DEVELOPMENT AND APPLICATIONS OF SEMI-STOCHASTIC
COUPLED-CLUSTER METHODS FOR OPEN-SHELL SYSTEMS AND ELECTRONIC
EXCITATIONS IN MOLECULES

By

Arnab Chakraborty

A DISSERTATION

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

Chemistry—Doctor of Philosophy

2024

ABSTRACT

It is well established that the exponential wave function ansatz of coupled-cluster (CC) theory

and its equation-of-motion (EOM) extension to excited states are among the most appealing

ways to describe the electronic structure of molecules. One of the key challenges in the

development of the CC and EOMCC methodologies is the incorporation of many-electron

correlation effects due to higher–than–two-body components of the cluster and EOM ex-

citation operators, needed to achieve a quantitative description, without running into the

usually prohibitive computational costs of the CC approach with singles, doubles, and triples

(CCSDT) and its excited-state EOMCCSDT extension, the CC method with singles, doubles,

triples, and quadruples (CCSDTQ) and its excited-state EOMCCSDTQ extension, etc., and

without resorting to perturbative CCSD(T)-type ideas that fail in multireference situations,

such as bond breaking and excited states dominated by two-electron transitions. One of the

most promising approaches in this area is the semi-stochastic CC(P ;Q) methodology that

identifies the most important higher–than–doubly excited determinants needed in the high-

level CC/EOMCC calculations with the help of the stochastic configuration interaction (CI)

and CC Quantum Monte Carlo (QMC) wave function propagations and uses the suitably

designed deterministic iterative and noniterative steps of the CC(P;Q) formalism to converge

the desired CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ, etc., energetics. In this dis-

sertation, we first discuss our recent work on extending the semi-stochastic, CIQMC-driven,

particle-conserving CC(P ;Q) framework to excited electronic states and open-shell systems.

We tested performance of the resulting methods by examining their ability to recover the

high-level CCSDT/EOMCCSDT energetics in calculations of the electronic excitation spec-

tra of the CH+, CH, and CNC molecules and singlet–triplet gaps in a few biradical systems,

including methylene, (HFH)−, cyclobutadiene, cyclopentadienyl cation, and trimethylen-

emethane. The second part of this dissertation focuses on an alternative way of determining

ground and excited states of open-shell systems within the EOMCC framework by turning to

the single and double electron attachment (EA) and single and double ionization potential

(IP) EOMCC schemes. By generating ground and excited states of open-shell species, such

as radicals and biradicals, with the help of suitably designed operators that can formally

add electrons to or remove electrons from the parent closed-shell cores (an operation pro-

ducing the appropriate multi-configurational reference space within a single-reference frame-

work, while relaxing the remaining electrons), the resulting EA/IP- and DEA/DIP-EOMCC

methods offer several advantages over the particle-conserving CC/EOMCC treatments, in-

cluding rigorous spin and symmetry adaptation of the calculated electronic states and the

ability of handling high- and low-spin states in an accurate and well-balanced manner. We

demonstrate how to utilize the stochastic CIQMC wave function propagations, which are of

particle-conserving character, in identifying the dominant higher-order 3-particle-2-hole (3p-

2h)/3-hole-2-particle (3h-2p) and 4-particle-2-hole (4p-2h)/4-hole-2-particle (4h-2p) compo-

nents of the respective particle-nonconserving electron attaching/ionizing operators, needed

to obtain a quantitative description, without having to resort to the previously exploited

user- and system-dependent active-space concepts. The effectiveness of the semi-stochastic,

CIQMC-driven, EA/IP/DEA/DIP-EOMCC approaches will be illustrated by examining the

adiabatic excitations in the C2N, CNC, N3, and NCO radicals and by revisiting the singlet–

triplet gaps in methylene and trimethylenemethane.

Copyright by
ARNAB CHAKRABORTY
2024

To my parents, teachers, and friends.

v

ACKNOWLEDGMENTS

First and foremost, I would like to thank my doctoral advisor, Professor Piotr Piecuch, for his

invaluable advice, guidance, and support throughout my Ph.D. journey. He introduced me

to the world of coupled-cluster and other many-body theories, which I sincerely appreciate.

Professor Piecuch’s readiness to engage in scientific discussions and his inspiring presence

have been fundamental to my academic growth. I am grateful to him for sharing his deep

knowledge, his patient mentoring, and his constant motivation to approach research with

extreme care.

I would also like to acknowledge the current and previous members of my guidance com-

mittee: Professor Katharine Hunt, Professor James Jackson, Professor Angela Wilson, and

Professor Benjamin Levine, for their support, time, and advice. I would like to thank Pro-

fessor Marcos Dantus for the opportunity to work on the extensive study of H3

+ generation

from the double ionization of organic halogens and pseudo halogens. This collaboration has

taught me how to communicate ideas to a diverse group of scientists. I would also like to

thank Professors Madhav Ranganathan, Srihari Keshavamurthy, and Nishanth Nair from the

Indian Institute of Technology Kanpur; Professor Satrajit Adhikari from the Indian Associ-

ation for the Cultivation of Science, and Professor Pinaki Chaudhury from the University of

Calcutta for sparking my interest in theoretical chemistry.

Furthermore, I would like to thank Dr. Kathryn Severin and Dr. Elizabeth McGaw, under

whom I have had so many wonderful semesters of teaching opportunities for CEM 395 and

CEM 495, and Professor Katharine Hunt, under whom I had the great opportunity to teach

CEM 484. These teaching assignments gave me the scope to develop my communication skills

and helped me understand the fundamental concepts of physical chemistry more deeply.

I am also grateful to the current and former Piecuch group members: Dr. Jun Shen, Dr.

Suhita Basumallick, Dr. J. Emiliano Deustua, Dr. Ilias Magoulas, Dr. Stephen Yuwono, Mr.

Karthik Gururangan, Mr. Tiange Deng, Ms. Swati Priyadarsini, and Mr. Agnibha Hanra.

Their support, help, and encouragement made my Ph.D. journey unforgettable.

vi

I would also like to thank all my friends from the East Lansing area. My Ph.D. experience

would not have been so enjoyable without you all.

I want to thank my parents, Mr. Hari Sadhan Chakraborty and Mrs. Mira Chakraborty,

for their continuous support throughout my life.

Thank you all!

vii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

CHAPTER 2

BACKGROUND INFORMATION . . . . . . . . . . . . . . . . . .

15

2.1 Single-Reference Coupled-Cluster Theory and Its Equation-of-Motion

Extension to Excited Electronic States

. . . . . . . . . . . . . . . . . . .
2.2 The CC(P;Q) Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Particle Nonconserving Equation-of-Motion Coupled-Cluster Theories:
The Single and Double Electron Attachment and Ionization Potential
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .

2.4 Overview of Configuration Interaction Quantum Monte Carlo

CHAPTER 3

THE SEMI-STOCHASTIC CC(P;Q) METHODOLOGY FOR
GROUND AND EXCITED STATES . . . . . . . . . . . . . . . .
3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Electronic Excitations in CH+, CH, and CNC . . . . . . . . . . . . . . .
3.3 Singlet–Triplet Gaps in Methylene, (HFH)−, Cyclobutadiene, Cyclopen-

15
19

22
30

34
34
39

tadienyl Cation, and Trimethylenemethane . . . . . . . . . . . . . . . . .

61

CHAPTER 4

THE SEMI-STOCHASTIC EXTENSIONS OF PARTICLE
NONCONSERVING EQUATION-OF-MOTION
COUPLED-CLUSTER THEORIES . . . . . . . . . . . . . . . . . 111
4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.2 Adiabatic Excitations in C2N, CNC, N3, and NCO . . . . . . . . . . . . 116
4.3 Singlet–Triplet Gaps in Methylene and Trimethylenemethane . . . . . . . 137

CHAPTER 5

CONCLUDING REMARKS AND FUTURE OUTLOOK . . . . . 155

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

viii

LIST OF TABLES

Table 3.1 Convergence of the CC(P )/EOMCC(P ) and CC(P;Q) energies toward
CCSDT/EOMCCSDT for CH+, calculated using the [5s3p1d/3s1p] ba-
sis set of Ref. [164], at the C–H internuclear distance R = Re = 2.13713
bohr. The P spaces used in the CC(P ) and EOMCC(P ) calculations
were defined as all singles, all doubles, and subsets of triples extracted
from i-FCIQMC propagations for the lowest states of the relevant sym-
metries. Each i-FCIQMC run was initiated by placing 1500 walkers on
the appropriate reference function [the RHF determinant for the 1Σ+
g
states, the 3σ → 1π state of the 1B1(C2v) symmetry for the 1Π states,
and the 3σ2 → 1π2 state of the 1A2(C2v) symmetry for the 1∆ states], set-
ting the initiator parameter na at 3, and the time step ∆τ at 0.0001 a.u.
The Q spaces used in constructing the CC(P;Q) corrections consisted of
the triples not captured by i-FCIQMC. Adapted from Ref. [100].

. . . . .

48

Table 3.2 Same as Table 3.1 for the stretched C–H internuclear distance R = 2Re =

4.27426 bohr. Adapted from Ref. [100].

. . . . . . . . . . . . . . . . . . .

49

Table 3.3 Convergence of the CC(P )/EOMCC(P ) and CC(P;Q) energies toward
CCSDT/EOMCCSDT for CH, calculated using the aug-cc-pVDZ basis
set. The P spaces used in the CC(P ) and EOMCC(P ) calculations were
defined as all singles, all doubles, and subsets of triples extracted from
i-FCIQMC propagations for the lowest states of the relevant symme-
tries. Each i-FCIQMC run was initiated by placing 1500 walkers on the
appropriate reference function [the ROHF 2B2(C2v) determinant for the
X 2Π state, the 1π → 4σ state of the 2A1(C2v) symmetry for the A 2∆
and C 2Σ+ states, and the 3σ → 1π state of the 2A2(C2v) symmetry for
the B 2Σ− state], setting the initiator parameter na at 3, and the time
step ∆τ at 0.0001 a.u. The Q spaces used in constructing the CC(P;Q)
corrections consisted of the triples not captured by i-FCIQMC. Adapted
from Ref. [100].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 3.4 Convergence of the CC(P )/EOMCC(P ) and CC(P;Q) energies toward
CCSDT/EOMCCSDT for CNC, calculated using DZP[4s2p1d] basis set.
The P spaces used in the CC(P ) and EOMCC(P ) calculations were
defined as all singles, all doubles, and subsets of triples extracted from
i-FCIQMC propagations for the lowest states of the relevant symmetries.
Each i-FCIQMC run was initiated by placing 1500 walkers on the appro-
priate reference function [the ROHF 2B2g(D2h) determinant for the X 2Πg
state and the 3σu → 1πg state of the 2B1u(D2h) symmetry for the A 2∆u
and B 2Σ+
u states], setting the initiator parameter na at 3, and the time
step ∆τ at 0.0001 a.u. The Q spaces used in constructing the CC(P;Q)
corrections consisted of the triples not captured by i-FCIQMC. Adapted
from Ref. [100].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

59

ix

Table 3.5 Convergence of the CC(P ) and CC(P ;Q) energies of the X 3B1 and A 1A1
states of methylene, as described by the aug-cc-pVTZ basis set, and of the
corresponding adiabatic singlet–triplet gaps toward their parent CCSDT
values. The geometries of the X 3B1 and A 1A1 states, optimized in the
FCI calculations using the TZ2P basis set, were taken from Ref. [181].
The P spaces used in the CC(P ) and CC(P ;Q) calculations were de-
fined as all singly and doubly excited determinants and subsets of triply
excited determinants extracted from the i-FCIQMC propagations with
δτ = 0.0001 a.u. The Q spaces used to determine the CC(P ;Q) correc-
tions consisted of the triply excited determinants not captured by the
corresponding i-FCIQMC runs. The i-FCIQMC calculations preceding
the CC(P ) and CC(P ;Q) steps were initiated by placing 1500 walkers on
the ROHF (X 3B1 state) and RHF (A 1A1 state) reference determinants
and the na parameter of the initiator algorithm was set at 3. In all post-
Hartree–Fock calculations, the lowest core orbital was kept frozen and
the spherical components of d and f orbitals were employed throughout.
Adapted from Ref. [102].

. . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 3.6 The total numbers of walkers, reported as percentages of the total walker
populations at 200000 MC iterations, characterizing the i-FCIQMC prop-
agations with δτ = 0.0001 a.u. that were needed to generate the CC(P)
and CC(P;Q) results for methylene reported in Table 3.5. Adapted from
Ref. [102].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 3.7 Convergence of the CC(P ) and CC(P ;Q) energies of the X 1Σ+

g state
of (HFH)−, as described by the 6-31G(d,p) basis set, at selected H–F
distances RH-F toward their parent CCSDT values. The P spaces used
in the CC(P ) and CC(P ;Q) calculations were defined as all singly and
doubly excited determinants and subsets of triply excited determinants
extracted from the i-FCIQMC propagations with δτ = 0.0001 a.u. The Q
spaces used to determine the CC(P ;Q) corrections consisted of the triply
excited determinants not captured by the corresponding i-FCIQMC runs.
The i-FCIQMC calculations preceding the CC(P ) and CC(P ;Q) steps
were initiated by placing 1500 walkers on the RHF reference determinant
and the na parameter of the initiator algorithm was set at 3. In all post-
Hartree–Fock calculations, the lowest core orbital was kept frozen and the
spherical components of d orbitals were employed throughout. Adapted
from Ref. [102].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

69

80

x

Table 3.8 Convergence of the CC(P ) and CC(P ;Q) energies of the A 3Σ+

u state of
(HFH)−, as described by the 6-31G(d,p) basis set, at selected H–F dis-
tances RH-F toward their parent CCSDT values. The P spaces used in the
CC(P ) and CC(P ;Q) calculations were defined as all singly and doubly
excited determinants and subsets of triply excited determinants extracted
from the i-FCIQMC propagations with δτ = 0.0001 a.u. The Q spaces
used to determine the CC(P ;Q) corrections consisted of the triply ex-
cited determinants not captured by the corresponding i-FCIQMC runs.
The i-FCIQMC calculations preceding the CC(P ) and CC(P ;Q) steps
were initiated by placing 1500 walkers on the ROHF reference determi-
nant and the na parameter of the initiator algorithm was set at 3. In all
post-Hartree–Fock calculations, the lowest core orbital was kept frozen
and the spherical components of d orbitals were employed throughout.
Adapted from Ref. [102].

. . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 3.9 Convergence of the CC(P ) and CC(P ;Q) singlet–triplet gaps of (HFH)−,
as described by the 6-31G(d,p) basis set, at selected H–F distances RH-F
toward their parent CCSDT values. The P spaces used in the CC(P )
and CC(P ;Q) calculations were defined as all singly and doubly excited
determinants and subsets of triply excited determinants extracted from
the i-FCIQMC propagations with δτ = 0.0001 a.u. The Q spaces used
to determine the CC(P ;Q) corrections consisted of the triply excited
determinants not captured by the corresponding i-FCIQMC runs. The
i-FCIQMC calculations preceding the CC(P ) and CC(P ;Q) steps were
initiated by placing 1500 walkers on the RHF (X 1Σ+
g state) and ROHF
(A 3Σ+
u state) reference determinants and the na parameter of the ini-
tiator algorithm was set at 3. In all post-Hartree–Fock calculations, the
lowest core orbital was kept frozen and the spherical components of d
orbitals were employed throughout. Adapted from Ref. [102].

. . . . . . .

Table 3.10 The total numbers of walkers, reported as percentages of the total walker
populations at 200000 MC iterations, characterizing the i-FCIQMC prop-
agations with δτ = 0.0001 a.u. that were needed to generate the CC(P)
and CC(P;Q) results for the X 1Σ+
u states of (HFH)− reported
in Tables 3.7 and 3.8. Adapted from Ref. [102]. . . . . . . . . . . . . . . .

g and A 3Σ+

xi

81

82

83

Table 3.11 Convergence of the CC(P ) and CC(P ;Q) energies of the X 1B1g and
A 3A2g states of cyclobutadiene, as described by the cc-pVDZ basis set,
and of the corresponding vertical singlet–triplet gaps toward their parent
CCSDT values. All calculations were performed at the D4h-symmetric
transition-state geometry of the X 1B1g state optimized in the MR-AQCC
calculations reported in Ref. [202]. The P spaces used in the CC(P ) and
CC(P ;Q) calculations were defined as all singly and doubly excited de-
terminants and subsets of triply excited determinants extracted from the
i-FCIQMC propagations with δτ = 0.0001 a.u. The Q spaces used to
determine the CC(P ;Q) corrections consisted of the triply excited de-
terminants not captured by the corresponding i-FCIQMC runs. The
i-FCIQMC calculations preceding the CC(P ) and CC(P ;Q) steps were
initiated by placing 1500 walkers on the RHF (X 1B1g state) and ROHF
(A 3A2g state) reference determinants and the na parameter of the ini-
tiator algorithm was set at 3. In all post-Hartree–Fock calculations, the
four lowest core orbitals were kept frozen and the spherical components
of d orbitals were employed throughout. Adapted from Ref. [102].

. . . .

Table 3.12 The total numbers of walkers, reported as percentages of the total walker
populations at 80000 MC iterations, characterizing the i-FCIQMC prop-
agations with δτ = 0.0001 a.u. that were needed to generate the CC(P )
and CC(P ;Q) results for cyclobutadiene reported in Table 3.11. Adapted
from Ref. [102].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 3.13 Convergence of the CC(P ) and CC(P ;Q) energies of the X 3A′

2 and
A 1E′
2 states of cyclopentadienyl cation, as described by the cc-pVDZ
basis set, and of the corresponding vertical singlet–triplet gaps toward
their parent CCSDT values. All calculations were performed at the
D5h-symmetric geometry of the X 3A′
2 state optimized using the unre-
stricted CCSD/cc-pVDZ approach reported in Ref. [203]. The P spaces
used in the CC(P ) and CC(P ;Q) calculations were defined as all singly
and doubly excited determinants and subsets of triply excited determi-
nants extracted from the i-CISDTQ-MC propagations with δτ = 0.0001
a.u. The Q spaces used to determine the CC(P ;Q) corrections con-
sisted of the triply excited determinants not captured by the correspond-
ing i-CISDTQ-MC runs. The i-CISDTQ-MC calculations preceding the
CC(P ) and CC(P ;Q) steps were initiated by placing 1500 walkers on the
ROHF (X 3A′
2 state) reference determinants and
In all post-
the na parameter of the initiator algorithm was set at 3.
Hartree–Fock calculations, the five lowest core orbitals were kept frozen
and the spherical components of d orbitals were employed throughout.
Adapted from Ref. [102].

2 state) and RHF (A 1E′

. . . . . . . . . . . . . . . . . . . . . . . . . . .

91

92

99

xii

Table 3.14 The total numbers of walkers, reported as percentages of the total walker
populations at 80000 MC iterations, characterizing the i-CISDTQ-MC
propagations with δτ = 0.0001 a.u. that were needed to generate the
CC(P ) and CC(P ;Q) results for the cyclopentadienyl cation reported in
Table 3.13. Adapted from Ref. [102]. . . . . . . . . . . . . . . . . . . . . . 100

Table 3.15 Convergence of the CC(P ) and CC(P ;Q) energies of the X 3A′

2 and
B 1A1 states of trimethylenemethane, as described by the cc-pVDZ basis
set, and of the corresponding adiabatic singlet–triplet gaps toward their
parent CCSDT values. The D3h- and C2v-symmetric geometries of the
2 and B 1A1 states, respectively, optimized in the SF-DFT/6-31G(d)
X 3A′
calculations, were taken from Ref. [231]. The P spaces used in the
CC(P ) and CC(P ;Q) calculations were defined as all singly and dou-
bly excited determinants and subsets of triply excited determinants ex-
tracted from the i-CISDTQ-MC propagations with δτ = 0.0001 a.u.
The Q spaces used to determine the CC(P ;Q) corrections consisted
of the triply excited determinants not captured by the corresponding
i-CISDTQ-MC runs. The i-CISDTQ-MC calculations preceding the
CC(P ) and CC(P ;Q) steps were initiated by placing 1500 walkers on
2 state) and RHF (B 1A1 state) reference determinants
the ROHF (X 3A′
and the na parameter of the initiator algorithm was set at 3. In all post-
Hartree–Fock calculations, the four lowest core orbitals were kept frozen
and the spherical components of d orbitals were employed throughout.
Adapted from Ref. [102].

. . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Table 3.16 The total numbers of walkers, reported as percentages of the total walker
populations at 80000 MC iterations, characterizing the i-CISDTQ-MC
propagations with δτ = 0.0001 a.u. that were needed to generate the
CC(P ) and CC(P ;Q) results for trimethylenemethane reported in Table
3.15. Adapted from Ref. [102].

. . . . . . . . . . . . . . . . . . . . . . . . 109

Table 4.1 Convergence of the EA-EOMCC(P ) energies [abbreviated as EA(P )] of
the X 2Π, A 2∆, B 2Σ−, and C 2Σ+ states of C2N, as described by the
DZP[4s2p1d] basis set of Refs. [167, 168], and of the corresponding adia-
batic excitation energies toward their parent EA-EOMCC(3p-2h) values.
The geometries of the X 2Π, A 2∆, B 2Σ−, and C 2Σ+ states, optimized in
the SAC-CI SDT-R/PS calculations using the same basis set, were taken
from Ref. [162]. The P spaces used in the EA-EOMCC(P ) calculations
were defined as all 1p and 2p-1h determinants and subsets of 3p-2h deter-
minants extracted from the i-FCIQMC propagations with δτ = 0.0001
a.u. The i-FCIQMC calculations preceding the EA-EOMCC(P ) steps
were initiated by placing 500 walkers on the ROHF reference determi-
nants of the corresponding states and the na parameter of the initiator
algorithm was set at 3. In all post-Hartree–Fock calculations, the lowest
core orbitals of the carbon and nitrogen atoms were kept frozen.

. . . . . 121

xiii

Table 4.2 Convergence of the EA-EOMCC(P ) energies [abbreviated as EA(P )]
of the X 2Πg, A 2∆u, and B 2Σ+
u states of CNC, as described by the
DZP[4s2p1d] basis set of Refs. [167, 168], and of the corresponding adia-
batic excitation energies toward their parent EA-EOMCC(3p-2h) values.
The geometries of the X 2Πg, A 2∆u, and B 2Σ+
u states, optimized in
the SAC-CI SDT-R/PS calculations using the same basis set, were taken
from Ref. [162]. The P spaces used in the EA-EOMCC(P ) calculations
were defined as all 1p and 2p-1h determinants and subsets of 3p-2h deter-
minants extracted from the i-FCIQMC propagations with δτ = 0.0001
a.u. The i-FCIQMC calculations preceding the EA-EOMCC(P ) steps
were initiated by placing 500 walkers on the ROHF reference determi-
nants of the corresponding states and the na parameter of the initiator
algorithm was set at 3. In all post-Hartree–Fock calculations, the lowest
core orbitals of the carbon and nitrogen atoms were kept frozen.

. . . . . 126

Table 4.3 Convergence of the IP-EOMCC(P ) energies [abbreviated as IP(P )] of the
X 2Πg and B 2Σ+
u states of N3, as described by the DZP[4s2p1d] basis
set of Refs. [167, 168], and of the corresponding adiabatic excitation
energy toward their parent IP-EOMCC(3h-2p) values. The geometries
of the X 2Πg and B 2Σ+
u states, optimized in the SAC-CI SDT-R/PS
calculations using the same basis set, were taken from Ref. [162]. The
P spaces used in the IP-EOMCC(P ) calculations were defined as all
1h and 2h-1p determinants and subsets of 3h-2p determinants extracted
from the i-FCIQMC propagations with δτ = 0.0001 a.u. The i-FCIQMC
calculations preceding the IP-EOMCC(P ) steps were initiated by placing
500 walkers on the ROHF reference determinants of the corresponding
states and the na parameter of the initiator algorithm was set at 3. In all
post-Hartree–Fock calculations, the lowest core orbitals of the nitrogen
atoms were kept frozen.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Table 4.4 Convergence of the IP-EOMCC(P ) energies [abbreviated as IP(P )] of the
X 2Π, A 2∆, and B 2Π states of NCO, as described by the DZP[4s2p1d]
basis set of Refs. [167, 168], and the corresponding adiabatic excitation
energies toward their parent IP-EOMCC(3h-2p) values. The geometries
of the X 2Π, A 2∆u, and B 2Π states, optimized in the SAC-CI SDT-
R/PS calculations using the same basis set, were taken from Ref. [162].
The P spaces used in the IP-EOMCC(P ) calculations were defined as all
1h and 2h-1p determinants and subsets of 3h-2p determinants extracted
from the i-FCIQMC propagations with δτ = 0.0001 a.u. The i-FCIQMC
calculations preceding the IP-EOMCC(P ) steps were initiated by placing
500 walkers on the ROHF reference determinants of the corresponding
states and the na parameter of the initiator algorithm was set at 3. In
all post-Hartree–Fock calculations, the lowest core orbitals of the carbon,
nitrogen, and oxygen atoms were kept frozen. . . . . . . . . . . . . . . . . 135

xiv

Table 4.5 Convergence of the DEA-EOMCC(P ) energies [abbreviated as DEA(P )]
of the X 3B1, A 1A1, B 1B1, and C 1A1 states of methylene, as described
by the TZ2P basis set of Ref. [180], and of the corresponding adiabatic
singlet–triplet gaps toward their parent DEA-EOMCC(4p-2h) values.
The geometries of the X 3B1, A 1A1, B 1B1, and C 1A1 states, optimized
in the FCI calculations using the TZ2P basis set, were taken from Ref.
[181]. The P spaces used in the DEA-EOMCC(P ) calculations were
defined as all 2p and 3p-1h determinants and subsets of 4p-2h determi-
nants extracted from the i-FCIQMC propagations with δτ = 0.0001 a.u.
The i-FCIQMC calculations were initiated by placing 500 walkers on the
corresponding ROHF reference determinant for the X 3B1 state and the
corresponding RHF reference determinants for the remaining states and
the na parameter of the initiator algorithm was set at 3. In all the post-
Hartree–Fock calculations, the lowest core orbital of the carbon atom
was kept frozen.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Table 4.6 Convergence of the DIP-EOMCC(P ) energies [abbreviated as DIP(P )]
of the X 3B1, A 1A1, B 1B1, and C 1A1 states of methylene, as described
by the TZ2P basis set of Ref. [180], and of the corresponding adiabatic
singlet–triplet gaps toward their parent DIP-EOMCC(4h-2p) values. The
geometries of the X 3B1, A 1A1, B 1B1, and C 1A1 states, optimized in the
FCI calculations using the TZ2P basis set, were taken from Ref. [181].
The P spaces used in the DIP-EOMCC(P ) calculations were defined as
all 2h and 3h-1p determinants and subsets of 4h-2p determinants ex-
tracted from the i-FCIQMC propagations with δτ = 0.0001 a.u. The
i-FCIQMC calculations were initiated by placing 500 walkers on the cor-
responding ROHF reference determinant for the X 3B1 state and the
corresponding RHF reference determinants for the remaining states and
the na parameter of the initiator algorithm was set at 3. In all the post-
Hartree–Fock calculations, the lowest core orbital of the carbon atom
was kept frozen.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

xv

Table 4.7 Convergence of the DEA-EOMCC(P ) and DIP-EOMCC(P ) energies [ab-
breviated as DEA(P ) and DIP(P ), respectively] of the X 3A′
2 and B 1A1
states of TMM, as described by the 6-31G(d) basis set of Ref. [178], and
of the corresponding adiabatic singlet–triplet (S–T) gaps toward their
parent DEA-EOMCC(4p-2h) and DIP-EOMCC(4h-2p) values. The ge-
ometries of the X 3A′
2 and B 1A1 states, optimized using the SF-DFT/6-
31G(d) calculations, were taken from Ref. [231]. The P spaces used
in the DEA-EOMCC(P ) calculations were defined as all 2p and 3p-1h
determinants and subsets of 4p-2h determinants extracted from the i-
CISDTQ-MC propagations with δτ = 0.0001 a.u. The P spaces used
in the DIP-EOMCC(P ) calculations were defined as all 2h and 3h-1p
determinants and subsets of 4h-2p determinants extracted from the i-
CISDTQ-MC propagations with δτ = 0.0001 a.u. In all the post-SCF
calculations, the lowest core orbitals of the carbon atoms were kept frozen
and the spherical components of the carbon d orbitals were employed
throughout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

xvi

LIST OF FIGURES

Figure 2.1 A schematic representation of the CIQMC algorithm.

|Φ⟩ represents
the reference determinant, while {|ΦS1⟩, |ΦS2⟩, |ΦS3⟩, |ΦS4⟩, · · · } denote
singly excited determinants, {|ΦD1⟩, |ΦD2⟩, |ΦD3⟩, |ΦD4⟩, · · · } doubly
excited determinants, {|ΦT1⟩, |ΦT2⟩, |ΦT3⟩, |ΦT4⟩, · · · } triples, {|ΦQ1⟩,
|ΦQ2⟩, |ΦQ3⟩, |ΦQ4⟩, · · · } quadruples, etc. The number of walkers on a
particular determinant is indicated as a superscript within parenthesis.
Green and red rectangles distinguish positive and negative walkers, re-
spectively, with darker shades indicating a higher number of walkers on
the determinants. Here, the simulation starts with one walker placed
on the reference determinant |Φ⟩. Panel (a) illustrates spawning steps,
panel (b) depicts the death of a walker on the determinant |ΦD4⟩ with a
gray box, panel (c) shows more spawning events and panel (d) displays
the annihilation step on |ΦQ3⟩ with an orange rectangular box. . . . . . .

Figure 3.1 A schematic illustration depicting the construction of P -spaces in CC(P)
and EOMCC(P) computations. Panel (a) showcases the stabilization
of correlation energy (green line) and the corresponding increase in the
total number of walkers is shown in panel (b) (red line). On the right
four snapshots from a QMC calculation are presented, featuring the lists
of determinants picked up by the QMC algorithm at various time steps
(green for 2000, orange for 20000, violet for 50000, and magenta for
100000 QMC iterations). It is evident that QMC deems some determi-
nants more important than others by placing more walkers on them. . . .

Figure 3.2 Convergence of the EOMCC(P ) [panels (a) and (c)] and CC(P;Q) [pan-
els (b) and (d)] energies toward EOMCCSDT for the three lowest-energy
excited states of the 1Σ+ symmetry, two lowest states of the 1Π sym-
metry, and two lowest 1∆ states of the CH+ ion, as described by the
[5s3p1d/3s1p] basis set of Ref. [164], at the C–H internuclear distance R
set at Re = 2.13713 bohr [panels (a) and (b)] and 2Re = 4.27426 bohr
[panels (c) and (d)]. Adapted from Ref. [100].

. . . . . . . . . . . . . . .

Figure 3.3 The distributions of the differences between the R(MC)

µ,3

amplitudes and
their EOMCCSDT counterparts resulting from the EOMCC(P ) com-
putations at (a) 4000, (b) 10,000, and (c) 50,000 MC iterations using
τ = 0.0001 a.u. for the 21Σ+ state of CH+ at R = 2Re with the analogous
distribution characterizing the Rµ,3 amplitudes obtained with the EOM-
CCSDt approach employing the 3σ, 1πx, 1πy, and 4σ active orbitals to
define the corresponding triples space [panel (d)]. All vectors Rµ needed
to construct panels (a)–(d) were normalized to unity. Adapted from Ref.
[100]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

36

50

51

xvii

Figure 3.4 Convergence of the EOMCC(P) and CC(P;Q) energies of the A 2∆
[panel (a)], B 2Σ− [panel (b)], and C 2Σ+ [panel (c)] states of CH toward
EOMCCSDT for the three lowest-energy excited states of CH calculated
as described by the aug-cc-pVDZ basis set. The geometries used are
the equilibrium C–H distances reported in Refs. [62, 68, 161], which are
1.1031 ˚A for the A 2∆ state [175], 1.1640 ˚A for the B 2Σ− state [176],
and 1.1143 ˚A for the C 2Σ+ state [177]. . . . . . . . . . . . . . . . . . . .

Figure 3.5 Convergence of the EOMCC(P) and CC(P;Q) energies of the A 2∆u
[panel (a)] and B 2Σ+
u [panel (b)] states of CNC toward EOMCCSDT
for the two lowest-energy doublet excited states of CNC calculated as
described by the DZP[4s2p1d] basis set. The geometries used are the
equilibrium C–N distances reported in Refs. [68,162], which are 1.256 ˚A
for the A 2∆u state and 1.259 ˚A for the B 2Σ+

u state.

. . . . . . . . . . .

Figure 3.6 Convergence of the CC(P ) and CC(P ;Q) energies of the X 3B1 [panel
(a)] and A 1A1 [panel (b)] states of methylene, as described by the aug-
cc-pVTZ basis set, and of the corresponding adiabatic singlet–triplet
gaps [panel (c)] toward their parent CCSDT values. The geometries of
the X 3B1 and A 1A1 states, optimized in the FCI calculations using the
TZ2P basis set, were taken from Ref. [181]. The P spaces consisted
of all singles and doubles and subsets of triples identified during the i-
FCIQMC propagations with δτ = 0.0001 a.u. and the Q spaces consisted
of the triples not captured by i-FCIQMC. Adapted from Ref. [102].

. . .

Figure 3.7 Comparison of convergences of the CC(P), CC(P;Q), and the underly-
ing i-FCIQMC calculations toward their respective limits for the X 3B1
and A 1A1 states of the CH2 molecule at their respective geometries op-
timized in the FCI calculations using the TZ2P basis set are taken from
Ref. [181].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.8 Total electronic energies of the X 1Σ+

g (open circles and solid line) and
A 3Σ+
u (filled circles and dotted line) states of (HFH)− with increase in
the H–F distance, from 1.5 ˚A to 4.0 ˚A, obtained from the FCI (red cir-
cles), CCSD (blue circles), and CCSDT (green circles) methods. Recre-
ated from the data reported in Refs. [66, 84, 88, 197]. . . . . . . . . . . .

xviii

56

60

69

70

84

Figure 3.9 Convergence of the CC(P ) and CC(P ;Q) energies of the X 1Σ+

g [pan-
els (a) and (b)] and A 3Σ+
u [panels (c) and (d)] states of (HFH)−, as
described by the 6-31G(d,p) basis set, and of the corresponding singlet–
triplet gaps [panels (e) and (f)] toward their parent CCSDT values. The
H–F distances RH-F used are 1.50 ˚A, 1.75 ˚A, 2.00 ˚A, 2.50 ˚A, and 4.00 ˚A.
The P spaces consisted of all singles and doubles and subsets of triples
identified during i-FCIQMC propagations with δτ = 0.0001 a.u. and the
Q spaces consisted of the triples not captured by i-FCIQMC. Adapted
from Ref. [102].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.10 π molecular orbital network of the cyclobutadiene molecule, obtained at
the HF/cc-pVDZ level, at the D4h-symmetric transition-state geometry
of the X 1B1g state optimized in the MR-AQCC calculations in Ref.
[202]. The orbital irreducible representations in the D4h symmetry are
shown in black and the corresponding labels in the C2v symmetry are
shown in the parenthesis in orange. This electronic configuration refers
to the triplet state.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.11 Convergence of the CC(P ) and CC(P ;Q) energies of the X 1B1g [panel
(a)] and A 3A2g [panel (b)] states of cyclobutadiene, as described by
the cc-pVDZ basis set, and of the corresponding vertical singlet–triplet
gaps [panel (c)] toward their parent CCSDT values. All calculations
were performed at the D4h-symmetric transition-state geometry of the
X 1B1g state optimized in the MR-AQCC calculations in Ref. [202]. The
P spaces consisted of all singles and doubles and subsets of triples iden-
tified during the i-FCIQMC propagations with δτ = 0.0001 a.u. and the
Q spaces consisted of the triples not captured by i-FCIQMC. Adapted
from Ref. [102].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

92

93

Figure 3.12 π molecular orbital network of the cyclopentadienyl cation molecule,
obtained at the HF/cc-pVDZ level, at the D5h-symmetric geometry of
the X 3A′
2 state optimized using the unrestricted CCSD/cc-pVDZ ap-
proach in Ref. [203]. The orbital irreducible representations in the D5h
symmetry are shown in black and the corresponding labels in the C2v
symmetry are shown in the parenthesis in orange. This electronic con-
figuration refers to the triplet state. . . . . . . . . . . . . . . . . . . . . . 100

xix

Figure 3.13 Convergence of the CC(P ) and CC(P ;Q) energies of the X 3A′

2 [panel
(a)] and A 1E′
2 [panel (b)] states of cyclopentadienyl cation, as described
by the cc-pVDZ basis set, and of the corresponding vertical singlet–
triplet gaps [panel (c)] toward their parent CCSDT values. All calcula-
tions were performed at the D5h-symmetric geometry of the X 3A′
2 state
optimized using the unrestricted CCSD/cc-pVDZ approach in Ref. [203].
The P spaces consisted of all singles and doubles and subsets of triples
identified during the i-CISDTQ-MC propagations with δτ = 0.0001 a.u.
and the Q spaces consisted of the triples not captured by i-CISDTQ-
MC. Adapted from Ref. [102].

. . . . . . . . . . . . . . . . . . . . . . . . 101

Figure 3.14 π molecular orbital network of the trimethylenemethane molecule, ob-
tained at the HF/cc-pVDZ level, at the D3h-symmetric geometry of
the X 3A′
2 state optimized in the SF-DFT/6-31G(d) calculations and
taken from Ref. [231]. The orbital irreducible representations in the
D3h symmetry are shown in black and the corresponding labels in the
C2v symmetry are shown in the parenthesis in orange. This electronic
configuration refers to the triplet state. . . . . . . . . . . . . . . . . . . . 109

Figure 3.15 Convergence of the CC(P ) and CC(P ;Q) energies of the X 3A′

2 [panel
(a)] and B 1A1 [panel (b)] states of trimethylenemethane, as described by
the cc-pVDZ basis set, and of the corresponding adiabatic singlet–triplet
gaps [panel (c)] toward their parent CCSDT values. The geometries of
the X 3A′
2 and B 1A1 states, optimized in the SF-DFT/6-31G(d) calcu-
lations, were taken from Ref. [231]. The P spaces consisted of all singles
and doubles and subsets of triples identified during the i-CISDTQ-MC
propagations with δτ = 0.0001 a.u. and the Q spaces consisted of the
triples not captured by i-CISDTQ-MC. Adapted from Ref. [102]. . . . . . 110

Figure 4.1 A schematic illustration depicting the construction of P -spaces in the
EA/IP/DEA/DIP-EOMCC(P ) computations. Panel (a) showcases the
stabilization of correlation energy (green line) and the corresponding
increase in the total number of walkers is shown in panel (b) (red line).
On the right four snapshots from a QMC calculation are presented,
featuring the lists of determinants picked up by the QMC algorithm
at various time steps (light blue for 2000, dark blue for 20000, violet
for 40000, and magenta for 100000 QMC iterations). It is evident that
QMC deems some determinants more important than others by placing
more walkers on them.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Figure 4.2 Convergence of (a) the EA-EOMCC(P ) energies of the X 2Π, A 2∆, B
2Σ−, and C 2Σ+ states of C2N, as described by the DZP[4s2p1d] basis
set, and (b) the corresponding adiabatic excitation energies toward their
parent EA-EOMCC(3p-2h) values.

. . . . . . . . . . . . . . . . . . . . . 122

xx

Figure 4.3 Convergence of (a) the EA-EOMCC(P ) energies of the X 2Πg, A 2∆u,
and B 2Σ+
u states of CNC, as described by the DZP[4s2p1d] basis set,
and (b) the corresponding adiabatic excitation energies toward their
parent EA-EOMCC(3p-2h) values.

. . . . . . . . . . . . . . . . . . . . . 127

Figure 4.4 Convergence of (a) the IP-EOMCC(P ) energies of the X 2Πg and B
2Σ+
u states of N3, as described by the DZP[4s2p1d] basis set, and (b)
the corresponding adiabatic excitation energy toward their parent IP-
EOMCC(3h-2p) values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Figure 4.5 Convergence of (a) the IP-EOMCC(P ) energies of the X 2Π, A 2∆, and
B 2Π states of NCO, as described by the DZP[4s2p1d] basis set, and
(b) the corresponding adiabatic excitation energies toward their parent
IP-EOMCC(3h-2p) values. . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Figure 4.6 Convergence of (a) DEA-EOMCC(P ) energies of X 3B1, A 1A1, B 1B1,
and C 1A1 states of methylene, as described by the TZ2P basis set, and
(b) of the corresponding adiabatic singlet–triplet gaps towards their
parent DEA-EOMCC(4p-2h) values.

. . . . . . . . . . . . . . . . . . . . 143

Figure 4.7 Convergence of (a) DIP-EOMCC(P ) energies of X 3B1, A 1A1, B 1B1,
and C 1A1 states of methylene, as described by the TZ2P basis set, and
(b) of the corresponding adiabatic singlet–triplet gaps towards their
parent DIP-EOMCC(4h-2p) values. . . . . . . . . . . . . . . . . . . . . . 147

Figure 4.8 Jahn-Teller distortion in the trimethylenemethane molecule. At the
geometry of the D3h-symmetric triplet ground state (shown in blue),
trimethylenemethane has a doubly degenerate singlet excited state. Due
to Jahn–Teller distortion these states split into an open-shell singlet
state A 1B1 (shown in green) and a multi-configurational singlet state
B 1A1 (shown in red). Although the A 1B1 state becomes the first ex-
cited state, it is not observed experimentally due to unfavorable Franck–
Condon factors. Thus, we calculate the singlet–triplet gap between the
ground triplet and the second excited singlet state B 1A1. . . . . . . . . . 153

Figure 4.9 Convergence of DEA-EOMCC(P ) energies of (a) X 3A′

2 and (b) B 1A1
states of TMM, as described by the 6-31G(d) basis set, and (c) of the
corresponding adiabatic singlet–triplet gap towards their parent DEA-
EOMCC(4p-2h) values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Figure 4.10 Convergence of DIP-EOMCC(P ) energies of (a) X 3A′

2 and (b) B 1A1
states of TMM, as described by the 6-31G(d) basis set, and (c) of the
corresponding adiabatic singlet–triplet gap towards their parent DIP-
EOMCC(4h-2p) values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

xxi

CHAPTER 1

INTRODUCTION

Electronic structure theory is a major branch of quantum mechanics that aims to elucidate

the behavior of electrons in atoms, molecules, and condensed matter systems. It provides

a fundamental framework for understanding and predicting the properties of matter at the

microscopic level. At its core, lies the time-independent electronic Schr¨odinger equation,

which, within the Born–Oppenheimer approximation, has the form

HΨµ(X; R) = Eµ(R)Ψµ(X; R),

(1.1)

where H is the electronic Hamiltonian, Ψµ, which depends on the coordinates of all electrons

X and parametrically on the coordinates of the nuclei R that provide the external poten-

tial for the motion of electrons, is the many-electron wave function characterizing the µth

electronic state, with µ = 0 denoting the ground state and µ > 0 designating excited states,

and Eµ that parametrically depends on the nuclear coordinates R is the corresponding total

electronic energy. In the non-relativistic description, the electronic Hamiltonian operator for

a system containing N electrons and M nuclei can be expressed in atomic units as

H =

N
X

(−

i=1

1
2

∇2

i −

M
X

A=1

ZA
riA

) +

N
X

i<j

1
rij

= Z + V,

(1.2)

where

with zi = − 1

2∇2

i −PM

A=1

Z =

N
X

i=1

zi,

(1.3)

ZA
riA

, is the one-body term that contains the kinetic energy of electrons

and the coulombic attraction between the electrons and the frozen nuclei (with ZA denoting

the nuclear charge associated with the Ath nucleus) and

V =

N
X

i<j

vij,

(1.4)

with vij = 1
rij

, is the two-body operator describing the electron-electron coulombic repulsions.

Although it has been nearly a century since the Schr¨odinger equation was proposed, we can

1

solve this equation analytically only for the one-electron systems, such as the hydrogen atom

or the H+

2 molecule. Nevertheless, due to the great theoretical and computational advances

in the last few decades, it is possible to obtain numerical solutions for the description of

increasingly large and complex molecular systems. However, an accurate determination of

the electronic spectra and potential energy surfaces (PESs) of closed-shell species and open-

shell systems, such as radicals and biradicals, along bond breaking coordinates, multiple

bond breaking, and excited states dominated by two-electron transitions continue to pose

a challenge. Much of this challenge originates from the fact that the above problems are

governed by a complicated combination of two different types of many-electron correlation

effects, including dynamical correlations, which result from the motion of electrons trying

to avoid each other, especially at shorter distances, due to repulsive electron-electron inter-

actions, and nondynamical or static correlations that result from electron entanglement at

longer electron-electron separations. One of the biggest challenges of electronic structure the-

ory is to design computationally affordable ways of describing dynamical and nondynamical

correlations accurately and in a balanced manner.

In principle, one can try to address this challenge by turning to the full configuration

interaction (FCI) approach, where the many-electron wave function is expressed as a linear

combination of all possible Slater determinants within a basis set of one-electron functions,

i.e., spin-orbitals. Inserting this expansion into the electronic Schr¨odinger equation results

in an eigenvalue problem, involving the matrix elements of H, which is equivalent to diago-

nalizing the Hamiltonian matrix in the many-electron Hilbert space spanned by the Slater

determinants. This procedure provides the exact numerical solution of the Schr¨odinger

equation in a basis, and the only remaining step then is to gradually increase the basis set

dimension to reach the complete basis set (CBS) limit, which provides the true, numerically

exact solution of the Schr¨odinger equation for the electronically bound states. Unfortunately,

this seemingly straightforward approach bears a dimensionality that scales factorially with

respect to the system size (i.e., number of electrons and the basis set dimension), render-

2

ing it impractical for most of the systems encountered in chemical sciences. With only a

few electrons in realistic basis sets, the computational cost of FCI diagonalization becomes

unmanageable even with the most powerful supercomputers available today.

Indeed, the

combinatorial scaling of the computational costs associated with diagonalizing the Hamil-

tonian matrix in the complete N -electron Hilbert space with the number of electrons and

one-electron basis functions is so severe that relying solely on the growth of the processing

speed of classical computers is far from being sufficient. Quantum computers might address

this issue, but devices in this category capable of producing noiseless FCI solutions for real-

istic electronic structure problems do not exist. Therefore, one of the main efforts in the area

of quantum chemistry method development has been to formulate computationally tractable

approaches capable of capturing FCI-quality many-electron correlation effects for a broad

spectrum of chemically relevant systems without running into the prohibitive computational

costs associated with the FCI approach.

Since constructing the many-electron wave functions using all possible Slater determi-

nants and numerically exact coefficients at these determinants is not feasible, the simplest

alternative, at least for the ground state, would be to use single determinants to represent

them. This leads to the Hartree–Fock (HF) approximation, which is one of the most well-

known independent particle models in which we apply the variational principle to optimize

a single Slater determinant via a self consistent-field (SCF) procedure. Although the HF

method is capable of recovering total ground-state electronic energies of molecular systems

to within about 1%, it is very far from being a good solution, since it neglects many-electron

correlation effects needed to describe covalent chemical bond breaking, electronic excitation,

electron attachment, and ionization phenomena, and the vast majority of molecular prop-

erties of interest in chemistry. It also does not capture important physical effects, such as

dispersion in van der Waals molecules, and is particularly bad in describing dissociations of

low-spin species into high-spin open-shell fragments. This can be illustrated with the exam-

ple of the single bond dissociation of F2 into two F atoms [1]. In this case, the restricted

3

HF (RHF) solution, which requires each spatial molecular orbital (MO) to be either empty

or doubly occupied by electrons of opposite spins, overbinds the dissociation energy by an

order of magnitude, forcing F2 to dissociate into F+ and F−, when it should dissociate into

neutral F atoms. On the other hand, the unrestricted HF (UHF) method, which uses dif-

ferent spatial orbitals for different spins, produces spin-symmetry-broken solutions without

binding the molecule at all. All of the above problems in the HF approximation originate

from the mean-field treatment of electron-electron interactions that results in essentially un-

correlated motion of electrons. However, despite all the deficiencies of the HF theory, the

HF determinants often serve as a good starting point for the so-called post-HF methods

that generate the remaining Slater determinants in the wave function as particle-hole (p-h)

excitation from them. In this way, we can capture the relevant many-electron correlation ef-

fects, whose strength in the ground state can be measured by the correlation energy, defined

as the difference between the exact or FCI and HF energies, and describe all of the above

phenomena that are driven by electron correlation.

Among all the correlated, post-HF, approaches that have been developed over the years

to provide accurate approximate solutions to the Schr¨odinger equation, methods based on

the coupled-cluster (CC) theory [2–7] and their suitable extensions to open-shell and excited

states are widely recognized as offering the best balance between accuracy and computational

cost, thus emerging as one of the most prominent alternatives to FCI [8, 9]. Historically,

the CC theory emerged as an infinite-order generalization of the finite-order many-body

perturbation theory (MBPT) by summing the linked wave function and connected energy

diagrams to infinite order, using the linked [10–13] and connected [12, 13] cluster theorems.

These two theorems provide the CC theory with some very important characteristics of the

exact FCI description. For example, CC approaches are size extensive, i.e., there is no loss

of accuracy with an increase of the system size, because, contrary to truncated CI methods,

only connected diagrams are present in the energy expressions and there are no unlinked

contributions in the CC wave functions. Also, the CC theory relies on an exponential

4

ansatz for the wave function, and this allows for the separability or size consistency of

the wave function in the non-interacting limit, provided the underlying reference state is

also separable, enabling the CC methods to deal with fragmentation phenomena. These

important properties, combined with the fast convergence of the truncated CC methods

toward the exact FCI limit in the majority of molecular applications, have established the

CC theory as the de facto standard for high-accuracy electronic structure calculations.

There are two distinct conceptual frameworks within the CC theory depending on the

dimensionality of the reference or model space used as the zeroth-order description of the

many-electron problem of interest. The oldest formalism, which uses a single Slater determi-

nant as the reference state and Fermi vacuum, defines the single-reference (SR) CC methods.

One can also consider multi-reference (MR) CC approaches where multi-dimensional model

spaces, spanned by multiple Slater determinants, provide the zeroth-order states. Since

many of the problems of interest, such as low-lying excited states dominated by two-electron

transitions, electronic spectra of radical and biradical species, and excited-state potentials

along bond stretching coordinates have an intrinsically MR character, one naturally tends

to think of resorting to MRCC theories [8, 9, 14–18] and other MR theories [19–23] that

are specifically designed to deal with these kinds of situations. Unfortunately, even the best

MRCC theories suffer from various mathematical and numerical problems, such as unphysi-

cal [14, 15, 24] and singular [14, 15, 25–28] solutions, intruder states [14, 15, 24], and intruder

solutions [15, 24]. Also, the multi-dimensional reference spaces grow factorially with respect

to the number of active electrons and orbitals, which makes it very difficult to perform

MRCC (in fact, any MR) calculations when the numbers of active orbitals and electrons

become larger, which is often the case when polyradical species and transition metal atoms

are involved or truncations used in MRCC and other MR methods are too severe. On the

other hand, as further elaborated below, the SRCC methods can be applied to most of the

MR situations relevant to chemistry, while being much easier to understand and apply com-

pared to their MR counterparts. Therefore, this dissertation work focuses on SRCC-type

5

ideas, where the relevant dynamical as well as nondynamical correlation effects are formally

recovered through conventional p-h excitations from a single Slater determinant defining the

Fermi vacuum.

The SRCC theory uses an exponential ansatz to construct the N -electron wave function,

|Ψ0⟩ = eT |Φ⟩, where |Φ⟩ is the reference determinant (typically, but not necessarily the HF

determinant) and T = PmA

n=1 Tn is the cluster operator, with Tn representing its n-particle-n-

hole (np-nh) component and mA ≤ N designating the truncation level. When mA = N , the

full CC method is obtained which is equivalent to the FCI approach, but for practical rea-

sons mA is almost always set at a much lower value resulting in the hierarchy of conventional

SRCC approximations, such as CC with singles (S) and doubles (D) or CCSD [29–32], where

mA is set to 2, CC with singles (S), doubles (D), and triples (T) or CCSDT [33–35], where

mA = 3, CC with singles (S), doubles (D), triples (T), and quadruples (Q) or CCSDTQ

[36–39], where mA = 4, and so on. One of the interesting facts regarding the CC theory is

that the use of an exponential wave operator enables the CC theory to include additional

excitations, not explicitly included in the calculations, through product or “disconnected”

excitations. In other words, even in low orders like CCSD, where only single and double

excitations out of the HF reference are included in the cluster operator T , the various prod-

uct terms that originate due to the expansion of the exponential operator in a Taylor series

lead to some triple, some quadruple, some pentuple, etc. excitations out of the reference

determinant. The CC formalism can also be extended to excited states with the help of, for

example, the equation-of-motion (EOM) CC methodology [40–44] or its linear response (LR)

CC [45–49] and symmetry-adapted-cluster (SAC) CI [50] counterparts. In this dissertation

research, the focus is on the EOMCC formalism that adopts the ansatz |Ψµ⟩ = RµeT |Φ⟩,

where Rµ = rµ,01 + PmA

n=1 Rµ,n is a linear, CI-like, excitation operator, rµ,0 and Rµ,n are the

zero- and n-body components of Rµ, and mA is the level truncation in Rµ, which usually

is, but does not have to be [44], the same as that used in T . Similar to the ground-state

SRCC theory, different values of mA result in different truncated EOMCC schemes, such

6

as EOMCCSD [41–43] when mA = 2, EOMCCSDT [51–53] when mA = 3, EOMCCSDTQ

[54, 55] when mA = 4, etc. The SRCC framework and its EOMCC extension are very effi-

cient in accounting for dynamical correlations, providing rapid convergence to the exact FCI

limit, but the basic CCSD and EOMCCSD approaches have problems in situations involv-

ing stronger nondynamical correlations that originate from electronic quasi-degeneracies and

MR character of the states of interest. One can deal with the latter situations, by incorpo-

rating higher–than–two-body components of the cluster and EOM excitation operators, as

is done in the high-level CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ, etc., methods.

The “only” problem is that the inclusion of higher–than–two-body components of the T and

Rµ operators in the calculations poses another challenge, namely, large, often prohibitive,

computational costs that limit the resulting approaches to smaller many-electron systems.

For example, the computational time scaling of CCSD/EOMCCSD is n2

on4

u, or N 6, where

no and nu are the numbers of correlated occupied and unoccupied orbitals, respectively, and

N is a measure of the system size, which is manageable for systems with about 100 corre-

lated electrons and, with the help of local correlation techniques, for systems with dozens

or hundreds of atoms, but CCSDT/EOMCCSDT and CCSDTQ/EOMCCSDTQ scale as

on5
n3

u (N 8) and n4

on6

u (N 10), respectively, limiting their use to systems with a few non-

hydrogen atoms. It is, therefore, important to come up with approximations within the CC

and EOMCC formalisms that would allow one to incorporate higher–than–two-body com-

ponents of the cluster and EOM excitation operators at small fractions of the computational

costs compared to their full CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ, etc., parents

and without much loss of accuracy, while avoiding failures of the perturbative CC/EOMCC

approximations of the CCSD(T) [56] and similar types in MR cases and situations charac-

terized by large and nonperturbative correlation effects.

Among the various methodologies developed to tackle the above problems, the completely

renormalized (CR) CC/EOMCC [57–70] and the active-space CC/EOMCC [38, 51, 52, 71–

80] approaches have become especially valuable. In the former case, one uses the formalism

7

of the method of moments of CC (MMCC) equations [57–68, 70, 80–82] to add a posteri-

ori, noniterative, and state-specific corrections to the energies obtained in the lower-order

CC/EOMCC calculations. For example, in the ground-state CR-CC(2,3) [63–66, 68] and

CR-CC(2,4) [64, 65, 67, 83, 84] approaches one corrects the CCSD energies for the cor-

relation effects due to the connected triply, in the former case, and triply and quadruply

excited clusters, in the latter case, which are disregarded in the initial CCSD calculation. In

the case of the excited-states, the CR-EOMCC methods, such as CR-EOMCC(2,3) [65, 68]

and δ-CR-EOMCC(2,3) [69], correct the total or vertical excitation energies obtained with

EOMCCSD. The CR-CC/EOMCC approaches can be very accurate, eliminating failures of

perturbative methods, such as CCSD(T) or EOMCCSD(T) [85, 86], in the examination of

bond breaking, excited states dominated by two-electron transitions, and other MR situa-

tions, but they do not allow for the relaxation of the lower-order cluster and EOM excitation

operator components, such as T1 and T2 or Rµ,1 and Rµ,2, in the presence of their higher-

order counterparts, such as T3 or Rµ,3, which may lead to substantial inaccuracies in certain

reactions involving biradical transition states, singlet–triplet gaps in some biradicals, and

certain cases of doubly excited states [87–89]. In the active-space CC/EOMCC methodolo-

gies, higher–than–two-body components of the cluster and EOM excitation operators are

obtained iteratively, so that the low-order components of T and Rµ can relax in the presence

of their higher-order counterparts, which are down-selected using small subsets of orbitals,

called active orbitals, to reduce computational costs compared to CCSDT/EOMCCSDT and

similar high-level approaches. For example, in CCSDt/EOMCCSDt [38, 51, 52, 72–77, 79]

and CCSDtq/EOMCCSDtq [38, 71, 73–76, 79], one considers all singles and doubles, treat-

ing T1, T2, Rµ,1 and Rµ,2 exactly, but only relatively small subsets of triples (t) or triples and

quadruples (tq), selected using active orbitals are included in the definitions of T3, T4, Rµ,3,

and Rµ,4. Although it is true that the inclusion of the leading higher–than–two-body com-

ponents of T and Rµ defined by active orbitals dramatically reduces the computational costs

compared to the parent approaches, such as CCSDT/EOMCCSDT or CCSDTQ/EOMCCS-

8

DTQ, the active-space CC/EOMCC methods are not free of problems. For example, al-

though the active-space CC/EOMCC approaches allow the lower-rank components of T and

Rµ to be relaxed in presence of the higher-rank ones, they miss some important higher-order

dynamical correlations because they neglect the higher–than–two-body T and Rµ ampli-

tudes outside the active-space definitions. Fortunately, this can be successfully addressed by

merging the CR and active-space CC/EOMCC methodologies with the help of the CC(P ;Q)

[70, 84, 87, 88] formalism, which is of main significance for the development of new ideas

pursued in this dissertation.

In a CC(P ;Q) computation, the CC/EOMCC energies obtained in the subspace of the

many-electron Hilbert space called the P -space, designated as H(P ), i.e., the energies ob-

tained with conventional (CCSD/EOMCCSD, CCSDT/EOMCCSDT, etc.) or unconven-

tional (e.g., CCSDt/EOMCCSDt, CCSDtq/EOMCCSDtq) truncations of the T and Rµ

operators, are corrected for the missing correlation effects captured with the help of the

complementary subspace of the many-electron Hilbert space called the Q space, denoted as

H(Q), using noniterative moment corrections similar to those used in the CR-CC/EOMCC

approaches. The conventional truncations in the iterative P -space CC/EOMCC calculations,

combined with similarly conventional definitions of the Q spaces based on the many-body

excitation ranks of the determinants included in them, give rise to the aforementioned CR-

CC/EOMCC methods, such as CR-CC(2,3), CR-CC(2,4), or CR-EOMCC(2,3), whereas the

unconventional truncations based on exploiting the active-space ideas yield the CC(t;3),

CC(t,q;3), CC(t,q;3,4), etc., hierarchy, where one first solves the CCSDt [CC(t;3)] or CCS-

Dtq [CC(t,q;3) and CC(t,q;3,4)] equations, which recover much of the nondynamical and

some dynamical correlations, while correcting the resulting energies for the missing triples

[CC(t;3) and CC(t,q;3)] or missing triples and quadruples [CC(t,q;3,4)]. These methods re-

produce the results obtained with their respective parent CC approaches, such as CCSDT, if

CC(t;3) is used, or CCSDTQ, if one uses CC(t,q;3) or CC(t,q;3,4), to within small fractions

of a millihartree and at small fractions of the computational effort [70, 84, 87–90], but due to

9

the selection of user- and system-dependent active orbitals, they are no longer black-box ap-

proaches. Naturally, to address this concern, the next development efforts should be directed

towards finding computationally efficient ways by which the selection of the H(P ) and H(Q)

spaces in the CC(P;Q) calculations is made fully automatic. In this way, we might achieve

the desired relaxation of the lower-rank components of T and Rµ in the presence of their

higher-rank counterparts without resorting to the user- and system-dependent concepts of

the active orbitals exploited in the CC(t;3), CC(t,q;3), CC(t,q;3,4), etc., hierarchy. Clearly,

this should be done in such a manner that the resulting CC(P ;Q) computations rapidly con-

verge the target CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ, etc., energetics, at small

fractions of computational costs of the parent approaches, even when higher–than–two-body

components of the T and Rµ operators become large and nonperturbative.

In search of the algorithms enabling an automated determination of the P and Q spaces

for CC(P;Q) computations, the Piecuch research group proposed a new class of hybrid

CC(P ;Q) methods that can loosely be categorized into three different approaches. The

first one is based on merging the deterministic CC(P ;Q) theory with the stochastic quantum

Monte Carlo (QMC) wave function propagations in the many-electron Hilbert space defining

the configuration interaction QMC (CIQMC) [91–94] and the CC Monte Carlo (CCMC) [95–

98] methods, resulting in the called the semi-stochastic or QMC-driven CC(P ;Q) schemes

[99–102]. In the second approach, the deterministic CC(P ;Q) framework is combined with

one of the selected-CI [103–119] methods, such as CIPSI [105, 116–118], resulting in the

selected-CI-driven CC(P ;Q) methodology [120]. These hybrid CC(P ;Q) schemes rely on

external sources, such as CIQMC, CCMC, and selected CI, for efficiently identifying the

leading determinants in many-electron wave functions in a black-box manner, which are

then used to construct the P and Q spaces for the CC(P ;Q) calculations. Very recently, a

third algorithm has been proposed where the automated design of the P and Q spaces for

CC(P;Q) calculations is accomplished using the generalized moments of the CC/EOMCC

equations. This algorithm is referred to as the adaptive CC(P ;Q) approach [121]. This

10

dissertation focuses on the semi-stochastic or CIQMC-driven CC(P ;Q) framework and one

of the main topics of interest is the extension and application of the semi-stochastic CC(P ;Q)

methodology to ground and excited states of open-shell systems, such as radicals, and other

interesting problems involving non-singlet electronic states, such as singlet–triplet gaps in

biradicals. However, before discussing the QMC-driven CC(P ;Q) methodology, we briefly

review the key concepts of QMC, as applied to electronic systems.

The history of QMC methods dates back to the pioneering work of Metropolis and Ulam

[122], which paved they way for solving deterministic problems, such as differential equations,

via stochastic sampling procedures [122–124]. This resulted in the development of methods

such as the variational Monte Carlo (VMC) [125, 126], where the parameters of a trial wave

function are optimized by using a stochastic sampling procedure to calculate the expectation

values of the Hamiltonian. In another approach, which is commonly referred to as diffusion

Monte Carlo (DMC) [127–129], one evolves a trial wave function according to the imaginary-

time form of the Schr¨odinger equation, which results in projecting the exact ground state at

the limit of infinite time, (see, e.g., Refs. [130–132] for further details). The DMC methods

are easily parallelizable across multiple nodes and operate in the real space of 3N particle

coordinates, removing any limitations arising from the use of finite basis sets encountered in

quantum chemistry, making them very attractive. Unfortunately, in the context of electronic

structure theory, they suffer from a major issue: the DMC methods are unaware of the nodal

structure of the target wave function and if the wave function propagation is run without any

constraints, it projects out the true mathematical ground-state of the spin-free Hamiltonian,

which is bosonic in nature, violating the Pauli exclusion principle and resulting in the so-

called “boson catastrophe” or “fermion sign problem”. A common way to mitigate this is

to use the “fixed-node approximation” [133–136], where nodal structures of approximate

wave functions obtained in some inexpensive quantum chemistry calculations are imposed

as constraints, but, as a result, the exactness of the DMC propagations in the limit of infinite

imaginary time is compromised.

11

In recent years, Full Configuration Interaction Quantum Monte Carlo (FCIQMC) and its

truncated variants [91–94] have emerged as methods to circumvent the fermion sign problem

inherent in DMC propagations. In these approaches, the imaginary-time propagation of wave

functions in real space is substituted by the propagation of CI expansions within the many-

electron Hilbert space defined by Slater determinants. FCIQMC and its truncated variants

aim to achieve convergence to their respective targets (e.g., FCI for FCIQMC, CISD for

CISD-MC, CISDT for CISDT-MC, CISDTQ for CISDTQ-MC, etc.) in the limit of infinite

imaginary time employing a walker population dynamics algorithm. This algorithm samples

the many-electron Hilbert space, allocating more walkers to determinants of higher impor-

tance and fewer walkers to those of lesser importance. While the original CIQMC algorithm

of Ref. [91] exhibits slow convergence towards its target CI solutions, recent advancements,

such as the initiator approximation [92] and its adaptive shift modifications [93, 94], have

been introduced to accelerate the convergence of CIQMC. Along with CIQMC, Coupled

Cluster Monte Carlo (CCMC) methods have been developed [95–98], substituting linear CI

expansions with the exponential CC ansatz when propagating wave functions in the many-

electron Hilbert space.

As all stochastic approaches, the CIQMC/CCMC algorithms suffer from large numerical

noise in the early stages of wave function propagations, but they are excellent in identifying

the leading determinants in the wave functions of interest. This observation forms the

basis of the semi-stochastic CC(P;Q) methods, initially proposed in Ref. [99] and further

developed in Refs. [100–102] (see, also Ref. [137]), where lists of Slater determinants extracted

from CIQMC/CCMC propagations are used to construct the P spaces for the underlying

CC(P ) and EOMCC(P ) computations, and then the remaining determinants relevant to the

CC/EOMCC target approach, such as CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ,

etc., not captured by CIQMC/CCMC are used to build the Q spaces defining the noniterative

CC(P ;Q) corrections. The purely stochastic approach to solving the electronic Schr¨odinger

equation using the CIQMC [91–94] and CCMC [95–98] algorithms is plagued by the noise

12

in energies and properties associated with the varying walker populations and this can only

be reduced by using very long propagation times, large walker populations, and additional

techniques such as blocking analysis. This limits the applicability of the purely stochastic

CIQMC and CCMC calculations to about 40-50 correlated electrons. On the other hand,

the semi-stochastic CC(P;Q) formalism of Refs. [99–102], which limits the use of CIQMC

or CCMC to short wave function propagations sufficient to the identify the dominant wave

function components that are subsequently handled by the deterministic CC(P )/EOMCC(P )

computations followed by the CC(P;Q) corrections to capture the remaining information,

eliminates the stochastic noise and, as shown in Refs. [99–102], enables us to reproduce

the target high-level CC/EOMCC energetics out of the early stages of QMC propagations.

The semi-stochastic CC(P;Q) formalism [99–102] and its underlying CC(P )/EOMCC(P )

counterpart [99–102, 137] have shown a lot of promise in the calculations for singlet ground

and excited states, including PESs along bond breaking coordinates [99–101], but prior to

my involvement, nothing has been done for open-shell systems and non-singlet electronic

states. In this dissertation, I will discuss our efforts on extending and applying the semi-

stochastic CC(P;Q) algorithm to ground and excited states of open-shell systems, such as

radicals, and other problems involving non-singlet electronic states, such as singlet–triplet

gaps in biradicals [100, 102].

Another way to tackle the challenging problem of open-shell systems, such as radicals and

biradicals, within the SRCC framework is to use the formalism of the electron-attachment

(EA) [78, 79, 138–142] and ionization-potential (IP) [78, 79, 140–146] EOMCC theories, and

their double electron-attachment (DEA) and double ionization-potential (DIP) extensions

[80, 147–157]. Here, the wave functions of the ground and excited states of (N ± 1)-electron

(EA/IP) and (N ± 2)-electron (DEA/DIP) open-shell systems are constructed using linear

EOM-type operators, which attach electrons to or remove electrons from the ground-state

CC wave functions of an N -electron closed-shell core. The use of a closed-shell reference

state ensures that the EA/IP and DEA/DIP EOMCC wave functions are automatically

13

orthogonally spin adapted and, thus, preserve the spin symmetry of the Hamiltonian, pro-

viding an elegant and powerful way of studying radicals, biradicals, and other systems that

differ by one or two electrons from the underlying closed-shell cores.

In analogy to the

previously discussed particle-conserving EOMCC schemes, where the reference and target

systems contain the same number of electrons, the EA/IP and DEA/DIP EOMCC method-

ologies also suffer from the fact that their lower level approximations, such as EA-EOMCCSD

and IP-EOMCCSD, truncated at the 2-particle-1-hole (2p-1h) and 2-hole-1-particle (2h-1p)

components, respectively, and their DEA/DIP counterparts truncated at 3p-1h and 3h-1p

terms [147–156] cannot accurately describe the electronic spectra of radicals and biradicals

[78–80, 140–142, 153–156]. To obtain accurate spectra of radicals and biradicals, one needs

to incorporate 3p-2h/3h-2p and 4p-2h/4h-2p terms in the respective EA/IP and DEA/DIP

EOMCC approaches. Unfortunately, as soon as we incorporate these terms, the calculations

become very (usually prohibitively) expensive, limiting their applicability to smaller many-

electron systems. In analogy to the particle conserving CC/EOMCC methods, this difficulty

can be addressed with the help of user- and system-dependent active orbitals to reduce com-

putational costs [78–80, 140–142, 154–156], but, once again, the question emerges if one could

automate the procedure of selecting the higher-rank many-body components of the relevant

Rµ-type electron-attaching or ionizing operators, such as the 3p-2h, 3h-2p, 4p-2h, and 4h-2p

terms. Answering this question using the aforementioned stochastic CIQMC ideas within the

EA/IP and DEA/DIP EOMCC frameworks has been one of my other method development

research projects and is the second main objective of this dissertation. The effectiveness and

utility of the semi-stochastic, CIQMC-driven, CC(P;Q) and EA/IP/DEA/DIP EOMCC ap-

proaches and computer codes developed in this dissertation research in applications involving

ground and excited states of open-shell species and lowest-energy singlet and triplet states

of biradical systems will be illustrated through several molecular examples.

14

CHAPTER 2

BACKGROUND INFORMATION

2.1 Single-Reference Coupled-Cluster Theory and Its Equation-of-Motion Ex-

tension to Excited Electronic States

As already mentioned in the Introduction, in the SRCC formalism, the exact ground-state

wave function of an N -electron system is expressed as

with

|Ψ0⟩ = eT |Φ⟩,

T =

N
X

n=1

Tn,

(2.1)

(2.2)

where |Φ⟩ is an independent-particle-model reference state that serves as the Fermi vacuum

(usually, a HF Slater determinant) and T is the cluster operator. The n-body or np-nh

component of the cluster operator is defined as

Tn = X

a1...anEa1...an
ti1...in
i1...in ,

i1<···<in
a1<···<an

(2.3)

where ti1...in

a1...an are the cluster amplitudes and Ea1...an

i1...in are the elementary np-nh excitation

operators defined as Ea1...an

i1...in = aa1 . . . aanain . . . ai1, with ap and ap representing the usual

creation and annihilation operators associated with spin-orbital |p⟩, respectively, which gen-

erate excited Slater determinants |Φa1...an

i1...in ⟩ when acting on |Φ⟩. Here, we use indices i1, i2 . . .

or i, j, . . . to denote the occupied spin-orbitals and indices a1, a2, . . . or a, b, . . . to denote

the unoccupied spin-orbitals in the reference determinant |Φ⟩. In practice, we truncate the

many-body expansion defining the cluster operator T at an excitation rank mA < N and

different choices of mA result in the conventional CC hierarchy. For example, mA = 2 results

in the basic CCSD approximation in which T is defined as T (CCSD) = T1 + T2, mA = 3

produces CCSDT [T (CCSDT) = T1 + T2 + T3], mA = 4 generates the CCSDTQ approach

[T (CCSDTQ) = T1 + T2 + T3 + T4], etc. To obtain the many-body components Tn and the

15

energy corresponding to a particular truncation of the cluster operator, denoted here as

method A, one first inserts the exponential form of the wave function, Eq. (2.1), into the

Schr¨odinger equation to get its connected-cluster form,

¯H|Φ⟩ = E0|Φ⟩,

(2.4)

where ¯H = e−T HeT = (HeT )C is the similarity-transformed Hamiltonian and the subscript

C designates the connected part of a given operator product. Next, Eq. (2.4) is projected

onto the set of excited determinants |Φa1...an

i1...in ⟩ defined by the level of truncation mA ≤ N and

T is replaced by T (A) = PmA

n=1 Tn to obtain the system of non-linear equations,

⟨Φa1...an

i1...in | ¯H (A)|Φ⟩ = 0,

i1 < · · · < in,

a1 < · · · < an, n = 1, . . . , mA,

(2.5)

which are solved iteratively to yield the relevant cluster amplitudes ti1...in

a1...an. The ground-

state SRCC energy, E(A)

0

, is obtained by calculating the expectation value of the similarity-

transformed Hamiltonian ¯H (A) = e−T (A)HeT (A) = (He−T (A))C with respect to the Fermi

vacuum |Φ⟩,

E(A)

0 = ⟨Φ| ¯H (A)|Φ⟩.

(2.6)

As an illustration, in the basic CCSD approach, where mA = 2, one must first solve the
ij | ¯H (CCSD)|Φ⟩ = 0, where ¯H (CCSD) =

i | ¯H (CCSD)|Φ⟩ = 0 and ⟨Φab

non-linear equations ⟨Φa

e−T1−T2HeT1+T2 = (HeT1+T2)C, to obtain the cluster amplitudes ti

a and tij

ab and, subsequently,

the CCSD energy is computed as E(CCSD)

0

= ⟨Φ| ¯H (CCSD)|Φ⟩.

The SRCC theory can be extended to excited states using the EOM formalism, where

one applies a linear, CI-like, excitation operator Rµ to the CC ground state defined by Eq.

(2.1). Hence, the EOMCC ansatz for the wave function of the µth excited state, |Ψµ⟩, is

where

|Ψµ⟩ = Rµ|Ψ0⟩ = RµeT |Φ⟩,

Rµ = rµ,01 +

N
X

n=1

Rµ,n,

16

(2.7)

(2.8)

with

Rµ,n = X

µ,a1...anEa1...an
ri1...in
i1...in

i1<···<in
a1<···<an

(2.9)

representing the n-body (np-nh) component of Rµ and 1 designating the unit operator (we

use the convention in which µ = 0 represents the ground state and µ > 0 designates excited

states). Typically, we truncate the excitation operator Rµ at the same level of truncation

as the cluster operator T and, as a result, we obtain the hierarchy of EOMCC approxima-

tions similar to the ground state problem. Thus, mA = 2 gives rise to the basic EOM-

CCSD approximation [R(EOMCCSD)

µ

= rµ,01 + Rµ,1 + Rµ,2], mA = 3 produces EOMCCSDT

[R(EOMCCSDT)

µ

= rµ,01 + Rµ,1 + Rµ,2 + Rµ,3], mA = 4 results in the EOMCCSDTQ approach

[R(EOMCCSDTQ)

µ

= rµ,01 + Rµ,1 + Rµ,2 + Rµ,3 + Rµ,4], etc. To obtain the equations for the

excitation amplitudes ri1...in

µ,a1...an and energies E(A)

µ , one inserts Eq. (2.7) into the Schr¨odinger

equation and projects the resulting equation onto the set of excited Slater determinants

|Φa1...an

i1...in ⟩ = Ea1...an

i1...in |Φ⟩ with mA ≤ N , to obtain an eigenvalue problem

⟨Φa1···an

i1···in |( ¯H (A)

openR(A)

µ,open)C|Φ⟩ = ω(A)

µ ri1···in

µ,a1···an,

(2.10)

i1 < · · · < in,

a1 < · · · < an, n = 1, . . . , mA,

where ¯H (A)

open = ¯H (A) − ¯H (A)

closed = ¯H (A) − E(A)

0

1 and R(A)

µ,open = R(A)

parts of ¯H (A) and R(A)

µ having external Fermion lines and ω(A)

µ = E(A)

µ − rµ,01 are the open
µ − E(A)

is the vertical

0

excitation energy corresponding to the excited state. After solving Eq. (2.10) (typically,

using the iterative Davidson-type diagonalization procedure generalized to non-Hermitian

eigenvalue problems) for amplitudes ri1...in

µ,a1...an and excitation energies ω(A)

µ , we determine the

coefficient rµ,0 entering Rµ, which is nonzero for excited states of the same symmetry as the

ground state, by using the expression

rµ,0 = ⟨Φ|( ¯H (A)

openR(A)

µ,open)C|Φ⟩/ω(A)
µ .

(2.11)

It is important to point out that the similarity-transformed Hamiltonian ¯H (A) is not

Hermitian. This implies that for the CC and EOMCC theories, the “bra” and “ket” eigen-

17

vectors of ¯H (A) corresponding to a given eigenvalue are completely different. Therefore, we

have to distinguish between the right or ket CC and EOMCC states defined by Eqs. 2.1 and

2.7, respectively, and their left or bra counterparts. The appropriate ansatz for the left CC

(µ = 0) and EOMCC (µ > 0) states matching their |Ψ0⟩ and |Ψµ⟩ is as follows:

⟨ ˜Ψµ| = ⟨Φ|Lµe−T ,

(2.12)

where Lµ is a hole-particle de-excitation operator that satisfies the biorthonormality condi-

tion

⟨ ˜Ψµ|Ψν⟩ = ⟨Φ|LµRν|Φ⟩ = δµν.

Here, δµν is the Kronecker delta and the de-excitation operator Lµ is defined as

where

Lµ = δµ01 +

N
X

n=1

Lµ,n,

Lµ,n = X

µ,i1...in(Ea1...an
la1...an

i1...in )†

i1<···<in
a1<···<an

(2.13)

(2.14)

(2.15)

is the n-body (np-nh) component of Lµ. In analogy to T and Rµ, we approximate Lµ by

L(A)

µ , in which the summation over n is truncated at the many-body rank mA, and determine

the amplitudes la1...an

µ,i1...in by solving the left eigenvalue problem

δµ0⟨Φ| ¯H (A)

open|Φa1...an

µ,i1...in⟩ + ⟨Φ|L(A)

µ,open

¯H (A)

open|Φa1...an

µ,i1...in⟩ = ω(A)

µ la1...an

µ,i1...in,

(2.16)

i1 < · · · < in,

a1 < · · · < an, n = 1, . . . , mA.

This left eigenvalue problem is crucial because it allows one to compute properties other

than energy within the CC and EOMCC formalisms. As it turns out, it is also essential

in defining robust noniterative energy corrections to lower-level CC/EOMCC theories, used

by the CR-CC(2,3), CR-EOMCC(2,3), and other biorthogonal CR-CC/EOMCC approaches

and their CC(P;Q) generalization, which we discuss next.

18

2.2 The CC(P;Q) Formalism

The CC(P;Q) framework [70, 84, 87, 88], a generalization of the biorthogonal MMCC [63–

65, 68, 70] formalism to unconventional truncations in T and Rµ, comprises two steps. In

the first step, abbreviated as CC(P) for the ground (µ = 0) state and EOMCC(P) for

excited (µ > 0) states, one solves the CC/EOMCC equations in the subspace of the N -

electron Hilbert space referred to as the P space, denoted as H(P ). This space is spanned by

the excited determinants |ΦK⟩ = EK|Φ⟩ which, together with the reference determinant |Φ⟩,

dominate the ground- and excited-state wave functions |Ψµ⟩ of interest (EK is the elementary

particle–hole excitation operator generating |ΦK⟩ from |Φ⟩; for the sake of brevity of this

description, we assume that ground and excited states have the same symmetry; excited

states having different symmetries than the ground state are addressed later). This is done

in a usual way adopted in all single-reference CC and EOMCC calculations, i.e., one starts

with obtaining the cluster operator

T (P ) = X

tKEK,

|ΦK ⟩∈H(P )

(2.17)

with tK representing the corresponding cluster amplitudes by solving the system of equations

⟨ΦK| ¯H (P )|Φ⟩ = 0,

|ΦK⟩ ∈ H(P ),

(2.18)

where ¯H (P ) = e−T (P )HeT (P ) = (HeT (P ))C is the relevant similarity-transformed Hamiltonian,

and determining the corresponding ground-state energy

E(P )

0 = ⟨Φ| ¯H (P )|Φ⟩.

(2.19)

Then, the similarity-transformed Hamiltonian ¯H (P ) is diagonalized in the P space H(P ) to

determine the excited-state EOMCC(P ) energies E(P )

µ

and the corresponding EOM excita-

tion and de-excitation operators,

µ = rµ,01 + X
R(P )

rµ,KEK

|ΦK ⟩∈H(P )

(2.20)

19

and

µ = δµ01 + X
L(P )

lµ,K(EK)†,

|ΦK ⟩∈H(P )

(2.21)

respectively, where rµ,K and lµ,K designate the relevant amplitudes, which define the EOMCC(P )

ket states

|Ψ(P )

µ ⟩ = R(P )

µ eT (P )|Φ⟩

and the CC(P )/EOMCC(P ) bra states

⟨ ˜Ψ(P )

µ | = ⟨Φ|L(P )

µ e−T (P )

(2.22)

(2.23)

satisfying ⟨ ˜Ψ(P )

µ |Ψ(P )

ν

⟩ = δµν. Once all of this is done, one proceeds to the second step, which

is the calculation of the noniterative corrections δµ(P ;Q) to the CC(P) and EOMCC(P)

energies E(P )

µ

that account for the remaining many-electron correlation effects of interest

captured with the help of the another subspace of the N -electron Hilbert space, referred

to as the Q space and designated as H(Q) [H(Q) ⊆ (H(0) ⊕ H(P ))⊥, where H(0) is a one-

dimensional subspace spanned by |Φ⟩]. The expression for these corrections is

where

when µ = 0, and

δµ(P ;Q) = X

ℓµ,K(P )Mµ,K(P ),

|ΦK ⟩∈H(Q)

M0,K(P ) = ⟨ΦK| ¯H (P )|Φ⟩,

Mµ,K(P ) = ⟨ΦK| ¯H (P )R(P )

µ |Φ⟩,

(2.24)

(2.25)

(2.26)

when µ > 0, are the generalized moments of the CC(P) and EOMCC(P) equations that

correspond to projections of the P space CC/EOMCC equations on to the complementary Q-

space determinants |ΦK⟩ ∈ H(Q) and the coefficients ℓµ,K(P ) multiplying moments Mµ,K(P )

in Eq. (2.24) are calculated with the expression

ℓµ,K(P ) = ⟨Φ|L(P )

µ

¯H (P )|ΦK⟩/Dµ,K(P ),

(2.27)

20

in which the Dµ,K(P ) denominators are given by

Dµ,K(P ) = E(P )

µ − ⟨ΦK| ¯H (P )|ΦK⟩

(2.28)

(one could replace the Epstein–Nesbet Dµ,K(P ) denominators entering Eq. (2.27) ℓµ,K(P )

by their Møller–Plesset analogs, but, as shown in the past, for example in Refs. [63, 64, 66,

68, 84, 88, 101], the Epstein–Nesbet form is generally more effective). The final CC(P;Q)

electronic energies for the ground (µ = 0) and excited (µ > 0) states are determined as

E(P +Q)
µ

= E(P )

µ + δµ(P ; Q).

(2.29)

Now, a question arises as to how to define the P and Q spaces entering the CC(P;Q)

framework to obtain accurate ground- and excited-state energetics that match the quality of

the target high-level CC/EOMCC calculations without incurring the substantial computa-

tional costs of methods, such as CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ, etc. The

simplest possibility is to rely on the conventional choices, where the P and Q spaces are

defined based on the many-body ranks of the excited determinants |ΦK⟩ included in them.

For example, if we want to correct the CCSD and EOMCCSD energies for triples using the

CC(P;Q) formulas, the P space H(P ) is spanned by the singly and doubly excited determi-

nants |Φa

i ⟩ and |Φab

ij ⟩, respectively, and the Q space H(Q) by the triply excited determinants

|Φabc

ijk⟩. As already alluded to above, the resulting CR-CC(2,3) and CR-EOMCC(2,3) correc-

tions to the CCSD and EOMCCSD energies have been very successful, but, by decoupling

the low-order Tn and Rµ,n components with n ≤ 2 from their higher-order, such as T3 and

Rµ,3 counterparts, they may not be as accurate as desired, for example in situations where

T3 and Rµ,3 components become large, resulting in substantial inaccuracies and difficulties

in balancing ground- and excited-state energies. One can address this problem by using

active orbitals to enrich the relevant P spaces with the dominant higher–than–doubly ex-

cited determinants, as in the CC(t;3), CC(t,q;3), CC(t,q;3,4), etc., methodologies mentioned

in the Introduction, but the resulting approaches are no longer computational black boxes.

The semi-stochastic CC(P;Q) approach to ground- and excited-state calculations [99–102],

21

which is described in detail in Chapter 3, exploits the CIQMC or CCMC propagations to

identify the leading higher–than–doubly excited determinants pertinent to the CC/EOMCC

calculations of interest in a black-box manner, while using corrections δµ(P ;Q) to capture

the remaining correlations that the CC(P)/EOMCC(P) energies at a given QMC propaga-

tion time do not describe, eliminates the above concerns. Since one of the objectives of this

dissertation project is to extend the semi-stochastic CC(P;Q) ideas to the EA/IP/DEA/DIP

EOMCC methods, these particle nonconserving approaches are discussed next.

2.3 Particle Nonconserving Equation-of-Motion Coupled-Cluster Theories: The
Single and Double Electron Attachment and Ionization Potential Methods

Radicals, biradicals, and other open-shell species and their low-lying excited states constitute

a major challenge to CC and other ab initio methods due to, in most cases, their inherent

multiconfigurational character that is difficult to capture using the low-rank approximations,

such as CCSD, EOMCCSD, and their linear response analogs. One can always think of using

genuine MRCC and other MR methodologies, but as pointed out in the Introduction, their

routine use is not straightforward and they have their own, often severe, challenges. The

failures of CCSD/EOMCCSD in the presence of electronic quasi-degeneracies in open-shell

systems imply the need to incorporate higher–than–double excitations in the CC/EOMCC

wave functions, but this results in several computational and formal difficulties. As already

discussed, the full inclusion of higher–than–two-body components in the cluster and EOM

excitation operators improves the results, but the associated computer costs become very

(often prohibitively) large as we increase the system size. One can think of resorting to the

active-space ideas, giving rise to methods such as CCSDt/EOMCCSDt, CCSDtq/EOMCCS-

Dtq etc., but with the usual spin-integrated spin-orbital formulation, neither these methods

nor any other particle conserving CC/EOMCC methodology properly account for the spin

symmetry of the calculated states of open-shell systems. There is, however, an interesting,

underappreciated, alternative discussed in this section and pursued in my doctoral work.

One can deal with the above problems by utilizing the flexibility of the EOMCC formal-

22

ism and resorting to the particle-nonconserving EA/IP and DEA/DIP EOMCC frameworks.

These approaches provide an elegant way of obtaining orthogonally spin-adapted results

for the ground and excited states of radicals and biradicals by diagonalizing the similarity-

transformed Hamiltonian of a closed-shell CC theory in the subspaces of the Fock space

obtained by adding one or two electrons to (EA, DEA) or removing one or two electrons

from (IP, DIP) a related closed-shell core.

In these kinds of methods, one starts with a CC ground-state wave function |Ψ0⟩ = eT |Φ⟩

of an N -electron closed-shell system for which the corresponding correlation energy is defined

as

∆E(N )

0 = E(N )

0 − ⟨Φ|H|Φ⟩ = (HN eT )C,closed,

(2.30)

where ⟨Φ|H|Φ⟩ is the reference energy, HN = H −⟨Φ|H|Φ⟩ is the Hamiltonian operator in the

normal ordered form relative to the N -electron Fermi vacuum |Φ⟩, the superscript (N ) indi-

cates the number of electrons in the closed-shell system, and T is the corresponding cluster

operator. In the EA and IP-EOMCC theories, the similarity-transformed Hamiltonian,

¯HN,open = (HN eT )C,open = e−T HN eT − ∆E(N )

0

,

(2.31)

corresponding to the N -electron closed-shell CC theory, is diagonalized in the (N +1)-electron

(EA) and (N − 1)-electron (IP) subspaces, H (N +1) and H (N −1), respectively, of the Fock

space. In doing so, the ground and excited-state wave functions of the target (N + 1)- and

(N − 1)-electron systems of interest are defined as

|Ψ(N ±1)
µ

⟩ = R(N ±1)
µ

|Ψ(N )
0

⟩ = R(N ±1)
µ

eT |Φ⟩,

(2.32)

where the cluster operator T is obtained by solving the usual ground-state CC equations for

the N -electron closed-shell system. R(N +1)

µ

is an electron-attaching operator defined as

R(N +1)
µ

=

MRX

n=0

Rµ,(n+1)p-nh,

(2.33)

23

where

Rµ,(n+1)p-nh = X

µ,aa1...anEaa1...an
ri1...in
i1...in

(n ≥ 1).

(2.34)

i1>···>in
a<a1<···<an

R(N −1)
µ

is an electron ionizing operator defined as

R(N −1)
µ

=

MRX

n=0

Rµ,(n+1)h-np,

with

Rµ,(n+1)h-np = X

µ,a1...anEa1...an
rii1...in

ii1...in (n ≥ 1).

i>i1>···in
a1<···<an

(2.35)

(2.36)

Here, MR = N in the exact theory and MR < N in approximate EA/IP EOMCC schemes.

Similarly to the particle-conserving EOMCC calculations, we insert Eq. (2.32) into the

Schr¨odinger equation to obtain the EOMCC-type eigenvalue problem for the |Ψ(N ±1)

µ

states

or the R(N ±1)

µ

operators that define them,

( ¯HN,openR(N ±1)

µ

)C|Φ⟩ = ω(N ±1)

µ

R(N ±1)
µ

|Φ⟩,

where

ω(N ±1)
µ

= E(N ±1)
µ

− E(N )
0

(2.37)

(2.38)

are the corresponding electron attaching (ω(N +1)

µ

) or electron ionizing (ω(N −1)

µ

) energies.

After solving this eigenvalue problem, one obtains the electron-attachment or ionization

energies directly. As the similarity-transformed Hamiltonian, ¯HN,open, of the underlying

closed-shell N -electron system commutes with S2 and Sz spin operators, the EA-EOMCC
and IP-EOMCC wave functions obtained by diagonalizing ¯HN,open in the appropriate sub-

spaces of H (N +1) and H (N −1) are automatically the eigenfunctions of both the S2 and Sz

operators and thus free from the spin-contamination issues present in the usual open-shell

EOMCC schemes employing unrestricted HF or restricted open-shell HF (ROHF) references.

24

The most basic approximations in the EA- and IP-EOMCC categories are the EA-

EOMCCSD or EA-EOMCC(2p-1h) and IP-EOMCCSD or IP-EOMCC(2h-1p) approaches,

where the cluster operator

T = T1 + T2

(2.39)

is obtained by solving the CCSD equations for the N -electron closed-shell system, the elec-

tron attaching operator R(N +1)

µ

defining the (N +1)-electron target states |Ψ(N +1)

µ

⟩ is given

by

R(N +1)
µ

= Rµ,1p + Rµ,2p-1h = X

rµ,aaa + X

a

j,a<b

rj
µ,abaaabaj,

(2.40)

and the ionizing operator R(N −1)

µ

needed to generate the (N -1)-electron states |Ψ(N −1)

µ

⟩ is

defined as

R(N −1)
µ

= Rµ,1h + Rµ,2h-1p = X

µai + X
ri

rij
µ,babajai.

(2.41)

i

i>j,b

In these two cases, we end up with diagonalizing the similarity-transformed Hamiltonian of

CCSD, (HN eT1+T2)C,open, in the space spanned by |Φa⟩ = aa|Φ⟩ and |Φab

j ⟩ = aaabaj|Φ⟩

determinants for EA-EOMCC(2p-1h) and the space spanned by the |Φi⟩ = ai|Φ⟩ and

|Φb

ij⟩ = abajai|Φ⟩ determinants when the IP-EOMCC(2h-1p) calculations are performed.

Although the EA-EOMCC(2p-1h) and IP-EOMCC(2h-1p) methods can be useful in obtain-

ing the lowest electron affinities and ionization potentials, they struggle to describe electron

attachment and ionization energies corresponding to excited states of anions and cations,

what is particularly relevant for this dissertation, the electronic excitations in radicals, es-

pecially when the target states have inherently multiconfigurational character. The good

news is that one can usually address this issue by incorporating the higher-rank 3p-2h/3h-2p

components in the R(N ±1)

µ

operator expressions i.e., by using the

and

R(N +1)
µ

= Rµ,1p + Rµ,2p-1h + Rµ,3p-2h,

R(N −1)
µ

= Rµ,1h + Rµ,2h-1p + Rµ,3h-2p

25

(2.42)

(2.43)

operators in the EA/IP-EOMCC computations. Now, there are two options: either one uses

Eqs. (2.42) and (2.43) for the electron attaching and ionizing operators and the similarity-

transformed Hamiltonian of CCSD, ¯H (CCSD)

N,open = (HN eT1+T2)C,open, resulting in the EA-EOMCC(3p-2h)

and IP-EOMCC(3h-2p) approaches, respectively, or, one keeps the same definitions of the

R(N ±1)
µ

operators as in Eqs. (2.42) and (2.43), but replaces ¯H (CCSD)

N,open by the similarity-

transformed Hamiltonian of the CCSDT theory, ¯H (CCSDT)

N,open = (HN eT1+T2+T3)C,open, resulting

in the EA-EOMCCSDT and IP-EOMCCSDT methods. Both classes of methods are similarly

effective at describing the electronic spectra of the (N ± 1) electron radical systems, so we

will focus on EA-EOMCC(3p-2h) and IP-EOMCC(3h-2p), which do not require solving the

CCSDT equations, but they may still be too expensive in practice due to the relatively high

computational costs of diagonalizing ¯H (CCSD)

N,open in spaces containing |Φabc

jk ⟩ = aaabacakaj|Φ⟩

and |Φbc

ijk⟩ = abacakajai|Φ⟩ determinants that scale as N 7 with the system size.

One way to deal with the problem of high costs of EA-EOMCC(3p-2h) and IP-EOMCC(3h-

2p) computations is to resort to the active-space ideas, similar to the ground-state CC and

excited-state EOMCC cases, by including only the leading components of the Rµ,3p-2h/Rµ,3h-2p

operators defined by active orbitals. These active-space components, defined in Refs. [78,

79, 140–142], will be denoted as rµ,3p-2h/rµ,3h-2p. Therefore, one incorporates all singly

and doubly excited amplitudes of the cluster operator T1 and T2, all attaching Rµ,1p and

Rµ,2p-1h amplitudes, and all electron ionizing Rµ,1h and Rµ,2h-1p amplitudes, but only small

subsets of 3p-2h/3h-2p amplitudes of R(N ±1)

µ

defined by active orbitals, giving rise to the

active-space EA/IP-EOMCC methods known as EA-EOMCCSDt and IP-EOMCCSDt or

EA-EOMCC(3p-2h){Nu} and IP-EOMCC(3h-2p){No}, where No and Nu are the numbers

of active occupied and active unoccupied orbitals in |Φ⟩, respectively [78, 79, 140–142]. In

the EA-EOMCCSDt = EA-EOMCC(3p-2h){Nu} approach, the cluster operator T of the

N -electron closed-shell core obtained with CCSD is defined as

T = T1 + T2

(2.44)

26

and the electron attaching operator given by Eq. (2.42) is replaced by

R(N +1)
µ

{Nu} = Rµ,1p + Rµ,2p-1h + rµ,3p-2h,

where

rµ,3p-2h = X

rjk
µ,AbcaAabacakaj.

j>k
A<b<c

(2.45)

(2.46)

In IP-EOMCCSDt = IP-EOMCC(3h-2p){No}, the same definition of the cluster operator is

used and the ionizing operator given by Eq. (2.43) is replaced by

with

R(N −1)
µ

{No} = Rµ,1h + Rµ,2h-1p + rµ,3h-2p,

rµ,3h-2p = X

rIjk
µ,bcabacakajaI.

I>j>k
b<c

(2.47)

(2.48)

The bold capital indices in the above expressions for rµ,3p-2h and rµ,3h-2p denote active

spin-orbitals in the respective categories (I for occupied and A for unoccupied). The EA-

EOMCCSDt = EA-EOMCC(3p-2h){Nu} and IP-EOMCCSDt = IP-EOMCC(3h-2p){No}

approaches are computationally much more affordable than their EA-EOMCC(3p-2h)/IP-

EOMCC(3h-2p) and EA-EOMCCSDT/IP-EOMCCSDT counterparts. They scale as Nun2

on4
u

and Non2

on4

u, respectively, in steps needed to diagonalize the similarity-transformed Hamil-

tonian, as opposed to n2

on5

u and n3

on4

u, respectively (No < no, Nu << nu) when 3p-2h/3h-2p

components are treated fully.

The EA/IP-EOMCC approaches can be extended to attach or remove more than one

electrons from a suitable closed-shell reference state, since the Rµ operator can be designed

to handle multiple electron attachments or removals, leading to the multiple electron attach-

ment or ionization potential EOMCC approaches. For instance, attaching or removing two

electrons from a closed-shell reference state resulting from CC calculations, while relaxing the

remaining electrons, gives rise to the DEA or DIP EOMCC approaches. In the DEA/DIP-

EOMCC schemes, the ground- and excited-state wave functions of the (N + 2)-electron (the

27

DEA case) or (N − 2)-electron (the DIP case) systems, where N is the number of electrons

in the underlying closed-shell species, are represented as

|Ψ(N ±2)
µ

⟩ = R(N ±2)
µ

eT |Φ⟩.

(2.49)

Here, once again, T is the cluster operator obtained from SRCC calculations for the N -

electron reference system (using the N -electron reference determinant |Φ⟩, which defines the

Fermi vacuum),

R(N +2)
µ

=

MRX

n=0

Rµ,(n+2)p-nh

is the double electron attachment operator, and

R(N −2)
µ

=

MRX

n=0

Rµ,(n+2)h-np

(2.50)

(2.51)

is the double ionizing operator, where MR ≤ N defines the truncation level. The individual

many-body components of the R(N ±2)

µ

operators are given by the expressions

Rµ,(n+2)p-nh = X

ri1...in
µ,aba1...anaaabaa1 . . . aanain . . . ai1,

(2.52)

i1>···>in
a<b<a1<···<an

in the DEA case, and

Rµ,(n+2)h-np = X

µ,a1...anaa1 . . . aanain . . . ai1ajai,
riji1...in

(2.53)

i>j>i1>···>in
a1<···<an

in the case of the DIP formalism. Similarly to the EA/IP-EOMCC approaches, inserting Eq.

(2.49) into the Schr¨odinger equation yields the corresponding DEA/DIP-EOMCC eigenvalue

problem

( ¯HN,openR(N ±2)

µ

)C|Φ⟩ = ω(N ±2)

µ

R(N ±2)
µ

|Φ⟩.

(2.54)

The DEA-EOMCC and DIP-EOMCC methods are particularly suitable for biradical sys-

tems, since they are capable of generating an appropriate multideterminantal reference space

by adding or removing two electrons from a closed-shell core, but in order to get quantita-

tive accuracies in most of the situations encountered in chemistry, especially for the singlet

28

manifolds and singlet–triplet gaps, one needs to incorporate the Rµ,4p-2h components, in the

DEA-EOMCC case, and the Rµ,4h-2p components, when the DIP-EOMCC methodology is

employed, in the corresponding R(N ±2)

µ

operators, giving rise to the DEA-EOMCC(4p-2h)

and DIP-EOMCC(4h-2p) approaches, respectively [80, 153–156]. Due to the enormous num-

bers of the 4p-2h/4h-2p amplitudes in the majority of realistic applications, the DEA/DIP-

EOMCC approaches with a full treatment of 4p-2h/4h-2p correlations are very expensive,

but this can again be dealt with by turning to the much more practical active-space DEA-

EOMCC(4p-2h) {Nu} and DIP-EOMCC(4h-2p) {No} approaches where the 4p-2h and 4h-2p

amplitudes are down-selected using active orbitals [80, 153–156]. This can be done by using

the following definitions of the R(N ±2)

µ

operators [80, 153–156]:

R(N +2)
µ

{Nu} = Rµ,2p + Rµ,3p-1h + rµ,4p-2h,

(2.55)

in the DEA-EOMCC(4p-2h){Nu} case, and

R(N −2)
µ

{No} = Rµ,2h + Rµ,3h-1p + rµ,4h-2p,

(2.56)

in the case of the DIP-EOMCC(4h-2p){No} approach, where

and

rµ,4p-2h = X

rkl
µ,ABcdaAaBacadalak

k>l
A<B<c<d

rµ,4h-2p = X

µ,cd acadalakaJaI,
rIJkl

I>J>k>l
c<d

(2.57)

(2.58)

with the capital bold indices representing active spin-orbitals in the respective occupied (I,

J) and unoccupied (A, B) categories.

As with all active-space CC/EOMCC methods, the introduction of active orbitals, which

are user and system dependent, into Eqs. (2.45)–(2.48) and Eqs. (2.55)–(2.58) means that the

resulting EA-EOMCC(3p-2h){Nu}, IP-EOMCC(3h-2p){No}, DEA-EOMCC(4p-2h){Nu}, and

DIP-EOMCC(4h-2p){No} methods are not black-box schemes. The novel semi-stochastic

29

EA/IP/DEA/DIP-EOMCC(P ) approaches to ground and excited states of radical and bi-

radical species, which we describe in Chapter 4, exploit the CIQMC propagations to auto-

matically identify the leading 3p-2h/3h-2p/4p-2h/4h-2p determinants pertinent to the cal-

culations of interest, and thus eliminate the above concerns. Before introducing these and

other semi-stochastic methods in Chapter 3, we provide a brief overview of the CIQMC

methodology exploited in such approaches.

2.4 Overview of Configuration Interaction Quantum Monte Carlo

In this section, we summarize the key ingredients of the FCIQMC methodology and its

truncated CIQMC counterparts. The FCIQMC approach introduced in Refs. [91, 92] shares

similarities with both the deterministic FCI and stochastic diffusion Monte Carlo (DMC)

methods. As mentioned in the Introduction, unlike the conventional DMC methodology,

FCIQMC does not suffer from the Fermion sign problem [158]. The basic idea of FCIQMC

is to propagate the imaginary-time Schr¨odinger equation written here(in atomic units),

∂|Ψ⟩
∂τ

= −H|Ψ⟩,

(2.59)

in the many-electron Hilbert space spanned by Slater determinants. It can be shown that

by starting from a trial wave function |Ψ(τ = 0)⟩, such as the HF Slater determinant, that

has a nonzero overlap with the exact FCI state |Ψ0⟩, and by applying the energy shift S

such that it approaches the FCI energy E0 as τ → ∞, in the infinite imaginary-time limit,

we project out |Ψ0⟩,

|Ψ0⟩ = lim
τ→∞
S→E0

e−τ (H−S)|Ψ(τ = 0)⟩.

(2.60)

In the many-electron Hilbert space, the propagated wave function |Ψ(τ )⟩ = e−τ (H−S)|Ψ(τ =

0)⟩ can be represented as a CI expansion

|Ψ(τ )⟩ = c0(τ )|Φ0⟩ + X

cK(τ )|ΦK⟩,

K

(2.61)

30

where the time-dependent CI coefficients at the Slater determinants |ΦK⟩ satisfy the system

of coupled differential equations [91]

∂cK(τ )
∂τ

= −(HKK − S)cK(τ ) − X

HKLcL(τ ),

(2.62)

L(̸=K)

where HKL = ⟨ΦL|H|ΦL⟩ are matrix elements of the Hamiltonian involving the various

Slater determinants |ΦK⟩. It can be readily demonstrated that in the τ → ∞ limit, with

S converging to E0, the system of equations given by Eq. (2.62) becomes equivalent to the

conventional CI eigenvalue problem

HKLcL(∞) = E0cK(∞).

(2.63)

X

L

Thus, when all possible Slater determinants are allowed in the wave function propagation de-

fined by Eq. (2.62), we obtain results equivalent to FCI. Furthermore, if we confine ourselves

to a subset of Slater determinants corresponding to one of the truncated CI methods, such

as CISD, CISDT, CISDTQ, etc., the τ = ∞ limit of Eq. (2.62) results in the corresponding

truncated CI solution. The main challenge with the system of equations represented by Eq.

(2.62) lies in the impracticality of achieving its direct deterministic integration. Thus, in the

spirit of DMC, the FCIQMC and truncated CIQMC schemes introduced in Refs.

[91] and

[92] employ a walker population dynamics to determine the time-dependent CI coefficients

cK(τ ) and, consequently, the energy E(τ ) corresponding to |Ψ(τ )⟩. The walkers, designated

here by α, are fictitious particles that populate Slater determinants and have signs sα = ±1,

and the CI coefficients cK(τ ) can be calculated as the signed sums of the walkers associated

with particular determinants |ΦK⟩, i.e.,

cK(τ ) ∼ NK = X

sαδKKα,

α

(2.64)

where |ΦKα⟩ designates the determinant on which the walker α is located and sα is the

associated sign. The resulting CIQMC algorithm defining the walker population dynamics

equivalent to Eq. (2.62) comprises three main steps that are executed at each time step τ of

the imaginary-time propagation:

31

1. The spawning step: For each walker α located on determinant |ΦKα⟩, we attempt to

spawn a new “child” walker on another determinant |ΦL⟩, selected through a normalized

probability

ps(L|Kα) =

∆τ |HKαL|
pgen(L|Kα)

,

(2.65)

which is compared to a number between 0 and 1 obtained with the random number

generator. In this step, if spawning occurs, the total number of walkers increases. Here,

∆τ represents the time step used in the imaginary-time propagation.

2. The diagonal death/cloning step: For each walker, we compute the probability

pd(Kα) = ∆τ (HKαKα − S).

(2.66)

If pd > 0, the walker is removed from the simulation, and if pd < 0, the walker is cloned

with the probability |pd|. The cloning (or birth) step is usually rare.

3. The annihilation step: Subsequent to the previous two steps, the annihilation step

is performed, wherein all pairs of walkers with opposite sign are removed from each

determinant.

A schematic representation of how the CIQMC algorithm works is depicted in Fig. 2.1.

After reaching a sufficiently large number of walkers, the correlation energy and the walker

populations stabilize using a suitable energy shift S. Since the propagation is performed

in the space of Slater determinants, the resulting wave function has the proper Fermionic

symmetry. As already alluded to above, if the spawning of walkers is restricted to a cer-

tain level of truncation, we can obtain truncated CIQMC approaches, such as CISDT-MC

and CISDTQ-MC to mention two representative examples, where spawning of walkers on

determinants higher than triples, in the CISDT-MC case, or higher than quadruples in the

CISDTQ-MC case, is forbidden. By replacing the time-dependent CI expansion, Eq. (2.61),

by the analogous CC ansatz, one obtains the CCMC approach introduced in Ref. [95] (see,

32

Figure 2.1 A schematic representation of the CIQMC algorithm. |Φ⟩ represents the reference
determinant, while {|ΦS1⟩, |ΦS2⟩, |ΦS3⟩, |ΦS4⟩, · · · } denote singly excited determinants,
{|ΦD1⟩, |ΦD2⟩, |ΦD3⟩, |ΦD4⟩, · · · } doubly excited determinants, {|ΦT1⟩, |ΦT2⟩, |ΦT3⟩, |ΦT4⟩,
· · · } triples, {|ΦQ1⟩, |ΦQ2⟩, |ΦQ3⟩, |ΦQ4⟩, · · · } quadruples, etc. The number of walkers
on a particular determinant is indicated as a superscript within parenthesis. Green and
red rectangles distinguish positive and negative walkers, respectively, with darker shades
indicating a higher number of walkers on the determinants. Here, the simulation starts with
one walker placed on the reference determinant |Φ⟩. Panel (a) illustrates spawning steps,
panel (b) depicts the death of a walker on the determinant |ΦD4⟩ with a gray box, panel (c)
shows more spawning events and panel (d) displays the annihilation step on |ΦQ3⟩ with an
orange rectangular box.

also, Refs. [96–98]). Several ideas have been explored to improve the convergence of the

QMC algorithm. One of them is the initiator CIQMC (i-CIQMC) approach of Ref. [92],

which we use in our work, and its i-CCMC counterpart, where spawning is allowed as long

as there is at least a minimum walker population, na, on the parent determinant. In the

next chapters, we discuss how we can take advantage of the CIQMC algorithm to accelerate

our CC(P;Q) and EA/IP/DEA/DIP-EOMCC computations, while making them fully au-

tomated and free from the user- and system-dependent active orbitals when identifying the

appropriate P spaces [CC(P;Q)] or the leading 3p-2h/3h-2p/4p-2h/4h-2p amplitudes.

33

<latexit sha1_base64="ExlCiW2eBMnmaPMLVFpcLjJchBo=">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</latexit>|i(1)|S1i(+11)|D1i(+11)|T1i(+11)|Q1i(+11)···|S2i(1)+1|D2i(1)+1|T2i(+11)|Q2i(+11)···|S3i(+11)|D3i(+11)|T3i(+11)|Q3i(+11)···|S4i(+11)|D4i(1)+1|T4i(+11)|Q4i(+11)···...............SpawningBirth & DeathAnnihilationSpawningPositive walkerNegative walker<latexit sha1_base64="HFqzazvHWN7SQuE/QrvlM2aFRXg=">AAAGJ3icnVRNb9NAEN0WG4r5Sos4cVlREaWqEtlpC1yQKpEDx0QkbaU4WOv1JlnVXlu7a2hk+cqJH8AViQt/hQsSIARH/gkbpxW1nVama1kaz8zbN2/GGjfyqZCm+Xtl9ZqmX7+xdtO4dfvO3Xu19Y0DEcYckwEO/ZAfuUgQnzIykFT65CjiBAWuTw7d4xfz+OEbwgUNWV/OIjIK0ITRMcVIKpezrj23XTKhLAmQ5PQkNWyVLaHdnVJoc8QmPoGvE1uSE5k0drfSFNbhvxQneeVYaTExqyp5O6WSpElj22paZVwnj2u0t8rQ7aZVwvWvyNcr8FmX8WEvlALatmHU4dmz+MpLb+eubC69syyhU8BdKL0OC9Ir8hWVt6t17D+E71S+Mad7p9oIihO/EqxXgC1KulhjUeJuvtdLh1Qi7eRhVTvTL7JVnWxltnOyy5PNItnPdqnlzS3DMGzCvLN14dQ2zZaZHVg2rFNjc3/j47sPew/ed53aN9sLcRwQJrGPhBhaZiRHCeKSYp+o9RMLEiF8jCZkqEyGAiJGSaYqhY+Vx4PjkKuXSZh5zyMSFAgxC1yVqeqbimJs7lwWG8Zy/GyUUBbFkjC8IBrHPpQhnC9N6FFOsPRnykCYU1UrxFPEEZZqtRqqCVZRctk4aLesJ629nuqGCRZnDTwEj0ADWOAp2AcvQRcMANY+aV+079oP/bP+Vf+p/1qkrq6cYu6D3NH//AWjIQgT</latexit>|i(4)|S1i(+11)|D1i(2)+1|T1i(+11)|Q1i(1)+1···|S2i(1)+1|D2i(2)+1|T2i(1)+1|Q2i(+11)···|S3i(+11)|D3i(1)+1|T3i(1)+1|Q3i(+11)···|S4i(2)+1|D4i(+11)|T4i(1)+1|Q4i(+11)···...............<latexit sha1_base64="InRYT54SSqdoYpxJ2AuwlUDmi+k=">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</latexit>|i(1)|S1i(+11)|D1i(1)+1|T1i(+11)|Q1i(1)+1···|S2i(1)+1|D2i(2)+1|T2i(+11)|Q2i(+11)···|S3i(+11)|D3i(+11)|T3i(1)+1|Q3i(+11)···|S4i(1)+1|D4i(0)+1|T4i(+11)|Q4i(+11)···...............<latexit sha1_base64="eoiubsoUyoy6rjq2hWr5xdJT+k4=">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</latexit>|i(3)|S1i(+11)|D1i(2)+1|T1i(+11)|Q1i(1)+1···|S2i(1)+1|D2i(2)+1|T2i(1)+1|Q2i(+11)···|S3i(+11)|D3i(1)+1|T3i(1)+1|Q3i(+11)···|S4i(2)+1|D4i(+11)|T4i(1)+1|Q4i(+11)···...............(a)(b)(d)(c)CHAPTER 3

THE SEMI-STOCHASTIC CC(P;Q) METHODOLOGY FOR GROUND AND
EXCITED STATES

In the previous chapters, we have discussed how the deterministic CC(P;Q) theory that

combined the CR-CC/EOMCC and active space ideas emerged as a very robust tool for

studying challenging chemical systems in a computationally affordable way. The only draw-

back, however, is the need to choose a suitable set of active orbitals that can generate an

ensemble of determinants, which define the proper active space, and this selection is system

and user dependent. On the other hand, the QMC methods based on propagating the CI

ansatz according to the imaginary time Schr¨odinger equation in the many electron Hilbert

space spanned by Slater determinants are very efficient in identifying the most important de-

terminants, based on probabilistic arguments, in an automated manner, but they spend a lot

of time balancing the corresponding coefficients. In this chapter we discuss a novel hybrid

approach, namely the semi-stochastic CC(P;Q) formalism, that merges the deterministic

CC(P;Q) framework with the stochastic wave function sampling of CIQMC methodologies

[99–102].

In particular, we focus on my work on extending the semi-stochastic CC(P;Q)

methodology to ground and excited states of open-shell systems and non-singlet excited

states. In doing so, we follow the results presented in Refs. [100] and [102], with a focus on

my contributions to these papers.

3.1 Theory

The CIQMC algorithm needs a very long propagation time to converge a stable wave func-

tion, but the leading determinants are identified much sooner, and on the other hand the

deterministic CC(P;Q) framework allows for very accurate energetics even in challenging

MR situations as long as the lower clusters, such as T1 and T2, are allowed to relax in the

presence of the higher order T3, T4, etc., clusters. This observation was utilized in Ref. [99],

where it was demonstrated that one could use the information about the leading determi-

nants captured during the early stages of i-CIQMC to create lists of determinants defining

34

P spaces for CC(P ) calculations and then use the deterministic, noniterative, CC(P;Q)

corrections to capture the correlations effects missing in the P -space CC(P ) calculations.

Later in Ref. [137], the semi-stochastic CC(P ) approach was extended to the excited state

EOMCC(P ) method that provided a fast convergence to target energetics in a very straight-

forward manner, without resorting to more-complex excited state i-CIQMC frameworks of

Refs. [159, 160]. In this work we discuss the extension of the semi-stochastic ground-state

CC(P;Q) [99] and EOMCC(P ) [137] approaches to open-shell systems and non-singlet ex-

cited states and the CC(P;Q) corrections to EOMCCC(P ), as described in Refs. [100] and

[102].

The key steps of the semi-stochastic CC(P;Q) [99–102, 137] algorithm for ground and

excited states that resulted from the merger of deterministic CC(P;Q) algorithm and the

stochastic CIQMC wave function sampling, are described below.

1. Initiate a CIQMC run for the ground state and, if the system of interest has spin,

spatial, or other symmetries, the analogous QMC propagation for the lowest state of

each irreducible representation (irrep) to be considered in the CC(P;Q) calculations by

placing a certain number of walkers on the appropriate reference function(s) |Φ⟩ (e.g.,

the restricted Hartree–Fock (RHF) or restricted open- shell Hartree–Fock (ROHF)

determinants).

2. At some propagation time τ > 0, i.e., after a certain number of CIQMC time steps,

called MC iterations, extract a list or, if states belonging to multiple irreps are targeted,

lists of determinants relevant to the desired CC(P;Q) computations from the QMC

propagation(s) initiated in step 1 to determine the P space or spaces needed to set

up the ground-state CC(P ) and excited state EOMCC(P ) calculations (for example,

cf. Fig. 3.1).

If the goal is to converge the CCSDT/EOMCCSDT-level energetics,

the P space for the CC(P ) calculations and the EOMCC(P ) calculations for excited

states belonging to the same irrep as the ground state is defined as all singly and

35

Figure 3.1 A schematic illustration depicting the construction of P -spaces in CC(P) and
EOMCC(P) computations. Panel (a) showcases the stabilization of correlation energy (green
line) and the corresponding increase in the total number of walkers is shown in panel (b)
(red line). On the right four snapshots from a QMC calculation are presented, featuring the
lists of determinants picked up by the QMC algorithm at various time steps (green for 2000,
orange for 20000, violet for 50000, and magenta for 100000 QMC iterations). It is evident
that QMC deems some determinants more important than others by placing more walkers
on them.

doubly excited determinants and a subset of triply excited determinants, where each

triply excited determinant in the subset is populated by a minimum of nP positive

or negative walkers (in this work, nP = 1). For the excited states belonging to other

irreps, the P space defining the CC(P ) problem is the same as that used in the case of

the ground state, but the lists of triply excited determinants defining the EOMCC(P )

diagonalizations are provided by the CIQMC propagations for the lowest-energy states

of these irreps. One proceeds in a similar way when the goal is to converge other types

of high-level CC/EOMCC energetics. For example, if we want to obtain the results of

the CCSDTQ/EOMCCSDTQ quality, we also have to extract the lists of quadruples,

in addition to the triples, from the CIQMC runs to define the corresponding P spaces.

36

(a)(b)# walkersdeterminants3. Solve the CC(P ) and EOMCC(P ) equations in the P space or spaces obtained in

the previous step. If we are targeting the CCSDT/EOMCCSDT-level energetics and

the excited states of interest belong to the same irrep as the ground state, we define

T (P ) = T1 + T2 + T (MC)

3

, R(P )

µ = rµ,01 + Rµ,1 + Rµ,2 + R(MC)

µ,3

, and L(P )

µ = δµ,01 + Lµ,1 +

Lµ,2 + L(MC)

µ,3

, where the list of triples T (MC)

3

, R(MC)
µ,3

, and L(MC)

µ,3

is extracted from the

ground-state CIQMC propagation at time τ . For the excited states belonging to other

irreps, we construct the similarity-transformed Hamiltonian ¯H (P ), to be diagonalized

in the EOMCC steps, in the same way as in the ground-state computations, but then

use the CIQMC propagations for the lowest states of these irreps to define the lists

of triples in R(MC)

µ,3

and L(MC)

µ,3

. We follow a similar procedure when targeting the

CCSDTQ/EOMCCSDTQ-level energetics in which case T (P ) = T1+T2+T (MC)
µ = δµ,01+Lµ,1+Lµ,2+L(MC)

µ = rµ,01+Rµ,1+Rµ,2+R(MC)

µ,3 +R(MC)

, and L(P )

R(P )

µ,4

3

µ,3 +L(MC)

µ,4

.

+T (MC)
4

,

4. Correct the CC(P ) and EOMCC(P ) energies for the missing correlations of interest

that were not captured by the CIQMC propagations at the time τ the lists of the

P -space excitations were created (the remaining triples if the goal is to recover the

CCSDT/EOMCCSDT energetics, the remaining triples and quadruples if one targets

CCSDTQ/EOMCCSDTQ, etc.) using the CC(P;Q) corrections δµ(P ;Q) defined by

Eq. (2.24).

5. Check the convergence of the resulting E(P +Q)

µ

energies calculated using Eq. (2.29) by

repeating steps 2–4 at some later CIQMC propagation time τ ′ < τ .

If the E(P +Q)

µ

energies do not change within a given convergence threshold, we can stop the calcula-

tions. One can also stop them if τ in steps 2–4 is chosen such that the stochastically

determined P space(s) contain sufficiently large fraction(s) of higher–than–doubly ex-

cited determinants relevant to the target CC/EOMCC level. Our unpublished tests

using the CC(P;Q)-based CC(t;3) corrections to the EOMCCSDt energies, the ground-

state semi-stochastic CC(P;Q) calculations reported in Ref. [99], and the excited-state

37

CC(P;Q) calculations using i-FCIQMC to generate the underlying P spaces performed

in this work indicate that one should be able to reach millihartree or sub-millihartree

accuracies relative to the parent CC/EOMCC computations, when the stochastically

determined P spaces contain as little as ∼5–10% and no more than ∼30–40% of higher–

than–double excitations of interest, although this may need further study.

Similarly to the semi-stochastic form of the ground-state CC(P;Q) methodology intro-

duced in Ref. [99], the above algorithm offers significant savings in the computational effort

compared to the fully deterministic, high-level, EOMCC approaches it targets. These savings

originate from three factors. First, the computational times associated with the early stages

of the i-CIQMC walker propagations are very short compared to the corresponding converged

runs. Second, the CC(P ) calculations and the subsequent EOMCC(P ) diagonalizations of-

fer significant speedups compared to their CC/EOMCC parents, when the corresponding

excitation manifolds contain small fractions of higher–than–doubly excited determinants.

For example, as pointed out in Refs. [99, 137], when the most expensive ⟨Φabc

⟨Φabc

ijk|[ ¯H (2), T3]|Φ⟩, where ¯H (2) = exp(−T1−T2)Hexp(T1+T2)) and ⟨Φabc

ijk|[H, T3]|Φ⟩ (or
ijk|[ ¯H (P ), Rµ,3]|Φ⟩ terms

in the CCSDT and EOMCCSDT equations are isolated and reprogrammed using techniques

similar to implementing selected CI approaches, combined with sparse matrix multiplica-

tion and index rearrangement routines (rather than conventional many-body diagrams that

assume continuous excitation manifolds labelled by occupied and unoccupied orbitals from

the respective ranges of indices; generally, the stochastically determined lists of excitations

do not form continuous manifolds that could be a priori identified), one can speed up their

determination by a factor of up to (D/d)2, where D is the number of all triples and d is the

number of triples included in the stochastically determined P space. Other terms, such as

⟨Φabc

ijk|[H, T2]|Φ⟩ and ⟨Φabc

ijk|[ ¯H (P ), Rµ,2]|Φ⟩ or ⟨Φab

ij |[H, T3]|Φ⟩ and ⟨Φab

ij |[ ¯H (P ), Rµ,3]|Φ⟩, when

treated in a similar manner, may offer additional speedups, on the order of (D/d), too. Our

current CC(P ) and EOMCC(P ) routines are not as efficient yet, but the speedups that scale

linearly with (D/d) in the most expensive ⟨Φabc

ijk|[H, T3]|Φ⟩ and ⟨Φabc

ijk|[ ¯H (P ), Rµ,3]|Φ⟩ contri-

38

butions are attainable. The third factor contributing to major savings in the computational

effort offered by the semi-stochastic CC(P;Q) approach is the observation that the deter-

mination of the noniterative correction δµ(P ;Q) for a given electronic state µ is much less

expensive than the time required to complete a single iteration of the target CC/EOMCC

calculation (in the case of the calculations aimed at the CCSDT/EOMCCSDT energetics, the

computational time associated with each δµ(P ;Q) scales no worse than ∼2n3

on4

u, as opposed

to the n3

on5

u scaling of every CCSDT and EOMCCSDT iteration).

Before going to the next section, we must discuss an interesting aspect of the semi-

stochastic CC(P )/EOMCC(P ) and CC(P;Q) methodologies. The CC(P ) and EOMCC(P )

energies at τ = 0 are identical to the energies obtained in the CCSD and EOMCCSD cal-

culations and that the corresponding τ = 0 CC(P;Q) corrections are equivalent to those

of CR-CC(2,3) (the ground state) and CR-EOMCC(2,3) (excited states). It should also be

noted that the CC(P ) and EOMCC(P ) energies at τ = ∞ are identical to the energies

obtained in the full CCSDT and EOMCCSDT calculations. The semi-stochastic CC(P;Q)

calculations recover the CCSDT and EOMCCSDT energetics in this limit too, although the

τ = ∞ values of the δµ(P ;Q) corrections are zero in this case, since the τ = ∞ P spaces

contain all the triples, i.e., the corresponding Q-space triples lists are empty. These relation-

ships between the semi-stochastic CC(P ), EOMCC(P ), and CC(P;Q) approaches and the

fully deterministic CCSD/EOMCCSD, CR-CC(2,3)/CREOMCC(2,3), and CCSDT/EOM-

CCSDT methodologies were helpful in examining the correctness of our codes. They also

point to the ability of the CC(P ), EOMCC(P ), and CC(P;Q) calculations driven by the

information extracted from CIQMC to offer a systematically improvable description as τ

approaches ∞.

3.2 Electronic Excitations in CH+, CH, and CNC

In order to explore the performance of the semi-stochastic CC(P;Q) approach to excited

states proposed in this work and examine, in particular, the ability of the noniterative

δµ(P ;Q) corrections to accelerate the convergence of the CIQMC-driven EOMCC(P ) calcu-

39

lations toward the desired EOMCC energetics, represented in this study by EOMCCSDT,

we carried out benchmark calculations for the frequently studied vertical excitations in CH+

ion at the equilibrium [Table 3.1 and Fig. 3.2(a) and (b)] and stretched [Table 3.2 and Fig.

3.2(c) and (d)] geometries, which were previously used to test the EOMCC(P) framework

[137] and which was a useful system to check the correctness of our codes, and the adiabatic

excitations in the challenging open-shell CH (Table 3.3) and CNC (Table 3.4) systems, which

have low lying excited states dominated by two-electron transitions that require at least the

EOMCCSDT theory level to obtain a reliable description [62, 68, 78, 79, 141, 161–163]. The

CH+ ion was described by the [5s3p1d/3s1p] basis set of Ref. [164] and we used the aug-

cc-pVDZ [165, 166] and DZP[4s2p1d] [167, 168] bases for the CH and CNC species, respec-

tively. Following Refs. [99, 137] (cf., also, Ref. [169]), we used the HANDE software package

[170, 171] to execute the stochastic i-FCIQMC runs, needed to generate the lists of triply

excited determinants included in the CC(P) and EOMCC(P) calculations. Our standalone

CC/EOMCC codes, interfaced with the RHF, ROHF, and integral routines in the GAMESS

program suite [172–174], were used to carry out the required CC(P ), EOMCC(P ), CC(P;Q),

and fully deterministic (CCSD/EOMCCSD and CCSDT/EOMCCSDT) computations (the

Q spaces used to construct the CC(P;Q) corrections to the CC(P ) and EOMCC(P ) energies

consisted of the triples not captured by the i-FCIQMC runs at the corresponding propaga-

tion times τ ). Each i-FCIQMC run was initiated by placing 1500 walkers on the relevant

reference function (see Tables 3.1–3.4 for the details) and we set the initiator parameter na at

3. All of the i-FCIQMC propagations used the time step τ of 0.0001 a.u. In the post-ROHF

computations for the CH and CNC species, the core electrons corresponding to the 1s shells

of the carbon and nitrogen atoms were kept frozen. In the case of CH+, we correlated all

electrons.

3.2.1 CH+

We begin our discussion of the numerical results with the CH+ ion, where we investigated

the three lowest excited states of the 1Σ+ symmetry (labelled as 2 1Σ+, 3 1Σ+, and 4 1Σ+;

40

the ground state is designated as 1 1Σ+), two lowest states of the 1Π symmetry (1 1Π and 2

1Π), and two lowest 1∆ states (1 1∆ and 2 1∆). While these results were obtained mainly

by Dr. Stephen Yuwono, with minor contributions from me, I will begin with them, since

they are good for setting the stage for the subsequent discussion of the electronic excitation

spectra of open-shell CH and CNC species and singlet–triplet gaps in biradicals, where I

was a lead contributor. Two C–H internuclear separations were considered, the equilibrium

distance R = Re = 2.13713 bohr [Table 3.1 and Fig. 3.2(a) and (b)] and the stretched R =

2Re geometry [Table 3.2 and Fig. 3.2(c) and (d)]. Following the semi-stochastic CC(P;Q)

algorithm, as described above, and our interest in converging the CCSDT/EOMCCSDT

energetics, the cluster and right and left EOM operators used in the calculations for the 1Σ+

states were approximated by T (P ) = T1 + T2 + T (MC)

3

, R(P )

µ = rµ,01 + Rµ,1 + Rµ,2 + R(MC)

µ,3

,

and L(P )

µ = δµ,01 + Lµ,1 + Lµ,2 + L(MC)

, respectively, where the list of triples defining the

three-body components T (MC)

, R(MC)
µ,3
state i-FCIQMC propagation at the same value of τ . The T (MC)

3

3

at a given time τ was obtained from the ground-

component of T (P ) used

µ,3
, L(MC)
µ,3

in the CC(P;Q) computations of the 1Π and 1∆ states, needed to determine the similarity-

transformed Hamiltonian ¯H (P ) to be diagonalized in the subsequent EOMCC steps, was

defined in the same way as in the case of the 1Σ+ states, but the lists of triples entering the

R(MC)
µ,3

component of R(P )

µ

and the L(MC)

µ,3

component of L(P )

µ were obtained differently. They

were extracted from the i-FCIQMC runs for the lowest states within the irreps of C2v relevant

to the symmetries of interest, meaning the 1B1 (C2v) component of 1 1Π for the 1Π states

and the 1A2 (C2v) component of 1 1∆ for the 1∆ states (C2v is the largest Abelian subgroup

of the true point group of CH+, C∞v; our codes cannot handle non-Abelian symmetries).

As implied by Eq. 2.24, the δµ(P ;Q) corrections to the CC(P ) and EOMCC(P ) energies at

a given time τ were computed using the Mµ,K(P ) and ℓµ,K(P ) amplitudes corresponding to

the triply excited determinants |ΦK⟩ not captured by i-FCIQMC at the same τ .

As pointed out in Refs. [51, 52, 137], the 2 1Σ+, 2 1Π, 1 1∆, and 2 2∆ states of CH+

at R = Re and all of the excited states of the stretched CH+/R = 2Re system, which we

41

calculated in this work, are characterized by substantial MR correlations that originate from

large two-electron excitation contributions (the 2 2∆ state at R = 2Re also has significant

triple excitations [51, 52, 137]).

It is therefore not surprising that the basic EOMCCSD

level, equivalent to the EOMCC(P) calculations at τ = 0, performs poorly for all of these

states, producing very large errors relative to EOMCCSDT that are about 12, 20, and 34–35

millihartree for the 2 1Σ+, 2 1Π, and both 1∆ states, respectively, at R = Re and ∼14–

144 millihartree when the excited state at R = 2Re are considered (see Tables 3.1 and

3.2). The EOMCCSD energies for the 3 1Σ+, 4 1Σ+, and 1Π, states at the equilibrium

geometry, which are dominated by one-electron transitions, are more accurate, but errors

on the order of 3–6 millihartree still remain. As shown in Tables 3.1 and 3.2, the CR-

EOMCC(2,3) triples correction to EOMCCSD, equivalent to the CC(P;Q) calculations at

τ = 0, offers substantial improvements, as exemplified by the small errors, on the order of

1–3 millihartree, for the majority of excited states of CH+ considered in this subsection, but

there are cases, especially the 4 1Σ+ and 2 1∆ states at R = Re, where the differences between

the CR-EOMCC(2,3) and parent EOMCCSDT energies, which are about 12 millihartree in

the former case and more than 63 millihartree in the case of the latter state, remain very

large. This is related to the substantial coupling of the one- and two-body components of the

cluster and EOM excitation and deexcitation operators with their three-body counterparts,

which the CR-EOMCC(2,3) corrections to EOMCCSD neglect. Our group’s older active-

space EOMCCSDt calculations for CH+ reported in Refs. [51, 52] and the more recent semi-

stochastic EOMCC(P) calculations for the same system described in Ref. [137] are indicating

that the incorporations of the leading triples in the relevant P spaces, which allows the one-

and two-body components of T , Rµ, and Lµ to relax in the presence of their three-body

counterparts, is the key to improve the results of the CR-EOMCC(2,3) calculations.

This is exactly what we observe in Tables 3.1 and 3.2 and Fig. 3.2. In agreement with

our previous work [137], by enriching the P spaces used in the CC(P) and EOMCC(P)

computations with the subsets of triples captured during i-FCIQMC propagations, the results

42

greatly improve, allowing us to reach the millihartree or sub-millihartree accuracy levels

for all the calculated excited states of CH+ at both nuclear geometries considered in this

work when the stochastically determined P spaces contain about 20–30% of all triples.

The CC(P;Q) corrections to the EOMCC(P) energies based on Eq. (2.24) accelerate the

convergence toward EOMCCSDT even further. As shown in Tables 3.1 and 3.2 and Fig. 3.2,

these corrections are so effective that we reach the millihartree or sub-millihartree accuracy

levels relative to the parent EOMCCSDT energetics almost instantaneously, i.e., out of early

stages of the i-FCIQMC propagations, when no more that 5–10% of all triples are included

in the relevant P spaces. This is true even when the highly complex 4 1Σ+ and 2 1∆ states

at R = Re, for which the EOMCCSD calculations produce the massive, ∼33 and ∼144

millihartree, errors, which remain large (about 13 and 63 millihartree, respectively) at the

CR-EOMCC(2,3) level. As shown in Table 3.2, the CC(P;Q) corrections to the EOMCC(P)

energies, which account for the missing triples that the i-FCIQMC propagations at a given

time τ did not capture, allow us to reach the sub-millihartree accuracy levels with less

than 5% (the 2 1∆ state) or ∼10% (the 4 1Σ+ state) of triples in the relevant P spaces.

The uncorrected EOMCC(P) calculations display the relatively fast convergence toward

EOMCCSDT as well, but they reach similar accuracies at later propagation time τ , when

about 15% (the 2 1∆ state) or 25% (the 4 1Σ+ state) of triples are captured by i-FCIQMC.

Obviously, the details of the rate of convergence of the semi-stochastic CC(P;Q) calculations

toward EOMCCSDT, especially when one wants to tighten it, depend on the specific excited

state being calculated, but, as shown in Tables 3.1 and 3.2, once about 20% of triples are

captured by the i-FCIQMC propagations, we recover the EOMCCSDT energetics for all the

calculated excited states of CH+ at both geometries examined in this study to within 0.1

millihartree or better.

Interestingly, there is a great deal of consistency between the behavior of the uncorrected

semi-stochastic EOMCC(P) approach, in which the lists of triples defining the relevant P

spaces are extracted from i-FCIQMC propagations, and the fully deterministic EOMCCSDt

43

calculations for CH+ reported in Refs. [51, 52], in which the leading triples were identified

using active orbitals. Indeed, once the stochastically determined P spaces extracted from

i-FCIQMC capture about 20–30% of all triples, which in the case of CH+ system examined

here is achieved after 50000 or fewer ∆τ = 0.0001 a.u. MC iterations, the energies resulting

from the EOMCC(P) computations become very similar to those obtained with the EOM-

CCSDt method using the active space that consists of the highest-energy occupied (3σ) and

three lowest-energy unoccupied (1πx, 1πy, and 4σ) orbitals. Following, the definitions of the

‘little t’ T3 and Rµ,3 operators adopted in EOMCCSDt, for the state symmetries considered

in this work, the active spaces consisting of 3σ, 1πx, 1πy, and 4σ valence orbitals amount

to about 26–29% of all triples included in the respective EOMCC diagonalization spaces

[51, 52]. This suggests that the types and values of the triply excited amplitudes defining

the Rµ,3 components of the EOM operators Rµ, which characterize the EOMCCSDt compu-

tations reported in Refs. [51, 52], and those that define the R(MC)

µ,3

components obtained in

the i-FCIQMC-driven EOMCC(P) calculations performed after 50000 MC iterations using

∆τ = 0.0001 a.u. should be similar too. This is illustrated in Fig. 3.3, where we compare the

distributions of the differences between the R(MC)

µ,3

amplitudes and their full EOMCCSDT

counterparts resulting from the EOMCC(P) computations at 4000 [Fig. 3.3(a)], 10000 [Fig.

3.3(b)], and 50000 [Fig. 3.3(c)] MC iterations for the 2 1Σ+ state of CH+ at R = 2Re with

the analogous distribution characterizing the Rµ,3 amplitudes obtained with the EOMCCSDt

approach using the 3σ, 1πx, 1πy, and 4σ active orbitals to define the corresponding triples

space [Fig. 3.3(d)]; all EOM vectors Rµ needed to construct Fig. 3.3, corresponding to the

EOMCC(P) EOMCCSDt, and EOMCCSDT calculations, were normalized to unity). As

shown in Fig. 3.3 [cf. Fig. 3.3(c) and 3.3(d)], the small differences between the R(MC)

µ,3

ampli-

tudes resulting from the EOMCC(P) calculations performed after 50000 MC iterations and

the Rµ,3 amplitudes obtained with EOMCCSDT, including their numerical values and dis-

tribution, closely resemble those characterizing the active-space EOMCCSDt computations

reported in Refs. [51, 52]. This is in perfect agreement with the small errors relative to EOM-

44

CCSDT characterizing the two calculations, which are 0.302 millihartree in the former case

(cf. Table 3.2) and 0.576 millihartree in the case of EOMCCSDt [51, 52]. When we start

using considerably smaller fractions of triples and, as a consequence, significantly smaller

P spaces in the EOMCC(P) calculations, which is what happens when the underlying i-

FCIQMC propagation is terminated too soon, the differences between the R(MC)

µ,3

amplitudes

resulting from the EOMCC(P) calculations and their EOMCCSDT counterparts, including

their values and distribution, and the errors in the EOMCC(P) energies relative to EOM-

CCSDT increase. This can be seen in Fig. 3.3, especially when one compares panel (a),

which corresponds to the EOMCC(P) calculations performed after 4000 MC iterations that

use only 7% of triples, with panel (d) representing EOMCCSDt, which uses a much larger

fraction of triple excitations (∼30%), and in Table 3.2, where the error in EOMCC(P) energy

of the 2 1Σ+ state of CH+ at R = 2Re relative to EOMCCSDT obtained after 4000 MC

iterations, of 4.263 millihartree, is ∼14 times larger than the analogous error obtained after

50000 MC steps.

The above analysis, which could be repeated for the remaining states of CH+, reach-

ing similar conclusions, has several interesting implications for the semi-stochastic CC(P;Q)

methodology pursued in this study, which will be examined by us in the future.

It sug-

gests, for example, that the CC(P)/EOMCC(P) and CC(P;Q) approaches using CIQMC

propagations to determine the lists of higher–than–double excitations in the corresponding P

spaces can be regarded as natural alternatives to the fully deterministic active-space EOMCC

methods, such as EOMCCSDt, and their CC(P;Q)-corrected counterparts, such as CC(t;3)

[70, 84, 87], whose performance in excited-state calculations will be an interesting thing for

a future study. It also suggests that the fractions of higher–than–double excitations used to

define the stochastically determined P spaces, needed to achieve high accuracies observed

in the semi-stochastic CC(P;Q) calculations discussed in this work, should decrease with

the basis set.

It was already observed in the previous ground-state semi-stochastic work

[99], and we anticipate that the same will remain true in the CIQMC-driven excited-state

45

CC(P;Q) calculations. While this remark requires a separate thorough study, beyond the

scope of this initial work on the excited-state CC(P;Q) we can rationalize it by referring

to the analogies between the semi-stochastic CC(P)/EOMCC(P) and CC(P;Q) approaches

and their deterministic CCSDt/EOMCCSDt and CC(t;3) counterparts. Indeed, the afore-

mentioned (D/d) ratio that controls the speedups offered by the CC(P)/EOMCC(P) and

CC(P;Q) calculations becomes (no/No)(nu/Nu) when the active-space CCSDt/EOMCCSDt

and CC(t;3) calculations, based on the ideas laid down in Refs. [51, 52, 70, 77, 79, 87], are

considered, where No and Nu are the numbers of active occupied and active unoccupied

orbitals, respectively, which either do not grow with the basis set or grow with it very slowly

compared to no and nu.

Finally, before moving to the next molecular example, we would like to point out that,

in analogy to the CC(P;Q)-based CC(t;3), CC(t,q;3), and CC(t,q;3,4) calculations using

active orbitals to define the underlying P spaces (see, e.g., Ref. [84]), one is better off by

using smaller P spaces in the semi-stochastic CC(P)/EOMCC(P) considerations, which can

be extracted out of the early stages of CIQMC propagations, and capturing the remain-

ing correlations using noniterative CC(P;Q) corrections, than by running long-time CIQMC

simulations to generate larger P spaces for the uncorrected CC(P)/EOMCC(P) calculations.

This can be seen in Table 3.1 and 3.2 for CH+ and in the remaining Tables 3.3 and 3.4 dis-

cussed in the next two subsections. We illustrate this remark by inspecting the EOMCC(P)

and CC(P;Q) calculations for the 4 1Σ+ state of CH+. As shown in Table 3.1, one needs

to capture about 50% of triples in the P space to reach 0.1 millihartree accuracy relative to

EOMCCSDT at R = Re using the uncorrected EOMCC(P) approach. When the CC(P;Q)

correction is employed, only 15% of triples are needed to reach the same accuracy level. At

the more challenging R = 2Re geometry (Table 3.2), one reaches a ∼0.1 millihartree accuracy

level with about 40% of triples in the P space when using the uncorrected EOMCC(P) ap-

proach. This fraction reduces to about 20%, without any accuracy loss, when the CC(P;Q)

correction is added to the EOMCC(P) energy. Based on the information provided in Sec-

46

tion 3.1, running the EOMCC(P) calculations with a smaller fraction of triples in the P

space offers much larger savings in the computational effort than the additional time spent

on determining the CC(P;Q) correction, which is, as pointed out above, considerably less

expensive than a single EOMCCSDT iteration. For example, in the pilot implementation of

the excited-state EOMCC(P) and CC(P;Q) approaches aimed at recovering EOMCCSDT

energetics, employed in this work, the uncorrected EOMCC(P) run using 50% of triples in

the P space, needed to reach a ∼0.1 millihartree accuracy relative to EOMCCSDT for the 4

1Σ+ state of CH+ at R = Re, is about twice as fast as the corresponding EOMCCSDT cal-

culation. The EOMCC(P) diagonalization that forms part of the analogous CC(P;Q) run,

which needs only 15% of triples in the P space to reach the same accuracy level, is about 6

times faster than EOMCCSDT. The noniterative CC(P;Q) correction is so inexpensive here

that one can largely ignore the computational costs associated with its determination in this

context [cf. Ref. [90] for the analogous comments made in the context of comparing costs of

the CCSDt computations with those of CC(t;3)].

47

Table 3.1 Convergence of the CC(P )/EOMCC(P ) and CC(P;Q) energies toward CCSDT/EOMCCSDT for CH+, calculated
using the [5s3p1d/3s1p] basis set of Ref. [164], at the C–H internuclear distance R = Re = 2.13713 bohr. The P spaces used in
the CC(P ) and EOMCC(P ) calculations were defined as all singles, all doubles, and subsets of triples extracted from i-FCIQMC
propagations for the lowest states of the relevant symmetries. Each i-FCIQMC run was initiated by placing 1500 walkers on
g states, the 3σ → 1π state of the 1B1(C2v) symmetry for
the appropriate reference function [the RHF determinant for the 1Σ+
the 1Π states, and the 3σ2 → 1π2 state of the 1A2(C2v) symmetry for the 1∆ states], setting the initiator parameter na at 3,
and the time step ∆τ at 0.0001 a.u. The Q spaces used in constructing the CC(P;Q) corrections consisted of the triples not
captured by i-FCIQMC. Adapted from Ref. [100].

MC iter. (103)

0d
2

4

6

8

10

50

100

150

200
∞e

P a
1.845

1.071

0.423

0.249

0.181

0.172

0.077

0.044

0.015

0.006

1 1Σ+
g
(P ;Q)b
0.063

%Tc
0

2 1Σ+
g
(P ;Q)b P a
1.373

3.856

3 1Σ+
g
(P ;Q)b P a
0.787

5.537

4 1Σ+
g
(P ;Q)b P a
0.954

3.080

P a
19.694

0.024

0.009

0.003

0.003

0.004

0.001

0.000

0.000

0.000

7

15

20

23

24

37

48

59

69

11.004

0.909

3.248

0.587

4.826

−4.469 0.772

5.474

4.712

1.371

1.572

0.755

0.277

0.085

0.024

0.090

1.893

0.047

1.980

0.100

0.513

0.111

1.268

0.046

1.077

0.068

0.213

0.112

0.643

0.067

0.702

0.075

0.170

0.061

0.295

0.044

0.385

0.026

0.118

0.026

0.139

0.037

0.208

0.032

0.053

0.009

0.007

0.013

0.155

0.017

0.021

0.005

0.058

0.006

0.041

0.007

0.008

0.002

0.014

0.002

0.002

0.003

0.004

1 1Π
(P ;Q)b
0.792

0.179

0.102

0.054

0.058

0.046

0.027

0.013

0.005

0.003

2 1Π

%Tc
0

P a
11.656

(P ;Q)b
2.805

P a
34.304

13

20

25

27

29

43

57

71

82

3.746

1.852

0.957

0.743

0.411

0.157

0.063

0.020

0.008

0.530

0.128

0.073

0.060

0.047

0.027

0.012

0.004

1.492

0.525

0.471

0.240

0.198

0.039

0.014

0.004

−0.001 0.003

1 1∆
(P ;Q)b
−0.499

0.151

0.051

0.028

0.021

0.017

0.008

0.005

0.002

0.002

2 1∆

%Tc
0

P a
34.685

(P ;Q)b
0.350

10

16

18

22

24

42

56

71

82

5.951

2.542

1.892

0.940

0.877

0.133

0.043

0.008

0.003

0.432

0.128

0.094

0.057

0.041

0.011

0.005

0.003

0.002

−38.019516

−37.702621

−37.522457

−37.386872

−37.900921

−37.498143

−37.762113

−37.402308

g ground state) and EOMCC(P ) (excited states) energies relative to the corresponding CCSDT and EOMCCSDT

aErrors in the CC(P ) (the 1 1Σ+
data, in millihartree.
bErrors in the CC(P;Q) energies relative to the corresponding CCSDT and EOMCCSDT data, in millihartree.
cThe %T values are the percentages of triples captured during the i-FCIQMC propagations for the lowest state of a given symmetry [the 1 1Σ+
1 1A1(C2v) ground state for the 1Σ+
state for the 1∆ states].
dThe CC(P ) and EOMCC(P ) energies at τ = 0.0 a.u. are identical to the energies obtained in the CCSD and EOMCCSD calculations. The τ = 0.0
a.u. CC(P;Q) energies are equivalent to the CR-CC(2,3) (the ground state) and the CR-EOMCC(2,3) (excited states) energies.
eThe CC(P ) and EOMCC(P ) energies at τ = ∞ a.u. are identical to the energies obtained in the CCSDT and EOMCCSDT calculations.

g =
g states, the 1B1(C2v) component of the 1 1Π state for the 1Π states, and the 1A2(C2v) component of the 1 1∆

48

Table 3.2 Same as Table 3.1 for the stretched C–H internuclear distance R = 2Re = 4.27426 bohr. Adapted from Ref. [100].

MC iter. (103)

0

2

4

6

8

10

50

100

150

200

∞

1 1Σ+
g
(P ;Q) %T

2 1Σ+
g
(P ;Q)

P

3 1Σ+
g
(P ;Q)

P

4 1Σ+
g

P

(P ;Q)

P

1 1Π
(P ;Q) %T

2 1Π

P

(P ;Q)

P

1 1∆
(P ;Q)

2 1∆

%T

P

(P ;Q)

0.012

0.031

0.015

0.002

0.004

0.003

0.000

0.000

0.000

0.000

0

3

7

11

12

14

26

39

52

63

17.140

1.646

19.929

−2.871 32.639

12.657 13.552

5.209

4.263

1.405

1.543

0.792

0.302

0.103

0.031

0.024

0.478

12.524

−2.079 33.400

14.297 1.398

−1.741 6.383

−0.760 12.671

2.178

0.712

0.047

0.065

0.094

0.002

0.003

0.000

0.000

1.352

1.173

0.613

0.339

0.119

0.035

0.019

0.051

0.020

0.047

0.007

0.006

0.003

5.870

4.406

2.331

0.457

0.110

0.076

0.593

0.409

0.699

0.436

0.342

0.227

0.013

0.061

0.011

0.013

0.006

0.005

0.000 −0.006

0.001

0.002

2.303

0.306

0.058

0.033

0.050

0.039

0.007

0.002

0.002

0.001

0

7

12

14

16

17

30

41

52

65

21.200

−1.429 44.495

−4.526

1.644

0.724

0.612

0.457

0.220

0.079

0.016

0.007

0.001

−0.060 1.372

0.050

0.031

0.451

0.422

−0.002 0.253

0.014

0.060

0.004

0.002

0.000

0.122

0.047

0.013

0.005

0.001

0.046

0.014

0.022

0.007

−0.001

0.005

0.004

0.001

0.000

0

6

9

12

13

14

26

36

47

57

144.414

−63.405

13.363

3.338

2.340

2.088

0.862

0.288

0.038

0.014

0.003

0.368

0.130

0.063

0.021

0.038

0.005

0.000

0.000

0.000

P

5.002

1.588

0.504

0.275

0.263

0.148

0.030

0.009

0.004

0.001

−37.900394

−37.704834

−37.650242

−37.495275

−37.879532

−37.702345

−37.714180

−37.494031

49

Figure 3.2 Convergence of the EOMCC(P ) [panels (a) and (c)] and CC(P;Q) [panels (b)
and (d)] energies toward EOMCCSDT for the three lowest-energy excited states of the 1Σ+
symmetry, two lowest states of the 1Π symmetry, and two lowest 1∆ states of the CH+ ion,
as described by the [5s3p1d/3s1p] basis set of Ref. [164], at the C–H internuclear distance R
set at Re = 2.13713 bohr [panels (a) and (b)] and 2Re = 4.27426 bohr [panels (c) and (d)].
Adapted from Ref. [100].

50

Figure 3.3 The distributions of the differences between the R(MC)
amplitudes and their
EOMCCSDT counterparts resulting from the EOMCC(P ) computations at (a) 4000, (b)
10,000, and (c) 50,000 MC iterations using τ = 0.0001 a.u. for the 2 1Σ+ state of CH+ at
R = 2Re with the analogous distribution characterizing the Rµ,3 amplitudes obtained with
the EOMCCSDt approach employing the 3σ, 1πx, 1πy, and 4σ active orbitals to define the
corresponding triples space [panel (d)]. All vectors Rµ needed to construct panels (a)–(d)
were normalized to unity. Adapted from Ref. [100].

µ,3

51

3.2.2 CH

This subsection and Section 3.3 focus on the results of semi-stochastic CC(P;Q) calcula-

tions for open-shell species and biradicals reported in Refs. [100] and [102] and obtained by

me. In this subsection, we discuss the results for the CH radical that I contributed to Ref.

[100].

Similar convergence patterns in the semi-stochastic EOMCC(P) and CC(P;Q) calcula-

tions are observed for the CH radical (see Table 3.3).

In this case, following our earlier

deterministic EOMCC work, including the CR-EOMCC [62, 68] and electron-attachment

(EA) EOMCC [68, 78, 141] approaches, and a wide range of EOMCC computations, in-

cluding the high EOMCCSDT and EOMCCSDTQ levels, published by Hirata [161], along

with the X 2Π ground state, we examined the three low-lying doublet excited states, des-

ignated as A 2∆, B 2Σ−, and C 2Σ+, which belong to different irreducible representations

than that of the ground state. In analogy to the aforementioned EOMCC studies of CH

[62, 68, 78, 141, 161], the relevant CC(P) (the X 2Π state) and EOMCC(P) (excited states)

electronic energies and their CC(P;Q) counterparts were determined at the corresponding

experimentally derived equilibrium C–H distances, which are 1.1197868 ˚A for the X 2Π state

[175], 1.031 ˚A for the A 2∆ state [175], 1.1640 ˚A for the B 2Σ− state [176], and 1,1143 ˚A for

the C 2Σ+ state [177] (cf. Table 3.3). SInce all of our CC(P)/EOMCC(P) and CC(P;Q)

calculations, starting from the τ = 0 CCSD/EOMCCSD and CR-EOMCC(2,3) levels and

ending up with the larger values of τ needed to examine the convergence toward the parent

CCSDT/EOMCCSDT energetics, were performed using the ROHF reference determinant,

we also computed the ROHF-based CCSDT/EOMCCSDT energies, which formally corre-

spond to the τ = ∞ CC(P)/EOMCC(P) and CC(P;Q) results. We had to do it, since the

previously published CCSDT/EOMCCSDT results [161] relied on the unrestricted Hartree–

Fock rather than the ROHF reference.

In analogy to CH+, the lists of triples defining the T (MC)

3

component of the cluster op-

erator T (P ) and the R(MC)

µ,3

and the L(MC)

µ,3

components of the EOM excitation and deexcita-

52

tion operators, R(P )

µ

and L(P )

µ , respectively, used in the CC(P), EOMCC(P), and CC(P;Q)

calculations for the CH radical, were extracted from the i-FCIQMC propagations for the

lowest-energy states of the relevant irreps of C2v, namely, the 2B2(C2v) component of the

X 2Π state, the lowest state of the 2A1(C2v) symmetry in the case of the A 2∆ and C 2Σ+

states, and the lowest 2A2(C2v) state when considering the B 2Σ− state (again, we used C2v,

which is the largest Abelian subgroup of the true point group of CH, C∞v).

As explained in the Piecuch group’s earlier papers [62, 68, 78, 141] and as shown in

Ref. [161], all three excited states of the CH radical considered here, especially B 2Σ− and

C 2Σ+, which are dominated by two-electron excitations (cf. the reduced excitation level

(REL) diagnostic values in Tables II and III of Ref. [68] or Table II of Ref. [62]), constitute a

significant challenge, requiring the full EOMCCSDT treatment to obtain a reliable adiabatic

excitation spectrum. This can be seen by inspecting the τ = 0 EOMCC(P) i.e., EOM-

CCSD, energies for the A 2∆, B 2Σ−, and C 2Σ+ states of CH shown in Table 3.3, which

are characterized by the ∼13, ∼39, and ∼44 millihartree errors relative to EOMCCSDT,

respectively. The CR-EOMCC(2,3) triples corrections to EOMCCSD, represented in Table

3.3 by the τ = 0 CC(P;Q) values, help, especially in the case of the C 2Σ+ state, but the

situation is far from ideal, since errors on the order of 8 and 5 millihartree for the A 2∆ and

B 2Σ− states, respectively, remain. The situation considerably improves when we turn to the

semi-stochasticCC(P;Q) calculations, which incorporate the leading triples in the relevant P

spaces by extracting them from the corresponding i-FCIQMC propagations and correct the

resulting energies for the remaining triple excitations that were not captured by i-FCIQMC

at a given time τ . As shown in Table 3.3, in the case of the A 2∆ and B 2Σ− states, which

are not only challenging to EOMCCSD, but also to CR-EOMCC(2,3), we can reach com-

fortable 1–2 millihartree errors relative to EOMCCSDT using the semi-stochasticCC(P;Q)

corrections developed in this work once the relevant P spaces contain about 20–40% of all

triples. With ∼50% triples in the same P space, the CC(P;Q) energies of the A 2∆ and

B 2Σ− states are within fractions of a millihartree from EOMCSDT. These are consider-

53

able improvements relative to the purely deterministic EOMCCSD and CR-EOMCC(2,3)

computations, which give ∼13–39 and ∼5–8 millihartree errors, respectively, for the same

two states, and the semi-stochasticEOMCC(P) calculations that reach 1–2 millihartree ac-

curacy levels with about 70–80% triples in the respective P spaces. In the case of the C

2Σ+ state, which is a major challenge to EOMCCSD, but not to CR-EOMCC(2,3), the

behavior of the EOMCC(P) and CC(P;Q) approaches is different, since the CC(P;Q) cor-

rections obtained with the help of some triples in the P space captured by i-FCIQMC are

no longer needed to obtain the well-converged energetics, i.e., the τ = 0 CC(P;Q) result,

where the P space is spanned by singles and doubles only, is sufficiently accurate, but it

is still interesting to observe that one can tighten the convergence further, reaching sta-

ble < 0.1 millihartree errors relative to EOMCCSDT with about 50% of all triples in the P

space. In analogy to the A 2∆ and B 2Σ− states, it is also interesting to observe a reasonably

smooth convergence of the uncorrected EOMCC(P) energies toward EOMCCSDT. It is clear

from the results presented in Table 3.3 that the CC(P;Q) corrections to the semi-stochastic

CC(P) and EOMCC(P) energies offer considerable speedups compared to the uncorrected

CC(P)/EOMCC(P)calculations, not only for the closed-shell molecules, such as CH+, but

also when examining open-shell species.

54

Table 3.3 Convergence of the CC(P )/EOMCC(P ) and CC(P;Q) energies toward CCS-
DT/EOMCCSDT for CH, calculated using the aug-cc-pVDZ basis set. The P spaces used
in the CC(P ) and EOMCC(P ) calculations were defined as all singles, all doubles, and sub-
sets of triples extracted from i-FCIQMC propagations for the lowest states of the relevant
symmetries. Each i-FCIQMC run was initiated by placing 1500 walkers on the appropriate
reference function [the ROHF 2B2(C2v) determinant for the X 2Π state, the 1π → 4σ state
of the 2A1(C2v) symmetry for the A 2∆ and C 2Σ+ states, and the 3σ → 1π state of the
2A2(C2v) symmetry for the B 2Σ− state], setting the initiator parameter na at 3, and the
time step ∆τ at 0.0001 a.u. The Q spaces used in constructing the CC(P;Q) corrections
consisted of the triples not captured by i-FCIQMC. Adapted from Ref. [100].

MC iter. (103)

0

2

4

6

8

10

12

14

16

18

20

50

100

150

200
∞e

X 2Π
(P ;Q)b
0.231

0.170

0.086

0.035

0.022

0.019

0.015

0.013

0.008

0.008

0.006

0.002

0.002

0.000

0.000

P a
2.987

2.405

1.413

0.883

0.603

0.495

0.445

0.389

0.309

0.292

0.243

0.150

0.055

0.025

0.010

P a

%Tc
0.0 13.474

13.8 13.009

41.7 10.907

58.9 10.119

66.8 7.764

72.6 6.987

76.5 6.640

77.5 7.040

79.2 6.047

80.3 4.646

82.2 3.809

89.1 1.367

95.3 0.177

A 2∆
(P ;Q)b
7.727

7.395

5.288

4.577

2.436

2.170

1.981

1.887

1.667

0.875

0.754

0.112

0.017

98.1 0.042

−0.003

%Tc
0.0

P a
38.620

9.8

10.602

19.3

27.2

34.6

38.1

42.3

45.7

48.3

49.8

52.6

74.1

91.7

98.0

7.066

3.452

2.309

1.965

1.832

1.180

1.303

1.349

0.796

0.298

0.144

0.010

B 2Σ−
(P ;Q)b
−4.954

−1.848

−1.259

−0.371

−0.149

−0.024

−0.081

0.030

0.012

P a

%Tc
0.0 43.992

18.5 40.700

38.9 31.017

53.2 26.364

61.4 20.545

64.8 17.180

69.5 16.929

72.2 13.114

C 2Σ+
(P ;Q)b
0.087

−0.689

−0.319

−0.508

−0.412

0.435

0.029

0.253

75.6 7.646

−0.041

−0.062

77.5 5.312

0.038

0.038

0.014

0.007

79.5 4.691

91.6 1.436

98.3 0.204

99.6 0.063

0.011

0.108

0.070

0.013

0.010

0.001

%Tc
0.0

9.8

19.7

28.8

34.3

38.3

42.5

45.1

48.7

50.1

52.2

74.0

91.3

98.2

99.7

99.2 0.007

0.001

99.7 −0.001

−0.001

99.9 0.010

−38.387749

−38.276770

−38.267544

−38.238205

aErrors in the CC(P ) (the X 2Π ground state) and EOMCC(P ) (excited states) energies relative to the
corresponding CCSDT and EOMCCSDT data, in millihartree, calculated at the experimentally obtained
equilibrium C–H distances used in Refs. [62, 68, 161], which are 1.1197868 ˚A for the X 2Π state [175], 1.1031
˚A for the A 2∆ state [175], 1.1640 ˚A for the B 2Σ− state [176], and 1.1143 ˚A for the C 2Σ+ state [177]. The
lowest-energy core orbital was frozen in all correlated calculations.
bErrors in the CC(P;Q) energies relative to the corresponding CCSDT and EOMCCSDT data, in milli-
hartree, calculated at the experimentally determined equilibrium C–H distances as used in Refs. [62, 68, 161]
(see footnote a for the C–H distances).
cThe %T values are the percentages of triples captured during the i-FCIQMC propagations for the lowest
state of a given symmetry [the 2B2(C2v) component of the X 2Π ground state, the lowest 2A1(C2v) state
for the A 2∆ and C 2Σ+ states, and the lowest 2A2(C2v) state for the B 2Σ− state].
dThe CC(P ) and EOMCC(P ) energies at τ = 0.0 a.u. are identical to the energies obtained in the CCSD
and EOMCCSD calculations. The τ = 0.0 a.u. CC(P;Q) energies are equivalent to the CR-CC(2,3) (the
ground state) and the CR-EOMCC(2,3) (excited states) energies.
eThe CC(P ) and EOMCC(P ) energies at τ = ∞ a.u. are identical to the energies obtained in the ROHF-
based CCSDT and EOMCCSDT calculations.

55

Figure 3.4 Convergence of the EOMCC(P) and CC(P;Q) energies of the A 2∆ [panel (a)],
B 2Σ− [panel (b)], and C 2Σ+ [panel (c)] states of CH toward EOMCCSDT for the three
lowest-energy excited states of CH calculated as described by the aug-cc-pVDZ basis set.
The geometries used are the equilibrium C–H distances reported in Refs. [62, 68, 161], which
are 1.1031 ˚A for the A 2∆ state [175], 1.1640 ˚A for the B 2Σ− state [176], and 1.1143 ˚A for
the C 2Σ+ state [177].

56

050100150200MC Iterations (103)−1012345678Error rel. (o EOMCCSDT (mE )A2Δ(a)EOMCC(P)CC(P;Q)050100150200MC I(era(ions (103)−1012345678B2Σ−(b)EOMCC(P)CC(P;Q)050100150200MC I(era(ions (103)−1012345678C2Σ+(c)EOMCC(P)CC(P;Q)3.2.3 CNC

Our last example, which is also the largest many-electron system considered in the present

study, is the linear, D∞h symmetric, CNC molecule. Following our earlier CR-CC(2,3)/CR-

EOMCC(2,3) and EA-EOMCC calculations for this challenging open-shell molecular species

[68, 162, 163], we considered the X 2Πg ground state and the two low-lying doublet excited

states, A 2∆u and B 2Σ+

u . The i-FCIQMC-driven CC(P) ground-state and EOMCC(P)

excited-state energies and the corresponding CC(P;Q) corrections, along with their deter-

ministic EOMCCSD, CR-EOMCC(2,3), and EOMCCSDT counterparts, were calculated us-

ing the equilibrium C–N distances optimized in Ref. [162] with EA-SAC-CI. They are 1.253

˚A for the X 2Πg state, 1.256 ˚A for the A 2∆u state, and 1.259 ˚A for the B 2Σ+

u state. As in

the case of the CH radical, we used the ROHF reference determinant. Following the com-

putational protocol adopted in this study, and in analogy to the CH+ and CH species, the

lists of triples defining the T MC

3

, R(MC)
µ,3

, and L(MC)

µ,3

components used in the semi-stochastic

CC(P),EOMCC(P), and CC(P;Q) calculations for CNC were obtained using the i-FCIQMC

propagations for the lowest-energy states of the relevant irreps of the largest Abelian sub-

group of D∞h, i.e., D2h, meaning the 2B2g (D2h) component of the X 2Πg state and the lowest

state of the 2B1u (D2h) symmetry in the case of the A 2∆u and B 2Σ+

u states.

As shown in Table 3.4 and in agreement with one of our previous studies [68], all three

states of CNC considered in this work, especially A 2∆u and B 2Σ+

u , are poorly described by

CCSD and EOMCCSD, which produce more than 18, 31, and 111 millihartree errors, respec-

tively, relative to the target EOMCCSDT energetics (see the τ = 0 CC(P) and EOMCC(P)

energies in Table 3.4). The excessively large, > 111 millihartree, error in the EOMCCSD

energy of the B 1Σ+

u state is related to its strongly multireference character dominated by two-

electron excitations (cf. the REL values characterizing the excited states of CNC in Table IV

of Ref. [68]). In the case of the ground state and the B 2Σ+

u excited state, the CR-CC(2,3) and

CR-EOMCC(2,3) corrections to CCSD and EOMCCSD seem to be quite effective, reducing

the large errors relative to CCSDT/EOMCCSDT observed in the CCSD and EOMCCSD

57

calculations to a sub-millihartree level, but the ∼16 millihartree error resulting from the

CR-EOMCC(2,3) calculations for the A 2∆u state, while considerably lower than the > 31

millihartree error obtained with EOMCCSD, is still rather large (see the τ = 0 CC(P;Q)

energies in Table 3.4). By incorporating the dominant triply excited determinants captured

by the i-FCIQMC propagations in the respective P spaces, the semi-stochasticCC(P) and

EOMCC(P) approaches help, allowing us to reach stable ∼1–2 millihartree accuracy levels

for the X 2Πg and A 2∆u states relative to the target CCSDT/EOMCCSDT energetics with

about 50–60% triples, but the CC(P;Q) corrections that account for the remaining triples,

missing in the i-FCIQMC wave functions, are considerably more effective. In the case of

the A 2∆u state, which poses problems to both EOMCCSD and CR-EOMCC(2,3), which

give about 31 and 16 millihartree errors relative to EOMCCSDT, respectively, we reach a

stable ∼1–2 millihartree accuracy level with about 30–40% triples in the corresponding P

space, as opposed to the aforementioned 50–60% needed in the uncorrected EOMCC(P)

run. The benefits of using the semi-stochastic CC(P;Q) vs. deterministic CR-EOMCC(2,3)

corrections for the X 2Πg and B 2Σ+

u states are less obvious, but it is encouraging to observe

the rapid convergence toward the target CCSDT and EOMCCSDT energetics in the former

calculations. In particular, they allow us to lower the 0.4–0.5 millihartree Errors obtained

with CR-EOMCC(2,3) to a 0.1 millihartree level with about 10% of all triples, identified

by i-FCIQMC, in the case of the X 2Πg state and with ∼30–40% triples in the P space

when the B 2Σ+

u state is considered. Once again, the CC(P;Q) corrections to the energies

resulting from the semi-stochastic CC(P) and EOMCC(P) calculations speed up the uncor-

rected CC(P)/EOMCC(P) computations, while allowing us to improve the CR-CC(2,3) and

CR-EOMCC(2,3) energetics by bringing them very close to the CCSDT and EOMCCSDT

levels at the fraction of the cost.

58

Table 3.4 Convergence of the CC(P )/EOMCC(P ) and CC(P;Q) energies toward CCS-
DT/EOMCCSDT for CNC, calculated using DZP[4s2p1d] basis set. The P spaces used in
the CC(P ) and EOMCC(P ) calculations were defined as all singles, all doubles, and sub-
sets of triples extracted from i-FCIQMC propagations for the lowest states of the relevant
symmetries. Each i-FCIQMC run was initiated by placing 1500 walkers on the appropriate
reference function [the ROHF 2B2g(D2h) determinant for the X 2Πg state and the 3σu → 1πg
state of the 2B1u(D2h) symmetry for the A 2∆u and B 2Σ+
u states], setting the initiator pa-
rameter na at 3, and the time step ∆τ at 0.0001 a.u. The Q spaces used in constructing
the CC(P;Q) corrections consisted of the triples not captured by i-FCIQMC. Adapted from
Ref. [100].

MC iter. (103)

0

2

4

6

8

10

12

14

16

18

20

50

100

150
∞e

X 2Πg
(P ;Q)b
−0.495

−0.043

−0.029

−0.011

−0.013

−0.006

−0.003

−0.005

−0.003

−0.003

−0.001

0.000

0.000

0.000

P a
18.458

10.331

4.424

2.824

1.818

1.306

1.092

0.911

0.820

0.651

0.610

0.077

0.002

0.000

P a

%Tc
0.0 31.157

13.2 18.835

33.2 10.637

44.1 7.555

49.9 6.181

53.3 5.187

56.5 4.162

58.7 3.529

60.6 3.106

62.5 2.510

63.9 2.395

79.7 0.172

94.5 0.002

99.3 0.000

A 2∆u
(P ;Q)b
16.017

9.114

5.717

4.199

3.090

2.441

1.778

1.418

1.149

0.811

0.785

0.058

0.001

0.000

P a

%Tc
0.0 111.307

6.5

81.493

16.1 53.677

22.7 35.539

27.5 26.767

30.8 21.337

34.0 17.056

37.0 12.843

39.5

41.7

44.4

70.9

92.3

99.1

9.197

8.879

7.548

0.732

0.005

0.000

B 2Σ+
u
(P ;Q)b
−0.433

−2.496

−2.526

−1.254

−0.864

−0.284

0.196

0.046

0.134

−0.034

0.151

0.055

0.003

0.000

%Tc
0.0

6.5

16.0

22.8

27.9

31.5

34.3

37.5

39.9

42.4

44.7

70.7

91.9

99.1

−130.421932

−130.276946

−130.252999

aErrors in the CC(P ) (X 2Πg state) and EOMCC(P ) (the remaining states) energies relative to the cor-
responding CCSDT and EOMCCSDT data, in millihartree, calculated at the experimentally obtained
equilibrium C–N distances optimized in Ref. [162], which are 1.253 ˚A for the X 2Πg state, 1.256 ˚A for
the A 2∆u state, and 1.259 ˚A for the B 2Σ+
u state. The three lowest-energy core orbital was frozen in all
correlated calculations.
bErrors in the CC(P;Q) energies relative to the corresponding CCSDT and EOMCCSDT data, in milli-
hartree, calculated at the equilibrium C–N distances optimized in Ref. [162] (see footnote a for these C–N
distances).
cThe %T values are the percentages of triples captured during the i-FCIQMC propagations for the lowest
state of a given symmetry [the 2B2g(D2h) component of the X 2Πg ground state and the lowest 2B1u(D2h)
state for the A 2∆ and B 2Σ+
dThe CC(P ) and EOMCC(P ) energies at τ = 0.0 a.u. are identical to the energies obtained in the CCSD
and EOMCCSD calculations. The τ = 0.0 a.u. CC(P;Q) energies are equivalent to the CR-CC(2,3) (the
ground state) and the CR-EOMCC(2,3) (excited states) energies.
eThe CC(P ) and EOMCC(P ) energies at τ = ∞ a.u. are identical to the energies obtained in the ROHF-
based CCSDT and EOMCCSDT calculations.

u states].

59

Figure 3.5 Convergence of the EOMCC(P) and CC(P;Q) energies of the A 2∆u [panel (a)]
and B 2Σ+
u [panel (b)] states of CNC toward EOMCCSDT for the two lowest-energy doublet
excited states of CNC calculated as described by the DZP[4s2p1d] basis set. The geometries
used are the equilibrium C–N distances reported in Refs. [68, 162], which are 1.256 ˚A for the
A 2∆u state and 1.259 ˚A for the B 2Σ+

u state.

60

050100150MC Iterations (103)−20246810121416Error re . to EOMCCSDT (mEh)A2Δu(a)EOMCC(P)CC(P;Q)050100150MC Iterations (103)−20246810121416B2Σ+u(b)EOMCC(P)CC(P;Q)3.3 Singlet–Triplet Gaps in Methylene, (HFH)−, Cyclobutadiene, Cyclopenta-

dienyl Cation, and Trimethylenemethane

In order to assess the performance of our semi-stochastic, CIQMC-driven, CC(P ;Q)

methodology in converging the full CCSDT data for the singlet–triplet gaps and the corre-

sponding singlet- and triplet-state energies of biradical systems, we applied it to methylene,

(HFH)−, cyclobutadiene, cyclopentadienyl cation, and trimethylenemethane. The results

discussed in this section, which were all generated by me as part of this dissertation project,

were reported in Ref. [102]. Following our earlier studies of the singlet–triplet gaps in the

same systems using the deterministic CC(P ;Q) [88] and DEA/DIP-EOMCC [80, 153–155]

approaches, we used the aug-cc-pVTZ basis set [165, 166] for methylene, the 6-31G(d,p) basis

[178, 179] for the (HFH)− ion, and the cc-pVDZ basis set [165] for cyclobutadiene, cyclopenta-

dienyl cation, and trimethylenemethane. In the case of methylene and trimethylenemethane,

we focused on the ability of the semi-stochastic CC(P ;Q) algorithm to converge the adiabatic

∆ES-T data obtained with CCSDT. When executing the semi-stochastic CC(P ;Q) calcula-

tions for (HFH)−, cyclobutadiene, and cyclopentadienyl cation, we focused on recovering the

CCSDT values of the vertical singlet–triplet gaps. Throughout this work, we define ∆ES-T

as ES − ET, where ES and ET are the electronic energies of the corresponding singlet and

triplet states, i.e., the positive ∆ES-T value implies that triplet is lower in energy.

All of the CC calculations reported in this section were performed using our group’s

standalone codes, interfaced with the RHF, ROHF, and integral transformation routines in

the GAMESS package [172, 173] which were originally developed in Refs. [70, 84, 87, 88],

and extended to the stochastically generated P spaces for the use in CC(P ) and CC(P ;Q)

computations in Refs. [99–101, 137]. The i-FCIQMC [methylene, (HFH)−, and cyclobuta-

diene] and i-CISDTQ-MC (cyclopentadienyl cation and trimethylenemethane) calculations,

needed to generate the lists of triples for the semi-stochastic CC(P ) and CC(P ;Q) runs, were

carried out with the HANDE software [170, 171]. Each of the i-FCIQMC and i-CISDTQ-MC

propagations was initiated by placing 1500 walkers on the relevant reference determinant.

61

The CIQMC time step δτ and the initiator parameter na were set at 0.0001 a.u. and 3,

respectively.

In all post-Hartree–Fock calculations, the core MOs correlating with the 1s

orbitals of the C and F atoms were kept frozen. If the true point group of the biradical sys-

tem of interest was not Abelian, we used its largest Abelian subgroup, since our CC codes

interfaced with GAMESS and the CIQMC routines in HANDE cannot handle non-Abelian

symmetries.

3.3.1 Methylene

We begin the discussion of our results by analyzing the performance of the semi-stochastic

CC(P ;Q) approach in converging the CCSDT energies of the ground (X 3B1) and first-

excited (A 1A1) states of the methylene/aug-cc-pVTZ system and the adiabatic gap between

them. The C2v-symmetric geometries of CH2 in the two states, optimized using FCI and the

[5s3p/3s] triple zeta basis set of Dunning [180] augmented with two sets of polarization func-

tions (TZ2P), were taken from Ref. [181]. The electronically nondegenerate triplet ground

state has a predominantly SR nature dominated by the (1a1)2(2a1)2(1b2)2(3a1)1(1b1)1 config-

uration, whereas the first-excited singlet state exhibits a significant MR character requiring a

linear combination of the (1a1)2(2a1)2(1b2)2(3a1)2 and doubly excited (1a1)2(2a1)2(1b2)2(1b1)2

closed-shell determinants for a proper zeroth-order description. Because of these fundamen-

tally different characteristics of the X 3B1 and A 1A1 states, a well-balanced and accurate

treatment of dynamical and nondynamical correlation effects is the key to a reliable descrip-

tion of the singlet–triplet gap in methylene. It is, therefore, unsurprising that one usually

resorts to methods of the MRCI [181–186] or MRCC [187–190] type, or to the high-level

SRCC theories that account for higher–than–doubly excited clusters in an iterative manner,

such as full CCSDT used in Refs. [88, 191], to accomplish this goal (for other examples

of high-level ab initio calculations for the X 3B1 and A 1A1 states of methylene, see Refs.

[80, 153, 154, 192] and references therein). The CCSDT results for the adiabatic singlet–

triplet gap in methylene, which are of interest in the present study, are indeed very accurate.

As shown, for example, in Ref. [88], the difference between the adiabatic ∆ES-T value ob-

62

tained in the CCSDT/TZ2P calculations and the corresponding FCI result of 11.14 kcal/mol

[181] is only 0.11 kcal/mol or 38 cm−1. As demonstrated in Ref. [88] as well, the purely elec-

tronic A 1A1 − X 3B1 separation resulting from the CCSDT computations using the aug-cc-

pVTZ basis employed in this work is only about 0.15 kcal/mol (∼50 cm−1) higher than the

experimentally derived value of 9.37 kcal/mol reported in Ref. [188], obtained by correcting

the vibrationless adiabatic singlet–triplet gap determined in Ref. [193] for the relativistic and

nonadiabatic (Born–Oppenheimer diagonal correction) effects estimated in Refs. [194] and

[195], respectively. It is, therefore, interesting to examine if the semi-stochastic CC(P ;Q)

approach advocated in this work is capable of reproducing the high-quality CCSDT/aug-

cc-pVTZ data for the X 3B1 and A 1A1 states of methylene and the adiabatic separation

between them.

The results of our FCIQMC-driven CC(P ;Q) calculations for the methylene/aug-cc-pVTZ

system, reported as errors relative to the parent CCSDT data, and their CC(P ) counterparts

are shown in Table 3.5 and Fig. 3.6. The reference determinants |Φ⟩ used to initiate the

i-FCIQMC propagations and to carry out the CC(P ), CC(P ;Q), CCSD, CR-CC(2,3), and

CCSDT calculations were the ROHF determinant in the case of the X 3B1 state and the

RHF determinant for the A 1A1 state. The subsets of triply excited determinants needed to

construct the P spaces used in the CC(P ) and CC(P ;Q) computations at various propagation

times τ were the Sz = 1 triples of the B1 symmetry captured during the i-FCIQMC run for

the X 3B1 state and the Sz = 0 triples of the A1 symmetry captured during the analogous run

for the A 1A1 state. Following the semi-stochastic CC(P ;Q) algorithm described in Section

3.1, the Q spaces needed to determine corrections δ(P ; Q) were defined as the remaining

triples not captured by i-FCIQMC.

Let us start our analysis by examining the CC(P ) and CC(P ;Q) data at τ = 0, where the

P spaces do not contain any triply excited determinants. As shown in Table 3.5, the CC(P )

energies of the X 3B1 and A 1A1 states at τ = 0, which are equivalent to those obtained

using conventional CCSD, are above their CCSDT [i.e., τ = ∞ CC(P )] counterparts by

63

4.187 and 5.918 millihartree, respectively. This translates into a 380 cm−1 or ∼11% error

in the adiabatic ∆ES-T value when compared to the 3328 cm−1 singlet–triplet gap obtained

with CCSDT. The situation improves when the CC(P ;Q) corrections δ(P ; Q) due to T3

correlation effects, calculated by placing all triply excited determinants in the respective

Q spaces, are added to the CC(P ) energies. The τ = 0 CC(P ;Q) or CR-CC(2,3) energy

characterizing the X 3B1 state is only 0.177 millihartree above the parent CCSDT value,

which is an error reduction relative to CCSDT compared to the underlying CC(P ) result

by a factor of ∼24. The δ(P ; Q) correction improves the τ = 0 CC(P ) energy of the more

challenging A 1A1 state as well, although the difference between the resulting CR-CC(2,3)

energy and its CCSDT counterpart, of 0.656 millihartree, is almost 4 times larger than the

analogous difference obtained for the X 3B1 state. As a result, the 105 cm−1 error relative

to CCSDT characterizing the adiabatic A 1A1 − X 3B1 separation obtained in the τ = 0

CC(P ;Q) or CR-CC(2,3) calculations, while considerably smaller than the 380 cm−1 obtained

in the underlying CC(P) (i.e., CCSD) runs, leaves room for further improvements. One can

improve the CR-CC(2,3) energies of the X 3B1 and A 1A1 states and the gap between them

by enriching the P spaces defining the CC(P ) calculations with the leading triply excited

determinants identified using active orbitals and correcting the resulting CCSDt energies

for the remaining T3 correlations that have not been captured by CCSDt [88], but our

objective here is to examine if one can accomplish the same, or improve the CC(t;3) results

reported in Ref. [88] even further, by turning to the more black-box semi-stochastic CC(P ;Q)

methodology, in which the dominant triply excited determinants are identified with CIQMC.

The results in Table 3.5 and Fig. 3.6 show that when the τ = 0 P spaces are augmented

with the subsets of triply excited determinants captured in the i-FCIQMC runs at τ > 0 and,

following the CC(P ;Q) recipe, the resulting CC(P ) energies are corrected for the remaining

T3 correlations, the convergence of the total electronic energies of the X 3B1 and A 1A1 states

and the adiabatic separation between them toward their CCSDT parents is rapid. We can

see this already in the early stages of the i-FCIQMC propagations. For example, at τ = 0.8

64

a.u., i.e., after only 8000 δτ = 0.0001 a.u. MC iterations, the errors in the CC(P ;Q) energies

of the X 3B1 and A 1A1 states and the corresponding ∆ES-T value relative to CCSDT are

0.049 millihartree, 0.106 millihartree, and 13 cm−1, respectively, substantially improving the

CR-CC(2,3) [i.e., τ = 0 CC(P ;Q)] calculations, which give 0.177 millihartree for the X 3B1

state, 0.656 millihartree for the A 1A1 state, and 105 cm−1 for ∆ES-T. This confirms our

expectation that the main source of errors in the CR-CC(2,3) computations, especially in

the case of the more MR A 1A1 state, which is characterized by larger T3 effects, is the use

of the unrelaxed T1 and T2 amplitudes obtained with CCSD in constructing the correction

due to triples. The FCIQMC-based CC(P ;Q) calculations at τ = 0.8 a.u., which use as

little as 16% of all triply excited determinants to define the P space for the X 3B1 state and

only 25% of all triples in the P space for the A 1A1 state, are also more accurate than the

CC(t;3) computations reported in Ref. [88], which produced the 0.130 millihartree, 0.409

millihartree, and 61 cm−1 errors relative to CCSDT for the X 3B1 and A 1A1 energies and

∆ES-T, respectively. This is all very promising, especially if we realize that the i-FCIQMC

propagations used to generate the lists of triples for our semi-stochastic CC(P;Q) runs, which

work so well, are very far from convergence when τ = 0.8 a.u. Indeed, as seen in Table 3.6,

the total numbers of walkers at 8000 δτ = 0.0001 a.u. MC iterations, which are 132689 in

the case of the X 3B1 state and 165564 for the A 1A1 state, represent tiny fractions, 2.17%

and 1.11%, respectively, of the total walker populations at τ = 20.0 a.u., where we stopped

our i-FCIQMC propagations (see Fig. 3.7 for a comparison of the rate of convergence of the

CC(P), CC(P;Q), and the underlying i-FCIQMC calculations).

As demonstrated in Table 3.5 and Fig. 3.6, the convergence of the energies of the X 3B1

and A 1A1 states and the gap between them resulting from the FCIQMC-driven CC(P;Q)

calculations remains fast at the larger propagation times τ as well. For example, if we allow

i-FCIQMC to populate the respective P spaces with about 26–38% of all triples, which

happens after 20000 δτ = 0.0001 a.u. MC iterations, the CC(P ;Q) energies of the X 3B1 and

A 1A1 states and the resulting singlet–triplet gap become practically indistinguishable from

65

the parent CCSDT data, with errors in the total electronic energies and ∆ES-T being only

∼20 microhartree and 2 cm−1, respectively. As shown in Table 3.6 of the, walker populations

characterizing the X 3B1 and A 1A1 states produced by i-FCIQMC at 20000 δτ = 0.0001 a.u.

MC time steps are still very small compared to the total numbers of walkers at τ = 20.0 a.u.,

where we terminated our i-FCIQMC propagations (4.11% for the X 3B1 state and 2.17% in

the case of the A 1A1 state). It is also interesting to note that the more MR A 1A1 state

requires a higher fraction of triply excited determinants in the P space than its SR X 3B1

counterpart to achieve similar accuracy levels in the semi-stochastic CC(P ;Q) computations

for both states. For example, the i-FCIQMC propagation has to capture about 25% of all

triples, for the inclusion in the P space, if we are to reduce errors relative to CCSDT in the

CC(P ;Q) calculations for the A 1A1 state to ∼0.1 millihartree. In the case of the X 3B1 state,

the analogous fraction of triples is about 10% (cf. Table 3.5). This highlights the importance

of balancing the SR triplet state with the more MR singlet state in obtaining accurate ∆ES-T

estimates, which is not a problem for the semi-stochastic CC(P;Q) methodology because the

underlying i-FCIQMC wave function sampling is very effective in identifying the dominant

higher–than–doubly excited determinants, to be included in the relevant P spaces, and the

δ(P ; Q) corrections to the CC(P ) energies take care of the remaining correlation effects of

interest.

Before concluding this subsection and discussing other molecular examples, we would

like to comment on the effectiveness of the noniterative corrections δ(P ; Q), adopted in

the CC(P ;Q) formalism, in accelerating convergence of the underlying CC(P ) calculations

toward CCSDT. The CC(P ) and CC(P ;Q) error curves shown in Fig. 3.6 illustrate this best.

It is clear from this figure that the CC(P ;Q) energies of the X 3B1 and A 1A1 states [Fig.

3.6 (a) and (b)] and the corresponding ∆ES-T values [Fig. 3.6 (c)] converge to the parent

CCSDT data much faster than in the case of the uncorrected CC(P ) computations. One

can see the same by inspecting the numerical data shown in Table 3.5. In this context, it is

worth commenting on the CC(P) and CC(P;Q) results obtained after 8000 MC iterations. In

66

that case, the CC(P;Q) calculations reduce the ∼2.4 millihartree errors relative to CCSDT

characterizing the CC(P ) energies of the X 3B1 and A 1A1 states to 0.1 millihartree or less,

which is a desired behavior, but the CC(P ;Q) ∆ES-T value is less accurate than that obtained

with the uncorrected CC(P). One should not read too much into this though. The fact that

the CC(P;Q) calculations at 8000 MC iterations increase the very small 3 cm−1 error obtained

with CC(P) to 13 cm−1 is a coincidence arising from the accidental cancellation of errors

in the CC(P) total electronic energies obtained at this particular propagation time. Indeed,

when the later stages of the i-FCIQMC propagations are considered, the differences between

the CC(P) and CCSDT values of ∆ES-T become increasingly negative, reaching −107 cm−1

at 50000 MC iterations, before eventually converging to the CCSDT limit, whereas the

corresponding CC(P;Q) results display a smooth behavior, rapidly approaching CCSDT. In

particular, they reduce the relatively large negative error value obtained for ∆ES-T in the

CC(P) calculations at 50000 MC iterations to a numerical 0 cm−1. This highlights, once

again, the ability of the CC(P;Q) corrections δ(P ; Q) to offer a well-balanced description of

the lowest singlet and triplet states in methylene, in addition to improving the individual

state energies.

67

Table 3.5 Convergence of the CC(P ) and CC(P ;Q) energies of the X 3B1 and A 1A1 states
of methylene, as described by the aug-cc-pVTZ basis set, and of the corresponding adiabatic
singlet–triplet gaps toward their parent CCSDT values. The geometries of the X 3B1 and
A 1A1 states, optimized in the FCI calculations using the TZ2P basis set, were taken from
Ref. [181]. The P spaces used in the CC(P ) and CC(P ;Q) calculations were defined as all
singly and doubly excited determinants and subsets of triply excited determinants extracted
from the i-FCIQMC propagations with δτ = 0.0001 a.u. The Q spaces used to determine the
CC(P ;Q) corrections consisted of the triply excited determinants not captured by the corre-
sponding i-FCIQMC runs. The i-FCIQMC calculations preceding the CC(P ) and CC(P ;Q)
steps were initiated by placing 1500 walkers on the ROHF (X 3B1 state) and RHF (A 1A1
state) reference determinants and the na parameter of the initiator algorithm was set at 3. In
all post-Hartree–Fock calculations, the lowest core orbital was kept frozen and the spherical
components of d and f orbitals were employed throughout. Adapted from Ref. [102].

MC Iterations
0
2000
4000
6000
8000
10000
20000
50000
100000
150000
200000
∞

P a
4.187d
3.948
3.281
2.749
2.428
2.192
1.703
1.133
0.532
0.218
0.076

X 3B1
(P ; Q)a %Tb
0.177e
0.162
0.111
0.072
0.049
0.038
0.018
0.005
0.000
0.000
0.000
−39.080575f

0
1.8
7.1
12.4
16.3
19.0
26.3
39.1
59.5
76.8
88.7

P a
5.918d
5.361
3.908
2.993
2.444
2.093
1.358
0.644
0.171
0.037
0.006

A 1A1
(P ; Q)a %Tb
0.656e
0.549
0.304
0.190
0.106
0.080
0.025
0.004
0.000
0.000
0.000
−39.065411f

A 1A1 − X 3B1
(P ; Q)c
P c
105e
380d
85
310
42
138
26
53
13
3
9
2
0
0
0
0

0
3.0
11.9
19.7
24.9
28.7 −22
37.7 −76
54.8 −107
76.5 −79
90.7 −40
97.2 −15

3328g

a Unless otherwise stated, all energies are reported as errors relative to CCSDT in millihartree.
b The %T values are the percentages of triples captured during the i-FCIQMC propagations (the Sz = 1
triply excited determinants of the B1 symmetry in the case of the X 3B1 state and the Sz = 0 triply excited
determinants of the A1 symmetry in the case of the A 1A1 state).
c Unless otherwise specified, the values of the singlet–triplet gap are reported as errors relative to CCSDT
in cm−1.
d Equivalent to CCSD.
e Equivalent to CR-CC(2,3) [the most complete variant of CR-CC(2,3) abbreviated sometimes as CR-
CC(2,3),D or CR-CC(2,3)D].
f Total CCSDT energy in hartree.
g The CCSDT singlet–triplet gap in cm−1.

68

Table 3.6 The total numbers of walkers, reported as percentages of the total walker popula-
tions at 200000 MC iterations, characterizing the i-FCIQMC propagations with δτ = 0.0001
a.u. that were needed to generate the CC(P) and CC(P;Q) results for methylene reported
in Table 3.5. Adapted from Ref. [102].

MC Iterations X 3B1 A 1A1
0.01a
0.19
0.51
0.83
1.11
1.35
2.17
4.64
13.13
36.96
100c

0
2000
4000
6000
8000
10000
20000
50000
100000
150000
200000

0.02a
0.39
1.00
1.65
2.17
2.58
4.11
7.69
18.59
43.10
100b

a The initial walker population, meaning 1500 walkers on the ROHF (X 3B1 state) and RHF (A 1A1 state)
reference determinants.
b The total number of walkers at 200000 MC iterations is 6118222.
c The total number of walkers at 200000 MC iterations is 14878766.

Figure 3.6 Convergence of the CC(P ) and CC(P ;Q) energies of the X 3B1 [panel (a)] and
A 1A1 [panel (b)] states of methylene, as described by the aug-cc-pVTZ basis set, and of the
corresponding adiabatic singlet–triplet gaps [panel (c)] toward their parent CCSDT values.
The geometries of the X 3B1 and A 1A1 states, optimized in the FCI calculations using the
TZ2P basis set, were taken from Ref. [181]. The P spaces consisted of all singles and doubles
and subsets of triples identified during the i-FCIQMC propagations with δτ = 0.0001 a.u.
and the Q spaces consisted of the triples not captured by i-FCIQMC. Adapted from Ref.
[102].

69

0501001502000246(a)X 3B1CC(P)CC(P;Q)050100150200MC Ite ations (103)0246E  o   el. to CCSDT (mEh)(b)A 1A1CC(P)CC(P;Q)050100150200MC Ite ations (103)−1000100200300400E  o   el. to CCSDT (cm−1)(c)ΔES–TCC(P)CC(P;Q)Figure 3.7 Comparison of convergences of the CC(P), CC(P;Q), and the underlying i-
FCIQMC calculations toward their respective limits for the X 3B1 and A 1A1 states of the
CH2 molecule at their respective geometries optimized in the FCI calculations using the
TZ2P basis set are taken from Ref. [181].

70

3.3.2

(HFH)−

Our next example is the linear, D∞h-symmetric, (HFH)− anion, a prototype magnetic

system in which unpaired spins of terminal hydrogen atoms couple to singlet and triplet

states via a polarizable diamagnetic bridge of F− [196]. The energies of the lowest two

electronic states of the (HFH)− system, including the singlet ground state X 1Σ+

g and the first-

excited triplet state A 3Σ+

u , and the vertical gap between them, which is proportional to the

magnetic exchange coupling constant J and which should approach zero as both H–F bonds

are stretched to infinity, were used in the past to test various quantum chemistry approaches

[66, 80, 84, 88, 153, 196–199]. Among them were methods developed in the Piecuch group,

including CR-CC(2,3) [66, 197], CR-CC(2,4)[84], the DIP-EOMCC approaches with full and

active-space treatments of 4h-2p correlations of top of CCSD [80, 153], and the active-orbital-

based CC(t;3), CC(t,q;3), and CC(t,q;3,4) hierarchy [84, 88]. Here, we test the alternative

to CC(t;3) offered by the semi-stochastic, FCIQMC-driven, CC(P ;Q) algorithm aimed at

the CCSDT energetics. As in our previous studies [66, 80, 84, 88, 153, 197], we used the

6-31G(d,p) basis set and several stretches of both H–F bonds, including RH-F = 1.50, 1.75,

2.00, 2.50, and 4.00 ˚A, where RH-F is the distance between the hydrogen and fluorine nuclei.

An accurate computation of the singlet–triplet gap in the (HFH)− system is compli-

cated by the fact that, unlike the A 3Σ+

u state, which is weakly correlated and well rep-

resented by a single ROHF determinant, its ground-state counterpart X 1Σ+

g displays a

substantial MR character that includes a significant contribution from the doubly excited

(HOMO)2 → (LUMO)2 determinant, in addition to the RHF configuration. The MR char-

acter of the X 1Σ+

g state, which is already noticeable at shorter H–F separations and which

substantially strengthens as RH-F increases, can be illustrated by the ratio of the FCI ex-

pansion coefficients at the (HOMO)2 → (LUMO)2 and RHF determinants or the equivalent

T2 cluster amplitude extracted from FCI, which increases, in absolute value, from 0.38 at

RH-F = 1.50 ˚A to 1.17 at RH-F = 4.00 ˚A, when the 6-31G(d,p) basis is employed [66, 197]

(the HOMO and LUMO have different symmetries, σg and σu, respectively, so that the

71

HOMO → LUMO T1 amplitude is zero). As a result of all this, it is difficult to balance the

lowest two states of the (HFH)− system in a single quantum chemistry calculation, especially

when the SRCC framework using the RHF reference determinant for the X 1Σ+

g state and the

ROHF reference for the A 3Σ+

u state is employed. Indeed, as shown in Refs. [66, 88, 197], the

differences between the energies obtained in the CCSD/6-31G(d,p) computations and their

FCI counterparts at RH-F = 1.50 ˚A are 12.674 millihartree for the X 1Σ+
millihartree when the A 3Σ+

g state and only 2.628
u state is considered. The analogous differences at RH-F = 2.00 ˚A

are 19.398 and 2.068 millihartree, respectively. The observed large discrepancies between the

errors in the CCSD energies for the X 1Σ+

g and A 3Σ+

u states translate into a poor descrip-

tion of the singlet–triplet gaps. One can see this by comparing the ∆ES-T values resulting

from the RHF/ROHF-based CCSD/6-31G(d,p) computations at RH-F = 1.50, 1.75, 2.00,

2.50, and 4.00 ˚A with the corresponding FCI data. CCSD/6-31G(d,p) gives −7320, −1838,

1656, 3605, and 230 cm−1, respectively, as opposed to −9525, −4911, −2147, −277, and 0

cm−1 obtained with FCI [66, 88, 197]. If we are to improve the CCSD results within the

SRCC framework, we must turn to higher-level theories, such as the CCSDT approach that

interests us in this study [84, 88, 198], CCSDTQ [84], or the DIP-EOMCC methodology,

especially after incorporating 4h-2p correlations [80, 153]. The CCSDT method is indeed

very accurate, reducing the 2205, 3073, 3804, 3882, and 230 cm−1 errors relative to FCI in

the ∆ES-T values obtained with CCSD/6-31G(d,p) at RH-F = 1.50, 1.75, 2.00, 2.50, and 4.00

˚A to 198, 270, 341, 420, and 58 cm−1, respectively [84, 88, 198]. It also greatly improves the

total electronic energies. Indeed, the differences between the CCSDT and FCI energies of

the X 1Σ+

g and A 3Σ+

u states in the entire RH-F = 1.50 − 4.00 ˚A region obtained using the

6-31G(d,p) basis set do not exceed 2.276 and 0.389 millihartree, respectively [84, 88]. The

analogous differences between the CCSD and FCI energies are as large as 20.546 millihartree

for the former state and 2.628 millihartree when the latter state is considered (see Fig. 3.8 for

a comparison of CCSD, CCSDT, and FCI energetics throughout the 1.500 ˚A–4.00˚A range).

One can reduce the remaining small errors in the CCSDT results even further or practi-

72

cally eliminate them by using CCSDTQ [84] or the DIP-EOMCC approaches with 4h-2p

contributions, [80, 153] but the objective of this study is to assess the performance of our

semi-stochastic CC(P ;Q) methodology in converging the CCSDT data.

The results of our FCIQMC-driven CC(P ;Q)/6-31G(d,p) computations for the X 1Σ+
g

and A 3Σ+

u states of the linear (HFH)− system at the H–F distances RH-F = 1.50, 1.75, 2.00,
2.50, and 4.00 ˚A and the corresponding ∆ES-T values, along with the underlying CC(P ) data,

are reported in Tables 3.7–3.9 and Fig. 3.9. In all of our CC(P ) and CC(P ;Q) computations

and the underlying i-FCIQMC runs for the D∞h-symmetric (HFH)− system, we used the
D2h Abelian subgroup of D∞h. In particular, the i-FCIQMC calculations for the X 1Σ+
A 3Σ+

u states were set up to converge the lowest-energy states of the 1Ag(D2h) and 3B1u

g and

(D2h) symmetries. As a result, the subsets of triply excited determinants used to construct

the P spaces for the subsequent CC(P ) and CC(P ;Q) computations for the X 1Σ+

g state at

the various RH-F and τ values considered in this work were defined as the Sz = 0 triples

of the Ag (D2h) symmetry captured in the underlying i-FCIQMC propagations. Similarly,

the subsets of triply excited determinants used to design the P spaces for the CC(P ) and

CC(P ;Q) calculations for the A 3Σ+

u state were the Sz = 1 triples of the B1u (D2h) symmetry

extracted from i-FCIQMC. In analogy to all other CC(P ;Q) computations performed in this

work, the Q spaces used to determine the δ(P ; Q) corrections to the CC(P ) energies were

defined as the remaining triples not captured by the respective i-FCIQMC runs.

As shown in Table 3.7, and in line with our earlier CC(P ;Q) work [88] and the above

remarks, the CC(P ) energies of the X 1Σ+

g state of (HFH)− obtained at τ = 0, which

are identical to those resulting from the conventional CCSD calculations reported in Refs.

[66, 88, 197], are characterized by large errors relative to their τ = ∞, i.e., CCSDT, parents.

Indeed, the differences between the τ = 0 and τ = ∞ CC(P ) energies for the X 1Σ+

g state
increase from 11.412 millihartree at RH-F = 1.50 ˚A to more than 17 millihartree at RH-F =

2.00 and 2.50 ˚A. These differences become smaller at large H–F separations, represented in

our calculations by RH-F = 4.00 ˚A, where the D∞h-symmetric (HFH)− system is essentially

73

dissociated into the stretched hydrogen molecule, which has only two electrons, so that

CCSD becomes exact, and the closed-shell fluoride ion, which has the electronic structure

of the neon atom and which is characterized by small Tn correlations with n > 2, but they

remain large when the RH-F values are smaller. This should be contrasted with the small,

∼1–2 millihartree, differences between the τ = 0 and τ = ∞ CC(P ) energies obtained at all

values of RH-F for the predominantly SR A 3Σ+
above, and as shown in Table 3.9, this imbalance in the description of the X 1Σ+

u state (see Table 3.8). As already alluded to
g and A 3Σ+

u

states by the CCSD, i.e., τ = 0 CC(P ), calculations gives rise to large errors in the resulting

∆ES-T values relative to their τ = ∞ (CCSDT) counterparts, which range from 2007 cm−1

to 3462 cm−1 in the RH-F = 1.50–2.50 ˚A region. Once again, these errors become small at

large H–F separations, such as RH-F = 4.00 ˚A used in this work, where (HFH)− is more or

less equivalent to the stretched H2 and F−, resulting in the nearly degenerate singlet and

triplet states and the 172 cm−1 difference between the τ = 0 and τ = ∞ CC(P ) values of

∆ES-T, but at shorter H–F distances they are large and comparable to or even larger than

the singlet–triplet gap values provided by CCSDT or FCI.

The situation dramatically changes, when the τ = 0 CC(P ) or CCSD energies are cor-

rected for T3 correlations with the help of the noniterative correction δ(P ; Q), as in the

τ = 0 CC(P ;Q) calculations, which are equivalent to the purely deterministic CR-CC(2,3)

runs reported in Refs. [66, 84, 88, 197]. As shown in Tables 3.7–3.9, the τ = 0 CC(P ;Q), i.e.,

CR-CC(2,3), energies of the X 1Σ+

g and A 3Σ+

u states at the various H–F distances considered

in this study and the gaps between them are substantially more accurate than their uncor-

rected CC(P ) (i.e., CCSD) counterparts. For example, the CR-CC(2,3) approach reduces

the large, more than 17 millihartree, errors in the CCSD energies of the X 1Σ+

g state relative
to their CCSDT [τ = ∞ CC(P ) or CC(P ;Q)] parents at RH-F = 2.00 and 2.50 ˚A to ∼1–3
millihartree. We see similarly significant improvements in the CCSD energies of the X 1Σ+
g
state by CR-CC(2,3) at other H–F distances, even at the “easiest” RH-F = 4.00 ˚A value,

where the triples correction δ(P ; Q) is capable of reducing the already small, 1.907 milli-

74

hartree, difference between the CCSD and CCSDT energies to the much smaller (in absolute

value) 0.291 millihartree (see Table 3.7). Consistent with our earlier studies, [66, 84, 88, 197]

the CR-CC(2,3) method performs even better when the weakly correlated A 3Σ+

u state is

examined, reducing the ∼1–2 millihartree errors in the underlying CCSD energetics relative

to our CCSDT target to about 0.2 millihartree (see Table 3.8). As a result of all of these

accuracy improvements, the singlet–triplet gap values obtained using CR-CC(2,3) are much

closer to their CCSDT parents than their CCSD counterparts, reducing the 2007, 2803,

3462, 3462, and 172 cm−1 errors relative to CCSDT obtained with CCSD at RH-F = 1.50,

1.75, 2.00, 2.50, and 4.00 ˚A, respectively, by factors ranging from 6 at RH-F = 2.50 ˚A to

72 at RH-F = 1.50 ˚A, but, as shown in Table 3.9 [see, also, Ref. [88], where one can find

a comparison of the CCSD, CR-CC(2,3), and CCSDT ∆ES-T data for additional H–F dis-

tances], the differences on the order of (−600)–(−300) cm−1 between the CR-CC(2,3) and

CCSDT singlet–triplet separations in the intermediate RH-F = 2.00–3.00 ˚A region remain.

The question arises if one can refine the CR-CC(2,3) results by enriching the P spaces used

in the CC(P ;Q) calculations, which in CR-CC(2,3) consist of only singles and doubles, with

the subsets of triply excited determinants identified by i-FCIQMC propagations.

As shown in Tables 3.7–3.9 and Fig. 3.9, once the leading triply excited determinants,

captured using i-FCIQMC at τ > 0, are included in the respective P spaces and the δ(P ; Q)

corrections due to the remaining T3 correlations are added to the energies obtained in the

CC(P ) calculations, the resulting CC(P ;Q) values of the X 1Σ+

g and A 3Σ+

u energies and

vertical gaps between them display very fast convergence toward their CCSDT counterparts.

This is already observed when the i-FCIQMC propagation times are short, engaging tiny

walker populations that are orders of magnitude smaller than those required to converge the

i-FCIQMC runs, and the fractions of the triply excited determinants captured by i-FCIQMC

are small. For example, after as few as 2000 δτ = 0.0001 a.u. MC iterations, where τ is only

0.2 a.u. and where, as shown in Table 3.10, the total walker populations characterizing the

underlying i-FCIQMC runs are 0.01–0.11% of the respective numbers of walkers at τ = 20.0

75

a.u. [the termination time for our i-FCIQMC propagations for (HFH)−], the differences

between the CC(P ;Q) and CCSDT energies obtained for the strongly correlated X 1Σ+
g
state are −0.035 millihartree for RH-F = 1.50 ˚A, −0.056 millihartree for RH-F = 1.75 ˚A,

−0.110 millihartree for RH-F = 2.00 ˚A, −0.583 millihartree for RH-F = 2.50 ˚A, and −0.025

millihartree for RH-F = 4.00 ˚A. In spite of using only about 10–30% of all triply excited
determinants in the underlying P spaces, the FCIQMC-based CC(P ;Q) energies of the X 1Σ+
g

state obtained after 2000 MC iterations reduce the errors relative to CCSDT characterizing

the CR-CC(2,3) [i.e., τ = 0 CC(P ;Q)] computations in the RH-F = 1.50–4.00 ˚A region by

factors ranging from 5 to 13 (see Table 3.7). In fact, with an exception of RH-F = 2.00 and

2.50 ˚A, they are much more accurate than the results produced by the purely deterministic

CC(t;3) analog of the semi-stochastic CC(P ;Q) methodology, reported in Refs. [84, 88]. One

can observe even more dramatic improvements over CR-CC(2,3) offered by the FCIQMC-

driven CC(P ;Q) approach, when the propagation time τ increases. For example, after 4000

MC iterations, where the i-FCIQMC propagations are still far from being converged (cf. the

total walker populations used by our i-FCIQMC runs relative to the termination time τ =

20.0 a.u. in Table 3.10) and the fractions of triples included in the stochastically determined

P spaces, which range from 12% at RH-F = 4.00 ˚A to 56% at RH-F = 1.50 ˚A, remain

relatively small, the differences between the CC(P ;Q) and CCSDT energies obtained for

the X 1Σ+

g state at RH-F = 1.50, 1.75, 2.00, 2.50, and 4.00 ˚A are −28, −9, −17, −50,

and −4 microhartree, respectively, reducing the errors relative to CCSDT that characterize

the corresponding CR-CC(2,3) calculations by factors ranging from 12 to 86 [2 to 61 when

compared to the CC(t;3) results reported in Refs. [84, 88]]. As shown in Table 3.8, the

performance of the FCIQMC-driven CC(P ;Q) approach becomes even more impressive when

the A 3Σ+

u state, which has a SR character, is examined. After 2000 δτ = 0.0001 a.u. MC

time steps, the errors in the CC(P ;Q) energies relative to their CCSDT parents obtained for

the A 3Σ+

u state at RH-F = 1.50, 1.75, 2.00, 2.50, and 4.00 ˚A are only −40, −24, −38, −29,

and −14 microhartree, respectively. After 4000 MC iterations, they become −10, −10, −12,

76

−9, and −2 microhartree, respectively. Once again, these are considerable improvements

compared to CR-CC(2,3) and CC(t;3) that both give errors on the order of −0.2 millihartree

[84, 88], especially if we realize that the fractions of triples captured by the i-FCIQMC runs

after 2000 and 4000 MC iterations are relatively small (5–28% and 5–49%, respectively)

and, as shown in Table 3.10, the corresponding numbers of walkers represent only about

1–2% of the total numbers of walkers at τ = 20.0 a.u., where we stopped our i-FCIQMC

propagations.

As a consequence of the small errors in the CC(P ;Q) total energies characterizing the

X 1Σ+

g and A 3Σ+

u states in the early stages of the i-FCIQMC propagations, the resulting

singlet–triplet gap values are very accurate as well. This is demonstrated in Table 3.9,

where one can see that after 2000 δτ = 0.0001 a.u. MC iterations, which is, as already

explained, a very short propagation time engaging tiny walker populations and small fractions

of triples, most of the differences between the CC(P ;Q) and CCSDT ∆ES-T values in the

RH-F = 1.50–4.00 ˚A region are on the order of a few reciprocal centimeter. The only

exception is the semi-stochastic CC(P ;Q) run at RH-F = 2.50 ˚A, where the −122 cm−1

error relative to CCSDT characterizing the singlet–triplet gap obtained after 2000 MC time

steps, while representing a five-fold error reduction compared to CR-CC(2,3), is comparable,

in magnitude, to the CCSDT value of ∆ES-T. This happens because the CC(P ;Q) energy
of the strongly correlated X 1Σ+

g state obtained after 2000 MC iterations at RH-F = 2.50

˚A differs from its CCSDT parent by −0.583 millihartree, whereas the analogous difference

between the CC(P ;Q) and CCSDT energies for its weakly correlated A 3Σ+

u companion is

only −29 microhartree. This is not a problem though, since by running i-FCIQMC a little

longer and capturing about 20% of all triply excited determinants in the relevant P spaces,

as is the case when 4000 δτ = 0.0001 a.u. MC time steps are considered, one reduces the

differences between the CC(P ;Q) and CCSDT energies of the X 1Σ+

g and A 3Σ+

u states to −50

and −9 microhartree, respectively (cf. Tables 3.7 and 3.8), so that the 122 cm−1 unsigned

error in the CC(P ;Q) value of ∆ES-T relative to CCSDT obtained after 2000 MC iterations

77

decreases to less than 10 cm−1. This is yet another illustration of the ability of the semi-

stochastic CC(P ;Q) methodology pursued in this work to balance the more MR singlet and

weakly correlated triplet states of biradical systems in a single computation at the fraction

of the cost of the parent high-level CC calculations. As shown in Table 3.9, at τ = 0.4 a.u.,

where the i-FCIQMC propagations are still far from being converged, the FCIQMC-driven

CC(P ;Q) calculations recover the CCSDT values of the singlet–triplet gaps in (HFH)− at

all H–F distances considered in this study to within a few reciprocal centimeter, reaching a

1–2 cm−1 or better accuracy after 6000 MC iterations.

Last, but not least, the results reported in Tables 3.7–3.9 and Fig. 3.9 also demonstrate

the remarkable efficiency of the δ(P ; Q) corrections in accelerating the convergence of the

CC(P ) energies of the X 1Σ+

g and A 3Σ+

u states and the vertical gaps between them toward

CCSDT, independent of the H–F distance considered. Let us, for example, compare the un-

corrected CC(P ) and corrected CC(P ;Q) energies of the X 1Σ+

g and A 3Σ+

u states of (HFH)−

at the five H–F separations considered in this work obtained after 2000 MC iterations. In

the case of the former, more MR, state, the CC(P ;Q) corrections δ(P ; Q) reduce the positive

2.601, 3.998, 3.511, 6.586, and 0.412 millihartree errors relative to CCSDT resulting from the

CC(P ) computations at RH-F = 1.50, 1.75, 2.00, 2.50, and 4.00 ˚A to the much smaller neg-

ative error values of −0.035, −0.056, −0.110, −0.583, and −0.025 millihartree, respectively.

When the latter state, which is characterized by much weaker correlations, is considered, the

0.995, 0.826, 0.834, 0.502, and 0.239 millihartree errors obtained with CC(P ) are reduced to

−40, −24, −38, −29, and −14 microhartree, respectively, when the CC(P ;Q) approach is

employed. It is interesting to notice that while the errors characterizing the CC(P ) calcu-

lations for the A 3Σ+

u state are generally much smaller than their X 1Σ+

g counterparts, and

the two states have a substantially different character, the error reductions offered by the

CC(P ;Q) corrections δ(P ; Q), by at least one order of magnitude, apply to both states. As

already alluded to above, and as shown in Table 3.9 and Fig. 3.9 (e) and (f), where we

examine the convergence of the CC(P ) and CC(P ;Q) ∆ES-T values toward their CCSDT

78

parents, the noniterative corrections δ(P ; Q) are also very effective in improving the balance

in the description of the X 1Σ+

g and A 3Σ+

u states by the CC(P ) approach and smoothing

the convergence of the resulting singlet–triplet gaps toward their CCSDT limits. This can

be illustrated by comparing the behavior of the error values relative to CCSDT character-

izing the CC(P ) calculations of ∆ES-T at RH-F = 2.00 ˚A with their CC(P ;Q) counterparts,

shown in Table 3.9. In the former case, the 3462 cm−1 error at τ = 0 decreases, in absolute

value, to 8 cm−1 at τ = 0.8 a.u. (8000 MC iterations), to increase to 17 cm−1 at τ = 2.0 a.u.

(20000 MC iterations), to decrease again to a numerical 0 cm−1 at τ = 20.0 a.u. (200000 MC

iterations). Once the CC(P ) energies of the X 1Σ+

g and A 3Σ+

u states are corrected using the

δ(P ; Q) corrections, the unsigned errors in the resulting CC(P ;Q) values of ∆ES-T relative

to their CCSDT parent monotonically and rapidly decrease, from 282 cm−1 at τ = 0 to a

numerical 0 cm−1 at τ ≥ 0.8 a.u. It is clear from Tables 3.7–3.9 and Fig. 3.9 that while both

the CC(P ) and CC(P ;Q) energies converge to the parent CCSDT limit, the latter energies

and the gaps between them converge to CCSDT a lot faster.

79

Table 3.7 Convergence of the CC(P ) and CC(P ;Q) energies of the X 1Σ+
g state of (HFH)−, as described by the 6-31G(d,p)
basis set, at selected H–F distances RH-F toward their parent CCSDT values. The P spaces used in the CC(P ) and CC(P ;Q)
calculations were defined as all singly and doubly excited determinants and subsets of triply excited determinants extracted
from the i-FCIQMC propagations with δτ = 0.0001 a.u. The Q spaces used to determine the CC(P ;Q) corrections consisted
of the triply excited determinants not captured by the corresponding i-FCIQMC runs. The i-FCIQMC calculations preceding
the CC(P ) and CC(P ;Q) steps were initiated by placing 1500 walkers on the RHF reference determinant and the na parameter
of the initiator algorithm was set at 3. In all post-Hartree–Fock calculations, the lowest core orbital was kept frozen and the
spherical components of d orbitals were employed throughout. Adapted from Ref. [102].

MC Iters.
0
2000
4000
6000
8000
10000
20000
50000
100000
150000
200000
∞

RH-F = 1.50 ˚A
P a

(P ; Q)a %Tb

0

11.412c −0.343d
2.601 −0.035 34.2
0.843 −0.028 56.1
0.595 −0.004 63.9
0.225 −0.003 68.6
0.258 −0.003 70.9
77.2
0.000
0.112
88.4
0.000
0.017
97.7
0.000
0.002
99.5
0.000
0.000
99.9
0.000
0.000

−100.588130e

RH-F = 1.75 ˚A

P a

(P ; Q)a %Tb

RH-F = 2.00 ˚A
P a

(P ; Q)a %Tb

RH-F = 2.50 ˚A
P a

(P ; Q)a %Tb P a

RH-F = 4.00 ˚A

(P ; Q)a %Tb

0

0

17.453c −1.455d
14.738c −0.686d
3.511 −0.110 22.6
3.998 −0.056 30.5
1.979 −0.017 40.5
1.078 −0.009 49.6
0.432 −0.010 46.7
0.434 −0.003 58.1
0.187 −0.003 50.2
0.477 −0.007 61.4
0.136 −0.003 54.5
0.161 −0.002 63.3
0.079 −0.002 61.1
0.056 −0.001 71.0
77.5
0.000
0.005
85.8
0.000
0.019
94.4
0.000
0.001
96.3
0.000
0.001
99.2
0.000
99.4
0.000
0.000
0.000
99.9
0.000
100.0 0.000
0.000
0.000
−100.576056e

−100.561110e

17.051c −2.800d
1.907c −0.291d
0
0
7.6
6.586 −0.583 15.2 0.412 −0.025
0.973 −0.050 25.6 0.141 −0.004 11.7
0.459 −0.012 30.2 0.076 −0.003 13.2
0.225 −0.003 33.9 0.037 −0.001 14.4
0.167
35.4 0.025 −0.001 15.3
0.042 −0.001 41.8 0.026 −0.001 19.0
58.8 0.002 −0.001 28.6
0.009
54.8
81.8 0.000
0.000
73.7
94.1 0.000
0.000
86.9
99.2 0.000
0.000

0.000
0.000
0.000
0.000

0.000
0.000
0.000
−100.525901e

−100.539783e

0.000

a Unless otherwise stated, all energies are reported as errors relative to CCSDT in millihartree.
b The %T values are the percentages of triples captured during the i-FCIQMC propagations [the Sz = 0 triply excited determinants of the Ag (D2h)
symmetry].
c Equivalent to CCSD.
d Equivalent to CR-CC(2,3) [the most complete variant of CR-CC(2,3) abbreviated sometimes as CR-CC(2,3),D or CR-CC(2,3)D].
e Total CCSDT energy in hartree.

80

Table 3.8 Convergence of the CC(P ) and CC(P ;Q) energies of the A 3Σ+
u state of (HFH)−, as described by the 6-31G(d,p)
basis set, at selected H–F distances RH-F toward their parent CCSDT values. The P spaces used in the CC(P ) and CC(P ;Q)
calculations were defined as all singly and doubly excited determinants and subsets of triply excited determinants extracted
from the i-FCIQMC propagations with δτ = 0.0001 a.u. The Q spaces used to determine the CC(P ;Q) corrections consisted of
the triply excited determinants not captured by the corresponding i-FCIQMC runs. The i-FCIQMC calculations preceding the
CC(P ) and CC(P ;Q) steps were initiated by placing 1500 walkers on the ROHF reference determinant and the na parameter
of the initiator algorithm was set at 3. In all post-Hartree–Fock calculations, the lowest core orbital was kept frozen and the
spherical components of d orbitals were employed throughout. Adapted from Ref. [102].

RH-F = 1.50 ˚A

RH-F = 1.75 ˚A

RH-F = 2.00 ˚A

RH-F = 2.50 ˚A

RH-F = 4.00 ˚A

MC Iters. P a

(P ; Q)a %Tb

P a

(P ; Q)a %Tb

P a

(P ; Q)a %Tb

P a

(P ; Q)a %Tb

P a

(P ; Q)a %Tb

0
2000
4000
6000
8000
10000
20000
50000
100000
150000
200000
∞

0

0

0

0

1.967c −0.181d

1.277c −0.167d

2.268c −0.217d
1.123c −0.180d
1.678c −0.172d
0.995 −0.040 27.8 0.826 −0.024 24.2 0.834 −0.038 19.1 0.502 −0.029 10.7 0.239 −0.014
0.456 −0.010 49.4 0.477 −0.010 41.7 0.475 −0.012 33.7 0.236 −0.009 17.2 0.079 −0.002
0.338 −0.005 56.4 0.266 −0.001 50.5 0.321 −0.005 41.2 0.174 −0.003 21.5 0.070 −0.003
0.290 −0.003 60.1 0.254 −0.003 54.2 0.225 −0.003 44.7 0.195 −0.006 23.9 0.064 −0.002
0.271 −0.003 61.1 0.267 −0.004 56.6 0.201 −0.002 45.7 0.064 −0.003 25.0 0.056 −0.002
0.201 −0.002 67.9 0.151 −0.001 62.1 0.157 −0.002 52.2 0.078 −0.003 28.6 0.025 −0.001
0.000
0.082
0.000
0.021
0.000
0.007
0.000
0.001
−100.526164e

76.3 0.069 −0.001 66.1 0.049 −0.001 37.4 0.012
52.9 0.002
82.8 0.014
89.4 0.015
68.6 0.001
92.9 0.002
95.8 0.003
81.8 0.000
97.1 0.000
98.4 0.001

0.000
0.000
0.000
0.000
−100.545633e

0.000
0.000
0.000
0.000
−100.554908e

0.000
0.000
0.000
−100.552882e

0.000
0.000
0.000
−100.540435e

80.0 0.056
91.8 0.016
96.7 0.003
98.8 0.001

0
4.5
5.4
5.9
6.0
6.4
7.4
8.4
11.7
16.8
23.8

a Unless otherwise stated, all energies are reported as errors relative to CCSDT in millihartree.
b The %T values are the percentages of triples captured during the i-FCIQMC propagations [the Sz = 1 triply excited determinants of the B1u (D2h)
symmetry].
c Equivalent to CCSD.
d Equivalent to CR-CC(2,3) [the most complete variant of CR-CC(2,3) abbreviated sometimes as CR-CC(2,3),D or CR-CC(2,3)D].
e Total CCSDT energy in hartree.

81

Table 3.9 Convergence of the CC(P ) and CC(P ;Q) singlet–triplet gaps of (HFH)−, as
described by the 6-31G(d,p) basis set, at selected H–F distances RH-F toward their parent
CCSDT values. The P spaces used in the CC(P ) and CC(P ;Q) calculations were defined
as all singly and doubly excited determinants and subsets of triply excited determinants
extracted from the i-FCIQMC propagations with δτ = 0.0001 a.u. The Q spaces used to
determine the CC(P ;Q) corrections consisted of the triply excited determinants not captured
by the corresponding i-FCIQMC runs. The i-FCIQMC calculations preceding the CC(P )
and CC(P ;Q) steps were initiated by placing 1500 walkers on the RHF (X 1Σ+
g state) and
ROHF (A 3Σ+
u state) reference determinants and the na parameter of the initiator algorithm
was set at 3. In all post-Hartree–Fock calculations, the lowest core orbital was kept frozen
and the spherical components of d orbitals were employed throughout. Adapted from Ref.
[102].

RH-F = 1.50 ˚A RH-F = 1.75 ˚A RH-F = 2.00 ˚A RH-F = 2.50 ˚A RH-F = 4.00 ˚A

MC Iters. P a

(P ; Q)a P a

2007b −28c
0
353
2000
85
4000
6000
56
8000 −14
10000 −3
20000 −20
50000 −14
100000 −4
150000 −2
200000
0
∞

1
−4
0
0
0
0
0
0
0
0
−9327d

(P ; Q)a P a

(P ; Q)a P a

(P ; Q)a
(P ; Q)a P a
2803b −111c 3462b −282c 3462b −578c 172b −24c
−2
696
0
132
0
37
0
49
0
−23
0
−21
0
−8
0
−3
0
−1
0
0
58d

1335 −122
−9
162
−2
62
1
7
0
23
1
−8
0
−9
0
−3
0
0
0
0
143d

−16
−1
−1
0
0
0
0
0
0
0
−1806d

−7
0
0
−1
0
0
0
0
0
0
−4641d

588
330
24
−8
−14
−17
−14
−3
−1
0

38
14
1
−6
−7
0
−2
−1
0
0

a Unless otherwise stated, all singlet–triplet gaps are reported as errors relative to CCSDT in cm−1.
b Equivalent to CCSD.
c Equivalent to CR-CC(2,3) [the most complete variant of CR-CC(2,3) abbreviated sometimes as CR-
CC(2,3),D or CR-CC(2,3)D].
d The CCSDT singlet–triplet gap in cm−1.

82

Table 3.10 The total numbers of walkers, reported as percentages of the total walker popula-
tions at 200000 MC iterations, characterizing the i-FCIQMC propagations with δτ = 0.0001
g and A 3Σ+
a.u. that were needed to generate the CC(P) and CC(P;Q) results for the X 1Σ+
states of (HFH)− reported in Tables 3.7 and 3.8. Adapted from Ref. [102].

u

MC Iters. X 1Σ+
0.02a
0
0.11
2000
0.20
4000
0.27
6000
0.32
8000
0.37
10000
0.55
20000
1.42
50000
100000
6.21
150000 25.44
100b
200000

RH-F = 1.50 ˚A RH-F = 1.75 ˚A RH-F = 2.00 ˚A RH-F = 2.50 ˚A RH-F = 4.00 ˚A
g A 3Σ+
0.13a
0.73
1.23
1.59
1.77
1.91
2.46
4.81
13.61
37.71
100g

g A 3Σ+
u X 1Σ+
u
0.59a
0.00a
1.74
0.01
2.45
0.02
2.97
0.02
3.20
0.03
3.29
0.03
4.09
0.06
6.83
0.21
1.61
16.64
12.45 41.09
100k
100j

u X 1Σ+
0.01a
0.03
0.05
0.07
0.08
0.09
0.15
0.47
2.88
17.11
100h

u X 1Σ+
0.01a
0.08
0.15
0.19
0.23
0.26
0.40
1.10
5.16
23.37
100d

g A 3Σ+
0.10a
0.64
1.14
1.46
1.66
1.79
2.38
4.59
12.87
36.28
100e

g A 3Σ+
0.24a
0.95
1.54
1.86
2.12
2.29
3.03
5.58
15.13
38.88
100i

g A 3Σ+
0.09a
0.59
1.08
1.36
1.55
1.68
2.19
4.19
12.24
35.16
100c

u X 1Σ+
0.01a
0.06
0.10
0.13
0.16
0.18
0.28
0.81
4.25
21.01
100f

g state) and ROHF (A 3Σ+

u state)

a The initial walker population, meaning 1500 walkers on the RHF (X 1Σ+
reference determinants.
b The total number of walkers at 200000 MC iterations is 9865967.
c The total number of walkers at 200000 MC iterations is 1749699.
d The total number of walkers at 200000 MC iterations is 12468454.
e The total number of walkers at 200000 MC iterations is 1431689.
f The total number of walkers at 200000 MC iterations is 15510033.
g The total number of walkers at 200000 MC iterations is 1123676.
h The total number of walkers at 200000 MC iterations is 24265207.
i The total number of walkers at 200000 MC iterations is 632102.
j The total number of walkers at 200000 MC iterations is 50189301.
k The total number of walkers at 200000 MC iterations is 254390.

83

Figure 3.8 Total electronic energies of the X 1Σ+
g (open circles and solid line) and A 3Σ+
u
(filled circles and dotted line) states of (HFH)− with increase in the H–F distance, from 1.5
˚A to 4.0 ˚A, obtained from the FCI (red circles), CCSD (blue circles), and CCSDT (green
circles) methods. Recreated from the data reported in Refs. [66, 84, 88, 197].

84

H-F distance increasesFigure 3.9 Convergence of the CC(P ) and CC(P ;Q) energies of the X 1Σ+
g [panels (a) and
(b)] and A 3Σ+
u [panels (c) and (d)] states of (HFH)−, as described by the 6-31G(d,p) basis
set, and of the corresponding singlet–triplet gaps [panels (e) and (f)] toward their parent
CCSDT values. The H–F distances RH-F used are 1.50 ˚A, 1.75 ˚A, 2.00 ˚A, 2.50 ˚A, and 4.00
˚A. The P spaces consisted of all singles and doubles and subsets of triples identified during
i-FCIQMC propagations with δτ = 0.0001 a.u. and the Q spaces consisted of the triples not
captured by i-FCIQMC. Adapted from Ref. [102].

85

05010015020001234Error rel. to CCSDT (mEh)(a)X 1Σ+g, CC(P)1.50 Å1.75 Å2.00 Å2.50 Å4.00 Å05010015020001234(b)X 1(+g, CC(P;Q)1.50 Å1.75 Å2.00 Å2.50 Å4.00 Å05010015020001234Error rel. to CCSDT (mEh)(c)A 3Σ+u, CC(P)1.50 Å1.75 Å2.00 Å2.50 Å4.00 Å05010015020001234(d)A 3(+u, CC(P;Q)1.50 Å1.75 Å2.00 Å2.50 Å4.00 Å050100150200MC Iterations (103)010203040Error rel. to CCSDT (c −1)(e)ΔES–T,ΔCC(P)1.50ΔÅ1.75ΔÅ2.00ΔÅ2.50ΔÅ4.00ΔÅ050100150200MCΔIterationsΔ(103)010203040(f)ΔES–T,ΔCC(P;Q)1.50ΔÅ1.75ΔÅ2.00ΔÅ2.50ΔÅ4.00ΔÅ3.3.3 Cyclobutadiene and cyclopentadienyl cation

We now proceed to the examination of the performance of the semi-stochastic CC(P ;Q)

algorithm in calculations involving medium-sized organic biradicals, starting from two pro-

totypical anti-aromatic systems, cyclobutadiene and cyclopentadienyl cation, both described

using the cc-pVDZ basis set. As in the rest of this chapter, we are mainly interested in how

efficient the CIQMC-driven CC(P ;Q) methodology is in recovering the CCSDT energies of

the lowest singlet and triplet states and gaps between them. In the case of cyclobutadiene

and cyclopentadienyl cation discussed in this subsection, we focus on examining vertical

singlet–triplet gaps.

We begin with the FCIQMC-driven CC(P ;Q) calculations for cyclobutadiene, in which

we adopted the D4h-symmetric geometry that represents the transition state for the au-

tomerization of cyclobutadiene proceeding on the lowest singlet potential, optimized with

the MR average-quadratic CC (MR-AQCC) approach [200, 201] using the cc-pVDZ basis in

Ref. [202]. We employed this geometry for two reasons. One of them is the fact that we used

the same geometry in our earlier CIQMC- and CCMC-based [99, 101], CIPSI-driven [120],

and active-orbital-based [87] CC(P ;Q) calculations for cyclobutadiene, when examining its

automerization. Because of this, we could verify the correctness of our FCIQMC-driven

CC(P ;Q) calculations for the lowest-energy singlet state, which is also the ground state of the

system. Another is the observation that the D4h-symmetric transition-state structure char-

acterizing the automerization of cyclobutadiene is practically identical to the D4h-symmetric

minimum on the lowest triplet surface.

Indeed, the MR-AQCC/cc-pVDZ C–C and C–H

bond lengths defining the transition state on the ground-state singlet potential differ from

those characterizing the triplet minimum optimized using unrestricted CCSD (UCCSD) in

Ref. [203] by less than 0.009 and 0.001 ˚A, respectively.

At the D4h-symmetric geometry used in our calculations, cyclobutadiene is characterized

by the delocalization of four π electrons over four π MOs, which gives rise to the close-lying

singlet and triplet states that require a highly accurate treatment of electron correlation

86

effects if we are to obtain a well-balanced description of the two states and the small energy

separation between them. One can understand this by examining the valence π network of the

D4h-symmetric cyclobutadiene species, which consists of the doubly occupied nondegenerate

a2u orbital, the doubly degenerate eg level, in which each component MO is occupied by a

single electron, and the nondegenerate b1u orbital, which in the zeroth-order description of the

lowest singlet and triplet states remains empty. The two valence electrons in the degenerate

eg shell can couple to a singlet or a triplet, resulting in the open-shell singlet ground state,

X 1B1g, which has a substantial MR character, and the first excited triplet state, A 3A2g,

which is predominantly SR in nature (see Fig. 3.10 for an illustration of the π MO network

of cyclobutadiene along with the triplet electronic configuration). In order to balance the

substantial nondynamical correlation effects, needed for an accurate description of the low-

spin X 1B1g state, with the dynamical correlations dominating its high-spin triplet A 3A2g

companion within a conventional, particle-conserving, SRCC framework and produce reliable

∆ES-T values for cyclobutadiene, which could compete with the high-accuracy ab initio data

reported in Refs. [154, 155, 202–208], one has to consider robust treatments of the connected

triply excited clusters, such as that offered by CCSDT [204, 208].

Indeed, full CCSDT,

which is the target of this investigation, produces high-quality results for the lowest singlet

and triplet states of the D4h-symmetric cyclobutadiene system and the energy separation

between them. For example, the ∆ES-T value obtained in the CCSDT/cc-pVDZ calculations

at the transition-state geometry used in the present study, of −4.8 kcal/mol, is practically

identical to the results of the state-of-the-art DEA-EOMCC computations including the

high-rank 4p-2h correlations on top of CCSD, reported in Refs. [154, 155], which give −5.0

kcal/mol when the cc-pVDZ basis set is employed (for similar recent observations regarding

the reliability of full CCSDT in generating virtually exact singlet–triplet gap values for

cyclobutadiene, see Ref. [208]). It is, therefore, interesting to explore if the semi-stochastic

CC(P ;Q) methodology investigated in this work is capable of converging the CCSDT results

for the X 1B1g and A 3A2g states of cyclobutadiene and vertical gap between them out of the

87

early stages of CIQMC propagations.

The results of our FCIQMC-driven CC(P ) and CC(P ;Q) computations for cyclobutadi-

ene are summarized in Table 3.11 and Fig. 3.11. In all of our calculations, starting with the

stochastic i-FCIQMC steps and ending with the deterministic CC(P ;Q) and CCSDT runs,

we used the D2h Abelian subgroup of the D4h point group characterizing the cyclobutadiene’s

geometry adopted in this work. Consequently, the i-FCIQMC propagations for the X 1B1g

and A 3A2g states were set up to converge the lowest states of the 1Ag (D2h) and 3B1g (D2h)

symmetries. Consistent with the CC(P ) and CC(P ;Q) runs that follow the i-FCIQMC steps

and the accompanying CCSD, CR-CC(2,3), and CCSDT computations, the reference de-

terminants used to initiate our i-FCIQMC propagations were the closed-shell, D2h-adapted,

RHF function obtained by placing two electrons on one of the eg valence orbitals for the

lowest-energy 1Ag (D2h) state and the high-spin ROHF determinant, adapted to D2h as well,

for the lowest 3B1g (D2h) state. As a result, the lists of triply excited determinants extracted

from the i-FCIQMC runs at the various propagation times τ > 0, needed to define the P

spaces for the CC(P ) and CC(P ;Q) computations, consisted of the Sz = 0 triples of the Ag

(D2h) symmetry for the X 1B1g state and the Sz = 1 triples of the B1g (D2h) symmetry in

the case of the A 3A2g state. Given our interest in converging the CCSDT energetics, the

Q spaces used to construct the δ(P ; Q) corrections consisted of the remaining triply excited

determinants, absent in the i-FCIQMC wave functions of the X 1B1g and A 3A2g states at a

given τ .

The results shown in Table 3.11 and Fig. 3.11 display several similarities with the previ-

ously discussed methylene and (HFH)− cases. One cannot, for example, obtain an accurate

description of the more MR singlet ground state and the energy separation between the

X 1B1g and A 3A2g states without incorporating the leading triply excited determinants in

the P space. Indeed, when the P space consists of only singly and doubly excited determi-

nants, as in the τ = 0 CC(P ) (i.e., CCSD) and CC(P ;Q) [i.e., CR-CC(2,3)] calculations,

one ends up with the enormous errors in the energies of the X 1B1g state relative to their

88

CCSDT parent, which are 47.979 millihartree in the former case and 14.636 millihartree

when the latter computation is considered. The τ = 0 CC(P ;Q) energy of the A 3A2g state

is a lot more accurate, reducing the large, 23.884 millihartree, error relative to CCSDT ob-

tained in the underlying CC(P ) calculation to −60 microhartree, but this does not help

too much. The corresponding CR-CC(2,3) triples correction to CCSD, which neglects the

coupling of the low-order T1 and T2 clusters with their higher-order T3 counterpart, is inca-

pable of offering a balanced description of the X 1B1g and A 3A2g states, so that the resulting

singlet–triplet gap is very poor. The 9.2 kcal/mol difference between the ∆ES-T values ob-

tained in the τ = 0 CC(P ;Q) or CR-CC(2,3) and CCSDT calculations is so large that the

X 1B1g − A 3A2g separation predicted by CR-CC(2,3) has a wrong sign compared to its −4.8

kcal/mol CCSDT counterpart, while being nearly identical in magnitude. This difference

becomes even larger when the uncorrected τ = 0 CC(P ), meaning CCSD, calculations are

considered (15.1 kcal/mol).

As shown in Table 3.11 and Fig. 3.11, the situation dramatically changes when the

P spaces used in the CC(P ) and CC(P ;Q) calculations are enriched with the subsets of

triply excited determinants captured by the i-FCIQMC propagations. The convergence of

the CC(P ;Q) energies of the X 1B1g and A 3A2g states, especially the former ones, and the

vertical separations between them is particularly impressive. For example, after as few as

6000 δτ = 0.0001 a.u. MC time steps and i-FCIQMC capturing less than 30% of all triples

in the P space, where, as demonstrated in Table 3.12, the walker population characterizing

the i-FCIQMC run for the X 1B1g state is only 0.02% of the total number of walkers at

τ = 8.0 a.u. (the termination time for our i-FCIQMC propagations for cyclobutadiene), the

CC(P ;Q) approach reduces the 14.636 millihartree difference between the CR-CC(2,3) and

CCSDT energies of the strongly correlated singlet ground state to 2.223 millihartree. While

the CR-CC(2,3) description of the A 3A2g state, which has a largely SR character, is already

excellent, the CC(P ;Q) calculation performed after 6000 MC iterations, which uses only 26%

of triples in the P space and a tiny walker population that amounts to 0.04% of all walkers at

89

τ = 8.0 a.u. in the underlying i-FCIQMC propagation, improves it too, reducing the small,

60 microhartree, unsigned difference between the CR-CC(2,3) and CCSDT energies to an

even smaller 51 microhartree. As a consequence of the above improvements, especially for

the X 1B1g state, the error relative to CCSDT characterizing the ∆ES-T value obtained in the

FCIQMC-driven CC(P ;Q) calculations after 6000 δτ = 0.0001 a.u. MC time steps, where

the underlying i-FCIQMC propagations are still in their early stages, is only 1.4 kcal/mol, as

opposed to 9.2 kcal/mol obtained at τ = 0 with CR-CC(2,3). The resulting X 1B1g − A 3A2g

energy separation, of −3.4 kcal/mol, has not only the correct sign, but is also very close

to the −4.8 kcal/mol value obtained with CCSDT. If we wait a little longer, by executing

the extra 2000 MC iterations, so that the i-FCIQMC propagations can capture 34%–39%

of all triply excited determinants, we can reduce the already small 2.223 millihartree, 51

microhartree, and 1.4 kcal/mol errors in the CC(P ;Q) energies of the X 1B1g and A 3A2g

states and separation between them relative to CCSDT, obtained after 6000 MC time steps,

to 0.835 millihartree, 31 microhartree, and 0.5 kcal/mol, respectively. It is clear from Table

3.11 and Fig. 3.11 that the convergence of the semi-stochastic CC(P ;Q) results for the lowest-

energy singlet and triplet states of cyclobutadiene, especially the X 1B1g energies and the

X 1B1g − A 3A2g gap values, which the τ = 0 CC(P ;Q) or CR-CC(2,3) calculations describe

poorly, toward CCSDT is very fast, even when the underlying i-FCIQMC propagations

are far from convergence.

It is also apparent from our calculations that the noniterative

corrections δ(P ; Q) play a significant role in accelerating convergence of the corresponding

CC(P ) energetics toward CCSDT. As shown, for example, in Table 3.11, the relatively large

differences between the uncorrected CC(P ) energies of the X 1B1g and A 3A2g states and

vertical gap between them obtained at τ = 0.8 a.u., i.e., after 8000 δτ = 0.0001 a.u. MC

iterations, and the corresponding CCSDT data, which exceed 11 and 7 millihartree and 3

kcal/mol, respectively, are reduced to 0.835 millihartree, 31 microhartree, and 0.5 kcal/mol,

when the CC(P ;Q) approach is employed. We can see similar improvements in the CC(P )

energies at other τ values.

90

Table 3.11 Convergence of the CC(P ) and CC(P ;Q) energies of the X 1B1g and A 3A2g states
of cyclobutadiene, as described by the cc-pVDZ basis set, and of the corresponding vertical
singlet–triplet gaps toward their parent CCSDT values. All calculations were performed
at the D4h-symmetric transition-state geometry of the X 1B1g state optimized in the MR-
AQCC calculations reported in Ref. [202]. The P spaces used in the CC(P ) and CC(P ;Q)
calculations were defined as all singly and doubly excited determinants and subsets of triply
excited determinants extracted from the i-FCIQMC propagations with δτ = 0.0001 a.u. The
Q spaces used to determine the CC(P ;Q) corrections consisted of the triply excited deter-
minants not captured by the corresponding i-FCIQMC runs. The i-FCIQMC calculations
preceding the CC(P ) and CC(P ;Q) steps were initiated by placing 1500 walkers on the RHF
(X 1B1g state) and ROHF (A 3A2g state) reference determinants and the na parameter of
the initiator algorithm was set at 3. In all post-Hartree–Fock calculations, the four lowest
core orbitals were kept frozen and the spherical components of d orbitals were employed
throughout. Adapted from Ref. [102].

MC Iterations
0
2000
4000
6000
8000
10000
20000
50000
80000
∞

P a
47.979d
40.663
27.235
17.188
11.207
8.299
2.030
0.049
0.001

X 1B1g
(P ; Q)a %Tb
14.636e
11.059
5.921
2.223
0.835
0.429
0.013
0.000
0.000

0
3.5
16.6
29.5
39.2
46.6
70.0
96.9
99.9

P a
23.884d
21.004
14.317
10.016
7.463
5.865
2.461
0.166
0.009

A 3A2g
(P ; Q)a %Tb
-0.060e
0.031
0.068
0.051
0.031
0.020
0.005
0.000
0.000

0
3.0
14.2
25.5
34.3
41.0
62.8
94.2
99.6

−154.232002f

−154.224380f

X 1B1g − A 3A2g
(P ; Q)c
P c
9.2e
15.1d
6.9
12.3
3.7
8.1
1.4
4.6
0.5
3.3
0.3
1.5
0.0
-0.3
0.0
-0.1
0.0
0.0

−4.8g

a Unless otherwise stated, all energies are reported as errors relative to CCSDT in millihartree.
b The %T values are the percentages of triples captured during the i-FCIQMC propagations [the Sz = 0
triply excited determinants of the Ag(D2h) symmetry in the case of the X 1B1g state and the Sz = 1 triply
excited determinants of the B1g (D2h) symmetry in the case of the A 3A2g state].
c Unless otherwise specified, the values of the singlet–triplet gaps are reported as errors relative to CCSDT
in kcal/mol.
d Equivalent to CCSD.
e Equivalent to CR-CC(2,3) [the most complete variant of CR-CC(2,3) abbreviated sometimes as CR-
CC(2,3),D or CR-CC(2,3)D].
f Total CCSDT energy in hartree.
g The CCSDT singlet–triplet gap in kcal/mol.

91

Table 3.12 The total numbers of walkers, reported as percentages of the total walker popu-
lations at 80000 MC iterations, characterizing the i-FCIQMC propagations with δτ = 0.0001
a.u. that were needed to generate the CC(P ) and CC(P ;Q) results for cyclobutadiene re-
ported in Table 3.11. Adapted from Ref. [102].

MC Iterations
0
2000
4000
6000
8000
10000
20000
50000
80000

X 1B1g
0.00a
0.00
0.01
0.02
0.03
0.05
0.16
3.81
100b

A 3A2g
0.00a
0.00
0.02
0.04
0.07
0.09
0.28
4.93
100c

a The initial walker population, meaning 1500 walkers on the RHF (X 1B1g state) and ROHF (A 3A2g state)
reference determinants.
b The total number of walkers at 80000 MC iterations is 8457504823.
c The total number of walkers at 80000 MC iterations is 4067481034.

Figure 3.10 π molecular orbital network of the cyclobutadiene molecule, obtained at the
HF/cc-pVDZ level, at the D4h-symmetric transition-state geometry of the X 1B1g state opti-
mized in the MR-AQCC calculations in Ref. [202]. The orbital irreducible representations in
the D4h symmetry are shown in black and the corresponding labels in the C2v symmetry are
shown in the parenthesis in orange. This electronic configuration refers to the triplet state.

92

Figure 3.11 Convergence of the CC(P ) and CC(P ;Q) energies of the X 1B1g [panel (a)] and
A 3A2g [panel (b)] states of cyclobutadiene, as described by the cc-pVDZ basis set, and of the
corresponding vertical singlet–triplet gaps [panel (c)] toward their parent CCSDT values. All
calculations were performed at the D4h-symmetric transition-state geometry of the X 1B1g
state optimized in the MR-AQCC calculations in Ref. [202]. The P spaces consisted of
all singles and doubles and subsets of triples identified during the i-FCIQMC propagations
with δτ = 0.0001 a.u. and the Q spaces consisted of the triples not captured by i-FCIQMC.
Adapted from Ref. [102].

Most of the observations regarding the performance of the semi-stochastic CC(P ;Q)

methodology and its CC(P ) counterpart remain valid when the larger cyclopentadienyl

cation, which is also the largest molecular system considered in our CC(P )/CC(P ;Q) work to

date, is examined. Following the previous DEA-EOMCC studies of cyclopentadienyl cation

from the Piecuch group [154, 155], where the effect of high-order 4p-2h correlations on the

singlet–triplet gap was investigated, we used the D5h-symmetric geometry corresponding to

a minimum on the lowest triplet surface obtained in the UCCSD/cc-pVDZ optimization in

Ref. [203]. At this geometry, cyclopentadienyl cation is characterized by the delocalization

of four π electrons over five π MOs, resulting in the doubly occupied nondegenerate a′′

2 or-

bital, the doubly degenerate e′′

1 shell, in which each component MO is occupied by a single

electron, and the doubly degenerate e′′

2 shell, which in the zeroth-order description of the

lowest-energy singlet and triplet states remains empty. In analogy to the previously discussed

cyclobutadiene system, the two electrons in the degenerate e′′

1 MOs can couple to a singlet

93

02040608001020304050(a)X 1B1gCC(P)CC(P;Q)020406080MC Iterations (103)01020304050Error re . to CCSDT (mEh)(b)A 3A2gCC(P)CC(P;Q)020406080MC Iterations (103)03691215Error re . to CCSDT (kca /mo )(c)ΔES–TCC(P)CC(P;Q)or triplet, but compared to cyclobutadiene, where the lowest-energy singlet state is also a

ground state, the state ordering in cyclopentadienyl cation is reversed, so that the lowest

triplet, designated as X 3A′

2, is the ground state and the lowest-energy singlet, denoted as

A 1E′

2, is the first excited state (see Fig. 3.12 for an illustration of the π MOs). Similar to

all other examples considered in this work, in order to obtain a well-balanced description

of the X 3A′

2 state, which has a SR character dominated by dynamical correlations, and its

A 1E′

2 companion, which is an open-shell singlet characterized by significant MR correlations,

and obtain an accurate value of ∆ES-T within a conventional SRCC framework, one must

turn to higher-level theories that can offer a robust treatment of Tn clusters with n > 2.

Otherwise, as shown in Ref. [203], and as confirmed in our calculations, the results can be

very poor. For example, the A 1E′

2 − X 3A′

2 separation in cyclopentadienyl cation result-

ing from the restricted CCSD calculations using the cc-pVDZ basis, which are equivalent

to our τ = 0 CC(P ) computations, is about 23 kcal/mol. This is in large disagreement

with the most accurate ab initio calculations of the singlet–triplet gap in cyclopentadienyl

cation performed to date using the DEA-EOMCC formalism including 3p-1h as well as 4p-2h

correlations on top of the CCSD treatment of the underlying closed-shell core, which give

about 14 kcal/mol when the cc-pVDZ basis set is employed [154, 155] (for the examples of

other high-level SRCC and MRCC calculations of the singlet–triplet gap in cyclopentadienyl

cation, see Ref. [203]; Ref. [155] also provides the well-converged MR perturbation theory

data, which agree with the state-of-the-art DEA-EOMCC computations reported in Refs.

[154, 155]). The restricted CCSDT approach, which is the target SRCC method in this study,

provides a much better description, reducing the approximately 9 kcal/mol error relative to

the most accurate DEA-EOMCC calculations with up to 4p-2h excitations reported in Refs.

[154, 155] obtained with restricted CCSD to less than 3 kcal/mol, when the cc-pVDZ basis

set is employed.

It would certainly be interesting to examine if the inclusion of higher–

than–triply excited clusters, such as T4, could further improve the CCSDT description of the

singlet–triplet gap in cyclopentadienyl cation, but in this work we focus on the ability of the

94

semi-stochastic, CIQMC-based, CC(P ;Q) methodology to improve the CR-CC(2,3) ∆ES-T

values and converge the results of CCSDT computations. We hope to return to the topic of

the role of T4 clusters in describing the singlet–triplet gap in cyclopentadienyl cation in one of

our future studies. It may be worth pointing out that the A 1E′

2 − X 3A′

2 gap obtained in the

restricted CCSDT/cc-pVDZ calculations, which give ∆ES-T = 16.7 kcal/mol, is in very good

agreement with the 16.1 kcal/mol resulting from the DEA-EOMCC/cc-pVDZ computations

truncated at 3p-1h excitations [154, 155].

The results of our CIQMC-driven CC(P ) and CC(P ;Q) computations for cyclopentadi-

enyl cation are reported in Table 3.13 and Fig. 3.13. As already alluded to above, to reduce

the computational costs of the CIQMC propagations preceding the CC(P ) and CC(P ;Q)

steps, especially in the later stages of the CIQMC runs that are included in Table 3.13 and

Fig. 3.13 for the completeness of our presentation, we replaced the i-FCIQMC algorithm,

which we exploited in our calculations for methylene, (HFH)−, and cyclobutadiene, by its

truncated i-CISDTQ-MC counterpart. It has been established in Ref. [101] that the replace-

ment of i-FCIQMC by i-CISDTQ-MC, when identifying the leading higher–than–doubly

excited determinants for the inclusion in the P spaces used in the semi-stochastic CC(P )

and CC(P ;Q) runs, has virtually no effect on the rate at which these runs converge the parent

SRCC energetics. In analogy to cyclobutadiene, all of our i-CISDTQ-MC, semi-stochastic

CC(P ) and CC(P ;Q), and deterministic CCSD, CR-CC(2,3), and CCSDT computations

utilized the largest Abelian subgroup of the D5h point group characterizing the cyclopenta-

dienyl cation’s structure examined in the present study, which is C2v. This means that in

setting up our calculations for the X 3A′

2 state, we treated it as the lowest state of the 3B2

(C2v) symmetry, whereas the doubly degenerate A 1E′

2 state was represented by its 1A1 (C2v)

component. Similar to cyclobutadiene, and to remain consistent with the CC(P ), CC(P ;Q),

and other SRCC runs for cyclopentadienyl cation carried out in this study, the reference

determinant used to initiate the i-CISDTQ-MC propagation for the lowest-energy 3B2 (C2v)

state was the triplet ROHF determinant. In the case of the 1A1 (C2v) component of the

95

A 1E′

2 state, we used the RHF determinant obtained by pairing the two valence electrons

in one of the e′′

1 MOs to initiate the corresponding i-CISDTQ-MC run. Consistent with

the above description, the subsets of triply excited determinants used to construct the P

spaces for the semi-stochastic CC(P ) and CC(P ;Q) computations for the X 3A′

2 state were

the Sz = 1 triples of the B2 (C2v) symmetry captured by i-CISDTQ-MC. In the case of

the A 1E′

2 state, represented, as explained above, by its 1A1 (C2v) component, we used the

Sz = 0 triples of the A1 (C2v) symmetry identified by the i-CISDTQ-MC propagation set up

to converge the lowest A1 (C2v) state. As usual, the corresponding Q spaces were spanned

by the remaining triply excited determinants that were not captured by the i-CISDTQ-MC

runs when the lists of P -space triples were created.

Our calculations for cyclopentadienyl cation, summarized in Table 3.13 and Fig. 3.13,

demonstrate that the CC(P ;Q) energies of the X 3A′

2 and A 1E′

2 states and vertical gaps

between them display fast convergence toward the respective CCSDT values with the propa-

gation time τ . This is particularly apparent in the case of the CC(P ;Q) energies of the more

MR A 1E′

2 state and the A 1E′

2 − X 3A′

2 separation, which cannot be accurately described if

the underlying P spaces contain only singly and doubly excited determinants. Indeed, the

CR-CC(2,3) energy of the A 1E′

2 state, which is equivalent to the τ = 0 CC(P ;Q) value, is

much more accurate than the result of the associated CC(P ) or CCSD calculation, which

produces the enormous error relative to CCSDT exceeding 38 millihartree, but the substan-

tial, > 6 millihartree, difference with the CCSDT energy remains. The situation for the SR

X 3A′

2 state, where the CR-CC(2,3) approach reduces the nearly 29 millihartree error relative

to CCSDT obtained in the CCSD calculations to ∼0.2 millihartree, is a lot better, but this

does not help the resulting ∆ES-T value, which differs from its CCSDT counterpart by almost

4 kcal/mol (almost a quarter of the CCSDT value of ∆ES-T). The discrepancy between the

errors in the CR-CC(2,3) energies of the X 3A′

2 and A 1E′

2 states is simply too large. Clearly,

one needs to incorporate some triples in the corresponding P spaces, especially in the case

of the more challenging A 1E′

2 state.

96

Once the τ = 0 P spaces are augmented with the leading triply excited determinants

identified by the i-CISDTQ-MC propagations and the noniterative corrections δ(P ; Q) are

added to the CC(P ) energies to estimate the effects of the remaining T3 correlations, we

observe smooth convergence of the resulting CC(P ;Q) energetics toward their respective

CCSDT limits. This includes significant improvements in the poor description of the A 1E′
2

state and the A 1E′

2 − X 3A′

2 separation by CR-CC(2,3). As shown in Table 3.13, already

after 10000 δτ = 0.0001 a.u. MC time steps, where the i-CISDTQ-MC propagations are still

in their infancy, capturing only 25–30% of all triples and using tiny walker populations, on

the order of 0.1–0.2% of the total numbers of walkers at τ = 8.0 a.u. (see Table 3.14), the

6.245 millihartree and 3.8 kcal/mol errors in the energy of the A 1E′

2 state and the ∆ES-T

value relative to CCSDT obtained with CR-CC(2,3) reduce in the CC(P ;Q) calculations

to 2.248 millihartree and 1.3 kcal/mol, respectively. By running i-CISDTQ-MC a little

longer and capturing about 50–60% of all triples in the relevant P spaces, as is the case

after 20000 MC iterations, where the walker populations compared to τ = 8.0 a.u. are still

tiny, the errors in the CC(P ;Q) values of the A 1E′

2 energy and ∆ES-T relative to their

CCSDT parents drop down by an order of magnitude compared to 10000 MC iterations, to

0.217 millihartree and 0.1 kcal/mol, respectively. Although the excellent description of the

predominantly SR X 3A′

2 state by the CR-CC(2,3) approach hardly needs any improvement,

the i-CISDTQ-MC-driven CC(P ;Q) calculations are helping here too, reducing the 0.245

millihartree difference between the CR-CC(2,3) and CCSDT energies to 0.108 millihartree

after 10000 MC iterations (26 microhartree when the number of MC iterations is increased

to 20000). As anticipated, the uncorrected CC(P ) energies of the X 3A′

2 and A 1E′

2 states

converge to the respective CCSDT limits too, but they do it at a much slower pace than their

CC(P ;Q) counterparts. A comparison of the results of the CC(P ) and CC(P ;Q) calculations

for the A 1E′

2 − X 3A′

2 gap shown in Table 3.13 and Fig. 3.13 (c) may create an impression

as if the noniterative corrections δ(P ; Q) offer very little, but this would be misleading. The

relatively fast convergence of the CC(P ) values of ∆ES-T toward their CCSDT parent in

97

the early stages of the underlying i-CISDTQ-MC propagations, which compares well with

that observed in the corresponding CC(P ;Q) computations, is a result of the fortuitous

cancellation of large errors characterizing the CC(P ) energies of the X 3A′

2 and A 1E′

2 states.

Since no other system examined in this study displays similar error cancellations, and since

costs of computing corrections δ(P ; Q), which offer major error reductions in the individual

CC(P ) energies, while accelerating their convergence toward the SRCC target, are low, we

recommend using the δ(P ; Q)-corrected CC(P ;Q) energetics.

98

2 and A 1E′

Table 3.13 Convergence of the CC(P ) and CC(P ;Q) energies of the X 3A′
2 states
of cyclopentadienyl cation, as described by the cc-pVDZ basis set, and of the correspond-
ing vertical singlet–triplet gaps toward their parent CCSDT values. All calculations were
performed at the D5h-symmetric geometry of the X 3A′
2 state optimized using the unre-
stricted CCSD/cc-pVDZ approach reported in Ref. [203]. The P spaces used in the CC(P )
and CC(P ;Q) calculations were defined as all singly and doubly excited determinants and
subsets of triply excited determinants extracted from the i-CISDTQ-MC propagations with
δτ = 0.0001 a.u. The Q spaces used to determine the CC(P ;Q) corrections consisted of the
triply excited determinants not captured by the corresponding i-CISDTQ-MC runs. The
i-CISDTQ-MC calculations preceding the CC(P ) and CC(P ;Q) steps were initiated by plac-
ing 1500 walkers on the ROHF (X 3A′
2 state) reference determinants
and the na parameter of the initiator algorithm was set at 3. In all post-Hartree–Fock cal-
culations, the five lowest core orbitals were kept frozen and the spherical components of d
orbitals were employed throughout. Adapted from Ref. [102].

2 state) and RHF (A 1E′

MC Iterations
0
2000
4000
6000
8000
10000
20000
50000
80000
∞

P a
28.840d
27.396
22.253
17.394
13.743
11.027
4.250
0.155
0.007

X 3A′
2
(P ; Q)a %Tb
0.245e
0.272
0.267
0.212
0.152
0.108
0.026
0.001
0.000

0
0.8
5.1
11.6
18.3
24.8
52.1
95.3
99.8

−192.615924f

P a
38.572d
35.598
27.946
21.124
16.042
12.947
3.964
0.060
0.001

A 1E′
2
(P ; Q)a %Tb
6.245e
5.948
5.078
3.971
2.756
2.248
0.217
0.001
0.000
−192.589235f

0
1.0
6.5
14.7
23.0
30.9
61.4
98.3
100.0

A 1E′
P c
6.1d
5.1
3.6
2.3
1.4
1.2
-0.2
-0.1
0.0

2 − X 3A′
2
(P ; Q)c
3.8e
3.6
3.0
2.4
1.6
1.3
0.1
0.0
0.0

16.7g

a Unless otherwise stated, all energies are reported as errors relative to CCSDT in millihartree.
b The %T values are the percentages of triples captured during the i-CISDTQ-MC propagations [the Sz = 1
triply excited determinants of the B2 (C2v) symmetry in the case of the X 3A′
2 state and the Sz = 0 triply
excited determinants of the A1 (C2v) symmetry in the case of the A 1E′
2 state].
c Unless otherwise specified, the values of the singlet–triplet gaps are reported as errors relative to CCSDT
in kcal/mol.
d Equivalent to CCSD.
e Equivalent to CR-CC(2,3) [the most complete variant of CR-CC(2,3) abbreviated sometimes as CR-
CC(2,3),D or CR-CC(2,3)D].
f Total CCSDT energy in hartree.
g The CCSDT singlet–triplet gap in kcal/mol.

99

Table 3.14 The total numbers of walkers, reported as percentages of the total walker
populations at 80000 MC iterations, characterizing the i-CISDTQ-MC propagations with
δτ = 0.0001 a.u. that were needed to generate the CC(P ) and CC(P ;Q) results for the
cyclopentadienyl cation reported in Table 3.13. Adapted from Ref. [102].

MC Iterations
0
2000
4000
6000
8000
10000
20000
50000
80000

X 3A′
2
0.00a
0.01
0.03
0.06
0.11
0.16
0.62
13.20
100b

A 1E′
2
0.00a
0.00
0.02
0.05
0.09
0.13
0.54
15.73
100c

a The initial walker population, meaning 1500 walkers on the ROHF (X 3A′
reference determinants.
b The total number of walkers at 80000 MC iterations is 7867091953.
c The total number of walkers at 80000 MC iterations is 11371381724.

2 state) and RHF (A 1E′

2 state)

Figure 3.12 π molecular orbital network of the cyclopentadienyl cation molecule, obtained at
the HF/cc-pVDZ level, at the D5h-symmetric geometry of the X 3A′
2 state optimized using the
unrestricted CCSD/cc-pVDZ approach in Ref. [203]. The orbital irreducible representations
in the D5h symmetry are shown in black and the corresponding labels in the C2v symmetry
are shown in the parenthesis in orange. This electronic configuration refers to the triplet
state.

100

Figure 3.13 Convergence of the CC(P ) and CC(P ;Q) energies of the X 3A′
2 [panel (a)] and
A 1E′
2 [panel (b)] states of cyclopentadienyl cation, as described by the cc-pVDZ basis set,
and of the corresponding vertical singlet–triplet gaps [panel (c)] toward their parent CCSDT
values. All calculations were performed at the D5h-symmetric geometry of the X 3A′
2 state
optimized using the unrestricted CCSD/cc-pVDZ approach in Ref. [203]. The P spaces
consisted of all singles and doubles and subsets of triples identified during the i-CISDTQ-
MC propagations with δτ = 0.0001 a.u. and the Q spaces consisted of the triples not captured
by i-CISDTQ-MC. Adapted from Ref. [102].

3.3.4 Trimethylenemethane

Our final example is trimethylenemethane, a fascinating non-Kekul´e hydrocarbon exam-

ined as early as in 1948 [209] and 1950 [210], in which four valence π electrons are delocalized

over four closely spaced π-type orbitals. Assuming the D3h symmetry, which is the symmetry

of the minimum-energy structure on the ground-state triplet surface of trimethylenemethane,

the four MOs of this system’s valence π network consist of the nondegenerate 1a′′

2 orbital,

the doubly degenerate 1e′′ shell, and the nondegenerate 2a′′

2 orbital. If one adopts the C2v

symmetry, relevant to the low-lying singlet states, which is also the largest Abelian sub-

group of D3h exploited in our CCSD, CR-CC(2,3), CCSDT, and CIQMC-driven CC(P ) and

CC(P ;Q) computations, the nondegenerate 1a′′

2 and 2a′′

2 orbitals in a D3h description become

the 1b1 and 3b1 MOs, respectively, whereas the degenerate 1e′′ shell splits into the 1a2 and

2b1 components (see Fig. 3.14 for an illustration of the π MOs).

The first experimental identification of trimethylenemethane dates back to 1966 [211],

101

0204060800122436(a)X 3A′2CC(P)CC(P;Q)020406080MC Iteratio s (103)0122436Error rel. to CCSDT (mEh)(b)A 1E′2CC(P)CC(P;Q)020406080MC Iteratio s (103)0.01.53.04.56.0Error rel. to CCSDT (kcal/mol)(c)ΔES–TCC(P)CC(P;Q)a definitive experimental verification, using electron paramagnetic resonance, of its triplet

ground state was accomplished already in 1976 [212], and the electronic structure of trimethylen-

emethane has been well understood for decades (cf., e.g., Ref. [213] and references therein),

but an accurate characterization of its triplet ground state and low-lying singlet states and

energy separations between them continues to present a significant challenge to quantum

chemistry approaches [80, 88, 153, 154, 214–239]. The D3h-symmetric triplet ground state,

designated as X 3A′

2 (in a C2v description adopted in this study, X 3B2), which is dominated

by the |{core}(1a′′

2)2(1e′′

1)1(1e′′

2)1| configuration (in C2v, |{core}(1b1)2(1a2)1(2b1)1|), is rela-

tively easy to describe, but the next two states, which are the nearly degenerate singlets sta-

bilized by the Jahn–Teller distortion that lifts their exact degeneracy in a D3h description, are

not. The lower of the two singlets, which is characterized by a Cs-symmetric minimum that

can be approximated by a twisted C2v structure and which is, therefore, usually designated

as the A 1B1 state, is an open-shell singlet that emerges from the |{core}(1b1)2(1a2)1(2b1)1|

configuration. The second singlet, labeled as the B 1A1 state, is a C2v-symmetric multi-

configurational state dominated by the |{core}(1b1)2(1a2)2| and |{core}(1b1)2(2b1)2| closed-

shell determinants. The A 1B1 state, although lower in energy compared to its B 1A1 coun-

terpart, has not been observed experimentally due to unfavorable Franck–Condon factors

[229, 240], so we do not consider it in this work. However, the second singlet, B 1A1, has

been detected in photoelectron spectroscopy experiments reported in Refs. [240, 241], which

located it at 16.1 ± 0.1 kcal/mol above the X 3A′

2 ground state. Thus, following our pre-

vious deterministic, active-orbital-based, CC(P ;Q) work [88] and the state-of-the-art DEA-

and DIP-EOMCC computations with up to 4p-2h and 4h-2p excitations reported in Refs.

[80, 153, 154], in carrying out the CIQMC-driven CC(P ) and CC(P ;Q) calculations dis-

cussed in this subsection and executing the accompanying CCSD, CR-CC(2,3), and CCSDT

runs, we focused on the D3h-symmetric triplet ground state, X 3A′

2, the C2v-symmetric B 1A1

singlet, and the adiabatic gap between them, adopting the geometries of the two states op-

timized using the spin-flip density functional theory (SF-DFT) and the 6-31G(d) basis in

102

Ref. [231]. In analogy to other organic biradicals discussed in this chapter, we employed the

cc-pVDZ basis set, so that the parent CCSDT computations, needed to judge the perfor-

mance of our semi-stochastic CC(P ) and CC(P ;Q) methods, and the more expensive CC(P )

and CC(P ;Q) calculations employing large, near-100%, fractions of triples in the relevant

P spaces (captured in the later stages of the underlying CIQMC propagations) were not

too difficult to execute on the computers available to us. As shown in our earlier deter-

ministic CC(P ;Q) work [88], in which we tested the active-orbital-based CC(t;3) method,

which recovers the CCSDT energetics to within small fractions of kilocalorie per mole, and as

confirmed by the authors of Ref. [237], who managed to perform the CCSDT/cc-pVTZ calcu-

lations, the use of a larger cc-pVTZ basis changes the adiabatic B 1A1 − X 3A′

2 gap by about

0.5–1 kcal/mol, i.e., the use the cc-pVDZ basis is sufficient to draw meaningful conclusions

regarding the performance of the semi-stochastic CC(P ) and CC(P ;Q) approaches.

While the main goal of this study is to examine the efficiency of the CIQMC-driven

CC(P ;Q) approaches in converging the CCSDT energetics, it is worth pointing out that the

parent CCSDT calculations using the ROHF reference determinant for the X 3A′

2 state and

the RHF reference for the more strongly correlated B 1A1 state, in spite of their SR character,
are capable of producing a reasonably accurate description of the adiabatic B 1A1 − X 3A′
separation in trimethylenemethane. Indeed, the purely electronic B 1A1 − X 3A′

2 gap, desig-

2

nated, in analogy to other singlet–triplet gaps considered in this work, as ∆ES-T, resulting

from the ROHF/RHF-based CCSDT/cc-pVDZ computations using the SF-DFT/6-31G(d)

geometries of the X 3A′

2 and B 1A1 states optimized in Ref. [231] is 21.7 kcal/mol [88] (cf.

Table 3.15). The corresponding experimentally derived result, obtained by subtracting the

zero-point vibrational energy correction ∆ZPVE resulting from the SF-DFT/6-31G(d) cal-

culations reported in Ref. [231] from the experimental B 1A1 − X 3A′

2 gap determined in

Refs. [240, 241], is 18.1 kcal/mol. The CCSDT/cc-pVDZ value of ∆ES-T is not as accurate
as the electronic B 1A1 − X 3A′

2 gaps generated in the high-level DEA- and DIP-EOMCC

calculations with the explicit inclusion of 4p-2h and 4h-2p correlations on top of CCSD,

103

which produce 18–19 kcal/mol [80, 153, 154], but it is certainly much better than 46.1, 24.4,

and 29.8 kcal/mol obtained with the ROHF/RHF-based CCSD, CCSD(T), and CR-CC(2,3)

methods, respectively, when the cc-pVDZ basis set is employed [88] [as demonstrated in

Ref. [88], the use of a larger cc-pVTZ basis makes the CCSD, CCSD(T), and CR-CC(2,3)

results even worse; the CCSD/cc-pVDZ and CR-CC(2,3)/cc-pVDZ values of ∆ES-T are in-

cluded in Table 3.15 as the τ = 0 CC(P ) and CC(P ;Q) data, respectively]. While much of

the 3.6 kcal/mol difference between the electronic B 1A1 − X 3A′

2 separation obtained in the

ROHF/RHF-based CCSDT/cc-pVDZ calculations and its experimentally derived estimate

of 18.1 kcal/mol determined in Ref. [231] is, most likely, a consequence of the neglect of

T4 clusters in the CCSDT approach, we should keep in mind that the latter estimate de-

pends on the source of the information about the ∆ZPVE correction. For example, if one

replaces the ∆ZPVE value obtained in the SF-DFT/6-31G(d) calculations reported in Ref.

[231] by its CCSD(T)/6-311++G(2d,2p) estimate and accounts for the core polarization

effects determined with the help of the CCSD(T)/cc-pCVQZ computations, combining the

resulting information with the experimental B 1A1 − X 3A′

2 separation determined in Refs.

[240, 241], the purely electronic, experimentally derived, adiabatic ∆ES-T gap increases to

19.4 kcal/mol [237], which differs from our CCSDT/cc-pVDZ result by 2.3 kcal/mol. On the

other hand, as shown in Ref. [237], the CCSDT value of the adiabatic B 1A1 − X 3A′

2 gap

increases with the basis set too, to 23.1 kcal/mol when the cc-pVTZ basis is employed, which

reinforces our view that without accounting for T4 correlations one cannot bring the results

of conventional SRCC computations to a close agreement with the experimentally derived

data. While the examination of the role of T4 clusters, basis set, geometries of the X 3A′

2

and B 1A1 states employed in the calculations, ∆ZPVE corrections, etc., would certainly be

interesting, it would also be outside the scope of the present study. Thus, in the remainder of

this subsection, we return to the analysis of the performance of the CIQMC-driven CC(P ;Q)

approach and its CC(P ) counterpart, especially their ability to converge the parent CCSDT

energetics when the cc-pVDZ basis is employed.

104

The results of our semi-stochastic CC(P )/cc-pVDZ and CC(P ;Q)/cc-pVDZ computa-

tions for the X 3A′

2 and B 1A1 states of trimethylenemethane and the adiabatic gap be-

tween them, along with the associated CCSD, CR-CC(2,3), and CCSDT data, are sum-

marized in Table 3.15 and Fig. 3.15. As in the case of cyclopentadienyl cation, to reduce

the computational costs of the underlying CIQMC propagations, especially in their later

stages, we resorted to the truncated i-CISDTQ-MC approach. In analogy to cyclobutadiene

and cyclopentadienyl cation, we terminated our i-CISDTQ-MC propagations after 80000

δτ = 0.0001 a.u. MC time steps, where the differences between the CC(P ;Q) and CCSDT

energies of the X 3A′

2 and B 1A1 states fall below 1 microhartree. Consistent with the CC(P ),

CC(P ;Q), and other SRCC calculations for trimethylenemethane reported in Table 3.15 and

Fig. 3.15, we used the ROHF determinant to initiate the i-CISDTQ-MC propagation for

the D3h-symmetric X 3A′

2 (in C2v, X 3B2) state and the RHF determinant to initiate the i-

CISDTQ-MC run for the C2v-symmetric B 1A1 state. The lists of triply excited determinants

captured by the i-CISDTQ-MC runs at the various times τ > 0, needed to construct the P

spaces for the CC(P ) and CC(P ;Q) computations, were the Sz = 1 triples of the B2 (C2v)

symmetry in the case of the X 3A′

2 state and the Sz = 0 triples of the A1 (C2v) symmetry

when considering the B 1A1 state. The remaining triples not captured by i-CISDTQ-MC

defined the corresponding Q spaces.

It is clear from the results presented in Table 3.15 and Fig. 3.15 that the semi-stochastic

CC(P ;Q) approach is very effective in converging the parent CCSDT energetics character-

izing the X 3A′

2 and B 1A1 states of trimethylenemethane and the adiabatic gap between

them. It offers substantial improvements in the results of the CR-CC(2,3) calculations in

the early stages of the underlying i-CISDTQ-MC propagations, especially when the multi-

configurational B 1A1 state and the adiabatic B 1A1 − X 3A′

2 separation ∆ES-T, which are

poorly described by CR-CC(2,3), are examined, while greatly accelerating the convergence

of the CC(P ) energies toward CCSDT. Indeed, after 6000 δτ = 0.0001 a.u. MC iterations,

which is a very short propagation time engaging only ∼0.1% of the total walker popula-

105

tions at τ = 8.0 a.u., where we terminated our i-CISDTQ-MC runs (cf. Table 3.16), and

i-CISDTQ-MC capturing as little as 14–17% of all triply excited determinants, the semi-

stochastic CC(P ;Q) methodology reduces the 13.370 millihartree difference between the

CR-CC(2,3) and CCSDT energies of the B 1A1 state and the 8.1 kcal/mol error in the CR-
CC(2,3) value of the B 1A1 − X 3A′

2 gap relative to CCSDT to 1.260 millihartree and 0.6

kcal/mol, respectively, which is a chemical accuracy regime. Interestingly, the i-CISDTQ-

MC-based CC(P ;Q) value of ∆ES-T obtained after 6000 MC iterations matches the quality of
the B 1A1 − X 3A′

2 gap resulting from the fully deterministic CC(P ;Q) calculations using the

CC(t;3) approach, which give a 0.5 kcal/mol error relative to CCSDT when the cc-pVDZ ba-

sis set is employed [88]. After the additional 4000 MC time steps, where the i-CISDTQ-MC

propagations for the X 3A′

2 and B 1A1 states are still very far from convergence and where

the fractions of triples captured by i-CISDTQ-MC increase to about 30%, the small errors

relative to CCSDT characterizing the i-CISDTQ-MC-based CC(P ;Q) values of the energy of

the B 1A1 state and ∆ES-T at τ = 0.6 a.u. drop down by factors of 4–6, to 0.314 millihartree

and 0.1 kcal/mol, respectively, illustrating how rapid the convergence of the CIQMC-driven

CC(P ;Q) calculations toward the parent SRCC data can be. While the CR-CC(2,3) de-

scription of the X 3A′

2 state, which has a SR character, is much better than in the case of

its strongly correlated B 1A1 counterpart, the semi-stochastic CC(P ;Q) computations offer

great improvements in this case too. They are, for example, capable of reducing the ∼0.4

millihartree difference between the CR-CC(2,3) and CCSDT energies to a 0.1 millihartree

level after 10000 MC iterations and i-CISDTQ-MC capturing less than 30% of all triples. In

analogy to all other molecular examples considered in this chapter, the uncorrected CC(P )

values of the energies of the X 3A′

2 and B 1A1 states and separation between them converge

to their CCSDT limits too, but it is clear from Table 3.15 and Fig. 3.15 that they do it at

a much slower rate than their CC(P ;Q) counterparts. This can be illustrated by comparing

the errors relative to CCSDT characterizing the CC(P ) and CC(P ;Q) energies of the X 3A′
2

and B 1A1 states and separation between them obtained after 6000 MC iterations. They

106

are more than 11 millihartree, about 21 millihartree, and almost 6 kcal/mol, respectively, in

the former case and only 0.253 millihartree, 1.260 millihartree, and 0.6 kcal/mol, when the

CC(P ) energies are corrected for the remaining T3 correlations using the CC(P ;Q) approach.

As explained in Section 3.1, the CC(P ) energies converge to CCSDT more slowly than their

δ(P ; Q)-corrected CC(P ;Q) counterparts, since the initial, τ = 0, CC(P ) calculation for a

given electronic state is equivalent to CCSD, where T3 = 0. The CIQMC-driven CC(P ;Q)

calculations start from CR-CC(2,3), which provides information about T3 clusters via non-

iterative corrections to CCSD. This once again emphasizes the benefits of using corrections

δ(P ; Q) in the context of the semi-stochastic CC(P ;Q) work.

107

Table 3.15 Convergence of the CC(P ) and CC(P ;Q) energies of the X 3A′
2 and B 1A1 states
of trimethylenemethane, as described by the cc-pVDZ basis set, and of the corresponding adi-
abatic singlet–triplet gaps toward their parent CCSDT values. The D3h- and C2v-symmetric
2 and B 1A1 states, respectively, optimized in the SF-DFT/6-31G(d)
geometries of the X 3A′
calculations, were taken from Ref. [231]. The P spaces used in the CC(P ) and CC(P ;Q)
calculations were defined as all singly and doubly excited determinants and subsets of triply
excited determinants extracted from the i-CISDTQ-MC propagations with δτ = 0.0001 a.u.
The Q spaces used to determine the CC(P ;Q) corrections consisted of the triply excited
determinants not captured by the corresponding i-CISDTQ-MC runs. The i-CISDTQ-MC
calculations preceding the CC(P ) and CC(P ;Q) steps were initiated by placing 1500 walk-
2 state) and RHF (B 1A1 state) reference determinants and the na
ers on the ROHF (X 3A′
parameter of the initiator algorithm was set at 3. In all post-Hartree–Fock calculations, the
four lowest core orbitals were kept frozen and the spherical components of d orbitals were
employed throughout. Adapted from Ref. [102].

MC Iterations
0
2000
4000
6000
8000
10000
20000
50000
80000
∞

P a
19.202d
17.975
14.462
11.319
9.066
7.429
3.294
0.213
0.012

X 3A′
2
(P ; Q)a %Tb
0.418e
0.422
0.357
0.253
0.173
0.123
0.031
0.001
0.000

0
1.1
6.6
14.1
21.3
27.9
52.3
92.8
99.5

P a
58.051d
50.012
32.925
20.628
14.601
10.680
2.675
0.061
0.002

B 1A1
(P ; Q)a %Tb
13.370e
9.362
3.236
1.260
0.649
0.314
0.028
0.000
0.000

0
1.2
7.7
16.8
25.5
33.1
61.1
97.1
99.9

−155.466242f

−155.431596f

B 1A1 − X 3A′
2
(P ; Q)c
P c
8.1e
24.4d
5.6
20.1
1.8
11.6
0.6
5.8
0.3
3.5
0.1
2.0
0.0
-0.4
0.0
-0.1
0.0
0.0

21.7g

a Unless otherwise stated, all energies are reported as errors relative to CCSDT in millihartree.
b The %T values are the percentages of triples captured during the i-CISDTQ-MC propagations [the Sz = 1
triply excited determinants of the B2 (C2v) symmetry in the case of the X 3A′
2 state and the Sz = 0 triply
excited determinants of the A1 symmetry in the case of the B 1A1 state].
c Unless otherwise specified, the values of the singlet–triplet gaps are reported as errors relative to CCSDT
in kcal/mol.
d Equivalent to CCSD.
e Equivalent to CR-CC(2,3) [the most complete variant of CR-CC(2,3) abbreviated sometimes as CR-
CC(2,3),D or CR-CC(2,3)D].
f Total CCSDT energy in hartree.
g The CCSDT singlet–triplet gap in kcal/mol.

108

Table 3.16 The total numbers of walkers, reported as percentages of the total walker
populations at 80000 MC iterations, characterizing the i-CISDTQ-MC propagations with
δτ = 0.0001 a.u. that were needed to generate the CC(P ) and CC(P ;Q) results for
trimethylenemethane reported in Table 3.15. Adapted from Ref. [102].

MC Iterations
0
2000
4000
6000
8000
10000
20000
50000
80000

X 3A′
2
0.00a
0.01
0.06
0.14
0.24
0.34
1.09
14.60
100b

B 1A1
0.00a
0.01
0.05
0.11
0.18
0.26
0.93
15.05
100c

a The initial walker population, meaning 1500 walkers on the ROHF (X 3A′
reference determinants.
b The total number of walkers at 80000 MC iterations is 2363904677.
c The total number of walkers at 80000 MC iterations is 3543757954.

2 state) and RHF (B 1A1 state)

Figure 3.14 π molecular orbital network of the trimethylenemethane molecule, obtained
at the HF/cc-pVDZ level, at the D3h-symmetric geometry of the X 3A′
2 state optimized
in the SF-DFT/6-31G(d) calculations and taken from Ref. [231] The orbital irreducible
representations in the D3h symmetry are shown in black and the corresponding labels in the
C2v symmetry are shown in the parenthesis in orange. This electronic configuration refers
to the triplet state.

109

Figure 3.15 Convergence of the CC(P ) and CC(P ;Q) energies of the X 3A′
2 [panel (a)] and
B 1A1 [panel (b)] states of trimethylenemethane, as described by the cc-pVDZ basis set, and
of the corresponding adiabatic singlet–triplet gaps [panel (c)] toward their parent CCSDT
2 and B 1A1 states, optimized in the SF-DFT/6-31G(d)
values. The geometries of the X 3A′
calculations, were taken from Ref. [231]. The P spaces consisted of all singles and doubles
and subsets of triples identified during the i-CISDTQ-MC propagations with δτ = 0.0001
a.u. and the Q spaces consisted of the triples not captured by i-CISDTQ-MC. Adapted from
Ref. [102].

110

02040608001224364860(a)X 3A′2CC(P)CC(P;Q)020406080MC Iterations (103)01224364860Error re . to CCSDT (mEh)(b)B 1A1CC(P)CC(P;Q)020406080MC Iterations (103)0510152025Error re . to CCSDT (kca /mo )(c)ΔES–TCC(P)CC(P;Q)CHAPTER 4

THE SEMI-STOCHASTIC EXTENSIONS OF PARTICLE
NONCONSERVING EQUATION-OF-MOTION COUPLED-CLUSTER
THEORIES

This chapter is based on the method development and programming work, followed by

benchmark computations, aimed at extending the semi-stochastic approaches to particle

nonconserving EOMCC schemes, which is subject to the manuscript in preparation [242],

in which I have played a lead role in every aspect of the work other than the proposal for

developing such ideas by my advisor Professor Piotr Piecuch and oversight from him.

As described in the Introduction, despite remarkable strides in computer hardware and

efficient software advancements, providing an accurate and reliable description of MR prob-

lems and open-shell systems remain a challenge. This includes excited states dominated by

two-electron transitions and electronic spectra of radical and biradical species, to name a

few. In the previous chapter, we have already discussed how we can use the semi-stochastic

CC(P;Q) approach to tackle these challenging situations, and in this chapter, we describe

the how the semi-stochastic ideas can be combined with the particle nonconserving EOMCC

formalisms to achieve highly accurate results for open-shell systems.

As previously discussed, the particle nonconserving EOMCC approaches offer a simple

and elegant way of describing the electronic spectra of open-shell systems by formally adding

electrons to (electron attachment) or by removing electrons from (electron ionization) the

nearest closed-shell reference core. Within this family of EOMCC methods, we find the

EA and and IP EOMCC approaches [44, 78, 79, 138–146, 243–248], where the reference

and the target systems differ by one electron (|Ntarget − Nreference| = 1), the DEA and DIP

EOMCC methodologies [80, 147–154, 157, 249, 250], where |Ntarget − Nreference| = 2, and

their higher order extensions. These approaches prove particularly advantageous in the

study of electronic spectra of open-shell systems because the operation of adding electrons

to or removing electrons from a closed-shell reference to attain a target system automatically

creates an appropriate multi-reference model space specific to the system of interest while

111

relaxing the remaining electrons.

In contrast to employing genuine MRCC approaches,

these methods, being formally single reference in nature, are considerably much simpler to

use. Furthermore, compared to traditional spin-integrated, spin-orbital implementations of

particle-conserving CC/EOMCC treatments employing unrestricted or restricted open-shell

reference determinants they offer distinct advantages, such as rigorous spin and symmetry

adaptation of the computed states and the capability to balance both high-spin and low-spin

states in an accurate and equally balanced footing.

Within this area, the EA-EOMCC(3p-2h) and IP-EOMCC(3h-2p) methods [78, 140–

142, 162, 163, 251–253], have demonstrated notable success in accurately describing open-

shell systems featuring one unpaired electron outside a closed-shell core i.e., they can effec-

tively treat the (1,0) and (0,1) sectors of the Fock space. Additionally, the DEA-EOMCC(4p-

2h) and DIP-EOMCC(4h-2p) methods [80, 153–156, 254–256] enable quantitative descrip-

tions of the (2,0) and (0,2) sectors of the Fock space, thereby facilitating the precise deter-

mination of singlet–triplet gaps in biradical systems. To facilitate the application of these

approaches without compromising on the accuracy, the Piecuch group has previously lever-

aged the active space ideas based on active orbitals to select the leading components in the

respective electron attaching or removing operators, which resulted in methods [78, 80, 140–

142, 153–156, 162, 163, 251–256] designated as EA-EOMCCSDt, IP-EOMCCSDt, DEA-

EOMCC(3p-1h,4p-2h){Nu}, DEA-EOMCC(4p-2h){Nu}, and DIP-EOMCC(4h-2p){No} (No

and Nu indicate the numbers of active occupied and unoccupied orbitals used to define the

active-space). While the aforementioned methodologies have proven to be very reliable in

describing the electronic spectra of radicals and determining singlet–triplet gaps in biradical

systems, the selection of an appropriate active-space relies heavily on the specific system un-

der study and the chemical intuition of the investigator. Notably, the active-space selection

can pose a significant challenge in large and complex molecular systems. In this work, we

propose an alternative to using active orbitals to select the dominant higher-order excita-

tions. Drawing inspiration from our group’s prior semi-stochastic CC(P )/EOMCC(P ) and

112

CC(P ;Q) work [99–102, 137], where the CIQMC wave function propagations [91–94, 257]

were fused with deterministic CC/EOMCC computations to provide high-level CC/EOMCC

energetics at a reduced computational cost and in a black-box manner. To be more spe-

cific, we extend the semi-stochastic particle-conserving EOMCC(P ) approach [100, 137] to

the particle nonconserving regime, resulting in the EA-EOMCC(P ), IP-EOMCC(P ), DEA-

EOMCC(P ), and DIP-EOMCC(P ) methodologies [242]. These methods utilize CIQMC

wave function propagations to stochastically select lists of dominant 3p-2h/3h-2p/4p-2h/4h-

2p components in the EOMCC electron-attaching and electron-removing operators in an au-

tomated fashion and subsequently solve the particle-nonconserving EOMCC equations based

on those stochastically determined lists. To validate the efficiency of these semi-stochastic

approaches, we examine the C2N, CNC, N3, and NCO radicals and methylene and TMM

biradicals.

4.1 Theory

Consistent with the previously developed semi-stochastic CC(P ) and EOMCC(P ) schemes

[99–102, 137], the CIQMC-driven EA/IP/DEA/DIP-EOMCC methods, abbreviated as EA-

EOMCC(P ), IP-EOMCC(P ), DEA-EOMCC(P ), and DIP-EOMCC(P ), respectively, utilize

stochastic QMC wave function propagations to select the dominant high-order correlations

in an automated black-box manner. The key algorithmic details of the semi-stochastic

EA/IP/DEA/DIP-EOMCC calculations are as follows [242]:

1. First, perform a CCSD calculation, for the (N ∓ 1)- or (N ∓ 2)-electron closed-shell

system in the case of the EA/IP-EOMCC, or DEA/DIP-EOMCC, respectively, to

obtain the cluster amplitudes and to construct the similarity transformed Hamiltonian.

2. Next, start a CIQMC propagation by placing a certain number of walkers on the

reference determinant pertaining to the N -electron target system, while using the one-

and two-electron integrals from the closed-shell reference.

3. Afterwards extract a list of the most important determinants relevant to the EA/IP

113

and DEA/DIP EOMCC theory of interest [e.g., 3p-2h excited determinants relative

to |Φ(N −1)⟩ for EA-EOMCC(3p-2h), 3h-2p excited determinants relative to |Φ(N +1)⟩ for

IP-EOMCC(3h-2p), 4p-2h excited determinants relative to |Φ(N −2)⟩ for DEA-EOMCC(4p-

2h), and 4h-2p excited determinants relative to |Φ(N +2)⟩ for DIP-EOMCC(4h-2p)] from

the CIQMC wave function at a given time τ to define the P spaces for the EA/IP-

EOMCC(P ) and DEA/DIP-EOMCC(P ) calculations. If the target approach is EA-
EOMCC(3p-2h), the P space (H (P )) is defined as all 1p (G1p), all 2p-1h (G2p-1h), and
a subset of 3p-2h (g3p-2h) excited determinants having at least nP (e.g., one) positive

Figure 4.1 A schematic illustration depicting the construction of P -spaces in the
EA/IP/DEA/DIP-EOMCC(P ) computations. Panel (a) showcases the stabilization of cor-
relation energy (green line) and the corresponding increase in the total number of walkers
is shown in panel (b) (red line). On the right four snapshots from a QMC calculation are
presented, featuring the lists of determinants picked up by the QMC algorithm at various
time steps (light blue for 2000, dark blue for 20000, violet for 40000, and magenta for 100000
QMC iterations). It is evident that QMC deems some determinants more important than
others by placing more walkers on them.

or negative walker, such that H (P ) = G1p ⊕ G2p-1h ⊕ g3p-2h. Similarly, define H (P ) =
G1h ⊕ G2h-1p ⊕ g3h-2p for IP-EOMCC(3h-2p), H (P ) = G2p ⊕ G3p-1h ⊕ g4p-2h for DEA-
EOMCC(4p-2h), and in case of DIP-EOMCC(4h-2p) use H (P ) = G2h ⊕ G3h-1p ⊕ g4h-2p

(see Fig. 4.1 for an illustration).

4. Then solve the semi-stochastic EA/IP-EOMCC and DEA/DIP-EOMCC equations in

114

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159  1  2  3  4  5  6  7  8  9  10  11  13  1425   159  1  2  3  4  5  6  9  10  11  12  13  15  18(a)(b)the stochastically determined P spaces, i.e., use R(+1)

µ = Rµ,1p + Rµ,2p-1h + R(MC)
µ,3p-2h

for EA-EOMCC(P ), R(−1)

µ = Rµ,1h + Rµ,2h-1p + R(MC)

Rµ,2p + Rµ,3p-1h + R(MC)

µ,4p-2h for DEA-EOMCC(P ), and R(−2)

µ,3h-2p for IP-EOMCC(P ), R(+2)
µ = Rµ,2h + Rµ,3h-1p + R(MC)
µ,4h-2p

µ =

for DIP-EOMCC(P ).

5. Finally, check the convergence of the resulting energies by repeating the steps 3 and 4

at some later CIQMC propagation time τ ′ > τ . If the energies do not change within

a given convergence threshold, we can stop the calculation. One can also stop the

calculation if τ in steps 3 and 4 is chosen such that the stochastically determined

P spaces contain sufficiently large fraction of higher order excitations relevant to the

target EOMCC level.

Before going to the next section, we must discuss an interesting aspect of the semi-

stochastic particle nonconserving EOMCC(P ) approaches. At τ = 0, the EA-EOMCC(P ),

IP-EOMCC(P ), DEA-EOMCC(P ), and DIP-EOMCC(P ) energies are identical to the en-

ergies obtained in the EA-EOMCC(2p-1h), IP-EOMCC(2h-1p), DEA-EOMCC(3p-1h), and

DIP-EOMCC(3h-1p) calculations. This is because at τ = 0 the P spaces in the respective cal-

culations do not contain any 3p-2h, 3h-2p, 4p-2h, or 4h-2p determinants. On the other hand,

when τ = ∞, the P spaces in the EA-EOMCC(P ), IP-EOMCC(P ), DEA-EOMCC(P ),

and DIP-EOMCC(P ) calculations contain all the 3p-2h, 3h-2p, 4p-2h, and 4h-2p determi-

nants, respectively, and, as a result, the semi-stochastic EA/IP/DEA/DIP-EOMCC calcu-

lations provide energetics identical to that of the EA-EOMCC(3p-2h), IP-EOMCC(3h-2p),

DEA-EOMCC(4p-2h), and DIP-EOMCC(4h-2p) approaches. This relationship between the

EA-EOMCC(P ), IP-EOMCC(P ), DEA-EOMCC(P ), and DIP-EOMCC(P ) methods and

the fully deterministic EA-EOMCC(3p-2h), IP-EOMCC(3h-2p), DEA-EOMCC(4p-2h), and

DIP-EOMCC(4h-2p) methodologies were helpful in examining the correctness of our codes.

They also point to the ability of these semi-stochastic particle nonconserving EOMCC ap-

proaches driven by the information extracted from CIQMC to offer a systematically improv-

115

able description as τ goes from 0 to ∞. In the next section, we examine the performance

of the semi-stochastic EA-EOMCC(P ) and IP-EOMCC(P ) approaches in converging the

high-level EA-EOMCC(3p-2h) and IP-EOMCC(3h-2p) methods and the section after that

illustrates the capability of their double electron attachment [DEA-EOMCC(P )] and double

ionization potential [DIP-EOMCC(P )] extensions to converge the DEA-EOMCC(4p-2h) and

DIP-EOMCC(4h-2p) methodologies.

4.2 Adiabatic Excitations in C2N, CNC, N3, and NCO

In order to demonstrate the ability of the semi-stochastic EA-EOMCC(P ) and IP-

EOMCC(P ) approaches in converging their high-level, fully-deterministic parents, EA-EOMCC(3p-

2h) and IP-EOMCC(3h-2p), we carried out benchmark calculations for the ground and a

few lowest-lying doublet states along with the corresponding adiabatic excitation energies

of the C2N [Table 4.1 and Fig. 4.2 ], CNC [Table 4.2 and Fig. 4.3], N3 [Table 4.3 and Fig.

4.4], and NCO [Table 4.4 and Fig. 4.5] radicals. All the systems utilized the DZP[4s2p1d]

basis set [167, 168] and employed geometries optimized with the SAC-CI SDT-R/PS method

combined with the DZP[4s2p1d] basis set, as reported in Ref. [162]. Following our previ-

ous semi-stochastic work [99–101, 137], we used the HANDE software package [170, 171] to

execute all our i-FCIQMC calculations needed to generate the lists of 3p-2h and 3h-2p de-

terminants entering the P spaces for the EA-EOMCC(P) and IP-EOMCC(P) calculations.

Our standalone CC/EOMCC codes, interfaced with the RHF, ROHF, and integral transfor-

mation routines available in the GAMESS software package [172–174], were used to carry out

the EA-EOMCC(P ), IP-EOMCC(P ), and the fully deterministic CCSD, EA-EOMCC(2p-

1h), EA-EOMCC(3p-2h), IP-EOMCC(2h-1p), and IP-EOMCC(3h-2p) computations. Each

i-FCIQMC run was initiated by placing 500 walkers on the relevant reference function (see

Tables 4.1–4.4 for the details), the initiator parameter na was set at 3, and all of the i-

FCIQMC propagations used a time step of τ = 0.0001 a.u. As the true point group of the

system of interest was not Abelian for all four molecules, namely C2N, CNC, N3, and NCO,

we utilized their largest Abelian subgroups in the calculations. This choice was necessary,

116

since all of our CC/EOMCC codes interfaced with GAMESS and the CIQMC routines in

HANDE can only handle Abelian symmetries. In all the post-HF computations, the core

electrons corresponding to the 1s shells of the carbon, nitrogen, and oxygen atoms were kept

frozen.

4.2.1 Application of the EA-EOMCC(P ) approach to C2N and CNC radicals

In this subsection, we investigate the performance of the semi-stochastic EA-EOMCC(P )

approach in converging the EA-EOMCC(3p-2h) total electronic and adiabatic excitation

energies of a few low-lying doublet states of C2N and CNC radicals. The reference wave

functions are obtained by performing CCSD calculations on the nearest closed-shell C2N+

and CNC+ cations using the DZP[4s2p1d] basis set [167, 168] and, as mentioned above, the

nuclear geometries of the ground and excited states of C2N and CNC, optimized using EA

SAC-CI-SDT-R/PS and DZP[4s2p1d] basis set, are taken from Ref. [162].

4.2.1.1 C2N

We begin our discussion by using the C2N radical, where we study the X 2Π ground state

and three low-lying valence doublet excited states, A 2∆, B 2Σ−, and C 2Σ+. Following

Ref. [162], we used the EA SAC-CI-SDT-R/PS with DZP[4s2p1d] optimized geometries,

which are RCC = 1.400 ˚A and RCN = 1.185 ˚A for the X 2Π state, RCC = 1.315 ˚A and

RCN = 1.207 ˚A for the A 2∆ state, RCC = 1.302 ˚A and RCN = 1.223 ˚A for the B 2Σ− state,

and RCC = 1.311 ˚A and RCN = 1.214 ˚A for the C 2Σ+ state. At these geometries, all the

three valence excited states studied here, are characterized by two-electron transitions as

well as significant 2p-1h components (the B 2Σ− also has non-negligible 3p-2h contributions)

in the wave functions relative to the reference C2N+ ion. Therefore, to properly describe

these states, an accurate treatment of 3p-2h excitations is required and this is evident from

Table 1 of Ref. [162] or Table IV of Ref. [68] (cf., also, Table 4.1 of this work), where the

basic EA-EOMCCSD or EA-EOMCC(2p-1h) approach predicts an incorrect state ordering,

i.e., suggesting the B 2Σ− state to be higher in energy than the C 2Σ+ state. Furthermore,

the A 2∆ − X 2Π, B 2Σ− − X 2Π, and C 2Σ+ − X 2Π adiabatic excitation energies obtained

117

from EA-EOMCC(2p-1h) (6.190 eV, 7.860 eV, and 6.722 eV) are ∼3–5 eV away from the

experimental results (2.636 eV, 2.779 eV, and 3.306 eV) reported in Ref. [258]. The higher

order EA-EOMCC(3p-2h) method that accounts for a complete treatment of 3p-2h terms

in the electron attaching operator R(+1)

µ

, reduces these errors to ∼1 eV or lower (3.055 eV,

3.678 eV, and 3.809 eV), necessitating the need to use at least the EA-EOMCC(3p-2h) level

of theory, if a realistic description is wanted. However, as mentioned before, EA-EOMCC(3p-

2h) has a scaling of N 7 and, hence, could become very expensive very quickly. It would,

therefore, be interesting to see if the semi-stochastic EA-EOMCC(P) approach is capable

of recovering the EA-EOMCC(3p-2h)-quality results in this challenging case of C2N with a

fraction of 3p-2h terms selected in a automated manner using the i-FCIQMC wave function

propagations.

Following the semi-stochastic EA-EOMCC(P ) algorithm, as described above, and our

interest in converging the EA-EOMCC(3p-2h) energetics of the ground and three lowest

lying valence excited states of C2N, the cluster operator and the electron attaching operator

were approximated as T = T1 + T2 and R(+1)

µ = Rµ,1p + Rµ,2p-1h + R(MC)

µ,3p-2h, where the lists

of 3p-2h determinants defining the R(MC)

µ,3p-2h components at a given time τ were obtained

from the corresponding i-FCIQMC propagations at the same value of τ , meaning the B1

(C2v) component of 2Π for the X 2Π state, the A1 (C2v) component of 2∆ for the A 2∆

state, A2 (C2v) for the B 2Σ− state, and the A1 (C2v) for the C 2Σ+ state (C2v is the largest

Abelian subgroup of the true point group of C2N, C∞v). The results of our EA-EOMCC(P )

computations are reported in Table 4.1 and Fig. 4.2.

As can be seen from Table 4.1, at τ = 0.0, the the EA-EOMCC(P ) results are far from

the EA-EOMCC(3p-2h) energetics, which is expected since, at this stage, the P spaces lack

any 3p-2h determinants, rendering these results equivalent to those obtained from the basic

EA-EOMCC(2p-1h) method. The ground state of the C2N radical, X 2Π, is dominated by

1p excitations relative to the C2N+ reference and, as a result, it is described reasonably well

by EA-EOMCCSD [EA-EOMCC(P |τ =0)]. Still errors of ∼5 millihartree remain relative to

118

EA-EOMCC(3p-2h). This error rapidly decreases to the 1–3 millihartree range as soon as

we start incorporating ∼20–30% 3p-2h determinants in the P space. Indeed, at 20000 MC

iterations and with about 30% of 3p-2h determinants the error in EA-EOMCC(P ) energy is

only 1.421 millihartree. The next state denoted as A 2∆, which is also the first excited state,

has significant 2p-1h contributions relative to the reference determinant and, therefore, the

effect of adding 3p-2h determinants in the P space is more prominent. For this state, the

EA-EOMCC(P |τ =0) approach (EA-EOMCCSD) has, massive, 120 millihartree errors relative

to the target EA-EOMCC(3p-2h) method. This huge error is brought down to less than 20

millihartree with 40% of 3p-2h components and at 20000 QMC iterations. If we let QMC run

a little longer until 50000 QMC iterations, it recovers ∼70% of 3p-2h determinants and the

errors in the total energies are reduced to less than 1 millihartree relative to EA-EOMCC(3p-

2h). If we look at the adiabatic excitation energies, they are converged to 0.2 eV or better

at a much earlier QMC time step. For example, already at 20000 Monte Carlo time steps

the A 2∆ − X 2Π excitation energy is less than 0.5 eV in error and after 50000 iterations the

error is only 0.013 eV. The next state of interest is B 2Σ− and this is also the most difficult

to describe state in this system.

It is characterized by significant 3p-2h contributions in

addition to dominant 2p-1h correlations, and as a result, the EA-EOMCC(2p-1h) approach

fails miserably to describe this state, producing ∼160 millihartree errors relative to EA-

EOMCC(3p-2h). As we start including 3p-2h determinants in the P space with the help of

CIQMC wave function propagations, the errors decrease at an unbelievable rate (see Fig.

4.2). For example, just after 4000 MC iterations and with only 4% of 3p-2h determinants in

the P space, the huge > 150 millihartree errors decrease to 12.217 millihartree. After 30000

MC iterations, the CIQMC wave function samplings select 22.8% of 3p-2h determinants

in the P space and the EA-EOMCC(P ) energy at this stage is only ∼1 millihartree from

EA-EOMCC(3p-2h). This extremely fast convergence in the total electronic energy is also

reflected in the rapid convergence of the B 2Σ− − X 2Π gap. At 4000 MC time steps this gap

is already within 0.2 eV from the EA-EOMCC(3p-2h) gap and as soon as we reach 30000

119

MC iterations this B 2Σ− − X 2Π gap is < 0.01 eV from the target. For the final C 2Σ+

state for this system in our study, the errors in the EA-EOMCC(2p-1h) energy is > 110

millihartree relative to EA-EOMCC(3p-2h). For this state also, with incorporation of 3p-2h

determinants in the P space, the errors decrease rapidly. At 10000 MC iterations and with

18% of 3p-2h determinants in the P space the errors in total electronic energies are reduced

to half the errors in EA-EOMCCSD. As we incorporate more and more 3p-2h terms in the

P space, the errors keep decreasing further. In terms of the C 2Σ+ − X 2Π gap, the results

are even more favorable. For example, the 2.913 eV error at τ = 0 reduces to 0.432 eV at

30000 MC iterations and this error decreases even further to less than 0.1 eV if we let QMC

run an additional 10000 iterations.

120

Table 4.1 Convergence of the EA-EOMCC(P ) energies [abbreviated as EA(P )] of the X 2Π, A 2∆, B 2Σ−, and C 2Σ+ states
of C2N, as described by the DZP[4s2p1d] basis set of Refs. [167, 168], and of the corresponding adiabatic excitation energies
toward their parent EA-EOMCC(3p-2h) values. The geometries of the X 2Π, A 2∆, B 2Σ−, and C 2Σ+ states, optimized in the
SAC-CI SDT-R/PS calculations using the same basis set, were taken from Ref. [162]. The P spaces used in the EA-EOMCC(P )
calculations were defined as all 1p and 2p-1h determinants and subsets of 3p-2h determinants extracted from the i-FCIQMC
propagations with δτ = 0.0001 a.u. The i-FCIQMC calculations preceding the EA-EOMCC(P ) steps were initiated by placing
500 walkers on the ROHF reference determinants of the corresponding states and the na parameter of the initiator algorithm
was set at 3. In all post-Hartree–Fock calculations, the lowest core orbitals of the carbon and nitrogen atoms were kept frozen.

MC
Iters.
0
4000
10000
20000
30000
40000
50000
60000
100000
∞

X 2Π
EA(P )a %(3p-2h)b
4.696d
5.634
2.686
1.421
0.883
0.679
0.428
0.253
0.071
−130.404919e

0
8.1
18.8
30.2
39.8
48.1
57.1
64.8
82.4

A 2∆
EA(P )a %(3p-2h)b
119.913d
104.423
56.413
18.608
6.901
4.193
0.907
0.582
0.011

0
8.1
21.6
38.3
50.1
60.5
68.0
74.1
90.1

−130.292647e

B 2Σ−
EA(P )a %(3p-2h)b
158.397d
12.217
5.812
2.407
1.212
0.757
0.306
0.405
0.045
−130.269764e

0
4.3
9.3
15.8
22.8
30.4
38.2
47.2
76.3

c

C 2Σ+

c ∆E2

EA(P )a %(3p-2h)b ∆E1
111.733d
0
7.1
98.803
17.7
66.416
31.0
33.336
42.7
16.752
52.6
9.944
60.9
4.400
67.7
3.028
85.1
0.180

Ad. Excit. Energy
c ∆E3
3.135d 4.182d 2.913d
2.535
0.179
2.688
1.734
0.085
1.462
0.868
0.027
0.468
0.164
0.432
0.009
0.252
0.002
0.096
0.013 −0.003 0.108
0.076
0.004
0.009
−0.002 −0.001 0.003
3.678f 3.809f
3.055f

−130.264924e

a Unless otherwise stated, all energies are reported as errors relative to EA-EOMCC(3p-2h) in millihartree.
b The %(3p-2h) values are the percentages of 3p-2h determinants captured during the i-FCIQMC propagations (the Sz = 1/2 3p-2h determinants of
the B1 symmetry in the case of the X 2Π state, the A1 symmetry in the case of the A 2∆ state, the A2 symmetry in the case of the B 2Σ− state,
and the A1 symmetry in the case of the C 2Σ+ state).
c Unless otherwise specified, the adiabatic excitation energies are reported as errors relative to EA-EOMCC(3p-2h) in eV; ∆E1 = A 2∆ − X 2Π,
∆E2 = B 2Σ− − X 2Π, and ∆E3 = C 2Σ+ − X 2Π.
d Equivalent to EA-EOMCC(2p-1h).
e Total EA-EOMCC(3p-2h) energy in hartree.
f The EA-EOMCC(3p-2h) adiabatic excitation energy in eV.

121

Figure 4.2 Convergence of (a) the EA-EOMCC(P ) energies of the X 2Π, A 2∆, B 2Σ−, and
C 2Σ+ states of C2N, as described by the DZP[4s2p1d] basis set, and (b) the corresponding
adiabatic excitation energies toward their parent EA-EOMCC(3p-2h) values.

122

020406080100MC Iterations (103)04080120160Error rel. to EA-EOMCC(3p-2h) ( Eh)(a)X 2ΠA 2ΔB 2Σ−C 2Σ+020406080100MC Iterations (103)012345Error rel. to EA-EOMCC(3p-2h) (eV)(b)A 2Δ− X 2ΠB 2Σ−− X 2ΠC 2Σ+− X 2Π4.2.1.2 CNC

The next example studied using the semi-stochastic EA-EOMCC(P ) approach is the linear,

D∞h symmetric, CNC radical, where we considered the ground X 2Πg and the two low-lying

A 2∆u and B 2Σ+

u doublet excited electronic states and the corresponding adiabatic excita-

tion energies, A 2∆u − X 2Πg and B 2Σ+

u − X 2Πg. Following Ref. [162], we used the EA
SAC-CI-SDT-R/PS with DZP[4s2p1d] optimized geometries, which are RCN = 1.253 ˚A for

the X 2Πg state, RCN = 1.256 ˚A for the A 2∆u, and RCN = 1.259 ˚A for the B 2Σ+

u state.

At these geometries, the X 2Πg state is dominated by 1p excitations out of the closed-shell

reference state CNC+, whereas the A 2∆u and B 2Σ+

u states are characterized by significant

two-electron excitations, resulting in large 2p-1h contributions in the corresponding wave

functions. Therefore, it is of no surprise that the effect of incorporating 3p-2h correlations

is more important in the for A 2∆u and B 2Σ+

u states, but not so much in case of the X 2Πg

state. So, in order to accurately describe the total electronic energies of the A 2∆u and B

2Σ+

u states, we need at least the EA-EOMCC(3p-2h) level of theory and this is reflected in

the A 2∆u − X 2Πg and B 2Σ+

u − X 2Πg adiabatic excitation energies. It can be seen from

Table 1 of Ref. [162] or Table IV of Ref. [68] (cf., also, Table 4.2 of this work) that the EA-

EOMCCSD predicted A 2∆u − X 2Πg and B 2Σ+

u − X 2Πg adiabatic excitation energies are

7.206 eV and 7.639 eV, respectively, which are as much as 3.3–3.5 eV away from the experi-

mentally obtained values of 3.761 eV and 4.315 eV [258]. When the EA-EOMCC calculations

include all possible 3p-2h correlations, the resulting EA-EOMCC(3p-2h) approach predicts

these excitation energies to be 4.105 eV and 4.718 eV, respectively, which deviate from the

experimental values by only 0.3–0.4 eV. The active-space EA-EOMCCSDt method also per-

form very well, not only improving the EA-EOMCCSD results, but also yielding results as

good as the target EA-EOMCC(3p-2h) methodology. Our target in this work is, therefore, to

reproduce the EA-EOMCC(3p-2h) energetics using the semi-stochastic, i-FCIQMC-driven,

EA-EOMCC(P ) approach. In all of our i-FCIQMC calculations for CNC, we used the D2h

Abelian subgroup of its true point group D∞h. In particular, following the computational

123

protocol described earlier, the underlying i-FCIQMC calculations required to select the lists

of 3p-2h determinants defining the R(MC)

3p-2h components in the EA-EOMCC(P ) calculations

for CNC were set to converge the lowest energy states of B3g, Au, and B1u symmetry for

the X 2Πg, A 2∆u, and B 2Σ+

u states at their respective equilibrium geometries. The results

of our semi-stochastic EA-EOMCC(P ) approach in recovering EA-EOMCC(3p-2h)-quality

energetics for the X 2Πg, A 2∆u, and B 2Σ+

u states of CNC and the corresponding A 2∆u −

X 2Πg and B 2Σ+

u − X 2Πg excitation energies are reported in Table 4.2 and Fig. 4.3 of this

subsection.

As previously mentioned, the first row of Table 4.2 refers to EA-EOMCC(P ) results

for τ = 0, which are equivalent to EA-EOMCCSD and as already discussed above, it pro-

vides a very poor description of the lowest-lying doublet states of CNC. As the QMC wave

function evolves in time, which in this context refers to accumulating more and more 3p-2h

determinants that can be used to enrich the P spaces in the semi-stochastic calculations, EA-

EOMCC(P ) systematically converges to our target EA-EOMCC(3p-2h). The X 2Πg state,

that is 4.881 millihartree away from EA-EOMCC(3p-2h) at τ = 0, narrows down to within

∼1 millihartree of EA-EOMCC(3p-2h) with only 16% of 3p-2h determinants in the P space

and this happens only at 20000 MC iterations. If the i-FCIQMC propagation is allowed to

run an additional 10000 MC iteration, the EA-EOMCC(P) energies are less than 1 milli-

hartree away from target and the percentage of 3p-2h determinants in this case is only 21%.

For the remaining two states, A 2∆u and B 2Σ+

u , the gradual increase in the 3p-2h determi-

nants in the P space, as dictated by the underlying i-FCIQMC wave function propagation,

is more pronounced. For the A 2∆u state, only at 4000 MC iterations and with only 5.5% of

3p-2h determinants in the P space, the ∼119 millihartree error in the EA-EOMCC(P |τ =0)

[EA-EOMCC(2p-1h] energy, relative to EA-EOMCC(3p-2h) steeply decreases to 12 milli-

hartree and this happens only at 4000 MC iterations. If we let the i-FCIQMC propagations

to run longer, the EA-EOMCC(P) energies converge to the target EA-EOMCC(3p-2h) in a

very rapid pace. For example, at 10000 MC iteration and with 12% of 3p-2h determinants

124

in the P space, the errors reduce to less than 5 millihartree and by the time 20%–30% 3p-2h

determinants are captured, EA-EOMCC(P) is only 1–2 millihartree away from the target. In

case of the A 2∆u − X 2Πg adiabatic excitation energy the situation is much more favorable.

Already at 4000 MC iteration, the errors are less than 0.2 eV relative to EA-EOMCC(3p-2h)

and after 10000 MC iterations the errors are always much lower than 0.1 eV. For the final,

and most difficult to describe, B 2Σ+

u state, at 10000 MC iterations and with 18% of 3p-2h

determinants, the errors in the EA-EOMCC(P) energies are reduced by almost a factor of

3 compared to EA-EOMCC(P |τ =0), which is ∼112 millihartree away from EA-EOMCC(3p-

2h). This error is further reduced to ∼5 millihartree or better when we include about 40%

or more 3p-2h determinants in the P space. Interestingly, the B 2Σ+

u − X 2Πg adiabatic

excitation energies are much more well behaved, and only at 20000 MC iterations and with

30% of 3p-2h determinants in the P space corresponding to the EA-EOMCC(P ) calculation

pertaining to the B 2Σ+

u state, this energy difference is converged to within 0.3 eV relative

to EA-EOMCC(3p-2h).

The two above examples demonstrate that the semi-stochastic EA-EOMCC(P ) method is

capable of rapidly converging the EA-EOMCC(3p-2h) energetics for the individual electronic

states and the corresponding adiabatic excitation energies of radicals with a very small

percentage of 3p-2h determinants in the respective P spaces.

125

Table 4.2 Convergence of the EA-EOMCC(P ) energies [abbreviated as EA(P )] of the X 2Πg, A 2∆u, and B 2Σ+
u states of CNC,
as described by the DZP[4s2p1d] basis set of Refs. [167, 168], and of the corresponding adiabatic excitation energies toward their
parent EA-EOMCC(3p-2h) values. The geometries of the X 2Πg, A 2∆u, and B 2Σ+
u states, optimized in the SAC-CI SDT-R/PS
calculations using the same basis set, were taken from Ref. [162]. The P spaces used in the EA-EOMCC(P ) calculations were
defined as all 1p and 2p-1h determinants and subsets of 3p-2h determinants extracted from the i-FCIQMC propagations with
δτ = 0.0001 a.u. The i-FCIQMC calculations preceding the EA-EOMCC(P ) steps were initiated by placing 500 walkers on the
ROHF reference determinants of the corresponding states and the na parameter of the initiator algorithm was set at 3. In all
post-Hartree–Fock calculations, the lowest core orbitals of the carbon and nitrogen atoms were kept frozen.

MC
Iters.
0
4000
10000
20000
30000
40000
50000
60000
100000
∞

X 2Πg
EA(P )a %(3p-2h)b
4.881d
4.591
2.419
1.370
0.817
0.572
0.369
0.264
0.040

0
5.1
11.1
16.0
20.9
24.5
28.8
32.0
40.9
−130.411530e

A 2∆u
EA(P )a %(3p-2h)b
118.827d
11.668
4.816
1.851
0.865
0.407
0.194
0.096
0.015

0
5.5
12.0
20.6
27.5
35.4
43.4
52.6
79.7
−130.260673e

B 2Σ+
u

Ad. Excit. Energy

c

c

EA(P )a %(3p-2h)b ∆E1
∆E2
2.921d
3.101d
112.241d
0
2.419
0.193
8.1
93.500
1.164
0.065
18.0
45.183
0.299
0.013
31.3
12.348
0.124
42.2
5.369
0.001
0.045
−0.004
51.1
2.221
0.036
−0.005
59.1
1.707
−0.005 −0.001
67.0
0.228
−0.001 −0.001
86.9
0.006
4.718f
4.105f
−130.238150e

a Unless otherwise stated, all energies are reported as errors relative to EA-EOMCC(3p-2h) in millihartree.
b The %(3p-2h) values are the percentages of 3p-2h determinants captured during the i-FCIQMC propagations (the Sz = 1/2 3p-2h determinants of
the B3g symmetry in the case of the X 2Πg state, the Au symmetry in the case of the A 2∆u state, the B1u symmetry in the case of the B 2Σ+
u
state).
c Unless otherwise specified, the adiabatic excitation energies are reported as errors relative to EA-EOMCC(3p-2h) in eV; ∆E1 = A 2∆u − X 2Πg and
∆E2 = B 2Σ+
u − X 2Πg.
d Equivalent to EA-EOMCC(2p-1h).
e Total EA-EOMCC(3p-2h) energy in hartree.
f The EA-EOMCC(3p-2h) adiabatic excitation energy in eV.

126

Figure 4.3 Convergence of (a) the EA-EOMCC(P ) energies of the X 2Πg, A 2∆u, and B
2Σ+
u states of CNC, as described by the DZP[4s2p1d] basis set, and (b) the corresponding
adiabatic excitation energies toward their parent EA-EOMCC(3p-2h) values.

127

020406080100MC Iterations (103)04080120160Error rel. to EA-EOMCC(3p-2h) ( Eh)(a)X 2ΠgA 2ΔuB 2Σ+u020406080100MC Iterations (103)012345Error rel. to EA-EOMCC(3p-2h) (eV)(b)A 2Δu− X 2ΠgB 2Σ+u− X 2Πg4.2.2 Application of the IP-EOMCC(P ) approach to N3 and NCO radicals

After examining the performance of the EA-EOMCC(P) methodology in converging the

high-level EA-EOMCC(3p-2h) energetics, we proceed to assess the capability of the IP-

EOMCC(P) approach in recovering the IP-EOMCC(3h-2h) energetics by applying the IP-

EOMCC(P) approach in computing the total electronic energies of the ground and a few

excited doublet states of N3 and NCO radicals and the corresponding adiabatic excitation

energies. The reference wave functions were obtained by performing CCSD calculations on

the N−

3 and NCO− anions using the DZP basis set [167, 168] and geometries obtained from

IP SAC-CI-SDT-R/PS / DZP optimizations as reported in Ref. [162].

4.2.2.1 N3

Our first system under investigation is the N3 radical, where we studied the X 2Πg and B

2Σ+

u states, along with the corresponding B 2Σ+

u − X 2Πg adiabatic excitation energy. For

this, we utilized the IP SAC-CI-SDT-R/PS (in combination with the DZP[4s2p1d] basis

set) optimized geometries obtained from Ref. [162]. These geometries are RN−N = 1.188 ˚A

for the X 2Πg state and RN−N = 1.185 ˚A for the B 2Σ+

u state. At these geometries both

the states are characterized by predominant 1h excitations from the reference wave function

N3

−, with the B 2Σ+

u state having some 2h-1p contributions. Consequently, the effect of

including higher-level correlations, such as 3h-2p, is less pronounced than in C2N and CNC.

Nevertheless, there are still noticeable improvements when going from IP-EOMCC(2h-1p)

to IP-EOMCC(3h-2p). As evidenced in Table 1 of Ref. [162] and Table V of Ref. [68] (cf.,

also Table 4.3 of this work), the lower-order IP-EOMCCSD approach estimates the B 2Σ+

u −

X 2Πg energy to be 4.640 eV, which exhibits a discrepancy of about 0.1 eV compared to

the experimental value of 4.555 eV reported in Ref. [258]. The IP-EOMCC(3h-2p) method,

that includes a complete treatment of 3h-2p components of R(+1)

µ

, reduces this error to

0.04 eV, marking a significant improvement. Therefore, it would be interesting to see how

effective the i-FCIQMC driven IP-EOMCC(P) approaches are in recovering IP-EOMCC(3h-

2p) energetics. In all the i-FCIQMC calculations the D2h Abelian subgroup of the true point

128

group of the molecule D∞h was utilized. The underlying i-FCIQMC propagations were set

to converge the lowest energy states of B3g (D2h) symmetry for the X 2Πg and the B1u (D2h)

symmetry in case of the B 2Σ+

u state at their respective equilibrium geometries. The results

of our semi-stochastic IP-EOMCC(P ) approach in recovering total electronic energies of the

X 2Πg and B 2Σ+

u states of N3 and the corresponding X 2Πg − B 2Σ+

u adiabatic excitation

energy is reported in Table 4.3 and Fig. 4.4.

From Table 4.3, it can be observed that at τ = 0, the IP-EOMCC(P ) calculated to-

tal electronic energies of the X 2Πg and B 2Σ+

u states are approximately 13-15 millihartree

away from the IP-EOMCC(3h-2p) energies. As soon as the accumulation of 3h-2p deter-

minants defining the R(MC)

µ,3h-2p component of the electron ionizing operator R(−1)

µ

begins, the

IP-EOMCC(P) energies quickly start converging towards the IP-EOMCC(3h-2p) energetics.

For the X 2Πg state, at just 4000 MC iterations and with 28.7% of 3h-2p determinants in the

P space, the IP-EOMCC(P) approach reduces the 13 millihartree error in the IP-EOMCCSD

method to 2.627 millihartree. By 10000 MC iterations, IP-EOMCC(P) is already within ∼1

millihartree relative to IP-EOMCC(3h-2p), with the underlying P space containing about

40% of 3h-2p terms. The situation is similar in case of the B 2Σ+

u state, where the ∼15

millihartree error in the IP-EOMCC(2h-1p) is reduced to 5 millihartree after only 4000 MC

iterations and with 32% of 3h-2p determinants. At 10000 MC iterations, this error is further

decreased to 1.340 millihartree, and at this state the underlying P space is populated by

48.1% 3h-2p determinants. In case of the X 2Πg − B 2Σ+

u gap, convergence is much easier

due to favorable cancellation of errors. Here, IP-EOMCC(2h-1p) already has a small er-

ror of 0.042 eV relative to IP-EOMCC(3h-2p), but the benefit of using the IP-EOMCC(P)

approach is still evident. At only 10000 MC iterations, this already small energy gap of

0.042 eV is further decreased to 0.006 eV, demonstrating the ability of the semi-stochastic

IP-EOMCC(P ) approach to improve the IP-EOMCCSD energetics, even when the scope of

improvement is not very large.

129

Table 4.3 Convergence of the IP-EOMCC(P ) energies [abbreviated as IP(P )] of the X 2Πg
and B 2Σ+
u states of N3, as described by the DZP[4s2p1d] basis set of Refs. [167, 168], and
of the corresponding adiabatic excitation energy toward their parent IP-EOMCC(3h-2p)
values. The geometries of the X 2Πg and B 2Σ+
u states, optimized in the SAC-CI SDT-R/PS
calculations using the same basis set, were taken from Ref. [162]. The P spaces used in
the IP-EOMCC(P ) calculations were defined as all 1h and 2h-1p determinants and subsets
of 3h-2p determinants extracted from the i-FCIQMC propagations with δτ = 0.0001 a.u.
The i-FCIQMC calculations preceding the IP-EOMCC(P ) steps were initiated by placing
500 walkers on the ROHF reference determinants of the corresponding states and the na
parameter of the initiator algorithm was set at 3. In all post-Hartree–Fock calculations, the
lowest core orbitals of the nitrogen atoms were kept frozen.

MC
Iters.
0
4000
10000
20000
30000
40000
50000
60000
100000
∞

X 2Πg
IP(P )a %(3h-2p)b
13.078d
2.627
1.127
0.488
0.271
0.189
0.104
0.082
0.052

0
28.7
41.4
53.8
61.3
68.4
72.5
75.9
84.3
−163.729333e

B 2Σ+
u
IP(P )a %(3h-2p)b
14.623d
5.035
1.340
0.343
0.134
0.036
0.012
0.004
0.000

0
31.9
48.1
61.1
68.2
73.9
77.2
80.1
85.7
−163.560374e

Ad. Excit. Energy
∆Ec
0.042d
0.066
0.006
−0.004
−0.004
−0.004
−0.003
−0.002
−0.001
4.598f

a Unless otherwise stated, all energies are reported as errors relative to IP-EOMCC(3h-2p) in millihartree.
b The %(3h-2p) values are the percentages of 3h-2p determinants captured during the i-FCIQMC propaga-
tions (the Sz = 1/2 3h-2p determinants of the B3g symmetry in the case of the X 2Πg state and the B1u
symmetry in the case of the B 2Σ+
c Unless otherwise specified, the adiabatic excitation energies are reported as errors relative to IP-
EOMCC(3h-2p) in eV; ∆E = B 2Σ+
d Equivalent to IP-EOMCC(2h-1p).
e Total IP-EOMCC(3h-2p) energy in hartree.
f The IP-EOMCC(3h-2p) adiabatic excitation energy in eV.

u − X 2Πg.

u state).

130

Figure 4.4 Convergence of (a) the IP-EOMCC(P ) energies of the X 2Πg and B 2Σ+
u states of
N3, as described by the DZP[4s2p1d] basis set, and (b) the corresponding adiabatic excitation
energy toward their parent IP-EOMCC(3h-2p) values.

131

020406080100MC Iterations (103)0481216Error rel. to IP-EOMCC(3h-2p) (mEh)(a)X 2ΠgB 2Σ+u020406080100MC Iterati ns (103)0.000.020.040.060.080.10Err r rel. t  IP-EOMCC(3h-2p) (eV)(b)B 2Σ+u− X 2Πg4.2.2.2 NCO

Our final example is the NCO radical, where we investigated the ground X 2Π state and

two low-lying valence excited doublet states A 2∆ and B 2Π and the corresponding A 2∆

− X 2Π and B 2Π − X 2Π excitation energies. Following Ref. [162], we employed the IP

SAC-CI-SDT-R/PS / DZP[4s2p1d] optimized geometries, which are RNC = 1.230 ˚A and

RCO = 1.193 ˚A for the X 2Π state, RNC = 1.191 ˚A and RCO = 1.190 ˚A for the A 2∆

state, and RNC = 1.220 ˚A and RCO = 1.309 ˚A for the B 2Π state. At these geometries, all

the low-lying states mentioned above are dominated by 1h excitations from the closed-shell

reference ion NCO− with some 2h-1p contributions being important for the B 2Π state. So,

the basic IP-EOMCCSD approach is fairly reasonable in this case, however, similar to the

case of N3, the effect of incorporating 3h-2p correlations using the Rµ,3h-2p component of the

electron ionizing operator R(−1)

µ

offers non-negligible improvements. This is evident from

Table 1 of Ref. [162] and Table V of Ref. [68] (see also, Table 4.4 of this work), where the

IP-EOMCCSD method predicts the A 2∆ − X 2Π and B 2Π − X 2Π excitation energies to

be 2.900 eV and 4.199 eV, respectively, which are ∼0.1–0.2 eV from the experimental results

of 2.821 eV and 3.937 eV reported in Ref. [258]. The IP-EOMCC(3h-2p) approach further

improves these results bringing them to within 0.04 eV or better relative to the experimental

values. Now, it would be interesting to see how the semi-stochastic, i-FCIQMC-driven, IP-

EOMCC(P) method performs in this case. All the results pertaining to the IP-EOMCC(P)

approach are reported in Table 4.4 and Fig. 4.5. In all the i-FCIQMC calculations we used the

C2v Abelian subgroup of the true point group of NCO, C∞v. In particular the underlying

i-FCIQMC calculations were set to converge the lowest energy states of B2, A1, and B2

symmetry for the X 2Π, A 2∆, and B 2Π states at the respective equilibrium geometries

reported above.

The first row of Table 4.4 contains IP-EOMCC(P) results for τ = 0, which is the start

of i-FCIQMC wave function propagations. At this stage, the P spaces utilized in the IP-

EOMCC(P) calculations do not contain any 3h-2p determinants, and, as described earlier,

132

this is equivalent to the IP-EOMCCSD method and for the X 2Π state, this energy is about

10 millihartree away from the high-level IP-EOMCC(3h-2p) energy. This error is readily

decreased to 3.503 millihartree as soon as the i-FCIQMC propagations complete 4000 steps

and recover 23% of 3h-2p determinants. When the i-FCIQMC propagations reach 10000

MC iterations, this error is further reduced to 1.862 millihartree and with an additional

10000 MC iterations the errors are in the sub-millihartree regime. In case of the A 2∆ state,

IP-EOMCC(2h-1p) has an error of 11 millihartree relative to IP-EOMCC(3h-2p). At 4000

MC iterations and with ∼24% of 3h-2p determinants in the P space, the IP-EOMCC(P)

approach brings this error down to 3 millihartree and after 10000 MC iterations the IP-

EOMCC(P) energy is in the sub-millihartree region relative to IP-EOMCC(3h-2p). At this

stage, the P space contains 37.4% of 3h-2p determinants pertaining to the A 2∆ state.

In case of the A 2∆ − X 2Π energy gap, the convergence is even faster. At the start of

the i-FCIQMC propagation, when the P spaces do not contain any 3h-2p determinants,

the error in the adiabatic excitation energy relative to IP-EOMCC(3h-2p) is only 0.036 eV.

However, as soon as we start including 3h-2p determinants in the P spaces pertaining to

the IP-EOMCC(P) computations, this error rapidly decreases as can be seen from Table 4.4

and Fig. 4.5. In case of the B 2Π state, which is the most difficult to describe between the

three states of NCO investigated here, the IP-EOMCCSD method produces an error of > 20

millihartree relative to the IP-EOMCC(3h-2p) method. At 4000 MC iterations and with

26% of 3h-2p determinants in the P space, the IP-EOMCC(P) energies reach within ∼5

millihartree to the IP-EOMCC(3h-2p) energetics. With an additional 6000 MC iterations,

this is further reduced to 1.964 millihartree and at this stage the P spaces contain ∼42%

of 3h-2p determinants. The B 2Π − X 2Π adiabatic excitation energies are much easier to

converge to the IP-EOMCC(3h-2p) results. The IP-EOMCCSD predicted energy gap, which

has an error of 0.288 eV is already within 0.05 eV at 4000 MC iterations and at 1000 MC

iterations this error is a mere 0.003 eV.

The two above examples demonstrate that the semi-stochastic IP-EOMCC(P ) method is

133

capable of rapidly converging the IP-EOMCC(3h-2p) energetics for the individual electronic

states as well as the corresponding adiabatic excitation energies of radicals with very small

percentages of 3h-2p determinants in the respective P spaces.

134

Table 4.4 Convergence of the IP-EOMCC(P ) energies [abbreviated as IP(P )] of the X 2Π, A 2∆, and B 2Π states of NCO,
as described by the DZP[4s2p1d] basis set of Refs. [167, 168], and the corresponding adiabatic excitation energies toward their
parent IP-EOMCC(3h-2p) values. The geometries of the X 2Π, A 2∆u, and B 2Π states, optimized in the SAC-CI SDT-R/PS
calculations using the same basis set, were taken from Ref. [162]. The P spaces used in the IP-EOMCC(P ) calculations were
defined as all 1h and 2h-1p determinants and subsets of 3h-2p determinants extracted from the i-FCIQMC propagations with
δτ = 0.0001 a.u. The i-FCIQMC calculations preceding the IP-EOMCC(P ) steps were initiated by placing 500 walkers on the
ROHF reference determinants of the corresponding states and the na parameter of the initiator algorithm was set at 3. In all
post-Hartree–Fock calculations, the lowest core orbitals of the carbon, nitrogen, and oxygen atoms were kept frozen.

MC
Iters.
0
4000
10000
20000
30000
40000
50000
60000
100000
∞

X 2Π
IP(P )a %(3h-2p)b
9.587d
3.503
1.862
0.973
0.428
0.329
0.164
0.097
0.011

0
23.2
34.9
46.6
55.0
62.5
68.1
72.4
81.8
−167.591596e

A 2∆
IP(P )a %(3h-2p)b
10.921d
3.198
0.991
0.359
0.153
0.102
0.032
0.015
0.000

0
23.6
37.4
52.5
61.4
68.0
72.6
76.1
84.1
−167.486358e

B 2Π

Ad. Excit. Energy

c

c

IP(P )a %(3h-2p)b ∆E1
∆E2
0.288d
0.036d
20.154d
0
0.049
−0.008
25.6
5.302
−0.024
41.5
1.964
0.003
−0.017 −0.001
54.9
0.943
−0.008
63.9
0.428
0.000
−0.006 −0.012
70.2
-0.106
−0.004 −0.004
74.8
0.003
0.001
−0.002
77.9
0.133
0.000
0.000
85.2
0.002
3.911f
2.864f
−167.447865e

a Unless otherwise stated, all energies are reported as errors relative to IP-EOMCC(3h-2p) in millihartree.
b The %(3h-2p) values are the percentages of 3h-2p determinants captured during the i-FCIQMC propagations (the Sz = 1/2 3h-2p determinants of
the B2 symmetry in the case of the X 2Π state, the A1 symmetry in the case of the A 2∆ state, and the B2 symmetry in the case of the B 2Π state).
c Unless otherwise specified, the adiabatic excitation energies are reported as errors relative to IP-EOMCC(3h-2p) in eV; ∆E1 = A 2∆ − X 2Π and
∆E2 = B 2Π − X 2Π.
d Equivalent to IP-EOMCC(2h-1p).
e Total IP-EOMCC(3h-2p) energy in hartree.
f The IP-EOMCC(3h-2p) adiabatic excitation energy in eV.

135

Figure 4.5 Convergence of (a) the IP-EOMCC(P ) energies of the X 2Π, A 2∆, and B
2Π states of NCO, as described by the DZP[4s2p1d] basis set, and (b) the corresponding
adiabatic excitation energies toward their parent IP-EOMCC(3h-2p) values.

136

020406080100MC Iterations (103)06121824Error rel. to IP-EOMCC(3h-2p) (mEh)(a)X 2ΠA 2ΔB 2Π020406080100MC Iterati ns (103)Δ0.10.00.10.20.30.4Err r rel. t  IP-EOMCC(3h-2p) (eV)(b)A 2Δ− X 2ΠB 2Π− X 2Π4.3 Singlet–Triplet Gaps in Methylene and Trimethylenemethane

After exploring the performance of the semi-stochastic EA-EOMCC(P) and IP-EOMCC(P)

methods, which describe the (1,0) and (0,1) sectors of the Fock space, we move on to

their extensions to the (2,0) and (0,2) sectors of the Fock space, which we refer to as the

semi-stochastic DEA- [DEA-EOMCC(P)] and DIP-EOMCC [DIP-EOMCC(P)] approaches.

These methods are particularly suitable for studying singlet and triplet states of biradical

systems and the corresponding singlet–triplet gaps. To explore the capability of the semi-

stochastic DEA-EOMCC(P) and DIP-EOMCC(P) approaches in converging their high-level

fully-deterministic parent DEA-EOMCC(4p-2h) and DIP-EOMCC(4h-2p) methods, we car-

ried out benchmark calculations for the lowest singlet and triplet states along with the

corresponding singlet–triplet gaps (∆ES-T = Esinglet − Etriplet) in the CH2 [Tables 4.5 and

4.6 and Figs. 4.6 and 4.7] and trimethylenemethane [Table 4.7 and Figs. 4.9 and 4.10]. We

utilized the TZ2P basis set [180] and FCI/TZ2P geometries as reported in Ref. [181] for the

CH2 molecule and for trimethylenemethane, we employed the 6-31G(d) basis set [178, 179]

with the SF-DFT/6-31G(d) geometries reported in Ref. [231]. For the smaller CH2 molecule

we used i-FCIQMC, while for the larger trimethylenemethane system we exploited the i-

CISDTQ-MC propagations to generate the lists of 4p-2h and 4h-2p determinants entering the

P spaces for the DEA-EOMCC(P) and DIP-EOMCC(P) calculations. Again, following our

previous semi-stochastic work [99–101, 137], we used the HANDE software package [170, 171]

to execute all our QMC calculations. Our standalone CC/EOMCC codes, interfaced with

the RHF, ROHF, and integral transformation routines available in the GAMESS software

package [172–174], were used to carry out the DEA-EOMCC(P ), DIP-EOMCC(P ), and the

fully deterministic CCSD, DEA-EOMCC(3p-1h), DEA-EOMCC(4p-2h), DIP-EOMCC(3h-

1p), and DIP-EOMCC(4h-2p) computations. Each i-FCIQMC and i-CISDTQ-MC run was

initiated by placing 500 walkers on the relevant reference function (see Tables 4.5–4.7 for the

details), the initiator parameter na was set at 3, and all of the i-FCIQMC and i-CISDTQMC

propagations used a time step of τ = 0.0001 a.u.

If the true point group of the system

137

of interest was not Abelian, which was the case for trimethylenemethane, we utilized the

largest Abelian subgroups in the calculations. This choice was necessary, since all of our

CC/EOMCC codes interfaced with GAMESS and the CIQMC routines in HANDE can only

handle Abelian symmetries. In all the post-HF computations, the core electrons correspond-

ing to the 1s shells of the carbon atom was kept frozen.

4.3.1 Methylene

The discussions for the semi-stochastic DEA- and DIP-EOMCC methods, DEA-EOMCC(P)

and DIP-EOMCC(P), begin with the results for total electronic energies of the X 3B1, A 1A1,

B 1B1, and C 1A1 states of methylene and the corresponding singlet–triplet gaps.

In the

DEA-EOMCC calculations we used the CH2+

2 dication as the reference and for the DIP-

EOMCC approach the CH2−

2 dianion was used as the reference. The geometries used were

the FCI/TZ2P geometries as reported in Ref. [181] for the CH2 molecule and throughout

this work we utilized the TZ2P basis set [180]. All the results of the DEA-EOMCC(P) and

DIP-EOMCC(P) calculations are reported in Tables 4.5 and 4.6 and Figs. 4.6 and 4.7. The

ground X 3B1 and the second excited state B 1B1 are characterized as having a SR char-

acter that can be well represented by high-spin triplet and open-shell singlet configurations

of (1a1)2(2a1)2(1b2)2(3a1)1(1b1)1 type. The first and the third excited states, A 1A1 and

C 1A1, have a significant MR character originating due to the mixing of the two configura-

tions (1a1)2(2a1)2(1b2)2(3a1)2 and (1a1)2(2a1)2(1b2)2(1b1)2, which makes it very challenging

for many electronic structure methods to provide an accurate description. As a result, in

order to obtain accurate results for the A 1A1 − X 3B1, B 1B1 − X 3B1, and C 1A1 − X 3B1

singlet–triplet gaps, one needs to use methods that can offer well balanced treatments of

both dynamical and nondynamical correlations. This makes CH2 a very popular example

to test the performance of newly developed electronic structure methods. The most basic

DEA-EOMCCSD [DEA-EOMCC(3p-1h)] and DIP-EOMCCSD [DIP-EOMCC(3h-1p)] fail

to provide an accurate description of the triplet ground and the low-lying singlet states of

methylene and the corresponding singlet–triplet gaps, and one needs to incorporate 4p-2h

138

and 4h-2p correlations for an accurate description. From Table 1 of Ref. [153] one can com-

pare the FCI, DEA-EOMCC(3p-1h), DEA-EOMCC(4p-2h), DIP-EOMCC(3h-1p), and DIP-

EOMCC(4h-2p) calculated singlet–triplet gaps A 1A1 − X 3B1, B 1B1 − X 3B1, and C 1A1

− X 3B1.

It is evident that the DEA-EOMCC(3p-1h) predicted gaps are −0.11, −1.89,

and −3.64 kcal/mol in error compared to the FCI singlet–triplet gaps of 11.14, 35.59, and

61.67 kcal/mol, respectively. The complete incorporation of 4p-2h correlations via the DEA-

EOMCC(4p-2h) methodology improves these results, especially for the second and third

gaps, bringing down the errors to 0.38, −0.02, and 0.21 kcal/mol compared to FCI. On the

other hand, the DIP-EOMCC(3h-1p) method predicts these singlet–triplet gaps to be −4.53,

−3.22, and −4.63 kcal/mol away from FCI. We see a significant improvement by using the

DIP-EOMCC(4h-2p) approach, which reduces these errors to only −0.44, −0.51, and −0.48

kcal/mol relative to FCI. So, it would be interesting to see how our DEA-EOMCC(P) and

DIP-EOMCC(P) approaches perform in this interesting case of methylene. We used the

i-FCIQMC method to extract the lists of 4p-2h determinants, in case of DEA-EOMCC(P),

and the 4h-2p determinants, in case of the DIP-EOMCC(P) approach. The i-FCIQMC

runs were initiated on the ROHF determinant of B1 symmetry for the X 3B1 state and

the RHF determinants of A1 symmetries for the A 1A1 and C 1A1 states, and the A2 sym-

metry B 1B1, and with the one- and two-body integrals obtained from the reference ionic

systems (CH2+
2

in case of DEA-EOMCC(P) and CH2−
2

for the DIP-EOMCC(P) case). In

the next paragraph, we explore the performance of the semi-stochastic, i-FCIQMC-driven,

DEA-EOMCC(P) approach and in the following paragraph we investigate the performance

of the DIP-EOMCC(P) approach.

From Table 4.5, one can notice that at τ = 0, the DEA-EOMCC(P) [equivalent to the

DEA-EOMCC(3p-1h)] energies are 14 millihartree in error compared to DEA-EOMCC(4p-

2h) for the X 3B1 state. As soon as we start incorporating 4p-2h determinants in the P spaces

defining the R(MC)

µ,4p-2h components of R(+2)

µ

operator in the DEA-EOMCC(P) calculations, the

energies rapidly converge to DEA-EOMCC(4p-2h). At 4000 MC iterations and with just

139

1.8% of 4p-2h determinants in the P space, the DEA-EOMCC(P) errors are already within

5 millihartree and as soon as the i-FCIQMC propagations select 7% of 4p-2h determinants,

the error in the DEA-EOMCC(P) energy becomes only ∼1 millihartree. This happens at

just 10000 MC iterations, and if we allow i-FCIQMC to complete 50000 MC steps, the

P spaces contain 15.3% of 4p-2h determinants and the error in energy of the X 3B1 state

reaches the sub-millihartree region. This fast convergence to DEA-EOMCC(4p-2h) remains

true even when we go to the first excited singlet state A 1A1, where the > 13 millihartree

error in the DEA-EOMCC(3p-1h) energies are quickly reduced to < 2 millihartree at only

10000 MC iterations and with just 4.3% of 4p-2h determinants in the P space. The situation

improves even more when the i-FCIQMC propagations reach 20000 MC time steps, where

with only 7.6% of 4p-2h determinants the total electronic energy is already in the sub-

millihartree regime.

If we look at the A 1A1 − X 3B1 singlet triplet gap, it can be seen

that the 0.487 kcal/mol error at τ = 0 is quickly reduced to 0.2 kcal/mol at already 10000

MC iterations and by the time we reach 50000 MC iterations, the error is less than 0.1

kcal/mol relative to DEA-EOMCC(4p-2h). For the next state B 1B1, which is mostly single

reference in nature, the error at τ = 0 is 11.129 millihartree. However, with only ∼5% and

∼8% of 4p-2h determinants in the P spaces and at just 10000 and 20000 MC iterations,

respectively, the ∼2 millihartree and ∼1 millihartree marks are reached by the errors in the

total electronic energy predicted by DEA-EOMCC(P). The situation in the corresponding B

1B1 − X 3B1 singlet–triplet gap is more favorable. For example, the 1.875 kcal/mol errors in

the DEA-EOMCC(P |τ =0) method is quickly brought down to 0.1 kcal/mol relative to DEA-

EOMCC(4p-2h) at only 10000 MC time steps, with additional iterations further improving

the results as seen in Table 4.5 and in Fig. 4.6.

In case of the most challenging C 1A1

state, the errors in the DEA-EOMCC(3p-1h) energies relative to DEA-EOMCC(4p-2h) is

7.987 millihartree. As the underlying i-FCIQMC wave function propagation gradually select

more and more 4p-2h determinants, this error quickly gets down to the sub-millihartree

regime. At 20000 MC iteration with 5% of 4p-2h determinants in the P space the DEA-

140

EOMCC(P) energies are about 3 millihartree away from the DEA-EOMCC(4p-2h) method

and after 50000 MC iterations and with 13% of 4p-2h determinants in the P space, these

errors are less than 1 millihartree. This is also reflected in the C 1A1 − X 3B1 singlet–

triplet gap. The 3.847 kcal/mol error at τ = 0 rapidly becomes ∼1 kcal/mol in absolute

value at 20000 MC iterations and by the time the i-FCIQMC propagations complete 50000

time steps, the absolute value of the error is only about 0.2 kcal/mol from the high-level

DEA-EOMCC(4p-2h) result.

It is interesting to note that, due to the different rates of

convergence of the low-lying states, the singlet–triplet gaps fluctuate before converging to

the DEA-EOMCC(4p-2h) results.

141

Table 4.5 Convergence of the DEA-EOMCC(P ) energies [abbreviated as DEA(P )] of the X 3B1, A 1A1, B 1B1, and C 1A1
states of methylene, as described by the TZ2P basis set of Ref. [180], and of the corresponding adiabatic singlet–triplet gaps
toward their parent DEA-EOMCC(4p-2h) values. The geometries of the X 3B1, A 1A1, B 1B1, and C 1A1 states, optimized
in the FCI calculations using the TZ2P basis set, were taken from Ref. [181]. The P spaces used in the DEA-EOMCC(P )
calculations were defined as all 2p and 3p-1h determinants and subsets of 4p-2h determinants extracted from the i-FCIQMC
propagations with δτ = 0.0001 a.u. The i-FCIQMC calculations were initiated by placing 500 walkers on the corresponding
ROHF reference determinant for the X 3B1 state and the corresponding RHF reference determinants for the remaining states
and the na parameter of the initiator algorithm was set at 3. In all the post-Hartree–Fock calculations, the lowest core orbital
of the carbon atom was kept frozen.

MC
Iters. DEA(P )a %(4p-2h)b DEA(P )a %(4p-2h)b DEA(P )a %(4p-2h)b DEA(P )a %(4p-2h)b ∆E1

A 1A1

C 1A1

X 3B1

B 1B1

0
4000
10000
20000
50000
100000
200000
∞

14.118d
4.863
2.252
1.323
0.448
0.079
0.001

0.0
1.8
4.4
7.0
15.3
38.2
85.2
−39.066449e

13.342d
4.611
1.953
0.996
0.316
0.028
0.000

0.0
1.5
4.3
7.6
20.1
51.9
88.9
−39.048089e

11.129d
4.477
2.085
1.163
0.394
0.050
0.000

0.0
2.0
4.8
7.9
18.3
48.5
89.3
−39.009764e

7.987d
6.344
4.794
3.161
0.701
0.078
0.000

0.0
1.3
3.3
5.1
13.2
40.7
88.0
−38.967833e

c ∆E2

Singlet–Triplet Gap
c
c ∆E3
0.487d 1.875d 3.847d
0.242 −0.929
0.158
0.105 −1.595
0.188
0.101 −1.154
0.205
0.034 −0.158
0.083
0.001
0.018
0.032
0.000
0.000
0.000
11.521f 35.570f 61.881f

a Unless otherwise stated, all energies are reported as errors relative to DEA-EOMCC(4p-2h) in millihartree.
b The %(4p-2h) values are the percentages of 4p-2h determinants captured during the i-FCIQMC propagations [the Sz = 1 4p-2h determinants of the
B1 symmetry in the case of the X 3B1 state, the Sz = 0 4p-2h determinants of the B1 symmetry in case of the B 1B1 state , and the Sz = 0 4p-2h
determinants of the A1 symmetry in the case of the A 1A1, and the A1 symmetry in case of the C 1A1 state].
c Unless otherwise specified, the singlet–triplet gaps are reported as errors relative to DEA-EOMCC(4p-2h) in kcal/mol; ∆E1 = A 1A1 − X 3B1,
∆E2 = B 1B1 − X 3B1, and ∆E3 = C 1A1 − X 3B1.
d Equivalent to DEA-EOMCC(3p-1h).
e Total DEA-EOMCC(4p-2h) energy in hartree.
f The DEA-EOMCC(4p-2h) singlet–triplet gap in kcal/mol.

142

Figure 4.6 Convergence of (a) DEA-EOMCC(P ) energies of X 3B1, A 1A1, B 1B1, and C
1A1 states of methylene, as described by the TZ2P basis set, and (b) of the corresponding
adiabatic singlet–triplet gaps towards their parent DEA-EOMCC(4p-2h) values.

143

04080120160200MC Iterations (103)0246810Error rel. to DEA-EOMCC(4p-2h) (mEh)(a)X 3B1A 1A1B 1B1C 1A104080120160200MC Iterations (103)−101234Error rel. to DEA-EOMCC(4p-2h) (kcal/mol)(b)A 1A1− X 3B1B 1B1− X 3B1C 1A1− X 3B1Table 4.6 contains the results of our semi-stochastic, i-FCIQMC-driven, DIP-EOMCC(P)

computations. At the start of the i-FCIQMC propagations, the total electronic energy of the

X 3B1 state calculated by the DIP-EOMCC(P) approach has an error of 26.147 millihartree

relative to DIP-EOMCC(4h-2p). This error is readily decreased to about 3 millihartree at

only 4000 MC iterations, where the P spaces contain about 35% of 4h-2p determinants in

the P space. Allowing the i-FCIQMC propagations to complete a total of 10000 and 20000

MC iterations results in errors of 1.357 and 0.766 millihartree, respectively. The percentage

of 4h-2p determinants in the P spaces at this stage are 43.4% and 50.3%, respectively. For

the first excited singlet state A 1A1, the 19.624 millihartree error reported at τ = 0 is sharply

decreased to 3 millihartree at just 4000 MC iteration and with ∼19% of 4h-2p determinants

in the P space. By the time the i-FCIQMC propagations reach 10000 MC time steps, this

error is already below 1 millihartree and this happens with 30% of 4h-2p determinants in

the P space. Similarly, the −4.093 kcal/mol error in the A 1A1 − X 3B1 singlet–triplet gap

narrows down to just −0.152 kcal/mol at 4000 MC iterations and by the time 50000 MC time

steps are completed, this error is only −0.028 millihartree away from the DIP-EOMCC(4h-

2p) value. The next state, which is an open-shell singlet state, is referred to as B 1B1, and in

this case the ∼22 millihartree errors in the DIP-EOMCC(3h-1p) energetics improves to only

3.681 millihartree at just 4000 MC iterations and with ∼33% of 4h-2p determinants in the

P space. Then with i-FCIQMC propagation it steadily converges to DIP-EOMCC(4h-2p).

By the time i-FCIQMC completes 20000 MC iterations, the errors in the total electronic

energies in this state falls in the sub-millihartree regime. At this stage, the B 1B1 − X

3B1 singlet–triplet gap is already converged to 0.032 kcal/mol compared to the target DIP-

EOMCC(4h-2p) values, and this is a significant improvement from the DIP-EOMCC(P |τ =0)

result of −2.714. The final state considered here is the C 1A1 state, which has a strong

multireference character as described above. In this case, the DIP-EOMCC(P) calculations

start with an error of 19.355 millihartree at τ = 0. After 4000 MC time steps and with 17.3%

of 4h-2p determinants in the P space, this error is down to 8.869 millihartree. It gradually

144

decreases to 3.480 millihartree at 20000 MC iterations before reaching the sub-millihartree

mark at 50000 MC iterations. At these stages, the percentages of the 4h-2p determinants are

36.6% and 51.7%, respectively. The convergence of the C 1A1 − X 3B1 singlet–triplet gap

mirrors this convergence pattern. The −4.149 kcal/mol error in this gap shrinks to 1.703

kcal/mol at 20000 MC iterations and by the time one reaches 50000 MC iterations the error

is only 0.149 kcal/mol (cf., Fig. 4.7 for a graphical representation).

145

Table 4.6 Convergence of the DIP-EOMCC(P ) energies [abbreviated as DIP(P )] of the X 3B1, A 1A1, B 1B1, and C 1A1 states
of methylene, as described by the TZ2P basis set of Ref. [180], and of the corresponding adiabatic singlet–triplet gaps toward
their parent DIP-EOMCC(4h-2p) values. The geometries of the X 3B1, A 1A1, B 1B1, and C 1A1 states, optimized in the FCI
calculations using the TZ2P basis set, were taken from Ref. [181]. The P spaces used in the DIP-EOMCC(P ) calculations
were defined as all 2h and 3h-1p determinants and subsets of 4h-2p determinants extracted from the i-FCIQMC propagations
with δτ = 0.0001 a.u. The i-FCIQMC calculations were initiated by placing 500 walkers on the corresponding ROHF reference
determinant for the X 3B1 state and the corresponding RHF reference determinants for the remaining states and the na parameter
of the initiator algorithm was set at 3. In all the post-Hartree–Fock calculations, the lowest core orbital of the carbon atom
was kept frozen.

MC
Iters. DIP(P )a %(4h-2p)b DIP(P )a %(4h-2p)b DIP(P )a %(4h-2p)b DIP(P )a %(4h-2p)b ∆E1

Singlet–Triplet Gap
c ∆E3

c ∆E2

c

A 1A1

C 1A1

X 3B1

B 1B1

0
4000
10000
20000
50000
100000
200000
∞

26.147d
3.289
1.357
0.766
0.131
0.009
0.000

0.0
34.6
43.4
50.3
59.5
68.1
70.6
−39.066449e

19.624d
3.046
0.946
0.524
0.085
0.005
0.000

0.0
19.4
29.8
39.8
55.5
63.8
65.2
−39.048089e

21.821d
3.681
1.853
0.818
0.168
0.009
0.000

0.0
32.9
42.0
48.6
59.0
68.1
70.6
−39.009764e

19.355d
8.869
5.741
3.480
0.369
0.057
0.000

0.0
17.3
28.2
36.6
51.7
62.8
65.2
−38.967833e

−4.093d −2.714d −4.149d
3.502
−0.152 0.246
2.751
−0.257 0.311
1.703
−0.152 0.032
0.149
−0.028 0.023
0.030
−0.003 0.000
0.000
0.000
0.000
10.701f 35.083f 61.191f

a Unless otherwise stated, all energies are reported as errors relative to DIP-EOMCC(4h-2p) in millihartree.
b The %(4h-2p) values are the percentages of 4h-2p determinants captured during the i-FCIQMC propagations [the Sz = 1 4h-2p determinants of the
B1 symmetry in the case of the X 3B1 state, the Sz = 0 4h-2p determinants of the B1 symmetry in case of the B 1B1 state , and the Sz = 0 4h-2p
determinants of the A1 symmetry in the case of the A 1A1, and the A1 symmetry in case of the C 1A1 state].
c Unless otherwise specified, the singlet–triplet gaps are reported as errors relative to DIP-EOMCC(4h-2p) in kcal/mol; ∆E1 = A 1A1 − X 3B1,
∆E2 = B 1B1 − X 3B1, and ∆E3 = C 1A1 − X 3B1.
d Equivalent to DIP-EOMCC(3h-1p).
e Total DIP-EOMCC(4h-2p) energy in hartree.
f The DIP-EOMCC(4h-2p) singlet–triplet gap in kcal/mol.

146

Figure 4.7 Convergence of (a) DIP-EOMCC(P ) energies of X 3B1, A 1A1, B 1B1, and C
1A1 states of methylene, as described by the TZ2P basis set, and (b) of the corresponding
adiabatic singlet–triplet gaps towards their parent DIP-EOMCC(4h-2p) values.

147

04080120160200MC Iterations (103)03691215Error rel. to DIP-EOMCC(4h-2p) (mEh)(a)X 3B1A 1A1B 1B1C 1A104080120160200MC Iterations (103)0246Error rel. to DIP-EOMCC(4h-2p) (kcal/mol)(b)A 1A1− X 3B1B 1B1− X 3B1C 1A1− X 3B14.3.2 Trimethylenemethane

Our final example is trimethylenemethane (TMM), a fascinating non-Kekul´e hydrocar-

bon, in which four valence π electrons are de-localized over four closely spaced π-type or-

bitals. Assuming D3h symmetry, which is the true point group symmetry of the minimum

energy structure of the ground-state triplet surface of trimethylenemethane, the four MOs

involved in the π-orbital network of trimethylenemethane consists of the nondegenerate 1a′′
2

orbital, the doubly degenerate 1e′′ orbitals, and the nondegenerate 2a′′

2 orbital. If, on the

other hand, the symmetry relevant to the low-lying singlet states, C2v is adopted, the 1a′′
2

and 2a′′

2 orbitals in the D3h description becomes 1b1 and 3b1, respectively, and the doubly

degenerate 1e′′ shell splits into the 1a2 and 2b1 components (see Fig. 3.14 for a pictorial

representation of the orbitals). Although the electronic structure of trimethylenemethane

has been well understood for decades (cf., e.g., Ref. [213] and references therein), an ac-

curate characterization of its triplet ground state and low-lying singlet states and the en-

ergy separation between them continues to represent a significant challenge to quantum

chemistry approaches [80, 88, 153, 154, 214–239]. The D3h-symmetric triplet ground state,

designated as X 3A′

2 (in a C2v description adopted in this study, X 3B2), which is domi-

nated by the |{core}(1a′′

2)2(1e′′

1)1(1e′′

2)1| configuration (in C2v, |{core}(1b1)2(1a2)1(2b1)1|), is

relatively easy to describe, but the next two states, which are the nearly degenerate sin-

glets, are not. These states undergo Jahn–Teller distortion that lifts their exact degeneracy

in a D3h description, splitting the two states. The lower of the two singlets, character-

ized by a Cs-symmetric minimum that can be approximated by a twisted C2v structure

and is usually designated as the open-shell singlet A 1B1 state and the second state is a

C2v-symmetric multi configurational state referred to as B 1A1. The first state emerges

from the |{core}(1b1)2(1a2)1(2b1)1| configuration, while the second one is dominated by the

|{core}(1b1)2(1a2)2| and |{core}(1b1)2(2b1)2| closed-shell determinants. The open-shell sin-

glet state A 1B1, after the Jahn–Teller distortion is lower in energy compared to the B 1A1

state, but it has not been observed experimentally due to unfavorable Franck–Condon fac-

148

tors [229, 240], so we do not consider it in this work. The second singlet state B 1A1, on

the other hand, has been detected in photoelectron spectroscopy experiments reported in

Refs. [240, 241], which located it at 16.1 ± 0.1 kcal/mol above the X 3A′

2 ground state. Thus,

following our group’s previous deterministic, active-orbital-based, CC(P ;Q) work [88], the

state-of-the-art DEA- and DIP-EOMCC computations with up to 4p-2h and 4h-2p excita-

tions reported in Refs. [80, 153, 154], and the semi-stochastic CC(P;Q) calculations reported

in Ref. [102], we focused on the D3h-symmetric X 3A′

2 ground state and the C2v-symmetric

B 1A1 singlet state and the adiabatic singlet–triplet gap between them (see, Fig. 4.8 for a

schematic illustration of the Jahn–Teller splitting in trimethylenemethane and the singlet–

triplet gap targeted). Similar to our previous work, we utilized the geometries optimized

using the the spin-flip density functional theory (SF-DFT) and the 6-31G(d) basis reported

in Ref. [231]. The purely electronic singlet–triplet gap derived from the experiments, ob-

tained by subtracting the zero-point vibrational energy correction ∆ZPVE resulting from the

SF-DFT/6-31G(d) calculations reported in Ref. [231] from the experimental B 1A1 − X 3A′

2

gap determined in Refs. [240, 241], is 18.1 kcal/mol. However, this estimate depends on

the source of the information about the ∆ZPVE correction. For example, if one replaces

the ∆ZPVE value obtained in the SF-DFT/6-31G(d) calculations reported in Ref. [231] by

its CCSD(T)/6-311++G(2d,2p) estimate and accounts for the core polarization effects de-

termined with the help of the CCSD(T)/cc-pCVQZ computations, combining the resulting

information with the experimental B 1A1 − X 3A′

2 separation determined in Refs. [240, 241],

the purely electronic, experimentally derived, adiabatic ∆ES-T gap increases to 19.4 kcal/mol

[237]. The DEA-EOMCC(4p-2h)/6-31G(d) and DIP-EOMCC(4h-2p)/6-31G(d) approaches

using the SF-DFT/6-31G(d) geometries predict this gap to be 19.856 kcal/mol and 19.907

kcal/mol, respectively. The remainder of this section describes how the DEA-EOMCC(P)

and DIP-EOMCC(P) employing the 6-31G(d) basis set perform in this challenging situation.

Following our previous semi-stochastic CC(P;Q) approach [102], we used the i-CISDTQ-MC

wave function propagation for constructing the P spaces. Based on our previous experience

149

in semi-stochastic methods, replacing FCIQMC propagations with CISDTQ-MC will not

affect the rate of convergence of our DEA-EOMCC(P) and DIP-EOMCC(P) approaches,

while offering additional computational savings in the QMC part.

In case of the DEA-

EOMCC calculations, we used the TMM2+ dication as the closed-shell reference and for

the DIP-EOMCC calculations, we used the TMM2− dianion as the reference determinant.

The leading 4p-2h (4h-2p) determinants captured during the i-CISDTQ-MC propagations

at various time steps τ are the determinants of B1 symmetry with Sz = 1 in the case of

the X 3A′

2 state, and the Sz = 0 A1 symmetric determinants in case of the B 1A1 state. For

the CIQMC propagations we used the ROHF reference for the X 3A′

2 state and the RHF

reference for B 1A1 state. The one- and two-body integrals used in the DEA-EOMCC (DIP-

EOMCC) calculations are extracted from the TMM2+ (TMM2−) system. All the results of

the semi-stochastic calculations are reported in Table 4.7 and Figs. 4.9 and 4.10. We first

discuss the results of our DEA-EOMCC(P) calculations in the next paragraph and then the

following paragraph contains the results of DIP-EOMCC(P) computations.

The results reported in Table 4.7 and Figure 4.9 demonstrate that, for both the X 3A′
2

and B 1A1 states, the large errors which range from 19 to 27 millihartree at τ = 0 get reduced

to half its value with just 1% of all 4p-2h determinants in the stochastically determined P

spaces captured by the i-CISDTQ-MC runs which happens at 10000 MC iteration. After the

additional 20000 MC time steps, which results in capturing 7–8% of all 4p-2h determinants,

the errors go down to ∼3 millihartree for both states. This error in the energies of the two

states reaches the sub-millihartree regime when the underlying P spaces contain 18–26% of

the 4p-2h determinants and this happens at 50000 MC iteration. For the B 1A1 − X 3A′
2

singlet–triplet gap, ∆ES-T, the ∼5 kcal/mol error relative to DEA-EOMCC(4p-2h) at τ = 0

quickly gets reduced to 1 kcal/mol at 10000 MC iterations. By the time we reach 20000

QMC iterations the singlet–triplet gap is only 0.349 kcal/mol away from the target 19.856

kcal/mol value obtained with DEA-EOMCC(4p-2h).

From Table 4.7 and Figure 4.10 we can see that the DIP-EOMCC(P |τ =0) calculations,

150

where the Rµ,4h-2p components are completely neglected, produce very large, 26–30 milli-

hartree, errors for both the states relative to the parent DIP-EOMCC(4h-2p) method. As

soon as we propagate a little in imaginary time i.e., after only 10000 QMC iterations the

DIP-EOMCC(P ) calculated energies for the X 3A′

2 and B 1A1 states are 4.259 and 9.124

millihartree away from their target values. This is a significant improvement, especially

considering the fact that the P spaces at this time step contain only 7.3% of 4h-2p determi-

nants for the X 3A′

2 state and 7.7% of 4h-2p determinants in case of the B 1A1 state. The

B 1A1 − X 3A′

2 gap at this point is ∼3 kcal/mol away from its DIP-EOMCC(4h-2p) predicted

target of 19.907 kcal/mol. After 30000 QMC iterations, where the P spaces for the singlet

and the triplet states contain about 30% and 16% of 4h-2p determinants respectively. At

this point the error in the singlet–triplet gap is less than 1 kcal/mol and the underlying

electronic states are only 1–3 millihartree in error.

151

Table 4.7 Convergence of the DEA-EOMCC(P ) and DIP-EOMCC(P ) energies [abbreviated as DEA(P ) and DIP(P ), respec-
2 and B 1A1 states of TMM, as described by the 6-31G(d) basis set of Ref. [178], and of the corresponding
tively] of the X 3A′
adiabatic singlet–triplet (S–T) gaps toward their parent DEA-EOMCC(4p-2h) and DIP-EOMCC(4h-2p) values. The geometries
2 and B 1A1 states, optimized using the SF-DFT/6-31G(d) calculations, were taken from Ref. [231]. The P spaces
of the X 3A′
used in the DEA-EOMCC(P ) calculations were defined as all 2p and 3p-1h determinants and subsets of 4p-2h determinants
extracted from the i-CISDTQ-MC propagations with δτ = 0.0001 a.u. The P spaces used in the DIP-EOMCC(P ) calculations
were defined as all 2h and 3h-1p determinants and subsets of 4h-2p determinants extracted from the i-CISDTQ-MC propaga-
tions with δτ = 0.0001 a.u. In all the post-SCF calculations, the lowest core orbitals of the carbon atoms were kept frozen and
the spherical components of the carbon d orbitals were employed throughout.

MC
Iters. DEA(P )a %(4p-2h)b DEA(P )a %(4p-2h)b

X 3A′
2

DEA-EOMCC(P )
B 1A1

0
4000
10000
20000
30000
40000
50000
80000
∞

19.933g
16.852
10.967
6.092
3.274
1.583
0.727
0.048

0.0
0.4
1.4
3.6
6.9
11.8
18.3
53.9
−155.399202i

27.755g
19.554
12.653
6.648
3.738
1.901
0.843
0.092

0.0
0.3
1.4
4.1
8.4
15.4
26.0
74.3
−155.367559i

S–T Gap
∆Ec
4.909g
1.695
1.058
0.349
0.291
0.200
0.073
0.028
19.856j

X 3A′
2

DIP-EOMCC(P )
B 1A1

DIP(P )d %(4h-2p)e DIP(P )d %(4h-2p)e
26.513h
8.053
4.259
1.747
2.649
1.526
1.110
0.007

0.0
3.8
7.3
13.9
15.9
22.3
28.8
54.7
−155.399528k

0.0
2.1
7.7
18.5
30.4
43.0
54.0
61.9
−155.367804k

29.721h
16.608
9.124
3.720
1.080
0.229
0.044
0.001

S–T Gap
∆Ef
2.013h
5.368
3.053
1.238
−0.985
−0.814
−0.669
−0.004
19.907l

2 state and the Sz = 0 4p-2h determinants of the A1 symmetry in case of the B 1A1 state].

a Unless otherwise stated, all energies are reported as errors relative to DEA-EOMCC(4p-2h) in millihartree.
b The %(4p-2h) values are the percentages of 4p-2h determinants captured during the i-CISDTQ-MC propagations [the Sz = 1 4p-2h determinants
of the B1 symmetry in case of the X 3A′
c Unless otherwise specified, the singlet–triplet gaps ∆E = B 1A1 − X 3A′
d Unless otherwise stated, all energies are reported as errors relative to DIP-EOMCC(4h-2p) in millihartree.
e The %(4h-2p) values are the percentages of 4h-2p determinants captured during the i-CISDTQ-MC propagations [the Sz = 1 4h-2p determinants
of the B1 symmetry in case of the X 3A′
f Unless otherwise specified, the singlet–triplet gaps ∆E = B 1A1 − X 3A′
g Equivalent to DEA-EOMCC(3p-1h).
h Equivalent to DIP-EOMCC(3h-1p).
i Total DEA-EOMCC(3p-1h) energy in hartree.
j The DEA-EOMCC(4p-2h) singlet–triplet gap in kcal/mol.
k Total DIP-EOMCC(3h-1p) energy in hartree.
l The DIP-EOMCC(4h-2p) singlet–triplet gap in kcal/mol.

2 state and the Sz = 0 4h-2p determinants of the A1 symmetry in case of the B 1A1 state].

2 are reported as errors relative to DEA-EOMCC(4p-2h) in kcal/mol.

2 are reported as errors relative to DIP-EOMCC(4h-2p) in kcal/mol.

152

Figure 4.8 Jahn-Teller distortion in the trimethylenemethane molecule. At the geometry of
the D3h-symmetric triplet ground state (shown in blue), trimethylenemethane has a doubly
degenerate singlet excited state. Due to Jahn–Teller distortion these states split into an
open-shell singlet state A 1B1 (shown in green) and a multi-configurational singlet state B
1A1 (shown in red). Although the A 1B1 state becomes the first excited state, it is not
observed experimentally due to unfavorable Franck–Condon factors. Thus, we calculate the
singlet–triplet gap between the ground triplet and the second excited singlet state B 1A1.

153

2 and (b) B 1A1 states
Figure 4.9 Convergence of DEA-EOMCC(P ) energies of (a) X 3A′
of TMM, as described by the 6-31G(d) basis set, and (c) of the corresponding adiabatic
singlet–triplet gap towards their parent DEA-EOMCC(4p-2h) values.

Figure 4.10 Convergence of DIP-EOMCC(P ) energies of (a) X 3A′
2 and (b) B 1A1 states
of TMM, as described by the 6-31G(d) basis set, and (c) of the corresponding adiabatic
singlet–triplet gap towards their parent DIP-EOMCC(4h-2p) values.

154

01224(a)X 3A′2020406080MC Iterations (103)01224Error rel. to DEA-EOMCC(4p-2h) (mEh)(b)B 1A1020406080MC Iterations (103)−20246Error rel. to DEA-EOMCC(4p-2h) (kcal/mol)(c)B 1A1−X 3A′201224(a)X 3A′2020406080MC Iterations (103)01224Error rel. to DIP-EOMCC(4h-2p) (mEh)(b)B 1A1020406080MC Iterations (103)−20246Error rel. to DIP-EOMCC(4h-2p) (kcal/mol)(c)B 1A1−X 3A′2CHAPTER 5

CONCLUDING REMARKS AND FUTURE OUTLOOK

In this dissertation, we have discussed some of the recent advances in the CC and EOMCC

theories to which the author of this thesis has had the opportunity to contribute during

his doctoral work in Professor Piotr Piecuch’s group. In particular, we have explored the

semi-stochastic, CIQMC-driven, CC(P;Q) framework and and its extension to ground and

excited states of open-shell systems. We have also discussed the semi-stochastic formulation

of the particle nonconserving EA, IP, DEA, and DIP EOMCC frameworks, again taking

advantage of CIQMC.

In the first part of this dissertation, we have discussed the CC theory as one of the

best ways of approaching the many-electron correlation problem in molecular systems in a

computationally tractable manner. After highlighting the advantages of the CC theory and

some of its key challenges, and discussing its extensions to excited and open-shell states via

the EOMCC formalism, we have focused on the CC(P;Q) methodology, where the flexibility

in defining the P and Q excitation spaces allows one to obtain highly accurate energet-

ics equivalent or very close to those obtained with the high-level CCSDT, CCSDTQ, etc.

methods and their EOMCC extensions at small fractions of the computational costs, even

when higher–than–two-body components of the cluster and EOM excitation operators be-

come large, nonperturbative, and strongly coupled to their lower-rank components. We have

also discussed the particle nonconserving EOMCC formalisms of the EA, IP, DEA, and DIP

EOMCC types that offer an elegant and orthogonally spin-adapted approach to open-shell

species within the SR framework. This includes problems involving electronic excitation

spectra of radicals and singlet–triplet gaps in biradicals. We have also briefly reviewed the

stochastic FCIQMC wave function propagation and sampling approach and its truncated

CIQMC counterparts.

In the second part of this dissertation, we have focused on the semi-stochastic CC(P;Q)

methodology that combines the flexible deterministic CC(P;Q) framework with the stochas-

155

tic CIQMC wave function propagations to automatically identify the P and Q spaces needed

in the CC(P;Q) calculations without any reference to the previously exploited user- and

system-dependent active-orbital concepts. By examining the excitation spectra of CH+,

CH, and CNC, we have demonstrated the ability of the semi-stochastic CC(P;Q) methodol-

ogy to converge ground and excited states, including challenging non-singlet excited states

of open-shell systems, out of the early stages of CIQMC propagations. Many of the excited

states examined in this dissertation are dominated by two-electron transitions, making them

difficult to describe by the majority of the existing quantum chemistry approaches. We have

also applied the semi-stochastic CC(P;Q) methodology to converge the CCSDT energet-

ics of the lowest singlet and triplet states of several biradical species, including methylene,

(HFH)−, cyclobutadiene, cyclopentadienyl cation, and trimethylenemethane, and the cor-

responding singlet–triplet gaps, which is another challenging problem for many quantum

chemistry methods because one has to balance the predominantly weakly correlated triplet

states with the multiconfigurational, often strongly correlated, singlet states.

In the third part of this dissertation, we have extended the semi-stochastic ideas to the

particle nonconserving EOMCC formalisms of the EA, IP, DEA, and DIP types. We have

demonstrated that by combining CIQMC wave function propagations with the determinis-

tic EA- and IP-EOMCC frameworks, one can converge the high-level EA-EOMCC(3p-2h)

and IP-EOMCC(3h-2p) energetics at small fractions of the computational costs out of the

early stages of CIQMC propagations, even in the presence of strong 3p-2h and 3h-2p cor-

relations. We have done this by studying the C2N, CNC, N3, and NCO radicals, where we

have calculated the low-lying doublet states and the associated adiabatic excitation energies.

We have then extended the semi-stochastic CIQMC-driven ideas to the DEA-EOMCC and

DIP-EOMCC formalisms, which has allowed us to converge the high-level DEA-EOMCC(4p-

2h) and DIP-EOMCC(4h-2p) energetics in an automated and efficient manner using small

fractions of 4p-2h and 4h-2p amplitudes, again identified in the early stages of the under-

lying CIQMC runs. To illustrate the efficiency of our semi-stochastic DEA-EOMCC(4p-2h)

156

and DIP-EOMCC(4h-2p) approaches, we have investigated the ground and three low-lying

singlet excited states of methylene, along with the corresponding singlet–triplet gaps, and

the trimethylenemethane biradical, where we have determined the ground triplet and the

low-lying and multiconfigurational singlet states and the corresponding singlet–triplet gap.

While this dissertation has explored the semi-stochastic CC(P;Q) approach aimed at

CCSDT/EOMCCSDT and the EA/IP/DEA/DIP EOMCC methodologies aimed at a highly

accurate description of 3p-2h/3h-2p/4p-2h/4h-2p correlations, showing a lot of promise, there

are still several areas that need to be examined. For example, recent work on the active

space [84] and QMC-driven [101] CC(P;Q) frameworks targeting the ground-state CCSDTQ

energetics have shown encouraging results, and hence it would be useful to extend the semi-

stochastic CC(P;Q) methodology investigated in this dissertation to target the EOMCCS-

DTQ energetics of excited states. The selected-CI-driven [120] and adaptive [121] CC(P;Q)

approaches have shown promising results in converging CCSDT energies as well, so it would

be interesting to extend them to target EOMCCSDT or even CCSDTQ/EOMCCSDTQ.

In the case of the particle nonconserving EOMCC approaches, the CC(P;Q)-type moment

corrections have not been implemented yet. It would, thus, be very beneficial to extend the

noniterative CC(P;Q) corrections to the particle nonconserving EOMCC schemes. Based

on the significant acceleration toward the desired high-level CC/EOMCC energetics these

corrections offer in the particle conserving cases, one may anticipate that they will be very

effective in the EA-EOMCC, IP-EOMCC, DEA-EOMCC, and DIP-EOMCC approaches as

well. Combining the particle nonconserving EOMCC formalisms with the selected-CI-based

or adaptive selections of higher-order correlations, examined in this work using CIQMC-

driven ideas, is another interesting aspect worth investigation.

In analogy to other post-SCF ab initio quantum chemistry approaches, in the longer-

term, one can also think of extending the applicability of the methods developed in this

dissertation to larger and more complex molecular systems containing hundreds of electrons

and dozens of non-hydrogen atoms using techniques such as the fragment molecular orbital

157

(FMO) approach [259–261], the effective fragment potential (EFP) embedding scheme [262]

and its merger with FMO abbreviated as EFMO [263–265], the cluster-in-molecule frame-

work [266–269], and, ultimately, the quantum mechanics/molecular mechanics [270–274] and

polarizable continuum [275, 276] models, to name a few examples. By doing so, especially

when combined with code parallelization across multiple multi-core nodes, at least some of

our semi-stochastic CC(P;Q) and EOMCC algorithms developed in this dissertation can be

made applicable to studies of molecular electronic excitation spectra and singlet–triplet gaps

in solution and condensed phases. Clearly, additional molecular applications of the semi-

stochastic CC(P;Q) and EA/IP/DEA/DIP EOMCC approaches investigated in this work

would be very useful too.

158

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