-EOMCC methodologies for excited-states are obtained in a similar manner to
their ground-state analogs, such as CR-CCSD(T) and CR-CCSD(TQ). For example, the
CR-EOMCCSD(T) method is obtained by replacing the wave function |Ψµ in Eq. (3.37)
with perturbative expressions resulting from an analysis of the EOMCCSDT equations, such
as

|Ψµ

= {Rµ,0 + (Rµ,1 + Rµ,0 T1 ) + [Rµ,2 + Rµ,1 T1
2
˜
+Rµ,0 (T2 + 1 T1 )] + [Rµ,3 + Rµ,2 T1
2
1 2
+Rµ,1 (T2 + 2 T1 )
1 3
+Rµ,0 (T1 T2 + 6 T1 )]}|Φ ,

(3.58)

where T1 and T2 are the singly and doubly excited clusters obtained in the CCSD calculations
and Rµ,0 , Rµ,1 , and Rµ,2 are the reference, singly excited, and doubly excited components of
(CCSD)

the EOMCCSD excitation operator Rµ

. The approximate triply excited components

of the EOMCC excitation operator, which enter Eq. (3.58) and which can be determined
40

through perturbative analysis of the EOMCCSDT equations, are defined as follows:

ijk

rµ,abc aa ab ac ak aj ai ,
˜

˜
Rµ,3 =

(3.59)

i<j<k,a<b<c

where
ijk

ijk

ijk

rµ,abc = Mµ,abc (2)/Dµ,abc ,
˜

(3.60)

ijk

with moments Mµ,abc (2) given by

Mabc (2) =
µ,ijk

¯ (CCSD) Rµ,2 )C |Φ + Φabc |[H (CCSD) (Rµ,1 + Rµ,2 )]C |Φ
¯
Φabc |(H2
ijk
ijk
3
(CCSD)

¯
+ Φabc |(H4
ijk

(CCSD)

Rµ,1 )C |Φ + rµ,0

(CCSD)

¯
Φabc |H3
ijk

|Φ ,

(3.61)

ijk

and the perturbative denominator Dµ,abc given by
(CCSD)

¯
− Φabc |H (CCSD) |Φabc
ijk
ijk

(CCSD)

¯ (CCSD)
− Φabc |Hopen |Φabc
ijk
ijk

(CCSD)

ijk

¯ (CCSD) |Φabc
− Φabc |H1
ijk
ijk

Dµ,abc = Eµ
= ωµ
= ωµ

(CCSD)

¯
− Φabc |H2
ijk

|Φabc
ijk

¯ (CCSD) |Φabc .
− Φabc |H3
ijk
ijk
(EOMCCSD)

Here, ωµ

(3.62)

represents the EOMCCSD vertical excitation energy defined by Eq.

(3.23),
(EOMCCSD)

ωµ

(EOMCCSD)

= Eµ

(CCSD)

− E0

,

(3.63)

¯ (CCSD) is the open part of H (CCSD) (all diagrams of H (CCSD) that have external lines),
¯
¯
Hopen

41

¯ (CCSD) is a k-body component of H (CCSD) . The variant of CR-EOMCCSD(T) de¯
and Hk
scribed here is variant CR-EOMCCSD(T),ID. While other variants exist [115], we do not
discuss them here for the sake of brevity. Relationships exist between the CR-EOMCC
methods and their conventional perturbative EOMCC counterparts, such as EOMCCSD(T)
˜
or EOMCCSD(T), and further details can be found elsewhere [115]. It should be emphasized that the CR-CCSD(T), CR-CCSD(TQ), and CR-EOMCCSD(T) approaches are not
only related to the conventional CCSD(T), CCSD(TQ), and EOMCCSD(T)-type methods in a straightforward manner, as described above, but they also have similar computer
costs. For example, in analogy to CCSD(T), the CR-CCSD(T) and CR-EOMCCSD(T) approaches have CPU steps that scale as n2 n4 (N 6 ) in the CCSD/EOMCCSD part and n3 n4
o u
o u
(N 7 ) in the triples correction parts. Similarly, in analogy to the factorized formulation of
CCSD(TQ) [108], the costs of CR-CCSD(TQ) scale as n2 n4 (N 6 ) in the CCSD part and
o u
n3 n4 + n2 n5 (N 7 ) in the (TQ) parts.
o u
o u

In the next section, an alternative, biorthogonal formulation of the MMCC equations is
presented and the left-eigenstate CR-CC and CR-EOMCC approaches, specifically the CRCC(2,3) and CR-EOMCC(2,3) methods that result from it, are discussed. These methods
have been shown to be even more accurate than the CR-CCSD(T), CR-EOMCCSD(T),
and other CR-CC/CR-EOMCC approaches derived out of Eq. (3.37), so that much of our
benchmarking effort of the CR-CC and CR-EOMCC methods presented in this thesis focuses
on the CR-CC(2,3) and CR-EOMCC(2,3) theories discussed in the next section.
42

3.1.3.2

Biorthogonal Formulation of the MMCC Equations and the CR-CC(2,3)
and CR-EOMCC(2,3) Methods

An alternative formulation of CR-CC methods that satisfies the property of size-extensivity
in the ground state, which the CR-CCSD(T) and CR-CCSD(TQ) approaches violate somewhat, was developed in Refs. [127–132]. The resulting approaches, such as CR-CC(2,3) and
CR-EOMCC(2,3) originate from the so-called biorthogonal MMCC formalism presented in
Refs. [127, 128]. The biorthogonal formulation of the MMCC formalism redefines the cor(A)

rection δµ

by introducing the following ansatz for the exact bra state, Ψµ |, entering Eq.

(3.37):
Ψµ | = Φ|Lµ e−T

(A)

,

(3.64)

where Lµ is a deexcitation operator defined as
N

Lµ =

Lµ,n ,

(3.65)

n=0

with
Lµ,n =
i1 <···<in ,a1 <···<an

a1 ...an
ℓµ,i ...in ai1 · · · ain aan · · · aa1 .
1

(3.66)

By substituting Eq. (3.64) into Eq. (3.37) [131], or by considering the appropriate asym(A)

metric energy expressions [127, 128], one can rewrite the δµ
43

correction in the following

way:

(A)
δµ

Nµ,A

Φ|Lµ,n Mµ,n (mA )|Φ

=
n=mA +1
Nµ,A

a ...a

=
n=mA +1 i1 <···<in ,a1 <···<an

i ...i

1 n
1 n
ℓµ,i ...in Mµ,a1 ...an (mA ),
1

(3.67)

where Mµ,n (mA ) has already been defined by Eqs. (3.39). Thus, for example, when one
wishes to obtain the CR-CC corrections to CCSD/EOMCCSD energies due to triple excitations, then A = CCSD and mA = 2, as before, and, the ground-state (µ = 0) CR-CC(2,3) or
excited-state (µ > 0) CR-EOMCC(2,3) energies, Eµ (2, 3), can be calculated in the following
manner:
(CCSD)

Eµ (2, 3) = Eµ

+ δµ (2, 3),

(3.68)

where
ijk

ℓabc Mµ,abc (2),
µ,ijk

δµ (2, 3) = Φ|Lµ,3 Mµ,3 (2)|Φ =

(3.69)

i<j<k,a<b<c
ijk

with Mµ,abc (2) representing the triply excited moments of the CCSD (µ = 0) or EOMCCSD
(µ > 0) equations defined by Eqs. (3.52) and (3.61), respectively.
Since the triply excited moments of the CCSD/EOMCCSD equations are already welldefined, the determination of the three-body amplitudes ℓabc entering Eq. (3.69) becomes
µ,ijk
the primary focus. Since the exact values of these amplitudes may not be obtained without solving the full CI problem for the exact bra state Ψµ |, a method for the approximate
determination of the ℓabc amplitudes has to be proposed. Following Refs. [127, 128], the
µ,ijk
derivation of the ℓabc amplitudes used in practical CR-CC(2,3) and CR-EOMCC(2,3) calµ,ijk
culations begins by defining the approximate form of the deexcitation operator Lµ which
44

parameterizes the full CI state Ψµ | as

(CCSD)

Lµ ≈ L µ

+ Lµ,3 ,

(3.70)

where Lµ,3 is the three-body component of Lµ of interest, which is expressed in the usual
way as
ℓabc ai aj ak ac ab aa ,
µ,ijk

Lµ,3 =

(3.71)

i<j<k,a<b<c

with ℓabc representing the desired triply deexcited amplitudes to be determined, and
µ,ijk
(CCSD)

Lµ

is the deexcitation operator that defines the left or bra CCSD/EOMCCSD states

via the equation [33, 34]
(CCSD) −T1 −T2
e
,

Ψµ | = Φ|Lµ

(3.72)

where
(CCSD)

Lµ

= δµ,0 1 + Lµ,1 + Lµ,2 ,

(3.73)

is obtained by solving the so-called left CCSD/EOMCCSD equations [33, 34] for the corresponding ℓa and ℓab amplitudes (the µ = 0 variant of the system of the left CCSD/
µ,i
µ,ij
EOMCCSD equations is equivalent to the system of the so-called “lambda” equations of the
analytic gradient CCSD theory [197]). The explicit, computationally tractable form of the
approximate ℓabc amplitudes, which enter the CR-CC(2,3) and CR-EOMCC(2,3) correcµ,ijk
tions, are then obtained by substituting the approximate expression for Lµ given by Eq.
(3.70) into the exact form of the simlarity-transformed bra eigenvalue problem for Lµ ,

¯
Φ|Lµ H (A) = Eµ Φ|Lµ ,

45

(3.74)

where A is set at CCSD (and mA at 2), and the resulting equation is right projected onto
the triply excited determinants |Φabc to obtain
ijk
(CCSD) ¯ (CCSD) abc
H
|Φijk

def
def ¯
Φlmn |H (CCSD) |Φabc ℓµ,lmn
ijk

+

Φ|Lµ

l<m<n,d<e<f
= Eµ ℓabc .
µ,ijk

(3.75)

The exact energy Eµ in Eq. (3.75) is then replaced by the corresponding CCSD/ EOMCCSD
(CCSD)

energy Eµ

¯
, and the triples-triples block of the matrix representing H (CCSD) in the

second term on the left-hand side of Eq. (3.75) is replaced by its diagonal, as in the EpsteinNesbet partitioning [198,199]. The result of all of these operations is the following formula for
the approximate ℓabc amplitudes in terms of the many-body components of the similarityµ,ijk
(CCSD)

transformed Hamiltonian of CCSD, one- and two-body components of Lµ
(CCSD)

µ > 0 case, the EOMCCSD excitation energies ωµ

, and in the

:

ijk

abc
ℓabc = Nµ,ijk /Dµ,abc ,
µ,ijk

(3.76)

ijk

abc
where the numerator Nµ,ijk and denominator Dµ,abc are defined as follows:

abc
Nµ,ijk =

=

(CCSD) ¯ (CCSD) abc
H
|Φijk

Φ|Lµ

(CCSD)

¯
Φ|[(Lµ,1 H2

(CCSD)

¯
+(Lµ,2 H2

)C ]|Φabc ,
ijk

46

(CCSD)

¯
)DC + (Lµ,2 H1

)DC
(3.77)

and

(CCSD)

¯
− Φabc |H (CCSD) |Φabc
ijk
ijk

(CCSD)

ijk

−

Dµ,abc = Eµ
= ωµ

3

(CCSD)

¯
Φabc |Hn
ijk

|Φabc .
ijk

(3.78)

n=1
ijk

Note that the denominator Dµ,abc used here is the same as that of the CR-EOM-CCSD(T)
approach (see Eq. 3.62). The CR-CC(2,3) (µ = 0) and CR-EOMCC(2,3) (µ > 0) approaches
ijk

are obtained by substituting Eqs. (3.52) and (3.61) for Mµ,abc (2) and Eq. (3.76) for ℓabc ,
µ,ijk
ijk

abc
where Nµ,ijk and Dµ,abc are given by Eqs. (3.77) and (3.78), respectively, into the triples

correction formula, Eq. (3.69), which is subsequently added to the CCSD/EOMCCSD energy
(CCSD)

Eµ

to obtain the total energy Eµ (2, 3), as in Eq. (3.68).

In both the CR-CC(2,3) and CR-EOMCC(2,3) theories, the exact treatment of the
ijk

ijk

Epstein-Nesbet-like denominator Dµ,abc , as in Eq. (3.78), where no terms in Dµ,abc are
neglected, characterizes the most complete variant of the CR-CC(2,3) and CR-EOMCC(2,3)
approaches designated as CR-CC(2,3),D or CR-EOMCC(2,3),D, respectively. By neglectijk

ing selected terms in Eq. (3.78) for Dµ,abc , we obtain approximate CR-CC(2,3) and CREOMCC(2,3) schemes. Let us focus on CR-EOMCC(2,3) for this discussion, which contains
CR-CC(2,3) as a special case corresponding to µ = 0. Variant C of the CR-EOMCC(2,3) theory, designated as the CR-EOMCC(2,3),C approach, is obtained by ignoring the three-body
¯ (CCSD) |Φabc term, while keeping the
¯
component of H (CCSD) in Eq. (3.78), i.e., the Φabc |H3
ijk
ijk
ijk
¯
contributions to Dµ,abc from the one- and two-body components of H (CCSD) intact. The

CR-EOMCC(2,3),B approach is obtained by ignoring the two- and three-body components
¯ (CCSD) |Φabc in
¯
of H (CCSD) in Eq. (3.78), leaving only the one-body contribution Φabc |H1
ijk
ijk
47

ijk

Dµ,abc . Finally, variant A of the CR-EOMCC(2,3) approach is obtained by replacing the
ijk

Epstein-Nesbet-like denominator Dµ,abc , Eq. (3.78), by the Møller-Plesset-like denominator
(CCSD)

for triple excitations, ωµ

− (ǫa + ǫb + ǫc − ǫi − ǫj − ǫk ), where ǫp ’s are the single-particle

energies associated with spin-orbitals p (diagonal elements of the Fock matrix). An analogous discussion may be made for the ground-state case resulting in the A, B, C, and D
variants of CR-CC(2,3).
In analogy to the CR-CCSD(T) approach discussed in the previous subsection, one can
extend the CR-CC(2,3) method to higher-than-triple excitations as in, for example, the
CR-CC(2,4) scheme [127, 128, 166]. The CR-CC(2,4) approach, when implemented fully,
combines the N 6 -type steps of CCSD with the N 7 (n3 n4 ) steps of CR-CC(2,3) needed to
o u
determine the triples correction, and the N 9 (n4 n5 ) steps needed to calculate the analogous
o u
correction due to quadruples. In order to address this CPU-time increase, Piecuch et al.
proposed the so-called CR-CC(2,3)+Q method, also tested in this thesis, where one calculates
the ground-state energy as follows [186]

E CR-CC(2,3)+Q = E CR-CC(2,3) + E CR-CCSD(TQ) − E CR-CCSD(T) ,

(3.79)

i.e., one adds the quadruples correction extracted from the CR-CCSD(TQ) calculations to
the CR-CC(2,3) energy. This has an advantage over CR-CC(2,4) in the fact that the CPUtime costs of the CR-CCSD(TQ) calculations in the quadruples correction part scale as N 7
(n2 n5 ) with the system size N , as opposed to the N 9 steps of CR-CC(2,4). As a result,
o u
the CR-CC(2,3)+Q approach is almost as affordable as the CR-CC(2,3) method itself, while
bringing information about connected quadruply excited clusters that become important in
multiple bond breaking situations [186, 187].
48

The CR-CC(2,3) approach is capable of breaking bonds and, unlike its CR-CCSD(T)
predecessor, is size-extensive. Unfortunately, the δµ (2, 3) corrections to the EOMCCSD energies, defining CR-EOMCCSD(2,3), violate the property of size-intensivity of the EOMCC
excitation energies [44, 115, 196]. Although this violation is often unimportant, it is useful
to consider the possibility of restoring size intensivity in CR-EOMCC(2,3). This aspect is
discussed next.

3.1.3.3

A Size-Intensive Variant of CR-EOMCC(2,3): The δ-CR- EOMCC(2,3)
Method

As shown in Refs. [132, 200], although the ground-state variants of CR-CC(2,3) are sizeextensive, their excited-state CR-EOMCC(2,3) analogs do not satisfy the property of sizeintensivity [44,126,196], i.e., the vertical excitation energy of a non-interacting system A+B,
in which fragment A is excited, resulting from the CR-EOMCC(2,3) calculations, is not the
same as that obtained for the isolated system A. The lack of size intensivity of the CREOMCC(2,3) excitation energies can be traced back to the presence of the size-extensive
contribution [200, 201]

ijk

(rµ,0 ℓabc − ℓabc ) M0,abc (2)
0,ijk
µ,ijk

βµ =

(3.80)

i<j<k,a<b<c

in the CR-EOMCC(2,3) vertical excitation energy

(CR-EOMCC(2,3))

ωµ

(CR-EOMCC(2,3))

= Eµ

49

(CR-CC(2,3))

− E0

.

(3.81)

Using the above equations for the CR-EOMCC(2,3) energies, particularly Eq. (3.80), we
can decompose the CR-EOMCC(2,3) excitation energy as follows [200, 201]:

(CR-EOMCC(2,3))

ωµ

(CCSD)

Here, ωµ

(CCSD)

= ωµ

+ αµ + βµ .

(3.82)

is the vertical excitation energy of EOMCCSD, Eq. (3.23),

˜ ijk
ℓabc Mµ,abc (2),
µ,ijk

αµ =

(3.83)

i<j<k,a<b<c

˜ ijk
¯
where Mµ,abc (2) = Φabc |H (CCSD) (Rµ,1 + Rµ,2 )|Φ is the contribution to the triply excited
ijk
ijk

moment Mµ,abc (2) of EOMCCSD due to the one- and two-body components of the EOM(CCSD)

CCSD excitation operator Rµ

, Eq. (3.20), and βµ is the quantity defined by Eq. (3.80).

Since the EOMCCSD approach is rigorously size intensive and, as shown in Refs. [132, 200],
(CCSD)

the αµ term is size intensive as well, the [ωµ
(CR-EOMCC(2,3))

excitation energy ωµ

+ αµ (2, 3)] part of the CR-EOMCC(2,3)

is a size-intensive quantity. Unfortunately, the βµ term

defined by Eq. (3.80), being a size-extensive contribution that does not cancel out, grows
(CR-EOMCC(2,3))

with the size of the system [132, 200], destroying the size intensivity of ωµ

.

In order to implement the rigorously size-intensive variant of CR-EOMCC(2,3), designated
as δ-CR-EOMCC(2, 3) [201], the problematic βµ term in Eq. (3.82) is simply neglected such
that the excitation energy is redefined as follows:

(δ -CR-EOMCC(2,3))

ωµ

50

(CCSD)

= ωµ

+ αµ ,

(3.84)

with αµ given by Eq. (3.83). The resulting δ-CR-EOMCC(2, 3) approach [201] provides a
size-intensive description of the excitation energies and, by defining the total energy of a
given electronic state µ, i.e., Eµ , as a sum of the size-extensive ground-state CR-CC(2,3)
(δ -CR-EOMCC(2,3))

energy and size-intensive excitation energy ωµ

(CR-CC(2,3))

Eµ = E0

(CCSD)

= Eµ

, Eq. (3.84), so that

(δ -CR-EOMCC(2,3))

+ ωµ

ijk

ℓabc M0,abc (2)
0,ijk

+

i<j<k,a<b<c
˜ ijk
+
ℓabc Mµ,abc (2).
µ,ijk
i<j<k,a<b<c

(3.85)

While the addition of noniterative corrections is one way to correct for higher-order excitations not included in lower-order CC/EOMCC approximations, in the next section yet
another inexpensive approach to account for higher-order effects in CC/EOMCC calculations
is considered.

3.1.4

The Active-Space Coupled-Cluster and Equation-of-Motion
Coupled-Cluster Approaches

In Sect. (3.1.3) methods for obtaining a partial account of the T3 and T4 clusters, and
their excited-state analogs, all based on noniterative MMCC corrections were considered.
Another practical way to account for higher-than-doubly excited clusters in the SRCC considerations is by exploiting the ideas originally presented in Refs. [54, 133–136, 144], where
the CCSDt and CCSDtq active-space CC equations were explored (see, also, Refs. [137–140]
and Ref. [101] for a review). The basic language of the active-space CC and EOMCC methods follows from MRCC theory, where one subpartitions the one-electron basis of occupied
51

and unoccupied spin-orbitals used in the conventional SRCC-style considerations into (i)
core or inactive occupied spin-orbitals, designated as i,j,..., (ii) active occupied spin-orbitals,
designated as I,J,..., (iii) active unoccupied spin-orbitals, designated as A,B,..., and (iv)
virtual or inactive unoccupied spin-orbitals, designated as a,b,.... The active-space methods
CCSDt and CCSDtq, for example, then restrict the higher-order cluster components, T3 or
T3 and T4 , respectively, which must be calculated, to a small subset of all triples and quadruples defined via active orbitals. This restriction greatly reduces the computer costs, when
compared to parent CCSDT and CCSDTQ approaches, as discussed in the Introduction.
2 2
The CPU-time determining steps of CCSDt and CCSDtq are No Nu n2 n4 and No Nu n2 n4 ,
o u
o u

respectively, where No and Nu are the numbers of active occupied and active unoccupied
orbitals, i.e., the costs of CCSDt and CCSDtq calculations scale as relatively small prefactors
times the costs of the corresponding CCSD calculations.

As an example of how the equations for the active-space methods are obtained, let us
take the CCSDtq method as an example. The conventional CCSDTQ equations, which must
ij

ijk

ijkl

be solved for the ti , tab , tabc , and tabcd amplitudes by projecting the connected cluster form
a
of the Schr¨dinger equation, Eq. (3.5), in which T (A) = T1 + T2 + T3 + T4 , on the singly,
o
doubly, triply, and quadruply excited determinants, can be written as follows:

Φa |CCSD + (HN T3 )C |Φ = 0,
i

(3.86)

Φab |CCSD + [HN (T3 + T1 T3 + T4 )]C |Φ = 0,
ij

(3.87)

1 2
Φabc |CCSD + [HN (T3 + T1 T3 + T4 + T1 T4 + T2 T3 + 2 T1 T3 )]C |Φ = 0,
ijk

52

(3.88)

2
Φabcd | CCSD + [HN (T3 + T1 T3 + T4 + T1 T4 + T2 T3 + 1 T1 T3
ijkl
2
1 2
1 3
2
+T2 T4 + 1 T3 + 2 T1 T4 + T1 T2 T3 + + 6 T1 T3 )]C |Φ = 0 ,
2

(3.89)

where CCSD designates all terms that contain T1 and T2 clusters only and terms that do
not contain cluster operators at all. Once the system of equations, Eqs. (3.86) – (3.89) is
solved, the energy is obtained using Eq. (3.14). The CCSDT method could be obtained
from the CCSDTQ approach described here by simply setting T4 = 0 and solving the system
of equations formed by Eqs. (3.86) – (3.88) only.

In order to introduce the active-space CCSDtq formalism, the T3 and T4 clusters must
be restricted to the internal and semi-internal excitations of the following types:

Ijk

tabC aa ab aC ak aj aI ,

(3.90)

tIJkl aa ab aC aD al ak aJ aI ,
abCD

(3.91)

t3 =
I>j>k,a>b>C

t4 =
I>J>k>l,a>b>C>D

where I and J are summed over only active occupied orbitals and C and D are summed
over only active unoccupied orbitals. The CCSDtq system of equations for the relevant ti ,
a
ij

Ijk

tab , tabC , and tIJkl amplitudes has the form of CCSDTQ equations in which T3 and T4 are
abCD
replaced by t3 and t4 , respectively. We obtain

Φa |CCSD + (HN t3 )C |Φ = 0,
i

(3.92)

Φab |CCSD + [HN (t3 + T1 t3 + t4 )]C |Φ = 0,
ij

(3.93)

2
ΦabC |CCSD + [HN (t3 + T1 t3 + t4 + T1 t4 + T2 t3 + 1 T1 t3 )]C |Φ = 0,
Ijk
2

53

(3.94)

1 2
ΦabCD | CCSD + [HN (t3 + T1 t3 + t4 + T1 t4 + T2 t3 + 2 T1 t3
IJkl
1 2
1 3
+T2 t4 + 1 t3 2 + 2 T1 t4 + T1 T2 t3 + 6 T1 t3 )]C |Φ = 0 ,
2

(3.95)

where t3 and t4 are defined by Eqs. (3.90) and (3.91), respectively. The CCSDt method
could be obtained from the CCSDtq equations described here by setting t4 = 0 and solving
the system of equations given by Eqs. (3.92) – (3.94) only.
The active-space methods of the CCSDt and CCSDtq types were shown to be effective for excited-state theories as well, for example, in Refs. [55–57], where the EOMCCSDt
approach, an active-space variant of EOMCCSDT was reported for the first time. While
it is rather straightforward to generalize the active-space methods to particle conserving
EOMCC theories, such as EOMCCSDT and EOMCCSDTQ, in this dissertation the focus
is on the active-space variants of particle non-conserving EOMCC theories, in particular,
the EA-EOMCCSD(3p-2h) and IP-EOMCCSD(3h-2p) methods. The full inclusion of the
(N +1)

Rµ,3p-2h and Rµ,3h-2p components of the Rµ

(N −1)

and Rµ

operators in the EA- and IP-

EOMCC calculations needed to obtain an accurate description of electronic excitations in
radicals comes at a high price, increasing the N 5 -like no n4 and n2 n3 operations defining the
u
o u
iterative diagonalization steps of the base EA-EOMCCSD(2p-1h) and IP-EOMCCSD(2h-1p)
schemes to the N 7 -like n2 n5 and n3 n4 steps, respectively. One way to retain the accuracy of
o u
o u
the higher-order EA- and IP-EOMCC schemes with the 3p-2h and 3h-2p excitations, while
avoiding this steep computer cost increase, is to use the active-space variants of the EA/IPEOMCC methods with higher-than 2p-1h/2h-1p excitations described in Refs. [147–149].
In analogy to the ground-state active-space CC approaches described above, in the activespace EA- and IP-EOMCC methods one divides the available orbitals of the N -electron
54

reference system into core, active occupied, active unoccupied, and virtual categories, and
(N +1)

uses active orbitals to define the electron attaching and ionizing operators Rµ
(N −1)

Rµ

and

, respectively. In particular, the active-space EA-EOMCCSD(3p-2h){Nu } approach

using Nu active unoccupied orbitals is obtained by replacing the 3p-2h component Rµ,3p-2h
(N +1)

of the electron attaching operator Rµ

, Eq. (3.28), by

jk

rAbc aA ab ac ak aj .

rµ,3p-2h =

(3.96)

j>k,A<b<c
jk

The relatively small set of the unknown amplitudes rAbc defining rµ,3p-2h , Eq. (3.96), in
which at least one of the three unoccupied spin-orbital indices is active, and the remainj

ing 1p and 2p-1h amplitudes ra and rab that enter the (N + 1)-electron wave functions
of the active-space EA-EOMCCSD(3p-2h){Nu } approach are obtained by diagonalizing the
similarity-transformed Hamiltonian of CCSD, Eq. (3.9), obtained in the ground-state CCSD
calculations for the N -electron reference system, in the subspace of the (N + 1)-electron
Hilbert space spanned by the |Φa , |Φab , and |ΦAbc determinants. Similarly, the activej
jk
space IP-EOMCCSD(3h-2p){No } approach using No active occupied orbitals is obtained by
(N −1)

replacing the 3h-2p component Rµ,3h-2p of the ionizing operator Rµ

Ijk

r bc ab ac ak aj aI ,

rµ,3h-2p =

, Eq. (3.29), by

(3.97)

I>j>k,b<c

Ijk

where the relatively small set of the unknown amplitudes r bc defining rµ,3h-2p , Eq. (3.97),
in which at least one of the three occupied spin-orbital indices is active, and the remaining 1h
ij

and 2h-1p amplitudes ri and r b that define the (N −1)-electron wave functions of the activespace IP-EOMCCSD(3h-2p){No } approach are obtained by diagonalizing the similarity55

transformed Hamiltonian obtained in the N -electron CCSD calculations, Eq. (3.9), in the
b
bc
subspace of the (N − 1)-electron Hilbert space spanned by the |Φi , |Φij , and |ΦIjk de-

terminants. In Sect. (3.2.5) the full and active-space variants of the EA-EOMCCSD(3p-2h)
and IP-EOMCCSD(3h-2p) methods are used to optimize the geometries of the ground and
low-lying excited states of four open-shell molecules, CNC, C2 N, NCO and N3 , and determine the corresponding adiabatic excitation energies. The results provided in that section
will demonstrate typical accuracies of the active-space EA- and IP-EOMCC methods as
compared with their parent approaches.

3.2

Applications

In this section, typical accuracies of the methods described in Sect. (3.1) are demonstrated
using various chemically relevant benchmarks and applications. For all of the studied systems, the relevant benchmark data are provided for comparison, originating either from
experimentally measured quantities or from quantum-chemical calculations performed at
or near the full CI level. In Sect. (3.2.1), the CR-CC(2,3) method will be used to calculate barrier heights for a large variety of simple chemical reactions, for which the activation barriers are well established and which are characterized by largely SR transition
states, to see if it can match the performance of CCSD(T) in cases where CCSD(T) is accurate. A more MR case is presented in Sect. (3.2.2) and then an extremely biradical case
is presented in Sect. (3.2.3) to demonstrate the relative performance of the conventional
CCSD(T) and CCSD(TQ) methods, as compared with the performance of the CR-CC(2,3)
and CR-CC(2,3)+Q theories including corrections for the connected triply and quadruply
excited clusters, which are specifically designed to handle such situations. In Sects. (3.2.4)
56

and (3.2.5), excited-state systems are considered in order to demonstrate accuracies of the
CR-EOMCC(2,3) method for calculating vertical excitations and the δ-CR-EOMCC(2,3)
approach for calculating adiabatic excitation energies. Finally, in Sect. (3.2.6) the advantages and accuracies of various levels of EA- and IP-EOMCC methods are discussed in more
challenging cases of many-electron excitations in open-shell systems. Although the results
presented in this section are chosen to demonstrate the strengths of the CC and EOMCC
methods, with particular attention paid to our CR-CC, CR-EOMCC, and active-space EAand IP-EOMCC approaches, the weaknesses of each method will be pointed out as well.

3.2.1

The DBH24 Benchmark Database for Thermochemical Kinetics

This section provides a systematic comparison of the performance of CCSD, CCSD(T), and
variants A-D of the CR-CC(2,3) theory for a diverse collection of reaction barrier heights.
Since it is rather inconvenient to test such methods on the typical large benchmark databases
such as NHTBH38/04 [202] or Database/3 [203], as calculating the necessary chemical species
can quickly become excessively time consuming when many methods and basis sets are examined, a representative benchmark database, DBH24 [204], was developed as a more feasible
alternative to these large databases, designed specifically with computationally more intensive ab initio methods in mind. DBH24 is composed of 24 barrier heights which were
determined to be the most statistically representative subset of all 38 of the forward and
reverse barrier heights of NHTBH38/04 and the 44 hydrogen-transfer barrier heights of
Database/3. DBH24 consists of four types of reactions, namely, hydrogen transfer (HT),
heavy-atom transfer (HAT), nucleophilic substitution (NS), and unimolecular and associ57

ation (UA) reactions. There are three reactions (six barrier heights because forward and
reverse reactions are considered) for each type of reaction in the DBH24 database. The six
barrier heights of each reaction are denoted as HATBH6, NSBH6, UABH6, and HTBH6,
respectively.
When the DBH24 benchmark database was introduced in Ref. [204], the performance of
many ab initio wave function and density functional theory methods was examined alongside some semi-empirical and composite approaches with the goal of determining the best
approaches (measured in terms of accuracy, consistency, and computational efficiency) for
the calculation of barrier heights for routine use in thermochemical kinetics. The benchmark
values in DBH24 were obtained using high-level theoretical methods, such as Weizmann-1
(W1) [205] or MRCI calculations, or, in a few cases, benchmark values were derived from
experimental data. The full list of reactions and forward and reverse barrier height benchmark values are given in Table (3.1). All calculations reported in Ref. [204] were based on
reactant, product, and transition structures optimized at the QCISD/MG3 level with the
spin-restricted formalism for closed-shell systems and the fully spin-unrestricted formalism
for open-shell systems. The effect of spin-orbit coupling was added to the energies of the Cl
and OH radicals, which lower their energies by 0.84 and 0.20 kcal/mol, respectively. Among
all of the single-level (i.e. not composite) wave function methods tested in Ref. [204], those
based on CC theory, especially CCSD(T), proved to be the most accurate.
Following this initial study, we extended the work reported in Ref. [204] in Ref. [206]
to see if we could objectively determine whether the CR-CC(2,3) method could statistically
outperform the CCSD(T) approach on the DBH24 benchmark set due to the improved form
of the triples correction in CR-CC(2,3). We also used the opportunity to test the depen58

Table 3.1: Representative barrier heights database DBH24 taken from Ref. [204].
database

reaction

Vf a

=

=
Vr a

HATBH6

H + N2 O → OH + N2
H + ClH → HCl + H
CH3 + FCl → CH3 F + Cl

18.14
18.00
7.43

83.22
18.00
61.01

NSBH6

Cl− · · · CH3 Cl → ClCH3 · · · Cl−
F− · · · CH3 Cl → FCH3 · · · Cl−
OH− + CH3 F → HOCH3 + F−

13.61
2.89
-2.78

13.61
29.62
17.33

UABH6

H + N2 → HN2
H + C2 H4 → CH3 CH2
HCN → HNC

14.69
1.72
48.16

10.72
41.75
33.11

HTBH6

OH + CH4 → CH3 + H2 O
H + OH → O + H2
H + H2 S → H2 + HS

6.7
10.7
3.6

19.6
13.1
17.3

a V= denotes forward barrier height and V= denotes reverse barrier height (in kcal/mol).
f
f

dence of the results on the quality of the basis set and the effect of freezing core orbitals.
Calculations were performed using the CCSD, CCSD(T), and CR-CC(2,3) approaches to calculate the forward and reverse barrier heights for all of the reactions included in the DBH24
database using five different basis sets of triple-zeta quality with and without applying the
frozen core approximation. The five basis sets used in our study [206] were MG3S [207], and
four correlation consistent basis sets, namely, aug-cc-pVTZ [208–210], aug-cc-pV(T+d) [211],
aug-cc-pCVTZ [208, 209, 212, 213], and aug-cc-pCV(T+d)Z. Note that MG3S is identical to
6-311+G(3d2f,2df,2p) for H-Si and is similar to 6-311+G(3d2f), but improved [214] for P-Ar.
The aug-cc-pV(T+d)Z basis set is the same as aug-cc-pVTZ except that it has a single extra
d function for the second row atoms from Al through Ar, and the other d functions of augcc-pVTZ are also optimized for these atoms. The aug-cc-pCV(T+d)Z basis set is same as
59

aug-cc-pCVTZ basis set except that all valence d functions are taken from aug-cc-pV(T+d)Z
plus two d functions describing inner shells are taken from aug-cc-pCVTZ. As in the original
study by Zheng et al. [204], the geometries used in our calculations [206] were optimized
using the QCISD/MG3 level and the energies of the Cl and OH radicals were corrected for
spin-orbit effects.
The entire set of reaction barrier heights for the DBH24 database, as calculated with
the CCSD, CCSD(T), and CR-CC(2,3),A-D approaches combined with the five triple-zeta
basis sets mentioned above is supplied in the Supporting Information to Ref. [206]. The
calculated mean signed errors (MSEs) and mean unsigned errors (MUEs) obtained from these
calculations, taken from Ref. [206], are reported in Tables (3.2) and (3.3), with Table (3.2)
collecting results from all-electron calculations with all orbitals are correlated and Table (3.3)
collecting results from calculations with the core orbitals frozen. It can be seen from these two
tables that the CCSD results are relatively poor, typically producing MUE values around or
above 2.0 kcal/mol, but the CCSD(T) and CR-CC(2,3),A-D methods significantly improve
the CCSD activation energies, especially in conjunction with the augmented correlation
consistent basis sets. The mean unsigned errors of CR-CC(2,3) and CCSD(T) with the MG3S
basis set are about 0.9-1.0 kcal/mol both when correlating all electrons and when correlating
only valence electrons, whereas the mean unsigned errors characterizing the CR-CC(2,3)
and CCSD(T) results for the augmented correlation consistent basis sets vary between 0.4
and 0.75 kcal/mol. We conclude from this that the MG3S basis set offers an economical
triple-zeta basis set alternative for systems which are too large for the correlation consistent
basis sets to be affordable.

60

Table 3.2: Mean signed error (MSE) and mean unsigned error (MUE) of coupled cluster methods calculated with all electrons
correlated compared against the DBH24 benchmark database (in kcal/mol).
HATBH6
Method
CCSD(full)
CCSD(T)(full)
CR-CC(2,3),A(full)
CR-CC(2,3),B(full)
CR-CC(2,3),C(full)
CR-CC(2,3),D(full)

CCSD(full)
CCSD(T)(full)

CCSD(full)
CCSD(T)(full)
CR-CC(2,3),A(full)
CR-CC(2,3),B(full)
CR-CC(2,3),C(full)
CR-CC(2,3),D(full)

CCSD(full)
CCSD(T)(full)

MSE
4.36
0.92
1.50
1.72
1.16
1.17

2.85
-0.72

3.61
-0.03
0.55
0.77
0.42
0.42

3.60
-0.05

MUE

NSBH6

UABH6

HTBH6

DBH24

MSE

MUE

MSE

MUE

MSE

MUE

MUE

4.36
1.24
1.61
1.77
1.35
1.35

2.29
-0.01
0.72
0.47
0.10
0.10

MG3S
2.29
0.74
0.54
0.57
0.59
0.60

1.76
0.70
0.93
0.96
0.82
0.82

1.76
0.70
0.93
0.96
0.82
0.82

2.55
0.93
1.13
1.19
0.98
0.98

2.55
1.04
1.14
1.19
1.04
1.04

2.74
0.93
1.06
1.12
0.95
0.95

2.85
0.84

aug-cc-pVTZ
1.83
1.83
1.28
-0.52
0.64
0.17

1.28
0.34

1.06
-0.72

1.06
0.72

1.76
0.64

3.61
0.61
0.84
0.97
0.76
0.76

aug-cc-pCVTZ
2.12
2.12
1.13
-0.26
0.46
0.01
0.04
0.34
0.23
0.23
0.34
0.26
-0.22
0.63
0.22
-0.22
0.63
0.22

1.13
0.28
0.38
0.41
0.43
0.43

1.75
-0.05
0.14
0.22
0.11
0.11

1.75
0.45
0.54
0.56
0.55
0.55

2.15
0.45
0.52
0.57
0.59
0.59

3.60
0.58

aug-cc-pCV(T+d)Z
2.15
2.15
1.13
-0.24
0.44
0.01

1.13
0.28

1.76
-0.05

1.76
0.45

2.16
0.44

61

Table 3.3: Mean signed error (MSE) and mean unsigned error (MUE) of coupled cluster methods calculated with frozen core
approximation compared against the DBH24 benchmark database (in kcal/mol).
HATBH6
Method
CCSD
CCSD(T)
CR-CC(2,3),A
CR-CC(2,3),B
CR-CC(2,3),C
CR-CC(2,3),D

MSE
4.43
1.06
1.63
1.85
1.28
1.29

MUE
4.43
1.37
1.76
1.91
1.49
1.49

NSBH6

UABH6

HTBH6

DBH24

MSE

MUE

MSE

MUE

MSE

MUE

MUE

2.03
-0.25
0.03
0.22
-0.17
-0.18

MG3S
2.03
0.94
0.75
0.63
0.83
0.83

1.58
0.53
0.76
0.80
0.66
0.65

1.58
0.53
0.76
0.80
0.66
0.65

2.62
1.04
1.23
1.29
1.08
1.08

2.62
1.10
1.23
1.29
1.10
1.10

2.67
0.98
1.12
1.16
1.02
1.02

aug-cc-pVTZ
1.66
1.03
0.68
-0.06
0.44
0.17
0.35
0.19
0.77
0.16
0.77
0.15

1.11
0.40
0.47
0.49
0.48
0.48

1.72
-0.04
0.15
0.23
0.11
0.11

1.72
0.57
0.62
0.65
0.64
0.64

2.01
0.64
0.68
0.70
0.75
0.75

1.11
0.40
0.47
0.49
0.48
0.48

1.69
-0.06
0.13
0.20
0.09
0.09

1.69
0.54
0.60
0.63
0.62
0.62

2.01
0.56
0.58
0.61
0.62
0.62

CCSD
CCSD(T)
CR-CC(2,3),A
CR-CC(2,3),B
CR-CC(2,3),C
CR-CC(2,3),D

3.54
0.01
0.58
0.80
0.46
0.46

3.54
0.91
1.20
1.32
1.13
1.13

1.66
-0.67
-0.39
-0.20
-0.70
-0.71

CCSD
CCSD(T)
CR-CC(2,3),A
CR-CC(2,3),B
CR-CC(2,3),C
CR-CC(2,3),D

3.41
-0.13
0.45
0.67
0.30
0.31

3.41
0.67
0.88
1.00
0.80
0.80

aug-cc-pV(T+d)Z
1.82
1.82
1.03
-0.53
0.62
-0.06
-0.24
0.39
0.17
-0.05
0.30
0.19
-0.53
0.60
0.16
-0.54
0.60
0.15

62

Although the CCSD(T)(full)/aug-cc-pCV(T+d)Z method gives the best results among all
the tested methods, with a mean unsigned error of only 0.44 kcal/mol, it is our opinion that
this is not the best combination of method and basis set for routine applications of the type of
the benchmark study examined here. There are several reasons in support of this conclusion.
First, it is clear by a comparison of Tables (3.2) and (3.3) that all-electron CCSD(T) and
CR-CC(2,3) calculations generally give only slightly better results than those produced by
frozen-core calculations using the MG3S, aug-cc-pVTZ, or aug-cc-pV(T+d)Z basis sets. As
an example, the CR-CC(2,3),D/MG3S calculations with all electrons correlated produced
MUEs ranging from 0.60 to 1.35 kcal/mol, while the same combination of method and basis
set produced MUEs of 0.65 to 1.49 kcal/mol under the frozen-core approximation. In general,
it is not recommended to use valence-optimized basis sets when including both core and
core-valence correlations, since this is not only more expensive, but also a potential source of
problems [215]. By examining MUEs in Table (3.2) corresponding to CCSD(T) calculations
using the aug-cc-pVTZ and aug-cc-pCVTZ basis sets, which have ranges from 0.34 to 0.84
kcal/mol and 0.28 to 0.61 kcal/mol, respectively, it is clear that the accuracy systematically
increases when the core-optimized basis sets are used in all-electron calculations. However,
since the results only improve slightly it is our opinion that employing the core-optimized
basis sets is not practical due to the increase in the number of basis functions composing
these basis sets and the additional correlated orbitals required to perform the corresponding
all-electron calculations, which, taken together, make the calculations significantly more
expensive then their frozen-core analogs. It is also clear from Tables (3.2) and (3.3) that
the overall accuracy of the CR-CC(2,3),A-D approaches is practically the same as that
of CCSD(T). In particular, since CR-CC(2,3),D is shown to reproduce the high accuracy

63

of CCSD(T) in the barrier height calculations for the reactions from the DBH24 database,
which are largely of the SR type, while also offering a significantly better performance in more
MR cases, as demonstrated in Sects. (3.2.2) and (3.2.3), we recommend it for applications
where an accurate treatment of triples is required, particularly when paired with the augcc-pV(T+d)Z basis set in the frozen-core approximation.
Additional unpublished work was performed addressing the question of whether the MSEs
and MUEs characterizing the CR-CC(2,3) calculations could be further reduced by saturating
the basis set. The increase in accuracy observed when switching from the aug-cc-pVTZ to
aug-cc-pCVTZ basis set can be interpreted in one of two ways. Either correlating the core
electrons has a significant effect on the accuracy of these calculations, or, since there is a
large disparity between the number of basis functions in the two sets, a large portion of the
errors reported in Ref. [206] were due to basis set incompleteness. To investigate this issue,
CR-CC(2,3) energies were calculated with the aug-cc-pV(Q+d)Z basis set for all relevant
species in the DBH24 benchmark database. This basis set was chosen because it follows augcc-pV(T+d)Z in the hierarchy of basis sets having the general form aug-cc-pV(X+d)Z, where
X is known as the cardinal number of the basis set, and, having obtained energies at the
aug-cc-pV(T+d)Z and aug-cc-pV(Q+d)Z basis set levels, it is then possible to extrapolate
the electronic correlation energy to the complete basis set (CBS) limit, ∆E∞ , using one of
the existing empirical laws defining the dependence of the electronic correlation energy ∆E
on X, such as
∆E(X) = ∆E∞ + AX −3 ,

(3.98)

where ∆E(X) is the correlation energy obtained with the aug-cc-pV(X+d)Z basis set and
∆E∞ and A are the parameters determined from fitting ∆E(X) to the calculated correla64

tion energies. Preference is always given to the aug-cc-pV(T+d)Z and aug-cc-pV(Q+d)Z
basis set data in these fits, rather than the easier to obtain aug-cc-pV(D+d)Z data, because
the aug-cc-pV(D+d)Z data are well known to produce rather poor fits to the function given
by Eq. (3.98). While the CBS-limit correlation energy can be obtained by the above extrapolation technique, the reference energy, which was provided by the RHF method for
the closed-shell DBH24 species and by the ROHF method for open-shell species, was explicitly calculated using a very large, nearly complete, basis set, i.e., the aug-cc-pV(6+d)Z
basis set. This has been shown to be a more accurate approach than extrapolation for obtaining approximate CBS-limit reference energies [216] and the calculations do not become
prohibitively expensive even for the largest molecules of interest in this study. Both the
fast (exponential) convergence behavior of the reference energy and the unreliable nature
of correlation energies resulting from correlation consistent DZ-type basis sets with cardinal
number 2 are thoroughly discussed and well illustrated for the H2 O system in Ref. [217].
Thus, the approximate CBS-limit total energy reported in this section is constructed as a
sum of the energy produced by an aug-cc-pV(6+d)Z reference calculation and a CBS-limit
correlation energy, obtained by entering the aug-cc-pV(T+d)Z and aug-cc-pV(Q+d)Z basis
set level correlation energies into the extrapolation formula given by Eq. (3.98).

65

=

=

Table 3.4: Errors resulting from CR-CC(2,3) calculations of forward and reverse barrier heights, Vf and Vr respectively,
=

=

reported as ǫ(Vf ) / ǫ(Vr )) at varying basis set levels compared against DBH24 benchmark values (in kcal/mol).

aug-cc-pV(D+d)Z
aug-cc-pV(T+d)Z
aug-cc-pV(Q+d)Z
CBS-limit

aug-cc-pV(D+d)Z
aug-cc-pV(T+d)Z
aug-cc-pV(Q+d)Z
CBS-limit

aug-cc-pV(D+d)Z
aug-cc-pV(T+d)Z
aug-cc-pV(Q+d)Z
CBS-limit

aug-cc-pV(D+d)Z
aug-cc-pV(T+d)Z
aug-cc-pV(Q+d)Z
CBS-limit

HATBH6
H + HCl → HCl + H
CH3 + FCl → CH3 F + Cl
1.92/1.92
-2.96/-0.98
0.24/0.24
-0.48/0.69
-0.10/-0.10
0.12/0.66
-0.41/-0.41
0.48/0.85
NSBH6
Cl− ...CH3 Cl → ClCH3 ...Cl− F− ...CH3 Cl → FCH3 ...Cl− OH− + CH3 F → HOCH3 + F−
-1.31/-1.31
-0.79/-2.16
-2.81/-1.58
-1.08/-1.08
-0.14/-1.08
0.19/-0.04
-0.93/-0.93
-0.01/-0.34
0.52/0.59
-0.93/-0.93
0.05/0.12
0.73/0.99
UABH6
H + N2 → HN2
H + C2 H4 → C2 H5
HCN → HNC
0.13/ 0.32
0.63/ 0.19
-1.92/-1.24
0.00/ 0.56
0.34/ 1.00
-0.67/-0.30
0.08/ 0.53
0.38/ 0.86
-0.50/-0.08
0.08/ 0.58
0.39/ 0.79
-0.36/ 0.04
HTBH6
OH + CH4 → CH3 + H2 O
H + OH → O + H2
H + H2 S → H2 + HS
0.15/-1.28
-0.88/ 0.93
0.44/ 0.00
0.20/-0.83
-0.76/ 0.67
0.28/ 0.95
0.19/-0.09
-0.24/ 0.31
0.43/ 0.74
0.16/ 0.31
0.05/ 0.02
0.52/ 0.56
H + N2 O → OH + N2
-1.17/3.84
-0.99/2.16
-0.37/1.85
0.07/1.69

66

The individual errors resulting from CR-CC(2,3) calculations for the various DBH24
reactions, when used in conjunction with the aug-cc-pV(X+Z)Z basis sets, the corresponding CBS-limit values, and the MSE and MUE values characterizing these calculations are
collected in Tables (3.4) and (3.5). Results produced by CR-CC(2,3)/aug-cc-pV(D+d)Z
calculations are only included to emphasize the convergence with the basis set. For every
reaction considered, improvements in the quality of basis set correlate with improvements
in accuracy. For many of these reactions it is clear that the basis set truncation was the
main source of the triple-zeta-level error in the calculations reported in Ref. [206]. As shown
in Table (3.5), the MSE reduces for both the forward and reverse barrier heights to well
beyond chemical accuracy as the CBS-limit of the CR-CC(2,3) energies is approached. The
MUEs seem to be converged at the aug-cc-pV(Q+d)Z basis set level since there is almost
no difference with the CBS-limit results. Until it is explicitly proven, it can be assumed
that additional contributions from the remaining triples, quadruples, and other higher-order
connected clusters contribute minimally to the barrier heights considered here and may be
disregarded when predicting the activation energies for similar reactions to within a fraction
of 1 kcal/mol. In Sects. (3.2.2) and (3.2.3) chemical systems are examined which require at
least a partial treatment of quadruple excitations in order to attain results within chemical
accuracy. Our interest is in determining if the CR-CC methods, such as CR-CC(2,3)+Q can
be used to accurately describe such situations when they arise.

3.2.2

Addition Reactions of Ethylene and Acetylene to Ozone

In the previous section it was shown that the CR-CC(2,3) method is capable of matching
the performance of CCSD(T), where both were able to achieve chemical accuracy in con67

Table 3.5: Mean signed errors (MSE) and mean unsigned errors (MUE) resulting from
=
=
CR-CC(2,3) calculations of all DBH24 forward and reverse barrier heights, Vf and Vr
=

=

respectively, reported as ǫ(Vf ) / ǫ(Vr )) at varying basis set levels (in kcal/mol).
MSE
aug-cc-pV(D+d)Z
aug-cc-pV(T+d)Z
aug-cc-pV(Q+d)Z
CBS-limit

-0.28 / 0.96
-0.35 / 0.71
-0.15 / 0.54
0.07 / 0.38

MUE
1.11
0.52
0.32
0.35

/
/
/
/

1.38
0.81
0.59
0.60

junction with medium basis sets when predicting simple thermochemical barrier heights of
the predominantly SR nature, and that the CR-CC(2,3) method could even achieve a subchemical level of accuracy in the CBS-limit. It was thus clear that for the species included
in the DBH24 benchmark database, the correlation effects included in the CCSD(T) and
CR-CC(2,3) methods were sufficient to reproduce benchmark data with excellent accuracy.
However, not all chemical systems can be described so accurately at the CCSD(T) and CRCC(2,3) levels. Some systems are more MR and as such, require a balanced description of
triply and quadruply excited clusters.
Ozone is one notorious example of a MR system [218–221]. Despite its well known closedshell singlet electronic structure, ozone exhibits a significant biradical character estimated to
be around 33% [222–225]. Ozone is a common reagent in organic chemistry for the generation
of ketones, aldehydes, epoxides, peroxides, anhydrides, and polymers via ozonolysis of alkenes
and alkynes [226–231] The mechanism of such processes is generally accepted to proceed by
initial formation of a van der Waals (vdW) complex followed by a concerted cycloaddition
transition state (TS) before finally reaching the cycloadduct configuration. In this section,
cycloadditions of ethylene and acetylene, shown in Fig. (3.1) to the 1,3 termini of ozone
68

C2H2 + O3 → vdW → TS → adduct

C2H4 + O3 → vdW → TS → adduct

Figure 3.1: Stationary points along the C2 H2 (top row) and C2 H4 (bottom row) ozonolysis
reaction pathways. In each row, the structures from left to right represent the van der Waals
minimum (vdW), transition state (TS), and the cycloadduct, respectively. The oxygen, carbon, and hydrogen atoms are represented by the red, yellow, and grey spheres, respectively.
For interpretation of the references to color in this and all other figures, the reader is referred
to the electronic version of this dissertation.

69

Table 3.6: Benchmark values for the C2 H2 and C2 H4 ozonolysis reaction pathways.
O3 +C2 H2
method

vdW

Wheeler et al.a
-1.85
b
CBS CCSDT+(2)Q
-1.88
c
CBS CCSD(T)+Q
-1.98
average = best estimate -1.90

O3 +C2 H4

TS cycloadduct

vdW

TS cycloadduct

7.74
7.90
7.58
7.74

-1.84
-1.94
-2.03
-1.94

3.43
3.50
3.18
3.37

-63.04
-63.91
-64.46
-63.80

-56.43
-57.15
-57.86
-57.15

a From Ref. [225], calculated as CCSDT/CBS plus corrections for core correlation (C), adi-

abatic Born-Oppenheimer terms (A), relativistic effects (R) and a correction for quadruple
excitations (+(2)Q ) estimated at the cc-pVDZ basis set level for C2 H2 and assumed to
be the same for C2 H4 . b CBS CCSDT/CBS+CAR from Ref. [225] plus quadruple excitation contributions calculated at the CCSDT(2)Q /cc-pVDZ level for both C2 H2 and C2 H4
separately [242]. c CBS CCSD(T)/CBS+CAR from Ref. [225] plus quadruple excitation
contributions calculated at the CCSD(TQ)/aug-cc-pVDZ level for both C2 H2 and C2 H4
separately [242].

are considered as a more challenging set of thermochemical barrier heights, which have
reported literature values from ordinarily reliable theoretical methods ranging from 2 to 18
kcal/mol for ethylene [232–237] and 5 to 22 kcal/mol for acetylene. [238–241] When such large
discrepancies are observed between various theoretical methods, it is typical that higher-order
effects play a significant role in the description of the species involved and, indeed, work by
Wheeler et al. [225] showed that there are non-negligible contributions to the energy from
quadruply excited clusters in the ozonolysis of C2 H2 . Table (3.6) collects the best known
computational benchmark values for stationary points along the C2 H2 and C2 H4 ozonolysis
reaction pathways, as reported in Refs. [225] and [242]. Following the strategy of Ref. [242],
the averages given in Table (3.6) are the values used for comparison in this discussion.
In a joint study published by the Piecuch and Truhlar groups in 2009 [242], the effect of
quadruples on the C2 H2 ozonolysis was reexamined, as treated with a larger basis set, and
70

the ozonolysis of C2 H4 was considered for the first time with quadruple excitations explicitly included. This discussion focuses on the performance of CC methods that start with an
iterative CCSD calculation and add noniterative corrections for connected higher-order excitations, including CCSD(T), CCSD(TQ), CR-CC(2,3), and CR-CC(2,3)+Q, although many
other CC and non-CC results, which are not discussed here, were reported in Ref. [242] as
well. Since calculating corrections to the correlation energy corresponding to quadruples at
the aug-cc-pVTZ basis set level was computationally too demanding, a composite method
was invented which treats most of the correlation energy with the aug-cc-pVTZ basis set but
treats the part of the correlation energy corresponding to quadruples at the aug-cc-pVDZ
level. For example, a CR-CC(2,3)/aug-cc-pVTZ calculation augmented by a quadruples
correction at the aug-cc-pVDZ level is designated as CR-CC(2,3)/aug-cc-pVTZ+Q(aug-ccpVDZ). All CR-CC(2,3) energies discussed in this section are the variant D values. The
underlying geometries representing stationary points for the reaction pathways were optimized with the M05 density functional [243] and MG3S basis set.

In Table (3.7) the selected conventional CC and CR-CC results are compared, reported
as relative energies with respect to the reactants, for each of the stationary points on the
reaction pathways. A specific subset of these results was selected to be presented here
which illustrates typical problems encountered at various levels of CC theory. We begin by
considering the CCSD-level results found using the aug-cc-pVDZ and aug-cc-pVTZ basis
sets, found in the first two rows of Table (3.7). It is clear that CCSD, when used with the
aug-cc-pVDZ and aug-cc-pVTZ basis sets, produces a proper qualitative description of the
two reaction pathways, however the overall accuracies, reported concisely as MUEs of all
considered species with respect to the benchmark values in the second to last column, are
71

quite poor, with reported values of 2.4 and 3.3 kcal/mol, respectively.
The CCSD(T) and CCSD(TQ) results reported here, generated in conjunction with
the aug-cc-pVDZ basis set, improve on the overall MUEs of CCSD, producing 1.7 and 1.5
kcal/mol, respectively, but both standard perturbative CC methods fail to give the correct
qualitative description of the ethylene ozonolysis reaction pathway, placing the transition
state -0.54 and -0.21 kcal/mol below the reactants, respectively. These results, predicting
no barrier to reaction at all, are unphysical, disagreeing with high-level theory and experiment [244]. In mechanistic studies such as this, the relative energetics between reactants
and the transition states are of particular interest, since the activation energy is often an
experimentally derivable quantity, so in the final column of Table (3.7) the MUEs for only
the transition states of both reactions are reported. Comparing CCSD(T) and CCSD(TQ)
to the other reported methodologies by this measure clearly shows their exceptionally poor
performance when used to describe barrier heights when either the reactants or transition
state have significant biradical character and the other species do not.
The situation looks much better when the CR-CC methods are used to describe the same
reaction pathways. By using the CR-CC(2,3) method rather than CCSD(T) in the same basis
set, a lowering of the total MUE from 1.7 to 0.8 kcal/mol is observed, and an appropriate
sign is obtained for the barrier in the ethylene ozonolysis. This is already considered subchemical accuracy, but not the best result yet. A logical next step toward increasing accuracy
is to increase the basis set from aug-cc-pVDZ to aug-cc-pVTZ. It is shown in Table 3.7 that
the same MUE of 0.8 kcal/mol is obtained when the aug-cc-pVTZ basis set is employed,
but the activation energies move toward the benchmark values. Further improvements are
observed when the CR-CC(2,3) results are corrected for quadruples. The incredibly small
72

total MUE produced by the CR-CC(2,3),D/aug-cc-pVTZ+Q(aug-cc-pVDZ) method (0.6
kcal/mol), and an ever smaller MUE for the activation energies (0.2 kcal/mol) show the
ability of the relatively inexpensive CR-CC approaches to produce excellent energetics, even
for difficult reaction pathways. In the next section, Sect. (3.2.3), an even more difficult
reaction profile is considered.

73

Table 3.7: Energetics of stationary points relative to reactants, in kcal/mol, produced at various levels of coupled-cluster
theory for the ozonolysis of ethylene and acetylene.
O3 +C2 H2

O3 +C2 H4
TS cycloadduct

MUE

MUE

Alla

BHsb

method

vdW

TS cycloadduct

vdW

CCSD/aug-cc-pVDZ
CCSD/aug-cc-pVTZ
CCSD(T)/aug-cc-pVDZ
CCSD(TQ)/aug-cc-pVDZ
CR-CC(2,3),D/aug-cc-pVDZ
CR-CC(2,3),D/aug-cc-pVTZ
CR-CC(2,3),D/aug-cc-pVTZ+Q(aug-cc-pVDZ)

-2.00
-1.76
-2.50
-2.48
-1.11
-0.58
-0.62

7.47
9.34
4.29
4.73
6.09
8.08
8.02

-69.90
-71.27
-64.21
-63.50
-63.46
-64.06
-63.29

-2.38
-1.89
-3.18
-3.17
-1.87
-0.68
-0.74

2.69
4.41
-0.54
-0.21
1.44
3.55
3.46

-64.22
-66.36
-57.58
-56.89
-57.10
-58.31
-57.52

2.4
3.3
1.7
1.5
0.8
0.8
0.6

0.5
1.3
3.7
3.3
1.8
0.3
0.2

Benchmark valuesc

-1.90

7.74

-63.80

-1.94

3.37

-57.15

—

—

a Average mean unsigned errors for all stationary points on the ethylene and acetylene ozonolysis reaction pathways. b Average
mean unsigned errors for the transition state barrier heights for ethylene and acetylene ozonolysis. c The best estimates from

Table (3.6) obtained in Ref. [242].

74

3.2.3

Mechanism of the Isomerization of Bicyclobutane to Butadiene

The pericyclic rearrangement of bicyclo[1.1.0]butane (abbreviated as bicbut) to trans-buta1,3-diene (abbreviated as t-but) was chosen as another test case for the CR-CC methods,
both for the intrinsic complexity of the associated isomerization pathways, which involve
polyatomic structures with a rapidly varying degree of biradical character that require an
accurate and balanced description of the dynamical and non-dynamical correlation effects,
as well as for the favorable size of the system, which has several atoms but is still not too
large, enabling calculations with larger basis sets to be performed and, in turn, allowing
convergence behavior with the size of the basis set to be examined. Let us recall that early
experimental studies suggest that the bicbut→t-but isomerization proceeds by concerted
conrotatory movement of the methylene groups [245, 246], as predicted by the WoodwardHoffman symmetry rules (see Figure (3.2)). Computational studies confirm this, predicting,
in addition, that near the end the reaction pathway passes through the gauche-buta-1,3-diene
configuration (abbreviated as g-but) before reaching the final t-but configuration [189,248].
Theoretical studies have also considered the concerted disrotatory [189, 248, 249] and nonconcerted [250] pathways, finding the concerted disrotatory TS to be ∼ 20 kcal/mol higher in
energy than the conrotatory TS and the non-concerted pathway to be much too high in energy to be even considered as a plausible mechanism. The conrotatory TS was found to have
a ∼ 24 % biradical character while the disrotatory TS was found to have a ∼ 90 % biradical
character, according to the high level electronic structure calculations reported in Ref. [189].
The current consensus on the mechanism for the isomerization of bicyclo[1.1.0]butane into
buta-1,3-diene is that both concerted pathways begin at bicbut and, after passing through
75

the corresponding conrotatory or disrotatory TS (con TS and dis TS, respectively), they
converge at the local minimum defining the intermediate g-but configuration (see Figure
(3.2)). The g-but intermediate isomerizes via a low-energy rotational barrier, defined by
the TS structure labeled as gt TS, before the final product, t-but, is reached. The conrotatory and disrotatory concerted pathways describing the isomerization of bicyclo[1.1.0]butane
into buta-1,3-diene, along with their available experimental activation [251] and reaction
enthalpies (the latter based on the enthalpies of formation of bicyclo[1.1.0]butane and buta1,3-diene reported in Ref. [252]; cf. Ref. [189]), and the theoretical enthalpy values obtained
in this work, are illustrated in Figure (3.2).

As demonstrated in Ref. [189], the CCSD(T) approach completely fails by placing the
disrotatory pathway defined by the strongly biradical dis TS about 20 kcal/mol below the
conrotatory pathway, contradicting experiment [245,246] and accurate MR calculations [248]
which state that the conrotatory pathway represents a true mechanism. At the same time,
the CASSCF and MCQDPT2 approaches, which correctly place the conrotatory pathway
below the disrotatory one, provide relatively poor energetics when compared to the available
experimental and more accurate electronic structure data [189], although CASSCF produces reasonable geometries of the corresponding stationary points. On the other hand, as
was also shown in Ref. [189], the CR-CC(2,3) approach provides an accurate and balanced
description of the conrotatory and disrotatory pathways describing the isomerization of bicyclo[1.1.0]butane to trans-buta-1,3-diene, and one of the primary goals of this thesis research
was to produce CR-CC(2,3) energies for this system which were converged with the basis
set [193]. To do this, the set of nuclear geometries optimized at the CASSCF(10,10)/ccpVDZ level of theory were taken from Ref. [189] and additional CR-CC(2,3) calculations
76

Figure 3.2:
The conrotatory and disrotatory pathways describing the isomerization
of bicyclo[1.1.0]butane to trans-buta-1,3-diene, along with the enthalpy values relative to the reactant at all stationary points obtained in the explicit CR-CC(2,3)/ccpVQZ//CASSCF(10,10)/cc-pVDZ calculations (numbers in roman) and the enthalpy values obtained with the PES extrapolation procedure using the CR-CC(2,3)/cc-pVDZ, CRCC(2,3)/cc-pVTZ, and RHF/cc-pVQZ energies at all stationary points resulting from the
CASSCF(10,10)/cc-pVDZ optimizations, and the CR-CC(2,3)/cc-pVQZ correlation energy
at the reactant geometry (numbers in bold italics). The available experimental enthalpy
values are in parentheses. All enthalpies are in kcal/mol.

77

Table 3.8: Relative enthalpies, in kcal/mol, with respect to reactant as calculated at various
levels of theory for the conrotatory and disrotatory bicbut → tbut isomerization pathways.
Method
CR-CC(2,3)/cc-pVTZa
CR-CC(2,3)/cc-pVQZb
CR-CC(2,3)/CBS limitb
CR-CC(2,3)+Q/cc-pVTZb
ACSE/6-311G**c
DMCd
Experimente

con TS

dis TS

g-but

gt-TS

tbut

41.1
41.3
41.5
40.8
41.2

66.1
67.1
67.5
67.0
55.7

-24.9
-24.8
-24.5
-24.9
-23.8

-22.1
-22.9
-21.6
-22.1
—

-27.9
-27.7
-27.4
-27.8
—

40.4(5)

58.6(5)

-25.2(5)

-22.2(5)

-27.9(5)

40.6(± 2.5)

—

—

—

-25.9(4)

a Taken from Ref. [189]. b The present study (CR-CC(2,3) results taken from Ref. [193]).
c Results obtained with the anti-Hermitian contracted Schr¨dinger equation (ACSE) apo
d Benchmark computational results obtained with the diffusion
proach taken from Ref. [253].
Quantum Monte Carlo (DMC) approach taken from Ref. [254]. e Benchmark experimental

results taken from Ref. [252].
were performed with the cc-pVQZ basis set and, together with the cc-pVTZ energies from
Ref. [189], CBS-limit extrapolated correlation energies were obtained using Eq. (3.98). These
and other results are reported in Table (3.8).
Enthalpy values in Table (3.8) are reported relative to reactants as calculated using
a variety of approaches. Comparison of the CR-CC(2,3)/cc-pVTZ results from Ref. [189]
with the CR-CC(2,3) results obtained through calculations at the cc-pVQZ and CBS-limit
levels, as originally reported in Ref. [193], confirms that small improvements were made by
saturating the basis set. Later in 2008, a paper was published challenging the accuracy of
these calculations. Calculations performed using the anti-Hermitian contracted Schr¨dinger
o
equation (ACSE) in Ref. [253], reported for comparison in Table (3.8), showed the dis TS
barrier to be much lower then CR-CC(2,3) calculations had predicted. Since the ACSE
method is good at accounting for non-dynamical correlation effects, we calculated the +Q
78

corrections at various basis set levels to assure that we were not neglecting nondynamical
correlation effects described by higher-order clusters. Since, as shown in Table (3.8), these
corrections adjusted the enthalpies by less then 1 kcal/mol, we assumed that our values for
the dis TS barrier were reliable.

An interesting recent development, which affects this discussion, came in 2010 when highquality diffusion Quantum Monte Carlo (DMC) results became available for the bicbut →
tbut isomerization system [254]. By comparing with this theoretical benchmark data it was
shown that CR-CC(2,3) and CR-CC(2,3)+Q enthalpy values at all reported basis sets are
within ∼ 1 kcal/mol for all stationary points except for the dis TS structure. There is a
rather large discrepancy at the dis TS stationary point with the highest quality CR-CC
data predicting a barrier of around 67 kcal/mol, while the DMC results predict a dis TS
barrier around 59 kcal/mol. At this point the issue emerged: what is behind the few kcal/mol
difference between the CR-CC and DMC results for the disrotatory transition state? Our
group finally found an answer a few weeks ago. Based on a still preliminary study using
a new theory being presently developed in our group, abbreviated CC(t;3), that represents
a merger of the active-space CCSDt approach and CR-CC(2,3) [255], it appears to be the
iterative treatment of the singly and doubly excited clusters in the presence of the leading
triple excitations which accounts for the remaining nondynamic and dynamic correlations
responsible for this energy lowering, producing a result in perfect agreement with DMC.
Thus, while species with extreme biradical character, such as dis TS, may sometimes be
difficult to describe to high accuracy, the CR-CC(2,3) method remains the key in obtaining
the desired accuracies, since CC(t;3) uses the CR-CC(2,3)-style corrections to correct the
CCSDt energies for the triples missing in the CCSDt calculations, while eliminating the
79

complete failure of CCSD(T).

Excluding the exceptionally difficult dis TS stationary point, which is not part of the
energetically favorable reaction profile for the bicbut → tbut isomerization, the CR-CC(2,3)
results with or without quadruples are within 1 kcal/mol of the DMC results. It also appears
that the best CR-CC results are within the DMC error bars. Meanwhile, the ACSE results
significanly underestimate the g-but barrier and cannot claim sub-chemical accuracy. We
speculate that the ACSE method does significanly better for the dis TS barrier, which has 90
% biradical character, but underestimates the correlation energies for the closed-shell species.
This is because the method is very good for describing nondynamical correlation energy, but
is less accurate in its description of dynamical correlation effects. On the other hand, unlike
CCSD(T), which describes dynamical correlation energy well and nondynamical correlation
energy poorly, it is at least a qualitatively predictive method. Meanwhile, the CR-CC(2,3)
method and its CR-CC(2,3)+Q and CC(t;3) extensions can, unlike the ACSE approach, give
a quantitatively correct description of the energically favorable con TS reaction profile and,
unlike the CCSD(T) method, predict the correct reaction channel.

The success of the ground-state CR-CC formalism in this and previous sections motivates
an investigation of the performance of the CR-EOMCC methods for excited states. This is
done in the next two sections, with the first considering the performance of CR-EOMCC(2,3)
for the description of low-lying excited-state PESs of the water molecule, particularly along
the bond-breaking O-H coordinate (in Sect. (3.2.4)), followed by a fully size-intensive description of vertical excitation energies and spectral shifts in weakly bound complexes (in
Sect. (3.2.5)).
80

3.2.4

Excited-State Potential Energy Surfaces for the Dissociation
of Water

The motivation for the work reported in the current section is to, for the first time, assess
the performance of the CR-EOMCC(2,3) approach in generating excited-state PESs along
a single-bond stretching dissociation channel. The water molecule was chosen as our target
system since, while there exist in the literature many experimental [256–264] and theoretical
[265–269] studies aimed at finding vertical excitation energies out of the ground state (see Ref.
[270] for a review of older work), fewer studies have focused on generating PESs other than
for the ground and first excited singlet states. Recently, however, a cut of the full groundstate PES corresponding to the asymmetric O-H bond-breaking dissociation was carefully
optimized by Li and Paldus using the CCSD theory in conjunction with a relatively large
cc-pVTZ basis set and then many low-lying excited-state PES cuts were calculated at the full
CI level using a cc-pVTZ basis stripped of polarization functions (referred to as the TZ basis)
[271]. Here, we intend to use the set of high-level optimized geometries and corresponding
full CI excited-state PES cuts reported in Ref. [271] as benchmark data for testing the
performance of the CR-CC(2,3) and CR-EOMCC(2,3) methods. The main objective is
to determine whether the CR-CC(2,3) and CR-EOMCC(2,3) methods can overcome the
deficiencies of the CCSD and EOMCCSD approaches for the PES cuts of interest, including
stretched O-H bond lengths and the H2 O → H + OH dissociation limit, where groups
of electronic states of water converge to become degenerate. Since values are available,
we will also take this opportunity to compare the PESs obtained in the CR-CC(2,3)/CREOMCC(2,3) calculations with those produced by leading MRCC or MRCC-like methods,
as taken from Ref. [271]. These include the N R-RMR-CCSD(T), N R-GMS-SU-CCSD, and
81

(N, M )-CCSD methods, all of which were mentioned in the Introduction.
Following Ref. [271], calculations were peformed using a small basis set referred to as
TZ, which, as mentioned above, is constructed by removing polarization functions from the
cc-pVTZ basis set of Dunning. The geometries associated with the asymmetric dissociation
were generated by designating the bond length of the stretched O-H bond and optimizing the
H-O-H angle and second O-H bond length at the CCSD/cc-pVTZ level. These geometrical
parameters were taken from Ref. [271]. All calculations of singlet ground and excited states
were performed using the efficient RHF-based EOMCCSD and CR-EOMCC(2,3) computer
codes developed earlier by our group and incorporated in the GAMESS program. Triplet
EOMCC calculations were performed using a new EOMCCSD code for ROHF references
developed as part of this thesis effort, to be included in a future official GAMESS distribution,
the implementation of which is discussed in Sect. (5.1.1). Open-shell ROHF-based CREOMCC(2,3) calculations were performed using the spin-orbital computer codes written by
our group that are loosely interfaced with GAMESS. In all calculations electrons in the
lowest occupied molecular orbital were frozen, spherical contaminants were dropped from
the atomic orbital basis sets, and the Cs point group was enforced, distinguishing states as
either A′ or A′′ symmetry. We use the convention for numbering states where the singlet
ground state is denoted in a usual way by the symbol X and excited states belonging to the
same irreducible representation, i.e. spin and spatial symmetry, are numbered sequentially
in order of increasing energy at the equilibrium nuclear configuration.
Ground-state X 1 A′ PES cuts are reported in Table (3.9), with internuclear separations
given in the first column and energies corresponding to various methods given in the remaining columns. The last two rows provide MUEs and non-parallelity errors (NPEs) relative
82

Table 3.9: Ground-state X 1 A′ energies for the asymmetric single-bond breaking of H2 O
at a series of the O-H bond lengths, R, in atomic units. The last two rows give the mean
unsigned errors (MUE) and non-parallelity errors (NPE) relative to the full CI PES obtained
in Ref. [271].
FCIa

7R-SUb

4R-RMR(T)c

CCSDd

CR-CC(2,3)d

1.3
1.6
1.809
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

-1.01567
-1.14894
-1.16847
-1.16417
-1.13058
-1.09255
-1.06129
-1.03889
-1.02451
-1.01967
-1.01603

2.87
3.20
3.50
3.83
4.65
5.73
7.15
8.84
10.69
11.61
12.48

0.08
0.10
0.13
0.16
0.20
0.20
0.15
0.13
0.19
0.24
0.29

2.84
3.17
3.46
3.77
4.59
5.72
7.21
8.97
10.81
11.66
12.43

-0.16
-0.16
-0.14
-0.12
-0.03
0.08
0.21
0.32
0.36
0.33
0.28

MUE
NPE

—
—

6.68
9.61

0.17
0.21

6.78
9.59

0.20
0.52

R (bohr)

a Full CI energies (E), reported as (E + 75) hartree, taken from Ref. [271]. b 7R-GMS-SU-

CCSD energies, in millihartree, reported as differences from full CI taken from Ref. [271]
c 4R-RMR-CCSD(T) energies, reported as differences from full CI taken from Ref. [271] d
This work. Reported as differences from full CI, in millihartree
to the full CI PES of Li and Paldus [271] for each data set. Let us first consider the errors
produced by the CCSD method, which range from 2.84 to 12.43 hartree, with MUE of 6.78
millihartree and NPE of 9.59 millihartree. As expected, errors for all methods increase significantly as the O-H bond is stretched, but it is seen that the addition of triple excitations
via the CR-CC(2,3) method reduces absolute errors relative to full CI to no greater than
0.36 millihartree for all points considered. The MUE of 0.20 millihartree and NPE of 0.52
millihartree produced by the black-box CR-CC(2,3) approach compare very well to the expert high-level 4R-RMR-CCSD(T) method, which yields a MUE of 0.17 millihartree and a
NPE of 0.21 millihartree.
83

Results for the low-lying singlet and triplet excited states of the A′ symmetry are collected in Tables (3.10) and (3.11), respectively. Each table is broken into three sections,
organizing results for the three reported excited states of a given symmetry under the headings 11 A′ , 21 A′ , and 31 A′ in Table (3.10) and 13 A′ , 23 A′ , and 33 A′ in Table (3.11). In
both tables it is clear that the EOMCCSD method does well for all calculated states in the
R = 1.3 − 2.0 bohr region, i.e., near the equilibrium geometry on the ground-state PES,
but clearly fails at geometries 2.4 bohr and beyond, producing errors relative to full CI of
20-35 millihartree. The CR-EOMCC(2,3) approach improves significantly upon the EOMCCSD results in the stretched geometry region of each PES cut, reducing the MUEs to no
more then 3.75 millihartree. To further assess the quality of the CR-EOMCC(2,3) results,
and better appreciate the performance of the “black-box” CR-EOMCC(2,3) theory, comparison with the MRCC results of Ref. [271] is made. The results obtained with the expert
7R-GMS-SU-CCSD approach are worse in all cases, with only a few exceptions. For the
31 A′ state, CR-EOMCC(2,3) does significantly better than 7R-GMS-SU-CCSD, while for
the 33 A′ state 7R-GMS-SU-CCSD does significantly better than CR-EOMCC(2,3). Overall,
it may be concluded that the “black-box” CR-EOMCC(2,3) method performs better than
the complicated, expert 7R-GMS-SU-CCSD approach for the A′ states reported in Tables
(3.10) and (3.11).

Now, moving to results for the A′′ symmetry states, which are presented for the two
lowest-lying singlet states in Table (3.12) and for the two lowest-lying triplet states in Table
(3.13), many of the same trends arise as in the case of the A′ states, including the failure
of EOMCCSD for stretched nuclear geometries and the great improvements offered by the
CR-EOMCC(2,3) method. For the 21 A′′ state results could not be obtained using the CR84

EOMCC(2,3) method for the O-H bond length R of 4.4 bohr, so for this one state, MUEs
and NPEs for all methods are calculated using only 10 points, from the R = 1.3 − 4.2 bohr
region. For all the A′′ symmetry states, the 4R-GMS-SU-CCSD and higher-quality (8,4)CCSD MRCC results can be found in Ref. [271], so both types of calculations are reported
in the tables. Comparing the performance of the CR-EOMCC(2,3) method with the (8,4)CCSD results for the 11 A′′ , 21 A′′ , 13 A′′ , and 23 A′′ states, it is clear that these methods
produce similar results in every case. In a few cases, CR-EOMCC(2,3) does slightly better
and in others (8,4)-CCSD is slightly better. For example, looking at the 21 A′′ state, CREOMCC(2,3) produces a MUE relative to full CI of 0.89 millihartree and a NPE of 2.66
millihartree, which is better than the values produced by (8,4)-CCSD, of 3.80 and 9.66
millihartree, respectively, but for the 23 A′′ state the (8,4)-CCSD method produces slightly
better values, with values for MUE and NPE of 2.16 and 1.67 millihartree, respectively, as
compared with the MUE of 3.19 millihartree and the NPE of 13.64 millihartree produced
by CR-EOMCC(2,3). It may be concluded from comparison with the N R-RMR-CCSD(T),
N R-GMS-SU-CCSD, and (N ,M )-CCSD results reported in Ref. [271] that the CR-CC(2,3)
and CR-EOMCC(2,3) methods produce results of the MRCC or better quality for all of the
low-lying PESs of water examined here.

85

Table 3.10: Same as Table (3.9) for the three lowest-lying 1 A′ states.

1 1 A′
R (bohr)

FCIa

2 1 A′

7R-SUb EOMSDc (2,3)d

FCIa

3 1 A′

7R-SUb EOMSDc (2,3)d

FCIa

7R-SUb EOMSDc (2,3)d

1.3
1.6
1.809
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

-0.62821
-0.77055
-0.79860
-0.80575
-0.81062
-0.81932
-0.82860
-0.83618
-0.84172
-0.84379
-0.84546

2.07
2.49
2.93
3.63
5.95
5.55
4.56
4.17
4.04
3.98
3.92

-0.71
-0.34
0.34
1.56
6.14
11.15
15.23
18.42
20.79
21.69
22.40

0.27
0.49
0.64
0.76
1.00
0.89
0.39
-0.34
-1.20
-1.64
-2.07

-0.54185
-0.86730
-0.71786
-0.72455
-0.71631
-0.72567
-0.73203
-0.73289
-0.72999
-0.72738
-0.72415

1.97
2.25
2.55
2.94
2.91
4.00
4.51
3.74
3.06
3.08
3.32

-1.49
-0.63
0.42
1.53
5.29
12.97
20.47
27.73
32.68
33.89
34.35

0.10
0.27
0.40
0.52
0.96
1.26
-0.46
-1.91
-4.57
-4.98
-4.92

-0.39097
-0.57861
-0.63701
-0.66800
-0.69672
-0.67863
-0.66287
-0.66382
-0.66878
-0.66943
-0.66889

3.20
3.12
3.36
3.83
4.47
4.72
8.51
19.90
28.01
29.03
28.68

0.64
1.29
2.17
3.34
5.46
6.82
27.97
26.69
32.64
34.21
-20.91

0.92
1.22
1.36
1.46
0.81
1.45
2.28
8.57
8.34
7.33
7.56

MUE
NPE

—
—

3.94
3.88

10.80
23.11

0.88
3.07

—
—

3.12
2.54

15.59
35.84

1.85
6.24

—
—

12.44
25.91

14.74
55.12

3.75
7.76

a Full CI energies (E), reported as (E + 75) hartree, taken from Ref. [271]. b 7R-GMS-SU-CCSD energies, reported as
differences from full CI, in millihartree, taken from Ref. [271]. c EOMCCSD energies, reported as differences from full CI, in
millihartree. d CR-EOMCC(2,3) energies, reported as differences from full CI, in millihartree.

86

Table 3.11: Same as Table (3.9) for the three lowest-lying 3 A′ states.

1 3 A′
R (bohr)

FCIa

2 3 A′

7R-SUb EOMSDc (2,3)d

FCIa

3 3 A′

7R-SUb EOMSDc (2,3)d

FCIa

7R-SUb EOMSDc (2,3)d

1.3
1.6
1.809
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

-0.65020
-0.79340
-0.82511
-0.84272
-0.89158
-0.93780
-0.96789
-0.98540
-0.99516
-0.99822
-1.00046

2.94
3.32
3.79
4.50
5.65
5.57
5.13
4.56
3.99
3.75
3.55

-0.93
-0.70
-0.22
0.91
3.76
4.75
4.89
4.70
4.35
4.15
3.94

0.30
0.51
0.65
0.81
1.26
1.41
1.41
1.42
1.45
1.46
3.70

-0.55577
-0.70872
-0.74823
-0.76263
-0.78565
-0.81625
-0.83261
-0.84103
-0.84553
-0.84693
-0.84797

2.92
3.76
4.68
5.49
5.68
4.96
4.26
3.79
3.48
3.37
3.28

-1.63
-0.59
0.62
1.47
4.85
8.94
12.71
16.07
18.81
19.91
20.82

0.18
0.42
0.67
0.92
1.22
1.08
0.82
0.34
-0.28
-0.62
-0.97

-0.44370
-0.61947
-0.67629
-0.71482
-0.73737
-0.70581
-0.67660
-0.65564
-0.64297
-0.63921
-0.63687

4.57
4.28
4.04
4.82
7.32
8.02
7.10
6.22
7.09
8.22
9.63

0.29
1.03
1.88
2.87
5.29
4.51
-6.01
12.83
19.65
33.47
34.58

0.84
1.21
1.40
1.30
1.41
1.46
3.81
3.87
4.77
18.41
17.25

MUE
NPE

—
—

4.36
2.71

2.78
5.82

1.03
1.16

—
—

4.32
2.76

7.42
21.54

0.70
1.84

—
—

6.07
4.18

7.58
39.48

3.75
17.57

a Full CI energies (E), reported as (E + 75) hartree, taken from Ref. [271]. b 7R-GMS-SU-CCSD energies, reported as
differences from full CI, in millihartree, taken from Ref. [271]. c EOMCCSD energies, reported as differences from full CI, in
millihartree. d CR-EOMCC(2,3) energies, reported as differences from full CI, in millihartree.

87

Table 3.12: Same as Table (3.9) for the two lowest-lying 1 A′′ states.
1 1 A′′

2 1 A′′

FCIa

4R-SUb

(8, 4)-SDc

EOMSDd

(2,3)e

FCIa

4R-SUb

(8, 4)-SDc

EOMSDd

(2,3)e

1.3
1.6
1.809
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

-0.70187
-0.85127
-0.88566
-0.90098
-0.92945
-0.95679
-0.97626
-0.98862
-0.99617
-0.99870
-1.00065

1.80
2.11
2.37
2.53
2.69
2.87
3.14
3.36
3.47
3.48
3.48

1.56
1.73
1.84
1.91
1.87
1.65
1.34
0.94
0.56
0.47
0.66

-0.71
-0.54
-0.22
0.62
4.44
8.51
11.88
14.43
16.15
16.73
17.16

0.28
0.53
0.62
0.62
0.63
0.54
0.08
-0.64
-1.48
-1.90
-2.30

-0.61400
-0.76861
-0.80694
-0.81992
-0.81008
-0.78177
-0.75430
-0.73276
-0.71846
-0.71372
-0.71037

1.89
2.21
2.46
2.78
3.50
4.47
6.21
9.12
13.52
16.02
18.04

1.53
1.66
1.77
1.88
1.98
2.03
2.75
5.15
8.89
10.32
11.19

-1.39
-0.67
0.18
0.88
1.01
0.70
0.78
1.57
3.43
4.78
6.35

0.21
0.37
0.44
0.47
0.41
0.40
0.58
1.04
2.08
2.87
N.C.

MUE
NPE

—
—

2.85
1.68

1.32
1.44

8.31
17.44

0.87
2.53

—
—

6.22
8.79

3.80
9.66

1.54
7.74

0.89
2.66

R (bohr)

a Full CI energies (E), reported as (E+75) hartree, taken from Ref. [271]. b 4R-GMS-SU-CCSD energies, reported as differences
from full CI, in millihartree, taken from Ref. [271]. c (8,4)-CCSD energies, reported as differences from full CI, in millihartree,
taken from Ref. [271]. d EOMCCSD energies, reported as differences from full CI, in millihartree. e CR-EOMCC(2,3) energies,

reported as differences from full CI, in millihartree.

88

Table 3.13: Same as Table (3.9) for the two lowest-lying 3 A′′ states.
1 3 A′′

2 3 A′′

FCIa

4R-SUb

(8, 4)-SDc

EOMSDd

(2,3)e

FCIa

4R-SUb

(8, 4)-SDc

EOMSDd

(2,3)e

1.3
1.6
1.809
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

-0.72170
-0.87187
-0.90726
-0.92363
-0.95148
-0.97420
-0.98837
-0.99640
-1.00091
-1.00236
-1.00345

2.77
3.18
3.60
4.08
4.87
4.72
4.31
4.00
3.81
3.72
3.61

1.88
2.09
2.29
2.52
2.89
2.95
3.00
3.20
3.43
3.47
3.25

-0.89
-0.73
-0.41
0.44
4.03
7.88
11.40
14.49
16.94
17.89
18.68

0.32
0.56
0.66
0.68
0.84
1.00
0.88
0.50
-0.05
-0.34
0.50

-0.62356
-0.78052
-0.82109
-0.83571
-0.82785
-0.80175
-0.77863
-0.76634
-0.76393
-0.76335
-0.76232

2.75
3.31
3.98
4.68
5.83
7.33
10.70
17.79
25.85
27.93
28.62

1.81
2.13
2.45
2.70
2.68
2.36
2.26
2.03
1.36
1.03
0.62

-1.19
-0.23
0.80
1.61
2.02
2.59
5.19
12.44
16.94
26.41
29.01

0.29
0.53
0.67
0.79
0.92
1.36
2.92
7.31
12.59
13.93
14.51

MUE
NPE

—
—

3.92
2.10

2.70
1.59

6.46
18.78

0.64
1.34

—
—

9.37
25.18

2.16
1.67

5.83
27.60

3.19
13.64

R (bohr)

a Full CI energies (E), reported as (E+75) hartree, taken from Ref. [271]. b 4R-GMS-SU-CCSD energies, reported as differences
from full CI, in millihartree, taken from Ref. [271]. c (8,4)-CCSD energies, reported as differences from full CI, in millihartree,
taken from Ref. [271]. d EOMCCSD energies, reported as differences from full CI, in millihartree. e CR-EOMCC(2,3) energies,

reported as differences from full CI, in millihartree.

89

A particularly weak aspect of the SR methods, when used to describe bond-breaking
PES cuts, is their erratic behavior as states become degenerate at the dissociation limit. In
order to assess the quality of the CC and EOMCC PES cuts considered in this report, a few
states have been plotted in Figure (3.3). The states were carefully chosen such that they all
converge with at least one other plotted state to become degenerate at the largest internuclear
separation. In the plot, these electronic states have been identified as those resulting from
combination of the doublet S ground-state of the hydrogen radical with the lowest-energy
2 Π, 2 Σ+ , and 2 Σ− states of the hydroxyl radical. It is quite clear from visual inspection

of Figure (3.3(a)) that the CCSD/EOMCCSD curves, which should converge to become
degenerate as the dissociation occurs, are for the most part still not falling on top of one
another as the O-H distance becomes large. This is especially evident for the highest-energy
pair of curves, 21 A′ and 33 A′′ , which are still separated by 20 millihartree at R = 4.4 bohr.
Significant improvement is seen in Figure (3.3(b)) where the CR-CC(2,3)/CR-EOMCC(2,3)
curves exhibit much better asymptotic behavior compared with the CCSD/EOMCCSD data,
and also give excellent agreement with the full CI potentials. However, the most chllenging
pair of curves for EOMCCSD, 21 A′ and 33 A′′ , also give CR-EOMCC(2,3) trouble, as we were
not able to obtain CR-EOMCC(2,3) results for the 21 A′ state at R = 4.4 bohr, although
the energies which were obtained for these two PESs showed dramatic overall improvement
over the corresponding CCSD/EOMCCSD results.

To summarize, in this section it has been shown that the CR-EOMCC(2,3) method is
capable of providing a highly accurate description of excited-state PESs of water involving
single bond-breaking. However, excited-state PESs are not directly observable quantities,
so next we move toward calculating spectroscopic properties which are observable. In the
90

next section, the utility of the size-intensive δ-CR-EOMCC(2,3) method in calculations of
vertical excitation energies and spectral shifts in an organic chromophore, resulting from the
introduction of hydrogen-bonded molecular environments, is examined.

3.2.5

Excitation Energies and Hydrogen-Bonding-Induced Spectral Shifts in Complexes of cis-7-Hydroxyquinoline

In spite of being relatively weak, non-covalent interactions with the environment, such as
hydrogen bonds, can qualitatively affect the electronic structure and properties of the embedded molecules. Accurately predicting the effect of a hydrogen-bonded environment on
the electronic structure of embedded molecules represents a challenge for computational
chemistry. Among such properties, electronic excitation energies are of great interest in view
of the common use of organic chromophores as probes in various environments [272–275].
Typically, hydrogen bonding results in shifts in the positions of the maxima of the absorption
and emission bands anywhere between a few hundred and about 3000 cm−1 [276]. Thus,
in order to be able to use computer modeling for interpretation of experimental data, the
intrinsic errors of the calculated shifts must be very small, on the order of 100 cm−1 or less.
We became involved in a joint project with the Wesolowski Geneva group aimed at comparing shifts in the π → π ∗ excitation energy of the cis-7-hydroquinoline (cis-7HQ) chromophore resulting from the formation of hydrogen-bonded complexes between cis-7HQ and a
number of small molecules [201]. Our main role was to provide reference excitation energies
using EOMCC methods, which were subsequently used to benchmark the so-called frozendensity embedding theory (FDET) approach, developed by Wesolowski et al. [277–281],
and the conventional supermolecular form of the time-dependent density functional theory
91

Figure 3.3: Cuts of the PESs for a few ground- and excited-states of a single-bond stretching
model of H2 O as obtained with (a) the CCSD/EOMCCSD and (b) the CR-CC(2,3)/CREOMCC(2,3) approaches, and the TZ basis set. Lines are used to represent the CC/EOMCC
data, while the corresponding full CI values are represented by points (which are identically
replicated in (a) and (b)).
92

(TDDFT), frequently used in spectral shift calculations. Since the main goal of this thesis is
to examine the performance of the CR-CC/CR-EOMCC and other CC/EOMCC approaches,
our discussion below focuses on a comparison of the EOMCCSD and CR-EOMCC(2,3)-level
results with the available experimental data for the π → π ∗ excitation energies and the
corresponding environment-induced spectral shifts in the 7HQ chromophore [276]. The environment molecules constituting the eight complexes involving 7HQ that were examined in
Ref. [201] included (i) a single water molecule, (ii) a single ammonia molecule, (iii) a water
dimer, (iv) a single molecule of methanol, (v) a single molecule of formic acid, (vi) a trimer
consisting of ammonia and two water molecules, (vii) a trimer consisting of ammonia, water,
and ammonia, and (viii) a trimer consisting of two ammonia and one water molecules (see
Figure 3.2.5).

In order to establish the appropriate level of EOMCC theory to serve as a reference for the
FDET and supermolecular TDDFT calculations reported in Ref. [201], we first examined the
dependence of the vertical excitation energies ωπ→π ∗ and environment-induced spectral shifts
∆ωπ→π ∗ on the basis set. Table (3.14) compares the results of the EOMCCSD calculations
obtained with the 6-31+G(d), [282–284] 6-31++G(d,p), [282–284] 6-311+G(d), [284,285] and
aug-cc-pVDZ basis sets, as well as with the [5s3p2d/3s2p] basis of Sadlej [286], designated
as POL, for the two smallest complexes, 7HQ · · · H2 O and 7HQ · · · NH3 , for which we could
afford the largest number of computations. The results in Table (3.14) indicate that although
the vertical excitation energies ωπ→π ∗ in the bare cis-7HQ system and its complexes with
the water and ammonia molecules vary with the basis set (for the basis sets tested here by as
much as about 600 cm−1 ), the environment-induced shifts ∆ωπ→π ∗ are almost insensitive to
the basis set choice, varying by at most 27 cm−1 between all basis sets reported. Although we
93

cis-7HQ · · · H2 O

cis-7HQ · · · NH3

cis-7HQ · · · (H2 O)2

cis-7HQ · · · CH3 OH

cis-7HQ · · · HCOOH

cis-7HQ · · · (NH3 -H2 O-H2 O)

cis-7HQ · · · (NH3 -H2 O-NH3 )

cis-7HQ · · · (NH3 -NH3 -H2 O)

Figure 3.4: The eight hydrogen-bonded complexes of the cis-7HQ molecule.
94

Table 3.14: The basis-set dependence of the vertical excitation energies ωπ→π ∗ and the
environment-induced shifts ∆ωπ→π ∗ (in cm−1 ) obtained with the EOMCCSD approach
corresponding to the lowest π → π ∗ transition in the cis-7HQ chromophore and its complexes
with the water and ammonia molecules.
∆ωπ→π ∗

ωπ→π ∗
Basis set

7HQ

7HQ · · · H2 O

6-31+G(d)
6-31++G(d,p)
6-311+G(d)
aug-cc-pVDZ
POL

35171
35120
35046
34707
34596

7HQ · · · NH3 7HQ · · · H2 O

34643
34597
34500
34182
34077

34396
34351
34263
33923
33819

-528
-523
-546
-525
-519

7HQ · · · NH3
-775
-769
-783
-784
-777

were unable to perform a similarly thorough analysis for other complexes due to prohibitive
computer costs, we were able to obtain the EOMCCSD ωπ→π ∗ and ∆ωπ→π ∗ values for
all of the complexes examined in this study using the 6-31+G(d) and 6-311+G(d) basis
sets. As shown in Table (3.15), the differences between the EOMCCSD/6-31+G(d) and
EOMCCSD/6-311+G(d) values of the environment-induced shifts ∆ωπ→π ∗ remain small
for all complexes of interest, ranging from 8 cm−1 in the 7HQ · · · NH3 case to 43 cm−1 in
the case of 7HQ · · · (H2 O)2 , or 1–3 %. Thus, we can conclude that the choice of the basis set,
although important for obtaining the converged ωπ→π ∗ values, is of little significance when
the environment-induced shifts in the vertical excitation energy corresponding to the lowest
π → π ∗ transition in the cis-7HQ chromophore are considered with the EOMCC methods.
Although the EOMCCSD approach is known to provide an accurate description of excited
states dominated by one-electron transitions, such as the π → π ∗ transition in cis-7HQ and
its complexes, there have been cases of similar states reported in the literature, where the
EOMCCSD level has not been sufficient to obtain high-quality results [287, 288]. Moreover,
our interest in this study is in the small energy differences defining the environment-induced
95

shifts ∆ωπ→π ∗ , which may be sensitive to the higher-order correlation effects neglected in
the EOMCCSD calculations. For this reason, we also examined the effect of triples corrections to EOMCCSD energies on the calculated ωπ→π ∗ and ∆ωπ→π ∗ values by performing
the δ-CR-EOMCC(2, 3) calculations with the 6-31+G(d) basis set. The results of these
calculations, shown in Table (3.15), indicate that triple excitations have a significant effect on the vertical excitation energies ωπ→π ∗ , reducing the 4000-5000 cm−1 differences
between the EOMCCSD and experimental data to no more than about 800 cm−1 , when
the δ-CR-EOMCC(2, 3),A/6-31+G(d) calculations are performed, and no more than about
500 cm−1 when the δ-CR-EOMCC(2, 3),D/6-31+G(d) approach is employed, while bringing
the ∆ωπ→π ∗ values closer to the experimentally observed shifts when compared with EOMCCSD values. Although the differences between the δ-CR-EOMCC(2, 3) and EOMCCSD
values of the environment-induced shifts ∆ωπ→π ∗ resulting from the calculations with the
6-31+G(d) basis set do not exceed 15–16 % of the EOMCCSD values, triples corrections
improve the EOMCCSD results and, as such, are useful for the generation of the reference
EOMCC data.

96

Table 3.15: The vertical excitation energies ωπ→π ∗ and the environment-induced shifts ∆ωπ→π ∗ (in cm−1 ) obtained with
the EOMCCSD/6-31+G(d), EOMCCSD/6-311+G(d), δ-CR-EOMCC(2, 3), A/6-31+G(d), and δ- CR-EOMCC(2, 3), D/631+G(d) approaches, and their composite EOMCC corresponding to the lowest π → π ∗ transition in the cis-7HQ chromophore
and its various complexes.

Environment

EOMCCSD/ EOMCCSD/ δ-CR-EOMCC(2, 3), A/ δ-CR-EOMCC(2, 3), D/
6-31+G(d) 6-311+G(d)
6-31+G(d)
6-31+G(d) EOMCC,Aa EOMCC,Db Exp.c

None
H2 O
NH3
2H2 O
CH3 OH
HCOOH
NH3 -H2 O-H2 O
NH3 -H2 O-NH3
NH3 -NH3 -H2 O

35171
34643
34396
33867
34830
34505
33381
33542
33302

35046
34500
34263
33699
34695
34371
33218
33385
33136

H2 O
NH3
2H2 O
CH3 OH
HCOOH
NH3 -H2 O-H2 O
NH3 -H2 O-NH3
NH3 -NH3 -H2 O

-528
-775
-1304
-341
-666
-1790
-1629
-1869

-546
-783
-1347
-351
-675
-1828
-1661
-1910

ωπ→π ∗
31103
30558
30291
29700
30717
30368
29171
29355
29088
∆ωπ→π ∗
-544
-812
-1403
-386
-734
-1932
-1748
-2014

30711
30199
29922
29378
30428
30056
28863
29036
28812

30977
30415
30157
29532
30582
30235
29008
29197
28922

30586
30056
29788
29210
30293
29922
28701
28879
28646

30830
30240
29925
29193
30363
29816
28340
28694
28348

-512
-789
-1333
-283
-655
-1847
-1675
-1899

-562
-820
-1446
-396
-743
-1969
-1780
-2055

-530
-797
-1376
-292
-664
-1885
-1707
-1940

-590
-905
-1637
-467
-1014
-2490
-2136
-2482

a Defined by Eq. (3.99), in which variant A of CR-EOMCC(2,3) is employed. b Defined by Eq. (3.99), in which variant D of
CR-EOMCC(2,3) is employed. c Obtained with the laser resonant two-photon ionization UV spectroscopy [276].

97

Ideally, one would like to perform the δ-CR-EOMCC(2, 3) calculations for basis sets
larger than 6-31+G(d), such as 6-311+G(d), but complexes of cis-7HQ examined in this
study were too large for performing such calculations on our computers. Thus, in the absence of the δ-CR-EOMCC(2, 3) larger basis set data and considering the fact that the
triples corrections to the environment-induced shifts ∆ωπ→π ∗ are relatively small when compared to the EOMCCSD ∆ωπ→π ∗ values, we have decided to combine the EOMCCSD/6311+G(d) results with the triples corrections to EOMCCSD energies extracted from the
δ-CR-EOMCC(2, 3)/6-31+G(d) calculations. The final EOMCC values of the vertical excitation energies ωπ→π ∗ were obtained using a composite approach, in which we augment the
EOMCCSD/6-311+G(d) results by the triples corrections to EOMCCSD energies extracted
from the δ-CR-EOMCC(2, 3)/6-31+G(d) calculations, as in the following formula:

ωπ→π ∗ (EOMCC) = ωπ→π ∗ (EOMCCSD/6-311+G(d))
+ [ωπ→π ∗ (δ-CR-EOMCC(2,3)/6-31+G(d))

(3.99)

− ωπ→π ∗ (EOMCCSD/6-31+G(d))] .
As shown in Table (3.15), the resulting composite EOMCC,A and EOMCC,D approaches
provide vertical excitation energies ωπ→π ∗ that are in excellent agreement with the experimental excitation energies, while offering further improvements in the environment-induced
shifts ∆ωπ→π ∗ when compared with the EOMCCSD/6-311+G(d) and δ-CR-EOMCC(2, 3)/631+G(d) calculations. Indeed, the EOMCC,A approach, which adds the triples correction
extracted from the δ-CR-EOMCC(2, 3),A/ 6-31+G(d) calculation to the EOMCCSD/6311+G(d) energy, gives errors in the calculated excitation energies ωπ→π ∗ relative to experiment that range between 147 cm−1 in the case of the bare cis-7HQ system and 668

98

cm−1 in the case of the 7HQ · · · (NH3 -H2 O-H2 O) complex, never exceeding 2 % of the
experimental excitation energies. The EOMCC,D approach, which adds the triples correction obtained in the δ-CR-EOMCC(2, 3),D/6-31+G(d) calculation to the EOMCCSD/6311+G(d) energy, gives errors in the calculated excitation energies ωπ→π ∗ relative to experiment that range between 17 cm−1 in the case of the 7HQ · · · (H2 O)2 complex and 361 cm−1
for 7HQ · · · (NH3 -H2 O-H2 O), or no more than 1 % of the experimental values. These results
should be compared to the much larger differences between the EOMCCSD/6-311+G(d)
and experimental excitation energies that range between 14 and 17 %. The complexationinduced spectral shifts ∆ωπ→π ∗ resulting from the EOMCC,A and EOMCC,D calculations
agree with their experimental counterparts to within 5–27 % or 15 % on average in the case
of EOMCC,A and 10–37 % or 22 % on average in the EOMCC,D case. The EOMCC,D
approach, while bringing the excitation energies ωπ→π ∗ to a closer agreement with experiment than the EOMCCSD/6-311+G(d) calculations, does not offer improvements in the
calculated shifts ∆ωπ→π ∗ . The composite EOMCC,A approach provides additional small
improvements in the calculated ∆ωπ→π ∗ values, reducing the 7–33 % errors relative to experiment obtained in the EOMCCSD/6-311+G(d) calculations to 5–27 %. The EOMCC,A
data, which was generated in this study, served as a computational benchmark for the
TDDFT and FDET results and it was successfully shown that the non-relaxed FDET model
represents a reasonable alternative to EOMCC methods for calculating environment-induced
spectral shifts of excitation energies, but with a much more favorable computational scaling than the EOMCC-type approaches. The non-relaxed FDET results for the π → π ∗
shifts reported in Refs. [201] agree with our best EOMCC,A and EOMCC,D data to within
about 100 cm−1 on average. This should be contrasted with the poor performance of the

99

supermolecular TDDFT appraoch that gives average errors in the calculated ∆ωπ→π ∗ values
relative to EOMCC,A or EOMCC,D on the order of 700 cm−1 [201]. In the next section the
performance of the EA- and IP-EOMCC methods for open-shell systems is considered.

3.2.6

Geometries and Adiabatic Excitation Energies of CNC, C2 N,
N3 , and NCO

The accurate determination of geometries and energetics of ground- and excited-state radicals is very difficult, both experimentally, as open-shell species are usually characterized by
short lifetimes, and computationally, due to the complexity of the many-electron correlation
problem in open-shell systems. This is particularly true for the SR CC and EOMCC methods.
Among the main reasons is the fact that the low-lying excited-states of radicals and other
open-shell species are often dominated by two-electron and other multi-electron transitions.
The goal of this section is the examination of the performance of the full and active-space
EA- and IP-EOMCC approaches with up to 3p-2h/3h-2p excitations in the calculations of
the ground and low-lying excited states of the challenging open-shell CNC, C2 N, NCO, and
N3 molecules [132, 289], as well as the prediction of the corresponding nuclear geometries
and adiabatic excitation energies, which was carried out jointly with Mr. Jared Hansen from
our group [290]. A series of EA-EOMCCSD(2p-1h), active-space EA-EOMCCSD(3p-2h){4},
and full EA-EOMCCSD(3p-2h) calculations were performed for the CNC and C2 N molecules
using the CCSD ground states of the CNC+ and C2 N+ closed-shell cations to provide the
reference wave functions. In addition to the DZP basis, [291] which was employed in a
previous study, [132, 289] the cc-pVDZ (all three methods), cc-pVTZ (all three methods),
and cc-pVQZ [the EA-EOMCCSD(2p-1h) and EA-EOMCCSD(3p-2h){4} approaches] ba100

sis sets were used. The full EA-EOMCCSD(3p-2h) calculations using the largest cc-pVQZ
basis set could not be performed due to the large computer costs of the relevant numerical geometry optimizations [the analytic gradients of the EA-EOMCCSD(3p-2h) approach
are not available]. The active orbital spaces for the EA-EOMCCSD(3p-2h){4} calculations
for CNC and C2 N consisted of the two lowest-energy pairs of unoccupied π molecular orbitals of CNC+ and C2 N+ , respectively. A series of IP-EOMCCSD(2h-1p), active-space
IP-EOMCCSD(3h-2p){2}, and full IP-EOMCCSD(3h-2p) calculations were performed for
the NCO and N3 molecules using the CCSD ground states of the NCO− and N3 − closedshell anions to provide the reference wave functions. Again, in addition to the previously
employed [132, 289] DZP basis, the cc-pVDZ (all three methods), cc-pVTZ (all three methods), and cc-pVQZ [the IP-EOMCCSD(2h-1p) and IP-EOMCCSD(3h-2p){2} approaches]
basis sets were used, and the full IP-EOMCCSD(3h-2p)/cc-pVQZ calculations were not
performed due to the large computer costs of the corresponding numerical geometry optimizations [the analytic gradients of the IP-EOMCCSD(3h-2p) method are not available].
The active orbital spaces for the IP-EOMCCSD(3h-2p){2} calculations for NCO and N3
consisted of the highest-energy pair of occupied orbitals of NCO− and N3 − , respectively.

Unlike in earlier work [132, 289], where the nuclear geometries of the ground and excited states of the CNC, C2 N, NCO, and N3 species were optimized using only one method
(SAC-CI-SDT-R/PS [292–294]) and one basis set (DZP), in each molecular case and for
each electronic state and basis set considered in the present work, the nuclear geometries
were optimized at the same level of the EA/IP-EOMCC theory and with the same basis
set as those used to evaluate the corresponding total and adiabatic excitation energies. The
geometry optimizations using the cc-pVXZ (X = D, T, and Q) basis sets were constrained
101

to linear geometries, since the analogous unconstrained optimizations using the DZP basis set and bent initial structures showed that the optimum geometries of the calculated
states of CNC, C2 N, NCO, and N3 are linear. The unconstrained optimizations with the
DZP basis set demonstrated that we can assume the D∞h (in practice, D2h ) symmetry for
each of the calculated states of CNC and N3 , and that we can use the C∞v (in practice,
C2v ) symmetry in the geometry optimizations for C2 N and NCO. In all post-RHF (CCSD
and EA/IP-EOMCC) calculations, the lowest-energy core orbitals correlating with the 1s
orbitals of the C and N atoms were kept frozen and the spherical components of the d, f ,
and g functions were employed throughout. In addition to the results of the finite basis
set calculations, the total and excitation energies obtained in the EA- and IP-EOMCC calculations for the CNC, C2 N, NCO, and N3 molecules were extrapolated to the CBS limit.
The CBS extrapolations were limited to the EA-EOMCCSD(2p-1h), IP-EOMCCSD(2h-1p),
active-space EA-EOMCCSD(3p-2h){4}, and active-space IP-EOMCCSD(3h-2p){2} calculations, since full EA-EOMCCSD(3p-2h) and IP-EOMCCSD(3h-2p) computations including
geometry optimizations were too expensive to be run using the cc-pVQZ basis set that would
be required to obtain reliable CBS-limit values. In all of the remaining cases, the complete
data sets corresponding to the cc-pVDZ, cc-pVTZ, and cc-pVQZ basis sets were available,
enabling reasonably meaningful CBS extrapolations. To verify the numerical stability of
our CBS extrapolations, two different extrapolation schemes, referred to as the CBS-A and
CBS-B approaches, were utilized.

In the CBS-A scheme, the CBS limit of the ground-state CCSD correlation energy of
the closed-shell N -electron reference system relevant to the EA/IP-EOMCC calculations
for the (N ± 1)-electron target species was determined using the well-known extrapolation
102

formula [217] given in Eq. (3.98) and the cc-pVTZ and cc-pVQZ data. Here, ∆E(X) in
Eq. (3.98) is the CCSD correlation energy obtained with the cc-pVXZ basis set, where
X represents the cardinal number of the basis set (X = 3 for cc-pVTZ and X = 4 for
cc-pVQZ), and ∆E∞ is the CCSD correlation energy in the CBS limit. The resulting
extrapolated CBS correlation energy ∆E∞ was then added to the RHF/cc-pV6Z energy
of the N -electron reference system, computed at the optimized geometry of the state of
interest resulting from the appropriate EA- or IP-EOMCC/cc-pVQZ calculations. Recalling
the well-known and previously discussed fact that the RHF energies converge exponentially
with the basis set, it is usually best to simply perform the calculations at the RHF level
with a very large correlation consistent basis set, such as cc-pV6Z, if such calculations are
affordable. As a check, the level of basis-set convergence was verified by comparing the
RHF/cc-pV5Z and RHF/cc-pV6Z data, obtaining differences of about 0.2 millihartree in all
of the examined cases. Once the CBS values of the RHF total and CCSD correlation energies
of the N -electron reference system were determined, the desired CBS limits of the groundand excited-state energies of the (N ± 1)-electron target species corresponding to the EAor IP-EOMCC calculations of interest were computed using the formula

EA/IP-EOMCC

Eµ,∞

RHF
CCSD
(N ± 1) = E6Z (N ) + ∆E0,∞ (N )
EA/IP-EOMCC

+Eµ,QZ
EA/IP-EOMCC

where Eµ,∞

(N ±1)

state |Ψµ

CCSD
(N ± 1) − E0,QZ (N ),

(3.100)

(N ± 1) is the final extrapolated energy of the (N ± 1)-electron

RHF
, E6Z (N ) is the ground-state RHF energy of the closed-shell N -electron

CCSD
reference system obtained with the cc-pV6Z basis set, ∆E0,∞ (N ) is the extrapolated

CCSD correlation energy of the N -electron reference system obtained using Eq. (3.98),
103

EA/IP-EOMCC

Eµ,QZ

(N ±1)

|Ψµ

(N ± 1) is the total EA/IP-EOMCC energy of the (N ± 1)-electron state

CCSD
obtained with the cc-pVQZ basis set, and E0,QZ (N ) is the total CCSD energy of

the N -electron reference system obtained using the cc-pVQZ basis set. This method of estimating the CBS values of the total electronic energies of the ground and excited states of the
CNC, C2 N, NCO, and N3 radicals is based on the assumption that the electron-attachment
(N ±1)

or ionization energies ωµ

obtained with the cc-pVQZ basis set are essentially converged

with respect to the basis set, so all one has to do is obtain the CBS limit of the CCSD
ground-state energy of the N -electron reference system and add the cc-pVQZ values of the
electron-attachment or ionization energies to estimate the CBS energies of the ground and
excited states of the corresponding (N ± 1)-electron target species.
In the second basis set extrapolation method, referred to as the CBS-B approach, the
total CBS energy of each (N ± 1)-electron target state of interest was directly determined
using the formula [212]

2

E(X) = E∞ + Be−(X−1) + Ce−(X−1) ,

(3.101)

and the cc-pVDZ, cc-pVTZ, and cc-pVQZ data. As in Eq. (3.98), the X variable number
entering Eq. (3.101) is again the cardinal number of the cc-pVXZ basis set (X = 2 for
cc-pVDZ, X = 3 for cc-pVTZ, and X = 4 for cc-pVQZ), E(X) is the total EA/IP-EOMCC
energy computed with the cc-pVXZ basis set, and E∞ is the desired CBS limit of the total
EA/IP-EOMCC energy for a given electronic state of the (N ± 1)-electron species. The
difference between the CBS-A and CBS-B extrapolation schemes lies in the fact that the
latter scheme extrapolates the total EA- or IP-EOMCC energy of each electronic state of the
(N ± 1)-electron target species separately, using Eq. (3.101), whereas the former approach
104

extrapolates the ground-state correlation energy of the N -electron reference system only
using Eq. (3.98) while making an assumption that the electron-attachment and ionization
energies resulting from the EA- and IP-EOMCC calculations converge faster with the basis
set than the total energies of the (N ± 1)-electron target species, as reflected in Eq. (3.100).
The results of our EA- and IP-EOMCC calculations, along with the available experimental data [295–297], are reported in Tables (3.16)-(3.21). The EA-EOM-CCSD(2p-1h),
EA-EOMCCSD(3p-2h){4}, and full EA-EOMCCSD(3p-2h) results for the CNC and C2 N
molecules are reported in Tables (3.16) and (3.17) for the total and adiabatic excitation energies, and (3.18) for the geometries. The IP-EOMCCSD(2h-1p), IP-EOMCCSD(3h-2p){2},
and full IP-EOMCCSD(3h-2p) results for the NCO and N3 molecules are reported in Tables
(3.19) and (3.20) for the total and adiabatic excitation energies, and (3.21) for the geometries.
Our discussion is divided into two parts.
We begin with the CNC molecule (see Tables (3.16) and (3.18)). For CNC, the EAEOMCC optimizations employing the DZP basis set produced results that deviate from
the previously reported [289] SAC-CI-SDT-R/PS optimized geometries and previously calculated [132, 289] EA-EOMCC adiabatic excitation energies using the SAC-CI-SDT-R/PS
geometries, all obtained with the same DZP basis, by 0.001-0.009 ˚ and 0.002-0.003 eV,
A
respectively, for all states and all methods considered here. Seeing that our present EAEOMCC optimizations employing the DZP basis set were able to reproduce the analogous
results of the SAC-CI-SDT-R/PS geometry optimizations and the EA-EOMCC excitation
energies at the SAC-CI-SDT-R/PS geometries reported in Ref. [289], the effect of the use
of the correlation-consistent basis sets of the cc-pVXZ quality on the calculated excitation
energies and geometries was investigated.
105

The ground X 2 Πg state of CNC is dominated by 1p excitations out of the ground state
of the closed-shell reference CNC+ ion but, the A2 ∆u and B 2 Σ+ excited states of the
u
same molecule exhibit a significant two-electron excitation character relative to the X 2 Πg
state. As shown in Table (3.16), the basic EA-EOM-CCSD(2p-1h) optimizations produced
adiabatic excitation energies that deviate from the experimental values by 3.379-4.022 eV for
the A2 ∆u and B 2 Σ+ states, demonstrating the same characteristically large errors compared
u
to experiment that are typically seen when the EA-EOMCCSD(2p-1h) approach is applied to
the excited states of radicals dominated by two-electron transitions [147–149, 298]. The full
EA-EOMCCSD(3p-2h) method improves these poor results, reducing the deviations from
experiment to 0.336-0.451 eV for both the A2 ∆u and B 2 Σ+ states, when the cc-pVDZ and
u
cc-pVTZ basis sets are employed. The reason for this considerable improvement in the data
over the EA-EOMCCSD(2p-1h) method is the explicit inclusion of the 3p-2h terms in the
(N +1)

Rµ

operator in the EA-EOMCCSD(3p-2h) calculations.

(N +1)

The inclusion of all 3p-2h components in the Rµ

operator is computationally de-

manding, particularly when one is interested in numerical gradient optimizations, such as
those performed in this work. Thus, it is of great significance to note that the active-space
EA-EOMCCSD(3p-2h){4} optimizations using the cc-pVDZ and cc-pVTZ basis sets, with
only four unoccupied orbitals in the active-space, which are only a few times more expensive
than the corresponding ground-state CCSD calculations and which require a small fraction
of the CPU time, disk, and memory when compared to the parent EA-EOMCCSD(3p-2h)
calculations, reproduce the full EA-EOMCCSD(3p-2h) optimization results to within 0.0190.039 eV for the adiabatic excitation energies and 0.8-3.2 millihartree for the total energies
of the ground and excited states of CNC examined in this study (see Table (3.16)). The
106

deviations of the EA-EOMCCSD(3p-2h){4} results from experiment are 0.312-0.422 eV for
all three correlation consistent basis sets used in the EA-EOMCCSD(3p-2h){4} optimizations. It is interesting and somewhat surprising to note the deviations from experiment
increase slightly with the size of the cc-pVXZ basis set for all three EA-EOMCC methods
exploited in this work. To help understand this behavior, the EA-EOMCCSD(2p-1h) and
EA-EOMCCSD(3p-2h){4} results were extrapolated to the CBS limit.

Examining the total energies of the ground and excited states of CNC shown in Table (3.16), it is clear that they are converging with the basis set in a systematic manner.
The CBS-A extrapolation scheme is based on the simplifying assumption that the electronattachment (or, in the IP case, ionization) energies are reasonably well converged with the
basis set, when the cc-pVQZ basis set is employed. The data in Table (3.16) show that this
is indeed a valid assumption, as the EA-EOMCCSD(2p-1h) and EA-EOMCCSD(3p-2h){4}
excitation energies do not significantly change when moving from the cc-pVTZ to cc-pVQZ
basis, the largest change being 0.109 eV for the less accurate EA-EOMCCSD(2p-1h) method
and only 0.003 eV for the active-space EA-EOMCCSD(3p-2h){4} approach. Moreover, the
CBS extrapolations resulting from the CBS-A and CBS-B schemes produce results that are
in good agreement with each other, especially for the higher-order EA-EOMCCSD(3p-2h){4}
method, where the differences in total energies do not exceed 1.3 millihartree, regardless of
the electronic state of CNC considered (recall that the CBS-B scheme extrapolates the total
energy of each state separately, without any simplifying assumptions). The CBS limits of
the excitation energies resulting from the CBS-A and CBS-B extrapolations obtained with
the EA-EOMCCSD(3p-2h){4} approach are essentially identical, deviating by 0.001 eV and
0.005 eV for the A2 ∆u and B 2 Σ+ states, respectively. Thus, it may be safely concluded that
u
107

the CBS EA-EOMCCSD(3p-2h){4} results are stable to approximately 1 millihartree for the
total energies and 0.005 eV for the adiabatic excitation energies, and can be regarded as converged with the basis set. This suggests that the 0.3–0.4 eV error relative to experiment
resulting from the full and active-space EA-EOMCCSD(3p-2h) calculations for CNC are due
to either 4p-3h excitations neglected in these calculations or errors in the experimental data.

In analogy to CNC, the ground X 2 Π state of C2 N is dominated by 1p excitations out of
the ground state of the closed-shell reference C2 N+ ion, but the low-lying A2 ∆, B 2 Σ− , and
C 2 Σ+ excited states have significant 2p-1h contributions demonstrating the rather complex
MR nature of their corresponding wave functions. The B 2 Σ− state also has non-negligible
3p-2h contributions, which make this state extremely difficult to describe by the methods used in this study. All of this causes major problems in the EA-EOMCCSD(2p-1h)
calculations. As shown in Table (3.17), even with a large cc-pVQZ basis set, the EAEOMCCSD(2p-1h) method incorrectly orders the excited states of C2 N, describing the C 2 Σ+
state as being lower in energy than the B 2 Σ− state. The errors in the EA-EOMCCSD(2p-1h)
results for the adiabatic excitation energies of C2 N relative to experiment are huge. Indeed,
our geometry optimizations using the EA-EOMCCSD(2p-1h) approach produce errors in
the calculated adiabatic excitation energies of C2 N relative to experiment of 3.507-3.969 eV
for the A2 ∆ state, 4.956-5.511 eV for the B 2 Σ− state, and 3.422-3.836 eV for the C 2 Σ+
state. As with CNC, the full inclusion of the 3p-2h components in the electron attaching
N
operator Rµ +1 significantly improves the adiabatic excitation energies relative to the disas-

trous EA-EOMCCSD(2p-1h) results, reducing the errors relative to experiment to at most
0.418 eV for the A2 ∆ state, at most 0.915 eV for the B 2 Σ− state, and at most 0.524 eV for
the C 2 Σ+ state when the cc-pVDZ and cc-pVTZ basis sets are employed, but the full EA108

EOMCCSD(3p-2h) are computationally demanding, particularly when larger basis sets have
to be examined. Thus, it is important to examine how well the considerably less expensive
active-space EA-EOMCCSD(3p-2h) approach works for the low-lying excited states of C2 N,
when the cc-pVxZ basis sets are employed.

As shown in Table (3.17), the results of the active-space EA-EOMCCSD(3p-2h){4} calculations are almost identical to those obtained with the parent EA-EOM-CCSD(3p-2h)
approach. The adiabatic excitation energies obtained with the full and active-space EAEOMCCSD(3p-2h) methods, where the latter approach uses only four unoccupied orbitals
in the active-space, calculated using the cc-pVDZ and cc-pVTZ basis sets, differ by 0.0230.053 eV for all states of C2 N examined here. The total energies obtained in the full EAEOMCCSD(3p-2h) and active-space EA-EOMCCSD(3p-2h){4} calculations employing the
cc-pVDZ and cc-pVTZ basis sets differ by 1.6-2.9 millihartree for the X 2 Π state, 0.5-1.1 millihartree for the A2 ∆ state, 0.5-0.9 millihartree for the B 2 Σ− state, and 0.7-1.4 millihartree for
the C 2 Σ+ state. Comparing the EA-EOMCCSD(3p-2h){4} results with experiment, it can
be seen that the adiabatic excitation energies resulting from the EA-EOMCCSD(3p-2h){4}
calculations using the cc-pVQZ basis set differ from the available experimental data by 0.340
eV, 0.844 eV, and 0.454 eV for the A2 ∆, B 2 Σ− , and C 2 Σ+ states, respectively, which is a
huge error reduction when compared to the corresponding EA-EOMCCSD(2p-1h)/cc-pVQZ
calculations that give the 3.969 eV, 5.511 eV, and 3.836 eV errors for the same three states,
in addition to wrong state ordering. The full EA-EOMCCSD(3p-2h) and active-space EAEOMCCSD(3p-2h){4} calculations produce the correct state ordering and relatively small
errors for the A2 ∆ and C 2 Σ+ states, but the discrepancy between the full and active-space
EA-EOMCCSD(3p-2h) results on the one hand and experiment on the other hand for the
109

B 2 Σ− state, on the order of 0.9 eV independent of the basis set, is a problem that needs to
be addressed.

The larger deviations with experiment observed in the full EA-EOMCCSD(3p-2h) and
active-space EA-EOMCCSD(3p-2h){4} calculations for the B 2 Σ− state, which do not seem
to be decreasing with the basis set and careful geometry optimizations performed in this
work, must be related to the presence of the non-negligible 3p-2h contributions in the B 2 Σ−
wave function, which indicate a highly MR character of this state that the EA-EOMCC
methods used in the present study cannot capture without incorporating higher-than-3p-2h
contributions in the EA-EOMCC considerations. As explained in Ref. [298], the presence
of significant 3p-2h contributions in the wave function requires an explicit consideration of
(N +1)

the 4p-3h and, perhaps, higher-than-4p-3h components of the Rµ

operator in the EA-

EOMCC calculations, neglected at the EA-EOMCCSD(3p-2h) level. The highly MR character of the B 2 Σ− state becomes clear when we examine the CASPT2 and CASSCF-based
MRCI calculations reported in Ref. [299]. These calculations are in reasonable agreement
with the results of our full and active-space EA-EOMCCSD(3p-2h) calculations for the A2 ∆
and C 2 Σ+ states, producing a 0.238 eV error for the A2 ∆ state and 0.219 eV error for the
C 2 Σ+ state when the CASPT2 approach is employed, but the CASPT2 and MRCI results
obtained in Ref. [299] for the B 2 Σ− state are considerably more accurate than those obtained here with the EA-EOMCCSD(3p-2h) theory levels. Indeed, the CASPT2 and MRCI
calculations for the B 2 Σ− state reported in Ref. [299] give errors of 0.225 eV and 0.250
eV, respectively, relative to experiment, as opposed to ∼ 0.9 eV obtained with the full EAEOMCCSD(3p-2h) and active-space EA-EOMCCSD(3p-2h){4} methods. Interestingly, the
MRCI approach improves the CASPT2 results for the A2 ∆ and C 2 Σ+ states as well, reduc110

ing the 0.238 eV and 0.219 eV errors obtained in the CASPT2 calculations to 0.060 eV and
0.058 eV, respectively [299], which suggests that the incorporation of the 4p-3h and, perhaps, some other higher-order excitations in the EA-EOMCC calculations may be necessary
to further improve the description of all three excited states of C2 N examined in this work.
Since the calculations reported in the present paper exclude the possibility that the basis
set or geometry optimizations may help the EA-EOMCCSD(3p-2h) results, the next logical
step is to examine the role of 4p-3h excitations in the EA-EOMCC calculations. One may
also have to examine whether the use of the full CCSDT approach rather than the CCSD
method in providing the ground-state wave function for the reference C2 N+ ion plays a role
here. These will be the topics of our group’s future work.

As shown in Table (3.17), the adiabatic excitation energies resulting from the EAEOMCCSD(2p-1h) and EA-EOMCCSD(3p-2h){4} calculations with the cc-pVQZ basis set
are reasonably well converged with the basis and, although the total energies of the individual electronic states of C2 N are not converged when the cc-pVQZ basis set is employed, they
behave in a systematic manner as we go from the cc-pVDZ to cc-pVQZ basis sets, facilitating the CBS extrapolations. Indeed, when going from the cc-pVTZ to the cc-pVQZ basis
sets, the changes in the EA-EOMCCSD(3p-2h){4} excitation energies are very small, at most
0.018 eV. The analogous changes in the EA-EOMCCSD(2p-1h) excitation energies are somewhat larger (at most 0.120 eV), but can still be viewed as reasonably stable considering the
complicated nature of the C2 N excited states that the EA-EOMCCSD(2p-1h) approach has
significant problems with. Overall, the simplifying assumption of the CBS-A extrapolation
scheme that one can treat the electron-attachment energies resulting from the EA-EOMCC
calculations with the cc-pVQZ basis set as essentially converged values remains valid for
111

C2 N, so we expect the CBS-A scheme to provide meaningful results. This can be verified
by comparing the CBS-A and CBS-B extrapolations. Comparing the EA-EOMCCSD(2p1h)
CBS-A and CBS-B values, the total energies differ by 1.9 millihartree for the X 2 Π state, 0.1
millihartree for the A2 ∆ state, 0.5 millihartree for the B 2 Σ− state, and 0.2 millihartree for
the C 2 Σ+ state. The adiabatic excitation energies resulting from the CBS-A and CBS-B
extrapolations of the EA-EOMCCSD(2p1h) data differ by 0.054 eV for the A2 ∆ state, 0.066
eV for the B 2 Σ− state, and 0.047 eV for the C 2 Σ+ state. The CBS-A and CBS-B results for
the EA-EOMCCSD(3p-2h){4} total energies differ by 6.0 millihartree for the X 2 Π state, 7.5
millihartree for the A2 ∆ state, 8.2 millihartree for the B 2 Σ− state, and 7.6 millihartree for
the C 2 Σ+ state. The differences in the adiabatic excitation energies obtained with the two
CBS extrapolation schemes, as applied to the EA-EOMCCSD(3p-2h){4} data, are 0.042 eV,
0.062 eV, and 0.045 eV for the A2 ∆, B 2 Σ− and C 2 Σ+ states, respectively. We can conclude
that our CBS EA-EOMCC results for the C2 N molecule are generally stable to within about
8 millihartree for the total energies and 0.060 eV for the adiabatic excitation energies. The
deviations of the CBS-A extrapolated EA-EOMCCSD(3p-2h){4} results from experiment
are 0.371 eV for the A2 ∆ state, 0.893 eV for the B 2 Σ− state, and 0.491 eV for the C 2 Σ+
state. The analogous CBS-B calculations employing the EA-EOMCCSD(3p-2h){4} data
give errors of 0.329 eV, 0.831 eV, and 0.446 eV, respectively. These results indicate once
again that higher-than-3p-2h excitations and, perhaps, methods better than CCSD for the
description of the ground state of the reference C2 N+ ion may have to be included in the
EA-EOMCC calculations for the low-lying states of the C2 N molecule, particularly in the
case of the B 2 Σ− state.

Having demonstrated the significance of higher than 2p-1h contributions for an accurate
112

description of the excitation energies in the CNC and C2 N molecules and having established
the ability of the active-space EA-EOMCCSD(3p-2h) approach to capture the most significant 3p-2h contributions with only a few active orbitals, independently of the basis set,
we turn now to the effectiveness of the EA-EOMCC schemes in describing the equilibrium
geometries of the ground and excited states of CNC and C2 N. As seen in Table (3.18), the
EA-EOMCCSD(2p-1h) level of theory gives the C–N bond lengths in CNC designated as
RC-N , which deviate from the corresponding experimental values by 0.003-0.015 ˚ for the
A
X 2 Πg state, 0.004-0.009 ˚ for the A2 ∆u state, and 0.001-0.016 ˚ for the B 2 Σ+ state, when
A
A
u
the cc-pVXZ basis sets with X = D, T, and Q are employed. The full EA-EOMCCSD(3p-2h)
approach employing the cc-pVDZ and cc-pVTZ basis sets produces RC-N values that deviate from experiment by 0.001-0.017 ˚, 0.003-0.012 ˚, and 0.004-0.011 ˚ for the X 2 Πg ,
A
A
A
A2 ∆u , and B 2 Σ+ states, respectively, i.e., results that are of equally high quality and not
u
much different than the low-order EA-EOMCCSD(2p-1h) data. The analogous active-space
EA-EOMCCSD(3p-2h){4} calculations give RC-N bond lengths that differ from experiment
by 0.001-0.017 ˚ in the case of the X 2 Πg state, 0.003-0.012 ˚ in the case of the A2 ∆u
A
A
state, and 0.005-0.014 ˚ when the B 2 Σ+ state is examined. All of this shows that not
A
u
only is the EA-EOMCCSD(3p-2h){4} approach able to reproduce the more computationally
demanding EA-EOMCCSD(3p-2h) results for the nuclear geometries of the low-lying states
of CNC, but that the high-level EA-EOMCCSD(3p-2h) values of RC-N and those obtained
with the the inexpensive EA-EOMCCSD(2p-1h) method differ only by 0.002-0.004 ˚ for the
A
X 2 Πg state and 0.001-0.003 ˚ for the A2 ∆u and B 2 Σ+ states, at least when the cc-pVDZ
A
u
and cc-pVTZ basis sets are employed. The active-space EA-EOMCCSD(3p-2h){4} approach
and the EA-EOMCCSD(2p-1h) method give RC-N values that differ by at most 0.004 ˚ for
A

113

all states of CNC and all basis sets examined in this work, confirming the observation that
it is sufficient to use the low-level EA-EOMCCSD(2p-1h) approach to obtain an accurate
description of the equilibrium geometries of the low-lying states of CNC.

Similar, but not entirely identical, remarks apply to the C2 N molecule. As shown in
Table (3.18), the EA-EOMCCSD(2p-1h) and EA-EOMCCSD(3p-2h){4} approaches employing the cc-pVDZ, cc-pVTZ, and cc-pVQZ basis sets give C–C and C–N bond lengths,
RC-C and RC-N , respectively, that differ by at most 0.065 ˚ when we compare the EAA
EOMCCSD(2p-1h) and the corresponding EA-EOMCCSD(3p-2h){4} data for all electronic
states of C2 N examined in this work, mostly because of the inability of the EA-EOMCCSD(2p-1h) approach to provide a highly accurate description of the excited-state geometries [the differences between the EA-EOMCCSD(2p-1h) and EA-EOMCCSD(3p-2h){4} geometries of the C2 N’s ground state are less than 0.004 ˚]. On the other hand, the differences
A
between the active-space EA-EOM-CCSD(3p-2h){4} and full EA-EOMCCSD(3p-2h) values
of RC-C and RC-N obtained with the cc-pVDZ and cc-pVTZ basis sets do not exceed 0.001 ˚,
A
confirming our earlier remarks about the ability of the active-space EA-EOMCCSD(3p-2h)
approach to capture essentially all correlation effects that are included in the full EA-EOMCCSD(3p-2h) calculations. We could not find any experimental data for the geometries of the
ground and excited states of C2 N, so we cannot comment on the accuracy of our RC-C and
RC-N values resulting from the EA-EOMCC calculations in any definitive manner, but, judging by the high quality of the EA-EOMCC results for the geometries of the low-lying states
of CNC, we can conclude that the geometries resulting from the full EA-EOMCCSD(3p-2h)
and active-space EA-EOMCCSD(3p-2h){4} calculations using the cc-pVTZ or cc-pVQZ basis sets are of the similarly high quality. The low-level EA-EOMCCSD(2p-1h) calculations
114

seem less accurate than in the CNC case, particularly when the excited states of C2 N are
examined, but they are still in reasonable agreement with the high-level full and active-space
EA-EOMCCSD(3p-2h) results.
The above discussion provides us with an important insight about the performance of the
EA-EOMCC methods. The EA-EOMCCSD(2p-1h) approach, while generally inadequate for
an accurate description of the excitation energies in open-shell systems, such as the CNC and
C2 N molecules examined in this work, is capable of providing reasonably accurate equilibrium
geometries, even for excited states that have a significant MR character. On the other hand,
it seems to be generally safer to use the active-space EA-EOMCCSD(3p-2h) approach in
geometry optimizations, particularly since it provides results that are virtually identical to
the corresponding full EA-EOMCCSD(3p-2h) data, both for the excitation energies and
nuclear geometries.
We now turn to the IP-EOMCC calculations for the NCO and N3 molecules, which
are summarized in Tables (3.19)–(3.21). For both molecules, the IP-EOMCC optimizations
employing the DZP basis set, carried out in the present work, produced results that are
very similar to the previously reported [132, 289] IP-EOMCC adiabatic excitation energies
calculated at the SAC-CI-SDT-R/PS optimized geometries, all obtained with the same DZP
basis as that used here. For example, the deviations between the adiabatic excitation energies
of NCO and N3 obtained in the present IP-EOMCC/DZP optimizations and the analogous
excitation energies reported in Refs. [132, 289], which used the geometries obtained with the
SAC-CI-SDT-R/PS approach, are 0.001-0.078 eV for all states and all methods considered
in this study. As in the case of the EA-EOMCC calculations, after seeing that our present
IP-EOMCC optimizations for the ground and excited states of NCO and N3 employing the
115

DZP basis set were able to reproduce the analogous results calculated at the geometries
obtained in the SAC-CI-SDT-R/PS calculations, reported in Refs. [132, 289], we moved to
the examination of the effect of the use of the correlation-consistent basis sets of the cc-pVXZ
quality on the calculated excitation energies and geometries.
Unlike the CNC and C2 N molecules, which are characterized by the presence of lowlying excited states with a significant MR character in their respective electronic spectra,
the low-lying states of NCO and N3 have a predominantly 1h excitation character relative to
the corresponding NCO− and N− reference ions, with only small contributions from higher3
than-1h excitations. As a result, it is much easier to describe the low-lying states of NCO
and N3 by the IP-EOMCC methods and already the basic IP-EOMCCSD(2h-1p) approach
performs quite well. For example, as shown in Table (3.19), the deviations from experiment
for the adiabatic excitations in NCO resulting from the IP-EOMCCSD(2h-1p) calculations
are only 0.007-0.098 eV for the A2 Σ+ state and 0.336-0.381 eV for the B 2 Π state when
the cc-pVXZ basis sets with X = D, T, and Q are employed. Inclusion of higher-order
(3h-2p) correlation effects through the full IP-EOMCCSD(3h-2p) method offers additional
improvements, reducing the overall deviations from experiment to 0.018-0.072 eV in the
A2 Σ+ case and 0.044-0.085 eV in the B 2 Π case, when the cc-pVDZ and cc-pVTZ basis sets
are employed.
The inexpensive active-space variant of IP-EOMCCSD(3h-2p) using only two active occupied orbitals, IP-EOMCCSD(3h-2p){2}, yields similar excitation energy values to those
from its more expensive parent scheme, with somewhat larger deviations from experiment
of 0.207-0.312 eV for the A2 Σ+ state and very small 0.063-0.139 eV deviations for the B 2 Π
state, confirming that one can essentially use any IP-EOMCC approach and obtain a reason116

able description of the low-lying states of NCO, but the deviations between the results of the
full and active-space IP-EOMCCSD(3h-2p) calculations for NCO are somewhat larger than
those observed in the EA-EOMCCSD(3p-2h) computations for CNC and C2 N. This is particularly true for the A2 Σ+ state, where they are 0.240 eV for the adiabatic excitation energy
and 11.0 millihartree for the total energy when the cc-pVDZ basis set is employed and 0.218
eV and 10.7 millihartree when the cc-pVTZ basis set is used. As pointed out in previous
work [289], these larger differences between the full and active-space IP-EOMCCSD(3h-2p)
results for the A2 Σ+ state of NCO are likely due to the small active-space used in the latter
calculations, which consists of only one pair of highest-energy occupied π orbitals of NCO− ,
and/or from changes in the character of molecular orbitals when going from the NCO− reference ion to the NCO target species. On the other hand, the overall agreement between
the full and active-space IP-EOMCCSD(3h-2p) results for NCO is rather good. For example, the differences between the full and active-space IP-EOMCCSD(3h-2p) results for the
adiabatic excitation energies corresponding to the B 2 Π state are only 0.019 eV when the
cc-pVDZ basis set is employed and 0.023 eV when the cc-pVTZ basis set is used. The differences between the total energies obtained in the full IP-EOMCCSD(3h-2p) and active-space
IP-EOMCCSD(3h-2p){2} calculations for the X 2 Π and B 2 Π states range between 2.1 and
3.5 millihartree when the cc-pVDZ and cc-pVTZ basis set are used, which is an excellent
agreement.

Many of the above observations remain valid when the IP-EOMCC methods are applied
to N3 . As shown in Table (3.20), the adiabatic excitation energies corresponding to the B 2 Σ+
u
state obtained with the IP-EOMCCSD(2h-1p) optimizations employing the cc-pVXZ basis
sets with X = D, T, and Q differ from the corresponding experimental value by 0.056-0.110
117

eV. Again, as in the NCO case, the full IP-EOMCCSD(3h-2p) approach reduces the already
small errors in the IP-EOMCCSD(2h-1p) results for the B 2 Σ+ state of N3 to the even smaller
u
0.023-0.049 eV range. The much less expensive active-space IP-EOMCCSD(3h-2p){2} calculations using the cc-pVXZ basis sets with X = D, T, and Q produce errors of 0.174-0.200
eV, which are larger than those obtained with full IP-EOMCCSD(3h-2p), but the general
agreement between the full and active-space IP-EOMCCSD(3h-2p) data is reasonable. Indeed, the total energies resulting from the full IP-EOMCCSD(3h-2p) and active-space IPEOMCCSD(3h-2p){2} calculations differ by only 1.4-1.7 millihartree in the X 2 Πg case and
6.9-7.3 millihartree in the case of the B 2 Σ+ state. The adiabatic excitation energies coru
responding to the B 2 Σ+ state obtained in the full and active-space IP-EOMCCSD(3h-2p)
u
calculations with the cc-pVDZ and cc-pVTZ basis sets differ by 0.151 eV, which is a reasonable agreement. Again, the somewhat larger differences between the full and active-space
IP-EOMCCSD(3h-2p) data for the B 2 Σ+ state compared to the analogous EA-EOMCC calu
culations for CNC and C2 N are likely due to the reasons cited above for the NCO molecule.

We now turn our attention to the numerical stability of our IP-EOMCC results for the
NCO and N3 molecules in the CBS limit. As in the EA-EOMCC calculations for CNC and
C2 N, the IP-EOMCC total energies of each state of NCO and N3 shown in Tables (3.19)
and (3.20) behave in a systematic manner, as we go from the cc-pVDZ to cc-pVQZ basis
sets, showing the initial signs of convergence; and the excitation energies obtained in the IPEOMCCSD(2h-1p) and IP-EOMCCSD(3h-2p){2} calculations with the cc-pVQZ basis set
can be regarded as reasonably well converged, which helps the validity of the CBS-A extrapolations. Indeed, the differences between the adiabatic excitation energies calculated with the
cc-pVTZ and cc-pVQZ basis sets at the IP-EOMCCSD(2h-1p) and IP-EOMCCSD(3h-2p){2}
118

levels of theory are 0.028-0.029 eV for the A2 Σ+ state of NCO, 0.020-0.031 eV for the B 2 Π
state of NCO, and 0.003 eV for the B 2 Σ+ state of N3 . It is, therefore, not surprising that
u
the CBS-A and CBS-B extrapolations for the ground and excited states of the NCO and N3
molecules summarized in Tables (3.19) and (3.20) are in good agreement. Indeed, the CBS-A
and CBS-B total energies obtained with the IP-EOMCCSD(2h-1p) data for NCO differ by
only 1.5 millihartree for the X 2 Π and A2 Σ+ states and 2.7 millihartree for the B 2 Π state.
The corresponding excitation energies resulting from both CBS extrapolations differ by 0.001
eV for the A2 Σ+ state and 0.034 eV for the B 2 Π state. In consequence, the CBS-A- and CBSB-extrapolated IP-EOMCCSD(2h-1p) excitation energies obtained for NCO differ from the
corresponding experimental values by 0.009-0.010 eV in the A2 Σ+ case and 0.393-0.427 eV in
the case of the B 2 Π state. Similar remarks apply to the IP-EOMCCSD(3h-2p){2} approach,
where the corresponding CBS-A- and CBS-B-extrapolated total energies differ by 2.5, 4.1,
and 3.0 millihartree for the X 2 Π, A2 Σ+ , and B 2 Π states, respectively, so that the differences
in the resulting CBS-A and CBS-B IP-EOMCCSD(3h-2p){2} excitation energies are 0.045
eV for the A2 Σ+ state and 0.014 eV for the B 2 Π state. As a consequence, the CBS-A- and
CBS-B-extrapolated IP-EOMCCSD(3h-2p){2} adiabatic excitation energies for NCO differ
from experiment by 0.189-0.234 eV for the A2 Σ+ state and 0.159-0.173 eV for the B 2 Π state.
Much of the above analysis applies to N3 . Indeed, although the CBS-A and CBS-B extrapolations applied to the IP-EOMCCSD(2h-1p) and IP-EOMCCSD(3h-2p){2} total energies
produce somewhat larger differences than in the case of NCO (7.6-8.4 millihartree in the case
of the X 2 Πg state and 6.6-8.6 millihartree in the case of the B 2 Σ+ state), the adiabatic exciu
tation energies resulting from both CBS extrapolations are very stable, to within 0.003 eV for
the IP-EOMCCSD(2h-1p) approach and 0.002 eV for the IP-EOMCCSD(3h-2p){2} method.

119

The CBS-A- and CBS-B-extrapolated IP-EOMCCSD(2h-1p) and IP-EOMCCSD(3h-2p){2}
adiabatic excitation energies corresponding to the B 2 Σ+ state of N3 are within 0.107-0.194
u
eV from experiment.

To conclude this discussion, we examine the performance of the IP-EOMCC methods
in describing the equilibrium geometries of the ground and low-lying excited states of the
NCO and N3 species. The results of our geometry optimizations for NCO and N3 are
summarized in Table (3.21). In the case of the X 2 Π state of the NCO molecule, the basic
IP-EOMCCSD(2h-1p) approach produces results that deviate from experiment by 0.0160.031 ˚ for the N–C bond length (designated as RN-C ) and 0.018-0.033 ˚ for the C–O
A
A
bond length (designated as RC-O ) when the cc-pVXZ basis sets with X = D, T, and Q
are employed. The same approach applied to the A2 Σ+ state of NCO gives errors of 0.0140.031 ˚ for RN-C and 0.018-0.031 ˚ for RC-O . The IP-EOMCCSD(3h-2p) results exhibit
A
A
very similar trends and accuracies, confirming the small role of higher-order contributions
neglected in IP-EOMCCSD(2h-1p) and present in IP-EOMCCSD(3h-2p). The differences
between the IP-EOMCCSD(3h-2p) and experimental values of RN-C are 0.024-0.038 ˚ for
A
the X 2 Π state and 0.021-0.035 ˚ for the A2 Σ+ state. The analogous differences for RC-O
A
are 0.015-0.025 ˚ for the X 2 Π state and 0.012-0.022 ˚ for the A2 Σ+ state. Although it may
A
A
very well be that higher-than-3h-2p contributions neglected in the IP-EOMCCSD(3h-2p)
calculations and high angular momentum functions that are not present in the cc-pVTZ
(or cc-pVQZ) basis sets are the sources of the above errors, it is also possible that the
experimental geometries of the X 2 Π and A2 Σ+ states of NCO reported in Ref. [295] might
be in some error too, since none of the states of NCO examined here is as challenging as
some of the states of CNC and C2 N. While there are unexplained differences between the
120

experimentally N–C and C–O bond lengths in the X 2 Π and A2 Σ+ states of NCO and our
theoretical predictions, it is of great interest to note that the differences between the results of
the geometry optimizations using the full and active-space IP-EOMCCSD(3h-2p) approaches
are virtually none. Indeed, there is no difference (to within 0.001 ˚) between the full IPA
EOMCCSD(3h-2p) and active-space IP-EOMCCSD(3h-2p){2} results for the N–C bond
length in the X 2 Π state and the corresponding C–O bond lengths differ by 0.002 ˚ only,
A
when the cc-pVDZ and cc-pVTZ are employed. In the case of the A2 Σ+ state, the differences
between the full IP-EOMCCSD(3h-2p) and active-space IP-EOMCCSD(3h-2p){2} values
of RN-C and RC-O are 0.004 ˚ and 0.001-0.002 ˚, respectively. In the case of the B 2 Π
A
A
state, these differences are 0.003 ˚ for RN-C and 0.006-0.008 ˚ for RC-O . The activeA
A
space IP-EOMCCSD(3h-2p){2} calculations are clearly capable of reproducing the parent
IP-EOMCCSD(3h-2p) data for the N–C and C–O bond lengths in the ground and excited
states of NCO to very high accuracy.

Much of the above discussion applies to N3 . The nearest-neighbor N–N bond lengths, designated as RN-N , resulting from the IP-EOMCCSD(2h-1p) calculations with the cc-pVXZ
basis sets with X = D, T, and Q, differ from the corresponding experimental data by
0.003-0.020 ˚ for the X 2 Πg state and 0.001-0.015 ˚ for the B 2 Σ+ state. The higher-order
A
A
u
IP-EOMCCSD(3h-2p) optimizations with the cc-pVDZ and cc-pVTZ basis sets produce similar results, errors of 0.002-0.013 ˚ for the X 2 Πg state and 0.007 ˚ for the B 2 Σ+ state. The
A
A
u
active-space IP-EOMCCSD(3h-2p){2} approach, for which we could also afford the calculations with the cc-pVQZ basis set, produces RN-N values that deviate from experiment
by 0.002-0.016 ˚ for the X 2 Πg state and 0.004-0.012 ˚ for the B 2 Σ+ state. Again, there
A
A
u
is a virtually perfect agreement between the expensive full IP-EOMCCSD(3h-2p) and in121

expensive active-space IP-EOMCCSD(3h-2p){2} calculations, where there is no difference
(to within 0.001 ˚) between the two sets of data in the case of he X 2 Πg state and a very
A
small, 0.002-0.003 ˚, difference between the full IP-EOMCCSD(3h-2p) and active-space IPA
EOMCCSD(3h-2p){2} values of RN-N in the case of the B 2 Σ+ state. As in the case of the
u
NCO molecule, the origin of the deviations between the IP-EOMCC calculations employing
basis sets as large as cc-pVQZ, which seem numerically quite stable, and experimental RN-N
values could lie in the higher-than-3h-2p correlations that we do not consider in this work
or in the significance of the high angular momentum functions absent in the cc-pVTZ and
cc-pVQZ bases, but one cannot exclude the possibility that the experimental data reported
in Ref. [295] may need to be revisited. As in the EA-EOMCC calculations for the CNC and
C2 N, it seems to us that the basic IP-EOMCCSD(2h-1p) method is capable of producing
optimized geometries of the ground- and excited-state NCO and N3 molecules that are comparable to those obtained with the computationally more demanding IP-EOMCCSD(3h-2p)
methods, which is a useful observation from the point of view of other applications of such
methods to geometry optimizations in other open-shell species.

122

Table 3.16: Total and adiabatic excitation energies for the ground and low-lying excited states of CNC, as obtained with the
different EA-EOMCC approaches using the DZP [4s2p1d] and cc-pVXZ (X = D, T, Q) basis sets and extrapolating to the
CBS limit.
Total Energy (hartree)
Method

Basis

EA-EOMCCSD(2p-1h)

DZP
x=D
x=T
x=Q
CBS-A
CBS-B
DZP
x=D
x=T
DZP
x=D
x=T
x=Q
CBS-A
CBS-B

EA-EOMCCSD(3p-2h)

EA-EOMCCSD(3p-2h){4}

Adiabatic Excitation Energy (eV)

X 2 Πg

A 2 ∆u

B 2 Σ+
u

A 2 ∆u -X 2 Πg

B 2 Σ+ -X 2 Πg
u

-130.406718
-130.402813
-130.502669
-130.534268
-130.551020
-130.552172
-130.411686
-130.408191
-130.510334
-130.409784
-130.406511
-130.507154
-130.538435
-130.554997
-130.556095

-130.141822
-130.136443
-130.220645
-130.248264
-130.264878
-130.264020
-130.260720
-130.257611
-130.358548
-130.259560
-130.256819
-130.356797
-130.388104
-130.404664
-130.405806

-130.125873
-130.120048
-130.204320
-130.232033
-130.248586
-130.247849
-130.238177
-130.234329
-130.335201
-130.236779
-130.233332
-130.333074
-130.364472
-130.380953
-130.382242

7.208
7.248
7.674
7.783
7.786
7.841
4.108
4.097
4.130
4.088
4.073
4.091
4.091
4.091
4.090
3.761

7.642
7.694
8.118
8.224
8.230
8.281
4.721
4.731
4.766
4.708
4.712
4.737
4.734
4.736
4.731
4.315

Experimenta
a Taken from Refs. [295, 296].

123

Table 3.17: Total and adiabatic excitation energies for the ground and low-lying excited states of C2 N, as obtained with the
different EA-EOMCC approaches using the DZP [4s2p1d] and cc-pVXZ (X = D, T, Q) basis sets and extrapolating to the
CBS limit.

Total Energy (hartree)
Method
EA-EOMCCSD(2p-1h)

Basis

DZP
x=D
x=T
x=Q
CBS-A
CBS-B
EA-EOMCCSD(3p-2h)
DZP
x=D
x=T
EA-EOMCCSD(3p-2h){4} DZP
x=D
x=T
x=Q
CBS-A
CBS-B
Experimenta

Adiabatic Excitation
Energy (eV)

X 2Π

A 2∆

B 2 Σ−

C 2 Σ+

A 2 ∆X 2Π

B 2 Σ− X 2Π

C 2 Σ+ X 2Π

-130.400501
-130.400086
-130.499176
-130.530280
-130.546521
-130.548409
-130.405260
-130.404842
-130.506456
-130.403651
-130.403260
-130.503555
-130.533052
-130.543559
-130.549517

-130.176452
-130.174345
-130.259914
-130.287558
-130.303831
-130.303738
-130.292989
-130.292610
-130.394642
-130.292385
-130.292089
-130.393547
-130.423692
-130.433071
-130.440555

-130.117500
-130.115824
-130.198940
-130.225643
-130.241794
-130.241271
-130.270231
-130.270337
-130.370688
-130.269696
-130.269870
-130.369756
-130.399922
-130.408605
-130.416854

-130.156651
-130.152828
-130.240134
-130.267832
-130.283903
-130.284090
-130.265181
-130.264086
-130.366299
-130.264361
-130.263371
-130.364858
-130.394864
-130.404035
-130.411633

6.097
6.143
6.511
6.605
6.604
6.658
3.055
3.054
3.043
3.028
3.025
2.993
2.976
3.007
2.965
2.636

7.701
7.735
8.170
8.290
8.292
8.358
3.674
3.660
3.694
3.645
3.630
3.641
3.623
3.672
3.610
2.779

6.635
6.728
7.049
7.142
7.146
7.193
3.812
3.830
3.814
3.791
3.807
3.774
3.760
3.797
3.752
3.306

a Taken from Refs. [295, 297].

124

Table 3.18: Comparison of the optimized equilibrium geometries for the low-lying states of CNC and C2 N, as obtained with
the EA-EOMCC and SAC-CI-SDT-R/PS approaches using the DZP[4s2p1d] and cc-pVXZ (X = D, T, Q) basis sets.

CNCa

C2 Nb

Method

Basis

X 2 Πg

A 2 ∆u

B 2 Σ+
u

SAC-CI-SDT-R/PS
EA-EOMCCSD(2p-1h)

DZP
DZP
x=D
x=T
x=Q
DZP
x=D
x=T
DZP
x=D
x=T
x=Q

1.253
1.259
1.260
1.242
1.239
1.261
1.262
1.246
1.262
1.262
1.246
1.242
1.245

1.256
1.258
1.258
1.245
1.241
1.262
1.261
1.246
1.262
1.261
1.246
1.243
1.249

1.259
1.260
1.260
1.247
1.243
1.264
1.263
1.248
1.264
1.264
1.249
1.245
1.259

EA-EOMCCSD(3p-2h)

EA-EOMCCSD(3p-2h){4}

Experimentc

X 2Π
(1.400,
(1.412,
(1.412,
(1.389,
(1.385,
(1.410,
(1.409,
(1.388,
(1.411,
(1.408,
(1.389,
(1.387,

1.185)
1.196)
1.193)
1.178)
1.174)
1.198)
1.195)
1.180)
1.197)
1.195)
1.180)
1.175)

A 2∆
(1.315,
(1.372,
(1.376,
(1.363,
(1.362,
(1.329,
(1.332,
(1.316,
(1.329,
(1.332,
(1.316,
(1.315,

1.207)
1.186)
1.182)
1.166)
1.162)
1.217)
1.212)
1.196)
1.217)
1.212)
1.196)
1.190)

B 2 Σ−
(1.302,
(1.372,
(1.376,
(1.361,
(1.360,
(1.308,
(1.313,
(1.297,
(1.308,
(1.313,
(1.297,
(1.295,

1.223)
1.190)
1.186)
1.170)
1.166)
1.241)
1.234)
1.215)
1.241)
1.234)
1.216)
1.210)

C 2 Σ+
(1.311,
(1.365,
(1.375,
(1.356,
(1.356,
(1.322,
(1.325,
(1.308,
(1.322,
(1.325,
(1.308,
(1.307,

1.214)
1.192)
1.188)
1.171)
1.167)
1.224)
1.220)
1.203)
1.224)
1.220)
1.203)
1.197)

a The R
b
A
C-N bond lengths in ˚. The D2h symmetry was employed. The numbers in parentheses report the RC-C and RC-N
A
bond lengths, respectively, in ˚. The C2v symmetry was employed. c Taken from [295–297].

125

Table 3.19: Total and adiabatic excitation energies for the ground and low-lying excited states of NCO, as obtained with the
different IP EOMCC approaches using the DZP [4s2p1d] and cc-pVXZ (X = D, T, Q) basis sets and extrapolating to the CBS
limit.
Total Energy (hartree)
Method

Basis

IP EOMCCSD(2h-1p)

DZP
x=D
x=T
x=Q
CBS-A
CBS-B
DZP
x=D
x=T
DZP
x=D
x=T
x=Q
CBS-A
CBS-B

IP EOMCCSD(3h-2p)

IP EOMCCSD(3h-2p){2}

Adiabatic Excitation Energy (eV)

X 2Π

A 2 Σ+

B 2Π

A 2 Σ+ -X 2 Π

B 2 Π-X 2 Π

-167.581951
-167.576116
-167.718401
-167.763112
-167.786913
-167.788412
-167.591701
-167.587630
-167.732789
-167.589579
-167.585489
-167.730109
-167.775958
-167.799482
-167.801944

-167.475380
-167.468912
-167.613444
-167.659168
-167.683604
-167.685072
-167.486508
-167.481331
-167.628442
-167.476255
-167.470340
-167.617771
-167.664698
-167.687223
-167.691316

-167.427707
-167.419125
-167.560443
-167.604432
-167.626529
-167.629275
-167.448441
-167.441319
-167.584981
-167.446490
-167.438481
-167.581463
-167.626155
-167.648439
-167.651415

2.900
2.919
2.856
2.828
2.811
2.812
2.862
2.893
2.839
3.081
3.133
3.057
3.028
3.055
3.010
2.821

4.197
4.273
4.298
4.318
4.364
4.330
3.898
3.981
4.022
3.891
4.000
4.045
4.076
4.110
4.096
3.937

Experimenta
a Taken from Ref. [295].

126

Table 3.20: Total and adiabatic excitation energies for the ground and low-lying excited states of N3 , as obtained with the
different IP EOMCC approaches using the DZP [4s2p1d] and cc-pVXZ (X = D, T, Q) basis sets and extrapolating to the CBS
limit.
Total Energy (hartree)
Method

Basis

IP EOMCCSD(2h-1p)

DZP
x=D
x=T
x=Q
CBS-A
CBS-B
DZP
x=D
x=T
DZP
x=D
x=T
x=Q
CBS-A
CBS-B

IP EOMCCSD(3h-2p)

IP EOMCCSD(3h-2p){2}

Adiabatic Excitation Energy (eV)

X 2 Πg

B 2 Σ+
u

B 2 Σ+ -X 2 Πg
u

-163.716374
-163.712083
-163.848768
-163.891293
-163.923747
-163.915306
-163.729782
-163.726673
-163.865416
-163.728362
-163.725315
-163.863766
-163.907325
-163.939566
-163.931977

-163.545829
-163.542627
-163.677460
-163.719861
-163.752426
-163.743856
-163.560803
-163.558437
-163.696218
-163.554434
-163.551533
-163.689041
-163.732710
-163.764109
-163.757469

4.641
4.611
4.662
4.665
4.662
4.665
4.598
4.578
4.604
4.733
4.729
4.755
4.752
4.747
4.749
4.555

Experimenta
a Taken from Ref. [295].

127

Table 3.21: Comparison of the optimized equilibrium geometries for the low-lying states of N3 and NCO, as obtained with the
IP EOMCC and SAC-CI-SDT-R/PS approaches using the DZP [4s2p1d] and cc-pVXZ (X = D, T, Q) basis sets.
N3 a

NCOb

Method

Basis

X 2 Πg

B 2 Σ+
u

SAC-CI-SDT-R/PS
IP EOMCCSD(2p-1h)

DZP
DZP
x=D
x=T
x=Q
DZP
x=D
x=T
DZP
x=D
x=T
x=Q

1.188
1.195
1.185
1.171
1.168
1.200
1.190
1.175
1.200
1.190
1.175
1.172
1.188

1.185
1.191
1.181
1.169
1.165
1.196
1.187
1.173
1.194
1.184
1.171
1.168
1.180

IP EOMCCSD(3p-2h)

IP EOMCCSD(3p-2h){2}

Experimentc

X 2Π
(1.230,
(1.232,
(1.231,
(1.219,
(1.216,
(1.240,
(1.238,
(1.224,
(1.240,
(1.238,
(1.224,
(1.222,
(1.200,

A 2 Σ+

B 2Π

1.193) (1.191, 1.190) (1.220, 1.309)
1.196) (1.197, 1.192) (1.225, 1.318)
1.188) (1.196, 1.184) (1.223, 1.313)
1.177) (1.182, 1.175) (1.206, 1.306)
1.173) (1.179, 1.171) (1.202, 1.304)
1.198) (1.200, 1.198) (1.235, 1.328)
1.191) (1.200, 1.190) (1.233, 1.322)
1.181) (1.186, 1.180) (1.216, 1.312)
1.196) (1.196, 1.196) (1.239, 1.319)
1.189) (1.196, 1.189) (1.236, 1.314)
1.179) (1.182, 1.178) (1.219, 1.306)
1.174) (1.179, 1.174) (1.214, 1.303)
1.206) (1.165, 1.202)

a The R
b
A
N-N bond lengths in ˚. The D2h symmetry was employed. The numbers in parentheses report the RN-C and RC-O
A
bond lengths, respectively, in ˚. The C2v symmetry was employed. c Taken from Ref. [295].

128

Chapter 4
Potential Energy Surface
Extrapolation Schemes

4.1

Motivation

The primary goal of this dissertation so far has been the examination of ab initio electronic
structure methods which allow for the efficient generation of highly accurate molecular PESs
or chemical reaction pathways. Unfortunately, under the constraints of current computing
capabilities and algorithms, the range of applicability of the electronic structure methods
discussed in Sect. (3) is limited to small- to medium-sized systems if local correlation, fragmentation, or other similar techniques are not exploited, and those have additional intrinsic
errors. This is not only because of characteristic steep scalings of computer costs of typical
high-accuracy ab initio methods with the system size, but also because of the enormous numbers of points typically associated with PESs of larger molecules. Under the conventional
procedure, one usually follows to obtain a PES, the calculation of mτ points is required,
129

where m is the number of nuclear geometries required to represent a one-dimensional PES
cut (typically, m is on the order of 10) and τ is the PES dimension (for the one-dimensional
PES cuts along the relevant intrinsic reaction coordinates, τ = 1; for the global multidimensional PESs, τ = 3M − 6, where M is the number of atomic nuclei in the molecular system
of interest). Meanwhile, the CPU time associated with the accurate ab initio electronic
structure calculation of a single point of the PES scales at least as kn4 , where the prefactor
u
k is a polynomial function of the number of occupied orbitals no . For most medium to large
systems this can cause even a single point energy calculation to become prohibitively expensive when using a basis set of realistic size. The total CPU time required to generate a PES
for a given molecular system scales as kmτ n4 , a scaling so poor that studies involving larger
u
polyatomic systems must be limited to small basis sets and small numbers of points on the
PES for the calculations to remain computationally feasible. The focus of this chapter is to
develop and test numerical techniques which can help reduce the enormous computer costs
associated with the conventional procedure for generating PESs.

Considerable progress has been made toward alleviating these large computer costs with
the proposal of an ab initio extrapolation scheme, described in Ref. [184], that predicts a PES
corresponding to a larger basis set from the results of smaller basis set calculations by scaling electron correlation energies. In the PES extrapolation scheme suggested in Ref. [184], a
universal correlation energy scaling factor is determined at a single nuclear geometry, called
a pivot geometry, over a series of basis sets of growing size. The scaling factor is then applied
to electron correlation energies calculated using smaller basis sets at the remaining geometries to obtain the entire PES at the desired (larger) basis set level. The original work [184],
as well as more recent effort by us [193, 300], and Varandas [301, 302] have demonstrated
130

the effectiveness of this procedure, where in each studied case one could generate the target
PESs to within, on average, fractions of a millihartree of the true calculated energies, while
effectively reducing the number of points which must be calculated on the high-level PES
of interest from mτ to one or a few. The general principles behind the PES extrapolation
methodology of Refs. [184, 193, 300] are outlined in Sect. (4.2), while Sect. (4.3) elaborates
on the the so-called single-level PES extrapolation schemes based on the ideas laid down
in Refs [184, 193, 300]. In Sect. (4.3.1) the details of the single-level PES extrapolation
scheme are described, while Sect. (4.3.2) deals with an extension allowing one to perform
PES extrapolations to the CBS-limit in an inexpensive way. Section (4.3.4) surveys a few
of the user-defined parameters, which must be chosen using the PES extrapolation schemes
examined in this work, and Sect. (4.3.3) demonstrates the performance of a few different
single-level PES extrapolation schemes in the first-ever practical application involving a complex polyatomic system reported in Refs. [193,300], namely the bicbut→t-but isomerization
examined in Sect. (3.2.3).

Despite the significant improvements in computational expense offered by the single-level
PES extrapolation schemes of Refs. [184, 193, 300], performing even a single high-level ab
initio calculation with a larger basis set may sometimes be too taxing. In all such cases, the
PES extrapolation scheme of Refs. [184, 193, 300] that requires one large-basis set high-level
ab initio calculation cannot be of much help. Rather then be forced to resort to using less
accurate methods to describe the associated PESs, an additional flexibility can be utilized
within the framework of the PES extrapolation scheme of Refs. [184, 193, 300] which enables
one to predict the correlation energy scaling factor for calculations using a higher-order
methodology from scaling factors calculated with lower-order methods. Using this so-called
131

dual-level PES extrapolation scheme, introduced for the first time in Sect. (4.4), one can
obtain a surface at the quality of a very accurate method and large basis set without having to
calculate even a single point at the target ab initio level of interest. This new approach allows
results to be obtained much more affordably for larger systems, as any explicit calculation
at the desired level of theory is completely circumvented. To demonstrate the potential
cost savings this implies, relative computational costs associated with performing typical
calculations at the HF and selected MBPT and CC levels are collected in Table (4.1). The
utility of replacing a CR-CC(2,3) calculation by, say, a CCSD or MP4SDQ calculation is
immediately apparent after examination of Table (4.1), as the computational effort can be
reduced by a factor of 7 or 40, respectively. If the dual-level PES extrapolation scheme could
be used to produce a PES which is virtually identical to the results of explicit large basis set
CR-CC(2,3) calculations using only the calculations performed with lower-level methods,
such as CCSD or MP4SDQ and small basis set CR-CC(2,3) computations, it would offer
incredible savings in the required computational effort.

To test the accuracy of the dual-level PES extrapolation scheme proposed in this thesis,
where a number of different lower-order ab initio methodologies combined with small basis
set CR-CC(2,3) calculations are used to approximate large basis set CR-CC(2,3) results,
several chemical systems were chosen for benchmark studies. The systems considered here
include the asymmetric stretch of the H2 O molecule, the bond stretching of the F2 and HCl
molecules, and, once again, the bicbut→t-but isomerization of Sect. (3.2.3). The three
single-bond breaking potential energy curves have been well studied and serve our purposes
particularly well because they have regions clearly dominated by dynamical correlation effects
near the equilibrium bond lengths and regions with significant nondynamical correlation
132

Table 4.1: Computer costs of typical ab initio wave function calculations at the aug-cc-pVTZ
basis set level, taken from Ref. [204]
Scalings of CPU stepsa
Method
CR-CC(2,3)
CCSD(T)
CCSD
MP4SDQ
MP3
MP2
HF

Iterative

Noniterative

CPU timeb

N6
N6
N6
—
—
—
N4

N7
N7
—
N6
N6
N5
—

574c
287
86
15
12
3
1

a N is a measure of the system size. b The CPU time for each method is reported as the time

required by an energy gradient calculation for phosphinomethanol divided by the computer
time characterizing the corresponding HF/aug-cc-pVTZ energy gradient calculation with
the same software on the same computer. Although such costs depend to some extent (for
example, 15%) on the machine, the program, and the computer load, they still provide a
useful indication of computer resource demand. c The cost of the CR-CC(2,3) method was
not measured explicitly, but rather approximated by doubling the CPU time of the CCSD(T)
calculation (the most expensive steps of CR-CC(2,3) are approximately twice as expensive
as those of CCSD(T)).

133

effects as relevant bond lengths approach the fully dissociated limits. The bicbut→t-but
isomerization is also reexamined, as it provides a good example of a reaction profile for a
polyatomic molecule which is composed of stationary points with strongly varying biradical
character. The general theory associated with the dual-level PES extrapolation scheme is
presented in Sect. (4.4.1) and results of the applications for the various di- and tri-atomic
systems and the isomerization pathways of bicyclo[1.1.0]butane to trans-butadiene are given
in Sects. (4.4.2) and (4.4.3), respectively. Finally, in Sect. (4.4.4), comparisons are made
regarding relative cost and accuracy under a number of different combinations of user-defined
choices required by the PES extrapolation schemes considered in this work.

4.2

Theory

The PES extrapolation scheme proposed in Ref. [184] and further developed in Refs. [193,
300–302] focuses on extrapolating the difference ∆E (A) between the total electronic energy,
E (A) obtained with some correlated approach A, and the base energy, E (base) , that one
should be able to calculate with any basis set. For most applications discussed in this
dissertation the base energy E (base) is set equivalent to the RHF reference energy E (RHF) ,
so that the extrapolated energy component ∆E (A) is the total correlation energy, but one
can envision other ways of decomposing the total energy E (A) into E (base) and ∆E (A) . For
example, E (base) could be the CASSCF energy and E (A) the MRCI energy, in which case we
describe the non-dynamical correlation effects exactly within the CASSCF approximation
and extrapolate the difference ∆E (A) = E (A) − E (base) describing dynamical correlations.
In this chapter, we focus on two specific choices of E (base) and ∆E (A) both related to the
choice of target method A as CR-CC(2,3), namely, (i) E (base) = E (RHF) and ∆E (A) =
134

E (CR-CC(2,3)) − E (RHF) ≡ ∆E (CR-CC(2,3)) , so that E (base) is the RHF energy, which is easy
to calculate, and ∆E (A) is the CR-CC(2,3) correlation energy, which is the expensive part we
want to extrapolate, and (ii) E (base) = E (CCSD) and ∆E (A) = E (CR-CC(2,3)) − E (CCSD) ≡
δ(2, 3), so that E (base) is the CCSD energy, which we can often calculate even when large
basis sets are employed, and ∆E (A) is the triples correction of CR-CC(2,3), which is the most
expensive component for the CR-CC(2,3) energy that we want to extrapolate. In addition to
the target method A, we introduce the auxiliary correlated approach B, which in single-level
PES extrapolation techniques of Refs. [184,193,300–302] equals A, and in dual-level schemes
is some other correlated method, less expensive than A. We use the auxiliary approach B
to determine the approximate correlation energy scaling factor, allowing us to rescale the
desired ∆E (A) energy part from smaller to larger basis sets, as described below.

Suppose a set of correlated PES calculations are performed with correlated methods A
and B, as described below, using smaller basis sets indexed by formal numbers m − 1 and
m. Let us designate the resulting PESs obtained for basis sets m − 1 and m using method
(A)

(A)

(B)

(B)

A as Em−1 (R) and Em (R), respectively, and using method B as Em−1 (R) and Em (R),
respectively. Using the PES extrapolation scheme that interests us here, the extrapolated
(A)

PES for the Em+1 (R) target (m + 1)-th basis set is obtained as
(A)

(base)

(B)

(A)

Em+1 (R) = Em+1 (R) + χm+1,m (R) ∆Em (R).

(4.1)

(base)

Here, Em+1 (R) is the energy of the base method calculated with the (m + 1)-th basis set,
(A)

(A)

(base)

∆Em (R) = Em (R)−Em

(R), R denotes the τ -dimensional vector of the nuclear space
(B)

coordinates defining the PES, and the scaling factor χm+1,m (R) is defined as [184, 193, 300–
135

302]


(B)

χm+1,m (R) = 1 + 

(B)

S(R)m,m−1 − 1
(B)
S(Re )m,m−1 − 1



 [S(Re )(B)

m+1,m − 1],

(4.2)

with
(B)

(B)

(B)

S(R)m,m−1 = ∆Em (R)/∆Em−1 (R).

(4.3)

(A)

Thus, the desired high-level PES Em+1 (R), obtained with the largest basis set m + 1, is
(B)

(B)

extrapolated from the PESs Em−1 (R) and Em (R) obtained in smaller basis set calculations,
(base)

(base)

the base energies Em−1 (R), Em

(base)

(R), and Em+1 (R), and a single correlated energy,

(B)

Em+1 (Re ), calculated at the pivot geometry Re .

It is easy to see that Eq. (4.1) represents the simplest mathematical expression one
(A)

can propose to extrapolate the energies Em+1 (R) from the smaller basis sets (m − 1) and
(B)

m to the larger basis set (m + 1). Indeed, the scaling function χm+1,m (R) satisfies the
(B)

following desirable properties: (i) χm+1,m (R) → 1 for all values of R when m → ∞, and
(B)

(B)

(B)

(A)

(base)

(ii) χm+1,m (Re ) =Em+1 (Re )/Em (Re ), so that when B = A, Em+1 (Re )=Em+1 (Re ) +
(A)

(B)

∆Em+1 (Re ), as one would like to have. Of course, we hope that χm+1,m (R) determined
using the information obtained with smaller basis sets (m − 1) and m, and the “correlation”
(B)

energy ∆Em+1 (Re ) obtained with target basis set (m + 1) at a reference (pivot) geometry
(A)

Re only is universal enough to extrapolate the ∆Em+1 (R) values at the remaining points
R on the PES, even when method B is an approximation to the target approach A.
136

4.3

Single-Level Potential Energy Surface Extrapolation Schemes

In the original proposal of the PES extrapolation scheme of Ref. [184], it was assumed that
(B)

(A)

the scaling factor χm+1,m (R) and the correlation energy it is applied to, ∆Em (R), were
required to be generated using the same electronic structure method. While this constraint,
i.e., that A = B in Eqs. (4.1)–(4.3), will be a helpful simplifying assumption for the initial
discussion of the PES extrapolation scheme considered here, it will be eventually shown
in Sect. (4.4), that the theory can work equally well with A = B, where B is only an
approximation to A, resulting in the so-called dual-level PES extrapolation scheme. In
this section, we begin a discussion of several ways to employ the basic equations defining
the PES extrapolation scheme, Eqs. (4.1)–(4.3), with the single-level PES extrapolation to
larger basis sets and to the CBS-limit in Sects. (4.3.1) and (4.3.2), repectively. Following
this, a discussion of the role of pivot geometries, Re , and base wave functions, Ψ(base) ,
which are behind base energies E (base) , is given in Sect. (4.3.3), and an application to the
bicyclobutane isomerization pathways of Sect. (3.2.3) is shown in Sect. (4.3.4).

4.3.1

Potential Energy Surface Extrapolation to Larger Basis Sets

Beginning with a simple example to demonstrate how the PES extrapolation scheme is
typically used, consider a case where A = B and E (base) = E (RHF) , in which a PES corresponding to the cc-pVQZ basis set (previously m + 1) is to be extrapolated from PESs
obtained from cc-pVDZ (m − 1) and cc-pVTZ (m) basis set calculations. In this example,
the following quantities must be collected to perform a single-level PES extrapolation: cor137

relation energies for every point calculated at the cc-pVDZ and cc-pVTZ basis set levels
using method A, base energies for every point calculated at the cc-pVDZ, cc-pVTZ, and
cc-pVQZ basis set levels, and a single correlated energy calculated with the cc-pVQZ basis
set at the pivot geometry Re . For simplicity, in all discussions in this dissertation except
for those involving CBS-limit extrapolation, the cc-pVDZ, cc-pVTZ, and cc-pVQZ or the
aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis set series are employed, meaning that
for all finite basis set extrapolations considered here, the index m used in Eqs. (4.1)–(4.3)
may be equated with the cardinal number 3 of the cc-pVXZ basis sets, corresponding to
cc-pVTZ or aug-cc-pVTZ, and thus m−1 and m+1 correspond to cc-pVDZ or aug-cc-pVDZ
and cc-pVQZ or aug-cc-pVQZ, respectively. Although PES extrapolations in this dissertation focus on extrapolating CR-CC(2,3) correlation energies across the (aug-)cc-pVXZ basis
sets of Dunning and co-workers, which are very systematic in terms of the dependence of
the angular momentum functions on the cardinal number X, it must be emphasized that
the PES extrapolation method is not tailored to fit any single class of electronic structure
approaches nor is it limited to any specific family of basis sets. This was demonstrated in
the original study, Ref. [184], using the MRCI approach and a series of Pople-type basis sets.

4.3.2

Potential Energy Surface Extrapolation to the Complete Basis Set Limit

The theory for PES extrapolation schemes presented in Sect. (4.2) is not limited to basis
sets immediately sequential in size either, as in, for example, the cc-pVDZ, cc-pVTZ, and
cc-pVQZ sequence. In fact, the target basis set m + 1 in Eqs. (4.1)–(4.3) could be of infinite
(A)

size enabling us to extrapolate the PES Em+1 (R), where (m + 1) represents the CBS limit,
138

from the finite basis set calculations. This requires knowledge of the CBS limits of the base
(base)

(base)

energies, Em+1 (R) ≡ E∞

(R), at each point R on the PES of interest and the CBS-limit
(base)

(base)

of the correlation energy at the pivot geometry Re , ∆Em+1 (R) ≡ ∆E∞

(R), which is

needed to define the χm+1,m (R) scaling factor in a situation where (m + 1) represents the
infinite basis set.

As an example, consider the PES extrapolation for a sequence of basis sets of cc-pVXZ
quality to the CBS-limit in a situation where the target method A is CR-CC(2,3) and where
(base)

the base energy originates from RHF calculations. The CBS-limit base PES Em+1 (R)
needed in Eq. (4.1) can be determined via relatively inexpensive RHF/cc-pV6Z level calculations (recall from Sect. (3.2.1) that at the RHF level, cc-pV6Z results are an excellent
approximation to the corresponding CBS limit). One can then determine the CBS limit
(A)

of the correlation energy ∆E∞ (Re ) at the pivot geometry Re using, for example, correlation energies obtained with the cc-pVTZ and cc-pVQZ basis sets and one of the CBS
(A)

extrapolation laws, such as Eq. (3.98). The resulting CBS-limit value of ∆E∞ (Re ) can be
(A)

(A)

used as ∆Em+1 (Re ) to determine the S(Re )m+1,m ratio and, ultimately, the scaling factor
(A)

χm+1,m (R) (recall that B = A here). Thus, the necessary quantities for extrapolation of
the CR-CC(2,3) PES to the CBS-limit are the RHF/cc-pVDZ and RHF/cc-pVTZ values
at all points R of interest, the CR-CC(2,3)/cc-pVDZ and CR-CC(2,3)/cc-pVTZ values at
(base)

all points R of interest, the CBS-limit base energies Em+1 (R) at all points R of interest,
obtained, for example, in the RHF/cc-pV6Z calculations, and a CBS-limit extrapolated cor(A)

relation energy ∆E∞ (Re ) corresponding to a single pivot geometry Re , which serves as
(A)

the ∆Em+1 (Re ) value in Eqs. (4.1)–(4.3). We could carry out a similar CBS PES extrapolation procedure for methods other than CR-CC(2,3) and other sequences of the correlation
139

consistent basis sets. An actual application of this procedure, as applied to the CR-CC(2,3)
approach, will be discussed in Sect. (4.3.4), but before we do this, we examine the role of
(base)

pivot geometries Re and base wave functions used to define Em+1 (R) in Eq. (4.1).

4.3.3

The Role of Pivot Geometries and Base Wave Functions in
Single-Level PES Extrapolations

A few inherent flexibilities of the PES extrapolation scheme should be be discussed before
moving on to applications. First, the pivot geometry, which appears in Eq. (4.2) as Re , can
be chosen as the equilibrium geometry, as the geometry corresponding to the reactants or
products, or in principle, as any other single point on the PES of interest. It is even possible to
choose more than one pivot geometry in PES extrapolations [301,302], although the benefits
of doing so may be minimal since, as was shown in Ref. [193], the accuracy of results does
not depend on the choice of the pivot geometry Re . This fact will be illustrated in the next
section. One also has a choice of generalizing Eqs. (4.1)–(4.3) to several pivots Re that may
provide an adequate sampling of the PES of interest for performing an extrapolation. The
idea of multiple pivots Re has been explored with considerable success, in Refs. [301, 302].
The other inherent flexibility which will be investigated numerically in the next section is
the choice of base wave function defining the base energies in Eqs. (4.1)–(4.3). In the specific
case of the PES calculations performed in Sect. (4.3.4), where the total electronic energies of
interest are those obtained using the CR-CC(2,3) approach, two alternative definitions of the
base energy E (base) are considered, E (base) = E (RHF) and E (base) = E (CCSD) . In the former
case, as already alluded to above, we decompose the CR-CC(2,3) energy inttoo the RHF base
energy and the CR-CC(2,3) total correlation energy, which we want to extrapolate across
140

the entire PES. In the latter case, the CR-CC(2,3) energy is decomposed into the CCSD
base energy and the triples correction of CR-CC(2,3) which we want to extrapolate across
the entire PES. The relative performance of the PES extrapolation procedure based on Eqs.
(4.1)–(4.3) under each of these two definitions of the base wave function is reported in the
next section.

4.3.4

Application to the Isomerization of Bicyclobutane to Butadiene

Since the excellent performance of the single-level PES extrapolation scheme had already
been established for di- and tri-atomic molecular PESs in Ref. [184], in 2008 we published a
study aimed at assessing the potential usefulness of the PES extrapolation scheme in studies
of chemical reaction pathways involving polyatomic molecules [193]. In that work, the PES
extrapolation scheme was applied to the stationary points defining the conrotatory and disrotatory paths characterizing the isomerization of bicyclo[1.1.0]butane to buta-1,3-diene, which
we discussed earlier in Sect. (3.2.3). There were a number of reasons for conducting the study
in Ref. [193]. The first reason was to examine the basis set dependence of the CR-CC(2,3)
results for both isomerization pathways, The second reason was to examine whether the PES
extrapolation scheme was capable of recovering the results of the laborious point-wise CRCC(2,3)/cc-pVQZ calculations from the PESs obtained in the CR-CC(2,3)/cc-pVDZ and
CR-CC(2,3)/cc-pVTZ calculations, the base energies (RHF or CCSD) obtained in the ccpVDZ, cc-pVTZ, and cc-pVQZ calculations, and a single CR-CC(2,3) energy obtained with
the cc-pVQZ basis set at the pivot geometry Re on the relevant polyatomic reaction pathway.
Finally, the third reason for the study in Ref. [193] was to demonstrate the effectiveness of
141

our PES extrapolation procedure to the CBS-limit.
As already mentioned, two choices of the base wave function for defining the ∆E values
that enter Eqs. (4.1)–(4.3) were considered, namely, the RHF energy (E (base) = E (RHF) )
and the CCSD energy (E (base) = E (CCSD) ). The results obtained with both choices of the
reference energy and for the pivot geometry defined by the bicbut reactant are collected
in Table (4.2). Different choices of the pivot geometry, Re , were also considered, each
corresponding to one of the stationary points that define the two isomerization pathways.
These results are shown in Tables (4.3)–(4.7). It can be seen in Tables (4.2)–(4.7) that
the resulting maximum differences between the calculated and extrapolated CR-CC(2,3)/ccpVQZ energies characterizing both pathways are virtually independent of the choice of the
pivot geometry Re . For this reason, in the following discussion we mainly focus on one
specific choice of Re , namely, the geometry of the bicbut reactant (see Figure 3.2), as
shown in Table (4.2), while mentioning numerical results produced using other choices of Re
very briefly.
As one can see in Table (4.2), independent of the choice of the base energies (RHF
or CCSD), there is virtually no difference between the calculated and extrapolated CRCC(2,3)/cc-pVQZ energies. The differences between the calculated and extrapolated CRCC(2,3)/cc-pVQZ energies characterizing both isomerization pathways do not exceed 0.631
millihartree when E (base) = E (RHF) , and 0.277 millihartree when E (base) = E (CCSD) , when
the geometry of the bicbut reactant is used as the pivot geometry Re . To appreciate the
small magnitude of these extrapolation errors, the changes in the total electronic energies
when going from the cc-pVTZ to cc-pVQZ basis sets are also included in Table (4.2), which
are values on the order of 42–44 millihartree. Similar remarks are true for the other pivot ge142

Table 4.2: The calculated CR-CC(2,3)/cc-pVDZ and CR-CC(2,3)/cc-pVTZ energies and the
calculated and extrapolated CR-CC(2,3)/cc-pVQZ energies at the stationary points defining
the conrotatory and disrotatory pathways characterizing the bicbut→t-but isomerization.
The bicbut reactant defines the pivot geometry for the PES extrapolations.
RHF Base
E2 a
bicbut
con TS
dis TS
g-but
gt TS
t-but

E3 a

-155.493284
-155.424792
-155.388352
-155.533086
-155.528145
-155.537788

-155.651348
-155.581666
-155.540034
-155.689327
-155.684308
-155.694080

(calc) a

E4

-155.695497
-155.625392
-155.582481
-155.733236
-155.728073
-155.738043

(extr) b

E4

-155.695497
-155.625147
-155.581850
-155.732754
-155.727663
-155.737480

∆E43 c

ǫ4 d

-44.149 0.000
-43.727 0.245
-42.447 0.631
-43.909 0.482
-43.764 0.410
-43.963 0.563

CCSD Base
E2 a
bicbut
con TS
dis TS
g-but
gt TS
t-but

E3 a

-155.493284
-155.424792
-155.388352
-155.533086
-155.528145
-155.537788

-155.651348
-155.581666
-155.540034
-155.689327
-155.684308
-155.694080

(calc) a

E4

-155.695497
-155.625392
-155.582481
-155.733236
-155.728073
-155.738043

(extr) b

E4

-155.695497
-155.625212
-155.582364
-155.732971
-155.727894
-155.737766

∆E43 c

ǫ4 d

-44.149 0.000
-43.727 0.180
-42.447 0.117
-43.909 0.265
-43.764 0.179
-43.963 0.277

a Total energies, in hartree, calculated at the CR-CC(2,3)/cc-pVDZ (E ), CR-CC(2,3)/cc2
pVTZ (E3 ), and CR-CC(2,3)/cc-pVQZ (E4 ) levels. b Total CR-CC(2,3)/cc-pVQZ ener-

gies, in hartree, resulting from the PES extrapolation procedure discussed in the text.
c Differences, in millihartree, between the actual CR-CC(2,3)/cc-pVQZ and CR-CC(2,3)/ccpVTZ energies. d Deviations, in millihartree, between the calculated and extrapolated CRCC(2,3)/cc-pVQZ energies.

143

Table 4.3: The calculated CR-CC(2,3)/cc-pVDZ and CR-CC(2,3)/cc-pVTZ energies and the
calculated and extrapolated CR-CC(2,3)/cc-pVQZ energies at the stationary points defining
the conrotatory and disrotatory pathways characterizing the bicbut→t-but isomerization.
The con TS transition state defines the pivot geometry for the PES extrapolations.
RHF Base
E2 a
bicbut
con TS
dis TS
g-but
gt TS
tbut

E3 a

-155.493284
-155.424792
-155.388352
-155.533086
-155.528145
-155.537788

-155.651348
-155.581666
-155.540034
-155.689327
-155.684308
-155.694080

(calc) a

(extr) b

E4

E4

-155.695497
-155.625392
-155.582481
-155.733236
-155.728073
-155.738043

-155.695750
-155.625392
-155.582077
-155.733001
-155.727908
-155.737726

∆E43 c

ǫ4 d

-44.149
-43.727
-42.447
-43.909
-43.764
-43.963

-0.253
0.000
0.404
0.235
0.164
0.317

∆E43 c

ǫ4 d

CCSD Base
E2 a
bicbut
con TS
dis TS
g-but
gt TS
tbut

E3 a

-155.493284
-155.424792
-155.388352
-155.533086
-155.528145
-155.537788

-155.651348
-155.581666
-155.540034
-155.689327
-155.684308
-155.694080

(calc) a

E4

-155.695497
-155.625392
-155.582481
-155.733236
-155.728073
-155.738043

(extr) b

E4

-155.695698
-155.625392
-155.582510
-155.7323159
-155.728082
-155.737954

-44.149 -0.202
-43.727 0.000
-42.447 -0.029
-43.909 0.077
-43.764 -0.009
-43.963 0.090

a Total energies, in hartree, calculated at the CR-CC(2,3)/cc-pVDZ (E ), CR-CC(2,3)/cc2
pVTZ (E3 ), and CR-CC(2,3)/cc-pVQZ (E4 ) levels. b Total CR-CC(2,3)/cc-pVQZ ener-

gies, in hartree, resulting from the PES extrapolation procedure discussed in the text.
c Differences, in millihartree, between the actual CR-CC(2,3)/cc-pVQZ and CR-CC(2,3)/ccpVTZ energies. d Deviations, in millihartree, between the calculated and extrapolated CRCC(2,3)/cc-pVQZ energies.

144

Table 4.4: The calculated CR-CC(2,3)/cc-pVDZ and CR-CC(2,3)/cc-pVTZ energies and the
calculated and extrapolated CR-CC(2,3)/cc-pVQZ energies at the stationary points defining
the conrotatory and disrotatory pathways characterizing the bicbut→t-but isomerization.
The dis TS transition state defines the pivot geometry for the PES extrapolations.
RHF Base
E2 a
bicbut -155.493284
con TS -155.424792
dis TS -155.388352
g-but
-155.533086
gt TS
-155.528145
tbut
-155.537788

E3 a
-155.651348
-155.581666
-155.540034
-155.689327
-155.684308
-155.694080

(calc) a

E4

-155.695497
-155.625392
-155.582481
-155.733236
-155.728073
-155.738043

(extr) b

E4

-155.696200
-155.625828
-155.582481
-155.733439
-155.728345
-155.738163

∆E43 c

ǫ4 d

-44.149 -0.704
-43.727 -0.436
-42.447 0.000
-43.909 -0.203
-43.764 -0.272
-43.963 -0.120

CCSD Base
E2 a

E3 a

bicbut -155.493284
con TS -155.424792
dis TS -155.388352
g-but
-155.533086
gt TS
-155.528145
tbut
-155.537788

(calc) a

E4

-155.651348
-155.581666
-155.540034
-155.689327
-155.684308
-155.694080

(extr) b

E4

-155.695497
-155.625392
-155.582481
-155.733236
-155.728073
-155.738043

∆E43 c

ǫ4 d

-155.695659
-155.625357
-155.582481
-155.733122
-155.728045
-155.737917

-44.149
-43.727
-42.447
-43.909
-43.764
-43.963

-0.162
0.036
0.000
0.114
0.028
0.127

a Total energies, in hartree, calculated at the CR-CC(2,3)/cc-pVDZ (E ), CR-CC(2,3)/cc2
pVTZ (E3 ), and CR-CC(2,3)/cc-pVQZ (E4 ) levels. b Total CR-CC(2,3)/cc-pVQZ ener-

gies, in hartree, resulting from the PES extrapolation procedure discussed in the text.
c Differences, in millihartree, between the actual CR-CC(2,3)/cc-pVQZ and CR-CC(2,3)/ccpVTZ energies. d Deviations, in millihartree, between the calculated and extrapolated CRCC(2,3)/cc-pVQZ energies.

145

Table 4.5: The calculated CR-CC(2,3)/cc-pVDZ and CR-CC(2,3)/cc-pVTZ energies and the
calculated and extrapolated CR-CC(2,3)/cc-pVQZ energies at the stationary points defining
the conrotatory and disrotatory pathways characterizing the bicbut→t-but isomerization.
The g-but intermediate defines the pivot geometry for the PES extrapolations.
RHF Base
E2 a
bicbut -155.493284
con TS -155.424792
dis TS -155.388352
g-but
-155.533086
gt TS
-155.528145
tbut
-155.537788

E3 a
-155.651348
-155.581666
-155.540034
-155.689327
-155.684308
-155.694080

(calc) a

E4

-155.695497
-155.625392
-155.582481
-155.733236
-155.728073
-155.738043

(extr) b

E4

-155.695992
-155.625626
-155.582294
-155.733236
-155.728142
-155.737960

∆E43 c

ǫ4 d

-44.149 -0.495
-43.727 -0.234
-42.447 0.187
-43.909 0.000
-43.764 -0.070
-43.963 0.083

CCSD Base
E2 a
bicbut -155.493284
con TS -155.424792
dis TS -155.388352
g-but
-155.533086
-155.528145
gt TS
tbut
-155.537788

E3 a
-155.651348
-155.581666
-155.540034
-155.689327
-155.684308
-155.694080

(calc) a

E4

-155.695497
-155.625392
-155.582481
-155.733236
-155.728073
-155.738043

(extr) b

E4

-155.695781
-155.625467
-155.582570
-155.733236
-155.728159
-155.738031

∆E43 c

ǫ4 d

-44.149 -0.285
-43.727 -0.074
-42.447 -0.089
-43.909 0.000
-43.764 -0.086
-43.963 0.012

a Total energies, in hartree, calculated at the CR-CC(2,3)/cc-pVDZ (E ), CR-CC(2,3)/cc2
pVTZ (E3 ), and CR-CC(2,3)/cc-pVQZ (E4 ) levels. b Total CR-CC(2,3)/cc-pVQZ ener-

gies, in hartree, resulting from the PES extrapolation procedure discussed in the text.
c Differences, in millihartree, between the actual CR-CC(2,3)/cc-pVQZ and CR-CC(2,3)/ccpVTZ energies. d Deviations, in millihartree, between the calculated and extrapolated CRCC(2,3)/cc-pVQZ energies.

146

Table 4.6: The calculated CR-CC(2,3)/cc-pVDZ and CR-CC(2,3)/cc-pVTZ energies and the
calculated and extrapolated CR-CC(2,3)/cc-pVQZ energies at the stationary points defining
the conrotatory and disrotatory pathways characterizing the bicbut→t-but isomerization.
The gt TS transition state defines the pivot geometry for the PES extrapolations.
RHF Base
E2 a
bicbut -155.493284
con TS -155.424792
dis TS -155.388352
g-but
-155.533086
gt TS
-155.528145
tbut
-155.537788

E3 a
-155.651348
-155.581666
-155.540034
-155.689327
-155.684308
-155.694080

(calc) a

E4

-155.695497
-155.625392
-155.582481
-155.733236
-155.728073
-155.738043

(extr) b

E4

-155.695920
-155.625557
-155.582229
-155.733166
-155.728073
-155.737891

∆E43 c

ǫ4 d

-44.149 -0.423
-43.727 -0.164
-42.447 0.252
-43.909 0.070
-43.764 0.000
-43.963 0.153

CCSD Base
E2 a
bicbut -155.493284
con TS -155.424792
dis TS -155.388352
g-but
-155.533086
-155.528145
gt TS
tbut
-155.537788

E3 a
-155.651348
-155.581666
-155.540034
-155.689327
-155.684308
-155.694080

(calc) a

E4

-155.695497
-155.625392
-155.582481
-155.733236
-155.728073
-155.738043

(extr) b

E4

-155.695689
-155.625384
-155.582503
-155.733150
-155.728073
-155.737945

∆E43 c

ǫ4 d

-44.149 -0.192
-43.727 0.009
-42.447 -0.022
-43.909 0.086
-43.764 0.000
-43.963 0.099

a Total energies, in hartree, calculated at the CR-CC(2,3)/cc-pVDZ (E ), CR-CC(2,3)/cc2
pVTZ (E3 ), and CR-CC(2,3)/cc-pVQZ (E4 ) levels. b Total CR-CC(2,3)/cc-pVQZ ener-

gies, in hartree, resulting from the PES extrapolation procedure discussed in the text.
c Differences, in millihartree, between the actual CR-CC(2,3)/cc-pVQZ and CR-CC(2,3)/ccpVTZ energies. d Deviations, in millihartree, between the calculated and extrapolated CRCC(2,3)/cc-pVQZ energies.

147

Table 4.7: The calculated CR-CC(2,3)/cc-pVDZ and CR-CC(2,3)/cc-pVTZ energies and the
calculated and extrapolated CR-CC(2,3)/cc-pVQZ energies at the stationary points defining
the conrotatory and disrotatory pathways characterizing the bicbut→t-but isomerization.
The t-but product defines the pivot geometry for the PES extrapolations.
RHF Base
E2 a
bicbut -155.493284
con TS -155.424792
dis TS -155.388352
g-but
-155.533086
gt TS
-155.528145
tbut
-155.537788

E3 a
-155.651348
-155.581666
-155.540034
-155.689327
-155.684308
-155.694080

(calc) a

E4

-155.695497
-155.625392
-155.582481
-155.733236
-155.728073
-155.738043

(extr) b

E4

-155.696077
-155.625709
-155.582370
-155.733319
-155.728225
-155.738043

∆E43 c

ǫ4 d

-44.149 -0.580
-43.727 -0.316
-42.447 0.110
-43.909 -0.083
-43.764 -0.152
-43.963 0.000

CCSD Base
E2 a
bicbut -155.493284
con TS -155.424792
dis TS -155.388352
g-but
-155.533086
-155.528145
gt TS
tbut
-155.537788

E3 a
-155.651348
-155.581666
-155.540034
-155.689327
-155.684308
-155.694080

(calc) a

E4

-155.695497
-155.625392
-155.582481
-155.733236
-155.728073
-155.738043

(extr) b

E4

-155.695795
-155.625479
-155.582579
-155.733248
-155.728172
-155.738043

∆E43 c

ǫ4 d

-44.149 -0.298
-43.727 -0.086
-42.447 -0.099
-43.909 -0.013
-43.764 -0.099
-43.963 0.000

a Total energies, in hartree, calculated at the CR-CC(2,3)/cc-pVDZ (E ), CR-CC(2,3)/cc2
pVTZ (E3 ), and CR-CC(2,3)/cc-pVQZ (E4 ) levels. b Total CR-CC(2,3)/cc-pVQZ ener-

gies, in hartree, resulting from the PES extrapolation procedure discussed in the text.
c Differences, in millihartree, between the actual CR-CC(2,3)/cc-pVQZ and CR-CC(2,3)/ccpVTZ energies. d Deviations, in millihartree, between the calculated and extrapolated CRCC(2,3)/cc-pVQZ energies.

148

ometries examined in Tables (4.3)–(4.7), which correspond to the remaining stationary points
along the conrotatory and disrotatory pathways defining the bicbut→t-but isomerization.
As shown in Tables (4.3)–(4.7), the differences between the calculated and extrapolated CRCC(2,3)/cc-pVQZ energies characterizing both isomerization pathways do not exceed 0.704
millihartree, when E (base) = E (RHF) , and 0.298 millihartree, when E (base) = E (CCSD) , and
are often much smaller, independent of the choice of the pivot geometry Re . Thus, the singlelevel PES extrapolation procedure, originally proposed in Ref. [184] and further developed
in Refs. [193,300], reproduces changes in the total energies when going from the cc-pVTZ to
cc-pVQZ basis sets to within 1.5 %, when the RHF energies are used as the base energies,
and 0.6 %, when the CCSD energies are employed instead to define the base energies E (base) .
As one might expect and as shown in Tables (4.2)–(4.7), the use of the CCSD wave function
in determining the base energies reduces the observed differences between the calculated and
extrapolated CR-CC(2,3)/cc-pVQZ energies. However, this is to be expected since the use
of the correlated base energies, such as those obtained in the CCSD calculations, reduces
the fraction of the correlation energy to be extrapolated, which in turn decreases the magnitude of the extrapolation errors. On the other hand, it is quite remarkable that the use of
uncorrelated RHF base energies in determining the ∆E values that enter Eqs. (4.1)–(4.3)
leads to extrapolated CR-CC(2,3)/cc-pVQZ PESs which are identical to the calculated CRCC(2,3)/cc-pVQZ PESs to within ∼ 0.7 millihartree, independent of the choice of Re . These
small differences between the calculated and extrapolated CR-CC(2,3)/cc-pVQZ electronic
energies are also reflected in the small differences between the calculated and extrapolated
CR-CC(2,3)/cc-pVQZ enthalpy values characterizing the six stationary points along the conrotatory and disrotatory pathways that define the isomerization of bicyclo[1.1.0]butane to

149

buta-1,3-diene, shown in Figure (3.2), which are on the order of 1 kcal/mol or, in most cases,
even less.

Returning now to the third reason for the study of the bicbut→t-but reaction published
in Ref. [193], this system was used to test the accuracy which may be obtained using singlelevel PES extrapolation techniques to obtain a PES at the CBS-limit. In the case in Ref.
[193], the cc-pVDZ basis set was chosen as basis level m−1, the cc-pVTZ basis set was chosen
as basis level m, and the infinite basis set limit was chosen as basis set m + 1 in Eqs. (4.1)–
(4.3). Thus, results of CR-CC(2,3) calculations at the cc-pVTZ and cc-pVQZ basis set levels
were used to determine the CBS limit of the CR-CC(2,3) correlation energy at Re using Eq.
(3.98), and the resulting CBS value of the CR-CC(2,3) correlation energy was subsequently
(A)

used to define ∆Em+1 (Re ) in Eqs. (4.1)–(4.3) (recall again that A = B in the present
(A)

considerations). Then, the ∆Em+1 (Re ) correlation energy at Re determined in this way and
(A)

the CR-CC(2,3)/cc-pVDZ and CR-CC(2,3)/cc-pVTZ correlation energies, ∆Em−1 (R) and
(A)

∆Em (R), respectively, at all geometries R of interest were used to obtain the scaling factor
(A)

χm+1,m (R) from the cc-pVTZ basis set to the CBS limit according to Eqs. (4.2) and (4.3).
(A)

Once the scaling factor χm+1,m (R) at each R was established, we used it to determine the
(A)

CBS limit of the CR-CC(2,3) PES, Em+1 (R), at all geometries of interest by multiplying
(A)

the CR-CC(2,3) correlation energies ∆Em (R) obtained with the cc-pVTZ basis set by
(A)

(base)

χm+1,m (R) and by adding the resulting energies to the base energies Em+1 (R) obtained
in the RHF/cc-pV6Z calculations which, as already explained, are practically equivalent to
the RHF energies in the CBS limit. By avoiding the point-wise CBS extrapolations of the
CR-CC(2,3) correlation energies from the cc-pVTZ and cc-pVQZ basis sets at all geometries
R (we had to perform the CBS extrapolation of the CR-CC(2,3) energy only at the pivot
150

geometry Re ) and by using the cc-pV6Z values of the RHF energies, we saved a lot of CPU
cycles, while producing the smooth CR-CC(2,3)-level PESs corresponding to the CBS limit,
since the base RHF PESs obtained with a large, cc-pV6Z basis set and the scaling factor
(A)

χm+1,m (R) are smooth functions of R.
Indeed, as shown in Table 4.8, the differences between the energies resulting from the
CBS extrapolation scheme that combines Eqs. (4.1)–(4.3) with Eq. (3.98) and the energies
resulting from the conventional point-wise CBS extrapolations using the CR-CC(2,3)/ccpVTZ and CR-CC(2,3)/cc-pVQZ data at each stationary point defining the conrotatory and
disrotatory pathways shown in Figure (3.2) do not exceed 1.092 millihartree, i.e., they are
on the same order as the intrinsic error of the CBS extrapolations based on Eq. (3.98). This
method provides a significant advantage over conventional CBS-limit extrapolation of PESs
when it is considered that combining Eqs. (4.1)–(4.3) with Eq. (3.98) requires only one CRCC(2,3)/cc-pVQZ calculation (at the pivot geometry), whereas the conventional point-wise
CBS extrapolation method requires the CR-CC(2,3)/cc-pVQZ calculations at each point on
the PES of interest, which represents an enormous increase in computer cost.

151

Table 4.8: The calculated CR-CC(2,3)/cc-pVDZ, CR-CC(2,3)/cc-pVTZ, and CR-CC(2,3)/cc-pVQZ energies and the CBS
values of the CR-CC(2,3) energies obtained using the point-wise extrapolations exploiting Eq. (3.98) and the CBS extrapolation
procedure combining Eqs. (4.1)–(4.3) with Eq. (3.98) discussed in the text at the stationary points defining the conrotatory
and disrotatory pathways characterizing the bicbut→t-but isomerization. The bicbut reactant was used to define the pivot
geometry for the PES extrapolations based on Eqs. (4.1)–(4.3).
E2 a
bicbut
con TS
dis TS
g-but
gt TS
t-but

E3 a

-155.493284
-155.424792
-155.388352
-155.533086
-155.528145
-155.537788

-155.651348
-155.581666
-155.540034
-155.689327
-155.684308
-155.694080

(calc) a

E4

-155.695497
-155.625392
-155.582481
-155.733236
-155.728073
-155.738043

(calc) b

E∞

-155.723202
-155.652777
-155.608902
-155.760795
-155.755491
-155.765622

(extr) c

E∞

-155.723202
-155.652353
-155.607810
-155.759962
-155.754781
-155.764647

∆E∞,3 d

ǫ∞ e

-71.854
-71.112
-68.868
-71.468
-71.182
-71.541

0.000
0.424
1.092
0.833
0.710
0.975

a Total energies, in hartree, calculated at the CR-CC(2,3)/cc-pVDZ (E ), CR-CC(2,3)/cc-pVTZ (E ), and CR-CC(2,3)/cc2
3
pVQZ (E4 ) levels. b Total energies, in hartree, obtained by adding the RHF/cc-pV6Z energies to the CBS correlation energies

resulting from the point-wise CBS extrapolations employing the CR-CC(2,3)/cc-pVTZ and CR-CC(2,3)/cc-pVQZ calculations and Eq. (3.98). c Total energies, in hartree, obtained by adding the RHF/cc-pV6Z energies to the CBS values of the
correlation energy resulting from the extrapolation procedure combining Eqs. (4.1)–(4.3) with Eq. (3.98), as discussed in the
text. d Differences, in millihartree, between the CR-CC(2,3)/CBS energies resulting from the point-wise CBS extrapolations
employing Eq. (3.98) and the corresponding CR-CC(2,3)/cc-pVTZ energies. e Differences, in millihartree, between the CRCC(2,3)/CBS energies resulting from the point-wise CBS extrapolations employing Eq. (3.98) and the CBS PES extrapolation
combining Eqs. (4.1)–(4.3) with Eq. (3.98).

152

4.4

Dual-Level Potential Energy Surface Extrapolation
Schemes

The main idea of the PES extrapolation scheme is to scale the difference, ∆E, between the
total electronic energy E, and the energy of some base wave function E (base) . The difference,
∆E, representing the correlation energy or some fraction of it, is scaled to the quality of
a larger basis set by applying a scaling factor which predicts the change in the correlation
energy with the size of the basis set at a particular nuclear configuration. Since the basis set
dependence of the correlation energy may be similar for related electronic structure methods,
after the original PES extrapolation scheme was proposed in Ref. [184] and further developed
and tested in Ref. [193], additional flexibility was introduced by allowing the scaling factor
to be generated using a method different from the method used to calculate the surface of
interest. This is reflected in Eqs. (4.1) – (4.3) by the designation of method A, corresponding
to the desired level of theory of the target PES, and method B, corresponding to the level
(B)

of theory used in the generation of the scaling factor, χm+1,m (R). A discussion of the ideas
behind the dual-level PES extrapolation scheme is given in Sect. (4.4.1). This is followed
by a few applications in Sects. (4.4.2) and (4.4.3), where the CR-CC(2,3) method is used to
calculate the PESs of interest using smaller basis sets, where computer costs remain low, in
combination with the dual-level PES extrapolation scheme, in which the correlation scaling
(B)

factor χm+1,m (R) is generated with lower-order methods, to obtain the PESs corresponding
to the massively more expensive large basis set CR-CC(2,3) calculations. The resulting PESs
are obtained using only a fraction of the computational resources required by the single-level
PES extrapolation scheme, not to mention the point-by-point CR-CC(2,3) computations
153

using larger bases. A detailed analysis of the huge computational savings which the singleand dual-level PES extrapolation schemes can offer will be presented in Sect. (4.4.4).

4.4.1

Using Lower-Order Methods to Produce Correlation Energy
Scaling Factors in High-Level Calculations

The dual-level extrapolation scheme presented here closely follows the original procedure,
proposed in Ref. [184] and described in Sect. (4.3), differing only in the definition of the
(B)

scaling factor χm+1,m (R), which may now be obtained with a different, less expensive,
methodology, B, than the target methodology A of interest. By allowing A and B to
(A)

differ, the approximate high-level PES Em+1 (R) can be generated by extrapolating the PES
(A)

(base)

(base)

Em (R) obtained with smaller basis set m, using the base surfaces Em−1 (R), Em
(base)

(B)

(R),

(B)

and Em+1 (R), the correlated surfaces Em−1 (R) and Em (R), and a single correlated
(B)

energy calculated at the pivot geometry, Em+1 (Re ), all obtained with method B, which is,
by choice, less expensive than the target method A. The most significant advantage over the
(A)

original PES extrapolation scheme, where χm+1,m (R) was used, is that the calculation of
(A)

the single high-level energy Em+1 (Re ), which accounts for the majority of the expense of the
original procedure of Ref. [184], is avoided entirely and replaced instead by the calculation
(B)

of Em+1 (Re ), using a quantum-chemistry method B which is less expensive than A.
The conventional low-order MBPT or CC approximations are among the approaches
which may be considered as lower-order methods for the extrapolations of the high-level
PESs resulting from higher-order CC calculations. The systematically improvable hierarchy
of MBPT approaches is especially appealing in this study because it provides a series of “built
in” lower-order methods that can facilitate an investigation of the lowest levels of correlation
154

treatment required to properly reproduce the correlation scaling factors for the higher-level
CC methodology, CR-CC(2,3). The lowest-level MBPT methods, including MP2, MP3, and
MP4X (X = D, DQ, and SDQ), are all much less expensive than the CR-CC(2,3) theory,
with formal noniterative scaling steps of only N 5 for MP2 and N 6 for MP3 and MP4SDQ.
For comparison, the CPU-time determining steps of CR-CC(2,3) scale as N 6 in the iterative
CCSD part and N 7 in the triples correction part. In fact, even CCSD offers considerable
savings compared to CR-CC(2,3), so this would be another candidate for determining the
correlation energy scaling factor for extrapolating the CR-CC(2,3) PES.

As an example, if, say A = CR-CC(2,3) and B = CCSD, a CR-CC(2,3)/cc-pVQZ
(CR-CC(2,3))
(R) may be obtained by extrapolating a CR-CC(2,3)/cc-pVTZ PES
PES ∆E4
(CCSD)
(CR-CC(2,3))
(R), a correlation energy scaling factor constructed
(R) by applying χ4,3
∆E3

from CCSD/cc-pVDZ energies, CCSD/cc-pVTZ energies, and a single CCSD/cc-pVQZ energy at the pivot geometry Re (rather than from the analogous set of energies obtained with
CR-CC(2,3), as would be the case in the previously discussed single-level PES extrapolation
(CCSD)

scheme) and adding the resulting χ4,3

(CR-CC(2,3))

(R)∆E3

(base)

culated RHF/cc-pVQZ PES that provides the base E4

(R) term to an explicitly cal-

(R) term in Eq. (4.1). Similarly,

if A = CR − CC(2, 3) and B = MP4SDQ, we can determine the CR-CC(2,3)/cc-pVQZ(CR-CC(2,3))

level PES E4
(MP4SDQ)

χ4,3

(MP4SDQ)

(R) by adding the χ4,3

(CR-CC(2,3))

(R)∆E3

(R) term, where

(R) is a correlation energy scaling factor obtained from the MP4SDQ calculation

using the cc-pVDZ and cc-pVTZ basis sets and a single MP4SDQ/cc-pVQZ point calcu(base)

lated at Re , to the RHF/cc-pVQZ base energy E4

(R). Other examples of the auxiliary

method B used in determining the correlation energy scaling factor for extrapolating the
CR-CC(2,3) PES examined in this work include the MP2, MP3, MP4D, and MP4DQ ap155

proaches. We apply the dual-level PES extrapolation procedure, as described above, to
predict the CR-CC(2,3)/cc-pVQZ or CR-CC(2,3)/aug-cc-pVQZ PESs characterizing a few
single-bond dissociations (Sect. (4.4.2)) and the previously examined bicyclobutane isomerization pathways (Sect. (4.4.3)).

4.4.2

Application to Single Bond-Breaking Potential Energy Curves

In this section, single bond-breaking potential energy curves of the H2 O, HCl, and F2
molecules are considered as benchmark cases to test the accuracy of the dual-level PES
extrapolation procedure. In all PES extrapolations discussed in this section, the pivot geometry Re is taken to be the equilibrium geometry, the base wavefunction is chosen to be the
RHF wavefunction, and the CR-CC(2,3)/aug-cc-pVTZ PESs are extrapolated to the CRCC(2,3)/aug-cc-pVQZ level by applying a correlation energy scaling factor generated from
the following quantities: RHF/aug-cc-pVDZ, RHF/aug-cc-pVTZ and RHF/aug-cc-pVQZ
base surfaces, B/aug-cc-pVDZ and B/aug-cc-pVTZ surfaces, and a single B/aug-cc-pVQZ
energy at the pivot geometry Re , where the lower-order method B is MP2, MP3, MP4X (X
= D, DQ, SDQ), or CCSD.
In Table (4.9), extrapolation errors are collected for selected points on the H2 O→ OH+H
dissociation curve. In the first two columns of this table, benchmark CR-CC(2,3)/aug-cc(CR-CC(2,3))

pVQZ energies, E4

(R) are given for selected points R on the potential energy

(CR-CC(2,3))
curve, along with the energy differences, ∆E4,3 (R), between E4
(R) and the CR(CR-CC(2,3))
(R). In the following columns, extrapolation
CC(2,3)/aug-cc-pVTZ energies E3
(CR-CC(2,3))

errors are reported, in millihartree, relative to the corresponding true E4
(B)

(R) val-

ues, organized according to the correlation energy scaling χ4,3 (R) that was used in their
156

generation. Each set of extrapolation errors should be compared to those reported in the
far right column of Table (4.9), which represent the errors found using the single-level formulation of the PES extrapolation scheme, appropriately designated by the scaling factor
(CR-CC(2,3))

χ4,3

(R) found in the corresponding column heading. While not all lower-order

methods can be used to accurately reproduce all points on the CR-CC(2,3)/aug-cc-pVQZ
PES, it is notable that relatively small extrapolation errors are observed when all of the
lower-order methods (except for MP2) are employed to generate the near-equilibrium region
of the H2 O→OH+H curve. This implies that MP3, the various variants of MP4, and CCSD
yield corrleation energies which have a very similar correlation energy scaling to the CRCC(2,3) energies in this region. If one is only concerned with the near-equilibrium region
of this bond-stretching curve, all of these methods except MP2 can be used successfully to
scale a CR-CC(2,3) correlation energy to the quality of a larger basis sets. Unfortunately,
the MP3 and various MP4X methods fail to produce the proper scaling at the largest internuclear distances, shown by steeply rising extrapolation errors beyond 2Re . As a specific
(MP4SDQ)

example, when χ4,3

(R) is used, the largest reported error in the extrapolation of the

CR-CC(2,3)/aug-cc-pVQZ PES in the region from 0.75Re − 2Re is only 0.338 millihartree,
which represents a recovery of about 98% of the correlation energy change when going from
(CR-CC(2,3))

aug-cc-pVTZ to aug-cc-pVQZ, identified in Table (4.9) as ∆E4,3

. However, at 4Re ,

(CR-CC(2,3))

the same error rises sharply to 4.798 millihartree, which is 26% of ∆E4,3

at this

geometry. It is quite remarkable, though, to observe such tiny errors while extrapolating
the CR-CC(2,3)/aug-cc-pVQZ PES in the relatively large R = 0.75Re − 2Re region, on the
order of small fractions of a millihartree, particularly when we realize that the CPU time of
the MP4SDQ calculations is approximately the same as the cost of a single CCSD iteration.

157

To predict an accurate PES at the highly stretched geometries, it is clear from the
extrapolation errors reported in Table (4.9) that CCSD is the only lower-order method
which accurately and consistently predicts the correct correlation energy scaling in both
(CCSD)

the equilibrium and bond-breaking regions of the CR-CC(2,3) surface. When χ4,3

(R)

is used, the maximum reported deviation from the true CR-CC(2,3)/aug-cc-pVQZ curve
in the R = 0.75Re − 4Re region is only 1.124 millihartree, with extrapolation errors in
the R = 0.75Re − 2Re region not exceeding 0.394 millihartree. For comparison, when the
CR-CC(2,3) methodology is used to generate the correlation energy scaling factor, as in
the single-level scheme, the largest error found for the extrapolated CR-CC(2,3)/aug-ccpVQZ H2 O PES is 0.760 millihartree in the R = 0.75Re − 2Re regioin and 0.314 millihartree when R does not exceed 2R. From these results it is clear that for the H2 O system,
(CCSD)

χ4,3

(CR-CC(2,3))

(R) and χ4,3

(R) are virtually identical for all reported geometries. The

final observation from Table (4.9) is that the MP2 method is found to consistently overesti(CR-CC(2,3))

mate χ4,3

(R), producing relatively large negative extrapolation errors on the order

of (-4)–(-2) millihartree, even in the near-equilibrium region. Thus, the MP2 method should
not be used in conjunction with the dual-level PES extrapolation scheme to describe the
scaling of CR-CC(2,3) with the basis set.

Analogous sets of extrapolation errors for the HCl and F2 bond-stretching surfaces are
reported in Tables (4.10) and (4.11), respectively, and similar trends are observed therein.
For both the HCl and F2 systems the MP3 and MP4X methods are again shown to provide
the proper correlation energy scaling in the equilibrium region, where large percentages
(CR-CC(2,3))

(between 93–100%) of ∆E4,3

(R) are consistently recovered and errors relative to

true CR-CC(2,3)/aug-cc-pVQZ calculations are on the order of a millihartree or less. Also,
158

as before, extrapolation errors grow rapidly as the internuclear separation approaches the
R > 2Re region, with CCSD appearing to be the only lower-order method which may be
used successfully at all internuclear separations. This is especially evident for the F2 curve
where results based on correlation energy scaling factors obtained with MP2, MP3, MP4D,
MP4DQ, and MP4SDQ diverge badly at 4Re from the true CR-CC(2,3)/aug-cc-pVQZ curve,
and while the use of CCSD instead leads to reasonable behavior. In fact, the maximum
reported extrapolation errors are again found to be very close at all internuclear distances
when the CCSD and CR-CC(2,3) correlation energy scaling factors are used to scale the CRCC(2,3)/aug-cc-pVTZ HCl and F2 PESs. For HCl, they are 1.891 and 1.664 millihartree,
(CR-CC(2,3))

respectively, each corresponding to a recovery of about 90% of ∆E4,3
(CCSD)

extrapolation errors found using χ4,3
(CR-CC(2,3))

∆E4,3

(R). For F2 ,

(R) show a recovery of 93–99% of the corresponding
(CR-CC(2,3))

(R) values, which may be compared to 95–100% obtained with χ4,3

159

.

Table 4.9: A comparison of calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies of the H2 O molecule in which
one of the two O-H bonds (R) is stretched, while keeping the other O-H bond at the equilibrium length and the H-O-H angle
fixed at 104.5 ◦ . The equilibrium geometry defines the pivot geometry Re and RHF defines the base wave function for the
PES extrapolations. In all post-RHF calculations, the lowest orbital, correlating with the 1s shell of the oxygen atom was kept
frozen.
(B)

ǫ4 (χ4,3 )d
R/Re a
0.75
0.90
1.00
1.10
1.25
1.50
2.00
3.00
4.00

E4 b

∆E4,3 c

MP2

MP3

MP4D

MP4DQ

MP4SDQ

CCSD

CR-CC(2, 3)

-76.259133
-76.351620
-76.363051
-76.355931
-76.329620
-76.276456
-76.199626
-76.162662
-76.161060

-22.676
-21.605
-21.059
-20.641
-20.278
-19.952
-19.168
-18.479
-18.303

-2.993
-2.759
-2.742
-2.723
-2.624
-2.639
-3.613
-4.086
-2.232

-0.763
-0.573
-0.576
-0.580
-0.509
-0.513
-1.217
-0.601
1.965

-0.611
-0.425
-0.428
-0.430
-0.349
-0.316
-0.788
0.729
3.688

-0.257
-0.046
-0.033
-0.019
0.080
0.130
-0.359
0.774
2.953

-0.073
0.121
0.131
0.145
0.253
0.338
-0.071
1.450
4.798

-0.186
0.031
0.054
0.085
0.225
0.394
0.294
0.912
1.124

-0.140
0.012
0.000
0.009
0.133
0.314
0.244
0.744
0.760

a The equilibrium value of R used here is R = 0.95785 ˚. b The calculated CR-CC(2,3)/aug-cc-pVQZ total energies in
A
e
c Differences, in millihartree, between the actual CR-CC(2,3)/aug-cc-pVQZ and CR-CC(2,3)/aug-cc-pVTZ energies.
hartree.
d Differences, in millihartree, between the calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies, where the latter
(B)

energies were generated by applying the correlation energy scaling factors χ4,3 (R) obtained with B = MP2, MP3, MP4D,
MP4DQ, MP4SDQ, CCSD, and CR-CC(2,3). The choice of B = CR-CC(2,3) is equivalent to the single-level PES extrapolation
scheme.

160

Table 4.10:
tions RH-Cl
function for
shells of Cl,

A comparison of calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies for several internuclear separaof the HCl molecule. The equilibrium geometry defines the pivot geometry Re and RHF defines the base wave
the PES extrapolations. In all post-RHF calculations, the lowest five orbitals, correlating with the 1s, 2s, and 2p
were kept frozen.
(B)

ǫ4 (χ4,3 )d
R/Re a
0.75
0.90
1.00
1.10
1.25
1.50
2.00
3.00
4.00

E4 b

∆E4,3 c

MP2

MP3

MP4D

MP4DQ

MP4SDQ

CCSD

CR-CC(2, 3)

-460.245772
-460.351216
-460.364178
-460.356856
-460.329720
-460.277589
-460.212229
-460.192428
-460.192532

-22.053
-20.979
-20.704
-20.550
-20.327
-19.823
-18.747
-18.345
-17.809

-4.017
-3.055
-2.610
-2.330
-2.178
-2.429
-3.755
-2.889
-0.558

-1.415
-0.537
-0.127
0.128
0.261
0.065
-0.828
0.849
2.958

-1.254
-0.387
0.017
0.268
0.407
0.276
-0.293
2.050
3.966

-0.834
0.033
0.439
0.692
0.834
0.697
0.055
1.675
1.670

-0.854
0.022
0.437
0.703
0.868
0.793
0.456
4.602
9.747

-0.896
-0.003
0.427
0.713
0.926
0.991
0.966
1.891
1.573

-1.273
-0.410
0.000
0.270
0.469
0.580
0.730
1.664
1.049

a The equilibrium value of R used here is R = 1.27455 ˚. b The calculated CR-CC(2,3)/aug-cc-pVQZ total energies in
A
e
hartree. c Differences, in millihartree, between the actual CR-CC(2,3)/aug-cc-pVQZ and CR-CC(2,3)/aug-cc-pVTZ energies.
d Differences, in millihartree, between the calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies, where the latter
(B)

energies were generated by applying scaling factors χ4,3 (R) obtained with B = MP2, MP3, MP4D, MP4DQ, MP4SDQ,
CCSD, and CR-CC(2,3). The choice of B=CR-CC(2,3) is equivalent to the single-level extrapolation scheme.

161

Table 4.11: A comparison of calculated and extrapolated CR-CC(2,3)/aug-ccpVQZ energies for several internuclear separations
RF-F of the F2 mole- cule. The equilibrium geometry defines the pivot geometry Re and RHF defines the base wave function
for the PES extrapolations. In all post-RHF calculations, the lowest two orbitals, correlating with the 1s shells of the F atoms,
were kept frozen.
(B)

ǫ4 (χ4,3 )d
R/Re a
0.75
0.90
1.00
1.10
1.25
1.50
2.00
3.00
4.00

E4 b

∆E4,3 c

MP2

MP3

MP4D

MP4DQ

MP4SDQ

CCSD

CR-CC(2, 3)

-199.196897
-199.349657
-199.364732
-199.356794
-199.333764
-199.307123
-199.296727
-199.297648
-199.297985

-56.537
-53.009
-51.743
-51.172
-50.464
-49.741
-49.300
-49.368
-49.422

-5.098
-3.991
-3.835
-3.871
-4.209
-4.370
-1.232
6.748
12.149

-2.516
-1.752
-1.737
-1.793
-2.010
-1.655
3.454
13.398
18.901

-1.915
-1.132
-1.028
-0.929
-0.775
0.587
8.189
18.543
22.880

-1.208
-0.444
-0.378
-0.349
-0.389
0.339
5.241
0.429
30.269

-0.689
0.065
0.191
0.284
0.313
1.119
6.107
4.979
31.933

-1.120
-0.197
0.036
0.236
0.396
0.944
2.453
3.239
3.585

-1.011
-0.258
0.000
0.227
0.455
0.965
2.146
2.460
2.673

a The equilibrium value of R used here is R = 0.988351 ˚. b The calculated CR-CC(2,3)/aug-cc-pVQZ total energies in
A
e
hartree. c Differences, in millihartree, between the actual CR-CC(2,3)/aug-cc-pVQZ and CR-CC(2,3)/aug-cc-pVTZ energies.
d Differences, in millihartree, between the calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies, where the latter
(B)

energies were generated by applying the correlation energy scaling factors χ4,3 (R) obtained with B = MP2, MP3, MP4D,
MP4DQ, MP4SDQ, CCSD, and CR-CC(2,3). The choice of B=CR-CC(2,3) is equivalent to the single-level PES extrapolation
scheme.

162

In summary, MP2 consistently fails to predict the proper information about the CRCC(2,3) correlation energy scaling, while MP3 typically does much better. This leads to the
conclusion that at least third-order correlation energy effects must be included for the proper
description of the correlation energy scaling in the near-equilibrium PES regions. In regions
further away from equilibrium, the third and even the partial fourth-order perturbative
corrections still cannot produce the proper correlation energy scaling factors that would be
compatible with those of CR-CC(2,3), which is a consequence of the inability of MBPT to
describe bond-breaking, but one can use the CCSD approach instead, which is qualitatively
correct at larger internuclear separations in single-bond breaking situations, providing a
reasonable estimate of the CR-CC(2,3) correlation energy scaling with the basis set.

4.4.3

Application to the Isomerization of Bicyclobutane to Butadiene

The results of dual-level PES extrapolations on the reaction profiles for the isomerization
of bicyclobutane to butadiene are given in Table (4.12). The format of this table is similar
to that of Tables (4.9)–(4.11), except that here the geometries of interest are the stationary points along the conrotatory and disrotatory pathways shown in Figure (3.2) and the
cc-pVXZ rather than aug-cc-pVXZ basis sets are employed throughout. From the results
presented in Table (4.12) it can be seen that the MP3 and various MP4 methods may be
successfully used to probe the CR-CC(2,3) correlation energy scaling for the energetically
favored conrotatory reaction profile, which consists entirely of species with correlation energy dominated by lower-order excitations, but in every case the MBPT methods produce a
significantly larger error for the highly biradical dis TS geometry. On the other hand, when
163

(CCSD)

χ4,3

(R) is used to extrapolate the CR-CC(2,3)/cc-pVQZ PES, the reported extrapo-

lation errors remain within a millihartree of the explictly calculated CR-CC(2,3)/cc-pVQZ
energy values at every stationary point, rivaling the sub-millihartree accuracies found using
(CR-CC(2,3))

the single-level PES extrapolation scheme where χ4,3

(R) is employed instead. This

is another case where a quasi-degeneracy, in this case resulting from the biradical nature of
the dis TS configuration, inhibits the MBPT methods from producing the correct correlation energy scaling information. It is clear, of the methods considered here, that only the
CCSD approach can offer a correlation energy scaling factor compatible with CR-CC(2,3),
although the extrapolation errors obtained with MP4SDQ, which are 1.950 millihartree or
∼ 1.5 kcal/mol for the strongly biradical dis TS structure (located over 60 kcal/mol above
the reactant) and less than 1 millihartree for the remaining structures, are excellent as well.
Also, once again, MP2 correlation energies do not contain sufficient information to model
the CR-CC(2,3) correlation energy scaling at any geometry.

164

Table 4.12: A comparison of calculated and extrapolated CR-CC(2,3)/cc-pVQZ energies at the stationary points defining the
conrotatory and disrotatory pathways characterizing the bicbut→t-but isomerization. The bicbut reactant defines the pivot
geometry Re and RHF defines the base wave function for the PES extrapolations. In all post-RHF calculations, the lowest
four orbitals, correlating with the 1s shells of the carbon atoms, were kept frozen.
(B)

ǫ4 (χ4,3 )c
Structure
bicbut
con TS
dis TS
g-but
gt TS
t-but

E4 a

∆E4,3 b

MP2

MP3

MP4D

MP4DQ

MP4SDQ

CCSD

CR-CC(2, 3)

-155.695497
-155.625392
-155.582481
-155.733236
-155.728073
-155.738043

-44.149
-43.727
-42.447
-43.909
-43.764
-43.963

-9.521
-10.169
-13.436
-9.730
-9.815
-9.614

-0.860
-1.578
-4.643
-0.879
-0.934
-0.786

-0.397
-0.993
-3.890
-0.325
-0.374
-0.234

0.845
0.292
-2.671
0.899
0.844
0.972

0.760
0.397
-1.950
0.906
0.837
0.987

0.582
0.466
0.096
0.927
0.876
1.008

0.000
0.245
0.631
0.482
0.410
0.563

a The calculated CR-CC(2,3)/cc-pVQZ total energies in hartree. b Differences, in millihartree, between the actual CRCC(2,3)/cc-pVQZ and CR-CC(2,3)/cc-pVTZ energies. c Differences, in millihartree, between the calculated and extrapolated

CR-CC(2,3)/cc-pVQZ energies, where the latter energies were generated by applying the correlation energy scaling factors
(B)

χ4,3 (R) obtained with B = MP2, MP3, MP4D, MP4DQ, MP4SDQ, CCSD, and CR-CC(2,3). The choice of B = CR-CC(2,3)
is equivalent to the single-level extrapolation scheme.

165

4.4.4

The Role of Pivot Geometries and Base Wave Functions in
Dual-Level PES Extrapolations

In Tables (4.9)–(4.12) it was clear that regardless of the correlation energy scaling factor
used in the PES extrapolation scheme based on Eqs. (4.1) – (4.3), the extrapolation errors
are largest in regions where the nature of the electron correlation effects differ most from
those found at the pivot geometry. To see how much the reported errors could be reduced by
employing additional pivot geometries, the same sets of PES extrapolations were considered,
but this time with all nuclear configurations treated as pivot geometries. In addition to
introducing a new extrapolation method which should yield improved accuracies, this approach provides a direct measure of the error introduced when lower-order correlation energy
scaling factors are employed to predict the results of higher-order calculations with a larger
basis set, since any error due to the earlier assumption of the approximate transferability
of the scaling factor from one geometry to another is eliminated. Additional calculations
required to perform these extrapolations, when compared to those required for the extrapolations of the previous section (done with a single pivot geometry), consist of the remaining
calculations required to obtain each PES using the aug-cc-pVQZ basis for the bond-breaking
curves or the cc-pVQZ basis for the bicbut→t-but isomerization pathways using the lowerlevel methodology B. This is still relatively inexpensive, since we never have to perform the
high-level CR-CC(2,3) calculation with the largest basis set employed at any geometry.
A comparison of Tables (4.9) and (4.13), (4.10) and (4.14), (4.11) and (4.15), and (4.12)
and (4.16) demonstrates the full extent of the accuracy which may be gained by using
additional pivot geometries for each of the systems considered in this study. The most
interesting detail to note in these comparisons is that when the same method is used to
166

predict the correlation energy scaling factor, increasing the number of pivot geometries does
not necessarily result in an overall improvement in the quality of the extrapolated surface.
Extrapolation errors are, for the most part, improved when the number of pivot geometries
is increased, especially when MP4SDQ and CCSD are used to generate χ, but there are no
significant benefits from switching from the previously discussed single-point approach to its
multi-point analog.

As stated before, the many options inherently included in the PES extrapolation scheme
based on Eqs. (4.1) – (4.3) allow one to tailor it to make it more accurate or more affordable,
as required by a given application. To demonstrate clearly and concisely the different levels
of accuracy and savings in the computer effort which may be obtained using different tiers
of the PES extrapolation scheme, the results of four different PES extrapolations examined
here, using the bicbut→t-but isomerization as an example, are collected in Table (4.17). We
recall that the goal of each extrapolation in this case is to predict the CR-CC(2,3)/cc-pVQZ
PES (as represented by six structures on the corresponding conrotatory and disrotatory
pathways) from the results of lower-level calculations. The most efficient PES extrapolation
considered for this table, where a lower-order MP4SDQ scaling factor is used to extrapolate
the CR-CC(2,3)/cc-pVTZ PES to the level of CR-CC(2,3)/cc-pVQZ calculations using one
pivot geometry (bicbut) and RHF base energies, requires only 5% of the CPU time of the
conventional method and the mean unsigned error (MUE) is slightly below 1 millihartree.
The MUE is reduced by 0.314 millihartree when the correlation energy scaling factor obtained
with MP4SDQ is replaced by that produced by CCSD, but the required CPU time is also
increased to 10% of that required by the conventional CR-CC(2,3)/cc-pVQZ calculations.
The results obtained with the original single-level PES extrapolation procedure are also
167

presented, where the time required to perform these calculations is 22% of that required
conventionally, but the mean unsigned error falls to below 0.4 millihartree. Finally, when
(CR-CC(2,3))

the base energy is obtained in CCSD calculations and χ4,3

(R) is used to scale

the remaining correlation energy, an MUE of 0.170 is attained, but now the computational
time savings amounts only to about one third of the conventional procedure. It can be seen
in Table (4.17) that regardless of the PES extrapolation approach used, the MUE remains
below 1 millihartree, which is a relatively insignificant loss in accuracy compared to the
conventional, point-wise CR-CC(2,3)/cc-pVQZ calculations. By using the dual-level PES
extrapolation scheme to extrapolate CR-CC(2,3)/cc-pVQZ energies, we have reduced the
time required to accurately construct a PES for this problematic polyatomic isomerization
by more than an order of magnitude. These computational savings would only grow larger
if another system were considered which contained a greater number of electrons or a larger
number of points on the PES were considered.

168

Table 4.13: A comparison of calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies of the H2 O molecule, in which
one of the two O-H bonds R is stretched, while keeping the other O-H bond at the equilibrium length and the H-O-H angle
fixed at 104.5 ◦ . Each geometry serves as its own pivot geometry and RHF defines the base wave function for the PES
extrapolations. In all post-RHF calculations, the lowest orbital, correlating with the 1s orbital of the oxygen atom, was kept
frozen.
(B)

ǫ4 (χ4,3 )d
R/Re a
0.75
0.90
1.00
1.10
1.25
1.50
2.00
3.00
4.00

E4 b

∆E4,3 c

MP2

MP3

MP4D

MP4DQ

MP4SDQ

CCSD

-76.259133
-76.351620
-76.363051
-76.355931
-76.329620
-76.276456
-76.199626
-76.162662
-76.161060

-22.676
-21.605
-21.059
-20.641
-20.278
-19.952
-19.168
-18.479
-18.303

-2.766
-2.737
-2.742
-2.769
-2.835
-3.072
-4.107
-5.201
-3.619

-0.504
-0.534
-0.576
-0.636
-0.733
-0.951
-1.651
-1.501
1.054

-0.360
-0.391
-0.428
-0.475
-0.538
-0.668
-1.073
0.009
3.143

-0.014
-0.016
-0.033
-0.060
-0.096
-0.190
-0.591
0.207
2.604

0.129
0.138
0.131
0.114
0.097
0.033
-0.301
0.856
4.400

0.059
0.063
0.054
0.041
0.034
0.021
-0.080
0.101
0.164

a The equilibrium value of R used here is R = 0.95785 ˚. b The calculated CR-CC(2,3)/aug-cc-pVQZ total energies in
A
e
c Differences, in millihartree, between the actual CR-CC(2,3)/aug-cc-pVQZ and CR-CC(2,3)/aug-cc-pVTZ energies.
hartree.
d Differences, in millihartree, between the calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies, with the latter
(B)

energies generated by applying the correlation energy scaling factors χ4,3 (R) obtained with B = MP2, MP3, MP4D, MP4DQ,
MP4SDQ, and CCSD.

169

Table 4.14: A comparison of calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies for several internuclear separations R of the HCl molecule. Each geometry serves as its own pivot geometry and RHF defines the base wave function for
the PES extrapolations. In all post-RHF calculations, the lowest five orbitals, correlating with the 1s, 2s, and 2p shells of Cl,
were kept frozen.
(B)

ǫ4 (χ4,3 )d
R/Re a
0.75
0.90
1.00
1.10
1.25
1.50
2.00
3.00
4.00

E4 b

∆E4,3 c

MP2

MP3

MP4D

MP4DQ

MP4SDQ

CCSD

-460.245772
-460.351216
-460.364178
-460.356856
-460.329720
-460.277589
-460.212229
-460.192428
-460.192532

-22.053
-20.979
-20.704
-20.550
-20.327
-19.823
-18.747
-18.345
-17.809

-2.707
-2.625
-2.610
-2.680
-2.885
-3.370
-5.151
-5.225
-2.748

-0.044
-0.087
-0.127
-0.220
-0.404
-0.739
-1.906
-0.655
2.016

0.123
0.056
0.017
-0.062
-0.204
-0.426
-1.204
0.927
3.643

0.516
0.466
0.439
0.373
0.249
0.046
-0.736
0.985
1.988

0.491
0.453
0.437
0.384
0.284
0.146
-0.265
4.375
11.510

0.447
0.424
0.427
0.401
0.362
0.386
0.336
0.933
0.533

a The equilibrium value of R used here is R = 1.27455 ˚. b The calcuated CR-CC(2,3)/aug-cc-pVQZ total energies in
A
e
hartree. c Differences, in millihartree, between the actual CR-CC(2,3)/aug-cc-pVQZ and CR-CC(2,3)/aug-cc-pVTZ energies.
d Differences, in millihartree, between the calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies, with the latter
(B)

energies generated by applying the correlation energy scaling factors χ4,3 (R) obtained with B = MP2, MP3, MP4D, MP4DQ,
MP4SDQ, and CCSD.

170

Table 4.15: A comparison of calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies for several internuclear separations R of the F2 molecule. Each geometry serves as its own pivot geometry and RHF defines the base wave function for
the PES extrapolations. In all post-RHF calculations, the lowest two orbitals, correlating with the 1s orbitals on the fluorine
atoms, were kept frozen.
(B)

ǫ4 (χ4,3 )d
R/Re a
0.75
0.90
1.00
1.10
1.25
1.50
2.00
3.00
4.00

E4 b

∆E4,3 c

MP2

MP3

MP4D

MP4DQ

MP4SDQ

CCSD

-199.196897
-199.349657
-199.364732
-199.356794
-199.333764
-199.307122
-199.296727
-199.297648
-199.297985

-56.537
-53.009
-51.743
-51.172
-50.464
-49.741
-49.300
-49.368
-49.422

-3.415
-3.570
-3.835
-4.116
-4.758
-5.496
-3.680
4.354
9.504

-1.068
-1.389
-1.737
-2.030
-2.539
-2.676
1.008
11.655
17.002

-0.764
-0.877
-1.028
-1.063
-1.062
-0.019
6.381
18.778
22.892

-0.128
-0.215
-0.378
-0.466
-0.646
-0.237
3.741
-4.021
30.905

0.310
0.290
0.191
0.176
0.107
0.664
4.751
2.203
32.328

0.190
0.149
0.036
0.009
-0.115
-0.119
0.250
0.723
0.876

a The equilibrium value of R used here is R = 0.988351 ˚. b The calculated CR-CC(2,3)/aug-cc-pVQZ total energies in
A
e
hartree. c Differences, in millihartree, between the actual CR-CC(2,3)/aug-cc-pVQZ and CR-CC(2,3)/aug-cc-pVTZ energies.
d Differences, in millihartree, between the calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies, with the latter
(B)

energies generated by applying scaling factors χ4,3 (R) obtained with B = MP2, MP3, MP4D, MP4DQ, MP4SDQ, and
CCSD.

171

Table 4.16: A comparison of calculated and extrapolated CR-CC(2,3)/aug-cc-pVQZ energies at the stationary points defining
the conrotatory and disrotatory pathways characterizing the bicbut→t-but isomerization. Each geometry serves as its own
pivot geometry and RHF defines the base wave function for the PES extrapolations. In all post-RHF calculations, the lowest
four orbitals, correlating with the 1s orbitals of the carbon atoms, were kept frozen.
(B)

ǫ4 (χ4,3 )c
Structure
bicbut
con TS
dis TS
gbut
gt TS
tbut

E4 a

∆E4,3 b

MP2

MP3

MP4D

MP4DQ

MP4SDQ

CCSD

-155.695497
-155.625392
-155.582481
-155.733236
-155.728073
-155.738043

-44.149
-43.727
-42.447
-43.909
-43.764
-43.963

-9.521
-10.455
-14.074
-10.245
-10.366
-10.220

-0.860
-1.614
-4.825
-1.009
-1.085
-1.007

-0.397
-1.036
-4.164
-0.473
-0.548
-0.472

0.845
0.322
-2.824
0.781
0.702
0.776

0.760
0.373
-2.131
0.728
0.652
0.729

0.582
0.295
-0.529
0.699
0.641
0.703

a The calculated CR-CC(2,3)/cc-pVQZ total energies in hartree. b Differences, in millihartree, between the actual CRCC(2,3)/cc-pVQZ and CR-CC(2,3)/cc-pVTZ energies. c Differences, in millihartree, between the calculated and extrapolated
(B)

CR-CC(2,3)/cc-pVQZ energies, with the latter energies generated by applying the correlation energy scaling factors χ4,3 (R)
obtained with B = MP2, MP3, MP4D, MP4DQ, MP4SDQ, and CCSD.

172

Table 4.17: A summary of the necessary calculations and the corresponding computer resources required to utilize different
tiers of the PES extrapolation scheme based on Eqs. (4.1) – (4.3) to scale the bicyclobutane isomerization pathway from the
CR-CC(2,3)/cc-pVTZ level of theory to the CR-CC(2,3)/cc-pVQZ level, along with the corresponding extrapolation errors.
The bicbut structure is used to provide the pivot geometry.
Base
Energya
RHF
RHF
RHF
CCSD

Correlation Energy cc-pVQZ Calculations Required c CPU Time

Mean Unsigned

Scaling Factorb

RHF

CCSD

CR-CC(2,3)

(t/tconv )d

Error (millihartree)e

MP4SDQ
CCSD
CR-CC(2, 3)
CR-CC(2, 3)

6
6
6
6

0
1
1
6

0
0
1
1

0.05
0.10
0.22
0.63

0.973
0.659
0.389
0.170

6

6

6

1

—

Conventional Calculationf

a The method used to generate the base energy. b The method B used to generate the correlation energy scaling factor χ(B) (R)
4,3
in Eq. (4.1). c The number of cc-pVQZ basis set calculations which must be performed using a given base wave function and
a given correlation energy scaling factor to extrapolate the CR-CC(2,3)/cc-pVQZ PES. d The CPU time needed to perform

the necessary calculations for each PES extrapolation type relative to the time needed to generate the true CR-CC(2,3)/ccpVQZ PES. e The mean unsigned error representing an extrapolated CR-CC(2,3)/cc-pVQZ reaction pathway generated using
the designated base energy and correlation energy scaling factor. f Characteristics of the true PES calculation, in which
each stationary point on the conrotatory and disrotatory pathways of the bicbut→t-but isomerization is calculated at teh
CR-CC(2,3)/cc-pVQZ level.

173

Chapter 5
Development of Computer Codes for
the GAMESS software Package
A large portion of the current doctoral effort was devoted to writing EOMCC computer
codes for the GAMESS software package. In this section, we discuss the highly efficient
GAMESS implementations of the open-shell EOMCCSD and IP-EOMCCSD(2h-1p) methods
developed as part of this thesis project, based on theory discussed in Sect. (3.1.2) and the
corresponding factorized equations, in terms of recursively generated intermediates that lead
to the vectorized computer codes through the use of fast matrix multiplication rountines
from the BLAS library. The open-shell EOMCCSD and IP-EOMCCSD(2h-1p) codes were
interfaced with previously existing ROHF and RHF/ROHF integral routines, respectively,
available in the GAMESS software package [194], as well as the CC programs and routines
for the generation of matrix elements of the similarity-transformed Hamiltonian of CCSD
originally developed for GAMESS by the Piecuch group at Michigan State University. In
Sect. (5.1), we begin our discussion of the implementation of these programs, with specific
174

details outlined for the open-shell EOMCCSD and IP-EOMCCSD(2h-1p) methods in Sects.
(5.1.1) and (5.1.2), respectively.

5.1

Key Details of Efficient Computer Implementations

Both the open-shell EOMCCSD and IP-EOMCCSD(2h-1p) codes must begin by solving the
usual CCSD equations for the ground-state of the N -electron reference system in order to
ij

obtain the singly and doubly excited cluster amplitudes, ti and tab , respectively. In both
a
cases, this is done using the general ROHF-based CCSD codes included in GAMESS that
work for closed- and open-shells, developed by the Piecuch group and described in [130].
ij

Following the CCSD calculation, the converged ti and tab amplitudes are used to contruct
a
the one- and two-body matrix elements of the CCSD similarity-transformed Hamiltonian
¯
¯ (CCSD) ¯ q
HN,open , hp and hrs , respectively, which define the one- and two-body components of
pq
¯ (CCSD)
HN,open within the second quantized formalism,
¯p
¯ (CCSD) = hq ap aq ,
H1

(5.1)

¯
¯ (CCSD) = 1 hrs N [ap aq as ar ],
H2
4 pq

(5.2)

and

respectively. N [. . .] is the normal product of the operators between the brackets and the
Einstein summation convention over repeated upper and lower indices is assumed throughout. The explicit equations defining these matrix elements in terms of the matrix elements
q

rs
of the Hamiltonian in the normal-ordered form fp = p|f |q and vpq = pq|v|rs − pq|v|sr ,
ij
and CCSD cluster amplitudes ti and tab , are given in Table (5.1). Once these common
a

175

initial steps are complete, the appropriate expressions for solving the EOMCCSD and IPEOMCCSD(2h-1p) eigenvalue problems have to be constructed. These steps, which are
specific to the open-shell EOMCCSD and IP-EOMCCSD(2h-1p) codes considered here, are
outlined in Sects. (5.1.1) and (5.1.2), respectively. Once the suitable EOMCCSD and IPEOMCCSD(2h-1p) equations are constructed, another common feature in our implementation of the EOMCCSD and IP-EOMCCSD(2h-1p) approaches is that we rely on the HiraoNakatsuji generalization [303] of the Davidson diagonalization algorithm [304] to solve the
resulting non-Hermitian eigenvalue problems, Eqs. (3.24) and (3.25) in Sect. (3.1.2.1).

5.1.1

Standard Equation-of-Motion Coupled-Cluster Theory with
Singles and Doubles for Open-Shell Systems

¯ (CCSD)
The left-hand sides of the EOMCCSD equations are calculated by projecting [HN,open
α
(Rµ,1 + Rµ,2 )]C |Φ onto the subspace of all singly- and doubly-excited determinants, Φaα |,
i
aβ

aβ bβ

aα bβ

aα α
Φi |, Φiα jbα |, Φi j |, and Φiα j |, to obtain the following expressions:
β
β β
β

α ¯ (CCSD)
Φaα |[HN,open (Rµ,1 + Rµ,2 )]C |Φ
i

i
mα
i
¯
¯
¯
= heα reα − hiαα raα + heαα raα mα
aα α
m
m
α eα
e

i e

m

β
m
¯ β i
¯i
¯α β
+hmβ raα mα + haα eαα reαα + haα mβ reβ
eβ
αm
β
i f

e f

m
i
1¯ α
1¯
− 2 hmααα ra α nα + 2 haα nα reα nα
n
α α α fα
α fα
iα fβ

mα nβ

eα fβ iα nβ

¯
¯
−hmα nβ ra f + haα nβ re f ,
α β
α β
176

(5.3)

Table 5.1: Explicit algebraic expressions for the one- and two-body matrix elements of
¯
¯ (CCSD) ¯ q
HN,open (hp and hrs , respectively) taken from Refs. [126, 132].
pq
Intermediate
¯
ha

i
j
¯
hi
¯
hb
a
¯
hbc
ai
¯ ka
hij
¯
hcd
ab
¯
hkl
ij
¯ jb
hia
¯
hic
ab

¯ jk
hia
′b
Ia
b
Ia
′jb

Iia

Expressiona
ae
fia + vim tm
e
ef mj
j
je
¯ j
fi + vim tm + 1 vmi tef + he te
e
2
i
b − hb tm
Ia ¯ m a
bc
bc
vai − vmi tm
a
ka
ea
vij + vij tk
e
cd
cd
cd
¯
vab + 1 vmn tmn − hcd tm + vbm tm
am b
a
2
ab
ef
ke
kl
¯
vij + 1 vij tkl − hle tk + vij tl
e
ij e
2
ef
′jb
jm
¯ jb
I − v eb tea − h tm

im
ia
im a
ic + v ec ti − hic tm + I ′ic tm − hc tim +
¯
¯
vab
ma b
m ab
ab e
mb a
ce tim − v ce tim + 1 hic tnm
¯
¯
hbm ae
am be
2 nm ab
jk
ke j
¯ jk
¯ je
via + hmi tm − via te + A jk him tkm
a
ae
¯ e tjk + I ′je tk − 1 v ef tjk
+hi ea
2 ai ef
ia e
b + v be tm
fa
am e
′b − 1 v eb tmn
Ia
2 mn ea
jb
eb j
via + via te

a Summation over repeated upper and lower indices is assumed. f q = p|f |q and v rs =
p
pq

pq|v|rs − pq|v|sr are the one- and two-body matrix elements of the Hamiltonian in the
ij
normal-ordered form (one- and two-electron integrals), and the ti and tab are the singly
a
and doubly excited cluster amplitudes defining the ground-state CCSD wave function of the
N -electron reference system. The antisymmetrizer Apq = 1 − (pq) operator is also used,
where (pq) is the transposition of indices p and q.

177

a

β ¯ (CCSD)
Φi |[HN,open (Rµ,1 + Rµ,2 )]C |Φ
β

e

i

i

m

e

i m

β
¯ β β ¯β
¯ β β β
= haβ reβ − hmβ raβ + hmβ raβ eβ
i m

i e

m

i e

β α
β
¯
¯β β
¯β α m
+heαα raβ eα + haβ mβ reβ + haβ mα reαα
m
iβ fβ

mβ nβ

eβ fβ iβ nβ

1¯
¯
− 1 hmβ nβ ra f + 2 haβ nβ re f
2
β β
β β
i f

e f

m n

i n

β α
¯ β α β α
¯β α
−hmβ nα ra f + haβ nα re f ,
β α
β α

α α ¯ (CCSD)
Φaα jbα |[HN,open (Rµ,1 + Rµ,2 )]C |Φ
i

(5.4)

m j
i j
¯
¯
= −Aiα jα hiαα ra α α + Aaα bα heα reα bα
m
aα
b
α α

α α

mα
¯e j i
¯ iα j
−Aaα bα hm α raα + Aiα jα haα bα reα
α α α
α bα
e f

i j

i j

m
1¯ α
1¯
+ 2 haα bα reα fα + 2 hmαα α ra α nα
n
α α α α
α bα
m j
¯ iα e
−Aiα jα Aab hmααα re α α
a
b

α α
i e
mβ jα
¯α β
+Aiα jα Aab haα mβ re b
β α

mβ
i j e
m
¯ iα e j
¯αα β
−hm α α reαα + ha b m reβ
α α β
α aα bα
m
¯ iα j f
− 1 Aaα bα hm α α ra α nα
2
α bα nα α fα
iα jα fβ

mα nβ

¯
−Aaα bα hm b n ra f
α α β α β

e j f
α α α α α
e j f i n
¯ αα β α β
−Aiα jα ha b n re f ,
α α β α β

i
1
¯
+ 2 Aiα jα haα bα nα reα nα
f

178

(5.5)

a b

β β ¯ (CCSD)
Φi j |[HN,open (Rµ,1 + Rµ,2 )]C |Φ
β β

iβ

mβ jβ

eβ iβ jβ

mβ

eβ jβ iβ

¯
¯
= −Aiβ jβ hmβ ra b + Aa b haβ re b
β β
β β
β β
iβ jβ

¯
¯
−Aa b hm b raβ + Aiβ jβ ha b reβ
β β
β β
β β
eβ fβ iβ jβ

iβ jβ

mβ nβ

1¯
¯
+ 1 ha b re f + 2 hmβ nβ ra b
2 β β β β
β β
iβ eβ

mβ jβ

iβ eα

mα jβ

¯
−Aa b Aiβ jβ hmβ aβ re b
β β
β β

¯
+Aa b Aiβ jβ haβ mα re b
α β
β β
i e j

m

i j e

β
¯β β α m
¯β β β
−hm a b reβ + ha b m reαα
β β α
β β β
iβ jβ fβ

mβ nβ

1
¯
− 2 Aa b hm b n ra f
β β
β β β β β
iβ jβ fα

mβ nα

¯
−Aa b hm b n ra f
β β
β β α β α

eβ jβ fβ iβ nβ

1
¯
+ 2 Aiβ jβ ha b n re f
β β β β β
eβ jβ fα iβ nα

¯
−Aiβ jβ ha b n re f ,
β β α β α

179

(5.6)

and

a b

α β
¯ (CCSD)
Φiα j |[HN,open (Rµ,1 + Rµ,2 )]C |Φ
β

j

m j

i j

i n

α β
α β
¯ β α β ¯
¯
= −hiαα ra b − hnβ ra b + heα re b
aα α
m
α β
α β
β
i j

i j

i j

f

n

¯ α β mα ¯ α β β
¯ β αβ
+hb ra f − hm b raα − haα nβ rb
α β
α β
β
β
e j

i f

j

¯ αβ i
¯α β β
+ha b reα + ha b rf
α
α β
α β β
eα fβ iα jβ

iα jβ

mα nβ

e j

i n

i f

m j

i f

n j

e j

¯
¯
+ha b re f + hmα nβ ra b
α β α β
α β

α β
¯ αβ α β ¯α β
−haα nβ re b − hm b ra f
α β α β
α β

¯α β β β ¯ αβ m
+haα nβ rf b + hm b reαα iα
aα
α β
β β
i e j

i j f

n

¯α αβ m
¯αβ β β
−hm a b reαα − ha n b rf
α β β
α β β β
m j

f j

i n

α β
¯ iα e
¯ β β α β
−hmααα re b − hb n ra f
a
α β
β β α β
i j f

i j e

n

¯αβ β β
¯αβ α m
+ha b m reαα + ha b n rf
α β β β
α β α
i j f

i j f

m n

iα jβ eβ

mβ nβ

α β
1¯ α β α m
¯αβ β
− 2 hm b n ra α nα − hm b n ra f
α β β α β
α β α α fα
iα jβ eα

mα nβ

1¯
¯
−haα nβ mα re b − 2 haα nβ mβ re b
α β
β β
e j f

e j f

i n

1¯ α β α i
¯ αβ β α β
+ 2 ha b n reα nα + ha b n re f
α β α α fα
α β β α β
iα fβ eα

jβ mα

¯
+ha b m rf e
α β α β α
iα fβ eβ

jβ mβ

1¯
+ 2 ha b m rf e ,
α β β β β

180

(5.7)

where the antisymmetrizer Apq = 1 − (pq) is used, with (pq) designating the transposition
of indices p and q, and

i j f
eα fα iα jβ
¯αβ α
hm b n = vmα nα te b ,
α β
α β α
i j f
eα fβ iα jβ
¯αβ β
hm b n = vmα nβ te b ,
α β β
α β
i j e

e f

i j

α β α β
¯αβ α
haα nβ mα = vmα nβ ta f ,
α
i j e
¯αβ β
haα nβ mβ
e j f

¯ αβ α
ha b n

α β α
eα jβ fβ
¯
ha b n
α β β

β
eβ fβ iα jβ
= vmβ nβ ta f ,
α β
eα fα mα jβ
= vmα nα ta b ,
α β
eα fβ

mα jβ

¯α β α
ha b m

i f e

eα fβ

iα nβ

i f e

eβ fβ

iα nβ

= vmα nβ ta b ,
α β

= vmα nβ ta b ,
α β α
α β

and

¯α β β
ha b m

= vmβ nβ ta b ,
α β β
α β

(5.8)

where the µ subscript was dropped from the r amplitudes for clarity. By substituting the
three-body components of the similarity transformed Hamiltonian of CCSD given in Eq.
(5.8) and factorizing the resulting equations, the open-shell EOMCCSD equations projected
on doubly excited determinants, Eqs. (5.5), (5.6), and (5.7), may be rewritten in the following

181

way:

aα α ¯ (CCSD)
Φiα jbα |[HN,open (Rµ,1 + Rµ,2 )]C |Φ

m j

i j

1¯
¯
= Aiα jα Aaα bα (− 1 hiαα ra α α + 2 heα reα bα
aα α α
2 m
α bα
mα
1 ¯e j i
¯ iα j
− 1 hm α raα + 2 haα bα reα
2
α bα
α α α
e f

i j

i j

m
1¯ α
¯
+ 1 haα bα reα fα + 8 hmαα α ra α nα
n
8 α α α α
α bα
i e
mβ jα
m j
¯α β
¯ e
−hiαααα re α α + haα mβ re b
m a
b
β α

α α

i j

m j

1
1
− 2 χmα ta α α + 2 χeα teα bα )
aα α α
iα
α bα

a b

β β ¯ (CCSD)
Φi j |[HN,open (Rµ,1 + Rµ,2 )]C |Φ
β β

iβ

mβ jβ

(5.9)

eβ iβ jβ

1¯
¯
= Aa b Aiβ jβ (− 2 hmβ ra b + 1 haβ re b
2
β β
β β
β β
iβ jβ

mβ

eβ jβ iβ

1¯
1¯
− 2 hm b raβ + 2 ha b reβ
β β
β β
eβ fβ iβ jβ

iβ jβ

mβ nβ

1¯
¯
+ 8 ha b re f + 1 hmβ nβ ra b
8
β β β β
β β
i e

m j

iβ

mβ jβ

i e

m j

α β
β β
¯β β
¯β α
−hmβ aβ re b + haβ mα re b
α β
β β
eβ iβ jβ

1
1
− 2 χmβ ta b + 2 χaβ te b )
β β
β β

182

(5.10)

and

a b

α β
¯ (CCSD)
Φiα j |[HN,open (Rµ,1 + Rµ,2 )]C |Φ
β

m j

j

i j

i n

α β
α β
¯ β α β ¯
¯
= −hiαα ra b − hnβ ra b + heα re b
aα α
m
α β
α β
β
i j

i j

i j

f

n

¯ α β mα ¯ α β β
¯ β αβ
+hb ra f − hm b raα − haα nβ rb
α β
α β
β
β
e j

i f

j

¯ αβ i
¯α β β
+ha b reα + ha b rf
α
α β
α β β
eα fβ iα jβ

iα jβ

mα nβ

e j

i n

i f

m j

i f

n j

e j

¯
¯
+ha b re f + hmα nβ ra b
α β α β
α β

α β
¯ αβ α β ¯α β
−haα nβ re b − hm b ra f
α β α β
α β

¯α β β β ¯ αβ m
+haα nβ rf b + hm b reαα iα
aα
α β
β β
m j

f j

i n

α β
¯ iα e
¯ β β α β
−hmααα re b − hb n ra f
a
α β
β β α β
mα jβ

iα jβ

−χiαα ta b + χeα te b
m
aα α
α β
β
jβ iα nβ

fβ iα jβ

−χnβ ta b + χb ta f
α β
β α β

(5.11)

where

iα fβ

i f

nβ

e f

α n
α n
χiαα = −vnα mα rf α + vmα nβ rf − vnα mα rf α tiα
m
α
α
α eα
α
β
eα fβ

nβ

eα fβ

e f

iα nβ

α α i
+vmα nβ rf tiα + 1 vmα nα reα nα + vmα nβ re f ,
eα
2
α β
α fα
β

eα fβ nβ

f e

(5.12)

e f

n
α
α n
χeα = −vaα nα rf α + vaα nβ rf + vnα mα rf α tmα
aα
α α
α
α aα
β
eα fβ

nβ

e f

eα fβ

mα nβ

1 α α m
−vmα nβ rf tmα − 2 vmα nα ra α nα − vmα nβ ra f ,
a
α β
α fα
β α

183

(5.13)

iβ

iβ fβ

iβ fα

nβ

eβ fβ

nβ iβ

n
χmβ = −vnβ mβ rf + vmβ nα rf α − vnβ mβ rf teβ
α
β
β
eβ fα

iβ

eβ fβ

iβ nβ

eβ fα

iβ nα

n
1
+vmβ nα rf α teβ + 2 vmβ nβ re f + vmβ nα re f ,
α
β β
β α

(5.14)

and

eβ

fβ eβ nβ

eβ fα

eβ fβ

nβ mβ

n
χaβ = −vaβ nβ rf + vaβ nα rf α + vnβ mβ rf taβ
α
β
β
eβ fα

mβ

eβ fβ

mβ nβ

eβ fα

mβ nα

n
−vmβ nα rf α taβ − 1 vmβ nβ ra f − vmβ nα ra f ,
2
α
β α
β β

(5.15)

which is the final form of the open-shell EOMCCSD equations used in the efficient vectorized
GAMESS code. Once the singly and doubly excited amplitudes defining the EOMCCSD
(CCSD)

ij

i
excitation operator, rµ,a and rµ,ab , respectively, and the vertical excitation energy ωµ

have been determined by solving Eqs. (3.24) (3.25), rµ,0 is calculated a posteriori from the
following expression:

(CCSD)
¯ (CCSD) (CCSD) + R(CCSD) )]C |Φ /ωµ
rµ,0 = Φ|[HN,open (Rµ,1
.
µ,2

5.1.2

(5.16)

Electron-Attached and Ionized Equation-of-Motion CoupledCluster Theories

The key difference between the open-shell EOMCC theory and EA- or IP-EOMCC is the sector of the Fock space the similarity-transformed Hamiltonian of CCSD is diagonalized within.
As an example, in the IP-EOMCC approaches we diagonalize the similarity-transformed
Hamiltonian obtained in calculations for an N -electron reference system in the sector of
the Fock space corresponding to (N − 1) electrons. Thus, the left-hand sides of the IP184

EOMCCSD(2h-1p) eigenvalue problem,

(CCSD) (2h-1p)

¯
(HN,open Rµ

(2h-1p) (2h-1p)
Rµ
|Φ

)C |Φ = ωµ

(5.17)

bβ

are obtained by projecting Eq. (5.17) onto all Φiβ |, Φi j |, and Φi bjα | determinants. In
β α
β β
this way the following equations are obtained:

¯ (CCSD) (2h-1p)
Φiβ |(HN,open RN,open )C |Φ

i
iβ mα
e
i m
¯β m
¯
¯ β β β
= −hmβ r β + heαα reα
+ hmβ reβ
m
iβ fα

mβ nα

¯
−hmβ nα rf

α

bβ
¯ (CCSD) (2h-1p)
Φi j |(HN,open RN,open )C |Φ
β β

−

jβ iβ nβ

¯
= −Aiβ jβ hnβ rb
β

1 hiβ fβ r mβ nβ ,
¯
2 mβ nβ fβ

(5.18)

fβ iβ jβ

¯
+ hb rf
β β

i j
i m
e j
¯β β m
¯ β α β α
−hm b r β + Aiβ jβ hm b reα
β α
β β
eβ jβ

iβ mβ

¯
−Aiβ jβ hb m reβ
β β
eβ fβ

i j
mβ nβ
¯β β
+ 1 hmβ nβ reβ
2

iβ jβ mβ nβ

− 1 vmβ nβ te b rf
2
β β β
eβ fα

iβ jβ mβ nα

−vmβ nα te b rf
β β α

185

,

(5.19)

and

¯ (CCSD) (2h-1p)
Φi bjα |(HN,open RN,open )C |Φ
α
β

i j
iβ nα
¯f β α
+ hb α rf
α
α α
i j
iβ mβ jα
¯β α m
¯
− hm b r β
−hmβ rb
α
β α
j

¯
= −hnα rb
α

eβ bα

iβ mβ

¯ α
− hb αm reα
α α

iβ eα

mβ jα

¯
+ hmβ nα rb

¯
−hm jα reβ
β

¯
−hm b reα
β α
eβ fβ

e j

iβ mα

iβ jα

mβ nα
α

iβ jα mβ nβ

1
− 2 vmβ nβ te b rf
β α β
eβ fα

iβ jα mβ nα

−vmβ nα te b rf
β α α

.

(5.20)

As in the case of EOMCCSD, the IP-EOMCCSD(2h-1p) equations are solved using the
Hirao-Nakatsuji algorithm [303].

186

Chapter 6
Summary and Concluding Remarks
In this dissertation, we addressed the problem of generating highly accurate potential energy
surfaces (PESs) for reactive processes by introducing and demonstrating the performance of
electronic structure methodologies that can provide a balanced description of chemical species
with varying levels of electronic degeneracy, but are also practical enough to be applied to a
wide range of chemical problems, as well as extrapolation techniques which facilitate the generation of PESs corresponding to high-level electronic structure calculations in a much more
efficient manner than that offered by conventional and laborious point-wise computations.
In particular, we examined the performance of two classes of coupled-cluster (CC) methods
which are capable of accounting for the diverse electron correlation effects encountered in the
majority of ground- and excited-state PES considerations. The first class of methods consisted of the size-extensive completely renormalized (CR) CC approaches for ground states
and their equation-of-motion (EOM) CC extensions for excited states, in which noniterative
corrections due to higher-order correlation effects are added to the energies obtained with
the standard CC and EOMCC approximations, such as CCSD or EOMCCSD, respectively.
187

We showed that the left-eigenstate CR-CC(2,3) and CR-EOMCC(2,3) methods that belong
to this category offer excellent performance for a diverse range of applications, including
a benchmark database of barrier heights for thermochemical kinetics, a pair of bimolecular association mechanisms involving the ozone molecule, competing intramolecular reaction
mechanisms describing the isomerization of bicyclobutane to butadiene, and the ground- and
excited-state PES cuts for the water molecule. When necessary, corrections for quadruple
excitations were also included via the CR-CC(2,3)+Q method which usually improved the
performance of the CR-CC(2,3) methods from chemical to sub-chemical accuracies for many
of the studied systems. A new variant of the CR-EOMCC(2,3) method was also presented
and discussed, namely, the δ-CR-EOMCC(2,3) approach that can provide a size-intensive
treatment of excitation energies. This method was applied to describe excitation energies
and hydrogen-bonding-induced spectral shifts in complexes of 7-Hydroxyquinoline with considerable success, helping to explain problems with time-dependent density functional theory.
The second class of methods considered here were the active-space variants of the electron
attached (EA) and ionized (IP) EOMCC theories. The EA- and IP-EOMCC approaches
were shown to be an excellent alternative to open-shell CC and EOMCC methods and their
perturbative extensions for describing open-shell molecular systems, providing spin-adapted
results while their active-space variants proved to be extremely efficient, significantly reducing the costs of the high-level parent EA- and IP-EOMCC approximations without sacrificing
accuracy. We also developed a general strategy for reducing the cost of generating PESs with
correlated electronic structure methods via the concept of correlation energy scaling. In order
to demonstrate typical accuracies one may expect when using the two types of PES extrapolation schemes presented here, namely, the single-level and dual-level schemes, a number

188

of benchmark applications were presented, such as the previously mentioned bicyclobutane
isomerization and single-bond breaking potential energy curves for the H2 O, HCl, and F2
molecules. The single-level extrapolation schemes were shown to reproduce PESs obtained
in laborious high-level point-by-point computations to within fractions of a millihartree in
most cases, even when used to extrapolate the PES to the CBS-limit. Meanwhile, the duallevel PES extrapolation schemes were shown to be capable of producing similar accuracies
at a tiny fraction of the computational cost of their single-level analogs. The insensitivity of
the results to the choice of pivot geometry and improvements in accuracy available when a
higher-order base wave function is chosen were also demonstrated. Finally, the development
of new open-shell EOMCCSD and IP-EOMCCSD(2h-1p) computer codes for the GAMESS
software package, along with the corresponding programmable equations, was discussed.

189

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