HIGH-LEVEL COUPLED-CLUSTER ENERGETICS USING
SELECTED-CONFIGURATION-INTERACTION-DRIVEN AND ADAPTIVE
MOMENT EXPANSIONS

By

Karthik Gururangan

A DISSERTATION

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

Chemistry—Doctor of Philosophy

2025

ABSTRACT

It is well established that size-extensive approaches based on the exponential wave function

ansatz of coupled-cluster (CC) theory and their extensions to open-shell, multiconfigura-

tional, and excited states are among the best ways of treating many-electron correlation ef-

fects. This is especially true in applications involving structural and spectroscopic properties

of molecular systems, chemical reaction pathways, noncovalent interactions, and photochem-

istry. In all these and similar cases, the CC hierarchy, including CCSD, CCSDT, CCSDTQ,

and so on, and its equation-of-motion (EOM) and linear-response extensions rapidly con-

verge to the exact, full configuration interaction (CI) limit. Unfortunately, the CCSDT,

CCSDTQ, and similar methods, needed to achieve a quantitative description, incur a steep

increase in computational costs, rendering their application to larger many-electron systems

prohibitively expensive. To address this challenge, this dissertation explores the development

of computationally practical alternative approaches based on the CC(P ;Q) and externally

corrected CC (ec-CC) frameworks, which allow us to study challenging chemical problems,

such as molecular bond breaking, biradicals, transition states, and excited states dominated

by two- and other many-electron excitations, while avoiding the well-known failures of per-

turbative CC approximations in the presence of electronic quasi-degeneracies.

The first part of this dissertation introduces two novel CC(P ;Q) approaches capable of

rapidly converging high-level CC/EOMCC energetics of the CCSDT/EOMCCSDT, CCS-

DTQ/EOMCCSDTQ, and similar types, at tiny fractions of the computational costs in an

automated fashion, even in cases of stronger correlations. The first methodology combines

the CC(P ;Q) moment expansions with the information provided by selected CI wave func-

tions obtained using the algorithm abbreviated as CIPSI to systematically recover the re-

sults corresponding to any desired level of CC/EOMCC theory. The second approach, called

adaptive CC(P ;Q), eliminates the reliance on active orbitals or external non-CC informa-

tion adopted in previous formulations of CC(P ;Q) by executing a sequence of CC(P ;Q)

calculations aimed at converging high-level CC/EOMCC energetics guided solely by the

mathematical structure of the moment expansions. We demonstrate the effectiveness of

both the CIPSI-driven and adaptive CC(P ;Q) methodologies through a number of molec-

ular applications aimed at recovering the full CCSDT/EOMCCSDT energetics when the

noniterative triples corrections to CCSD/EOMCCSD struggle or fail. For the CIPSI-driven

approach, we examine the dissociation of F2, the automerization of cyclobutadiene, and

the vertical excitation spectrum of CH+. The adaptive CC(P ;Q) approach is tested on the

stretched F2 and F+

2 molecules, the automerization of cyclobutadiene, singlet–triplet gaps in

organic biradicals, the degenerate Cope rearrangement of bullvalene, and the ground- and

excited-state potential energy surfaces of water along the O–H bond-breaking coordinate. To

illustrate the computational advantages of the adaptive CC(P ;Q) approach, as implemented

in the open-source CCpy software package, we also discuss the CPU timings characterizing

our calculations for cyclobutadiene and CnH2n+2 linear alkanes with n = 1–8.

In the final part of this dissertation, we implement and test a new family of ec-CC

approaches designed to recover the exact, full CI, energetics. These methods leverage infor-

mation about higher-order correlations provided by selected CI wave functions and adopt

moment expansions, similar to those used in CC(P ;Q), to account for missing higher-order

correlation effects. In this new class of ec-CC approaches, termed ec-CC-II, one solves CCSD-

like equations for the one- and two-body clusters in the presence of their three-body (T3) and

four-body (T4) counterparts extracted from the underlying CI wave function, after discarding

T3 and T4 amplitudes corresponding to CI coefficients that are zero. In this dissertation, we

focus on the ec-CC-II approach using T3 and T4 clusters extracted from CIPSI calculations,

along with its ec-CC-II3 extension, which corrects the ec-CC-II energetics for missing T3 cor-

relation effects using the appropriately defined CC(P ;Q)-like moment expansions. To assess

the performance of the CIPSI-based ec-CC-II and ec-CC-II3 methodologies, we apply them

to the challenging symmetric dissociation in water, where even high-level CC methods, such

as CCSDTQ, struggle to achieve a full-CI-level description.

Copyright by
KARTHIK GURURANGAN
2025

This thesis is dedicated to my grandparents, Gururangan Gopalaswamy, Vasantha
Gururangan, Savitiri Subramanian, and C.N. Subramanian, whose unwavering love and
support have been a constant presence throughout my life.

v

ACKNOWLEDGMENTS

To begin with, I would like to thank my doctoral advisor, Professor Piotr Piecuch. Since my

undergraduate days, I have been drawn to the idea of working as a theorist, and I am deeply

grateful to Professor Piecuch for giving me the opportunity to become one by introducing me

to many-body theory and providing challenging problems for me to work on, while offering

constant guidance and intellectual inspiration along the way. Above all, I am thankful for

his emphasis on precision and attention to detail in science, as well as for demonstrating

the work ethic and discipline required to excel at the highest level. Under his mentorship, I

was able to explore a variety of topics in coupled-cluster theory, which allowed me to grow

significantly as a scientist and gain confidence in tackling complex problems.

I would also like to thank the remaining members of my guidance committee, namely,

Professor Katharine C. Hunt, Professor Angela K. Wilson, and Professor Warren F. Beck,

for their support, advice, and patience over the course of my doctoral studies.

I would like to express my sincere gratitude to Professor Elad Harel for introducing me

to physical chemistry and giving me the opportunity to work on spectroscopic simulations

as an M.S. student in his laboratory. Professor Harel helped foster my interest in quantum

mechanics and played a pivotal role in shaping the beginning of my graduate journey. In

particular, he instilled in me an active approach to learning, where understanding comes

through implementation, a principle that proved invaluable throughout my doctoral studies.

I would like to thank our collaborator, Professor Achintya Kumar Dutta from IIT Mum-

bai, for many helpful discussions about cost-reduced and relativistic coupled-cluster methods.

I also want to thank Dr. Anthony Scemama, and the rest of the team at the CNRS in Univer-

sit´e Paul Sabatier (Toulouse III), for developing the CIPSI code in Quantum Package, which

was used extensively throughout my research. I am especially grateful to Dr. Scemama for

several helpful discussions regarding the design of efficient sparse coupled-cluster codes.

I would also like to extend my thanks to both former and current members of the Piecuch

group, with whom I have had the pleasure of working, including Dr. Jun Shen, Dr. J. Emiliano

vi

Deustua, Dr. Ilias Magoulas, Dr. Stephen Yuwono, Dr. Arnab Chakraborty, Ms. Swati S.

Priyadarsini, Mr. Tiange Deng, and Mr. Agnibha Hanra.

I feel fortunate that we have

remained good friends over the years. My sincere gratitude goes to Dr. J. Emiliano Deustua,

my first mentor, for helping me navigate the beginning of graduate school and introducing me

to scientific programming. I shudder to think how differently my doctoral studies may have

turned out if I did not pick the office in Room 87, and I am grateful that we have remained

friends and collaborators to this day. I am also deeply thankful to Dr. Ilias Magoulas for

his invaluable collaboration during my first project on externally corrected coupled-cluster

methods and for always providing kind and thoughtful answers to all of my questions. His

enthusiasm for spin-adaptation and diatomic spectroscopy is truly contagious. Finally, I

would like to express my profound gratitude to Dr. Stephen Yuwono for his friendship and

for engaging in countless discussions, ranging from excited states and Wigner–Wittmer rules

to the occasional brain teaser about many-body theory. I have learned a lot from him.

My experience in graduate school would not have been the same without the support

of my friends. Among others, I would like to thank Samanta Dalchand, Elizabeth Dalc-

hand, and her husband, James Tavornwattana, Siddharth Moghe, Ashvin Nair, Saavan Patel,

Aditya Pradhan, Siddharth Ramakrishnan, Jackson Rigley, Mustafa Sohail, and Shawn Zhao

(n´e Irgen-Gioro) for enriching my life with many good memories, experiences, and laughs. I

would also like to thank my friends at Michigan State University, especially Nathan Jansen

and Mejdi Mogannam, for their support through the thick and thin of graduate school.

Above all, I am most grateful to my family. To my parents, Raghu Gururangan and Gay-

atri Gururangan, grandparents, Gururangan Gopalaswamy, Vasantha Gururangan, Savitiri

Subramanian, and C.N. Subramanian (deceased), brother, Kapil Gururangan, and his wife,

Charisma Hooda, thank you all for your unending love and support from near and far while

I completed my doctoral studies. To Naomi, words cannot do justice to how much you have

done for me over the last six years. Thank you for giving me the strength to keep going,

especially during the hardest times.

vii

TABLE OF CONTENTS

CHAPTER 1

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

1

CHAPTER 2

THEORETICAL BACKGROUND . . . . . . . . . . . . . . . . .

16

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

Extension to Excited Electronic States

. . . . . . . . . . . . . . . . . . .
2.2 CC(P;Q) Moment Expansions . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Overview of Selected Configuration Interaction . . . . . . . . . . . . . . .
2.4 Externally Corrected Coupled-Cluster Methodology . . . . . . . . . . . .

CHAPTER 3

MERGING THE CC(P;Q) FORMALISM WITH SELECTED
CONFIGURATION INTERACTION . . . . . . . . . . . . . . . .
3.1 Theoretical and Algorithmic Details . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Numerical Examples

16
25
30
34

42
43
49

CHAPTER 4

THE ADAPTIVE CC(P;Q) FRAMEWORK . . . . . . . . . . . .
4.1 Theory and Algorithmic Details . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Numerical Examples
4.3 Computational Performance of Adaptive CC(P;Q)

67
68
76
. . . . . . . . . . . . 108

CHAPTER 5

EXTERNALLY CORRECTED COUPLED-CLUSTER METHODS
USING CIPSI AND MOMENT EXPANSIONS . . . . . . . . . . . 116
5.1 Theory and Algorithmic Details . . . . . . . . . . . . . . . . . . . . . . . 116
5.2 Symmetric Dissociation in the Water Molecule . . . . . . . . . . . . . . . 118

CHAPTER 6

CONCLUDING REMARKS AND FUTURE OUTLOOK . . . . . 123

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

APPENDIX A

CCPY: AN OPEN-SOURCE COUPLED-CLUSTER PACKAGE
IMPLEMENTED IN PYTHON . . . . . . . . . . . . . . . . . . . 149

APPENDIX B

DERIVATION OF THE CC(P;Q) MOMENT EXPANSIONS . . 158

APPENDIX C

EFFICIENT ALGORITHM FOR THE CC(P) AND EOMCC(P)
APPROACHES AIMED AT CONVERGING CCSDT AND EOM-
CCSDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

APPENDIX D

CC/EOMCC AND FCI ENERGIES FOR THE GROUND- AND
EXCITED-STATE POTENTIAL CUTS OF WATER . . . . . . . 188

viii

CHAPTER 1

INTRODUCTION

The primary objective of quantum chemistry is to solve the many-electron Schr¨odinger equa-

tion for a molecular system. The time-independent Schr¨odinger equation, H|Ψµ⟩ = Eµ|Ψµ⟩,

represents an eigenvalue problem that determines the electronic wave functions |Ψµ⟩ and

their corresponding energies Eµ, where µ = 0 denotes the ground state and µ > 0 represents

excited states. The nonrelativistic electronic Hamiltonian, expressed in atomic units,

H =

N
X

i=1

−

1
2

∇2

i −

M
X

A=1

ZA
|ri − RA|

!

+ X

i<j

1
|ri − rj|

,

(1.1)

describes a system consisting of N electrons (with spatial coordinates ri) and M nuclei

(with fixed coordinates RA and charges ZA) interacting electrostatically within the Born–

Oppenheimer approximation. Extensions of Eq. (1.1), which include the scalar-relativistic

and spin-orbit coupling effects, can be derived starting from the one-electron Dirac equation

(see Ref. [1] and references therein for further details), but in this dissertation, we focus

on a nonrelativistic description of chemical systems. From a many-body perspective, the

electronic Hamiltonian consists of one-body terms, which include the electronic kinetic energy

and electron-nuclear attraction, and two-body terms arising from electron-electron repulsion.

Although these interactions appear simple, the presence of two-body terms in Eq. (1.1)

couples the coordinates of all electrons, rendering the many-electron problem analytically

intractable. As a result, numerical approaches have long been recognized as the most effective

means of solving the many-electron Schr¨odinger equation.

One of the earliest breakthroughs in solving the many-electron problem came in 1927, just

one year after the Schr¨odinger equation was discovered, when Heitler and London explained

covalent bonding in H2 using a valence-bond ansatz [2]. Shortly thereafter, Slater introduced

a determinantal representation for many-fermion states [3] using an antisymmetrized product

of single-particle functions, which are called molecular spinorbitals. Building on this idea,

Fock applied the variational theorem in conjunction with a Slater determinant ansatz for the

1

 
many-electron wave function, leading to the formulation of the Hartree–Fock (HF) method

[4]. In the HF approach, the spinorbitals are determined as eigenfunctions of an effective one-

electron operator called the Fock operator, which incorporates electron-electron interactions

in an approximate fashion. Initially, the HF equations were numerically intractable for all

but the simplest systems. However, in 1951, Hall [5] and Roothaan [6] introduced a linear

combination of atomic orbitals ansatz for the spinorbitals in the HF method, recasting the

complex HF equations into a simple matrix eigenvalue problem within a basis of atomic

orbitals. In the Roothaan–Hall formulation, the HF equations are solved by diagonalizing

the Fock matrix in this basis, yielding the expansion coefficients and orbital energies that

define the resulting spinorbitals, which are iteratively determined in a self-consistent field

(SCF) procedure. While the HF method typically recovers most of the total electronic energy,

it is well known [7] that the many-electron correlation effects missing from the HF mean-

field solution are essential for accurately describing chemical transformations and properties.

Consequently, a central challenge in quantum chemistry is the development of more advanced

methodologies that systematically recover these crucial post-HF correlation effects.

Today, electronic structure calculations of this type are pivotal in investigations across

chemistry, physics, and materials science, with a number of widely used computational codes

making these techniques readily accessible to both theoreticians and experimentalists. How-

ever, despite nearly a century of advances, many contemporary quantum chemical methods

struggle to accurately describe quasi-degenerate electronic states in instances involving mul-

tiple bond breaking, biradical and polyradical species, and excited states dominated by

two- and other many-electron transitions. A defining feature of these challenging systems is

the presence of significant multireference (MR), or nondynamical, correlation effects, where

multiple quasi-degenerate Slater determinants contribute substantially to the electronic wave

function. Such effects commonly arise in bond-breaking processes, open-shell singlet states,

and various other low-spin configurations. In addition to capturing MR correlations, it is

also essential to account for dynamical correlation effects, which are associated with the

2

motion of electrons avoiding one another due to Coulombic repulsion, and typically man-

ifest themselves in numerous Slater determinants with small weight in the electronic wave

function. In practice, achieving highly accurate descriptions of molecular potential energy

surfaces (PESs), properties, and electronic excitation spectra requires a balanced and accu-

rate treatment of both types of correlation.

In principle, all of the aforementioned challenges are addressed by resorting to the full

configuration interaction (FCI) approach, which provides exact solutions to the electronic

Schr¨odinger equation, described using a finite basis of spinorbitals, by diagonalizing the

Hamiltonian within the full many-electron Hilbert space. The numerically exact energies

Eµ and wave functions |Ψµ⟩ can then be obtained by performing FCI calculations using

a single-particle basis set that is large enough to converge the complete basis set (CBS)

limit, or by extrapolating the CBS limit from results obtained in smaller basis sets. Un-

fortunately, the dimensionality of the many-electron Hilbert space, which for an N -electron

system characterized by K orbitals and total spin S is given by the Weyl–Paldus formula

[8, 9], 2S+1
K+1

(cid:16) K+1
N/2−S

(cid:17)(cid:16) K+1

(cid:17)

N/2+S+1

, grows rapidly with the number of electrons and orbitals, ren-

dering the FCI approach inapplicable to systems with more than a few correlated electrons,

even when smaller basis sets are employed. For example, in the case of a benzene molecule

described using the cc-pVDZ basis set [10], which contains 30 correlated electrons and 108

orbitals (using the frozen-core approximation), the singlet FCI space contains ∼ 1034 con-

figurations. To put this number in perspective, the largest FCI calculation to date was

performed for the propane molecule in a minimum STO-3G basis set [11], which involved

only 1.31 × 1012 determinants [12]. Needless to say, the routine use of FCI in realistic chem-

ical problems is impossible given our current computational capabilities. Thus, the central

activity in quantum chemistry, which is also a focus of this dissertation, is the development

of practical approximations to FCI that provide an accurate description of a wide range of

many-electron correlation effects without incurring prohibitive computational costs.

Many of the most successful quantum chemical methodologies in this category are based

3

on the exponential wave function ansatz [13, 14] of the coupled-cluster (CC) theory [15–20],

which offers excellent balance between accuracy and computational cost along with several

other desirable mathematical properties. The CC theory initially emerged as a generaliza-

tion of the many-body perturbation theory (MBPT) expansion, where linked wave function

and connected energy diagrams are summed to infinite order, in accordance with the linked

[13, 14, 21, 22] and connected [13, 14] cluster theorems. These two theorems ensure that the

resulting CC methods satisfy several important theoretical conditions. First, by summing

only connected energy diagrams, CC approaches remain size extensive at all levels of approx-

imation, ensuring that accuracy is preserved as the system size increases. Furthermore, due

to the exponential form of the wave operator, the CC wave function is size consistent (or sep-

arable) in the noninteracting limit if the underlying reference state also separates, which is

useful when describing fragmentation phenomena. Most importantly, the CC wave function

captures higher-order correlations via products of lower-rank excitation operators. As long

as the number of strongly correlated electrons is not too large, this structure ensures rapid

convergence toward the exact, FCI, limit while keeping computational costs manageable.

Within the CC framework, there exists two different formalisms depending on the di-

mensionality of the model space used to provide a zeroth-order description of the many-

electron wave function. If the model space is one-dimensional, spanned by a single Slater

determinant or configuration state function serving as the Fermi vacuum, the resulting CC

methodology is classified as single-reference (SR). In the SRCC theory, which is historically

the oldest type of CC formalism, the remaining determinants or configurations entering the

many-electron wave function are generated via particle–hole excitations out of the Fermi

vacuum. In practice, the Fermi vacuum in SRCC considerations can be obtained using a

number of black-box independent particle models (IPMs), including the HF wave function

in the restricted (RHF), unrestricted (UHF), or restricted open-shell (ROHF) forms, Br¨uck-

ner theory [23–26], or even Kohn–Sham density functional theory (DFT) [27, 28]. This

allows the SRCC calculations to be performed with minimal user-specified input. In con-

4

trast, a MRCC theory adopts a multidimensional model space spanned by several Slater

determinants or configuration state functions, which are chosen to capture the dominant

multiconfigurational content of the target wave functions. The remaining, mostly dynam-

ical, correlations are described using particle–hole excitations out of the individual Fermi

vacua, as in the Hilbert-space MRCC formalism. Alternatively, one can turn to the Fock

space MRCC theory and construct the relevant quasi-degenerate wave functions by starting

from a closed-shell vacuum state and adding (removing) electrons to (from) it. Given our

interest in describing quasi-degenerate electronic states, which often arise when studying

bond dissociations, open shells, and electronic excited states, it may seem more appropriate

to adopt an MRCC formalism. Unfortunately, there exists no unique parameterization for

MRCC wave functions or the wave operators that generate them, and as a result, several

MRCC and MRCC-like ideas exist (see Refs. [29–34] for reviews), which broadly belong to

either the state-universal, valence-universal, or state-selective category. The various MRCC

schemes also require the user-defined active spaces, which must be relatively small in order to

keep the computational cost of these methods manageable, but poor choices may still result

in loss of accuracy. Furthermore, due to mathematical complexities associated with their

numerical stability and convergence [35–41], the genuine MRCC-based methodologies are

currently unable to compete with the ease of use and wider applicability characterizing their

SRCC counterparts. Thus, many highly successful strategies for designing CC approaches

for MR problems adopt the SRCC perspective, recovering nondynamical correlations dy-

namically via inclusion of higher-rank particle–hole excitations out of the Fermi vacuum,

possibly augmented by particle-nonconserving operators (see, for example, Refs. [42–72]).

In the SRCC formalism, the exact N -electron ground-state wave function is expressed

as |Ψ0⟩ = eT |Φ⟩, where |Φ⟩ is a reference determinant serving as a Fermi vacuum, which is

typically the HF state, and T = PN

n=1 Tn is the cluster operator. The many-body components

of the cluster operator, Tn, act on |Φ⟩ to generate the connected contributions to the ground-

state wave function corresponding to n-tuply excited determinants, while products of the Tn

5

clusters resulting from the exponential eT produce the remaining disconnected but linked

contributions to |Ψ0⟩. The CC ansatz is formally exact (i.e., equivalent to FCI) when all

many-body components of the cluster operator are included in the definition of T , however,

in practice, the cluster operator is truncated at a lower many-body rank to generate the

standard hierarchy of CC approximations. The most basic and practical level of CC theory,

obtained when T is truncated at T2, is the CC method with singles and doubles (CCSD)

[73–76], which is characterized by computational steps that scale as n2

on4

u, where no (nu) is

the number of correlated occupied (unoccupied) orbitals in |Φ⟩, or as N 6 with the system

size N . The next two levels, namely the CC approach with singles, doubles, and triples,

abbreviated as CCSDT [77–80], obtained when T is truncated at T3, and the CC method

with singles, doubles, triples, and quadruples, abbreviated as CCSDTQ [81–84], in which T

is truncated at T4, involve the n3

on5

u (N 8) and n4

on6

u (N 10) steps, respectively. Higher-level

CC approaches that include Tn clusters with n > 4 can be analogously defined and come

with increasingly larger computational costs scaling as nn

o nn+2
u

(N 2n+2).

The ground-state CC formulation can also be extended to describe excited electronic

states using the equation-of-motion (EOM) CC formalism [85–89] or its linear-response (LR)

CC [90–98] and symmetry-adapted cluster CI (SAC-CI) [99] analogs. In the EOMCC frame-

work, which is the approach adopted in this dissertation, the exact excited states of the

N -electron system are defined as |Ψµ⟩ = RµeT |Φ⟩, where Rµ = rµ,01 + PN

n=1 Rµ,n (µ > 0) is

a linear excitation operator (1 is an identity operator). The EOMCC framework provides a

transparent route for computing increasingly accurate and systematically improvable excited-

state energetics and properties. The standard series of EOMCC approximations obtained

by truncating Rµ and T at a given many-body rank include the basic EOMCCSD method

[87–89], defined when Rµ (T ) is truncated at Rµ,2 (T2), and its higher-level EOMCCSDT

[100–104] and EOMCCSDTQ [102, 103, 105, 106] counterparts that include the Rµ,3 (T3) and

Rµ,4 (T4) components of Rµ (T ), respectively. The aforementioned EOMCC approximations

involve the computational steps that scale the same as their ground-state counterparts. In

6

analogy to the ground-state case, the approximate EOMCC wave functions retain key the-

oretical features, including size-extensive total energies and size-intensive vertical transition

energies, provided that only one fragment is excited in a noninteracting limit.

It is well established that in the majority of chemical applications, including molecules

near equilibrium geometries, chemical bond rearrangements involving smaller numbers of

strongly correlated electrons, noncovalent interactions, and photochemistry, the conven-

tional CCSD/EOMCCSD, CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ, etc. hierarchy

rapidly converges the exact, FCI, limit such that when one reaches the CCSDT/EOMCCSDT

or CCSDTQ/EOMCCSDTQ levels, the relevant many-electron correlation effects are typ-

ically well described [30, 32]. While the CCSD/EOMCCSD methods are computationally

practical approaches, they tend to provide only a qualitative level of accuracy for weakly cor-

related systems, and often struggle to produce meaningful results in the presence of stronger

correlations, such as those arising from multiconfigurational ground states or excited states

dominated by two-electron transitions. In order to accurately treat MR problems using the

SRCC framework, one needs to include the higher–than–two-body components of the T and

Rµ operators, as in the CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ, and higher-order

approaches. Sadly, the routine usage of the more powerful SRCC techniques, such as CCS-

DT/EOMCCSDT and CCSDTQ/EOMCCSDTQ, is hindered by the respective N 8 and N 10

computational steps, which limits their applicability to few-atom systems and smaller basis

sets. Thus, a key effort in the field of SRCC method development focuses on designing ac-

curate and reliable approximations to CCSDT/EOMCCSDT and CCSDTQ/EOMCCSDTQ

that are capable of reducing the above costs, while being more robust than perturbative

approaches of the CCSD(T) [107, 108], CCSDT-n [109–111], and CC3 [112] types and their

extensions to higher orders [113–116] and excited electronic states [117–121], which are all

known to fail in more MR situations [30, 32, 100, 122–125].

In order to accurately approximate high-level CC/EOMCC energetics with reduced com-

putational costs, even in the presence of substantial electronic quasi-degeneracies and MR

7

correlation effects, we can turn to the CC(P ;Q) framework [126–138]. The CC(P ;Q) for-

malism is based on correcting lower-level CC/EOMCC energetics for missing higher-order

correlation effects in a general fashion using the noniterative and nonperturbative energy ex-

pansions derived using the method of moments of CC equations (MMCC) [122, 123, 126, 139–

150]. In order to do this, each CC(P ;Q) calculation begins by defining one subspace of the

many-electron Hilbert space called the P space, which contains the excited Slater determi-

nants that, in addition to the reference |Φ⟩, dominate the ground or excited state of interest.

The complementary Q space is spanned by a subset of excited determinants absent from the

P space, which are used to correct the CC/EOMCC calculations in the P space [henceforth

abbreviated as CC(P ) for the ground state and EOMCC(P ) for excited states] for the miss-

ing correlation effects of interest. By varying the definitions of the P and Q spaces entering

the CC(P ;Q) framework, we are able to devise a variety of noniterative corrections to both

conventional and unconventional CC/EOMCC calculations. To date, work from our group

has established five different ways of choosing the underlying P and Q spaces, each one

resulting in a different type of CC(P ;Q) methodology.

In the first, conventional, form of CC(P ;Q), the determinants entering the P and Q

spaces are selected purely on the basis of many-body rank. The resulting CC(P ;Q) method-

ologies are equivalent to the left-eigenstate completely renormalized (CR) CC and EOMCC

approaches [146–149, 151–153], which correct the energies obtained in standard CC/EOMCC

calculations (e.g., CCSD/EOMCCSD, CCSDT/EOMCCSDT, etc.) for the missing correla-

tion effects associated with selected Tn and Rµ,n components of T and Rµ. For example, using

the CR-CC(2,3) [146–149], CR-CC(2,3)+Q [153, 154], or CR-CC(2,4) [146, 147, 153, 155]

corrections to CCSD, one can approximately account for the many-electron correlation effects

arising from connected triples or connected triples and quadruples absent in the initial CCSD

computation, respectively. In the case of excited states, one can use the EOM extensions of

the CR-CC formalism, including the CR-EOMCC(2,3) [148, 151] triples correction to EOM-

CCSD or its rigorously size-intensive δ-CR-EOMCC(2,3) modification [152], to account for

8

missing higher-order correlation effects resulting from the neglect of connected triples in the

preceding EOMCCSD calculation. The CR CC/EOMCC triples corrections can be accurate,

even when perturbative CCSD(T)-like approaches fail, such as in problems featuring single

bond breaking [146, 147, 149, 156–158] or excited states dominated by two-electron transi-

tions [159–161], and they can be performed at the similar noniterative n3

on4

u, or N 7, cost

characterizing the CCSD(T) method. That being said, the typical CR CC/EOMCC triples

and quadruples corrections to CCSD/EOMCCSD are not entirely robust in the presence of

stronger nondynamical correlation effects that may arise when studying more challenging

bond rearrangements [126, 129, 134], some biradical or polyradical species [127, 128, 134–

137], and certain excited-state potentials along bond-breaking coordinates [133, 138, 161].

Generally speaking, the aforementioned cases cause problems for any methodology based

on noniteratively correcting the CCSD/EOMCCSD energetics for missing higher-order cor-

relation effects due to the presence of substantial coupling between the lower-order cluster

and excitations components, such as T1 and T2 or Rµ,1 and Rµ,2, and their Tn and Rµ,n

counterparts with n > 2 [126–138].

One can address the concern of coupling the lower- and higher-order components of the T

and Rµ operators by incorporating the leading subsets of higher–than–doubly excited deter-

minants relevant to the problem into the P space and correcting the resulting CC/EOMCC

energetics for the remaining correlations of interest using the complementary Q space. By

doing this, we allow the lower-rank components of the cluster and excitation operators, such

as T1, T2, Rµ,1, and Rµ,2, to relax in the presence of the leading contributions to their higher–

than–two-body Tn and Rµ,n counterparts in the CC(P )/EOMCC(P ) calculations prior to

determining the CC(P ;Q) corrections.

The second variant of the CC(P ;Q) theory employs active orbitals to select chemically

important subsets of higher–than–doubly excited determinants that are included, in addition

to all singly and doubly excited determinants, in the P space, resulting in truncated T and

Rµ operators that are equivalent to those adopted in active-space CC/EOMCC method-

9

ologies [46, 66, 84, 100, 101, 124, 125, 162–169]. The hierarchy of active-orbital-based

CC(P ;Q) schemes includes approaches like CC(t;3), CC(t,q;3), CC(t,q;3,4), and CC(q;4)

[126–129, 131, 132, 138], where the active-space CCSDt/EOMCCSDt [84, 100, 101, 125, 162–

164, 166–168] [in the case of CC(t;3)], CCSDtq/EOMCCSDtq [84, 164, 167, 169] [in the cases

of CC(t,q;3) or CC(t,q;3,4)], or CCSDTq/EOMCCSDTq [131, 132] [in the case of CC(q;4)]

equations are first solved to produce the energies that account for the bulk of the nondynam-

ical and some dynamical correlations, which are then corrected for the remaining dynamical

correlation effects due to missing triples [in the cases of CC(t;3) and CC(t,q;3)], quadruples

[in the case of CC(q;4)], or triples and quadruples [in the case of CC(t,q;3,4)]. As demon-

strated in the past [126–129, 131, 132, 138], the active-orbital-based CC(P ;Q) approaches

can recover the energetics of high-level CC/EOMCC methods, such as CCSDT/EOMCCSDT

in the case of CC(t;3), or CCSDTQ/EOMCCSDTQ in the case of CC(t,q;3), CC(t,q;3,4),

or CC(q;4), to within sub-millihartree accuracy using fractions of the computational effort

associated with the parent CC/EOMCC calculations. Although the active-orbital-based

CC(P ;Q) undoubtedly provides a robust strategy for obtaining high-level CC/EOMCC en-

ergetics, it relies on user-selected active orbitals to define the underlying P spaces, which

means that the corresponding CC(P ;Q) calculations cannot be performed in a black-box

fashion. Therefore, one may wonder if there is a more easy-to-use form of the CC(P ;Q)

theory that preserves the effectiveness of the CC(t;3), CC(t,q;3), CC(t,q;3,4), CC(q,4), etc.

hierarchy and the improvements it offers over the CR CC/EOMCC corrections, while avoid-

ing the use of user- and system-dependent active orbitals to capture the coupling between

the T1, T2, Rµ,1, and Rµ,2 components and their T3, T4, Rµ,3, Rµ,4, etc. counterparts.

To address this challenge, a third variant of CC(P ;Q) was introduced within the last few

years, which automatically determines the P and Q spaces using information extracted from

stochastic Quantum Monte Carlo (QMC) wave function propagations of the CI [170–174] or

CC [175–178] types in the many-electron Hilbert space. In the resulting hybrid CC(P ;Q)

approaches, broadly classified as semi-stochastic CC(P ;Q) [130, 133, 134, 136], the P space

10

associated with a given CIQMC or CC Monte Carlo (CCMC) wave function is defined as

all singly and doubly excited determinants and a list of stochastically determined higher–

than–doubly excited determinants present in the CIQMC or CCMC wave function, thereby

removing the need to manually determine the appropriate P space using active orbitals.

As in the fully deterministic active-orbital-based CC(P ;Q) approaches, such as CC(t;3),

CC(t,q;3), CC(t,q;3,4), and CC(q;4), the companion Q space defining the moment correc-

tion spans the remaining higher–than–doubly excited determinants absent in the underlying

CIQMC or CCMC wave function such that the resulting semi-stochastic CC(P ;Q) approach

converges to a high-level target CC method. For example, if the P space is set up to con-

tain all singly and doubly excited determinants and the list of triply excited determinants

extracted from the stochastic CIQMC or CCMC wave function, and the Q space spans the

remaining triply excited determinants absent from the underlying CIQMC or CCMC wave

function at a given propagation time, then the resulting semi-stochastic CC(P ;Q) approach

converges the full CCSDT energetics, much like the fully deterministic CC(t;3) method.

Similarly, if the P space is augmented to include all triply and quadruply excited determi-

nants present in the underlying CIQMC or CCMC wave function and the Q space spans the

remaining triply and quadruply excited determinants, the semi-stochastic CC(P ;Q) compu-

tations converge the CCSDTQ energetics, analogous to the CC(t,q;3,4) approach. Previous

work has demonstrated that the semi-stochastic CC(P ;Q) approach is capable of converging

high-level energetics of the CCSDT [130, 133, 134, 136], CCSDTQ [134], and EOMCCSDT

[133] types out of the early stages of CIQMC or CCMC wave function propagations, with

minimal reliance on user- and system-dependent inputs. More importantly in the context of

this dissertation, the semi-stochastic CC(P ;Q) represents a major intellectual advancement

by introducing the idea of sampling determinants in the many-electron Hilbert space to

identify the important components of the CC/EOMCC ansatz. Naturally, this provokes one

to ask whether there exists alternative, potentially better, methods for probing the Hilbert

space to obtain the pertinent information about leading higher–than–pair correlations.

11

Answering this question leads us to the first main topic of this dissertation. In Chapter 3,

we introduce a fourth variety of CC(P ;Q) capable of converging any high-level CC/EOMCC

method of interest, called CIPSI-driven CC(P ;Q) [135]. The CIPSI-driven CC(P ;Q) method-

ology replaces the QMC framework originally used to define the P and Q spaces in the semi-

stochastic CC(P ;Q) computations with the selected CI [179–182] algorithm known as the CI

method using perturbative selections made iteratively, or CIPSI [181, 183, 184]. The selected

CI approaches, which originated from the pioneering efforts of the late 1960s and early 1970s,

have enjoyed a significant renewal of interest in recent years, as their modern implementa-

tions have demonstrated the ability to capture the bulk of many-electron correlation effects in

a computationally efficient and conceptually straightforward manner [183–195]. For our pur-

poses, selected CI represents an appealing alternative to the QMC framework as a provider

of the lists of higher–than–doubly excited determinants within CC(P ;Q) considerations due

to its ability to construct a CI wave function using a well-defined, systematic sequence of

Hamiltonian diagonalizations, resulting in the associated and similarly systematic sequence

of P and Q spaces that are subject to little or no stochastic noise. Using the examples of

bond breaking in the fluorine molecule, automerization of cyclobutadiene, and the vertical

excitation spectrum of CH+, we demonstrate that the CIPSI-driven CC(P ;Q) approach is

capable of rapidly converging the high-level CCSDT and EOMCCSDT energetics in a com-

pletely black-box fashion using small fractions of the computational costs associated with

the parent CCSDT/EOMCCSDT calculations, while eliminating the noise due to random

sampling associated with CIQMC or CCMC propagations. In fact, the results collected to

date indicate that the CIPSI-driven CC(P ;Q) method can be even more efficient that its

semi-stochastic predecessor in recovering the near-CCSDT/EOMCCSDT data, employing

P spaces derived from CI wave functions that are smaller and more compact than their

stochastically determined counterparts thanks, in part, to the more robust perturbative se-

lection mechanism adopted in CIPSI. Just as how the semi-stochastic CC(P ;Q) methodology

led us to consider alternative approaches for sampling the Hilbert space, the CIPSI-driven

12

CC(P ;Q) approach invites us to scrutinize the way in which we take advantage of sparsity

in the many-electron wave function. While CIPSI and other selected CI methods have long

exploited the sparsity of the FCI wave function using perturbative arguments to identify the

subset of determinants important to the CI expansion, it is tempting to consider whether

the underlying CC/EOMCC framework adopted in CC(P ;Q) can be used to do the same,

providing us with an intrinsic mechanism for selecting the dominant determinants entering

our P and Q spaces without turning to non-CC considerations.

In exploring this idea, we arrive at the final form of the CC(P ;Q) methodology, which we

currently believe to be the most promising and powerful version, called adaptive CC(P ;Q)

[137, 138]. The adaptive CC(P ;Q) approach takes us to an entirely new level by free-

ing us from the user-defined active orbitals and non-CC (CIQMC, CIPSI) or stochastic

(CIQMC, CCMC) concepts adopted in the previous CC(P ;Q) work. In Chapter 4 of this

dissertation, we introduce the adaptive CC(P ;Q) framework capable of converging high-level

CC/EOMCC energetics of the CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ, and simi-

lar types, even in cases of stronger correlations, such as those characterizing significant bond

rearrangements, biradical species, and larger regions of PESs, where higher–than–two-body

components of the cluster operator T and excitation operators Rµ become large, in a fully

automated, black-box, manner and at small fractions of the computational costs. The key

idea driving this new approach is an adaptive selection of the leading determinants or ex-

citation amplitude types needed to define the Tn and, in the case of excited states, Rµ,n,

components with n > 2, which takes advantage of the intrinsic structure of the CC(P ;Q)

moment expansions and the associated a posteriori energy corrections to capture the re-

maining correlations of interest, without any reference to the user- and system-dependent

or non-CC concepts. After presenting the general theoretical and algorithmic details be-

hind the adaptive CC(P ;Q) framework in Chapter 4, we provide a comprehensive set of

numerical tests designed to demonstrate its efficiency in converging the high-level CCSDT

and EOMCCSDT energetics using tiny fractions of the computational effort. The molecular

13

examples included in our studies are (i) the significantly stretched F2 and F+

2 molecules,

(ii) the automerization of cyclobutadiene, (iii) the singlet–triplet gaps in the cyclobutadiene,

cyclopentadienyl cation, and trimethylenemethane biradicals, (iv) the degenerate Cope rear-

rangement in bullvalene, and (v) the ground- and excited-state PESs of the water molecule

along its O–H bond-breaking coordinate. All of these examples are characterized by large

and highly nonperturbative T3 and Rµ,3 effects, which cannot always be handled using non-

iterative triples corrections to the CCSD/EOMCCSD energetics. Finally, to highlight the

computational benefits offered by the adaptive CC(P ;Q) approach, we conclude Chapter

4 with a discussion of the CPU timings characterizing calculations for cyclobutadiene and

CnH2n+2 linear alkanes with n = 1—8.

While the CC(P ;Q) formalism provides a flexible and efficient framework for devising

robust schemes aimed at converging any well-defined level of CC/EOMCC theory, there

exists another intriguing route to the study of MR problems within the SRCC framework

based on approaching the FCI-level ground-state energetics directly using the externally

corrected (ec) CC approaches [195–214]. The ec-CC formalism is based on the observation

that, for Hamiltonians containing up to two-body interactions, the CC correlation energy is

only dependent on the T1 and T2 components of the cluster operator, and the CC amplitude

equations projected onto singly and doubly excited determinants do not engage higher-rank

components of the cluster operator Tn with n > 4. Thus, by solving the CC amplitude

equations projected on the singly and doubly excited determinants for the T1 and T2 clusters

in the presence of their exact three- and four-body counterparts extracted from FCI, with

the help of the well-known cluster-analysis relations [215], one obtains the exact T1 and T2

clusters and, as a consequence, the exact, FCI energy. This suggests that using external non-

CC wave functions capable of generating an accurate representation of T3 and T4 clusters,

and subsequently solving for T1 and T2 in their presence, should not only result in correlation

energies that improve upon those obtained with CCSD, where T3 and T4 are zero, but also

substantially improve the results of calculations used to provide the T3 and T4 values.

14

In reality, and as revealed by our recent mathematical analysis of the ec-CC framework

in Ref. [214], this is not always the case. In fact, one must be careful in choosing the external

non-CC source entering the ec-CC considerations in order to obtain a useful methodology.

The external sources of T3 and T4 clusters adopted in the ec-CC methods developed prior to

our recent study in Ref. [214] include projected unrestricted HF (PUHF) wave functions used

to describe strongly correlated systems involved in modeling the metal-insulator transition

[196, 197, 202, 210], and wave functions designed to capture nondynamical correlation effects

in ab initio calculations, including those resulting from valence-bond [198–200], complete-

active-space self-consistent-field (CASSCF) [201, 204, 205, 207], MRCI [203, 206, 207, 209],

perturbatively selected CI [208], FCIQMC [195, 211], adaptive CI [212], density-matrix renor-

malization group (DMRG) [213], and heat-bath CI [213] computations. While the majority

of these non-CC sources lead to useful ec-CC methodologies, there are instances [212] where

the ec-CC computations do not improve the underlying CI energetics at all. This paradox

was resolved in Ref. [214], which showed that when using truncated CI wave functions as

sources of T3 and T4 clusters in ec-CC considerations, one must adopt the so-called ec-CC-II

framework, in which the T3 and T4 amplitudes associated with triply and quadruply excited

determinants absent in the underlying CI state are discarded a posteriori.

In Chapter 5, we report a new type of ec-CC-II approach based on incorporating the

information about T3 and T4 clusters obtained from CIPSI calculations into the ec-CC equa-

tions [214]. In addition to testing the CIPSI-driven ec-CC-II approach, we also introduce its

ec-CC-II3 extension that noniteratively corrects the ec-CC-II calculations for the effects of

missing T3 correlations using CC(P ;Q)-like moment expansions adapted to the ec-CC con-

text. Using the challenging symmetric dissociation of water as an example, we demonstrate

that the new generation of CIPSI-based ec-CC-II approaches, especially ec-CC-II3, is com-

petitive with high-level CC and CI methods, such as CCSDTQ and CISDTQPH, in providing

near-FCI energetics, even improving upon CCSDTQ when electronic degeneracies become

very strong, using computational steps that are more similar to CCSD and CR-CC(2,3).

15

CHAPTER 2

THEORETICAL BACKGROUND

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

tension to Excited Electronic States

As mentioned in the Introduction, the exact ground state of an N -electron system within

the SRCC theory is expressed as

|Ψ0⟩ = eT |Φ⟩.

Here, |Φ⟩ is the reference determinant defining the Fermi vacuum and

is the cluster operator, with

T =

N
X

n=1

Tn

Tn = X

a1···anEa1...an
ti1···in
i1...in

i1<...<in
a1<...<an

(2.1)

(2.2)

(2.3)

designating its n-body (n-particle–n-hole, or np-nh) component, in which ti1...in

a1...an are the

corresponding cluster amplitudes and Ea1...an

i1...in = aa1 . . . aanain . . . ai1 are the elementary np-nh

excitation operators that generate the excited determinants |Φa1...an

i1...in ⟩ when acting on |Φ⟩,

with ap (ap) representing the usual fermionic creation (annihilation) operator associated

with the spinorbital |p⟩. After inserting the CC ansatz, Eq. (2.1), into the Schr¨odinger

equation and multiplying on the left by e−T , we obtain the connected cluster form of the

Schr¨odinger equation,

H|Φ⟩ = E0|Φ⟩,

(2.4)

in which the exact ground-state energy E0 and reference determinant |Φ⟩ form an eigenpair

of the similarity-transformed Hamiltonian

H = e−T HeT = (HeT )C,

(2.5)

16

where subscript C denotes the connected operator product (in the sense of diagrammatic

MBPT). From Eq. (2.4), it follows that the exact, full CC, ground state satisfies

⟨Φa1...an

i1...in |H|Φ⟩ = 0,

i1 < . . . < in, a1 < . . . < an, n = 1, . . . , N,

with corresponding energy obtained by projecting Eq. (2.4) onto the Fermi vacuum

E0 = ⟨Φ|H|Φ⟩ = ⟨Φ|H|Φ⟩ + ⟨Φ|[H(T1 + T2 + 1

2T 2

1 )]C|Φ⟩.

(2.6)

(2.7)

If we do not apply any truncations to the CC ansatz, then Eqs. (2.4)–(2.7) represent a

projective formulation of the FCI eigenvalue problem.

In the case of approximate CC wave functions, we follow the prescription put forward

by ˇC´ıˇzek [17] and carefully adapt the formulas given in Eqs. (2.5)–(2.7) to obtain a solvable

system of equations applicable to truncated forms of T . In particular, if we define the cluster

operator truncated at many-body rank mA as

T (A) =

mAX

n=1

Tn,

(2.8)

then the system of equations used to determine the cluster amplitudes ti1...in

a1...an, n ≤ mA,

characterizing T (A) is given by the subset of projections considered in Eq. (2.6) corresponding

to the content of the truncated cluster operator,

⟨Φa1...an

i1...in |H (A)|Φ⟩ = 0,

i1 < . . . < in, a1 < . . . < an, n = 1, . . . , mA,

(2.9)

where

H (A) = e−T (A)HeT (A) = (HeT (A))C

(2.10)

is the associated Hamiltonian transformed with T (A). Once ti1...in

a1...an are obtained by solving the

corresponding CC amplitude equations, Eq. (2.9), the ground-state CC energy is determined

a posteriori as

0 = ⟨Φ|H (A)|Φ⟩.
E(A)

(2.11)

17

With the projective formulation summarized in Eqs. (2.8)–(2.11), we can carry out any stan-

dard CC approximation based on truncating T at a given many-body rank, including CCSD

(mA = 2), CCSDT (mA = 3), CCSDTQ (mA = 4), and so on. The working expressions for

the amplitude [Eq. (2.9)] and energy [Eq. (2.11)] equations can be derived in a straightfor-

ward manner using the generalized Wick’s Theorem or, more elegantly, using the techniques

of time-independent many-body diagrammatics [17, 18, 216, 217]. In recent decades, the use

of automated coding strategies [218–223] for deriving and implementing the CC equations

has also become increasingly popular.

Equation (2.9) represents a system of coupled nonlinear equations involving a large num-

ber of variables, namely, the cluster amplitudes. In the first applications of the CC theory,

the amplitude equations were solved using a quadratically convergent Newton–Raphson op-

timization algorithm [17]. In later years, the reduced linear equation [224] method became

the preferred approach, with quasi-linearization of nonlinear terms containing T2 used to

improve convergence in highly degenerate problems [225]. In many modern CC codes, in-

cluding those relevant to this dissertation incorporated in GAMESS [226, 227] and CCpy

[228] (see Appendix A of this dissertation for information about the CCpy software), Eq.

(2.9) is solved using the Jacobi method (an inexact Newton approach) augmented by con-

vergence accelerators, such as the direct inversion of the iterative subspace (DIIS) algorithm

[229–231], to help extrapolate the solution, and energy denominator shifts for obtaining con-

vergence in quasi-degenerate situations. Alternative techniques for solving Eq. (2.9) include

the CROP algorithm [232], quasi-Newton [233] and Newton–Krylov [234] methods, and ma-

trix eigenvalue formulations of the CC amplitude system based on dressed CI [235, 236].

Before moving on, it is worth emphasizing that for an approximate CC state, the eigenvalue

relationship in Eq. (2.4) no longer holds. While the projections of H (A)|Φ⟩ onto excited

Slater determinants corresponding to the content of T (A) are equal to zero, as implied by

Eq. (2.9), the projections onto excitations absent in T (A) are nonzero, and as we will see in

Section 2.2, these remaining projections ultimately help define the noniterative corrections

18

formulated within the CC(P ;Q) framework.

As alluded to in the Introduction, we can also study excited electronic states within the

SRCC theory using the EOMCC formalism, or alternatively, its LR CC and SAC-CI analogs.

In the EOMCC theory, the excited states of an N -electron system are represented as

|Ψ(A)

µ ⟩ = R(A)

µ eT (A)|Φ⟩,

where the linear excitation operator truncated at many-body rank mA is

R(A)

µ = rµ,01 +

mAX

n=1

Rµ,n,

with constituent n-body components

Rµ,n = X

µ,a1···anEa1...an
r i1···in
i1...in .

i1<...<in
a1<...<an

(2.12)

(2.13)

(2.14)

To ensure size-intensivity of the resulting vertical excitation energies, the truncation mA in

R(A)
µ

is the same as that used for T (A). The scalar rµ,0 represents the contribution of the

reference state |Φ⟩ to the µth electronic state |Ψ(A)

to describe the ground state, its weight in |Ψ(A)

µ ⟩. Given that |Φ⟩ is typically optimized
µ ⟩ should be small, if |Ψ(A)

0 ⟩ and |Ψ(A)
µ ⟩

belong to the same symmetry, or strictly zero, if they belong to different symmetries. For

convenience, we define R(A)

0 = 1 so that Eq. (2.12) can be used to express the CC ansatz

for the ground state (µ = 0) and its EOMCC extension for excited electronic states (µ > 0)

in a unified manner. As in the ground-state case, the basic EOMCC hierarchy is obtained

by varying the truncation in T and Rµ according to mA, resulting in EOMCCSD (mA = 2),

EOMCCSDT (mA = 3), EOMCCSDTQ (mA = 4), and so on. Given the linear nature of the

EOMCC ansatz for excited states, the equations used to determine the amplitudes r i1···in
µ,a1···an

and energies E(A)

µ

are obtained in a straightforward fashion by inserting Eq. (2.12) into

the Schr¨odinger equation. After some slight algebraic manipulation, we obtain the matrix

eigenvalue equation

(H (A)

openR(A)

µ,open)C|Φ⟩ = ω(A)

µ R(A)

µ,open|Φ⟩,

(2.15)

19

where ω(A)

µ = E(A)

state and H (A)

µ − E(A)
open ≡ H (A) − E(A)

0

is the vertical excitation energy characterizing the µth excited
µ − rµ,01 refer to the diagrams in H (A)

0 1 and R(A)

µ,open ≡ R(A)

and R(A)

µ

that contain external fermion lines, respectively. The EOMCC eigenvalue prob-

lem given in Eq. (2.15) amounts to diagonalizing of the similarity-transformed Hamiltonian

resulting from the ground-state CC computation in the subspace of the Hilbert space corre-

sponding to the content of R(A)

µ,open. In practice, the solutions of Eq. (2.15) are obtained using

the Hirao–Nakatsuji generalization [237] of the Davidson diagonalization algorithm [238] to

non-Hermitian Hamiltonians. The key quantities that must be computed when solving the

EOMCC eigenvalue problem using iterative diagonalization routines are the projections of

Eq. (2.15) onto the relevant subsets of excited determinants,

µ,a1...an = ⟨Φa1...an
σ i1...in

i1...in |(H (A)

openR(A)

µ,open)C|Φ⟩,

i1 < . . . < in,

a1 < . . . < an, n = 1, . . . , mA,

(2.16)

which are analogs of the σ-vectors used in CI computations. As in the case of the ground-

state CC amplitude equations, the working expressions for Eq. (2.16) can be derived using

standard algebraic or diagrammatic many-body techniques. After solving Eq. (2.15) to

determine the excitation amplitudes r i1···in

µ,a1···an and vertical excitation energies ω(A)

µ , the scalar

part of R(A)

µ

is obtained a posteriori as

rµ,0 = ⟨Φ|(H (A)

openR(A)

µ,open)C|Φ⟩/ω(A)
µ .

(2.17)

It is important to point out that the similarity-transformed Hamiltonian Eq. (2.10) is

not Hermitian [i.e., (H (A))† ̸= H (A)], which means that the bra ⟨ ˜Ψ(A)

µ | and its associated ket

state |Ψ(A)

µ ⟩ are not trivially related to each other by taking the adjoint. In other words,

⟨ ˜Ψ(A)

µ | ̸= (|Ψ(A)

µ ⟩)†. Instead, the correct parameterization of the CC/EOMCC bra state is

⟨ ˜Ψ(A)

µ | = ⟨Φ|L(A)

µ e−T (A),

(2.18)

which satisfies the biorthonormality condition

⟨ ˜Ψ(A)

µ |Ψ(A)

ν ⟩ = ⟨Φ|L(A)

µ R(A)
ν

|Φ⟩ = δµ,ν.

(2.19)

20

The hole–particle deexcitation operator

with many-body components

L(A)

µ = δµ,01 +

mAX

n=1

Lµ,n,

Lµ,n = X

l a1···an
µ,i1···in (Ea1...an

i1...in )†,

i1<...<in
a1<...<an

is determined by solving the companion left-eigenstate CC/EOMCC equations

⟨Φ| L(A)

µ H (A) = E(A)

µ ⟨Φ|L(A)
µ

corresponding to the linear system

(2.20)

(2.21)

(2.22)

⟨Φ| L(A)

µ H (A) (cid:12)

(cid:12)Φa1...an
(cid:12)

i1...in

E

= E(A)

µ l a1...an
µ,i1...in ,

i1 < . . . < in,

a1 < . . . < an, n = 1, . . . , mA.

(2.23)

For the ground state (µ = 0), Eq. (2.23) is solved similarly to the CC amplitude equations

using the Jacobi algorithm with DIIS acceleration, since the energy E(A)

0

is already known.

For excited states (µ > 0), it is still possible to solve Eq. (2.23) using the Jacobi algorithm

for a given E(A)

µ

and imposing the normalization given by Eq. (2.19) every iteration, but

this approach often leads to poor convergence. Instead, it is easier to solve Eq. (2.22) for

µ > 0 in the same way as Eq. (2.15) using the Hirao–Nakatsuji diagonalization algorithm,

normalizing the resulting L(A)

µ

operator according to Eq. (2.19) at the end. Alternatively,

Eqs. (2.15) and (2.22) can be solved simultaneously to obtain R(A)

µ

and L(A)

µ

together.

As discussed in the Introduction, the higher-level CC/EOMCC methodologies, such as

CCSDT/EOMCCSDT and CCSDTQ/EOMCCSDTQ, are capable of providing accurate ap-

proximations to the exact, FCI, solution in the majority of chemical problems of interest. We

illustrate this point in Figures 2.1 and 2.2 for single bond breaking in hydrogen fluoride and

double bond breaking in water, respectively. In both cases, CCSD fails to provide a quanti-

tative description in either the equilibrium or dissociation regions of the PESs. In contrast,

21

for single bond breaking in hydrogen fluoride (Figure 2.1), the RHF-based CCSDT approach

is extremely accurate, while its even higher-level CCSDTQ counterpart is virtually exact.

Even in the notoriously challenging case of symmetric O–H dissociation in water (Figure

2.2), the RHF-based CCSDT method remains highly accurate for O–H bond distances of up

to and including 2Re. As the O–H bonds fully dissociate, the CCSDT calculations using the

RHF determinant begin to struggle, but RHF-based CCSDTQ continues to deliver a very

accurate description of the entire dissociation process. This highlights the importance of

including higher-order excitations to capture stronger nondynamical correlation effects.

Given the limitations of CCSD and the breakdown of CCSDT in the asymptotic region for

double bond dissociation, one might consider adopting the symmetry-broken UHF reference

to avoid the failures of RHF at larger internuclear distances. However, this is not a robust

solution. For single bond breaking in hydrogen fluoride, replacing the RHF determinant with

UHF helps reduce the errors obtained using CCSD in the asymptotic region due to mixing the

singlet state with solutions of higher multiplicity, but the errors characterizing intermediate

H–F distances increase as a consequence, leaving the overall CCSD surface unreliable. When

we employ higher-level approximations like CCSDT or CCSDTQ, the choice of reference

becomes less relevant. Indeed, this might be anticipated when one is already so close to FCI.

For symmetric dissociation in water, while the UHF reference improves the description of the

asymptotic region for all CC methods, it again compromises the accuracy in the intermediate

region around 2Re due to significant spin contamination, which propagates into the correlated

CC state. It is important to recall that symmetric O–H bond dissociation cleaves water into

O(3P) and two H(2S) fragments, introducing severe nondynamical correlations associated

with the recoupling of four unpaired electrons into a singlet state. From this perspective,

it is remarkable that CCSDT remains reliable in this crucial recoupling region and that

CCSDTQ can describe the entire bond-breaking potential. This reinforces the idea that

higher-level SRCC methods are capable of handling strong nondynamical correlations in

chemical problems, making them competitive with more sophisticated MR treatments.

22

Figure 2.1 Errors, in millihartree, characterizing the RHF-based [(a)] and UHF-based [(b)]
all-electron CC calculations relative to FCI (taken from Ref. [140]) for bond breaking in
the HF molecule, as described using the DZ basis set [239]. The H–F bond length RH−F is
increased from equilibrium, RH−F = Re, where Re = 1.7832 bohr, to 2Re, 3Re, and 5Re.

Figure 2.2 Errors, in millihartree, characterizing the RHF-based [(a)] and UHF-based [(b)]
all-electron CC calculations relative to FCI (taken from Ref. [240]) for the symmetric dis-
sociation of water, as described using the cc-pVDZ basis set [10]. Following Ref. [240], the
H–O–H bond angle is fixed at 110.6 degree and the O–H bonds are simultaneously stretched
by factors of 2 and 3 relative to the equilibrium O–H bond length of Re = 1.8435 bohr.

23

Unfortunately, the computational cost associated with solving the CC/EOMCC equations

[Eqs. (2.9) and (2.15)] using high-level truncations in T and Rµ is prohibitively expensive for

most practical applications (e.g., N 8 for CCSDT/EOMCCSDT, N 10 for CCSDTQ/EOM-

CCSDTQ, and so on). In order to reduce these costs, standard methodologies like CCSD(T),

CCSDT-n, CC3, and their higher-order [113–116] and excited-state [117–121] extensions di-

rectly approximate the parent CC/EOMCC equations using perturbative arguments. Such

strategies may be justified for nondegenerate ground states or when describing excited states

dominated by one-electron transitions, but they will generally fail in situations characterized

by MR correlations and electronic quasi-degeneracies, which lead to nonperturbative Tn and

Rµ,n effects, even when the high-level CC and EOMCC theories remain well-behaved.

This breakdown of perturbative CC/EOMCC approximations is illustrated in Figure 2.3,

which compares the ground-state PEC of the F2 molecule, obtained using CCSD(T), and the

excited-state PECs of the lowest-lying 1∆ state of CH+, resulting from EOMCCSD(T)(a)*

and CC3 calculations, with the corresponding parent CCSDT/EOMCCSDT surfaces. For

both of these examples, the PESs obtained using perturbative CC/EOMCC approaches ex-

hibit inaccurate, or even unphysical, behavior, especially at larger internuclear separations,

while the full CCSDT/EOMCCSDT calculations remain highly accurate (cf. Ref. [241] and

Ref. [101] for numerical results demonstrating the accuracy of CCSDT and EOMCCSDT for

the ground state of F2 and lowest 1∆ state of CH+, respectively). Notably, however, the non-

perturbative CR-CC(2,3) [Figure 2.3(a)] and CR-EOMCC(2,3) [Figure 2.3(b)] approaches

faithfully track their respective parent bond-breaking potentials obtained with CCSDT and

EOMCCSDT. In effect, introducing perturbative approximations can undermine the accu-

racy of high-level CC/EOMCC methodologies, and due to the abundance of such perturba-

tive CC/EOMCC approaches developed in the literature and available in popular codes, this

has led some practitioners of quantum chemistry to incorrectly believe that SR CC/EOMCC

approaches are inherently unsuitable for describing multiconfigurational states. Given our

desire to accurately describe MR correlations effects, we forego perturbative approaches and

24

Figure 2.3 A comparison of perturbative and nonperturbative CC/EOMCC approximations
for the PESs describing F–F bond breaking in the X 1Σ+
g state of F2 [(a)] and C–H bond
breaking in the lowest-lying 1∆ state of CH+ [(b)]. Panel (a) shows the CCSD (blue),
CCSD(T) (green), CR-CC(2,3) (red), and parent CCSDT (black) calculations. Panel (b)
presents their excited-state counterparts, including EOMCCSD (blue), CR-EOMCC(2,3)
(red), and EOMCCSDT (black), alongside the perturbative EOMCCSD(T)(a)* (green) and
CC3 (magenta) approaches. The cc-pVDZ basis set [10] was employed for F2 and the
[5s3p1d/3s1p] basis set of Ref. [242] was used for CH+. The lowest-energy orbitals cor-
relating with 1s shells of F atoms were frozen in all post-RHF steps.

instead turn to the CC(P ;Q) formalism as a robust and practical alternative for accurately

approximating high-level CC/EOMCC methodologies.

2.2 CC(P;Q) Moment Expansions

The CC(P ;Q) formalism takes a different approach compared to standard perturbative

methodologies.

Instead of approximating the CC/EOMCC equations en masse to reduce

the computational costs, we solve the parent CC/EOMCC equations in a smaller subspace

of the Hilbert space and correct the resulting energetics for the missing correlation effects

in a nonperturbative fashion. Thus, our motivation for turning to CC(P ;Q) is not only to

avoid the failures resulting from perturbation theory, but also to reduce the costs of obtaining

high-level CC/EOMCC energetics by only solving a small fraction of the parent CC/EOMCC

equations in the iterative steps and using the appropriately designed corrections to handle

the remaining correlations. In order to partition the CC/EOMCC equations in a general way

25

and obtain the compatible noniterative energy corrections, we must adopt the mathematical

formulation of the CC(P ;Q) theory, which was originally reported in Refs. [126, 127].

Each CC(P ;Q) calculation begins by identifying two disjoint subspaces of the N -electron

Hilbert space, referred to as the P and Q spaces. The former space, designated as H (P ), is

spanned by the excited determinants |ΦK⟩ = EK|Φ⟩, where EK is the elementary particle–

hole excitation operator generating |ΦK⟩ from |Φ⟩, which, together with |Φ⟩, dominate

the ground state |Ψ0⟩ or the ground (µ = 0) and excited (µ > 0) states |Ψµ⟩ of the N -

electron system of interest. The latter space, designated as H (Q) [H (Q) ⊆ (H (0) ⊕ H (P ))⊥,

where H (0) is a one-dimensional subspace spanned by the reference determinant |Φ⟩], is

used to construct the noniterative corrections δµ(P ; Q) to the energies E(P )

µ

obtained in the

CC(P )/EOMCC(P ) calculations due to the missing higher-order correlation effects.

Once the P and Q spaces are defined, the first step in the CC(P ;Q) procedure is to

perform the iterative CC(P ) and, if excited states are also desired, EOMCC(P ) calculations.

In a nutshell, the CC(P )/EOMCC(P ) equations are almost identical to their counterparts

presented in the previous section for CC/EOMCC approximations truncated at many-body

rank mA, except that the T , Rµ, and Lµ operators in CC(P )/EOMCC(P ) adopt a more

general truncation based on an arbitrary subset of excited determinants spanning the P

space. With this in mind, we begin by solving the CC(P ) equations,

⟨ΦK|H (P )|Φ⟩ = 0,

|ΦK⟩ ∈ H (P ),

(2.24)

with H (P ) = e−T (P )HeT (P ), to determine the truncated form of the cluster operator T corre-

sponding to the content of the P space,

T (P ) = X

tKEK,

|ΦK ⟩∈H (P )

where tKs are the relevant cluster amplitudes, and the CC(P ) ground-state energy

0 = ⟨Φ|H (P )|Φ⟩.
E(P )

26

(2.25)

(2.26)

We also determine the companion hole–particle deexcitation operator

0 = 1 + X
L(P )

l0,K(EK)†,

|ΦK ⟩∈H (P )

(2.27)

which defines the bra ground state ⟨ ˜Ψ(P )

0
⟩ = eT (P )|Φ⟩ satisfying the normalization condition ⟨ ˜Ψ(P )

0 e−T (P ) associated with the CC(P ) ket
|Ψ(P )
0

⟩ = 1, by solving

| = ⟨Φ|L(P )

state |Ψ(P )

0

0

the linear system

⟨Φ| L(P )

0 H (P ) |ΦK⟩ = E(P )

0

l0,K,

|ΦK⟩ ∈ H (P ),

(2.28)

where E(P )

0

is the CC(P ) ground-state energy obtained with Eq. (2.26). If there is inter-

est in the ground- as well as excited-state energetics, we follow the CC(P ) calculations by

the diagonalization of the similarity-transformed Hamiltonian H (P ) in the P space to ob-

tain the EOMCC(P ) energies E(P )

µ

and the corresponding linear excitation and deexcitation

operators,

and

µ = rµ,01 + X
R(P )

rµ,KEK

|ΦK ⟩∈H (P )

µ = δµ,01 + X
L(P )

lµ,K(EK)†,

|ΦK ⟩∈H (P )

(2.29)

(2.30)

respectively, where the amplitudes rµ,K, along with rµ,0, define the EOMCC(P ) ket states

|Ψ(P )

µ ⟩ = R(P )

µ eT (P )|Φ⟩ and their left-eigenstate lµ,K counterparts define the EOMCC(P ) bra

states ⟨ ˜Ψ(P )

µ | = ⟨Φ|L(P )

µ e−T (P ), which as before, satisfy the biorthonormality condition

⟨ ˜Ψ(P )

µ |Ψ(P )

ν

⟩ = ⟨Φ|L(P )

µ R(P )
ν

|Φ⟩ = δµ,ν.

(2.31)

Once T (P ), L(P )

0

, and E(P )

0

and, in the case of excited states, R(P )

µ , L(P )

µ , and E(P )

µ

are

obtained, we proceed to the final step, which is the determination of the CC(P ;Q) energies

E(P +Q)
µ

= E(P )

µ + δµ(P ; Q),

(2.32)

27

where corrections δµ(P ; Q) are given by

δµ(P ; Q) = X

ℓµ,K(P ) Mµ,K(P ).

(2.33)

|ΦK ⟩∈H (Q)

The quantities Mµ,K(P ) entering Eq. (2.33) represent the generalized moments of the P -

space CC/EOMCC equations, which are

M0,K(P ) = ⟨ΦK|H (P )|Φ⟩

in the case of the ground state (µ = 0) [139, 140, 243] and

Mµ,K(P ) = ⟨ΦK|H (P )R(P )

µ |Φ⟩

(2.34)

(2.35)

for excited states (µ > 0) [142, 143, 244]. Equations (2.34) and (2.35) respectively correspond

to projections of the CC(P ) and EOMCC(P ) equations on the Q-space determinants |ΦK⟩ ∈

H (Q). The coefficients ℓµ,K(P ) that multiply moments Mµ,K(P ) in Eq. (2.33) are obtained

using the Epstein–Nesbet-like formula

ℓµ,K(P ) = ⟨Φ|L(P )

µ H (P )|ΦK⟩/D(P )
µ,K,

where

D(P )

µ,K = E(P )

µ − ⟨ΦK|H (P )|ΦK⟩.

(2.36)

(2.37)

One can replace the denominators D(P )

µ,K in Eq. (2.36) by their Møller–Plesset-like analogs,

but, as shown in Refs. [128–130, 134, 146, 147, 149, 151], the Epstein–Nesbet-like form of

D(P )

µ,K results in a more accurate description. The formulas for the CC(P ;Q) moment expan-

sions summarized in Eqs. (2.32)–(2.37) based on the left-eigenstate of the CC(P )/EOMCC(P )

problem can be obtained using the general recipes provided in Refs. [126, 146, 147]. A deriva-

tion of Eqs. (2.32)–(2.37) along these lines is provided in Appendix B of this thesis.

The CC(P ;Q) formalism can be viewed as a generalization of the left-eigenstate CR

CC/EOMCC moment expansions and, as such, includes the aforementioned CR-CC(2,3)

28

approach and its higher-order [129, 131, 153–155] and excited-state [148, 151, 152] exten-

sions, but its key advantage, which the previous moment expansions did not have, is the

possibility of making unconventional choices of the P and Q spaces and relaxing the lower-

order Tn and Rµ,n components with n ≤ 2 in the presence of their higher–than–two-body

counterparts, such as T3, T4, Rµ,3, and Rµ,4, which the CCSD(T), CR-CC(2,3), and other

triples or higher-order corrections to CCSD or EOMCCSD cannot do. In this regard, several

creative strategies for selecting the higher–than–doubly excited determinants entering the

P and Q spaces in CC(P ;Q) considerations have been employed, including (i) using active

orbitals to select subsets of higher–than–doubly excited determinants, as in the CC(t;3),

CC(t,q;3), CC(t,q;3,4), CC(q;4), etc. hierarchy, [126–129, 131, 132], (ii) relying on CIQM-

C/CCMC wave function propagations in the many-electron Hilbert space [170–178], adopted

in the semi-stochastic CC(P ;Q) theories [130, 133, 134, 136], (iii) extracting information from

the more deterministic sequence of CIPSI Hamiltonian diagonalizations [181, 183, 184], as in

the CIPSI-driven CC(P ;Q) framework [135], or (iv) automatically constructing the P and

Q spaces based on the information contained within the CC(P ;Q) moment expansions, Eq.

(2.33), using the adaptive CC(P ;Q) approach [137, 138].

All of these different variants of CC(P ;Q) offer unique advantages in converging the high-

level CC/EOMCC energetics using the same general recipe defined by Eqs. (2.24)–(2.37).

The only difference between them is in the method used to partition the manifold of higher–

than–doubles excitations between the iterative and noniterative steps of the CC(P ;Q) calcu-

lation. Each form of CC(P ;Q) also offers major savings the computational effort compared

to the parent full CC/EOMCC calculations, which were discussed, along with illustrative

timings, in Refs. [131, 137] [see Section 2.4 for further illustration of the computational ben-

efits offered by CC(P ;Q) methods]. With an exception of the standard CR CC/EOMCC

methods and the active-orbital-based CC(P ;Q) approaches, the various black-box CC(P ;Q)

methods, including the semi-stochastic, CIPSI-driven, and adaptive varieties, require that

one develops a new strategy for solving the CC(P )/EOMCC(P ) equations and computing

29

the noniterative CC(P ;Q) corrections that is capable of handling the potentially spotty sub-

sets of higher–than–doubly excited determinants in the P and Q spaces, which may not form

continuous manifolds labeled by occupied and unoccupied orbitals from the respective ranges

of indices. A detailed discussion of our novel algorithm for solving the CC(P )/EOMCC(P )

equations along with more precise comments about the resulting computational costs is pro-

vided in Appendix C of this thesis.

In the present context, the CC(P ;Q) methodologies

that interest us most are the CIPSI-driven and adaptive varieties, which are the subjects

of Chapter 3 and 4, respectively.

In the next section, we briefly review some of the key

details about selected CI methodologies, and in particular, the CIPSI algorithm adopted in

the hybrid CC(P ;Q) and ec-CC approaches examined in this dissertation.

2.3 Overview of Selected Configuration Interaction

The selected CI approaches, which date back to the late 1960s and early 1970s [179–

182], seek to exploit sparsity in the exact, FCI, wave function. As pointed out long ago,

a significant fraction of the determinants entering the FCI state have negligible impact on

the corresponding energy, and thus, one can construct approximate solutions with near-FCI

energies by correctly identifying which configurations to include in the diagonalization and

which ones to exclude. This basic observation from the early days of quantum chemistry

not only motivated the pioneering selected CI approaches, including the work of Whitten

and Hackmeyer [179], Bender and Davidson [180], the MR doubles CI (MRDCI) scheme of

Buenker and Peyerimhoff [182], and the CIPSI algorithm introduced by Malrieu and cowork-

ers [181], but it has also inspired a renewed interest in selected CI, with an ever-growing list of

methods based on this simple premise being developed in recent years, including the modern

reformulation of CIPSI [183, 184], heat-bath CI [191–193, 245], adaptive CI [185, 186], adap-

tive sampling CI [187, 188], iterative CI [189, 190], Monte Carlo CI [246], coordinate descent

CI [247], and even machine-learning-driven CI techniques [248, 249]. All of these selected CI

schemes carry out a sequence Hamiltonian diagonalizations in increasingly large, iteratively

defined, subspaces of the many-electron Hilbert space, differing primarily in the way that

30

determinants are selected for inclusion in the variational CI calculation. The majority of

selected CI algorithms, including the CIPSI approach of interest to this dissertation, rely on

perturbative arguments to identify determinants spanning the diagonalization spaces.

In the CIPSI approach, we seek to obtain an approximation to the FCI wave function by

constructing a sequence of subspaces V (k)

int , where k = 0, 1, 2, . . . enumerates the consecutive

CIPSI iterations, in which we diagonalize the Hamiltonian to determine the corresponding

wave functions |Ψ(CIPSI)

k

⟩ = P

|ΦI ⟩∈V (k)
int

c(k)
I

|ΦI⟩ and energies Evar,k. The initial subspace V (0)
int

can be one-dimensional, if the CIPSI calculations are started from a single determinant, such

as the RHF wave function, which is often good enough for ground-state CIPSI calculations,

or a relevant singly or doubly excited determinant, if one is interested in targeting specific

excited states. Alternatively, a multidimensional V (0)
int

space can be used, which may be

constructed with the help of a multideterminantal state generated in some preliminary trun-

cated CI computation. Once the initial subspace V (0)
int
are constructed via a recursive process in which V (k+1)

int

is defined, the remaining subspaces

is obtained by augmenting V (k)

int with

a subset of the leading singly and doubly excited determinants out of V (k)
int

identified with

the help of MBPT expansions. Thus, if V (k)
ext

is the space of all singly and doubly excited

determinants out of the state |Ψ(CIPSI)

k

⟩ obtained in the corresponding V (k)

int , then for each

determinant |Φα⟩ ∈ V (k)

ext , we evaluate the second-order MBPT correction

α,k = |⟨Φα|H|Ψ(CIPSI)
e(2)

k

⟩|2/(Evar,k − ⟨Φα|H|Φα⟩)

(2.38)

and use the resulting e(2)

to the determinants |ΦI⟩ already in V (k)

α,k values to decide which determinants from V (k)

ext should be added
int to construct the next diagonalization space V (k+1)

int

.

We can also use the e(2)

α,k values to calculate the perturbatively corrected CIPSI energies

Evar,k + ∆E(2)

k , where

∆E(2)

k = X

e(2)
α,k,

|Φα⟩∈V (k)
ext

and, after further manipulations, their Evar,k+∆E(2)

r,k counterparts, in which ∆E(2)

k

by its renormalized ∆E(2)

r,k form introduced in Ref. [184].

31

(2.39)

is replaced

In the modern implementation of CIPSI, formulated in Refs. [183, 184] and available in

the Quantum Package software [184], which was used for all CIPSI calculations presented in

this dissertation, the process of enlarging V (k)

int to generate V (k+1)

int

is executed in the following

manner. First, prior to examining the e(2)

α,k corrections, one stochastically samples the most

important singly and doubly excited determinants out of |Ψ(CIPSI)

k

⟩, so that not all singles

and doubles from V (k)

int are included in the accompanying V (k)

ext space, only the sampled ones.

Next, one arranges the sampled determinants |Φα⟩ ∈ V (k)
ext

in descending order according to

their |e(2)

α,k| values calculated using Eq. (2.38). The process of enlarging the current subspace

to construct the V (k+1)

V (k)
int
starts from the determinants |Φα⟩ characterized by the largest |e(2)

int

space for the subsequent Hamiltonian diagonalization, which

α,k| contributions, moving

toward those that have smaller |e(2)

α,k| values, continues until the number of determinants in

V (k+1)
int

exceeds the dimension of V (k)

int multiplied by a user-defined factor f > 1. In all of

the CIPSI calculations considered in this dissertation, we used f = 2, which is the default

in Quantum Package. In practice, a typical dimension of V (k+1)

int

is slightly larger than f

times the dimension of V (k)

int , since the CIPSI algorithm adds extra determinants to V (k+1)

int

to ensure that the resulting |Ψ(CIPSI)

k+1

⟩ wave function is an eigenstate of the total spin S2 and

Sz operators [250].

The final wave function |Ψ(CIPSI)⟩ of a given CIPSI run and the associated variational

(Evar) and perturbatively corrected [Evar + ∆E(2) and Evar + ∆E(2)

r

] energies are obtained

by terminating the above procedure in one of the following two ways: (i) stopping at the

first iteration k for which the second-order MBPT correction |∆E(2)

k | falls below a user-

defined threshold η, indicating that the CIPSI wave function is within a tolerable distance

from FCI, or (ii) stopping at the first iteration k for which the number of determinants

in the corresponding V (k)
int

space exceeds a user-defined input parameter Ndet(in). Within

the context of this dissertation, our primary interest in using CIPSI is for sampling the

many-electron Hilbert space to provide useful information for our CIPSI-based CC(P ;Q) or

ec-CC considerations rather than converging the FCI energetics, so we elect to terminate our

32

CIPSI calculations using the latter size-based termination option [to ensure that termination

according to (i) is not possible, or at least extremely unlikely, we always set η = 10−6 hartree].

As a result of setting the input parameter f at 2, the sizes of the final wave functions |Ψ(CIPSI)⟩

produced by our CIPSI runs, denoted as Ndet(out), were always between Ndet(in) and 2Ndet(in).

To give a feel for how the CIPSI calculations are performed in this dissertation, we provide

an example CIPSI run for the ground state of the water molecule in Figure 2.4. In particular,

we varied the input parameter Ndet(in) in a roughly semi-logarithmic manner, starting at 1

(which returns the RHF state in this example) and ending at a larger value, in this case

100,000. Based on Figure 2.4, we can make a few interesting observations about the behavior

of the CIPSI algorithm. First, as expected, both the variational and perturbatively corrected

CIPSI energies converge toward the FCI limit as Ndet(in) and Ndet(out) increase. In fact, we

can truly appreciate the sparsity in the FCI wave function when we compare the range of

Ndet(out) values shown in Figure 2.4, which is minuscule in comparison to the 451,681,246

Sz = 0 determinants of the A1(C2v) symmetry entering the FCI solution. Furthermore,

we can see how the perturbative selection scheme adopted in CIPSI influences which types

determinants are included in the CI expansions. Based on simple MBPT arguments, we

expect that doubly excited determinants form the leading contributions to the FCI expansion

when based on an RHF reference, and thus, it is not surprising that CIPSI accumulates

almost all doubly excited determinants in the problem when it reaches near-FCI accuracy.

After doubly excited determinants, singly, triply, and quadruply excited determinants are

picked up next, consistent with their corresponding 4th-order contributions to the energy in

perturbation theory. We should keep in mind that our perturbative assessment of the results

in Figure 2.4 are only valid when the reference state |Φ⟩ represents the leading approximation

to the FCI solution. The general strength of CIPSI lies in its ability to explore the many-

electron Hilbert space in a well-defined and objective manner, adapting the content of the

CI wave function to capture the dominant correlations entering the problem. We adopted

CIPSI throughout this dissertation, as opposed to alternative selected CI schemes, because

33

the CIPSI algorithm and its implementation in the Quantum Package code, consistently

proves to be among the most accurate ones [194]. Although we employ CIPSI in the context

of both CC(P ;Q) and ec-CC considerations, the latter serves as our strongest motivation for

choosing the most accurate selected CI method. Indeed, in the ec-CC approaches, we rely

heavily on the quality of the CI wave function, as we will discuss in the next section.

Figure 2.4 An example CIPSI run performed for the ground state of the water molecule,
as described using the cc-pVDZ basis set, at the equilibrium C2v-symmetric geometry with
H–O–H bond angle of 110.6 degrees and O–H bond length of 1.84345 bohr, also considered
in Ref. [240]. The CIPSI calculations were carried out using the protocol described in
Section 2.3 in which the input parameter Ndet(in) was varied in a roughly semi-logarithmic
manner, taking on the values 1, 10, 100, 1,000, 5,000, 10,000, 50,000, and 100,000. The left
panel shows the convergence of the variational (Evar, in red) and perturbatively corrected
(Evar + ∆E(2), in black) CIPSI energies energies toward the FCI limit as a function of the
number of determinants, Ndet(out), included in the resulting CIPSI wave function. The right
panel shows the corresponding evolution in the fractions of singly (%S, in cyan), doubly
(%D, in blue), triply (%T, in green), and quadruply (%Q, in magenta) excited determinants
captured by the CIPSI algorithm.

2.4 Externally Corrected Coupled-Cluster Methodology

As discussed in the Introduction, one of the intriguing ways of improving the results of

SRCC calculations in MR and strongly correlated situations, which is based on combining

34

the CC and non-CC (e.g., CI) concepts, is the ec-CC framework [195–214] (see Ref. [210] for a

review). The ec-CC approaches are based on the observation that for electronic Hamiltonians

containing up to two-body interactions, the CC amplitude equations projected on the singly

and doubly excited determinants in the absence of any approximations in T ,

⟨Φa

i |[HN

(cid:16)

1 + T1 + T2 + 1

2T 2

1 + T3 + T1T2 + 1

6T 3

1

(cid:17)

]C|Φ⟩ = 0

(2.40)

and

⟨Φab

ij |[HN

(cid:16)

1 + T1 + T2 + 1

2T 2

1 + T3 + T1T2 + 1

6T 3

1 + T4 + 1

2T 2

2 + 1

2T 2

1 T2 + T1T3 + 1

24T 4

1

(cid:17)

]C|Φ⟩ = 0,

(2.41)

respectively, where HN = H − ⟨Φ|H|Φ⟩ denotes H in the normal-ordered form with respect

to |Φ⟩, do not engage the higher-rank Tn components of the cluster operator T with n > 4.

Thus, by solving Eqs. (2.40) and (2.41) for T1 and T2 in the presence of the exact T3 and T4

amplitudes extracted from FCI using the cluster-analysis relations,

T (ext)
1

= C1

T (ext)
2

= C2 −

1
2

C 2
1

T (ext)
3

T (ext)
4

= C3 − C1C2 +

= C4 − C1C3 −

1
3
1
2

C 3
1

C 2

2 + C 2

1 C2 −

1
4

C 4
1 ,

(2.42)

where Cn represents the n-body component of the CI wave function expansion and T (ext)

n

represents the Tn component of T extracted from the CI wave function, produces the exact T1

and T2, and as a consequence, the exact, full CI energy. An explicit form of Eq. (2.42) relating

the cluster amplitudes ti

a, tij

ab, tijk

abc, and tijkl

abcd to their ci

a, cij

ab, cijk

abc, and cijkl

abcd counterparts, which

respectively characterize the C1, C2, C3, and C4 components extracted from CI, is given in

Table 2.1. To date, the ec-CC framework has been combined with non-CC wave functions

corresponding to PUHF [196, 197, 202, 210], valence-bond [198–200], CASSCF and CASCI

[201, 204, 205, 207], MRCI [203, 206, 207, 209], perturbatively selected CI [208], FCIQMC

[195, 211], adaptive CI [212], DMRG [213], heat-bath CI [213], and CIPSI [214] calculations,

35

all of which are designed to provide T3 and T4 clusters that accurately capture the strong

nondynamical correlations that are difficult to describe using low-level CC calculations alone.

In general, we expect that the energies resulting from ec-CC calculations performed using

well-behaved approximate T3 and T4 will improve upon CCSD, in which T3 and T4 are zero,

and the non-CC source from which T3 and T4 are obtained. While the successful applications

of the ec-CC framework do validate this premise, there are also cases where the subsequent

ec-CC computations offer no improvements to the underlying non-CC computations [212].

, and
resulting from cluster analysis [Eq. (2.42)] of the underlying ground-state CI wave

Table 2.1 The programmable expressions for the amplitudes in T (ext)
T (ext)
4
function |Ψ(CI)⟩ characterized by excitation components Cn.

, T (ext)
2

, T (ext)
3

1

T (ext)
n

Amplitude

ti
a
tij
ab
tijk
abc
tijkl
abcd

Expressiona
ci
a
cij
ab − A ijci
acjk

acj

b

cijkl
abcd − A i/jklAa/bcdci

acjkl

cijk
abc − A i/jkAa/bcci
bcd − A ij/klcij

bc + 2A ijkci
bck
c
cd + 2A ijA ij/klAab/cdci

abckl

acj

acj

bckl

cd − 6A ijklci

acj

bck

c cl
d

aThe contravariant antisymmetrizers in the expressions are defined as A pq = 1 − (pq), A pqr = A p/qrA qr,
and A pqrs = A pq/rsA pqA rs, with A p/qr = 1 − (pq) − (pr), A p/qrs = 1 − (pq) − (pr) − (ps), and
A pq/rs = 1 − (pr) − (ps) − (qr) − (qs) + (pr)(qs). The covariant antisymmetrizers Apq, Apqr, Apqrs,
Ap/qr, Ap/qrs, and Apq/rs are identical to their contravariant counterparts.

In order to gain a deeper understanding of this paradoxical phenomena, we have carried

out a detailed theoretical investigation of the ec-CC equations in Ref. [214] and identified

two families of ec-CC approaches within the ec-CC framework, called ec-CC-I and ec-CC-II,

which differ in the way the cluster-analysis relations, Eq. (2.42), are carried out.

In the

ec-CC-I approach, one obtains T (ext)

3

and T (ext)
4

by application of Eq. (2.42) directly, allowing

for the potential introduction of completely disconnected components into T (ext)

3

and T (ext)
4

in the form of T (ext)

3

= −C1C2 + 1

3C 3

1 and T (ext)

4

= −C1C3 − 1

2C 2

2 + C 2

1 C2 − 1

4C 4

1 , respectively.

Solving for T1 and T2 in the presence of these disconnected terms not only results in strongly

non-extensive energies, but may also produce poorer energetics that do not even improve

upon those obtained using the underlying CI wave function (see the results reported in Refs.

[212] and [214] for a demonstration). One can largely remove disconnected terms in the

36

ec-CC equations by following the ec-CC-II protocol, in which T1 and T2 clusters are solved

in the presence of T (ext)

3

and T (ext)
4

extracted from CI using Eq. (2.42) after eliminating T (ext)

3

and T (ext)
4

components that do not have the companion C3 and C4 amplitudes. As shown in

Ref. [214], the ec-CC-II method generally results in energetics that are more accurate than

those obtained using ec-CC-I.

Depending on whether we adopt the ec-CC-I or ec-CC-II framework, we can identify the

different classes of CC and non-CC wave functions that are solutions to Eqs. (2.40) and

(2.41), when T3 and T4 are obtained from Eq. (2.42), in the form of two theorems. These

two theorems, which are proven both algebraically and with a diagrammatic analysis in

Appendices A and B of Ref. [214], help one predict whether a given ec-CC approach will

improve upon the underlying non-CC energetics. They are given as follows:

1. The ec-CC-I calculation in which T3 and T4 are obtained by cluster analysis of the

CI wave function that describes singles and doubles fully, as in the CISD, CISDT,

CISDTQ, etc., or any other CI method that provides a complete treatment of the

single and double excitation manifolds, returns back the underlying CI energy.

2. The ec-CC-II calculation in which T3 and T4 are obtained by cluster analysis of the

CI wave function that captures all singles, doubles, triples, and quadruples, as in CIS-

DTQ, CISDTQP, CISDTQPH, etc., or any other CI method that provides a complete

treatment of the single, double, triple, and quadruple excitation manifolds, returns

back the underlying CI energy.

When designing ec-CC methods, the above two theorems are of crucial importance. First,

it makes clear that the main shortcoming of the ec-CC-I approach is that it cannot benefit

from modern selected-CI-based approaches, which are easily capable of providing a numer-

ically complete treatment of all singly and doubly excited determinants in the problem.

Indeed, in CIPSI calculations, the singly (for non-HF references) and doubly excited deter-

minants are, by design, the first excitation manifolds that are explored before attempting to

37

examine higher-order correlation effects (see Figure 2.2). In contrast, the ec-CC-II method

appears to be much more promising for all selected-CI-based ec-CC considerations because

it is rather uncommon to use CI wave functions that provide a complete treatment of all

excitation manifolds through quadruples for subsequent ec-CC computations. Based on the

above theorems, we have concluded that the truncated CI wave functions that are best suited

for the ec-CC calculations are those that efficiently sample the many-electron Hilbert space,

without saturating the lower-rank excitation manifolds, especially the excitations through

quadruples, too rapidly, while adjusting the singly through quadruply excited CI amplitudes

to the dominant higher–than–quadruply excited contributions. With these theorems in mind,

we have identified the FCIQMC [170–174] and modern CIPSI [183, 184, 194] schemes, which

are characterized by tempered growth of the wave function, as optimal non-CC sources of

T3 and T4 clusters for subsequent ec-CC-II considerations. In this dissertation, we focus on

the CIPSI-based ec-CC-II approach, which is developed and tested in Chapter 5.

Inspired by the RMR-CCSD(T) [209] and ACI-CCSD(T) [212] approaches, and drawing

upon our previous experience in devising CC(P ;Q) moment expansions, we have also intro-

duced a new type of triples correction to the ec-CC-II energetics, abbreviated as ec-CC-II3,

which captures the missing, generally higher-order T3 correlation effects not captured by

the CI wave function |Ψ(CI)⟩ using the CC(P ;Q)-like moment expansions. In formulating

the ec-CC-II3 triples correction to the ec-CC-II energies, we have adapted the basic equa-

tions defining the noniterative CC(P ;Q) corrections, namely Eqs. (2.32)–(2.37). To be more

specific, in the ec-CC-II3 approach, we determine the total energies

E(3)

II = EII + δ3,

(2.43)

where EII is the energy resulting from the underlying ec-CC-II calculation, and the correction

δ3 due to the three-body clusters missing from |Ψ(CI)⟩

δ3 = X
|Φabc

ijk ⟩ /∈|Ψ(CI)⟩

ijk(2) Mijk
ℓabc

abc(2),

(2.44)

38

in which the generalized moments of the ec-CC-II equations are given by

Mijk

abc(2) = ⟨Φabc

ijk|H(2)|Φ⟩,

(2.45)

with

H(2) = e−T1−T2HeT1+T2 = (HeT1+T2)C,

(2.46)

and the coefficients ℓabc

ijk(2) multiplying moments Mijk

abc(2) in Eq. (2.44) are

ℓabc
ijk(2) = ⟨Φ|(Λ1 + Λ2)H(2)|Φabc

ijk⟩/(EII − ⟨Φabc

ijk|H|Φabc

ijk⟩).

(2.47)

The amplitudes λa

i and λab

ij characterizing the one- and two-body deexcitation operators Λ1

and Λ2, respectively, are obtained by solving the corresponding left eigenstate system

⟨Φ|(1 + Λ1 + Λ2)H(2)|Φa

i ⟩ = EIIλa
i

⟨Φ|(1 + Λ1 + Λ2)H(2)|Φab

ij ⟩ = EIIλab
ij .

(2.48)

One particularly appealing aspect of the ec-CC-II and ec-CC-II3 approaches is that they

only require solving for the T1 and T2 clusters in the iterative steps of the ec-CC calculations,

with their T (ext)

3

and T (ext)
4

counterparts frozen at their values resulting from cluster analysis

of the underlying CI wave function. This makes the ec-CC equations very similar to CCSD

and easily applicable to larger calculations. In fact, the ec-CC system given in Eqs. (2.40) and

(2.41) only differs from its CCSD counterpart by the presence of ⟨Φa

i |(HN T (ext)

3

)C|Φ⟩ in the

projection onto singly excited determinants and ⟨Φab

ij |(HN T (ext)

3

)C|Φ⟩, ⟨Φab

ij |(HN T1T (ext)

3

)C|Φ⟩,

and ⟨Φab

ij |(HN T (ext)

4

)C|Φ⟩ in the projection on doubly excited determinants. With the excep-

tion of ⟨Φab

ij |(HN T1T (ext)

3

)C|Φ⟩, the other three additional terms depend only on the values of

T (ext)
3

and T (ext)
4

, which means that they can be precomputed prior to performing the ec-CC

iterations. We will now briefly discuss the computational implementation of these terms in

the ec-CC-II system.

In order to evaluate ⟨Φa

i |(HN T (ext)

3

)C|Φ⟩, ⟨Φab

ij |(HN T (ext)

3

)C|Φ⟩, and ⟨Φab

ij |(HN T (ext)

4

)C|Φ⟩,

we first note that the disconnected contribution corresponding to each of these terms is

39

zero. Thus, we may formally remove the connected operator product constraint and insert

a resolution of identity within the many-electron Hilbert space between HN and T (ext)

3

or

T (ext)
4

to obtain

and

⟨Φa

i |HN T (ext)

3

⟨Φab

ij |HN T (ext)

3

|Φ⟩ = X
|Φcde

klm⟩∈|Ψ(CI)⟩

⟨Φa

i |HN |Φcde

klm⟩⟨Φcde

klm|T (ext)

3

|Φ⟩,

(2.49)

|Φ⟩ = X
|Φcde

klm⟩∈|Ψ(CI)⟩

⟨Φab

ij |HN |Φcde

klm⟩⟨Φcde

klm|T (ext)

3

|Φ⟩,

(2.50)

⟨Φab

ij |HN T (ext)

4

|Φ⟩ = X

|Φcdef

klmn⟩∈|Ψ(CI)⟩

⟨Φab

ij |HN |Φcdef

klmn⟩⟨Φcdef

klmn|T (ext)

4

|Φ⟩,

(2.51)

where in accordance with the ec-CC-II approach, we have enforced that T (ext)

3

and T (ext)
4

are

zero for all triply and quadruply excited determinants for which there does not exist an asso-

ciated C3 or C4 amplitude, respectively. Since the T (ext)

3

and T (ext)
4

operators resulting from

cluster analysis of |Ψ(CI)⟩ are only as large as their corresponding C3 or C4 counterparts, we

can assume that these objects can fit into memory as a single vector of amplitudes alongside

an associated list of triply or quadruply excited determinants. If the latter assumption is not

true, this would imply that we must be saturating the triples and quadruples manifolds in

our CI calculation, which would render the ec-CC-II calculations useless anyway. Therefore,

our computation of Eqs. (2.49) and (2.50) can be easily executed by looping over all triply

excited determinants (|Φcde

klm⟩) in |Ψ(CI)⟩, and for each one, looping over all singly (|Φa

i ⟩)

and doubly (|Φab

ij ⟩) excited determinants and evaluating the matrix elements ⟨Φa

and ⟨Φab

ij |HN |Φcde

klm⟩, which are then multiplied by the cluster amplitude tklm

i |HN |Φbcd
jkl⟩
cde entering T (ext)

3

and accumulated into the corresponding projection onto singly or doubly excited determi-

nants. Similarly, Eq. (2.51) is evaluated by looping over all quadruply excited determinants

(|Φcdef

klmn⟩) captured by CI, and for each one, looping over all doubly excited determinants

(|Φab

ij ⟩) and multiplying the matrix elements ⟨Φab

ij |HN |Φcdef

klmn⟩ with the corresponding tklmn
cdef

amplitude in T (ext)

4

and adding the result to the projection onto doubly excited determi-

nants. The relevant matrix elements of HN entering Eqs. (2.49)–(2.51) are straightforward

40

to evaluate using Wick’s Theorem or many-body diagrams. Unfortunately, the additional

⟨Φab

ij |(HN T1T (ext)

3

)C|Φ⟩ term in Eq. (2.41) cannot be handled using similar one-time precom-

putation. Due to the presence of T1 clusters, this terms must be recomputed each iteration,

making the computational costs of our ec-CC-II iterations asymptotically (d/D)n3

on4

u, where

d is the number of triply excited determinants captured by |Ψ(CI)⟩ out of the total number

of determinants D spanning the triples manifold. Fortunately, for smaller values of (d/D),

the iterative ec-CC-II steps have costs that are approximately the same as that of CCSD.

For higher fractions of triply excited determinants in the underlying CI state, the costs may

scale as large as N 7 with the system size N . It is also worth pointing out that earlier ec-CC

studies [212] demonstrated that one can approximate the quantity ⟨Φab

ij |(HN T1T (ext)

3

)C|Φ⟩ en-

tering the ec-CC-II equations by its ⟨Φab

ij |(HN T (ext)

1

T (ext)
3

)C|Φ⟩ counterpart obtained from the

cluster analysis of |Ψ(CI)⟩ without significant loss of accuracy. Although we have not pursued

such approximations, they would undoubtedly be helpful in improving the computational

efficiency if our existing ec-CC-II codes. Finally, the triples correction δ3, Eq. (2.44), can be

determined using simple CR-CC(2,3)-like [or CCSD(T)-like] expressions involving T1, T2, Λ1,

and Λ2, which involve noniterative computational steps that scale as n3

on4

u. Therefore, under

most circumstances, the ec-CC steps of the ec-CC-II and ec-CC-II3 calculations should be

comparable to those characterizing the CCSD and CR-CC(2,3) methods, respectively.

41

CHAPTER 3

MERGING THE CC(P;Q) FORMALISM WITH SELECTED
CONFIGURATION INTERACTION

In the previous chapter, we have introduced the framework of CC(P;Q) moment expansions

and how its flexibility allows us to devise computationally tractable schemes for accurately

capturing higher-order correlation effects in the CC/EOMCC calculations associated with

the Tn and Rµ,n components with n > 2, even in cases where these higher–than–two-body

components of T and Rµ become large, nonperturbative, and strongly coupled to their lower-

rank T1, T2, Rµ,1, and Rµ,2 counterparts and conventional noniterative corrections of the

CR-CC or CR-EOMCC types fail. The original active-orbital-based CC(P;Q) approaches,

including the CC(t;3), CC(t,q;3,4), CC(q;4), etc. hierarchy [126–129, 131, 132, 138], are

capable of recovering high-level CC/EOMCC energies of the CCSDT/EOMCCSDT, CCS-

DTQ/EOMCCSDTQ, and so on, types by including a subset of leading higher–than–doubly

excited determinants in the underlying P spaces selected with the help of active orbitals

and correcting the resulting CCSDt, CCSDtq, CCSDTq, etc. calculations for missing cor-

relations using the suitably defined CC(P;Q) moment expansions. However, as mentioned

in the Introduction, such schemes rely on the user- and system-dependent active orbitals,

rendering these methods no longer computational black boxes. Subsequent work explored

the combination of the deterministic CC(P;Q) formalism with stochastic wave function sam-

pling procedures in which the higher–than–doubly excited determinants included in the P

spaces defining the CC(P;Q) calculations are automatically identified using the information

extracted CIQMC or CCMC calculations. The resulting semi-stochastic CC(P;Q) method-

ology [130, 133, 134] allows us to recover high-level CC/EOMCC energetics in a black-fox

fashion at small fractions of the computational costs with the help of relatively early-stage

CIQMC/CCMC calculations.

Inspired by the success of the semi-stochastic CC(P;Q) approach, in this chapter, we

present CIPSI-driven CC(P;Q) approach, which was originally introduced in Ref. [135], and

42

represents a more deterministic version of the semi-stochastic CC(P;Q) scheme.

Instead

of relying on stochastic wave function propagations of the CIQMC or CCMC types, in the

CIPSI-driven CC(P;Q) approach, the subsets of leading higher–than–doubly excited de-

terminants entering the P spaces are identified using the information extracted from the

sequences of Hamiltonian diagonalization defining the selected CI algorithm abbreviated as

CIPSI [181, 183, 184]. The primary advantages of using CIPSI instead of CIQMC/CCMC as

a provider of the lists of higher–than–doubly excited determinants are (i) the stochastic noise

associated with random sampling of the many-electron Hilbert space is largely or entirely

eliminated, (ii) the perturbative selection scheme driving the CIPSI algorithm is typically

more reliable than CIQMC/CCMC in identifying the leading higher–than–doubly excited

determinants in the problem, resulting in smaller-sized P spaces used in the CC(P;Q) com-

putations, and (iii) the CIPSI Hamiltonian diagonalizations provide easy and natural access

to excited-state wave functions, which are significantly more challenging to describe using

CIQMC/CCMC calculations. After summarizing the key algorithmic details used in our

CIPSI-driven CC(P;Q) calculations aimed at converging high-level CC/EOMCC energetics

for ground and excited electronic states, we demonstrate the usefulness of the CIPSI-driven

CC(P;Q) approach on a few molecular examples, including the dissociation of F2 and the

automerization of cyclobutadiene as well as the vertical excitation spectrum of the CH+

molecule, where we recover the electronic energies corresponding to the full CCSDT and

EOMCCSDT calculations based on the information extracted from compact CI wave func-

tions originating from relatively inexpensive Hamiltonian diagonalizations.

3.1 Theoretical and Algorithmic Details

Having already discussed the key ingredients of the CC(P;Q) and CIPSI formalisms

relevant to this work in Chapter 2, we can describe the CIPSI-driven CC(P;Q) algorithm

aimed at converging the high-level CC and EOMCC energetics, in which the higher–than–

doubly excited determinants entering the P spaces used in the CC(P;Q) calculations are

generated in an automated fashion with the help of CIPSI Hamiltonian diagonalizations. In

43

setting up and executing the CIPSI-driven CC(P;Q) calculations, we employ the following

procedure:

1. Given a reference state |Φ⟩, which in all of the CIPSI-driven CC(P;Q) calculations

reported in Figure 3.1 and Tables 3.1–3.5 was the RHF determinant (for open-shell

states, we would use the high-spin ROHF reference), choose an input parameter Ndet(in),

used to terminate the CIPSI wave function growth, and execute a CIPSI run to generate

the ground state or, if desired, the ground- and excited-state wave functions |Ψ(CIPSI)⟩

spanned by Ndet(out) determinants. If the goal of the CIPSI-driven CC(P;Q) calculation

is to determine the ground-state energy or the energies of several electronic states

belonging to the same irreducible representation (irrep) as the ground state, one can

initiate the corresponding CIPSI run from the reference determinant |Φ⟩. If, in addition

to the ground state, one also wishes to determine excited states belonging to different

irreps as the ground state, one can generate the initial CIPSI subspace V (0)

int with the

help of some preliminary (e.g., CIS) calculation.

2. After the CIPSI run defined by a given Ndet(in) value is completed, extract a list or,

if multiple electronic states belonging to multiple irreps are targeted, lists of higher–

than–doubly excited determinants relevant to the target CC/EOMCC theory level

from the ground-state or ground- and excited-state wave functions |Ψ(CIPSI)⟩ to define

the P space(s) needed to set up the ground-state CC(P) or ground- and excited-state

CC(P)/EOMCC(P) calculations. If the goal is to converge the CCSDT/EOMCCSDT-

level energetics, the P space for the CC(P) calculations and the EOMCC(P) calcula-

tions for excited states belonging to the same irrep as the ground state can be defined

as all singly and doubly excited determinants plus the triply excited determinants con-

tained in the corresponding CIPSI ground state |Ψ(CIPSI)⟩. For excited states belonging

to a different irrep than the ground state, the P space is defined to contain all singly

and doubly excited determinants in addition to the triply excited determinants con-

44

tained in the lowest-energy CIPSI wave function of the target irrep. In particular, in

our current implementation, a distinct P space is constructed for each irrep associated

with the point group of the molecule.

3. Solve the CC(P) and, if excited states are targeted, EOMCC(P) equations in the P

spaces or P spaces obtained in the previous step.

If we are targeting the CCSDT-

and EOMCCSDT-level energetics we define T (P ) = T1 + T2 + T (CIPSI)

3

, R(P )

µ = rµ,01 +

Rµ,1 + Rµ,2 + R(CIPSI)

µ,3

, and L(P )

of triples in T (CIPSI)

3

, R(CIPSI)
µ,3

µ = δµ,01 + Lµ,1 + Lµ,2 + L(CIPSI)
, and L(CIPSI)

µ,3

µ,3

is/are extracted from the ground-state or

, where the list, or lists,

ground- and excited-state wave function(s) |Ψ(CIPSI)⟩, as summarized in point 2. For

the excited states belonging to irreps other than that of the ground state, we construct

the similarity-transformed Hamiltonian H (P ), to be diagonalized in the EOMCC(P)

calculations, in the same way as in the ground-state computations, but then use the ap-

propriate irrep-specific list of triply excited determinants to define R(CIPSI)

µ,3

and L(CIPSI)
µ,3

.

We follow a similar procedure when targeting the CCSDTQ/EOMCCSDTQ-level en-

ergetics, in which case T (P ) = T1 + T2 + T (CIPSI)

3

+ T (CIPSI)
4

, R(P )

R(CIPSI)
µ,3

+ R(CIPSI)
µ,4

, and L(P )

µ = δµ,01 + Lµ,1 + Lµ,2 + L(CIPSI)

µ,3

µ = rµ,01 + Rµ,1 + Rµ,2 +
+ L(CIPSI)
µ,4

.

4. Use the information obtained in Step 3 to determine corrections δµ(P ; Q), Eq. (2.33),

which describe the remaining correlations of interest that were not captured by the

CIPSI-based CC(P ) and EOMCC(P ) calculations.

If the goal is to converge the

CCSDT/EOMCCSDT energetics, define the Q space(s) needed to calculate correc-

tions δµ(P ; Q) as the remaining triply excited determinants that are not contained

in corresponding P space(s) considered in points 2 and 3. If the target approach is

CCSDTQ/EOMCCSDTQ, define the relevant Q space(s) as the triply and quadruply

excited determinants absent in corresponding P space(s). Add the resulting corrections

δµ(P ; Q) to E(P )

µ

to obtain the CC(P ;Q) energies E(P +Q)

µ

, Eq. (2.32).

5. To check convergence, repeat Steps 1–4 for a larger value of Ndet(in). The CIPSI-

45

driven CC(P ;Q) calculations can be regarded as converged if the difference between

consecutive E(P +Q)

µ

energies falls below a user-defined threshold.

In analogy to the

semi-stochastic CIQMC/CCMC-based semi-stochastic CC(P ;Q) framework of Refs.

[130, 133, 134], one can also stop if the fraction(s) of higher–than–doubly excited

determinants contained in the final CIPSI state(s) |Ψ(CIPSI)⟩ is (are) sufficiently large

to produce the desired accuracy level.

The above algorithm can be carried out to perform the CIPSI-driven CC(P ;Q) compu-

tations aimed at converging the results corresponding to any high-level CC/EOMCC theory,

such as CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ, and so on. In our initial studies

to date exploring the CIPSI-driven CC(P ;Q) methodology, we have focused our efforts on the

CIPSI-based CC(P;Q) approach aimed at converging the CCSDT energetics for the ground

state and EOMCCSDT energetics for excited states. The resulting codes, which have been

incorporated in the open-source CCpy software package available on GitHub [228] and in-

terfaced with the RHF/ROHF and integral transformation routines in GAMESS [227, 251],

are capable of parsing the lists of triply excited determinants extracted from the ground-

or ground- and excited-state CIPSI wave functions |Ψ(CIPSI)⟩, generated with the Quantum

Package 2.0 code [184], to set up the relevant P space(s) as described in Step 2 of the above

CIPSI-driven CC(P ;Q) algorithm. The corresponding Q space(s) is/are automatically de-

fined as the remaining triples absent in the |Ψ(CIPSI)⟩ wave function(s).

By design, as the input parameter Ndet(in) used to terminate CIPSI runs increases, pro-

ducing longer and longer CI expansions to represent wave functions |Ψ(CIPSI)⟩, the CC(P ;Q)

energies E(P +Q)

µ

approach their CCSDT/EOMCCSDT parents. The underlying CC(P ) and

EOMCC(P ) calculations converge the CCSDT/EOMCCSDT energetics too, but, as further

elaborated on in our numerical examples, by ignoring the triples that were not captured

by CIPSI, they do it at a much slower rate. In examining the convergence of the CIPSI-

driven CC(P )/EOMCC(P ) and CC(P ;Q) energies toward CCSDT/EOMCCSDT, in Tables

3.1–3.5 and Fig. 3.1, we sampled the Ndet(in) values in a roughly semi-logarithmic manner,

46

starting from Ndet(in) = 1. Since all of the calculations reported in this work adopted RHF

determinants as reference functions, the |Ψ(CIPSI)⟩ state becomes the RHF determinant and

the resulting CC(P )/EOMCC(P ) and CC(P ;Q) energies become identical to those obtained

in the RHF-based CCSD/EOMCCSD and CR-CC(2,3)/CR-EOMCC(2,3) calculations, re-

spectively, when Ndet(in) = 1. Thus, we can regard the Ndet(in) input variable defining CIPSI

computations as the parameter connecting CCSD/EOMCCSD [in the CC(P )/EOMCC(P )

case] or CR-CC(2,3)/CR-EOMCC(2,3) [in the case of CC(P ;Q) runs] with CCSDT/EOM-

CCSDT. In the case of the CIPSI-driven CC(P ) calculations discussed in this chapter,

T (P ) = T1 + T2 + T (CIPSI)

3

, where T (CIPSI)

3

= P

ijk ⟩∈|Ψ(CIPSI)⟩ tijk

abc Eabc

|Φabc

ijk denotes the T3 operator

defined using the lists of triply excited determinants |Φabc

ijk⟩ contained in the final ground-state

CIPSI wave function |Ψ(CIPSI)⟩ (we use the usual notation in which i, j, k and a, b, c designate

the occupied and unoccupied spin-orbitals in |Φ⟩, respectively, and Eabc

ijk is the elementary

triple excitation operator generating |Φabc

ploying the CIPSI-based EOMCC(P ) approach, we use R(P )

ijk⟩ from |Φ⟩). For the excited-state applications em-
µ = rµ,01 + Rµ,1 + Rµ,2 + R(CIPSI)

µ,3

and L(P )

µ = δµ,01 + Lµ,1 + Lµ,2 + L(CIPSI)

µ,3

, where R(CIPSI)

µ,3

and L(CIPSI)
µ,3

are defined analogously

to T (CIPSI)
3

using the appropriate list of triply excited determinants. The noniterative cor-

rections δµ(P ; Q), Eq.

(2.33), which in the case of the CIPSI-driven CC(P ;Q) approach

developed in this work captures the T3 and Rµ,3 effects not described by T (CIPSI)

3

and R(CIPSI)
µ,3

and which involves the summation over the remaining triply excited determinants that are

not included in |Ψ(CIPSI)⟩, is determined using the appropriate form of the CC(P ;Q) moment

expansions, i.e., δµ(P ; Q) = P

ijk ⟩ /∈|Ψ(CIPSI)⟩ ℓabc
|Φabc

ijk(µ) Mijk

abc(µ).

In addition to rapidly converging the parent CCSDT and EOMCCSDT energetics, as

demonstrated in Section 3.2, and in analogy to the active-orbital-based [126–129] and semi-

stochastic [130, 133, 134] CC(P ;Q) approaches, the CIPSI-driven CC(P ;Q) methodology

examined in this work offers significant savings in the computational effort compared to

full CCSDT and EOMCCSDT. This is largely related to the fact that, as shown in our

calculations for F2, cyclobutadiene, and CH+ reported in Section 3.2, the convergence of the

47

CIPSI-driven CC(P ;Q) energies toward their respective CCSDT or EOMCCSDT parents

with the wave function termination parameter Ndet(in), with the number of determinants

used to generate the final CIPSI state Ndet(out), and with the fractions of triples in the P

space captured by the CIPSI algorithm is very fast.

Indeed, the CPU times associated

with the CIPSI runs using smaller Ndet(in) values, resulting in smaller diagonalization spaces,

are relatively short compared to the converged CIPSI computations. Next, as explained in

detail in Refs.

[130, 133, 134], the CC(P )/EOMCC(P ) calculations using small fractions

of triples in the P space, which is all one needs to converge the CCSDT- or EOMCCSDT-

level energetics in the CIPSI-driven CC(P ;Q) runs, are much faster than the corresponding

CCSDT or EOMCCSDT computations. Furthermore, as also explained in Refs. [130, 133,

134], the computational cost of determining the CC(P ;Q) correction δµ(P ; Q) is less than

the cost of a single iteration of CCSDT/EOMCCSDT.

In examining the CIPSI-driven CC(P )/EOMCC(P ) and CC(P ;Q) energies shown in

Tables 3.1–3.5 and Fig. 3.1, we are primarily interested in how fast they converge toward

their parent CCSDT/EOMCCSDT values as Ndet(in) and the fraction of triples in the P space

increase. In the case of our calculations for F2 and the automerization of cyclobutadiene, it

is also helpful to examine the convergence of the Evar, Evar +∆E(2), and Evar +∆E(2)

r

energies

characterizing the ground-state CIPSI wave function |Ψ(CIPSI)⟩. In reporting the variational

and perturbatively corrected ground-state energies for F2 (Tables 3.1 and 3.2 and Figure

3.1) and cyclobutadiene (Table 3.3) obtained in the corresponding CIPSI computations, we

do what is often done in CIPSI calculations (see, e.g., Refs. [184, 194]) and compare them

to their counterparts obtained by extrapolating the data obtained in the CIPSI runs defined

by the largest Ndet(in) values to the FCI limit. Specifically, following the procedure used

in Ref. [194], we performed a linear fit of the last four Evar,k + ∆E(2)

r,k energies leading to

the final |Ψ(CIPSI)⟩ state obtained for the largest value of Ndet(in) in a given CIPSI sequence,

plotted against the corresponding ∆E(2)

r,k corrections, and extrapolated the resulting line to

the ∆E(2)

r,k = 0 limit.

48

3.2 Numerical Examples

3.2.1 Dissociation of F2

Our first example is the frequently studied dissociation of the fluorine molecule, as de-

scribed by the cc-pVDZ basis set [10]. We chose this example, since it is well established

that the CCSDT approach provides an accurate description of bond breaking in F2 (see,

e.g., Refs. [146, 147, 241, 252]) and since we previously used it to benchmark the CC(P ;Q)-

based CC(t;3) approach and the semi-stochastic CC(P ;Q) methods driven by the CIQMC

and CCMC propagations. The results of our calculations for the F2/cc-pVDZ system, in

which the F–F bond length R was stretched from its equilibrium, Re = 2.66816 bohr, value,

where electron correlation effects are largely dynamical in nature, to 1.5Re, 2Re, and 5Re,

where they gain an increasingly nondynamical character, are summarized in Table 3.1 and

Fig. 3.1. The complexity of electron correlations in F2 manifests itself in the rapidly growing

magnitude of T3 contributions as the F–F distance increases, as exemplified by the unsigned

differences between the CCSDT and CCSD energies, which are 9.485 millihartree at R = Re,

32.424 millihartree at R = 1.5Re, 45.638 millihartree at R = 2Re, and 49.816 millihartree at

R = 5Re, when the cc-pVDZ basis set is employed. The T3 correlations grow with R so fast

that in the R = 2Re–5Re region, they become larger than the depth of the CCSDT potential

(estimated at ∼44 millihartree when the CCSDT energy at R = Re is subtracted from its

R = 5Re counterpart) and highly nonperturbative. The latter feature of T3 contributions in

the stretched F2 molecule can be seen by examining the errors relative to CCSDT obtained

in the CCSD(T) calculations at R = 1.5Re, 2Re, and 5Re, which are −5.711, −23.596, and

−39.348 millihartree, respectively, when the cc-pVDZ basis set is used. As shown in Table

3.1 [see the Ndet(in) = 1 CC(P ;Q) energies], the CR-CC(2,3) triples correction to CCSD helps,

reducing the large errors characterizing CCSD(T) to 1.735 millihartree at R = 1.5Re, 1.862

millihartree at R = 2Re, and 1.613 millihartree at R = 5Re, which are much more accept-

able, but, as demonstrated in our earlier active-orbital-based and semi-stochastic CC(P ;Q)

studies [126, 130, 134], further error reduction requires the relaxation of T1 and T2 clusters

49

in the presence of the dominant T3 contributions. This is precisely what the CIPSI-driven

CC(P ;Q) methodology, where we use CIPSI runs to identify the leading triple excitations

for the inclusion in the P space, allows us to do.

Indeed, with as little as 1,006–1,442 Sz = 0 determinants of the Ag(D2h) symmetry

in the final Hamiltonian diagonalization spaces (we used D2h group, which is the largest

Abelian subgroup of the D∞h symmetry group of F2, in our calculations), generated by

the inexpensive Ndet(in) = 1, 000 CIPSI runs at R = 1.5Re, 2Re, and 5Re, which capture

very small fractions, on the order of 0.1–0.2 %, of all triples, the errors in the resulting

CC(P ;Q) energies relative to CCSDT are 0.202 millihartree at R = 1.5Re, 0.132 millihartree

at R = 2Re, and 0.144 millihartree at R = 5Re. This is an approximately tenfold error

reduction compared to the CR-CC(2,3) calculations, in which T1 and T2 clusters, obtained

with CCSD, are decoupled from T3, and an improvement of the faulty CCSD(T) energetics

by orders of magnitude. As explained in detail in Refs. [130, 133, 134] within the context of

the CIQMC/CCMC-driven CC(P ;Q) approaches, with the fractions of triples in the relevant

P spaces being so small, the post-CIPSI steps of the CC(P ;Q) calculations are not much

more expensive than the CCSD-based CR-CC(2,3) computations and a lot faster than the

corresponding CCSDT computations. The CC(P ;Q) calculations using Ndet(in) = 1, 000 do

not offer any improvements over CR-CC(2,3) at the equilibrium geometry, since the final

diagonalization space of the underlying CIPSI run does not yet contain any triply excited

determinants, and the CR-CC(2,3) energy at R = Re is already very accurate anyway, but

with the relatively small additional effort corresponding to Ndet(in) = 10, 000, which results in

10,150 Sz = 0 determinants of the Ag(D2h) symmetry in the final CIPSI diagonalization space

and only 1.2 % of all triples in the P space, the unsigned error in the CC(P ;Q) energy relative

to its CCSDT parent substantially decreases, from 0.240 millihartree, when Ndet(in) ≤ 1, 000,

to 67 microhartree, when Ndet(in) is set at 10,000. The use of Ndet(in) = 10, 000 for the

remaining three geometries considered in Table 3.1 and Fig. 3.1 produces similarly compact

|Ψ(CIPSI)⟩ wave functions, spanned by 11,578–19,957 determinants, similarly small fractions

50

of triples in the corresponding P spaces, ranging from 1.5 % at R = 1.5Re to 2.2 % at

R = 5Re, and even smaller errors in the CC(P ;Q) energies relative to CCSDT.

It is clear from Table 3.1 and Fig. 3.1 that the convergence of the CIPSI-driven CC(P ;Q)

energies toward CCSDT with the wave function termination parameter Ndet(in), with the

number of determinants used to generate the final CIPSI state |Ψ(CIPSI)⟩ [Ndet(out)], and with

the fraction of triples in the P space captured by the CIPSI procedure is very fast. The uncor-

rected CC(P ) energies converge to CCSDT too, but they do it at a considerably slower rate

than their CC(P ;Q) counterparts. For example, the CIPSI-driven CC(P ) calculations reduce

the 9.485, 32.424, 45.638, and 49.816 millihartree errors relative to CCSDT obtained with

CCSD to 1.419, 0.991, 0.922, and 0.764 millihartree, respectively, when Ndet(in) = 50, 000,

which translates in the Ndet(out) values ranging between 65,172 and 92,682 and about 5–9

% of all triples included in the underlying P spaces, but the errors characterizing the cor-

responding CC(P ;Q) energies are already at the level of 20–30 microhartree at this stage,

which is obviously a substantial improvement over the CC(P ) results. It is also worth notic-

ing that the convergence of the CIPSI-driven CC(P ) and CC(P ;Q) energies toward their

CCSDT parents with Ndet(in) [or Ndet(out)] is considerably faster than the convergence of

the corresponding variational and perturbatively corrected CIPSI energies toward the ex-

trapolated Evar + ∆E(2)

r

values. This is in line with the above observations that indicate

that the CIPSI-driven CC(P ;Q) calculations are capable of recovering the parent CCSDT

energetics, even when electronic quasi-degeneracies and T3 clusters become significant, out

of the unconverged CIPSI runs that rely on relatively small diagonalization spaces. We ob-

served similar patterns when comparing the semi-stochastic, CIQMC- and CCMC-driven,

CC(P )/EOMCC(P ) and CC(P ;Q) calculations with the underlying CIQMC/CCMC prop-

agations [130, 133, 134, 253].

In analogy to the previously considered deterministic, active-orbital-based [126, 127, 129,

131, 132] and semi-stochastic, CIQMC/CCMC-based [130, 134] CC(P ;Q) studies, the con-

vergence of the CIPSI-driven CC(P ;Q) computations toward the parent CCSDT energetics

51

remains equally rapid when we use basis sets larger than cc-pVDZ. This is illustrated in

Table 3.2, where we show the results of the CIPSI-driven CC(P ;Q) calculations for the F2

molecule at R = 2Re using the cc-pVTZ basis set [10]. As pointed out in Ref. [134], and in

analogy to the cc-pVDZ basis, the T3 contribution characterizing the stretched F2/cc-pVTZ

system in which the internuclear separation is set at twice the equilibrium bond length,

estimated by forming the difference between the CCSDT and CCSD energies at −62.819

millihartree, is not only very large, but also larger, in absolute value, than the corresponding

CCSDT dissociation energy, which is about 57 millihartree, when the CCSDT energy at Re

is subtracted from its 5Re counterpart. It is also highly nonperturbative at the same time, as

demonstrated by the −26.354 millihartree error relative to CCSDT obtained with CCSD(T).

Again, the CR-CC(2,3) triples correction to CCSD, equivalent to the Ndet(in) = 1 CC(P ;Q)

calculation in Table 3.2, works a lot better than CCSD(T), but the 4.254 millihartree er-

ror relative to CCSDT remains. With as little as 5,118 Sz = 0 determinants of the Ag(D2h)

symmetry in the final diagonalization space obtained by the nearly effortless Ndet(in) = 5, 000

CIPSI run, which captures 0.03 % of all triples, the difference between the CC(P ;Q) and

CCSDT energies decreases to 0.345 millihartree, and with the help of the Ndet(in) = 50, 000

CIPSI calculation, which is still relatively inexpensive, resulting in 82,001 Sz = 0 Ag(D2h)-

symmetric determinants in the final diagonalization space and less than 1 % of the triples in

the P space, the error in the CC(P ;Q) energy relative to its CCSDT parent reduces to less

than 0.1 millihartree. Similarly to the cc-pVDZ basis, the convergence of the CIPSI-driven

CC(P ;Q) energies toward CCSDT with Ndet(in), Ndet(out), and the fraction of triples in the

P space captured by the CIPSI algorithm is not only fast, when the larger cc-pVTZ basis

set is employed, but also much faster than in the case of the uncorrected CC(P ) calcula-

tions. Once again, as Ndet(in) increases, the rate of convergence of the CIPSI-driven CC(P )

and CC(P ;Q) energies toward their CCSDT parent is higher than those characterizing the

corresponding variational and perturbatively corrected CIPSI energies in their attempt to

recover the extrapolated Evar + ∆E(2)

r

energy.

52

Figure 3.1 Convergence of the CIPSI-based CC(P ) (red lines and circles) and CC(P ;Q)
(black lines and squares) energies toward their CCSDT parents as functions of the actual
numbers of determinants, Ndet(out), defining the sizes of the final wave functions |Ψ(CIPSI)⟩
generated in the underlying CIPSI runs, for the F2/cc-pVDZ molecule in which the F–F
bond length R was set at (a) Re, (b) 1.5Re, (c) 2Re, and (d) 5Re, where Re = 2.66816
bohr is the equilibrium geometry. The P spaces used in the CC(P ) calculations consisted
of all singles and doubles and subsets of triples contained in the final |Ψ(CIPSI)⟩ states of
the underlying CIPSI runs, whereas the Q spaces needed to compute the corresponding
CC(P ;Q) corrections δ(P ; Q) were defined as the remaining triples absent in |Ψ(CIPSI)⟩. The
insets show the percentages of triples captured by the CIPSI runs as functions of Ndet(out).
Adapted from Ref. [135].

53

Table 3.1 Convergence of the CIPSI-based CC(P ) and CC(P ;Q) energies toward CCSDT,
alongside the variational and perturbatively corrected CIPSI energies, for the F2/cc-pVDZ
molecule in which the F–F bond length R was set at Re, 1.5Re, 2Re, and 5Re, where
Re = 2.66816 bohr is the equilibrium geometry. In all post-RHF calculations, the two lowest-
lying core orbitals were frozen and the Cartesian components of d orbitals were employed
throughout. Each CIPSI run was initiated from the RHF reference determinant and the
MBPT-based stopping parameter η and selection factor f were set at 10−6 hartree and 2,
respectively. Adapted from Ref. [135].

Evar + ∆E(2)a Evar + ∆E(2)

R/Re
1.0

1.5

2.0

5.0

Ndet(in) / Ndet(out)
1 / 1
1,000 / 1,266
5,000 / 5,072
10,000 / 10,150
50,000 / 81,288
100,000 / 162,430
500,000 / 649,849
1,000,000 / 1,300,305
5,000,000 / 5,187,150

1 / 1
1,000 / 1,442
5,000 / 5,773
10,000 / 11,578
50,000 / 92,682
100,000 / 185,350
500,000 / 742,754
1,000,000 / 1,484,218
5,000,000 / 5,907,228

1 / 1
1,000 / 1,006
5,000 / 8,118
10,000 / 16,291
50,000 / 65,172
100,000 / 130,448
500,000 / 521,578
1,000,000 / 1,043,539
5,000,000 / 8,190,854

1 / 1
1,000 / 1,241
5,000 / 9,977
10,000 / 19,957
50,000 / 79,866
100,000 / 159,668
500,000 / 639,593
1,000,000 / 1,278,976
5,000,000 / 5,099,863

% T
0
0
0.4
1.2
7.9
14.5
34.3
42.2
85.1

0
0.1
0.7
1.5
8.8
13.9
30.8
37.1
74.3

0
0.1
1.1
2.1
5.2
8.4
19.8
28.0
72.8

0
0.2
1.2
2.2
4.6
7.6
18.0
22.0
46.1

a

Evar
418.057c
65.926
23.596
19.197
11.282
9.222
5.630
4.816
1.161

541.109c
77.306
21.091
17.333
10.879
9.243
5.586
4.795
1.165

640.056c
105.265
17.355
14.555
11.064
9.410
5.929
4.820
0.764

730.244c
70.879
14.531
12.550
9.025
7.495
4.391
3.682
0.675

−94.150d
1.480
−0.133(0)
0.045(2)
0.133(1)
0.138(1)
0.092(1)
0.072(0)
0.015(2)

−130.718d
5.218
0.811(2)
0.811(2)
0.762(1)
0.632(1)
0.391(1)
0.330(0)
0.079(2)

−159.482d
5.589
0.787(1)
0.860(1)
0.800(1)
0.655(1)
0.375(1)
0.306(0)
0.047(1)

−183.276d
6.966
1.033(0)
1.039(0)
0.764(1)
0.580(1)
0.276(0)
0.238(0)
0.041(1)

r
−12.651
2.079
−0.069(0)
0.084(2)
0.145(1)
0.146(1)
0.095(1)
0.074(0)
0.016(2)

137.819
5.948
0.856(2)
0.839(2)
0.771(1)
0.639(1)
0.393(1)
0.332(0)
0.079(2)

289.080
7.036
0.815(1)
0.878(1)
0.810(1)
0.662(1)
0.378(1)
0.308(0)
0.047(1)

430.051
7.491
1.050(0)
1.050(0)
0.770(1)
0.584(1)
0.277(0)
0.239(0)
0.041(1)

a CC(P )b CC(P;Q)b

9.485e
9.485
4.031
3.010
1.419
0.983
0.519
0.464
0.023

32.424e
16.835
2.490
1.892
0.991
0.727
0.390
0.362
0.028

45.638e
21.727
1.725
1.338
0.922
0.695
0.400
0.314
0.009

49.816e
5.154
1.489
1.156
0.764
0.584
0.294
0.259
0.009

−0.240f
−0.240
−0.129
−0.067
−0.031
−0.020
−0.009
−0.008
−0.001

1.735f
0.202
0.009
0.028
0.033
0.023
0.005
0.004
−0.000

1.862f
0.132
−0.003
0.012
0.015
0.009
0.005
0.002
−0.000

1.613f
0.144
0.029
0.029
0.022
0.013
0.003
0.003
−0.000

r

r

energy found using a linear fit based on the last four Evar,k +∆E(2)

aEnergies at each internuclear separation are reported as errors, in millihartree, relative to the extrapolated
Evar +∆E(2)
r,k values leading to the largest
CIPSI wave function obtained with Ndet(in) = 5, 000, 000 following the procedure used in Ref. [194]. These
extrapolated Evar + ∆E(2)
energies at R = Re, 1.5Re, 2Re, and 5Re are −199.104422(6), −199.069043(1),
−199.060152(8), and −199.059647(11) hartree, respectively.
bEnergies reported as errors relative to CCSDT, in millihartree. The total CCSDT energies at R = Re,
1.5Re, 2Re, and 5Re are −199.102796, −199.065882, −199.058201, and −199.058586 hartree, respectively.
cEquivalent to RHF.
dEquivalent to the second-order MBPT energy using the Epstein–Nesbet denominator.
eEquivalent to CCSD.
fEquivalent to CR-CC(2,3).

54

Table 3.2 Convergence of the CIPSI-based CC(P ) and CC(P ;Q) energies toward CCSDT,
alongside the variational and perturbatively corrected CIPSI energies, for the F2/cc-pVTZ
molecule in which the F–F bond length R was fixed at 2Re, where Re = 2.66816 bohr is the
equilibrium geometry. In all post-RHF calculations, the two lowest-lying core orbitals were
frozen and the spherical components of d and f orbitals were employed throughout. Each
CIPSI run was initiated from the RHF reference determinant and the MBPT-based stopping
parameter η and selection factor f were set at 10−6 hartree and 2, respectively. Adapted
from Ref. [135].

Evar + ∆E(2)a Evar + ∆E(2)

Ndet(in) / Ndet(out)
1 / 1
10 / 18
100 / 156
1,000 / 1,277
5,000 / 5,118
10,000 / 10,239
50,000 / 82,001
100,000 / 163,866
500,000 / 655,859
1,000,000 / 1,311,633
5,000,000 / 5,253,775

% T
0
0
0.00
0.01
0.03
0.06
0.84
1.58
3.75
5.58
13.3

a

Evar
758.849c
441.567
393.749
253.172
123.591
73.122
29.674
27.002
22.301
20.244
14.499

−165.740d
−0.554
6.420
13.595(0)
10.874(1)
7.202(5)
3.371(2)
3.068(2)
2.517(1)
2.292(1)
1.645(1)

r
340.460
31.337
28.790
20.323
12.149(1)
7.636(5)
3.428(2)
3.113(2)
2.547(1)
2.316(1)
1.657(1)

a CC(P )b CC(P;Q)b

62.819e
62.819
58.891
42.564
18.036
11.439
4.898
4.157
3.111
2.739
1.866

4.254f
4.254
3.683
1.579
0.345
0.198
0.061
0.049
0.014
0.009
−0.015

energy found using
r,k values leading to the largest CIPSI wave function obtained
energy

aEnergies are reported as errors, in millihartree, relative to the extrapolated Evar +∆E(2)
a linear fit based on the last four Evar,k + ∆E(2)
with Ndet(in) = 5, 000, 000 following the procedure used in Ref. [194]. The extrapolated Evar + ∆E(2)
is −199.242119(59) hartree.
bEnergies reported as errors relative to CCSDT, in millihartree. The total CCSDT energy is −199.238344
hartree..
cEquivalent to RHF.
dEquivalent to the second-order MBPT energy using the Epstein–Nesbet denominator.
eEquivalent to CCSD.
fEquivalent to CR-CC(2,3).

r

r

55

3.2.2 Automerization of Cyclobutadiene

A more demanding test, shown in Table 3.3, is the frequently examined [127, 130, 134, 212,

254–268] automerization of cyclobutadiene. In this case, one of the key challenges is an accu-

rate determination of the activation barrier, which requires a well-balanced description of the

nondegenerate, rectangle-shaped, closed-shell reactant (or the equivalent product) species, in

which electron correlation effects are largely dynamical in nature, and the quasi-degenerate,

square-shaped, transition state characterized by substantial nondynamical correlations asso-

ciated with its strongly diradical character. Experimental estimates of the activation barrier

for the automerization of cyclobutadiene, which range from 1.6 to 10 kcal/mol [256, 257], are

not very precise, but the most accurate single- and multireference ab initio computations,

compiled, for example, in Refs. [127, 255, 268], place it in the 6–10 kcal/mol range. This, in

particular, applies to the CCSDT approach [127, 254], which is of the primary interest in the

present study. Indeed, if we, for example, use the reactant and transition-state geometries

obtained with the multireference average-quadratic CC (MR-AQCC) approach [269, 270] in

Ref. [262] and the cc-pVDZ basis set, the CCSDT value of the activation energy characteriz-

ing the automerization of cyclobutadiene becomes 7.627 kcal/mol [127], in good agreement

with the best ab initio calculations carried out to date. By adopting the same geometries

and basis set in this initial benchmark study of the CIPSI-driven CC(P ;Q) methodology,

we can examine if the CC(P ;Q) calculations using the P spaces constructed with the help

of CIPSI runs are capable of converging this result. The main challenge here is that the T3

effects, estimated as the difference between the CCSDT and CCSD energies, are not only

very large, but also hard to balance. When the cc-pVDZ basis set used in this study is

employed, they are −26.827 millihartree for the reactant and −47.979 millihartree for the

transition state. Furthermore, in the case of the transition state, the coupling of the lower-

rank T1 and T2 clusters with their higher-rank T3 counterpart is so large that none of the

noniterative triples corrections to CCSD provide a reasonable description of the activation

barrier [127, 254, 255]. This, in particular, applies to the CR-CC(2,3) approach, equivalent

56

to the Ndet(in) = 1 CC(P ;Q) calculation, which produces an activation barrier exceeding 16

kcal/mol, when the cc-pVDZ basis set is employed, instead of less than 8 kcal/mol obtained

with CCSDT (see Table 3.3). The failure of CR-CC(2,3) to provide an accurate description

of the activation energy is a consequence of its inability to accurately describe the tran-

sition state.

Indeed, the difference between the CR-CC(2,3) and CCSDT energies at the

transition-state geometry is 14.636 millihartree, when the cc-pVDZ basis set is employed,

as opposed to only 0.848 millihartree obtained for the reactant. As discussed in detail in

Refs. [127, 255], other triples corrections to CCSD, including the widely used CCSD(T)

approach, face similar problems.

It was previously demonstrated in Refs. [127, 130, 134]

that the deterministic CC(P ;Q)-based CC(t;3) approach and the semi-stochastic CC(P ;Q)

calculations using CIQMC and CCMC are capable of accurately approximating the CCSDT

energies of the reactant and transition-state species and the CCSDT activation barrier, so

it is interesting to explore if the CIPSI-driven CC(P ;Q) methodology can do the same.

As shown in Table 3.3, the CC(P ;Q) calculations using CIPSI to identify the dominant

triply excited determinants for the inclusion in the P space are very efficient in converging the

CCSDT energetics. With the final diagonalization spaces spanned by a little over 110,000

Sz = 0 determinants of the Ag(D2h) symmetry (we used D2h group for both the D2h-

symmetric reactant and the D4h-symmetric transition state in our calculations), generated in

the relatively inexpensive CIPSI runs defined by Ndet(in) = 100, 000 that capture 0.1 % of all

triples, the 0.848 millihartree, 14.636 millihartree, and 8.653 kcal/mol errors in the reactant,

transition-state, and activation energies relative to CCSDT obtained with CR-CC(2,3) are

reduced by factors of 2–4, to 0.382 millihartree, 3.507 millihartree, and 1.961 kcal/mol,

respectively, when the CC(P ;Q) approach is employed. When Ndet(in) is increased to 500,000,

resulting in about 890,000–900,000 Sz = 0 determinants of the Ag(D2h) symmetry in the final

diagonalization spaces used by CIPSI and 1.0–1.2 % of the triples in the resulting P spaces,

the errors in the CC(P ;Q) reactant, transition-state, and activation energies relative to

CCSDT become 0.267 millihartree, 0.432 millihartree, and 0.104 kcal/mol. Clearly, these are

57

great improvements compared to the initial Ndet(in) = 1, i.e., CR-CC(2,3), values, especially

if we realize that with the fractions of triples being so small, the post-CIPSI steps of the

CC(P ;Q) computations are not only a lot faster than the parent CCSDT runs, but also not

much more expensive than the corresponding CR-CC(2,3) calculations, as elaborated on in

Refs. [130, 133, 134].

In analogy to the previously discussed case of bond breaking in F2, the convergence of the

CIPSI-driven CC(P ;Q) energies toward CCSDT for the reactant and transition-state species

defining the automerization of cyclobutadiene with Ndet(in), Ndet(out), and the fractions of

triples in the relevant P spaces captured by the underlying CIPSI runs is not only very fast,

but also significantly faster than that characterizing the uncorrected CC(P ) calculations.

For each of the two species, the CC(P ) energies converge toward their CCSDT parent in

a steady fashion, but, as shown in Table 3.3, their convergence is rather slow, emphasizing

the importance of correcting the results of the CC(P ) calculations for the missing triple

excitations not captured by the CIPSI runs using smaller diagonalization spaces. Similarly to

the previously examined active-orbital-based [126, 127, 129, 131, 132] and CIQMC/CCMC-

based [130, 134] CC(P ;Q) approaches, the moment correction δ0(P ; Q), defined by Eq. (2.33),

is very effective in this regard. Another interesting observation, which can be made based on

the results presented in Table 3.3, is that while the CC(P ) energies for the individual reactant

and transition-state species converge toward their CCSDT parent values in a steady fashion,

the corresponding activation barrier values behave in a less systematic manner, oscillating

between about −1 and 1 kcal/mol when Ndet(in) = 500, 000–15, 000, 000. One might try to

eliminate this behavior, which is a consequence of a different character of the many-electron

correlation effects in the reactant and transition-state species, by merging the P spaces

used to perform the CC(P ) calculations for the two structures, but, as shown in Table

3.3, the CC(P ;Q) correction δ0(P ; Q), which is highly effective in capturing the missing T3

correlations, takes care of this problem too. As Ndet(in), Ndet(out), and the fractions of triples

in the P spaces used in the CC(P ) calculations for the reactant and transition state increase,

58

the CC(P ;Q) values of the activation barrier converge toward its CCSDT parent rapidly and

in a smooth manner, eliminating, at least to a large extent, the need to equalize the P spaces

used in the underlying CC(P ) steps. As in the case of bond breaking in the fluorine molecule,

the convergence of the CIPSI-driven CC(P ) and CC(P ;Q) energies toward their CCSDT

parents with Ndet(in)/Ndet(out) is considerably faster than the convergence of the variational

and perturbatively corrected CIPSI energies toward the extrapolated Evar+∆E(2)

r values, but

we must keep in mind that the calculated CCSDT and extrapolated Evar + ∆E(2)

r

energies,

while representing the respective parent limits for the CC(P ;Q) and CIPSI calculations,

are fundamentally different quantities, especially when higher–than–triply excited cluster

components, which are not considered in the present calculations, become significant. As

one might anticipate, the Ndet(in) values needed to accurately represent the CCSDT energies

of the reactant and transition-state species of cyclobutadiene by the CIPSI-driven CC(P ;Q)

approach are considerably larger than those used in the analogous CC(P ;Q) calculations for

the smaller F2 system, but they are orders of magnitude smaller than the values of Ndet(in)

required to obtain the similarly well converged Evar + ∆E(2)

r

energetics in the underlying

CIPSI runs.

59

Table 3.3 Convergence of the CIPSI-based CC(P ) and CC(P ;Q) energies toward CCSDT,
alongside the variational and perturbatively corrected CIPSI energies, for the reactant (R)
and transition-state (TS) species involved in the automerization of cyclobutadiene, as de-
scribed by the spherical cc-pVDZ basis set, and for the corresponding barrier height. In all
post-RHF calculations, the four lowest-lying core orbitals were frozen. Each CIPSI run was
initiated from the RHF reference determinant and the MBPT-based stopping parameter η
and selection factor f were set at 10−6 hartree and 2, respectively. Adapted from Ref. [135].

Species
R

TS

Barrier

Evar + ∆E(2)a Evar + ∆E(2)

a CC(P )b CC(P;Q)b

Ndet(in) / Ndet(out)
1 / 1
50,000 / 55,653
100,000 / 111,321
500,000 / 890,582
1,000,000 / 1,781,910
5,000,000 / 7,125,208
10,000,000 / 14,253,131
15,000,000 / 28,493,873

1 / 1
50,000 / 56,225
100,000 / 112,481
500,000 / 899,770
1,000,000 / 1,800,183
5,000,000 / 7,195,780
10,000,000 / 14,400,744
15,000,000 / 28,793,512

% T
0
0.0
0.1
1.2
2.0
7.9
11.8
25.8

0
0.0
0.1
1.0
1.6
5.4
9.7
15.2

a

Evar
598.120c
121.880
109.688
93.413
89.989
78.122
73.250
60.872

632.707c
146.895
130.832
93.288
89.049
78.472
71.784
63.375

−83.736d
26.065(182)
23.819(163)
19.049(141)
18.322(137)
16.311(123)
15.514(115)
12.842(96)

−102.816d
45.357(180)
36.716(183)
18.106(139)
17.458(142)
15.587(124)
14.397(114)
12.587(102)

r
120.809
28.096(178)
25.397(160)
20.167(139)
19.348(135)
17.045(122)
16.146(114)
13.260(95)

282.246
47.696(176)
38.673(179)
19.251(137)
18.482(140)
16.346(123)
15.016(113)
13.058(101)

26.827e
25.468
22.132
16.260
15.359
10.794
9.632
4.817

47.979e
42.132
31.723
14.742
13.645
10.720
8.358
7.080

1 / 1 ; 1
50,000 / 55,653 ; 56,225
100,000 / 111,321 ; 112,481
500,000 / 890,582 ; 899,770
1,000,000 / 1,781,910 ; 1,800,183
5,000,000 / 7,125,208 ; 7,195,780
10,000,000 / 14,253,131 ; 14,400,744
15,000,000 / 28,493,873 ; 28,793,512 25.8 ; 15.2

13.274e
21.703c
0 ; 0
10.457
15.697
0.0 ; 0.0
13.268
6.018
0.1 ; 0.1
−0.079 −0.592(124) −0.574(122) −0.953
1.2 ; 1.0
−0.590 −0.542(124) −0.544(122) −1.075
2.0 ; 1.6
7.9 ; 5.4
−0.454(110) −0.439(109) −0.047
0.220
11.8 ; 9.7 −0.920 −0.701(102) −0.710(100) −0.800
1.420

−11.973d
12.106(161)
8.093(154)

101.303
12.299(157)
8.331(151)

−0.127(87)

−0.159(88)

1.571

0.848f
0.678
0.382
0.267
0.251
0.150
0.127
0.046

14.636f
9.563
3.507
0.432
0.412
0.260
0.155
0.108

8.653f
5.576
1.961
0.104
0.101
0.069
0.017
0.039

r

aEnergies reported as errors, in millihartree, relative to the extrapolated Evar + ∆E(2)
energy found using a
linear fit based on the last four Evar,k + ∆E(2)
r,k values leading to the largest CIPSI wave function obtained
with Ndet(in) = 15, 000, 000 following the procedure used in Ref. [194]. These extrapolated Evar + ∆E(2)
energies for the R and TS species are −154.249292(314) and −154.235342(321) hartree, respectively. The
barrier heights are reported as errors, in kcal/mol, relative to the reference value of 8.753(0) kcal/mol
obtained using the extrapolated Evar + ∆E(2)
bEnergies reported as errors relative to CCSDT, in millihartree. The total CCSDT energies of the R and
TS species are −154.244157 and −154.232002 hartree, respectively. Barrier heights are reported as errors
in kcal/mol relative to the CCSDT value of 7.627 kcal/mol.
cEquivalent to RHF.
dEquivalent to the second-order MBPT energy using the Epstein–Nesbet denominator.
eEquivalent to CCSD.
fEquivalent to CR-CC(2,3).

energies of the R and TS species.

r

r

60

3.2.3 Electronic Excitation Spectrum of CH+

The previous examples of bond breaking in F2 and automerization of cyclobutadiene con-

vincingly demonstrate that the CIPSI-driven CC(P ;Q) approach is capable of rapidly and

efficiently recovering the CCSDT energetics using small fractions of triply excited determi-

nants included in the underlying P spaces, which are identified with the help of relatively

inexpensive CIPSI wave calculations characterized by small Ndet(in), Ndet(out), and % T val-

ues. In this section, we will examine if the same remains true in an excited-state application,

where the goal is to converge the EOMCCSDT energetics for the electronic excitation spec-

trum of CH+ using the CIPSI-driven CC(P ;Q) methodology.

Following the analogous study of the CH+ molecule carried out in Refs. [133, 253] using

the semi-stochastic CC(P ;Q) approach, we employed the [5s3p1d/3s1p] basis set of Ref. [242]

and performed all-electron calculations for the ground and three lowest-energy singlet ex-

cited states of the Σ+(C∞v) symmetry, denoted as n 1Σ+, n = 1–4, where n = 1 corresponds

to the ground state and n = 2–4 represent excited states, the two lowest singlet Π(C∞v)

states, labeled 1 1Π and 2 1Π, and the two lowest singlet ∆(C∞v) states, denoted as 1 1∆ and

2 1∆. As discussed in points 1 and 2 of the algorithm outlined in Section 3.1, our current

implementation of the CIPSI-driven CC(P ;Q) approach constructs separate P spaces for

each irrep of the molecular point group by extracting the list of triply excited determinants

contained in the corresponding CIPSI wave function for the lowest-energy state of a given

spatial symmetry. Accordingly, the P (and Q) spaces defining the CC(P ;Q) calculations for

the 1Σ+, 1Π, and 1∆ states of CH+ were constructed using the information obtained from

the CIPSI computations describing the 1 Σ+, 1 1Π, and 1 1∆ states of CH+, respectively. Our

calculations were performed using the C2v subgroup of the full non-Abelian C∞v symmetry

describing the CH+ molecule, and as a result, the underlying CIPSI runs for the 11Σ+ ground

state were initiated in a usual fashion using the RHF determinant |Φ⟩, whereas the CIPSI

calculations for the 1 1Π and 1 1∆ states of CH+ were carried out by defining the initial

V (0)

int subspace containing a singly excited determinant of the B1(C2v) symmetry describing

61

the 3σ → 1π excitation and a doubly excited determinant of the A2(C2v) symmetry charac-

terizing the 3σ2 → 1π2 transition, respectively. The results of our CIPSI-driven CC(P ;Q)

calculations for the CH+ molecule in the ground-state equilibrium geometry, in which the

C–H bond length was set to R = Re = 2.13713 bohr, as well as the stretched geometry

with internuclear separation of R = 2Re = 4.27426 bohr are reported in Tables 3.4 and 3.5,

respectively.

We begin by discussing the CIPSI-driven CC(P ;Q) calculations for the ground and low-

lying excited states of the CH+ molecule at the equilibrium structure, shown in Table 3.4.

When the C–H bond length is set to R = Re, the 1 1Σ+ ground state is nondegenerate and

characterized by predominantly dynamical correlations. Similarly, most of the excited states

are dominated by one-electron excitations, except for the 2 1Σ+, 1 1∆, and 2 1∆ states of

CH+, which contain significant contributions from two-electron transitions corresponding to

3σ2 → 1π2 in the case of 21Σ+ and 11∆ and 2σ 3σ → 1π2 in the case of 21∆, and which require

an accurate treatment of the T3 and Rµ,3 correlation effects. As indicated by the Ndet(in) = 1

data in Table 3.4, the CR-CC(2,3) (for the 11Σ+ ground state) and CR-EOMCC(2,3) (excited

states) calculations are very accurate in describing the ground and excited states of CH+

at its equilibrium structure, and with the exception of the 2 1Σ+ and 2 1Π states, result in

energies that are within 1 millihartree of the parent CCSDT and EOMCCSDT data. Even

for the 2 1Σ+ and 2 1Π states of CH+, the CR-EOMCC(2,3) calculations are characterized by

rather modest errors of 1.373 and 2.805 millihartree relative to EOMCCSDT, respectively.

In analogy to the previously examples studying the F2 molecule and cyclobutadiene, the

CC(P ;Q) calculations based on the tiny Ndet(in) = 1, 000 CIPSI runs, which capture 1.4

% of Sz = 0 triply excited determinants of the A1(C2v) symmetry for the 1Σ+ states and

3.9 % of triply excited determinants of the Sz = 0 B1(C2v) symmetry for the 1Π states,

substantially improve the CR-EOMCC(2,3) results, reducing the 1.373 and 2.805 millihartree

errors relative to EOMCCSDT obtained with CR-EOMCC(2,3) for the 2 1Σ+ and 2 1Π states

of CH+ to 0.465 and 0.478 millihartree, respectively. The convergence of the CIPSI-driven

62

CC(P ;Q) calculations toward CCSDT/EOMCCSDT is so rapid that when the wave function

input parameter Ndet(in) is set to 5,000, which is minuscule when compared to the 31,912,

27,180, or 22,012, Sz = 0 triply excited determinants of the Σ+(C∞v), Π(C∞v), and ∆(C∞v)

symmetries included in the full CCSDT/EOMCCSDT calculations, respectively, the errors

in the resulting CC(P ;Q) energetics relative to CCSDT/EOMCCSDT range between 0.003

and 0.128 millihartree. As in the ground-state case, it is worth noting that the CIPSI-based

CC(P ;Q) calculations provide the CCSDT- and EOMCCSDT-level energetics using small

fractions of triply excited determinants in the underlying P spaces identified with the help

of extremely small CIPSI computations, which result in massive reductions in computational

effort compared to the parent CCSDT/EOMCCSDT methodologies.

Similar remarks about the performance of the CIPSI-driven CC(P ;Q) calculations in

converging the CCSDT and EOMCCSDT energetics apply when examining the results shown

in Table 3.5 for the CH+ molecule in the significantly stretched geometry. In analogy to

the case of bond breaking in F2, as the C–H bond length increases from R = Re to 2Re,

the strength of the T3 and Rµ,3 correlations increases substantially.

In particular, all of

the excited states of CH+ considered in this work acquire significant contributions from two-

electron excitations, which cannot be described in the basic EOMCCSD calculations. Indeed,

as seen in the Ndet(in) = 1 data reported in Table 3.5, the errors relative to EOMCCSDT

obtained with EOMCCSD for the 21Σ+, 31Σ+, 41Σ+, 11Π, 21Π, 11∆, and 21∆ states of CH+

are 17.140, 19.930, 32.639, 13.522, 21.200, 44.495, and 144.414 millihartree, respectively.

Evidently, when the C–H bond is stretched, none of the low-lying singlet excited states

of CH+ considered in this study can be quantitatively studied using the basic EOMCCSD

method, and for the 1∆ states, EOMCCSD cannot provide a qualitative description. In order

to cope with the substantial contributions from two-electron excitations in these excited

states, the T3 and Rµ,3 effects must be incorporated. As expected, the noniterative CR-

EOMCC(2,3) corrections help improve the poor performance of EOMCCSD, reducing the

17.140, 19.930, 32.639, 13.522, 21.200, 44.495, and 144.414 millihartree errors relative to

63

EOMCCSDT obtained using EOMCCSD for the 2 1Σ+, 3 1Σ+, 4 1Σ+, 1 1Π, 2 1Π, 1 1∆,

and 2 1∆ states of CH+ to 1.646, −2.869, 12.657, 2.304, −1.428, −4.524, and −63.405

millihartree, respectively. Although the noniterative CR-EOMCC(2,3) triples correction

provides a significant improvement over EOMCCSD, it is clear that ignoring the coupling

between the T3 and Rµ,3 components and their lower-rank T1, T2, Rµ,1, and Rµ,2 counterparts

in this case results in substantial errors when describing the 4 1Σ+ and 2 1∆ states of CH+

in addition to non-negligible errors for the remaining states.

When the C–H bond length is stretched, the CIPSI-driven CC(P ;Q) approach becomes

an indispensable and extremely efficient scheme for providing a highly accurate description

of the vertical excitation spectrum of CH+. As in the case of R = Re, one does not have to

perform much additional work to improve the CR-CC(2,3) and CR-EOMCC(2,3) results us-

ing the CIPSI-based CC(P ;Q) calculations. Again, when the CIPSI input parameter Ndet(in)

is set to 1,000, the large 12.657 and −63.405 millihartree errors relative to EOMCCSDT

obtained CR-EOMCC(2,3) for the 4 1Σ+ and 2 1∆ states of CH+ are reduced by nearly

a hundred-fold to 0.999 and 0.855 millihartree, respectively. At the same time, the 1.646,

−2.869, 2.304, −1.428, and −4.524 millihartree differences between the CR-EOMCC(2,3)

and EOMCCSDT energetics for the 2 1Σ+, 3 1Σ+, 1 1Π, 2 1Π, and 1 1∆ states of CH+ are

reduced to 1.095, −1.282, 0.047, 0.095, and 0.047 millihartree, respectively, when the CIPSI-

driven CC(P ;Q) calculations based on the Ndet(in) = 1, 000 CIPSI wave functions are em-

ployed. We note, once again, that the CIPSI wave functions computed using Ndet(in) = 1, 000

are tiny, containing just 1.8 % of the Sz = 0 A1(C2v)-symmetric triply excited determinants,

3.2 % of the Sz = 0 B1(C2v)-symmetric triply excited determinants, and 3.7 % of the Sz = 0

A2(C2v)-symmetric triply excited determinants, which means that the CC(P ;Q) steps are

not much more expensive than their CR-CC(2,3)/CR-EOMCC(2,3) counterparts and the

CIPSI diagonalizations, involving so few determinants, add negligible additional computa-

tional cost to the calculations. Similar to the R = Re case, increasing Ndet(in) from 1,000 to

5,000 provides a near-perfect description of the parent CCSDT/EOMCCSDT energetics.

64

Table 3.4 Convergence of the CIPSI-based CC(P )/EOMCC(P ) and CC(P;Q) energies toward CCSDT/EOMCCSDT for CH+,
calculated using the [5s3p1d/3s1p] basis set of Ref. [242], 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 CIPSI wave functions for the lowest states of the relevant symmetries. Each CIPSI run was initiated using the V (0)
int space
spanned by the appropriate determinant [the RHF state for the 1Σ+ states, the 3σ → 1π singly excited determinant of the
B1(C2v) symmetry for the 1Π states, and the 3σ2 → 1π2 doubly excited determinant of the A2(C2v) symmetry for the 1∆ states],
and the MBPT-based stopping parameter η and selection factor f were set at 10−6 hartree and 2, respectively. The Q spaces
used in constructing the CC(P;Q) corrections consisted of the triples not captured by CIPSI.

Ndet(in)

1d
1,000

P a
1.845

0.619

5,000

0.100

10,000 0.034

50,000 0.002

100,000 0.001

1 1Σ+
(P ;Q)b
0.063

0.017

0.003

0.001

0.000

0.000

2 1Σ+

3 1Σ+

4 1Σ+

%Tc
0

P a
19.694

1.4

8.5

9.744

1.351

16.3 0.404

(P ;Q)b
1.373

0.465

0.086

0.020

P a
3.856

2.275

0.200

0.031

(P ;Q)b
0.787

0.192

0.060

0.019

P a
5.537

1.793

0.310

0.149

(P ;Q)b P a
0.954

3.080

0.118

0.636

0.012

0.128

0.008

0.035

47.3 0.052

0.001 −0.078

0.001 −0.005

0.001

0.002

68.9 0.012

0.000 −0.107

−0.002 −0.007

0.001

0.000

1 1Π
(P ;Q)b
0.792

0.123

0.033

0.012

0.000

0.000

2 1Π

%Tc
0

P a
11.656

(P ;Q)b
2.805

P a
34.304

3.9

3.699

13.9 2.043

23.3 1.188

51.1 0.063

70.9 0.019

0.478

0.157

0.040

0.002

0.001

2.040

0.077

0.022

0.001

0.000

1 1∆
(P ;Q)b
−0.499

0.160

0.015

0.005

0.000

0.000

2 1∆

%Tc
0

P a
34.685

2.6 10.027

17.0 1.012

26.9 0.312

56.2 0.130

73.3 0.011

(P ;Q)b
0.350

0.425

0.056

0.013

0.003

0.000

aErrors, in millihartree, characterizing the CC(P ) (the 1 1Σ+ ground state) and EOMCC(P ) (excited states) energies relative to the corresponding
CCSDT and EOMCCSDT data, which are −38.019516, −37.702621, −37.522457, −37.386872, −37.900921, −37.498143, −37.762113, and −37.402308
hartree for the 1 1Σ+, 2 1Σ+, 3 1Σ+, 4 1Σ+, 1 1Π, 2 1Π, 1 1∆, and 2 1∆ states of CH+, respectively.
bErrors, in millihartree, in the CC(P ;Q) energies relative to the corresponding CCSDT and EOMCCSDT data provided in footnote (a).
bThe %T values are the percentages of triply excited determinants contained in the CIPSI wave function for the lowest state of a given symmetry
[the 1 1Σ+ = 1 1A1(C2v) ground state for the 1Σ+ states, the 1B1(C2v) component of the 1 1Π state for the 1Π states, and the 1A2(C2v) component
of the 1 1∆ state for the 1∆ states].
cThe CC(P ) and EOMCC(P ) energies at Ndet(in) = 1 are identical to the energies obtained in the CCSD and EOMCCSD calculations. The Ndet(in) = 1
CC(P;Q) energies are equivalent to the CR-CC(2,3) (the ground state) and the CR-EOMCC(2,3) (excited states) energies.

65

Table 3.5 Same as Table 3.4 for the stretched C–H internuclear distance R = 2Re = 4.27426 bohr.

Ndet(in)

1d
1,000

P a
5.002

0.434

5,000

0.060

10,000 0.015

50,000 0.001

100,000 0.000

1 1Σ+
(P ;Q)b
0.012

0.006

0.002

0.001

0.000

0.000

2 1Σ+

3 1Σ+

4 1Σ+

%Tc
0

P a
17.140

(P ;Q)b
1.646

P a
19.930

P a

(P ;Q)b
−2.869 32.639

P a

(P ;Q)b
12.657 13.552

1.8

7.7

7.456

0.852

15.6 0.333

40.4 0.016

57.3 0.004

1.095

10.848

−1.282 9.609

0.033

0.010

0.000

0.000

0.890

0.332

0.018

0.005

0.011

0.012

0.001

0.001

1.860

0.741

0.046

0.018

0.999

0.109

0.054

0.001

0.000

0.452

0.052

0.013

0.001

0.000

1 1Π
(P ;Q)b
2.304

0.047

0.006

0.002

0.000

0.000

2 1Π

%Tc
0

P a
21.200

P a

(P ;Q)b
−1.428 44.495

3.2

1.758

11.0 0.159

19.2 0.043

44.7 0.004

61.2 0.002

0.095

0.012

0.003

0.001

0.000

0.430

0.047

0.013

0.001

0.001

1 1∆
(P ;Q)b
−4.524

0.047

0.009

0.004

0.001

0.001

2 1∆

P a
144.414

(P ;Q)b
−63.405

8.160

0.401

0.123

0.006

0.002

0.855

0.010

0.002

0.000

0.000

%Tc
0

3.7

11.0

18.9

41.5

58.6

aErrors, in millihartree, characterizing the CC(P ) (the 1 1Σ+ ground state) and EOMCC(P ) (excited states) energies relative to the corresponding
CCSDT and EOMCCSDT data, which are −37.900394, −37.704834, −37.650242, −37.495275, −37.879532, −37.702345, −37.714180, and −37.494031
hartree for the 1 1Σ+, 2 1Σ+, 3 1Σ+, 4 1Σ+, 1 1Π, 2 1Π, 1 1∆, and 2 1∆ states of CH+, respectively.
bErrors, in millihartree, in the CC(P ;Q) energies relative to the corresponding CCSDT and EOMCCSDT data provided in footnote (a).
bThe %T values are the percentages of triply excited determinants contained in the CIPSI wave function for the lowest state of a given symmetry
[the 1 1Σ+ = 1 1A1(C2v) ground state for the 1Σ+ states, the 1B1(C2v) component of the 1 1Π state for the 1Π states, and the 1A2(C2v) component
of the 1 1∆ state for the 1∆ states].
cThe CC(P ) and EOMCC(P ) energies at Ndet(in) = 1 are identical to the energies obtained in the CCSD and EOMCCSD calculations. The Ndet(in) = 1
CC(P;Q) energies are equivalent to the CR-CC(2,3) (the ground state) and the CR-EOMCC(2,3) (excited states) energies.

66

CHAPTER 4

THE ADAPTIVE CC(P;Q) FRAMEWORK

In Chapter 3, we introduced a new formulation of CC(P ;Q) in which the content of the P and

Q spaces is automatically generated with the help of the information provided by external

CIPSI calculations. Similar to its semi-stochastic CC(P ;Q) counterpart, which exploited the

CIQMC or CCMC wave function propagations in order to identify the higher–than–doubly

excited determinants spanning the P and Q spaces, the CIPSI-driven CC(P ;Q) approach

successfully replaces the user- and system-dependent selection of active orbitals used in the

original active-space CC(P ;Q) hierarchy with more black-box non-CC considerations.

In general, however, the reliance on non-CC sources of information in the CC(P ;Q)

framework is not entirely ideal for several reasons. First of all, from a purely practical point

of view, it less convenient to have our CC(P ;Q) calculations based on interfacing multiple

independent computational modules, as this hinders the accessibility of these methods for

non-expert users. On a more fundamental level, the linear wave function ansatz adopted in

the CIPSI and CIQMC approaches does not ensure size-extensivity by construction, unlike

the CC/EOMCC calculations. Thus, one may wonder how reliable CI-based calculations

are in providing information about the importance of the connected contributions to higher-

rank excitations when studying larger many-electron systems (especially when attempting

to recover the thermodynamic limit, where all truncated CI approximations fail). Finally,

for most problems of chemical interest, any Hilbert space sampling procedure starting from

a one-dimensional initial subspace will likely spend significant time during its early stages

sampling the singly (for non-HF references) and doubly excited determinants in the many-

electron Hilbert space. This is a waste of time from the point of view of our CC(P ;Q)

calculations, which often include all singly and doubly excited determinants in the P spaces

a priori. Needless to say, it would be better to directly explore the higher-rank excitation

manifolds of interest using the information contained within the lower-level CC/EOMCC

wave functions present from the very beginning of the CC(P ;Q) calculations.

67

In this chapter, we aim to solve the above-mentioned challenges by formulating a novel

adaptive CC(P ;Q) approach capable of rapidly converging high-level CC/EOMCC energetics

of the CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ, and similar types in a fully auto-

mated fashion, even in cases of stronger correlations, which is free from the use of active

orbitals and information provided by the non-CC or stochastic sources exploited in the pre-

vious variants of CC(P ;Q). The key idea, which is discussed in Section 4.1, is an adaptive

selection of the excitation manifolds defining higher–than–two-body components of the clus-

ter and excitation operators driven by the CC(P ;Q) moment expansions. The benefits of

the resulting adaptive CC(P ;Q) methodology are illustrated in Section 4.2 using a number

of molecular examples, including the significantly stretched F2 and F+

2 molecules and au-

tomerization of cyclobutadiene (Section 4.2.1), the singlet–triplet gaps in organic biradicals

(Section 4.2.2), the degenerate Cope rearrangement in bullvalene (Section 4.2.3), and the

ground- and excited-state PESs of the water molecule along its O–H bond-breaking coordi-

nate (Section 4.2.4), where the goal is to recover the full CCSDT and EOMCCSDT energetics

when the noniterative triples corrections to CCSD/EOMCCSD struggle to provide a highly

accurate description, or fail entirely. Along with demonstrating the rapid convergence to-

ward the parent CCSDT/EOMCCSDT results, we discuss the CPU timings in Section 4.3 to

highlight vast reductions in the computational effort relative to the high-level CC/EOMCC

calculations offered by the adaptive CC(P ;Q) calculations.

4.1 Theory and Algorithmic Details

4.1.1 Adaptive CC(P ;Q): General considerations

We now proceed to describing the adaptive CC(P ;Q) methodology applicable to con-

verging any high-level CC or EOMCC theory of interest in a fully automated fashion. In

developing the adaptive CC(P ;Q) approach, we have been inspired by the CIPSI algorithm

[181, 183, 184], and especially its modern implementation in Refs. [183, 184]. As a result,

many of the elements entering our adaptive CC(P ;Q) approach have direct analogs in the

CIPSI framework, and because of this strong similarity, we will first briefly recapitulate the

68

key elements in CIPSI, which were discussed in Section 2.3.

We recall that the main idea of CIPSI is a series of Hamiltonian diagonalizations in in-

creasingly large, iteratively defined, subspaces of the many-electron Hilbert space, which are

followed by correcting the resulting energies using expressions originating from the second-

order many-body perturbation theory to estimate the remaining correlations. Adopting the

notation used in Section 2.3, if V (k)

int , where k = 0, 1, 2, . . . enumerates the consecutive CIPSI

iterations, designates the current diagonalization subspace, the V (k+1)

int

space for the subse-

quent Hamiltonian diagonalization is constructed by arranging the candidate determinants

|Φα⟩ from outside V (k)
int

for a potential inclusion in V (k+1)

int
the absolute values of the perturbative corrections e(2)

in descending order according to

α,k associated with them, starting from

the |Φα⟩s characterized by the largest |e(2)

α,k| contributions, moving toward those that have

smaller |e(2)

α,k| values, and continuing until the number of determinants in V (k+1)
desired dimension (in the CIPSI algorithm of Refs. [183, 184], until the dimension of V (k+1)

reaches a

int

int

exceeds that of V (k)

int by the user-defined factor f > 1). We can adopt a similar strategy in

designing P spaces for the CC(P;Q) computations.

Indeed, within the CC(P ;Q) framework, we can interpret the moment expansions Eq.

(2.33) as a sum of contributions δµ,K(P ; Q) due to the individual determinants from the Q

space, |ΦK⟩ ∈ H (Q), evaluated as

δµ,K(P ; Q) = ℓµ,K(P ) Mµ,K(P ),

(4.1)

which, in analogy to the perturbative e(2)

α,k corrections that measure the significance of the can-

didate determinants |Φα⟩ in CIPSI, determine the importance of the Q-space determinants

|ΦK⟩. One can, therefore, propose an adaptive, self-improving, CC(P;Q) scheme, in which

we construct an approximation to the high-level CC/EOMCC approach of interest (in the

numerical examples in Section 4.2, CCSDT and EOMCCSDT) by a series of CC(P;Q) calcu-

lations using increasingly large, iteratively defined, P spaces H (P )(k), where k = 0, 1, 2, . . .

enumerates the consecutive CC(P;Q) computations, with the corresponding Q subspaces

H (Q)(k) being defined as complementary excitation spaces, such that H (P )(k) ⊕ H (Q)(k) is

69

always equivalent to the entire excitation manifold appropriate for the CC/EOMCC method

we are targeting, independent of k (when targeting CCSDT/EOMCCSDT, all singly, doubly,

and triply excited determinants, when targeting CCSDTQ/EOMCCSDT, all singly, doubly,

triply, and quadruply excited determinants, etc.). With this iterative scheme in mind, Eq.

(2.33) adapted to the kth P space H (P )(k) and its complementary Q space H (Q)(k) con-

taining the excited determinants of interest not included in H (P )(k) can be written as

where

δµ(P (k); Q(k)) = X

δµ,K(P (k), Q(k)),

|ΦK ⟩∈H (Q)(k)

δµ,K(P (k), Q(k)) = ℓµ,K(P (k)) Mµ,K(P (k))

(4.2)

(4.3)

is the contribution to δµ(P (k); Q(k)) that corresponds to a given Q-space determinant |ΦK⟩ ∈

H (Q)(k). For clarity of this description, the P and Q symbols seen in Eqs. (4.2) and (4.3),

which represent the H (P )(k) and H (Q)(k) spaces used in the kth iteration of the adaptive

CC(P;Q) procedure, are labeled with the additional superscript (k).

The initial P space H (P )(0) can be a conveniently chosen zeroth-order excitation man-

ifold, such as the space of singly and doubly excited determinants, |Φa

i ⟩ and |Φab

ij ⟩, respec-

tively, and the remaining subspaces are constructed via a recursive process analogous to

that used in CIPSI, where the P space H (P )(k + 1) is obtained by augmenting its H (P )(k)

predecessor with a subset of the leading Q-space determinants |ΦK⟩ ∈ H (Q)(k) [when target-

ing CCSDT/EOMCCSDT, the leading triply excited determinants |Φabc

ijk⟩ outside H (P )(k),

when targeting CCSDTQ/EOMCCSDT, the leading triply and quadruply excited deter-

minants outside H (P )(k), etc.]

identified with the help of corrections δµ(P (k); Q(k)), Eq.

(4.2). In analogy to CIPSI, one can enlarge the current subspace H (P )(k) to construct the

H (P )(k + 1) space for the subsequent CC(P;Q) computation by arranging the candidate de-

terminants |ΦK⟩ ∈ H (Q)(k) in descending order according to the δµ,K(P (k); Q(k)) corrections

associated with them, starting from the |ΦK⟩s characterized by the largest |δµ,K(P (k); Q(k))|

70

contributions, moving toward those that have smaller |δµ,K(P (k); Q(k))| values, and contin-

uing until the total number of the Q-space determinants in H (P )(k + 1) reaches a certain

fraction of all higher-rank determinants relevant to the CC/EOMCC approach of interest

(e.g., triples when targeting CCSDT/EOMCCSDT or triples and quadruples when target-

ing CCSDTQ/EOMCCSDT). The adaptive CC(P ;Q) algorithm is also shown graphically in

Figure 4.1 below.

Clearly, the adaptive CC(P;Q) procedure, as described above, guarantees convergence

toward the high-level CC/EOMCC theory, but, following the computational cost analysis of

the CC(P;Q) methods [130, 133, 134] (cf., also, Refs. [126–129, 131, 132, 135, 136]), in order

to be an attractive approach, it has to be capable of recovering the target CC/EOMCC

energetics to a very good accuracy with small fractions of higher-rank determinants relevant

to the CC/EOMCC method of interest in the underlying P spaces.

In order to test the

validity and efficiency of this novel, self-driven CC(P ;Q) model, we focused on implementing

the adaptive CC(P ;Q) approach that aims at converging the CC/EOMCC calculations with

a full treatment of triples.

71

Figure 4.1 Schematic depiction of the adaptive CC(P ;Q) algorithm described in Section
4.2.1, which is generally applicable to converging the energetics corresponding to any high-
level CC/EOMCC calculation of interest. The recursive procedure begins with an initial
definition of the P space, which we always take to be spanned by all singly and doubly
excited determinants, and the corresponding Q space appropriately defined for the target
level of CC/EOMCC theory (e.g., triply excited determinants if the goal is to converge
CCSDT/EOMCCSDT or triply and quadruply excited determinants if one is interested in
converging CCSDT/EOMCCSDTQ, etc.), where the sum of P and Q spaces remains fixed
for all calculations executed during iterative adaptive CC(P ;Q) algorithm. Once the initial
P and Q spaces are defined, the adaptive CC(P ;Q) calculation is carried out following the
cyclic sequence of steps shown in this figure.

72

4.1.2 Adaptive CC(P ;Q) approaches aimed at CCSDT/EOMCCSDT and their

computational implementation

Our adaptive CC(P ;Q) methodology aimed at converging CCSDT/EOMCCSDT applies

Eqs. (4.2) and (4.3) to a situation where the kth P space H (P )(k) is spanned by all singly

and doubly excited determinants and a subset of triply excited determinants identified by the

adaptive CC(P;Q) algorithm and the associated Q space H (Q)(k) consists of the remaining

triples not included in H (P )(k), i.e.,

δµ(P (k); Q(k)) = X

δ(k)
ijk,abc(µ),

|Φabc

ijk ⟩∈H (Q)(k)

where

δ(k)
ijk,abc(µ) = ℓabc

µ,ijk(P (k)) Mijk

µ,abc(P (k))

(4.4)

(4.5)

is the individual energy correction corresponding to a given triply excited determinant

|Φabc

ijk⟩ ∈ H (Q)(k). The ℓabc

µ,ijk(P (k)) and Mijk

µ,abc(P (k)) quantities in Eq. (4.5) are the co-

efficients ℓµ,K(P ) and moments Mµ,K(P ) entering the CC(P;Q) correction δµ(P ; Q), Eq.

(2.33), adapted to the above definitions of the H (P )(k) and H (Q)(k) spaces.

In the adaptive CC(P;Q) code that we incorporated in the CCpy package available on

GitHub [228], the k = 0 P space H (P )(0), used to initiate the calculations, is defined as

all singly and doubly excited determinants and the corresponding Q space H (Q)(0) consists

of all triples. This means that the k = 0 CC(P )/EOMCC(P ) and CC(P;Q) energies are

respectively identical to those obtained with CCSD and CR-CC(2,3) for the ground state

and EOMCCSD and CR-EOMCC(2,3) for excited states. We then follow the recursive

procedure described in Section 4.1 by moving more and more triply excited determinants

from the Q to P spaces. We enlarge the kth subspace H (P )(k), which consists of all singles

and doubles and, when k > 0, the subset of triples identified in the previous iteration, to

construct the H (P )(k + 1) space for the subsequent CC(P;Q) computation by arranging

the candidate triply excited determinants |Φabc

ijk⟩ ∈ H (Q)(k) in descending order according

to the δ(k)

ijk,abc(µ) corrections associated with them, starting from the triples characterized

73

by the largest |δ(k)

ijk,abc(µ)| contributions, moving toward those that have smaller |δ(k)

ijk,abc(µ)|

values, and continuing until the total number of triply excited determinants in H (P )(k + 1)

represents an increase of the number of triples in H (P )(k) by fAd%, where fAd is a user-

defined growth rate (analogous to the selection factor f used in CIPSI). When enlarging the

H (P )(k), k = 0, 1, 2, . . . , spaces in our numerical tests to date, we have found that setting 0 <

fAd ≤ 1 results in optimal performance, allowing us to use as small fractions of triples in the

consecutive CC(P )/EOMCC(P ) runs as possible, so that the iterative CC(P )/EOMCC(P )

steps preceding the determination of the δµ(P (k); Q(k)) corrections are much less expensive

than those of the CCSDT or EOMCCSDT targets, while still providing rapid convergence

within respect to k. In medium-sized systems containing ∼20–30 correlated electrons and

∼100 orbitals, an upper bound of fAd = 1 is reasonable, since the numbers of triply excited

determinants used to enlarge the P spaces are not too large. When we examine even larger

systems, such as the bullvalene molecule studied in Section 4.2.3, which contains 50 correlated

electrons and 440 orbitals, 1% of the triples manifold corresponds to more than 109 triply

excited determinants. In these cases, we have also found success by setting the growth rate in

an automated fashion according to fAd = (NS + ND)/NT , where NS, ND, and NT represent

the numbers of singly, doubly, and triply excited determinants relevant to the electronic state

of interest, respectively. This choice of fAd grows the P spaces using the size of the CCSD

problem as an increment, which ensures that the resulting CC(P )/EOMCC(P ) and CC(P ;Q)

calculations do not become overwhelmed by the number of triply excited determinants.

In the adaptive CC(P;Q) computations reported in this work, we distinguish between the

relaxed and unrelaxed schemes. In the relaxed variant of the adaptive CC(P;Q), we solve the

CC(P)/EOMCC(P) equations for the singly, doubly, and triply excited cluster amplitudes

corresponding to the content of each H (P )(k) space and recompute the corresponding triples

corrections δ(k)

ijk,abc(µ) accordingly, increasing the number of triples, when going from H (P )(k)
to H (P )(k + 1), by fAd%. In the unrelaxed CC(P;Q) approach, we simply pick a particular

fraction of triples for inclusion in the P space (say, 3fAd% or 5fAd%) that have the largest

74

initial |δ(0)

ijk,abc(µ)| values determined using the T1, T2, Rµ,1, and Rµ,2 amplitudes obtained

with CCSD/EOMCCSD [i.e., the largest absolute values of the δ(0)

ijk,abc(µ) contributions to

the triples correction of CR-CC(2,3) or CR-EOMCC(2,3)] and solve the CC(P)/EOMCC(P)

equations for the singly, doubly, and triply excited cluster amplitudes with this particular

fraction of triples, adding the correction due to the missing T3 and Rµ,3 correlations using

Eq.

(4.2). The relaxed variant of the adaptive CC(P;Q) approach has an advantage of

approaching the target CCSDT/EOMCCSDT energetics as k → ∞, but the unrelaxed

scheme, which does not require an iterative construction of multiple H (P )(k) spaces, is less

expensive.

It is also worth mentioning that when implementing the adaptive CC(P;Q) approach

aimed at converging the ground-state CCSDT energetics, we have the option of invoking

the so-called two-body approximation, which was successfully used in some of the ear-

lier CC(P;Q)-related work [126–129, 214].

In the two-body approximation, the ground-

state moments Mijk

0,abc(P (k)) and coefficients ℓabc

0,ijk(P (k)) entering Eq. (4.5) are replaced by

ijk|H (P )(2)|Φ⟩ and ⟨Φ|(1 + L0,1 + L0,2) H (P )(2)|Φabc

⟨Φabc
ijk⟩), re-
spectively, where H (P )(2) = e−T1−T2HeT1+T2, with T1 and T2 designating the one- and

ijk|H (P )(2)|Φabc

ijk⟩/(E(P )

0 − ⟨Φabc

two-body components of the cluster operator T (P ) obtained in the CC(P ) calculations

in H (P )(k), is an approximation to the true similarity-transformed Hamiltonian H (P ) =

e−T1−T2−T (P )

3 HeT1+T2+T (P )

3

. The one- and two-body deexcitation operators L0,1 and L0,2,

which enter the formula for the ℓabc

0,ijk(P (k)) coefficients, are obtained by solving the left
eigenvalue problem involving H (P )(2) in the space spanned by singly and doubly excited de-

terminants [this step in which the ground-state left CC(P ) eigenvalue problem is replaced by

its simpler CCSD-like counterpart is not possible in the EOMCC(P ) case since the right and

left EOMCC eigenvectors must be obtained within the same subspace of the many-electron

Hilbert space]. Thus, in analogy to T (P ), the three-body component L(P )

0,3 of the deexcitation

operator L(P )

0

corresponding to the P space H (P )(k) is neglected. The two-body approxima-

tion avoids the more complex computational steps associated with the use of the complete

75

form of H (P ) in determining the δ(k)

ijk,abc(0) corrections, while preserving the philosophy of the

CC(P;Q) algorithm and accounting for the relaxation of T1, T2, L0,1, and L0,2 in the presence

of the three-body components of T (P ) obtained in the CC(P ) calculations in H (P )(k).

4.2 Numerical Examples

4.2.1 Bond Breaking in F2 and F+

2 and Automerization of Cyclobutadiene

In order to illustrate the benefits offered by the adaptive CC(P ;Q) methodology, we begin

by discussing our initial test of the relaxed and unrelaxed adaptive CC(P;Q) approaches

based on Eqs. (4.2) and (4.5), especially their ability to recover the parent CCSDT energetics

when the noniterative triples corrections to CCSD struggle, using the significantly stretched

F2 and F+

2 molecules and the reactant (R) and transition-state (TS) species involved in the

automerization of cyclobutadiene, along with the corresponding barrier height, as examples.

With the exception of F+

2 , these systems were examined earlier in Chapter 3 within the

context of the CIPSI-driven CC(P ;Q) calculations. In analogy to Sections 3.2.1 and 3.2.2,

we employed the cc-pVTZ basis set [10] to describe the fluorine molecule and its cation (Table

4.1), in which the respective F–F bond lengths r were stretched to 2re, where re represents

the equilibrium geometry (2.66816 bohr for F2 and 2.49822 bohr for F+

2 ), and the cc-pVDZ

basis [10] for cyclobutadiene (Table 4.2), where the geometries of the D2h-symmetric R and

D4h-symmetric TS species correspond to the structures taken from Ref. [262] optimized using

the MR-AQCC approach.

We chose the stretched F2 and F+

2 molecules and the automerization of cyclobutadiene

as initial examples because (i) all of these systems feature large, and in the case of F+

2 and

the TS species of cyclobutadiene, nonperturbative, T3 correlations and represent meaningful

tests for our CC(P ;Q)-based approximations to CCSDT (see the discussions in Refs. [126,

127, 134, 135, 137] for more precise remarks), and (ii) it allows us to maintain consistency

with previous studies on the active-orbital-based CC(t;3) [126, 127], CIQMC-/CCMC-based

semi-stochastic CC(P ;Q) [130, 134], and the CIPSI-driven CC(P ;Q) approach [135] (cf.

Sections 3.2.1 and 3.2.2), which also considered these examples. Here, we show that the

76

fully automated, ”black-box”, adaptive CC(P;Q) methodology, as implemented in our CCpy

package [228], is equally, and oftentimes more, efficient than its active-orbital-based, semi-

stochastic, and CIPSI-based CC(P ;Q) predecessors in improving the CR-CC(2,3) results and

generating the CCSDT-quality data at small fractions of the computational effort associated

with the conventional CCSDT calculations. In all of our adaptive CC(P ;Q) calculations for

F2, F+

2 , and cyclobutadiene, the two-body approximation was invoked in constructing the

CC(P ;Q) moment expansions and the core orbitals correlating with the 1s shells of fluorine

and carbon were kept frozen in the post-SCF steps.

Indeed, as shown in Tables 4.1 and 4.2, the convergence of the adaptive CC(P;Q) calcu-

lations toward the parent CCSDT energetics is very fast. This includes the more challenging

multireference situations created by the stretched F2 and F+

2 molecules and the TS structure

of cyclobutadiene, where, T3 correlations are large, nonperturbative, and difficult to capture,

resulting in failures of methods like CCSD(T) and, in the case of the latter two systems, of

CR-CC(2,3), as well as the weakly correlated cyclobutadiene R species, which the CCSD(T)

and CR-CC(2,3) methods can handle (see Refs. [127, 137]), although not perfectly. The

relaxed variant of the adaptive CC(P;Q) algorithm is generally most accurate. For the most

demanding cases of the stretched F+

2 and TS species of cyclobutadiene, where the coupling

of the lower-rank T1 and T2 clusters with their higher-rank T3 counterpart is the largest,

it reduces the 10.971 and 14.636 millihartree errors relative to CCSDT obtained with CR-

CC(2,3) and the similarly large errors obtained with CCSD(T) to a 0.1 millihartree level

using as little as 2–3% of all triply excited determinants in the underlying P spaces. With

only 2% of all triples in the P space, the difference between the activation energies character-

izing the automerization of cyclobutadiene obtained with the relaxed variant of the adaptive

CC(P;Q) approach and full CCSDT is less than 0.1 kcal/mol, as opposed to the orders of

magnitude larger, 8.653 kcal/mol, errors relative to CCSDT resulting from the CR-CC(2,3)

calculations. The rate with which the energies resulting from the adaptive CC(P;Q) calcu-

lations based on the relaxed algorithm approach the parent CCSDT energetics, observed in

77

Tables 4.1 and 4.2, is certainly most encouraging.

As one might anticipate, and as confirmed in Tables 4.1 and 4.2, the unrelaxed variant

of the adaptive CC(P;Q) methodology, in which one picks a particular fraction of triples

for inclusion in the P space based on their contributions to the CR-CC(2,3) correction to

CCSD, is less accurate for the stretched F+

2 molecule, the TS structure of cyclobutadiene, and

the associated barrier height than its relaxed counterpart, which reaches a desired fraction

of triply excited determinants in the P space through a sequence of recursively generated

subspaces, but the results of the unrelaxed CC(P;Q) calculations, especially given their sim-

plicity and lower computational costs, are excellent too. As shown in Tables 4.1 and 4.2,

with only 2–3% of all triply excited determinants in the underlying P spaces, the adaptive

CC(P;Q) computations based on the unrelaxed algorithm reduce the 10.971 millihartree,

14.636 millihartree, and 8.653 kcal/mol unsigned errors relative to CCSDT characterizing

the CR-CC(2,3) calculations for the r = 2re F+

2 system, the TS species involved in the

automerization of cyclobutadiene, and the corresponding barrier height, respectively, to a

chemical accuracy (1 millihartree or 1 kcal/mol) level. Compared to the adaptive CC(P;Q)

calculations using the relaxed scheme, the convergence rate toward the parent CCSDT ener-

getics characterizing the unrelaxed approach is slower, but the fact that one can obtain such

high accuracies with just a few percent of all triples in the underlying P spaces, when the

CR-CC(2,3) corrections to CCSD fail or struggle, is encouraging. It is worth noticing that

with only 1% of all triply excited determinants in the relevant P spaces, which is the smallest

fraction of triples considered in this work, the adaptive CC(P;Q) computations reduce the

10.971 millihartree, 14.636 millihartree, and 8.653 kcal/mol errors obtained with CR-CC(2,3)

for the challenging r = 2re F+

2 system, the TS structure of cyclobutadiene, and the activa-

tion energy characterizing the automerization of cyclobutadiene relative to CCSDT to 2.173

millihartree, 0.601 millihartree, and 0.412 kcal/mol, respectively. Given the fact that 1% is

also the incremental fraction of triples used to enlarge the H (P )(k) spaces in the relaxed

CC(P;Q) calculations reported in this study, the relaxed and unrelaxed CC(P;Q) compu-

78

tations using the leading 1% of all triply excited determinants identified by the adaptive

CC(P;Q) algorithm are equivalent.

While the P spaces containing only 1% of all triply excited determinants may not be rich

enough to bring the errors characterizing the adaptive CC(P;Q) calculations for the stretched

F+

2 ion and the TS structure of cyclobutadiene relative to CCSDT to a 0.1 millihartree

level, they are sufficient for achieving high accuracies of this type in the adaptive CC(P;Q)

computations for the stretched fluorine molecule and the cyclobutadiene R species. As

shown in Tables 4.1 and 4.2, the adaptive CC(P;Q) calculations for the r = 2re F2 system

and the R structure of cyclobutadiene using only 1% of all triply excited determinants in

the underlying P spaces reduce the 4.254 and 0.848 millihartree differences between the

CR-CC(2,3) and CCSDT energies to about 60 microhartree. Clearly, these are dramatic

improvements, especially given the small computational effort involved. Furthermore, unlike

in the r = 2re F+

2 and cyclobutadiene TS systems, the results of the adaptive CC(P;Q)

computations for the stretched fluorine molecule and the R species of cyclobutadiene using

larger fractions of triples in the corresponding P spaces do not change much when the

relaxed algorithm is replaced by its simpler unrelaxed counterpart. We can rationalize these

observations as follows. In the case of the stretched F2 molecule, T3 correlations are large and

nonperturbative, so that one is much better off by using the CR-CC(2,3) triples correction

to CCSD instead of perturbative approaches like CCSD(T), but the coupling of T1 and

T2 clusters with T3 is not as well pronounced as in the stretched F+

2 and TS species of

cyclobutadiene and can, therefore, be captured by injecting a tiny fraction of the leading

triply excited determinants into the CC(P) calculations preceding the determination of the

noniterative δ0(P ; Q) correction. As a result, the incorporation of a larger fraction of triples in

the underlying P space and the iterative construction of the P space via the relaxed algorithm

are not necessary to obtain high accuracies in the adaptive CC(P;Q) computations for the

stretched F2. In the case of the R species of cyclobutadiene, T3 correlations are relatively

small, perturbative, and largely decoupled from those captured by T1 and T2 clusters and

79

their powers entering the CC wave function ansatz, making the use of larger fractions of

triples in the P space and the iteratively constructed H (P )(k) spaces less necessary.

Table 4.1 Convergence of the energies resulting from the relaxed (Rel.) and unrelaxed (Un-
rel.) variants of the adaptive CC(P;Q) approach and the underlying CC(P) computations
toward CCSDT for the F2 and F+
2 molecules described by the cc-pVTZ basis set, in which
the F–F bond lengths r were fixed at 2re, with re representing the relevant equilibrium ge-
ometries (2.66816 bohr for F2 and 2.49822 bohr for F+
2 ). The %T values are the percentages
of the triply excited determinants of the Sz = 0 A1g(D2h), for F2, and Sz = 1/2 B3g(D2h),
for F+
2 , symmetries identified by the adaptive CC(P ;Q) algorithm and included, alongside
all singles and doubles, in the respective P spaces. The Q spaces used in computing the
CC(P;Q) corrections were defined as the remaining triples not included in the associated P
spaces. In increasing the numbers of triply excited determinants in the P spaces employed
in the relaxed calculations, a 1% growth rate was assumed throughout. In all post-RHF (F2)
and post-ROHF (F+
2 ) calculations, the two lowest core orbitals were kept frozen. Adapted
from Ref. [137].

F2

% T CC(P) CC(P;Q) CC(P) CC(P;Q)

Unrel.a
62.819b
3.076
2.103
1.586
1.243
1.009

0
1d
2
3
4
5
100

Unrel.a
4.254c
0.063
0.089
0.104
0.098
0.105

Rel.a
62.819b
3.076
2.052
1.539
1.212
0.985
−199.238344e

Rel.a
4.254c
0.063
0.057
0.070
0.071
0.080

CC(P)
Unrel.a
76.291b
6.071
4.061
2.970
2.100
1.680

F+
2

CC(P;Q)
Unrel.a
10.971c
2.173
1.560
1.146
0.684
0.549

CC(P)
Rel.a
76.291b
6.071
2.599
1.707
1.330
1.077

−198.606409e

CC(P;Q)
Rel.a
10.971c
2.173
0.191
−0.026
−0.007
0.010

aThe CC(P) and CC(P;Q) energies are reported as errors relative to CCSDT in millihartree.
bEquivalent to CCSD.
cEquivalent to CR-CC(2,3).
dFor %T = 1, the CC(P) and CC(P;Q) energies obtained in the relaxed and unrelaxed calculations are
identical.
eTotal CCSDT energy in hartree.

80

Table 4.2 Convergence of the energies resulting from the relaxed (Rel.) and unrelaxed (Un-
rel.) variants of the adaptive CC(P;Q) approach and the underlying CC(P) computations
toward CCSDT for the reactant (R) and transition-state (TS) structures involved in the
automerization of cyclobutadiene, as described by the cc-pVDZ basis set, optimized in the
MR-AQCC calculations reported in Ref. [262], along with the corresponding barrier heights.
The %T values are the percentages of the Sz = 0 triply excited determinants of the A1g(D2h)
symmetry identified by the adaptive CC(P ;Q) algorithm and included, alongside all singles
and doubles, in the respective P spaces. In analogy to F2 and F+
2 , the Q spaces adopted
in computing the CC(P;Q) corrections consisted of the triply excited determinants not in-
cluded in the associated P spaces and in increasing the numbers of triples in the P spaces
used in the relaxed calculations, a 1% growth rate was assumed throughout. In all post-RHF
calculations, the four lowest core orbitals were kept frozen. Adapted from Ref. [137].

R

TS

Barrier Height

%T CC(P) CC(P;Q) CC(P) CC(P;Q) CC(P) CC(P;Q) CC(P) CC(P;Q) CC(P;Q) CC(P;Q)

Unrel.a Unrel.a
26.827c
0.848d
12.622 −0.055
10.143 −0.013
0.016
8.610
0.037
7.501
0.050
6.637

Rel.a
Rel.a
26.827c
0.848d
12.622 −0.055
10.121 −0.030
−0.002
8.588
0.022
7.482
6.620
0.035
−154.244157f

0
1e
2
3
4
5
100

Unrel.a
47.979c
12.622
10.180
8.643
7.571
6.722

Unrel.a
14.636d
0.601
0.598
0.561
0.568
0.559

Rel.a
47.979c
12.622
9.250
7.821
6.827
6.052
−154.232002f

Rel.a
14.636d
0.601
−0.169
−0.111
−0.040
0.010

Unrel.b
8.653d
0.412
0.384
0.343
0.334
0.320

Rel.b
8.653d
0.412
−0.087
−0.068
−0.038
−0.016

7.627g

aThe CC(P) and CC(P;Q) energies of the R and TS species are reported as errors relative to CCSDT in
millihartree.
bThe CC(P;Q) values of the barrier height are reported as errors relative to CCSDT in kcal/mol.
cEquivalent to CCSD.
dEquivalent to CR-CC(2,3).
eFor %T = 1, the CC(P) and CC(P;Q) energies obtained in the relaxed and unrelaxed calculations are
identical.
fTotal CCSDT energy in hartree.
gThe CCSDT barrier height in kcal/mol.

81

4.2.2 Singlet–Triplet Gaps in Organic Biradicals

Our next application of the adaptive CC(P ;Q) approach demonstrates its usefulness in

converging the CCSDT data characterizing the total energies of the lowest-lying singlet and

triplet states, along with the corresponding singlet–triplet gaps, in three organic biradicals,

namely, cyclobutadiene, cyclopentadienyl cation, and trimethylenemethane. The nature of

the lowest-lying singlet and triplet states in biradicals, along with the associated ∆ES−T

values (throughout this work, we define ∆ES−T as ES − ET, where ES and ET are the

electronic energies of the relevant singlet and triplet states), is a fascinating problem from

both an experimental and theoretical perspective, as biradicals often play key roles as reac-

tion intermediates in thermal and photochemical pathways [271–277] as well as functional

materials used in molecular magnets [278–280], battery electrodes [281], and organic photo-

voltaics [282–285]. The accurate computational determination of the singlet–triplet gaps in

biradicals remains, however, a difficult task because it requires balancing the strong nondy-

namical many-electron correlation effects characterizing the low-spin singlet states with the

predominantly dynamical correlations associated with their high-spin triplet counterparts

[128, 136, 149, 265, 286–298].

In this section, we explore the performance of both the relaxed and unrelaxed adaptive

CC(P ;Q) approaches in recovering the CCSDT-level energetics of the lowest singlet and

triplet states of cyclobutadiene, cyclopentadienyl cation, and trimethylenemethane, as well

as the corresponding singlet–triplet gaps. Our choice of molecules was motivated by the

recent semi-stochastic CC(P ;Q) study in Ref. [136], where it was demonstrated that the

CIQMC-based CC(P ;Q) approach was rather effective in converging the results for these

three systems obtained with CCSDT calculations using fractions of the computational ef-

fort. Therefore, it is interesting to examine whether the newly developed adaptive CC(P ;Q)

approach is capable of doing the same, but in a fully automated fashion that is free from

the active-orbital-based or external QMC and selected CI considerations adopted in the ear-

lier CC(P ;Q) work. Following Ref. [136], our CC calculations for the lowest-energy singlet

82

states of cyclobutadiene, cyclopentadienyl cation, and trimethylenemethan were carried out

using the respective Sz = 0 RHF determinants as the Fermi vacuum, whereas their lowest

triplet counterparts were determined using CC computations employing the corresponding

high-spin Sz = 1 ROHF determinants as the reference function. As in the previous study

discussed in Section 4.2.1, we are especially interested in how effective the adaptive CC(P ;Q)

calculations are in improving the CR-CC(2,3) energetics and converging the parent CCSDT

data as a function of the numbers of triply excited determinants included in the underlying P

spaces. The results of our adaptive CC(P ;Q) computations for the lowest-lying singlet and

triplet states of cyclobutadiene, cyclopentadienyl cation, and trimethylenemethane, along

with the corresponding vertical (cyclobutadiene and cyclopentadienyl cation) or adiabatic

(trimethylenemethane) singlet–triplet gap values are reported in Tables 4.3–4.5. As in our

adaptive CC(P ;Q) calculations for F2, F+

2 , and cyclobutadiene discussed in the previous

section, we employed the two-body approximation to evaluate the triples correction δ0(P ; Q)

and the lowest-energy orbitals correlating with the 1s shells of carbon atoms were frozen in

all post-RHF/ROHF steps.

We begin our discussion by pointing out that the CR-CC(2,3) approach, represented by

the % T = 1 data in Tables 4.3–4.4, is generally unable to provide accurate ∆ES−T values for

the three biradicals considered in this study. As shown in Tables 4.3–4.4, the errors in the

singlet–triplet gaps for cyclobutadiene, cyclopentadienyl cation, and trimethylenemethane

relative to full CCSDT obtained using CR-CC(2,3) are 9.222, 3.765, and 8.128 kcal/mol,

respectively. The error is most pronounced in the case of cyclobutadiene, where the CR-

CC(2,3) calculations are unable to correctly identify its lowest-energy singlet state as the

ground state (the determination of the ground state of cyclobutadiene has a long history in

quantum chemistry; see, e.g., Refs. [299–303] for some of the earliest ab initio calculations

predicting the X 1B1g ground state, which were confirmed in many subsequent theoretical

studies, such as those reported in Refs. [136, 254, 262, 288, 293–295, 297, 298, 304–306]).

For all three biradicals considered here, we should note that the failure in CR-CC(2,3) is al-

83

most entirely due to its inability to accurately describe the relevant open-shell singlet state.

Indeed, when we compare the errors relative to CCSDT obtained using CR-CC(2,3) for

the lowest-lying singlet states of cyclobutadiene (X 1B1g), cyclopentadienyl cation (A 1E′

2),

and trimethylenemethane (B 1A1), which are 14.636, 6.245, and 13.371 millihartree, respec-

tively, to the much smaller −0.060, 0.245, and 0.418 millihartree errors characterizing their

A 3A2g, X 3A′

2, and X 3A′

2 counterparts, it becomes clear that the large errors in the singlet–

triplet gap values obtained using CR-CC(2,3) arise from the inability of CR-CC(2,3) to

properly balance the description of the lowest high-spin triplet states, which are of largely

SR character and dominated by dynamical correlations, with the stronger MR correlations

characterizing their two-reference open-shell singlet counterparts. Failure of CR-CC(2,3)

[126, 127, 136] and, as demonstrated, for example, in Refs. [127, 255], of other nonitera-

tive triples corrections to CCSD, including CCSD(T) [127, 255], to accurately describe the

lowest-energy singlet states of biradicals is, in significant part, a consequence of the inability

of all such methods to capture the coupling of the lower-rank T1 and T2 clusters with T3,

which can become large, nonperturbative, and significant enough to substantially alter T1

and T2 amplitudes compared to their CCSD values. This means that to improve the de-

scription of the lowest-energy singlet state in cyclobutadiene, cyclopentadienyl cation, and

trimethylenemethane, and obtain accurate ∆ES−T values, one should relax the T1 and T2

clusters, adjusting them to the presence of T3 correlations, prior to determining noniterative

triples corrections. The adaptive CC(P ;Q) methodology allows us to do this in a computa-

tionally efficient manner, avoiding expensive CCSDT iterations, by incorporating the leading

triply excited determinants identified with the help of the δ0(P ; Q) corrections, Eqs. (4.2)

and (4.5), into the underlying P spaces and correcting the resulting CC(P ) energies for the

remaining T3 effects using the CC(P ;Q) moment expansions.

When we move on to examining the adaptive CC(P ;Q) calculations that include the

leading 1 % of triply excited determinants in the underlying P spaces, reported in Table

4.3–4.5, which for the sake of brevity in this discussion is abbreviated as CC(P ;Q)[%T =

84

1], we immediately see the benefits of relaxing T1 and T2 clusters in the presence of lead-

ing T3 correlations. The large 14.636, 6.245, and 13.371 millihartree differences between

the CR-CC(2,3) and CCSDT energetics for the X 1B1g state of cyclobutadiene, the A 1E′
2

state of cyclopentadienyl cation, and the B 1A1 state of trimethylenemethane are respec-

tively reduced to 0.601, 1.322, and 0.171 millihartree when the CR-CC(2,3) calculations are

replaced by their adaptive CC(P ;Q)[%T = 1] counterparts. At the same time, the errors

relative to CCSDT characterizing the adaptive CC(P ;Q)[%T = 1] calculations for the cor-

responding lowest-energy triplet states, which are −0.187 millihartree for the A 3A2g state

of cyclobutadiene, 0.031 millihartree for the X 3A′

2 state of cyclopentadienyl cation, and

0.253 millihartree for the X 3A′

2 state of trimethylenemethane, are roughly the same (cy-

clobutadiene) or improved (cyclopentadienyl cation and trimethylenemethane) compared to

the already very accurate results obtained using CR-CC(2,3). As a result of relaxing the

T1 and T2 clusters in the presence of higher-order correlation effects prior to determining

the CC(P ;Q) triples corrections in the calculations for the lowest-energy singlet states of cy-

clobutadiene, cyclopentadienyl cation, and trimethylenemethane, the corresponding ∆ES−T

values obtained with the adaptive CC(P ;Q)[%T = 1] approach are substantially more ac-

curate than their counterparts obtained in the CR-CC(2,3) calculations. In particular, the

adaptive CC(P ;Q)[%T = 1] calculations reduce the 9.222, 3.765, and 8.128 kcal/mol errors

relative to full CCSDT characterizing the singlet–triplet gaps of cyclobutadiene, cyclopen-

tadienyl cation, and trimethylenemethane computed with CR-CC(2,3) to 0.495, 0.810, and

−0.052 kcal/mol, respectively. In a similar fashion to our study testing the adaptive CC(P ;Q)

approach on bond breaking in F2 and F+

2 and automerization of cyclobutadiene, we observe

that the least expensive CC(P ;Q) calculations considered in our study, which use just 1%

of triply excited determinants in the underlying P spaces, are capable of reducing the er-

rors obtained using CR-CC(2,3) and providing a chemically accurate description (within 1

millihartree or 1 kcal/mol) relative to the parent CCSDT approach.

Naturally, we can study the convergence of the adaptive CC(P ;Q) calculations toward

85

the full CCSDT limit as the fractions of triply excited determinants entering the correspond-

ing P spaces are increased. As shown in Tables 4.3–4.5, the relaxed variant of the adaptive

CC(P ;Q) calculations using 2–5% of triply excited determinants in the underlying P spaces

(similarly abbreviated as CC(P ;Q)[%T = n], for n = 2–5) demonstrably converge the parent

CCSDT results. Focusing on the description of the ∆ES−T values, the 0.495 kcal/mol error

in the singlet–triplet gap of cyclobutadiene relative to CCSDT obtained with the adaptive

CC(P ;Q)[%T = 1] calculations are reduced to −0.046, −0.040, −0.015, and 0.003 kcal/mol

when the fractions of triply excited determinants entering the relaxed adaptive CC(P ;Q) are

increased to 2% , 3%, 4%, and 5%, respectively. In a similar fashion, the 0.810 kcal/mol

error in the ∆ES−T value relative to CCSDT obtained with the adaptive CC(P ;Q)[%T =

1] calculations are reduced to −0.076, −0.128, −0.111, and 0.096 kcal/mol in the relaxed

adaptive CC(P ;Q) computations employing 2% , 3%, 4%, and 5% of triply excited deter-

minants, respectively. In the case of trimethylenemethane, the singlet–triplet gap obtained

using the initial adaptive CC(P ;Q)[%T = 1] calculations was already very accurate, but it is

reassuring to observe that the −0.052 kcal/mol error relative to CCSDT obtained using the

CC(P ;Q) calculations with 1% of triply excited determinants remains to within (−0.133)–

(−0.067) kcal/mol of the CCSDT data when the fractions of triply excited determinants are

increased from 1% to 2–5%.

It is interesting to note, however, that the relaxed adaptive CC(P ;Q)[%T = n] calcu-

lations with n > 1 are more accurate than their unrelaxed counterparts when describing

the lowest-lying singlet states of cyclobutadiene, cyclopentadienyl cation, and trimethylen-

emethane. Indeed, when we compare the differences between the total energies of the lowest-

lying singlet states of cyclobutadiene, cyclopentadienyl cation, and trimethylenemethane

obtained in the adaptive CC(P ;Q)[%T = 1] calculations and CCSDT with their counter-

parts corresponding to the unrelaxed adaptive CC(P ;Q)[%T = 5] approach, we see that

the unrelaxed selection scheme is not quite as effective as the relaxed variant in presence of

stronger MR correlations. For example, the 0.601 millihartree error relative to CCSDT ob-

86

tained in the adaptive CC(P ;Q)[%T = 1] calculations for the X 1B1g state of cyclobutadiene

becomes 0.559 millihartree when the unrelaxed CC(P ;Q)[%T = 5] approach is employed.

Similarly minor changes are observed for the A 1E′

2 state cyclopentadienyl cation, in which

the 1.322 millihartree differences between the energy obtained in the adaptive CC(P ;Q)[%T

= 1] and CCSDT become 0.860 millihartree when 5% of triply excited determinants are

used in the underlying P spaces defining the unrelaxed CC(P ;Q) methodology. In the case

of trimethylenemethane, the 0.171 millihartree error relative to CCSDT resulting from the

initial adaptive CC(P ;Q)[%T = 1] computations characterizing the B 1A1 state actually

increases slightly to 0.245 millihartree when the CC(P ;Q)[%T = 1] approach is replaced

with the unrelaxed CC(P ;Q)[%T = 5] method. Thus, we can conclude that the relaxed

adaptive CC(P ;Q) model is more effective than its unrelaxed counterpart in converging the

full CCSDT description of the lowest-energy singlet and triplet state, along with the gap

between them, for the cyclobutadiene, cyclopentadienyl cation, and trimethylenemethane

biradicals, especially if a precision beyond standard chemical accuracy is desired. Although

the better performance of the relaxed adaptive CC(P ;Q) algorithm is to be expected (and

also demonstrated in our calculations discussed in Section 4.2.1), given that the relaxed

adaptive CC(P ;Q) calculations de facto determine the lists of leading higher–than–doubly

excited determinants entering the CC(P ;Q) steps using a more accurate description of many-

electron correlation effects than their unrelaxed counterparts, one should keep in mind that

this improvement in accuracy comes with higher computational costs, since, as mentioned in

Section 4.1, the relaxed adaptive CC(P ;Q) model requires an iterative construction of mul-

tiple H (P )(k) spaces. Therefore, while both the unrelaxed and relaxed adaptive CC(P ;Q)

models are highly efficient and viable tools for studying the electronic structure of organic

biradicals, the specific choice of algorithm should be carefully selected by considering the

balance of computational costs with the desired accuracy levels for the application of interest.

87

Table 4.3 Convergence of the energies resulting from the relaxed (Rel.) and unrelaxed
(Unrel.) variants of the adaptive CC(P;Q) approach and the underlying CC(P) computa-
tions for the X 1B1g and A 3A2g states of cyclobutadiene, as described by the cc-pVDZ basis
set, and of the corresponding singlet-triplet gap, toward their parent CCSDT values. All
calculations were performed at the D4h-symmetric transition-state geometry on the X 1B1g
potential optimized in the MR-AQCC calculations reported in Ref. [262]. The %T values
are the percentages of the triply excited determinants of the Sz = 0 A1g(D2h) symmetry,
for the X 1B1g state, and the Sz = 1 B1g(D2h), for the A 3A2g state, identified by the adap-
tive CC(P ;Q) algorithm and included, alongside all singles and doubles, in the respective P
spaces. In computing the CC(P;Q) corrections , the Q space consisted of the triply excited
determinants not included in the associated P spaces and in increasing the numbers of triples
in the P spaces used in the relaxed calculations, a 1% growth rate was assumed throughout.
In all post-RHF/ROHF calculations, the four lowest core orbitals were kept frozen.

%T CC(P) CC(P;Q) CC(P) CC(P;Q) CC(P) CC(P;Q) CC(P) CC(P;Q) CC(P;Q) CC(P;Q)

X 1B1g

A 3A2g

X 1B1g − A 3A2g

Unrel.a Unrel.a
14.636d
47.979c
0.601
12.622
0.598
10.180
0.561
8.643
0.568
7.571
0.559
6.722

Rel.a
47.979c
12.622
0.250
7.821
6.827
6.052
−154.232002f

0
1
2
3
4
5
100

Rel.a
14.636d
0.601
−0.169
−0.111
−0.040
0.010

Rel.a

Unrel.a
Unrel.a
23.884c −0.060d
−0.187
11.565
−0.097
9.290
−0.048
7.865
−0.018
6.833
0.003
6.029

Rel.a
23.884c −0.060d
−0.187
11.565
−0.095
9.292
−0.046
7.867
−0.016
6.835
0.005
6.031
−154.224380f

Unrel.b
9.222d
0.495
0.436
0.383
0.368
0.349

Rel.b
9.222d
0.495
−0.046
−0.040
−0.015
0.003

−4.783g

aThe CC(P) and CC(P;Q) energies of the X 1B1g and A 3A2g states of cyclobutadiene are reported as errors
relative to CCSDT in millihartree.
bThe CC(P;Q) values of the singlet–triplet gap are reported as errors relative to CCSDT in kcal/mol.
cEquivalent to CCSD.
dEquivalent to CR-CC(2,3).
eFor %T = 1, the CC(P) and CC(P;Q) energies obtained in the relaxed and unrelaxed calculations are
identical.
fTotal CCSDT energy in hartree.
gThe CCSDT singlet–triplet gap in kcal/mol.

88

2 and A 1E′

Table 4.4 Convergence of the energies resulting from the relaxed (Rel.) and unrelaxed (Un-
rel.) variants of the adaptive CC(P;Q) approach and the underlying CC(P) computations
for the X 3A′
2 states of cyclopentadienyl cation, as described by the cc-pVDZ basis
set, and of the corresponding singlet–triplet gap toward their parent CCSDT values. All cal-
culations were performed at the D5h-symmetric geometry of the X 3A′
2 state optimized using
the unrestricted CCSD/cc-pVDZ approach in Ref. [288]. The %T values are the percentages
of the triply excited determinants of the Sz = 1 B1(C2v) symmetry, for the X 3A′
2 state,
and the Sz = 0 A1(C2v), for the A 1E′
2 state, identified by the adaptive CC(P ;Q) algorithm
and included, alongside all singles and doubles, in the respective P spaces. In computing
the CC(P;Q) corrections consisted of the triply excited determinants not included in the
associated P spaces and in increasing the numbers of triples in the P spaces used in the
relaxed calculations, a 1% growth rate was assumed throughout. In all post-RHF/ROHF
calculations, the five lowest core orbitals were kept frozen.

%T CC(P) CC(P;Q) CC(P) CC(P;Q) CC(P) CC(P;Q) CC(P) CC(P;Q) CC(P;Q) CC(P;Q)

X 3A′
2

A 1E′
2

A 1E′

2 − X 3A′
2

Unrel.a Unrel.a
0.245d
28.840c
0.031
12.614
0.084
10.026
0.107
8.436
0.119
7.296
0.145
7.461

Rel.a
28.840c
12.614
10.026
8.437
7.298
6.420
−192.615924f

0
1e
2
3
4
5
100

Rel.a
0.245d
0.031
0.086
0.110
0.123
0.129

Unrel.a
38.572c
14.613
11.630
9.856
8.597
7.619

Unrel.a
6.245d
1.322
1.085
0.984
0.918
0.860

Rel.a
38.572c
14.613
10.418
8.704
7.562
6.681
−192.589235f

Rel.a
6.245d
1.322
−0.035
−0.094
−0.054
−0.023

Unrel.b
3.765d
0.810
0.628
0.550
0.501
0.448

Rel.b
3.765d
0.810
−0.076
−0.128
−0.111
−0.096

16.747g

2 and A 1E′

2 states of cyclopentadienyl cation are reported

aThe CC(P) and CC(P;Q) energies of the X 3A′
as errors relative to CCSDT in millihartree.
bThe CC(P;Q) values of the singlet–triplet gap are reported as errors relative to CCSDT in kcal/mol.
cEquivalent to CCSD.
dEquivalent to CR-CC(2,3).
eFor %T = 1, the CC(P) and CC(P;Q) energies obtained in the relaxed and unrelaxed calculations are
identical.
fTotal CCSDT energy in hartree.
gThe CCSDT singlet–triplet gap in kcal/mol.

89

Table 4.5 Convergence of the energies resulting from the relaxed (Rel.) and unrelaxed (Un-
rel.) variants of the adaptive CC(P;Q) approach and the underlying CC(P) computations
2 and B 1A1 states of trimethylenemethane, as described by the cc-pVDZ basis
for the X 3A′
set, and of the corresponding adiabatic singlet–triplet gap toward their parent CCSDT val-
ues. The D3h- and C2v-symmetric geometries of the X 3A′
2 and B 1A1 states, respectively,
optimized in the SF-DFT/6-31G(d) calculations, were taken from Ref. [286]. The %T values
are the percentages of the triply excited determinants of the Sz = 1 B1(C2v) symmetry,
for the X 3A′
2 state, identified by the adap-
tive CC(P ;Q) algorithm and included, alongside all singles and doubles, in the respective P
spaces. In computing the CC(P;Q) corrections consisted of the triply excited determinants
not included in the associated P spaces and in increasing the numbers of triples in the P
spaces used in the relaxed calculations, a 1% growth rate was assumed throughout. In all
post-RHF/ROHF calculations, the four lowest core orbitals were kept frozen.

2 state, and the Sz = 0 A1(C2v), for the A 1E′

%T CC(P) CC(P;Q) CC(P) CC(P;Q) CC(P) CC(P;Q) CC(P) CC(P;Q) CC(P;Q) CC(P;Q)

X 3A′
2

B 1A1

B 1A1 − X 3A′
2

Unrel.a Unrel.a
0.418d
19.202c
0.253
9.630
0.241
7.729
0.227
6.536
0.215
5.672
0.204
5.000

Rel.a
19.202c
9.630
7.729
6.536
5.673
5.001
−155.466242f

0
1e
2
3
4
5
100

Rel.a
0.418d
0.253
0.242
0.229
0.218
0.206

Unrel.a
58.052c
9.487
7.506
6.309
5.456
4.801

Unrel.a
13.371d
0.171
0.207
0.229
0.240
0.245

Rel.a
58.052c
9.487
7.255
6.080
5.250
4.616
−155.431596f

Rel.a
13.371d
0.171
0.031
0.064
0.085
0.099

Unrel.b
8.128d
−0.052
−0.021
0.001
0.015
0.026

Rel.b
8.128d
−0.052
−0.133
−0.104
−0.083
−0.067

21.740g

2 and B 1A1 states of trimethylenemethane are reported as

aThe CC(P) and CC(P;Q) energies of the X 3A′
errors relative to CCSDT in millihartree.
bThe CC(P;Q) values of the singlet–triplet gap are reported as errors relative to CCSDT in kcal/mol.
cEquivalent to CCSD.
dEquivalent to CR-CC(2,3).
eFor %T = 1, the CC(P) and CC(P;Q) energies obtained in the relaxed and unrelaxed calculations are
identical.
fTotal CCSDT energy in hartree.
gThe CCSDT singlet–triplet gap in kcal/mol.

90

4.2.3 The Cope Rearrangement of Bullvalene

As a final example to demonstrate the capabilities of our adaptive CC(P ;Q) approach

in converging the CCSDT energetics, we applied it to study the degenerate Cope rearrange-

ment in bullvalene (C10H10), depicted in Figure 4.2. First predicted by Doering and Roth

Figure 4.2 Schematic of the Cope rearrangement reaction in bullvalene.

[307] and synthesized by Schr¨oder in 1963 [308], bullvalene is a well-known fluxional molecule

in which rapid carbon-carbon bond rearrangements occur on the timescale of typical NMR

measurements [309, 310]. The Cope rearrangement of bullvalene constantly shuffles nuclei be-

tween equivalent reactant (R) and product conformers by passing through a typical aromatic

pericylic transition-state species (TS) [311]. As one might expect, the Cope rearrangement

in bullvalene has been the subject of numerous experimental [312–319] as well as theoretical

[320–325] studies. In particular, gas-phase NMR measurements of bullvalene estimate the

activation barrier for isomerization to be 13.8 ± 0.2 kcal/mol [312], while standard DFT

calculations [320] and thermochemical protocols like CBS-QB3 [323] predict barrier heights

of 11.3 and 12.5 kcal/mol, respectively. As we can expect after examining the automerization

of cyclobutadiene, lower-level quantum chemical methods that cannot capture and balance

the correlation effects characterizing the R and TS species are likely to struggle.

In this regard, we highlight two recent theoretical studies, which served as the inspira-

tion for the present work, examining the R and TS structures of bullvalene with the goal of

treating many-electron correlations at a higher level compared to previous attempts. In the

most recent study, Lesiuk performed SVD-CCSDT+ calculations augmented with pertur-

bative quadruples corrections to obtain a barrier height of 14.6 kcal/mol [325], which is in

line with an earlier prediction of 14.9 kcal/mol made by Karton [324] using a combination of

91

CCSD(T) calculations with post-CCSD(T) correlation effects estimated using a composite

scheme. While the barrier heights resulting from these two studies differ from the gas-phase

NMR measurements by ∼1 kcal/mol, both Lesiuk and Karton demonstrated that the vast

majority of the correlation effects are due to the T3 clusters (cf., also, the results of the

CCSD calculations provided in Table 4.6). Given that both the SVD-CCSDT method [326]

and the CCSD(T) approach rely on perturbative treatments of T3 [while this is obvious in

the case of CCSD(T), one should not forget that the SVD-CCSDT approach describes the

three-body clusters within a truncated basis resulting from singular value decomposition of

a low-order CC3-like approximation to T3], it is interesting to examine whether nonpertur-

bative CC(P ;Q) calculations can help resolve this discrepancy.

Our adaptive CC(P ;Q) calculations for the R and TS structures of bullvalene, along with

the corresponding barrier heights, are presented in Table 4.6. The C1-symmetric geometries

characterizing the R and TS species were taken from Ref. [325], where they were optimized

using the B3LYP-D3/pc-2 approach. All calculations for bullvalene employed the cc-pVTZ

[10] basis set and the lowest-energy orbitals correlating with the 1s shells of carbon atoms

were frozen in all post-RHF steps. Using the cc-pVTZ basis, the bullvalene molecule contains

50 correlated electrons and 440 orbitals and represents the largest system studied using the

adaptive CC(P ;Q) method to date.

In order to efficiently handle a single-particle basis

containing over 400 orbitals, we employed the Cholesky decompositions (CD) of the electron

repulsion integrals, which allows us to factorize the two-electron integrals as a product of

three-center integrals in an automated fashion. A key feature of the CD approach is that it

approximates the two-electron integrals with arbitrary precision according to a user-specified

energy tolerance parameter. In particular, the two-electron integrals resulting from the CD

are identical to their canonical (i.e., non-CD-based) counterparts up to n digits when a

tolerance of 10−n hartree is used. Our CD-based CC(P ;Q) calculations used a tolerance

of 10−7 hartree in the CD step in order to ensure that the energies resulting from our

calculations remain accurate relative to their canonical counterparts through at least seven

92

digits. Another modification that we had to make due to the larger size of this system was

in the choice of the growth rate used to increase the numbers of triply excited determinants

in the P spaces entering our adaptive CC(P ;Q) calculations. In our previous studies, we

adopted a default growth rate of 1%, but in the bullvalene molecule using the cc-pVTZ basis

set, a 1% increment results in the inclusion of roughly 5 × 109 triply excited determinants

in each iteration, which leads to excessive computation and storage costs. To mitigate this

issue, we used an alternative strategy, in which we set the growth increment for %T equal to

the ratio between the total size of the CCSD problem (e.g., the total number of all singly and

doubly excited determinants) and the total size of the triples manifold. For the bullvalene

molecule in the cc-pVTZ basis set, there are 20,250 singly excited determinants, 151,601,625

doubly excited determinants, and 547,548,876,000 triply excited determinants of the Sz = 0

symmetry, resulting in our chosen growth increment of 0.03%.

Before discussing the results of our adaptive CC(P ;Q) calculations for bullvalene sum-

marized in Table 4.6, we mention that, unlike in previous examples, we could not perform

the parent CCSDT calculations. Therefore, in evaluating the results of our CC(P ;Q) com-

putations for the isomerization barrier height, we treat the experimentally determined value

of 13.8 ± 0.2 kcal/mol as a reliable benchmark result. We begin our discussion by pointing

out that the CCSD calculations shown in Table 4.6 are entirely inadequate for predicting the

barrier for Cope rearrangement in bullvalene. As a result of neglecting T3 correlations, the

18.4 kcal/mol barrier predicted with CCSD lies far from the experimental range of 13.6–14.0

kcal/mol. On the other hand, the CR-CC(2,3) calculations predict a much improved barrier

height of 14.5 kcal/mol, which still differs from experiment by 0.5–0.9 kcal/mol. Interestingly,

the 14.5 kcal/mol barrier height predicted by the CR-CC(2,3) approach is very similar to

the corresponding 14.6 and 14.9 kcal/mol reaction barriers obtained using the SVD-CCSDT

approach with perturbative quadruples corrections and composite CCSD(T)/CCSDT(Q)

calculations of Refs. [325] and [324], respectively. However, as emphasized in the previous

examples, the CR-CC(2,3) method may result in errors due to ignoring the coupling be-

93

tween the T1, T2, and T3 clusters. Indeed, when the CR-CC(2,3) calculations are replaced

by CC(P ;Q)[%T = 0.03], we reduce the 14.6 kcal/mol barrier predicted using CR-CC(2,3)

by 0.8 kcal/mol to obtain 13.8 kcal/mol with CC(P ;Q)[%T = 0.03], which matches perfectly

with the experimentally determined mean barrier height. The agreement between the bar-

rier height resulting from the CC(P ;Q)[%T = 0.03] calculations and experiment is certainly

impressive, especially considering that inclusion of just 0.03% of triply excited determinants

in the underlying P spaces makes our CC(P ;Q) calculations computationally practical (far

more practical than full CCSDT, which could not be carried out). By examining the total

energies reported in Table 4.6, we find that the TS structure is more sensitive to the in-

clusion of coupling between the T1, T2, and T3 clusters compared to the R species. When

we move from CR-CC(2,3) to CC(P ;Q)[%T = 0.03], the total energy of the TS changes by

2.917 millihartree compared to 1.778 millihartree in the R species. Furthermore, as we in-

crease the fractions of triply excited determinants entering the P spaces used in our adaptive

CC(P ;Q) calculations from 0.03% to 0.09%, we find that the resulting barrier heights remain

at 13.8 kcal/mol. To lend further confidence to the reliability of our results, we also see that

the differences in total energies for the R and TS species obtained in the CC(P ;Q)[%T =

0.06] and CC(P ;Q)[%T = 0.09] calculations are just 0.186 millihartree, in the case of the R,

and 0.154 millihartree, in the case of the TS. Thus, we have reason to believe that we have

converged the parent CCSDT limit to within a fraction of a millihartree.

Although the nearly perfect agreement between our adaptive CC(P ;Q) calculations and

the gas-phase NMR data is most encouraging, we cannot claim that our calculations are of

strictly higher quality than those reported in Refs. [324, 325]. While we have treated the

T3 effects more completely at the CCSDT level, which is an improvement over the previous

two studies, we have not made any attempt to describe the T4 correlations, which were

approximately included in Refs. [324, 325]. Future work aimed at treating T4 clusters within

the adaptive CC(P ;Q) calculations, either fully as in CCSDTQ or approximately as in CR-

CC(3,4), will be required to verify the accuracy of the present results.

94

Table 4.6 Total energies, in hartree, of the R and TS structures involved in the Cope
rearrangement of bullvalene obtained using the adaptive CC(P ;Q) calculations, alongside
the corresponding barrier heights, in kcal/mol. The geometries of the C1-symmetric R and
TS structures of bullvalene were taken from Ref. [325], where they were optimized using the
B3LYP-D3/pc-2 approach. The lowest orbitals correlating with the 1s shells of carbon atoms
were frozen in all post-RHF steps and the cc-pVTZ basis set was employed throughout. In
carrying out the CD of the two-electron integrals, we used a tolerance parameter 10−7 hartree
to ensure that the energies resulting from our CC(P ;Q) calculations are identical to their
canonical counterparts through at least seven digits.

R

TS

Barrier Heighta,b

%Tc
0

CC(P)

CC(P;Q)

CC(P)

CC(P;Q)

−386.182025d −386.262955e −386.151090d −386.238226e
−386.184958 −386.241143
−386.191629 −386.241416
−386.195949 −386.241570

0.03 −386.201340 −386.264733
0.06 −386.206525 −386.265043
0.09 −386.210004 −386.265229

CC(P)
18.4d
9.3
8.3
7.8

CC(P;Q)
14.5e
13.8
13.8
13.8

aThe barrier heights, in kcal/mol, corresponding to the difference between the energies of the TS and R
structures of bullvalene. In reporting barrier heights, we have included the zero-point vibrational energy
correction of −1.0 kcal/mol obtained in Ref. [325] (cf., also, Ref. [324]).
bThe experimentally determined barrier height for the Cope rearrangement of bullvalene in the gas phase is
13.8 ± 0.2 kcal/mol [312].
cThe %T values reflect the numbers of Sz = 0 triply excited determinants identified by the relaxed variant
of the adaptive CC(P ;Q) algorithm that are included, in addition to all singly and doubly excited determi-
nants, in the underlying P spaces. The increment of 0.03%, used to increase the numbers of triply excited
determinants entering the P spaces, corresponds to the ratio between the sum of the total number of Sz = 0
singly and doubly excited determinants to the total number of Sz = 0 triply excited determinants.
dEquivalent to CCSD.
eEquivalent to CR-CC(2,3).

95

4.2.4 Ground- and Excited-State Potential Surfaces for O–H Bond-Breaking in

Water

In the previous three sections, we have carried out studies designed to test the effective-

ness of the adaptive CC(P ;Q) algorithm in converging the ground-state CCSDT energet-

ics for chemically relevant examples involving bond breaking, isomerization reactions, and

single–triplet gaps in biradicals. Our last example, which is inspired by Refs. [161, 327],

focuses on converging the ground- and excited-state PESs of the water molecule along the

O–H bond-breaking coordinate corresponding to the H2O → H + OH dissociation obtained

with CCSDT/EOMCCSDT. As we will show below, the CCSDT and EOMCCSDT water

potentials are nearly exact, closely matching the FCI data reported in Refs. [161, 327],

but they cannot be accurately approximated using any of the more recently developed

triples corrections to EOMCCSD [142, 148, 151, 152, 159, 161, 328, 329], including the

left-eigenstate CR-EOMCC(2,3) approach [126, 148, 151] or its rigorously size-intensive

δ-CR-EOMCC(2,3) modification [152, 159] that rely on the moment energy expansions

[126, 139, 140, 142, 143, 146–148, 151], especially when examining stretched regions of certain

excited-state water potentials when T3 and Rµ,3 correlations become larger, nonperturbative,

and strongly coupled to their lower-rank T1, T2, Rµ,1, and Rµ,2 counterparts [126, 127, 133].

In this section, we compare the ground- and excited-state PESs of the water molecule

along the O–H bond-breaking coordinate obtained using three different forms of CC(P ;Q),

including (i) the conventional CR-CC(2,3)/CR-EOMCC(2,3) triples corrections to CCS-

D/EOMCCSD, (ii) the active-orbital-based CC(t;3) approach based on correcting the under-

lying CCSDt/EOMCCSDt calculations for missing T3 and Rµ,3 effects, and (iii) the adaptive

CC(P ;Q) approach employing the relaxed algorithm, relative to the parent CCSDT/EOM-

CCSDT surfaces. We show that both the CC(t;3) and adaptive CC(P ;Q) calculations pro-

vide results that closely approximate the full CCSDT/EOMCCSDT data for the PESs of

water using significantly reduced computational costs while improving the CR-CC(2,3) and

CR-EOMCC(2,3) energetics in stretched regions of the O–H bond-breaking potentials. In

96

addition, we show that the adaptive CC(P ;Q) calculations produce PESs of water that are

nearly identical to their active-orbital-based CC(t;3) counterparts, while using much smaller

fractions of triply excited determinants in the underlying P spaces.

The nuclear geometries needed to construct the ground- and excited-state PESs of the

water molecule along the H2O → H + OH dissociation path, obtained by considering 11

values of the O–H bond separation ROH ranging from 1.3 to 4.4 bohr, with the remaining

O–H bond length and H–O–H bond angle optimized using the CCSD/cc-pVTZ method,

were taken from Ref. [327]. As pointed out in Refs. [161, 327], the resulting dissociation

pathway on the ground-state PES has several desirable characteristics. For example, the

equilibrium values of ROH and H–O–H bond angle resulting from the CCSD/cc-pVTZ geom-

etry optimization, of 1.809 bohr and 103.9 degree, respectively, are in good agreement with

experiment and, as ROH increases, the second O–H bond length approaches its equilibrium

value in the ground-state OH radical and the angular potential flattens. For each water

structure corresponding to a particular value of ROH, we carried out the CR-CC(2,3)/CR-

EOMCC(2,3), active-orbital-based CC(P ;Q) [i.e., CC(t;3)], adaptive CC(P ;Q), and parent

CCSDT/EOMCCSDT calculations for the four lowest A′(Cs)-symmetric singlet states, which

include the ground state X 1A′ and n 1A′ excited states with n = 1–3, three lowest A′(Cs)-

symmetric triplet states (denoted as n 3A′, n = 1–3), two lowest A′′(Cs)-symmetric singlet

states (denoted as n 1A′′, n = 1, 2), and three lowest triplet states of the A′′(Cs) symmetry

(denoted as n 3A′′, n = 1–3). To enable comparisons of our CCSDT/EOMCCSDT data with

the FCI and MRCC energetics obtained in Refs. [327] (FCI and MRCC) and [161] (FCI),

all calculations reported in this work were performed with the TZ basis set of Ref. [327],

used in Ref. [161] as well. The RHF determinant, used in our CC/EOMCC calculations as a

reference, consisted of four a′(Cs)-symmetric and one a′′(Cs)-symmetric orbitals. The lowest-

energy orbital correlating with 1s shell of oxygen was frozen in post-RHF steps. To define

the CC(t;3) and underlying CCSDt/EOMCCSDt calculations, the three highest occupied

and two lowest unoccupied orbitals in the RHF reference correlating with 2p shell of oxygen

97

and 1s shells of hydrogens were treated as active. Two of the three active occupied and both

active unoccupied orbitals were of the a′(Cs) symmetry. The third active occupied orbital

was a′′(Cs)-symmetric. The adaptive CC(P ;Q) calculations employed the relaxed algorithm,

discussed in Section 4.1, in which we assumed a 1% growth rate in the numbers of triply

excited determinants entering the underlying P spaces. The percentages of triply excited

determinants characterizing our adaptive CC(P ), EOMCC(P ), and CC(P ;Q) calculations

were defined as fractions of the Sz = 0 triples of the A′(Cs) (1A′ and 3A′ states) or A′′(Cs)

(1A′′ and 3A′′ states) symmetry identified with the adaptive CC(P ;Q) algorithm. The CCSD,

EOMCCSD, CR-CC(2,3), CR-EOMCC(2,3), CCSDt, EOMCCSDt, CC(t;3), CCSDT, and

EOMCCSDT computations were performed using our in-house CC/EOMCC codes inter-

faced with the RHF and integral transformation routines in GAMESS [227]. The adaptive

CC(P ;Q) calculations were executed using our recently developed CCpy package available

on GitHub [228] that can efficiently handle the potentially spotty subsets of triply excited

determinants entering the underlying CC(P ) and EOMCC(P ) computations, which may not

form continuous manifolds labeled by occupied and unoccupied orbitals from the respective

ranges of indices, to achieve the desired speedups compared to CCSDT/EOMCCSDT (see

Section 4.3 and Appendix C for further information).

The results of our CC/EOMCC calculations are shown in Tables 4.7–4.9, Fig. 4.3, and

Tables D.1–D.13 in Appendix D. Table 4.7 summarizes the mean unsigned error (MUE)

and nonparallelity error (NPE) values characterizing the CCSDT and EOMCCSDT po-

tential cuts of water computed in this work relative to their FCI counterparts obtained

in Refs. [161, 327]. The MUE and NPE values characterizing the CCSD/EOMCCSD,

CR-CC(2,3)/CR-EOMCC(2,3), adaptive CC(P )/EOMCC(P ) and CC(P ;Q), CCSDt/EOM-

CCSDt, and CC(t;3) PESs relative to their CCSDT/EOMCCSDT parents are reported in

Tables 4.8 and 4.9, respectively. The ground- and excited-state PES cuts of water corre-

sponding to the H2O → H + OH dissociation channels that correlate with the X 2Π ground

state and the lowest-energy 2Σ+ and 2Σ− states of OH obtained with CCSD/EOMCCSD,

98

CR-CC(2,3)/CR-EOMCC(2,3), adaptive CC(P )/EOMCC(P ) and CC(P ;Q), CCSDt/EOM-

CCSDt, CC(t;3), and CCSDT/EOMCCSDT are shown in Fig. 4.3. Tables D.2–D.13 in

Appendix D provide total electronic energies of all the calculated states.

Accuracy of the CCSDT and EOMCCSDT Approaches

Given that all three variants of the CC(P ;Q) methodology tested in this work aim at

converging or accurately approximating the CCSDT/EOMCCSDT energetics, it is important

to assess how well the CCSDT and EOMCCSDT approaches perform in describing the

ground- and excited-state PESs of water along the O–H bond-breaking coordinate relative

to their exact, FCI, counterparts. As shown in Table 4.7, the CCSDT method provides a

highly accurate description of the X 1A′ state, with the MUE and NPE values relative to

the ground-state FCI potential being as small as 0.63 and 1.14 millihartree, respectively.

EOMCCSDT is similarly accurate in describing the excited-state potentials, with only three

– out of 11 examined in this study – characterized by the MUE and NPE values exceeding

1 and 3 millihartree, respectively, but even in this case, which includes the 2 1A′, 3 1A′, and

3 3A′ states, the MUE and NPE values, of 1.28 and 3.19 millihartree for the 2 1A′ PES, 1.53

and 4.48 millihartree for the 3 1A′ PES, and 1.41 and 5.50 millihartree for the 3 3A′ PES,

remain rather small, especially when we realize that these three states are located hundreds

of millihartrees above the ground state (see Tables D.1 in Appendix D). Most importantly,

the EOMCCSDT approach accurately approximates the FCI water potentials in the highly

stretched ROH ≥ 2.4 bohr region, where all excited states of interest in this study acquire a

substantial MR character that EOMCCSD cannot capture. When compared with the MRCC

results reported in Ref. [327], we observe that the SR CCSDT and EOMCCSDT methods

can also rival, or even outperform, the state-of-the-art MR treatments. For example, for the

1 3A′, 1 1A′, 2 3A′′, 2 1A′′, 3 1A′, and 3 3A′ states, the EOMCCSDT NPE values relative to FCI

are lower – sometimes substantially – than those resulting from the nR-GMS-SU-CCSD and

(N, M )-CCSD calculations. For the X 1A′, 11A′′, 13A′′, 23A′, and 21A′ potentials, the CCSDT

and EOMCCSDT NPE values relative to FCI are within ∼1 millihartree from the best MRCC

99

results reported in Ref. [327]. It is clear from these comparisons that we can treat the ground-

and excited-state potentials of water along the O–H bond-breaking coordinate obtained with

CCSDT and EOMCCSDT as highly accurate benchmarks for evaluating performance of the

different CC(P ;Q) approaches explored in this work.

Table 4.7 The MUE and NPE values, in millihartree, relative to full CI characterizing
the ground-state CCSDT and excited-state EOMCCSDT potentials of the water molecule,
as described by the TZ basis set of Ref. [327], along the O–H bond-breaking coordinate
corresponding to the H2O → H + OH dissociation. Adapted from Ref. [138].

X 1A′
0.63
1.14

1 1A′′
0.78
1.85

1 3A′
0.28
0.83

1 3A′′
0.90
2.14

1 1A′
0.92
2.13

2 3A′
0.93
2.33

2 3A′′
0.43
1.35

2 1A′
1.28
3.19

2 1A′′
0.59
0.57

3 3A′′
0.63
2.67

3 1A′
1.53
4.48

3 3A′
1.41
5.50

MUE
NPE

Performance of Different CC(P ;Q) Approaches Relative to CCSDT and EOM-
CCSDT

We begin by discussing the first variant of the CC(P ;Q) methodology of interest in the

present study corresponding to the CR-CC(2,3) and CR-EOMCC(2,3) triples corrections

to CCSD and EOMCCSD (also examined in Ref. [161]). In the vicinity of the equilibrium

geometry on the ground-state PES, all CC/EOMCC approaches, including CR-CC(2,3)/CR-

EOMCC(2,3), and even CCSD/EOMCCSD, perform well.

Indeed, the errors relative to

CCSDT in the X 1A′ potential computed with CCSD in the ROH = 1.3–2.0 bohr region range

between 2.771 and 3.562 millihartree. CR-CC(2,3) reduces them to ∼0.2–0.3 millihartree.

For the excited-state potentials in the ROH = 1.3–2.0 bohr region, the largest error obtained

with EOMCCSD relative to EOMCCSDT, which occurs when the high-lying 3 1A′ state is

considered, is 3.392 millihartree. The largest error characterizing CR-EOMCC(2,3) in the

same region, encountered when the 3 3A′′ PES is examined, is 1.660 millihartree. For nearly

all excited states considered in this work, the differences between the CR-EOMCC(2,3) and

EOMCCSDT potentials in the ROH = 1.3–2.0 bohr region are about 1 millihartree, making

them virtually identical around the equilibrium geometry. The CR-EOMCC(2,3) PESs are

also more parallel to their EOMCCSDT counterparts than the corresponding EOMCCSD

surfaces.

100

The situation changes when we move toward larger ROH values, where all electronic states

of water considered in this study develop a strong MR character, resulting in failures of the

CCSD and EOMCCSD methods, especially when the X 1A′, 1 1A′′, 1 3A′′, 1 1A′, 2 3A′, 2 3A′′,

2 1A′, 3 3A′′, 3 1A′, and 3 3A′ potentials are examined. These failures become particularly

dramatic for the 2 3A′′, 2 1A′, 3 3A′′, and 3 1A′ states, where errors relative to EOMCCSDT

resulting from the EOMCCSD calculations in the ROH = 2.8–4.4 bohr region become as large

as 28.470, 32.540, 65.442, and 32.035 millihartree, respectively, although no state considered

in our computations is accurately described when T3 and Rµ,3 correlations are neglected and

ROH > 2.4 bohr. The CR-CC(2,3) and CR-EOMCC(2,3) triples corrections reduce the ex-

cessive errors characterizing the X 1A′, 1 1A′′, 1 3A′′, 1 1A′, 2 3A′, 2 3A′′, 2 1A′, 3 3A′′, 3 1A′, and

3 3A′ potentials obtained with CCSD/EOMCCSD at larger values of ROH and the associated

MUEs and NPEs shown in Tables 4.8 and 4.9, which are 6.154–28.262 and 8.453–64.663

millihartree, respectively, in the CCSD/EOMCCSD case and 0.543–6.075 and 0.692–42.170

millihartree when the CR-CC(2,3)/CR-EOMCC(2,3) data are examined, but several prob-

lems remain, especially when dealing with the 2 3A′′ and 3 3A′′ states. In the former case,

the CR-EOMCC(2,3) approach produces large, 11.891–13.976 millihartree, errors relative

to EOMCCSDT in the ROH = 4.0–4.4 bohr region, resulting in the qualitatively incorrect

asymptotic behavior of the CR-EOMCC(2,3) 2 3A′′ potential and the NPE of 13.043 milli-

hartree. The CR-EOMCC(2,3) PES for the 3 3A′′ state presents an even more distressing

situation, with a massive, 37.018 millihartree, error relative to EOMCCSDT at ROH = 2.8

bohr associated with a bump in the 3 3A′′ potential seen in Fig. 4.3(b) and the even larger

NPE value of 42.170 millihartree. These failures of CR-EOMCC(2,3) are consistent with

our earlier observations [126, 127, 130, 133, 135, 137] that none of the triples corrections to

CCSD/EOMCCSD can provide accurate results when T3 and Rµ,3 correlations become sub-

stantial and strongly coupled to their lower-rank counterparts, as is the case when examining

the 2 3A′′ and 3 3A′′ potentials at larger ROH values.

The above challenges encountered in the CR-EOMCC(2,3) calculations can be addressed

101

by turning to the CC(t;3) method. As explained in the Introduction and in Chapter 2,

CC(t;3) is a CC(P ;Q) approach in which energies obtained in the CCSDt/EOMCCSDt

calculations that incorporate the leading triply excited determinants identified with the help

of active orbitals in the P spaces used in the iterative CC(P )/EOMCC(P ) steps are corrected

for the missing, mostly dynamical, T3 and Rµ,3 correlations using Eq. (2.33). The numerical

results in Tables 4.8 and 4.9, with further details provided by Tables D.2–D.13 in Appendix

D, and the potential curves shown in Fig. 4.3(h) demonstrate that the CC(t;3) method

readily addresses the shortcomings of the CR-EOMCC(2,3) approach and its ground-state

CR-CC(2,3) counterpart. This becomes particularly clear when examining the 2 3A′′ and

3 3A′′ potentials that are poorly described by CR-EOMCC(2,3). For example, the CC(t;3)

calculations reduce the large, 11.891–13.976 millihartree, errors relative to EOMCCSDT

obtained with CR-EOMCC(2,3) for the 23A′′ PES in the ROH = 4.0–4.4 bohr region to 0.481–

0.625 millihartree. The massive error of 37.018 millihartree produced by CR-EOMCC(2,3)

for the 3 3A′′ state at ROH = 2.8 bohr is reduced in the CC(t;3) calculations by more than

two orders of magnitude, to 0.133 millihartree, eliminating the unphysical bump in the CR-

EOMCC(2,3) potential for this state in the ROH = 2.4–3.2 bohr region altogether. The

major improvements offered by CC(t;3) in describing the 2 3A′′ and 3 3A′′ PESs are also

reflected in the corresponding NPE values relative to EOMCCSDT, which decrease from

13.043 and 42.170 millihartree obtained with CR-EOMCC(2,3) to the minuscule 0.255 and

0.608 millihartree, respectively.

In general, the CC(P ;Q)-based CC(t;3) method provides

a highly accurate description of the ground and excited states of water along the O–H

bond-breaking coordinate, greatly improving the CR-CC(2,3)/CR-EOMCC(2,3) energetics

and closely matching the nearly exact CCSDT/EOMCCSDT potentials at small fractions

of the computational costs associated with CCSDT/EOMCCSDT. This is manifested by

the tiny MUE and NPE values relative to CCSDT/EOMCCSDT characterizing the CC(t;3)

calculations for the 12 electronic states of water reported in this work, which range from

0.166 to 0.951 millihartree for MUEs and 0.110 to 0.608 millihartree for NPEs (see Tables

102

4.8 and 4.9). By capturing the missing T3 and Rµ,3 correlations, the CC(t;3) corrections are

also very effective in improving the underlying CCSDt/EOMCCSDt potentials, especially in

reducing the MUE values relative to CCSDT/EOMCCSDT characterizing the CCSDt and

EOMCCSDt calculations, from 1.477–2.420 millihartree in CCSDt/EOMCCSDt to fractions

of a millihartree in CC(t;3).

We conclude by commenting on the results of our adaptive CC(P ;Q) calculations, which

are based on the same basic principles as those employed in the CC(t;3) considerations, but

do not rely on active orbitals to obtain accurate results, allowing us to converge the high-

level CCSDT and EOMCCSDT energetics in an entirely black-box fashion. As shown in

Tables 4.8 and 4.9, panels (d) and (f) of Fig. 4.3, and Tables D.2–D.13 in Appendix D, the

adaptive CC(P ;Q) approach is remarkably effective in accurately approximating the CCS-

DT/EOMCCSDT water potentials. Already with a tiny 1% of triply excited determinants

in the underlying P spaces, the adaptive CC(P ;Q) calculations offer major improvements in

the CR-CC(2,3)/CR-EOMCC(2,3) data. They reduce the MUE and NPE values of 0.543

and 0.692 millihartree, respectively, relative to CCSDT, obtained with CR-CC(2,3) for the

ground-state potential, to less than 0.3 millihartree. The improvements in the description

of the 11 excited states of water considered in this work offered by the adaptive CC(P ;Q)

approach using the leading 1% of triply excited determinants in the underlying P spaces,

which, as in Section 4.2.2, will be abbreviated as CC(P ;Q)[%T = 1], are similarly impressive,

especially when we realize that in the case of the Cs-symmetric water structures and the TZ

basis set used in our calculations, 1% of triply excited determinants amounts to only about

300 T3 and Rµ,3 amplitudes, as opposed to 31,832 A′(Cs)-symmetric and 32,232 A′′(Cs)-

symmetric Sz = 0 triples used by full CCSDT/EOMCCSDT. These improvements can be

best seen when comparing the 2 3A′′ and 3 3A′′ potentials obtained with CR-EOMCC(2,3)

with their CC(P ;Q)[%T = 1] counterparts. In the case of the 2 3A′′ PES, the MUE and

NPE values of 5.033 and 13.043 millihartree relative to EOMCCSDT resulting from the CR-

EOMCC(2,3) computations are reduced to 1.177 and 1.081 millihartree, respectively, when

103

the CC(P ;Q)[%T = 1] approach is employed, helping to alleviate the inaccurate behavior

of CR-EOMCC(2,3) in the asymptotic part of the 2 3A′′ potential. The poor description

of the 3 3A′′ PES by the CR-EOMCC(2,3) method becomes much more reasonable in the

CC(P ;Q)[%T = 1] calculations as well. The 37.018 millihartree error relative to EOM-

CCSDT obtained with CR-EOMCC(2,3) at ROH = 2.8 bohr reduces to 6.740 millihartree

when the adaptive CC(P ;Q)[%T = 1] approach is employed. The MUE and NPE values

characterizing the CR-EOMCC(2,3) 3 3A′′ potential, of 6.075 and 42.170 millihartree, respec-

tively, decrease in the CC(P ;Q)[%T = 1] calculations to 1.949 and 5.933 millihartree and,

as shown in Fig. 4.3 (d), the unphysical bump in the CR-EOMCC(2,3) PES for this state

in the ROH = 2.4–3.2 bohr region disappears. The generally small MUE and NPE values

characterizing the CC(P ;Q)[%T = 1] calculations for the 12 PESs of water considered in

this study, relative to their CCSDT/EOMCCSDT parents, which range from 0.275 to 1.949

millihartree for MUEs and 0.216 to 5.933 millihartree for NPEs, are certainly encouraging.

As shown in Tables 4.8 and 4.9, Fig. 4.3(f), and Tables D.2–D.13 in Appendix D, the sit-

uation gets even better when the fraction of triply excited determinants included in the P

spaces defining the adaptive CC(P ;Q) calculations increases to 2%. The MUEs and NPEs

relative to CCSDT/EOMCCSDT characterizing the resulting CC(P ;Q)[%T = 2] compu-

tations for the 12 potential cuts of water examined in this work reduce to 0.197–0.993 and

0.133–1.572 millihartree, respectively. In the case of the 2 3A′′ and 3 3A′′ potentials that cause

major troubles to the CR-EOMCC(2,3) method, the already small MUEs obtained in the

CC(P ;Q)[%T = 1] calculations, of 1.177 and 1.949 millihartree, decrease to 0.737 and 0.993

millihartree, respectively, when the fraction of triply excited determinants incorporated in

the underlying P spaces grows from 1% to 2%. The corresponding NPE values decrease from

1.081 and 5.933 millihartree in the CC(P ;Q)[%T = 1] case to 0.237 and 1.572 millihartree,

when the CC(P ;Q)[%T = 2] approach is employed. For some electronic potentials of water,

the NPEs characterizing the adaptive CC(P ;Q)[%T = 2] computations are slightly larger

than those obtained with the CC(t;3) approach, but the overall performance of the adaptive

104

CC(P ;Q)[%T = 2] and CC(t;3) methods is very similar. This is promising for the future

applications of the adaptive CC(P ;Q) framework since typical CC(t;3) computations, in ad-

dition to requiring the user to select active orbitals, employ considerably larger fractions of

triply excited determinants in the iterative steps of the underlying CC(P ;Q) algorithm than

their black-box adaptive CC(P ;Q) counterparts. For example, the CC(t;3) calculations re-

ported in this work used about 38% of all triples in the iterative CCSDt/EOMCCSDt steps,

as opposed to only 2% used in the adaptive CC(P ;Q) computations.

Table 4.8 The MUE values, in millihartree, relative to CCSDT/EOMCCSDT characterizing
the ground- and excited-state potential cuts of the water molecule, as described by the
TZ basis set of Ref. [327], along the O–H bond-breaking coordinate corresponding to the
H2O → H + OH dissociation obtained with the different CC(P )/EOMCC(P ) and CC(P ;Q)
approaches. Adapted from Ref. [138].

%T = 1e

%T = 2f

State
X 1A′
1 1A′′
1 3A′
1 3A′′
1 1A′
2 3A′
2 3A′′
2 1A′
2 1A′′
3 3A′′
3 1A′
3 3A′

CCSDa CR(2,3)b CCSDtc CC(t;3)d CC(P ) CC(P ;Q) CC(P ) CC(P ;Q)
6.154
7.666
2.819
7.693
10.103
8.846
9.258
14.460
2.337
28.262
13.547
8.305

0.166
0.599
0.951
0.627
0.589
0.646
0.590
0.718
0.553
0.542
0.804
0.854

2.176
3.159
2.061
3.237
3.578
3.379
5.384
3.230
3.461
8.111
5.151
3.794

0.543
1.517
1.137
1.080
1.544
1.225
5.033
2.988
1.750
6.075
2.410
2.481

2.047
1.683
1.741
1.658
1.602
1.682
1.565
2.276
1.477
2.420
1.688
1.656

0.275
0.755
0.793
0.854
0.590
0.665
1.177
0.782
0.870
1.949
1.152
1.048

0.197
0.622
0.616
0.665
0.494
0.530
0.737
0.618
0.579
0.993
0.643
0.676

1.616
2.271
1.532
2.339
2.420
2.322
3.532
2.359
2.535
4.973
3.120
2.671

a CCSD for the ground state and EOMCCSD for excited states.
b CR-CC(2,3) for the ground state and CR-EOMCC(2,3) for excited states.
c CCSDt/EOMCCSDt calculations using the active space consisting of the three highest occupied and two
lowest unoccupied RHF orbitals.
d CC(t;3) calculations using the active space consisting of the three highest occupied and two lowest unoc-
cupied RHF orbitals.
e CC(P )/EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited
determinants and 1% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
f CC(P )/EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited
determinants and 2% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.

105

Table 4.9 The NPE values, in millihartree, relative to CCSDT/EOMCCSDT characterizing
the ground- and excited-state potential cuts of the water molecule, as described by the
TZ basis set of Ref. [327], along the O–H bond-breaking coordinate corresponding to the
H2O → H + OH dissociation obtained with the different CC(P )/EOMCC(P ) and CC(P ;Q)
approaches. Adapted from Ref. [138].

%T = 1e

%T = 2f

State
X 1A′
1 1A′′
1 3A′
1 3A′′
1 1A′
2 3A′
2 3A′′
2 1A′
2 1A′′
3 3A′′
3 1A′
3 3A′

CCSDa CR(2,3)b CCSDtc CC(t;3)d CC(P ) CC(P ;Q) CC(P ) CC(P ;Q)
8.453
16.093
5.004
17.445
21.043
20.127
29.017
33.260
7.971
64.663
30.846
21.562

0.692
4.592
0.442
3.303
4.637
3.712
13.043
8.330
3.983
42.170
4.757
5.670

1.072
1.569
0.801
2.070
2.692
2.264
6.852
2.216
3.129
23.376
6.019
3.149

0.801
1.304
1.005
1.360
1.380
1.233
2.761
1.434
0.837
10.679
2.862
1.413

0.110
0.275
0.492
0.240
0.244
0.257
0.255
0.271
0.263
0.608
0.603
0.392

0.687
0.446
0.452
0.447
0.387
0.826
0.572
2.231
0.601
3.601
0.463
0.712

0.216
0.433
0.593
0.430
0.313
0.246
1.081
0.761
0.611
5.933
2.207
1.574

0.159
0.198
0.281
0.133
0.168
0.218
0.237
0.508
0.636
1.572
0.993
0.582

a CCSD for the ground state and EOMCCSD for excited states.
b CR-CC(2,3) for the ground state and CR-EOMCC(2,3) for excited states.
c CCSDt/EOMCCSDt calculations using the active space consisting of the three highest occupied and two
lowest unoccupied RHF orbitals.
d CC(t;3) calculations using the active space consisting of the three highest occupied and two lowest unoc-
cupied RHF orbitals.
e CC(P )/EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited
determinants and 1% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
f CC(P )/EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited
determinants and 2% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.

106

Figure 4.3 A comparison of the potential cuts of water, plotted as functions of ROH and
corresponding to the H2O → H + OH dissociation channels that correlate with the X 2Π
ground state and the lowest-energy 2Σ+ and 2Σ− states of OH, obtained with (a) CCS-
D/EOMCCSD, (b) CR-CC(2,3)/CR-EOMCC(2,3), (c) CC(P )/EOMCC(P )[%T = 1], (d)
CC(P ;Q)[%T = 1], (e) CC(P )/EOMCC(P )[%T = 2], (f) CC(P ;Q)[%T = 2], (g) CCS-
Dt/EOMCCSDt, and (h) CC(t;3) with their full CCSDT/EOMCCSDT counterparts. The
splined CCSD/EOMCCSD, CR-CC(2,3)/CR-EOMCC(2,3), adaptive CC(P )/EOMCC(P )
and CC(P ;Q), CCSDt/EOMCCSDt, and CC(t;3) data are represented by the solid and
dashed lines, whereas the open circles, squares, triangles, and inverted triangles correspond
to the parent CCSDT/EOMCCSDT energetics.
107

1.52.02.53.03.54.04.5−76.2−76.0−75.8−75.6−75.4Energy / hartree1.52.02.53.03.54.04.5−76.2−76.0−75.8−75.6−75.41.52.02.53.03.54.04.5−76.2−76.0−75.8−75.6−75.4Energy / hartree1.52.02.53.03.54.04.5−76.2−76.0−75.8−75.6−75.41.52.02.53.03.54.04.5−76.2−76.0−75.8−75.6−75.4Energy / hartree1.52.02.53.03.54.04.5−76.2−76.0−75.8−75.6−75.41.52.02.53.03.54.04.5ROH / bohr−76.2−76.0−75.8−75.6−75.4Energy / hartree1.52.02.53.03.54.04.5ROH / bohr−76.2−76.0−75.8−75.6−75.4(a)(b)(c)(d)(e)(f)(g)(h)X 1A'1 1A''1 1A'1 3A''1 3A'2 1A''2 3A'3 3A''4.3 Computational Performance of Adaptive CC(P;Q)

In presenting our numerical results of the adaptive CC(P ;Q) methodology in Section 4.2,

we have emphasized that the CC(P ;Q) calculations rapidly converge their CCSDT/EOM-

CCSDT parents, such that only a tiny fraction of triply excited determinants are needed in

the underlying P spaces in order to obtain energetics that are within a fraction of a milli-

hartree of the full CCSDT/EOMCCSDT data. As one might anticipate, the use of small sub-

sets of leading higher–than–doubly excited determinants in the iterative CC(P )/EOMCC(P )

steps and CC(P ;Q) corrections provides enormous computational savings compared to the

full CCSDT and EOMCCSDT approaches. In this section, we elaborate on these computa-

tional benefits offered by the adaptive CC(P ;Q) approach by discussing the single-core CPU

timings characterizing our adaptive CC(P ;Q) calculations for the TS structure involved in

the automerization of cyclobutadiene, previously examined in Section 4.2.1, and for the se-

ries of CnH2n+2 linear alkanes with n = 1–8. As in the case of all of the adaptive CC(P ;Q)

calculations performed in this work, we used the implementation included in the open-source

CCpy package available on GitHub [228], which is based on the Numpy library and aug-

mented by hand-coded Fortran routines in the computationally intensive parts [in particular,

we use Fortran for the iterative CC(P )/EOMCC(P ) algorithm described in Appendix C].

All reported timings presented in this section correspond to single-core runs on the Precision

7920 workstation from Dell equipped with Intel Xeon Silver 4114 2.2 GHz processor boards.

In presenting the CPU timings, we have ignored the computational times associated with

the execution of the integral, SCF, and integral transformation routines used to generate

the one- and two-electron molecular integrals in the RHF basis, which was carried out using

GAMESS, and the integral sorting operations preceding the CC steps performed using CCpy.

4.3.1 Timings for Calculations of Cyclobutadiene

We will begin by examining the computational timings characterizing our adaptive CC(P ;Q)

calculations for the TS structure of cyclobutadiene, as described by the cc-pVDZ basis set,

examined in Section 4.2.1. To facilitate our discussion, we focus on the unrelaxed CC(P ;Q)

108

calculations using 1%, 3%, and 5% of the Sz = 0 triply excited determinants of the A1g(D2h)

symmetry, which, as shown in Table 4.2, reduce the 14.636 millihartree errors relative to

CCSDT obtained with CR-CC(2,3) to as little as 0.601, 0.561, and 0.559 millihartree, re-

spectively, when the unrelaxed CC(P;Q) approach is employed. To get useful insights, in

addition to the total CPU times, we report the timings associated with the three key stages

of the adaptive CC(P;Q) calculations.

In the case of the unrelaxed CC(P;Q) algorithm

considered in Table 4.10, these three key stages include (i) the P space determination, which

consists of the initial CR-CC(2,3) run followed by the analysis of the δ(0)

ijk,abc(0) contributions

to the resulting triples correction to CCSD, needed to identify a desired fraction of triply

excited determinants for inclusion in the subsequent CC(P) computation, (ii) the iterative

CC(P) calculation using the P space H (P )(1) consisting of all singly and doubly excited

determinants and a subset of triply excited determinants identified in stage (i), and (iii) the

determination of the noniterative δ0(P ; Q) correction to the CC(P) energy obtained in stage

(ii) to capture the remaining T3 correlations with the help of the complementary Q space

H (Q)(1) using Eq. (4.2) in which we set k at 1.

The illustrative CPU timings included in Table 4.10, combined with the previously dis-

cussed energetics compared to CCSDT/EOMCCSDT, clearly demonstrate the enormous

benefit offered by the adaptive CC(P;Q) methodology proposed in this study. Indeed, the

adaptive CC(P;Q) runs using 1–5% of triples in the underlying P spaces accelerate the

parent CCSDT computations for the cyclobutadiene/cc-pVDZ system in its TS geometry

with only minimal loss of accuracy by factors on the order of 67–81, reducing the 14.636

millihartree errors relative to CCSDT obtained with the CR-CC(2,3) triples corrections to

CCSD to small fractions of a millihartree.

109

Table 4.10 Computational timings characterizing the various CC calculations for the
cyclobutadiene/cc-pVDZ system in the transition-state (TS) geometry optimized with MR-
AQCC in Ref. [262], including CCSD, CR-CC(2,3), and CCSDT and the unrelaxed variants
of the adaptive CC(P;Q) approach, abbreviated as CC(P;Q)[%T = x], where x = 1, 3, and
5, which used the leading x percent of triply excited determinants, identified on the basis of
the largest δ(0)
ijk,abc(0) contributions to the CR-CC(2,3) triples correction, in addition to all
singles and doubles, in constructing the respective P spaces. The Q spaces used to determine
the CC(P;Q)[%T = x] corrections to the CC(P)[%T = x] energies consisted of the remaining
(100 − x) percent of triples not included in the corresponding P spaces. In all post-RHF
calculations, the four lowest core orbitals were kept frozen. Adapted from Ref. [137].

Method

CCSD
CR-CC(2,3)
CC(P ;Q)[%T = 1]
CC(P ;Q)[%T = 3]
CC(P ;Q)[%T = 5]
CCSDT

P Space Determinationb
−
−
1.9
1.9
2.0
−

CPU Timea

Iterative Stepsc Noniterative Stepsd Total
0.5
1.1
5.6
6.2
6.8
456

0.5
0.8
3.4
4.0
4.5
456

−
0.3
0.3
0.3
0.3
−

a All reported timings, in CPU minutes, correspond to single-core runs on the Precision 7920 workstation from
Dell equipped with Intel Xeon Silver 4114 2.2 GHz processor boards. No advantage of the D4h symmetry
of the TS structure of cyclobutadiene or its D2h Abelian subgroup was taken in the post-RHF steps. The
computational times associated with the execution of the integral, RHF, and integral transformation and
sorting routines preceding the CC steps are ignored.
b The timings associated with determining the P spaces for the unrelaxed CC(P;Q)[%T = x] computations,
where x = 1, 3, and 5, include the times required to execute the corresponding initial CR-CC(2,3) runs plus
the times spent on analyzing the δ(0)
ijk,abc(0) contributions to the resulting triples corrections to CCSD needed
to identify the top x percent of triply excited determinants. The timings associated with constructing the P
spaces for the iterative steps of the CCSD, CR-CC(2,3), and CCSDT calculations are not reported, since in
these four cases these spaces are a priori known (consisting of all singly and doubly excited determinants in
the case of the CCSD and CR-CC(2,3) methods, and of all singly, doubly, and triply excited determinants
in the CCSDT case).
c In executing the iterative steps of each CC calculation, a convergence threshold of 10−7 hartree was
assumed. The timings corresponding to the iterative steps include the times required to construct and solve
the relevant CC amplitude equations and, in the case of CR-CC(2,3) and CC(P;Q), the times needed to
construct and solve the companion left eigenstate problems involving the respective similarity-transformed
Hamiltonians [using, in the case of CC(P;Q), the two-body approximation discussed in the text].
d In the language of Q spaces adopted by the CC(P;Q) formalism, the computational times required to
determine the noniterative triples corrections of CR-CC(2,3) and CC(P;Q) correspond to all triples in the
case of CR-CC(2,3) and the remaining (100 − x) percent of triples not included in the relevant P spaces in
the case of CC(P;Q)[%T = x] (x = 1, 3, and 5).

110

The computational timings collected in Table 4.10 allow us to draw a few other conclu-

sions. While we will continue working on improving our codes, the results in Table 4.10

already show a reasonably high efficiency of our current implementation of the adaptive

CC(P;Q) approach in the CCpy package. This is particularly true in the case of the most

challenging CC(P) routines which, in analogy to the previously formulated semi-stochastic

[130, 133, 134, 136] and CIPSI-driven [135] CC(P ;Q) approaches, are used to solve the CC

equations in the P spaces containing spotty subsets of triply excited determinants that do

not form continuous manifolds labeled by occupied and unoccupied orbitals from the respec-

tive ranges of indices assumed in conventional CC programming. General remarks about the

key algorithmic ingredients that must be considered when coding the CC(P;Q) approaches,

in order to benefit from the potentially enormous speedups offered by methods in this cate-

gory when the excitation manifolds used in the iterative CC(P) steps [and their excited-state

EOMCC(P ) counterparts] are small and spotty, were provided Refs. [130, 133, 134] (cf., also,

Ref. [253]), while Appendix C of this document covers the specific strategies adopted in our

current implementation of the adaptive CC(P;Q) method, as coded in the CCpy package,

in greater detail. One of the most useful consequences of our current implementation of the

adaptive CC(P;Q) approach aimed at accurately approximating the full CCSDT energetics,

in addition to the major savings in the computational effort compared to CCSDT, is the

observation that the CPU timings characterizing the CC(P) calculations preceding the de-

termination of the noniterative δ0(P, Q) corrections grow slowly with the fraction of triply

excited determinants included in the underlying P spaces (see Table 4.10). This is consis-

tent with the computational cost analysis of the CC(P;Q) methods using small fractions of

higher–than–doubly excited cluster and excitation amplitudes in the iterative CC(P) and

EOMCC(P ) steps in Refs. [130, 133, 134, 253].

Last, but not least, it is interesting to observe in Table 4.10 that the CPU timings

characterizing the adaptive CC(P;Q) computations capable of providing the near-CCSDT

energetics, while larger than those required by its CR-CC(2,3) counterpart, are not very far

111

removed from the costs of the CR-CC(2,3) calculations, which are generally less robust and

much less accurate, especially when T3 correlations are larger and nonperturbative and the

coupling between the lower-rank T1 and T2 and higher-rank T3 clusters becomes significant.

This is mainly because the CPU times needed to complete the iterative CC(P) steps, when

the fraction of triples incorporated in the P space is as tiny as 1–3%, which is, as shown in

Table 4.2, enough to recover the CCSDT energetics to within fractions of a millihartree, even

when CR-CC(2,3) fails or struggles, are only a few times longer than those characterizing

CCSD. This is not too surprising. When the fraction of triples in the P space is tiny, the

noniterative triples corrections of the adaptive CC(P;Q) and CR-CC(2,3) calculations have

virtually identical costs, since the corresponding Q spaces, which are spanned by either all

[CR-CC(2,3)] or nearly all [CC(P;Q)] triply excited determinants, are very similar. One can,

therefore, anticipate that much of the difference between the CPU timings characterizing the

CR-CC(2,3) and adaptive CC(P;Q) runs, especially if we put aside the P space determination

steps, is driven by the time spent on the CC(P) iterations, and this is exactly what we see

in Table 4.10. The storage requirements characterizing the adaptive CC(P;Q) calculations

are not far removed from those required by their CR-CC(2,3) counterpart either, if the

fractions of triples in the underlying P spaces are tiny, since the vectors representing the

T (P ) = T1 + T2 + T (P )

3

operators are not much longer than those used to store T1 + T2. In

summary, it is encouraging that with the introduction of the adaptive CC(P;Q) methodology,

which is only a few times more expensive than the CR-CC(2,3) approach, we can accelerate

the full CCSDT computations by orders of magnitude with only minimal loss of accuracy,

even when T3 effects are highly nonperturbative, causing CCSD(T) to fail, and even when

the coupling among T1, T2, and T3 components of the cluster operator becomes so large that

the CR-CC(2,3) correction to CCSD is no longer reliable.

112

4.3.2 Computational Scalability of Adaptive CC(P ;Q) for the Series CnH2n+2

Linear Alkanes

The previous example showcasing the computational timings of our adaptive CC(P ;Q)

calculations for cyclobutadiene help us appreciate how much we can reduce the CPU effort

of the parent CCSDT calculations without sacrificing accuracy.

Indeed, the fact that we

are able to reduce the 6–7 hours of CPU time required for CCSDT to just 5–7 minutes is

most encouraging. In this next example, we examine the computational performance of the

adaptive CC(P ;Q) approach using the series of CnH2n+2 linear alkanes, with n = 1–8, as

described using the cc-pVDZ basis set [10]. For the larger alkanes with n > 5, the CCSDT

calculations are no longer affordable on our well above-average computational resources.

The geometries for the CnH2n+2, with n = 1–8, molecules were taken from Ref. [326], where

they were optimized using B3LYP/cc-pVTZ method. For each molecule CnH2n+2, the CC

calculations involve 6n + 2 correlated electrons and 23n + 10 correlated orbitals.

In all

of our calculations, we have frozen the lowest-lying orbitals correlating with the 1s shells

of carbon atoms and the two-body approximation for the ground-state δ0(P ; Q) correction

was employed throughout. We present our timings for the CC(P ;Q)[%T = 1] calculations

(using an growth increment of 1%, there is no distinction between the relaxed and unrelaxed

calculations in this case), which are presented in Figure 4.4 using the same decomposition

adopted in Table 4.10 in terms of the preparatory P space determination phase, the iterative

CC(P ) and left-CCSD-like calculations, and the noniterative CC(P ;Q) correction steps

Upon examining Figure 4.4, the benefits offered by the adaptive CC(P ;Q)[%T = 1]

calculations is evident. In particular, the CPU timings required for the CCSDT calculations

increase dramatically as the length of the alkane chain increases. While the CPU timings

of 0.3, 20.7, and 133.7 minutes taken by the CCSDT calculations for methane, ethane, and

propane are not excessively large, they are still between one and two orders of magnitude

larger than the 0.04, 0.5, and 3.77 minutes required by their CC(P ;Q)[%T = 1] counterparts.

When we examine the butane molecule, the CCSDT calculations consume 1,252.1 minutes

113

Figure 4.4 Comparison of the computational timings (in CPU hours) characterizing the post-
RHF steps of the parent CCSDT and adaptive CC(P ;Q)[%T = 1] calculations, decomposed
into the timings for P space determination, iterative steps, and noniterative steps, for the
series of CnH2n+2 linear alkanes, with n = 1–8, described by the cc-pVDZ basis, using the
geometries optimized with B3LYP/cc-pVTZ in Ref.
[326]. The timings presented in this
figure correspond to single-core runs on the Precision 7920 workstation from Dell equipped
with Intel Xeon Silver 4114 2.2 GHz processor boards. In reporting timings, we have ignored
the computational times associated with the execution of the integral, SCF, and integral
transformation routines used to generate the one- and two-electron molecular integrals in
the RHF basis, and the integral sorting operations.

of CPU time, which is roughly 21 hours. In comparison, the CC(P ;Q)[%T = 1] calculations

for butane complete in just 21.0 minutes. In fact, within one day, which is approximately

the time required to perform the CCSDT calculation for butane, the adaptive CC(P ;Q)

method can be used to obtain results for systems as large as heptane, which consumes 833.1

minutes of CPU time. Within two days of CPU time (specifically, 2153.9 CPU minutes), the

CC(P ;Q)[%T = 1] calculations can be completed for a system as large as octane. Therefore,

this computational scalability test demonstrates that the adaptive CC(P ;Q) method can be

applied to systems far larger compared to those that can be studied using the high-level

CCSDT approach.

It is also interesting to note that our larger-scale computational scalability test confirms

114

12345678n020406080100120Timing / CPU hoursIterative StepsP Space DeterminationNoniterative StepsTotal for Adaptive CC(P;Q)CCSDTmany of the conclusions originally drawn from the calculations on the TS structure of cy-

clobutadiene, examined in Section 4.3.1. In particular, we observe that the total CPU time

of the CC(P :Q)[%T = 1] calculations is dominated by the time required to execute the

underlying iterative CC(P ) step. For the series of eight alkanes studies here, the fraction

of CPU time spent on the iterative steps grows from 50–60% of the total computational

timing for n = 1–4 to 68–80% when n = 5–8, with the remaining time mostly taken up

by the P space determination step. The increase in the fraction of time spent in the it-

erative portion of the calculation with n reflects not only the fact that the computational

steps involved in the CC(P ) calculation grow faster with the system size than either the

P space determination or noniterative CC(P ;Q) steps, but also the iterative nature of the

CC(P ) computation. For larger systems, we can expect that the total timing for our adap-

tive CC(P ;Q) calculations will be almost entirely determined by the time required for the

iterative CC(P ) step, and it will become increasingly more important that we continue im-

proving our CC(P ) and EOMCC(P ) algorithms in order to take better advantage of modern

computational accelerators.

115

CHAPTER 5

EXTERNALLY CORRECTED COUPLED-CLUSTER METHODS USING
CIPSI AND MOMENT EXPANSIONS

In Chapters 3 and 4, we presented two novel strategies for obtaining high-level CC/EOMCC

energetics by exploiting the flexibility of the CC(P ;Q) moment expansions to judiciously

include the most important higher–than–doubly excited determinants in the P spaces used

in the CC(P )/EOMCC(P ) calculations prior to determining corrections for the missing

higher-order correlation effects. In this way, we are able to accurately recover the energet-

ics corresponding to any level of CC/EOMCC theory of interest, including, for example,

CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ, and so on, at small fractions of the com-

putational costs, which, as shown in the last section, are on the order of the basic CCSD and

CR-CC(2,3) calculations. The final topic pursued in this thesis seeks to tackle an even loftier

goal: converging the exact, FCI, energetics directly at a cost comparable to that of CCSD.

In our attempt to address this challenge, we rely on the ec-CC framework, introduced in

Section 2.4, and develop a new family of ec-CC approaches in which we solve for the T1 and

T2 clusters in the presence of their T3 and T4 counterparts extracted from CIPSI wave func-

tions and correct the resulting energetics for missing higher-order correlation effects using

CC(P ;Q)-like moment expansions.

5.1 Theory and Algorithmic Details

We begin by recalling the basic premise underlying all ec-CC schemes, namely that for

Hamiltonians containing up to two-body interactions, the CC amplitude equations [Eq. (2.9)]

projected onto singly and doubly excited determinants in the absence of any approximations,

i.e., ⟨Φa

i |H|Φ⟩ = 0 and ⟨Φab

ij |H|Φ⟩ = 0, respectively, where H = e−T HeT , do not engage any

terms involving the Tn components of the cluster operator with n > 4. As alluded to in

Section 2.4, solving Eqs. (2.40) and (2.41) for T1 and T2 in the presence of the exact T3 and

T4 amplitudes extracted from full CI using the cluster-analysis relations, Eq. (2.42), produces

the exact T1 and T2, as well as the resulting full CI energy.

116

In this chapter, we focus on the ec-CC approaches based on CIPSI wave functions [214].

As discussed in Section 2.4, when combining the ec-CC framework with modern selected CI

wave functions, such as CIPSI, one should adopt the ec-CC-II methodology, which removes

the explicitly disconnected diagrams from the cluster analysis of the CI state, as opposed

to its ec-CC-I counterpart. Doing so allows us to benefit from the efficient exploration

of the many-electron Hilbert space performed by algorithms like CIPSI and gives us the

opportunity to improve upon the CIPSI calculations as long as the CIPSI wave function

does not saturate the singles through quadruples excitation manifolds. As we will see in our

numerical results discussed in the next section, we can make excellent use of this window

of opportunity and help accelerate the convergence of the CIPSI wave functions toward the

FCI limit in a computationally efficient manner.

In performing the CIPSI-driven ec-CC

calculations reported in this work, we carried out the following protocol:

1. Given a reference state |Φ⟩, which in our calculations reported in Table 5.1 was the

RHF determinant, choose an input parameter Ndet(in), used to terminate the CIPSI

wave function growth, and execute a CIPSI run to generate the ground state wave

function |Ψ(CIPSI)⟩ spanned by Ndet(out) determinants.

2. After the CIPSI run defined by a given Ndet(in) value is completed, extract the C1, C2,

C3, and C4 components of the CIPSI wave function |Ψ(CIPSI)⟩.

3. Perform the cluster analysis of |Ψ(CIPSI)⟩ according to Eq. (2.42) to obtain the corre-

sponding T (ext)

3

and T (ext)

4

.

In the case of the CIPSI-driven ec-CC calculations per-

formed in this study, we adhere to the ec-CC-II protocol, in which we remove all T (ext)

3

and T (ext)
4

amplitudes resulting from Eq. (2.42) that do not have the corresponding C3

and C4 coefficient.

4. Solve the ec-CC equations, Eq. (2.40) and Eq. (2.41), to obtain the T1 and T2 operators

relaxed in the presence of T (ext)

3

and T (ext)
4

extracted from CIPSI in the preceding step.

117

After the ec-CC equations are solved to obtain T1 and T2, solve the companion left-

CCSD-like system Eq. (2.47) to determine the deexcitation operators Λ1 and Λ2.

5. Using T1, T2, Λ1, and Λ2, determine correction δ3 to the ec-CC-II energy due to T3

correlations missing from the CIPSI wave function according to Eq. (2.44).

6. To check convergence, repeat Steps 1–5 for a larger value of Ndet(in). The CIPSI-driven

ec-CC calculations can be regarded as converged if the difference between consecutive

ec-CC-II3 energies falls below a user-defined threshold.

In the next section, we perform a test to measure how effective the newly developed

CIPSI-driven ec-CC-II and ec-CC-II3 approaches are in recovering the exact, FCI, energetics

for the challenging problem of symmetric dissociation in water.

5.2 Symmetric Dissociation in the Water Molecule

The results of our CIPSI-based all-electron ec-CC-II and ec-CC-II3 calculations for the

C2v-symmetric double bond dissociation of the water molecule, as described by the cc-pVDZ

basis set [10], are summarized in Table 5.1. The molecular geometry, taken from Ref. [240],

simultaneously stretches both O–H bonds without changing the ∠H–O–H angle. In addition

to the equilibrium geometry, designated as R = Re, we also considered two stretches of the

O–H bonds by factors of 2 and 3, designated as R = 2Re and R = 3Re, respectively. To

maintain consistency with the study in Ref. [240], we correlated all electrons in the problem

and used spherical components of d orbitals throughout. In all CIPSI and ec-CC calculations,

the RHF determinant was used as a reference.

We use the double bond dissociation of water described within the cc-pVDZ basis set to

showcase the performance of the CIPSI-driven ec-CC-II and ec-CC-II3 approaches since it is

small enough to perform full CI, while still posing significant challenges for many quantum

chemical approaches at stretched geometries due to substantial non-dynamical correlation

effects, resulting in large triply and quadruply excited CI and CC amplitudes. In particular,

examining the results in Ref. [214] indicates that CCSD produces significant errors relative to

118

full CI of 3.744, 22.034, and 10.849 millihartree at R = Re, 2Re, and 3Re, respectively. The

errors at equilibrium arise from the neglect of dynamical correlations described by T3 and T4,

and as expected, CCSDT and CCSTDQ produce nearly exact results, with errors of 0.493

and 0.019 millihartree relative to full CI, respectively. Once the two O–H bonds are stretched

to 2Re, CCSD is failing, and CCSDT, lacking T4 correlations, overshoots full CI by 1.403

millihartree, while CCSDTQ remains practically exact. The most challenging MR geometry

occurs at the large 3Re stretch, where CCSDT completely fails, overshooting full CI by

a massive 40.126 millihartree, and even CCSDTQ produces a sizeable −4.733 millihartree

error relative to full CI. Given the failure of even the powerful CCSDTQ method, it is

interesting to see if the ec-CC-II and ec-CC-II3 methodologies, which are formally much less

expensive than CCSDTQ, can outperform it and reach the full-CI–level energetics even in

these challenging MR geometries.

The results in Table 5.1 demonstrate that both the CIPSI-driven ec-CC-II and ec-CC-II3,

particularly the latter method, offer significant improvements over the underlying variational

CIPSI computations and achieve a fast convergence toward full CI, even in the challenging

3Re geometry. As shown in Table 5.1, using only 5,000 – 6,000 Sz = 0 determinants of the

A1(C2v) symmetry in the CIPSI diagonalization spaces, corresponding to the small Ndet(in) =

5, 000 CIPSI run that captures approximately 40–50% of singles, 20–70% of doubles, 1–2%

of triples, and 0.2% of quadruples, the ec-CC-II approach reduces the 8.445, 34.996, and

28.755 millihartree errors relative to full CI obtained using the CIPSI method at R = Re,

2Re, and 3Re, respectively, to 2.445, 4.102, and 4.161 millihartree. This also represents

an improvement over the 3.744, 22.034, and 10.849 millihartree errors obtained using the

CCSD method, where T3 and T4 correlations are neglected. Meanwhile, the triples-corrected

ec-CC-II3 approach greatly reduces the errors obtained using ec-CC-II to 0.012, 0.226, and

1.507 millihartree at each geometry, respectively. As a matter of fact, compared to the

truncated CI and CC energies found in Ref. [214], the ec-CC-II3 energies are substantially

more accurate than the CCSDT, CISDTQ, and even CISDTQP calculations, each of which

119

are considerably more expensive to perform. For the largest stretch of 3Re, the ec-CC-II3

method is actually more accurate than even CCSDTQ, which is remarkable, considering that

the ec-CC-II and ec-CC-II3 methods have the storage requirements and computational steps

that are similar to CCSD and CR-CC(2,3), respectively. It is also worthwhile to note that

the ec-CC-I approach, which retains the problematic fully disconnected terms in T (CIPSI)

3

and

T (CIPSI)
4

, produces errors relative to full CI of 8.123, 12.707, and 9.546 millihartree at each

geometry. These energies are very similar to those produced using CIPSI alone, which is a

consequence of CIPSI saturating the single and double excitation manifolds, resulting in ec-

CC-I returning energies very close to CIPSI, in accordance with Theorem 1. Increasing the

value of Ndet(in) to 50,000, produces larger CIPSI wave functions, containing around 80,000–

90,000 determinants of the Sz = 0 A1(C2v) symmetry. Using this CIPSI wave function in the

ec-CC-I approach results in essentially no change to the CIPSI energetics, indicating that

the single and double excitation manifolds are numerically complete. On the other hand,

only about 20% of triples and 4% of quadruples are captured using these wave functions,

hence the ec-CC-II method is still able to improve upon CIPSI, reducing its 2.612, 2.436,

and 0.906 millihartree errors relative to full CI at R = Re, 2Re, and 3Re, respectively, to

0.626, 0.788, and 0.341 millihartree, respectively. The ec-CC-II3 method, which corrects for

missing T3 correlations, reduces the errors obtained using ec-CC-II at Re and 2Re to 0.168

and 0.515 millihartree, respectively, while unable to further correct the ec-CC-II results at

3Re, producing a 0.358 millihartree error relative to full CI. For values of Ndet(in) equal to

50,000 and larger, the ec-CC-II3 appears ineffective at accelerating the convergence of the

ec-CC-II energetics toward full CI, which is likely due to the fact that it does not take

into account the missing T4 correlations, which are substantial, particularly in the highly

stretched 3Re geometry.

Looking at the overall picture emerging from the results in Table 5.1, it is quite clear that

the ec-CC-II and ec-CC-II3, particularly the latter ones, offer a rapid convergence to within a

millihartree of full CI using relatively small CIPSI diagonalization spaces. They also converge

120

to full CI much faster than the variational CIPSI energies. The efficacy of CIPSI-driven ec-

CC-II and ec-CC-II3 reinforces the view that the selected CI approaches like CIPSI, which are

capable of efficiently sampling the many-electron Hilbert space through a tempered evolution

of the wave function, without saturating the lower-rank excitation manifolds, are of the most

use in ec-CC considerations. We also mention that, while the convergence of the ec-CC-II

and ec-CC-II3 approaches is impressive, it is at times outperformed by the perturbatively

corrected CIPSI energies, indicating that at least for the small problems studied so far,

the CIPSI and other selected CI models are powerful methods in their own right. To gain

further insight into the utility of CIPSI-driven ec-CC-II and ec-CC-II3 approaches, further

testing is required, particularly on larger systems described with bigger basis sets, where

the perturbatively corrected CIPSI approach may begin to struggle in converging the exact,

FCI, limit.

121

Table 5.1 Convergence of the energies resulting from the all-electron CIPSI calculations initiated from the RHF wave function
and the corresponding CIPSI-based ec-CC energies toward FCI for the H2O molecule, as described by the cc-pVDZ basis set,
at the equilbrium and two displaced geometries in which both O–H bonds are stretched by factors of 2 and 3. a Adapted from
Ref. [214].

Ndet(in) / Ndet(out)

%Sb

%Db

%Tb

%Qb

1 / 1
1,000 / 1,299
5,000 / 5,216
10,000 / 10,448
50,000 / 83,762
100,000 / 167,425
500,000 / 665,840
1,000,000 / 1,308,003

1 / 1
1,000 / 1,399
5,000 / 5,664
10,000 / 11,350
50,000 / 90,880
100,000 / 181,579
500,000 / 718,316
1,000,000 / 1,390,678

1 / 1
1,000 / 1,437
5,000 / 5,793
10,000 / 11,603
50,000 / 92,707
100,000 / 184,903
500,000 / 703,445
1,000,000 / 1,215,321

0
15.2
51.5
60.6
93.9
93.9
100
100

0
36.4
54.5
57.6
93.9
100
100
100

0
30.3
39.4
54.5
87.9
90.9
97.0
97.0

0
37.9
73.8
80.4
94.7
97.8
99.7
99.9

0
18.1
43.1
55.6
85.3
91.6
97.7
99.3

0
8.8
21.4
28.6
70.8
78.5
89.3
92.8

0
0
1.0
3.1
22.1
34.5
69.0
82.5

0
0.3
1.8
3.6
21.6
31.7
59.6
73.2

0
0.3
1.8
3.5
14.9
22.7
46.0
56.3

0
0.0
0.2
0.4
4.9
10.0
31.9
45.6

0
0.0
0.2
0.4
3.7
6.8
21.4
31.9

0
0.1
0.2
0.4
2.5
4.4
13.5
19.5

r

−21.684
−0.024
0.108
0.190
0.090
0.064
0.008
0.004

−42.098
−0.109
0.098
0.184
0.089
0.064
0.008
0.004

CIPSIc
Evar Evar + ∆E(2) Evar + ∆E(2)
R = Re
217.822d
21.589
8.445
6.587
2.612
1.743
0.435
0.229
R = 2Re
363.956d
34.996
13.817
9.011
2.436
1.418
0.273
0.137
R = 3Re
567.554d
28.755
9.919
5.258
0.906
0.483
0.071
0.029

−180.621
1.884
0.605
0.375
0.084
0.046
0.009
0.003

−227.583
1.010
0.337
0.157
0.031
0.011
−0.001
−0.001

105.169
2.139
0.638
0.388
0.085
0.046
0.009
0.003

420.893
1.183
0.354
0.161
0.031
0.011
−0.001
−0.001

I

3.744e
11.019
8.123
6.441
2.610
1.742
0.435
0.229

22.034e
26.505
12.707
8.653
2.429
1.417
0.273
0.137

10.849e
22.107
9.546
5.093
0.902
0.483
0.071
0.029

ec-CCc
II

3.744e
3.637
2.455
1.887
0.626
0.410
0.120
0.069

22.034e
9.639
4.102
2.677
0.788
0.467
0.147
0.074

II3

0.344f
0.236
0.012
0.063
0.168
0.190
0.105
0.066

−0.548f
−0.454
0.226
0.321
0.515
0.356
0.138
0.072

10.849e −40.556f
−1.934
1.507
0.699
0.358
0.228
0.050
0.017

8.467
4.161
1.687
0.341
0.177
0.049
0.017

aThe equilibrium geometry, R = Re, and the geometries that represent a simultaneous stretching of both O–H bonds by factors of 2 and 3 without
changing the ∠(H–O–H) angle were taken from Ref. [240].
b%S, %D, %T, and %Q are, respectively, the percentages of the singly, doubly, triply, and quadruply excited Sz = 0 determinants of A1 symmetry
captured during the CIPSI computations.
cErrors relative to FCI in millihartree (see Table I of Ref. [214] for the FCI energies).
dEquivalent to RHF.
eEquivalent to CCSD.
eEquivalent to CR-CC(2,3).

122

CHAPTER 6

CONCLUDING REMARKS AND FUTURE OUTLOOK

In this dissertation, we explored recent advances in the development of new SR CC and

EOMCC methodologies aimed at addressing multiconfigurational problems in chemistry.

The primary goal was to introduce black-box approaches capable of determining molecular

PESs involving significant bond rearrangements, biradical transition states, singlet–triplet

gaps, and excited states dominated by both one- and many-electron transitions with lower

computational costs compared to other methods that aim at similar precision. Specifically,

we introduced, developed, and tested several novel and practical ab initio quantum chem-

istry methods: the CIPSI-driven CC(P ;Q), the adaptive CC(P ;Q), and the CIPSI-based

ec-CC-II and ec-CC-II3 schemes. The first two methodologies exploit the flexibility of the

CC(P ;Q) framework to automatically converge high-level CC/EOMCC energetics, such as

CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ, etc., at small fractions of the computa-

tional costs, while the latter ec-CC-based approaches offer an intriguing route to converging

exact, FCI, energetics directly, using computational steps akin to CCSD or CR-CC(2,3).

In the first two chapters of this dissertation, we introduced the many-electron Schr¨odinger

equation and discussed the computational challenges associated with obtaining numerically

exact solutions using the FCI method. We highlighted the practical necessity of resorting

to approximate methods and emphasized the preeminent role of SRCC theory in ab initio

quantum chemistry, as well as its EOM extension to excited electronic states, which pro-

vide a hierarchy of increasingly accurate approximations that allows us to converge FCI

or near-FCI descriptions for most chemical applications using polynomial computational

steps. After presenting the necessary theoretical background on the SR CC and EOMCC

formalisms, we demonstrated that higher-level CC/EOMCC approximations, such as CCS-

DT/EOMCCSDT and CCSDTQ/EOMCCSDTQ, remain robust in the presence of electronic

quasi-degeneracies, closely approximating the exact FCI energetics. This was illustrated us-

ing bond-breaking in HF and the challenging symmetric dissociation of water. We then

123

argued that, while high-level methods like CCSDT/EOMCCSDT and CCSDTQ/EOMCCS-

DTQ can accurately handle significant MR correlations, traditional perturbative approxima-

tions, such as CCSD(T), rob us of this benefit and are inadequate for studying chemical bond

rearrangements, open-shell systems like biradicals, and excited states dominated by multi-

electron transitions. To better describe MR problems within the SRCC framework, we moved

away from perturbation theory and adopted the more powerful CC(P ;Q) framework. This

formalism allows us to partition the manifold of higher-than-doubly excited determinants

in high-level CC/EOMCC theories, selecting a small subset of important determinants for

inclusion in the P space, which is used to define the cluster and excitation operators and

perform the iterative CC(P )/EOMCC(P ) step, and correcting the resulting energetics for

the remaining correlations of interest captured with the help of the complementary Q space

using the nonperturbative CC(P ;Q) moment energy expansions.

In Chapters 3 and 4 of this thesis, we introduced two novel CC(P ;Q) methodologies

based on this idea, which are capable of converging high-level CC/EOMCC energetics in an

automated fashion, even in situations where higher–than–pair correlations are large, nonper-

turbative, and difficult to capture with noniterative corrections to lower-level CC/EOMCC

calculations. The first approach combined CC(P ;Q) moment expansions with the informa-

tion provided by CIPSI wave functions, which were used to identify the subsets of higher-

than-doubly excited determinants entering the underlying P and Q spaces. The result-

ing CIPSI-driven CC(P ;Q) approach was implemented to converge the CCSDT and EOM-

CCSDT energetics and tested on several molecular examples, including dissociation of F2, the

automerization of cyclobutadiene, and the vertical excitation spectrum of CH+. In all cases,

we demonstrated that the CIPSI-driven CC(P ;Q) method successfully recovered the parent

CCSDT/EOMCCSDT results with tiny fractions of the computational costs using P spaces

containing a relatively small number of triply excited determinants identified with the help

of compact CI wave function expansions. Inspired by the success of the quasi-perturbative

CIPSI algorithm, the second CC(P ;Q) approach introduced in this dissertation aimed at

124

creating a CC-based analog of CIPSI, which we called adaptive CC(P ;Q). In this approach,

the excitation manifolds defining the higher–than–two-body components of the T and Rµ

operators are adaptively selected based on the information contained within the CC(P ;Q)

moment expansions themselves. This novel methodology represents a practical and powerful

approach for converging high-level CC/EOMCC energetics, such as CCSDT/EOMCCSDT,

CCSDTQ/EOMCCSDTQ, and similar types, freeing us from the reliance on active orbitals

and external information from non-CC or stochastic sources used in previous formulations

of CC(P ;Q). The effectiveness of the adaptive CC(P ;Q) methodology was demonstrated in

numerous applications, where the goal was to recover the CCSDT/EOMCCSDT energet-

ics when noniterative triples corrections to CCSD/EOMCCSD struggle or fail. These tests

included the significantly stretched F2 and F+

2 molecules, the automerization of cyclobuta-

diene, singlet–triplet gaps in several organic biradicals, the degenerate Cope rearrangement

of bullvalene, and the ground- and excited-state PESs of the water molecule along its O–H

bond-breaking coordinate. In addition to demonstrating rapid convergence toward the par-

ent CC results, we discussed the CPU timings for the TS of cyclobutadiene and for a series of

CnH2n+2 linear alkanes for n = 1–8 to highlight the significant reductions in computational

effort relative to CCSDT achieved by the adaptive CC(P ;Q) calculations.

In the final section of this dissertation, we introduced a new generation of ec-CC ap-

proaches designed to recover the near-FCI energetics, even in the presence of stronger MR

correlation effects, by leveraging information about T3 and T4 clusters extracted from CIPSI,

combined with CC(P ;Q)-like moment corrections. The resulting ec-CC-II methodology,

along with its ec-CC-II3 extension that accounts for missing T3 correlations, was tested on

the symmetric double dissociation of water, where we demonstrated that the CIPSI-driven

ec-CC calculations, particularly ec-CC-II3, can achieve accuracies on the order of 0.1 milli-

hartree relative to FCI, using relatively inexpensive CIPSI runs with small diagonalization

spaces. These results successfully compete with high-level CC and CI methods, such as

CCSDTQ and CISDTQPH, at a fraction of the computational cost, and they improve upon

125

CCSDTQ in the dissociation regime when electronic degeneracies become particularly strong.

The ideas and methodologies developed in this dissertation open several avenues for

future work. Regarding the CIPSI-based CC(P ;Q) methodologies, one potential direction

is to replace the CIPSI algorithm with alternative selected CI schemes, such as heat-bath

CI. While heat-bath CI calculations are generally less accurate than CIPSI (as seen by

comparing the heat-bath CI results in Ref. [195] with their counterparts obtained using

CIPSI in Ref. [194]), they offer the advantage of being computationally less expensive due

to using a simplified selection scheme. Since our focus on selected CI within the context

of CC(P ;Q) is primarily to identify the leading higher–than–doubly excited determinants

for setting up CC/EOMCC calculations, rather than converging full FCI energetics, we may

benefit from accelerating the preliminary CI steps, even if the resulting wave CI functions are

less accurate. Another promising area for future work is the incorporation of state-specific P

spaces in selected-CI-driven CC(P ;Q) calculations. This would allow us to tailor the P spaces

to better capture the many-electron correlation effects specific to each state of interest. Since

selected CI diagonalizations provide both ground- and excited-state wave functions, which

can be parsed to identify leading higher–than–doubly excited determinants, constructing

the necessary state-specific P spaces and integrating them with our existing CIPSI-driven

CC(P ;Q) codes is straightforward. This approach has the potential to further accelerate

convergence of high-level CC/EOMCC energetics. Lastly, we aim to extend the capabilities

of our CIPSI-driven CC(P ;Q) codes to converge even higher-level CC/EOMCC energetics,

such as CCSDTQ/EOMCCSDTQ. Work in the latter direction is already underway.

In the same vein, we anticipate significant future work focused on extending the capa-

bilities of our adaptive CC(P ;Q) codes to converge higher levels of CC/EOMCC theory,

including CCSDTQ/EOMCCSDTQ, and eventually, possibly, the exact FCI limit. Achiev-

ing the latter would involve performing adaptive CC(P ;Q) calculations without placing re-

strictions on the determinants entering our moment energy expansions. This untruncated

adaptive CC(P ;Q) approach would take advantage of the original purpose of MMCC expan-

126

sions as corrections toward FCI, ultimately transforming adaptive CC(P ;Q) into a form of

selected CC. The primary obstacle to realizing this selected CC approach is developing a

CC(P )/EOMCC(P ) code capable of handling any list of higher–than–doubly excited deter-

minants in an efficient manner. Most codes that are used to perform high-level CC/EOMCC

calculations introduce computational intermediates in order to recast the equations in a form

that is quasi-linear in the higher–than–two-body components of the cluster operator. Op-

timally factorized CC/EOMCC equations of this kind are straightforward to obtain if one

can limit the maximum excitation rank in the T and Rµ operators a priori, such as when

targeting CCSDT/EOMCCSDT or CCSDTQ/EOMCCSDTQ. However, if the structure of

the CC/EOMCC equations is left unspecified until the list of higher–than–doubly excited

determinants is provided, it becomes challenging to design a code that leverages intermedi-

ates without incurring excessive CPU or memory costs. To our knowledge, the only robust

method for performing general CC calculations of this type rely on less efficient string-based

CI-like approaches [102, 330–332]. One potential brute-force solution could involve con-

structing a library of factorized CC/EOMCC equations up to a high excitation rank and

dynamically assembling the appropriate CC(P )/EOMCC(P ) code on the fly. However, this

approach requires further investigation to fully understand its feasibility.

As we move toward higher levels of CC/EOMCC theory, it is critical that we continue

improving our adaptive CC(P ;Q) codes from a technical perspective.

In particular, we

believe that developing new algorithms to reduce the computational cost of filtering and

selecting subsets of higher–than–doubly excited determinants entering the P spaces during

the evaluation of CC(P ;Q) corrections will become increasingly important. One techni-

cal optimization of this type involves adopting stochastic selection algorithms, which have

proven to be very successful in accelerating CIPSI computations (see Ref. [183] for de-

tails). To make progress, one may draw inspiration from the recently developed stochas-

tic closed-shell CCSD(T) method [333], in which the triples correction δ(T ) is decomposed

as δ(T ) = P

a<b<c δabc, where δabc = P

i<j<k ℓ abc

0,ijk M ijk

0,abc, with M ijk

0,abc = ⟨Φabc

ijk|(HT2)C|Φ⟩

127

and ℓ abc

0,ijk = (M ijk

0,abc)†/Dijk

abc. A Monte Carlo procedure based on sampling triplets of un-

occupied spinorbitals (a, b, c), with a < b < c, is employed to estimate δ(T ) as δ(MC)

(T ) =

P

1
Ntrials

(a,b,c) n(a,b,c)δabc, where Ntrials is the number of Monte Carlo samples and n(a,b,c) is

the number of times that the triplet (a, b, c) is sampled. For each triplet (a, b, c), the cor-

responding correction δabc is computed and stored, allowing reuse of previously computed

δabc values in subsequent trials, and in the limit that Ntrials → ∞, the Monte Carlo esti-

mate δ(MC)

(T )

converges to the deterministic value δ(T ). To accelerate the convergence, the

authors of Ref. [333] sample the triplets (a, b, c) according to the probability distribution

P ∝ max(ϵ, ∆(a,b,c)), with ∆(a,b,c) ∝ (f a

a + f b

b + f c

c )−1, where f p

p is the energy of spinorbital

|p⟩, which places more weight on corrections δabc associated with virtual orbitals near the

Fermi level. The arbitrary constant ϵ (set to 0.2 hartree in Ref. [333]) is introduced to

ensure that P does not diverge. In a similar fashion, one could extend this stochastic algo-

rithm to the CC(P ;Q) framework by decomposing the moment energy expansion, Eq. (2.33),

into a sum over corrections due to higher–than–doubly excited determinants in the Q space,

δµ(P ; Q) = δµ,3 +δµ,4 +. . ., where δµ,n = P

|Φa1...an
i1...in
sponding to n-tuply excited determinants in the Q space, with n > 2. The corrections δµ,n are

⟩∈H (Q) δi1...in,a1...an(µ) is the correction corre-

approximated using the Monte Carlo estimator δ(MC)

µ,n = 1

Ntrials

P

(a1,...,an) n(a1,...,an)δa1...an(µ),

where δa1...an(µ) = P

|Φa1...an
i1...in

⟩∈H (Q) ℓ a1...an

µ,i1...in (P ) M i1...in

sample tuples (a1, . . . , an) associated with determinants |Φa1...an

distribution P ∝ max(ϵ, ∆(a1,...,an)), with ∆(a1,...,an) ∝ (Pn

µ,a1...an(P ). Following Ref. [333], one could
i1...in ⟩ ∈ H (Q) according to the
i=1 f ai
)−1, which corresponds to
ai

the Møller–Plesset energy denominator, or one could use the more accurate Epstein–Nesbet

form. This approach would not only reduce the computational cost of evaluating the cor-

rections associated with higher–than–doubly excited determinants in the Q space, but also

stochastically filter the subsets of determinants considered for potential inclusion in the P

spaces, thereby improving the efficiency of the adaptive CC(P ;Q) method.

In the longer term, we also propose combining the adaptive CC(P ;Q) methodology with

the cluster-in-molecule (CIM) local correlation framework [334–339], including its single-

128

environment [340] and multilevel [341] extensions, which has previously been used to develop

local CCD, CCSD, CCSD(T), and CR-CC(2,3) approaches. These CIM-based CC methods

achieve computational costs that scale linearly with the system size, while maintaining sub-

kcal/mol accuracy relative to canonical calculations, successfully extending the applicability

of CC approaches to much larger molecules containing hundreds of atoms. Undoubtedly,

obtaining high-level CC energetics corresponding to CCSDT, CCSDTQ, and similar methods

for large chemical systems remains a key goal for future SRCC development work. We believe

that the adaptive CC(P ;Q) approach introduced in this dissertation is particularly well-

suited for this purpose, as it is fully automated, does not require the user-defined selection

of active orbitals, and operates independently of information from external calculations.

Finally, regarding ec-CC approaches, we envision that future work should begin with ex-

tensive testing of the existing CIPSI-driven ec-CC-II and ec-CC-II3 methods to evaluate their

ability to converge FCI in larger many-electron systems described using more realistic basis

sets. This will provide invaluable insight into how CIPSI-based ec-CC approaches perform

in more practical scenarios, where obtaining FCI is unfeasible. In particular, this will help

us understand how to efficiently extract high-quality T3 and T4 clusters from CIPSI without

being hindered by the growing cost of Hamiltonian diagonalizations. In this context, detailed

studies varying the selection factor f (as initially explored in Ref. [214]) and investigating

well-chosen CI expansions for initiating CIPSI runs could be highly beneficial. Methodolog-

ically, one might consider correcting the ec-CC-II energetics for the missing T4 correlation

effects to accelerate convergence toward FCI. In fact, we have already developed the ec-

CC-II3,4 methodology, in which the energies from ec-CC-II calculations are corrected for the

missing T3 and T4 effects using the CC(P ;Q)-like moment expansions discussed in Section

2.4. Good initial tests for the new ec-CC-II3,4 method include the symmetric dissociation

of water in a larger basis set, where well-converged and extrapolated CIPSI energetics can

serve as near-FCI benchmarks, and the ground state of benzene, for which highly accurate

FCI estimates were obtained in Ref. [195].

129

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APPENDIX A

CCPY: AN OPEN-SOURCE COUPLED-CLUSTER PACKAGE
IMPLEMENTED IN PYTHON

In this appendix, we provide information about the CCpy package available on GitHub

[228], which is a user-friendly and flexible open-source Python-driven software package for

performing CC and EOMCC calculations for many-electron systems. The CCpy package

was originally developed to supplement the research reported in this dissertation, and in

particular, it contains the functionality required to reproduce the results of the calculations

reported in chapters 3–5 of this dissertation. In recent months, CCpy has also been incor-

porated in the PySCF ecosystem as an official extension module. CCpy offers a rich and

diverse selection of CC approaches, which includes a wide range of methods belonging to the

conventional, LR, spin-flip (SF), active-orbital-based, CR, externally corrected, approximate

coupled-pair (ACP), and CC(P ;Q) categories as well as their EOM extensions aimed at de-

scribing electronically excited, singly and doubly electron attached (EA, DEA), and singly

and doubly ionized (IP, DIP) states of many-electron systems.

CCpy is mainly written in Python, to ensure ease of use and extensibility, and uses Fortran

and Numpy libraries within the computationally critical routines to maintain good perfor-

mance relative to established production-level CC/EOMCC codes, such as those distributed

in GAMESS [227, 251]. The CCpy software package focuses on executing post-SCF steps

of the CC/EOMCC calculations and currently uses interfaces with either PySCF [342, 343]

or GAMESS to perform the initial HF computations and obtain the transformed one- and

two-electron integrals in the molecular orbital basis that are used to set up the correlated

CC calculations. A general interface that can be used to initialize CCpy calculations using

RHF/ROHF-like reference state information and one- and two-electron integrals provided by

an FCIDUMP file is also included. In the remainder of this appendix, we aim to categorize

and provide a brief overview of the different types of CC/EOMCC calculations that can be

performed using CCpy.

149

Ground-State CC Methodologies

CCpy provides a range of conventional CC methods defined by truncating the cluster

operator T at a specific many-body rank. Standard CC approaches, including CCD, CCSD,

CCSDT, and CCSDTQ, are available for RHF, UHF, and ROHF references. As emphasized

throughout this document, the higher-level CCSDT and CCSDTQ methods are included to

give the user access to powerful, albeit computationally expensive, approaches for obtaining

quantitative descriptions for correlation in many-electron systems. While these are useful

for benchmarking purposes, CCpy offers more practical options for obtaining high-level CC

energetics using active-space CC methodologies and the more general CC(P ;Q) formalism.

A distinctive feature of CCpy is its general CC(P ) module, which performs CC calcu-

lations in a P space spanned by all singly and doubly excited determinants along with an

arbitrary subset of triply excited determinants. Like the other CC options, the CC(P ) imple-

mentation is compatible with RHF, UHF, and ROHF references. The CC(P ) method offers

a huge amount of flexibility in defining CC calculations that include T3 clusters. While the

CC(P ) module can be used to carry out CCSDT computations (in a fashion that takes ad-

vantage of permutational and molecular point group symmetry), its primary strength lies in

allowing to user to perform various kinds of nonperturbative approximations to CCSDT. For

example, through the general CC(P ) code, CCpy supports active-space CCSDt calculations

along with its lower-cost CCSDt(II) and CCSDt(III) analogs introduced in Ref. [126]. Auxil-

iary routines for generating the appropriate lists of triply excited determinants corresponding

to the active-space CCSDt-type calculations are included in CCpy as well. Other varieties of

CC(P ) calculations, including those entering the semi-stochastic, CIPSI-based, and adaptive

CC(P ;Q) approaches, will be discussed in the later section on CC(P ;Q) methods.

EOMCC Approaches for Studying Electronic Excited States

A number of EOMCC approaches for describing electronically excited states are available

in CCpy as well. Within the standard EOMCC hierarchy, CCpy includes the basic EOM-

CCSD approach as well as its higher-level EOMCCSDT counterpart. As discussed earlier

150

in this dissertation, the EOMCCSDT approach is capable of providing an accurate treat-

ment of many-electron correlation effects needed to describe both singly and doubly excited

states accurately, but at significant computational cost. CCpy provides cheaper, yet robust,

alternatives to the EOMCCSDT calculations via the general EOMCC(P ) module, which

solves the EOMCCSDT-like equations in a P space spanned by all singly and doubly excited

determinants and a subset of triply excited determinants. With the help of the previously

mentioned functions that generate the lists of triply excited determinants corresponding to

active-space truncations schemes, CCpy can perform the EOMCCSDt calculations in ad-

dition to the analogously defined EOMCCSDt(II) and EOMCCSDt(III) counterparts. As

in the case of the ground-state CC methods, all particle-conserving EOMCC options are

compatible with RHF, UHF, and ROHF references. CCpy also includes two SF-EOMCC

methods, namely SF-EOMCCSD [344] and SF-EOMCC(2,3) [345], which can be used to

describe certain low-spin ground and excited states by applying the spin-flipping excitation

operator truncated at the two-body (SF-EOMCCSD) or three-body [SF-EOMCC(2,3)] level

to a high-spin (e.g., ROHF- or UHF-based) CCSD reference wave function.

When performing EOMCC calculations, the choice of the guess used to initiate the non-

Hermitian variant of the Davidson algorithm is often critical, especially when attempting

to describe states dominated by two-electron transitions. CCpy follows the philosophy of

GAMESS and provides the user with both the EOMCCS and active-space EOMCCSd guess

routines. While the former performs a typical CIS-like diagonalization, which is appropriate

for describing singly excited states, the latter also includes a subset of valence double exci-

tations needed to locate low-lying doubly excited states. The EOMCCSd guess is also often

helpful in improving convergence for certain excited states. Both the EOMCCS and EOM-

CCSd guess routines can be used to target states according to irrep and spin multiplicity.

EOMCC Approaches for Studying Electron Attachment and Ionization

CCpy includes a rather wide selection of particle-nonconserving EOMCC approaches of

the EA/IP- and DEA/DIP-EOMCC types [42–72], which are versatile tools for studying

151

open-shell systems. In the EA/IP-EOMCC methodology, the ground and excited states of a

target (N +1)-electron (EA) or (N −1)-electron (IP) system are determined by diagonalizing

the similarity-transformed CC Hamiltonian of the underlying N -electron closed-shell core in

the appropriate subspace of the Fock space spanned by (N ± 1)-electron excited Slater deter-

minants. Such approaches can be used to determine EAs and IPs of closed-shell molecules

as well as electronic spectra and PESs of radicals. Similarly, the DEA/DIP-EOMCC ap-

proaches diagonalize the N -electron core Hamiltonian in the relevant subspace containing

(N + 2)-electron (DEA) or (N − 2)-electron (DIP) excitations. The DEA/DIP-EOMCC

approaches are especially well-suited to describing the MR correlations characterizing birad-

icals and PESs along single bond-breaking coordinates. Unlike the particle-conserving or SF

CC/EOMCC methods, the EA/IP- and DEA/DIP-EOMCC approaches provide a rigorously

spin- and symmetry-adapted description of open-shell electronic states. To preserve this key

benefit, the EA/IP- and DEA/DIP-EOMCC methods included in CCpy are designed to

work with RHF references only. Although the EA/IP/DEA/DIP-EOMCC codes in CCpy

can be used with any RHF/ROHF-like set of orbitals, they are currently set up to use the

RHF orbitals of the underlying N -electron ground state. Orbitals of the target system can

be used, but this currently requires manual modification of the input scripts.

CCpy includes the basic IP-EOMCCSD(2h-1p) [44, 49, 50, 52, 56–58], EA-EOMCCSD(2p-

1h) [42–44, 51, 52, 57, 58], DEA-EOMCCSD(3p-1h) [57, 58, 62, 63, 70], and DIP-EOMCCSD(3h-

1p) [57–62, 64, 65] approaches, which respectively treat excitations up to the 2p-1h, 2h-1p,

3p-1h, and 3h-1p level on top of the CCSD description of the underlying N -electron closed-

shell core. In order to obtain a highly accurate description of the target attached or ionized

system, it is necessary to include the higher–than–two-body correlations corresponding to the

3p-2h (in the case of EA-EOMCC), 3h-2p (in the case of IP-EOMCC), 4p-2h (in the case of

DEA-EOMCC), or 4h-2p (in the case of DIP-EOMCC) excitations on top of CCSD. There-

fore, CCpy includes the EA-EOMCCSD(3p-2h) [44, 46–48], IP-EOMCCSD(3h-2p) [44, 46–

48], DEA-EOMCCSD(4p-2h) [66–69], and DIP-EOMCCSD(4h-2p) [66, 67] approaches that

152

allow one to do this as well. As in the previous cases, the higher-level EA/IP/DEA/DIP-

EOMCC methodologies come with much higher computational costs in the diagonalization

steps, which scale as N 7 (in the case of EA/IP-EOMCC methods) or N 8 (in the case

of DEA/DIP-EOMCC methods) with the system size. In order to provide options to help

reduce these costs while maintaining high accuracy, CCpy also provides a few routines for per-

forming general particle-nonconserving EOMCC(P ) calculations that diagonalize the CCSD

similarity-transformed Hamiltonian in a flexibly defined subspace of the Fock space. In par-

ticular, the EA-EOMCC(P ) option performs diagonalization in the subspace spanned by all

1p and 2p-1h determinants and a subset of 3p-2h determinants. Similarly, the IP-EOMCC(P )

option executes the diagonalization in a subspace containing all 1h, 2h-1p, and a subset of

3h-2p excitations. Likewise, the DEA-EOMCC(P ) routine diagonalizes the CCSD Hamilto-

nian in the (N + 2)-electron subspace spanned by all 2p and 3p-1h excitations and a general

selection of 4p-2h excitations. With the help of these P -space EOMCC routines, CCpy

can perform the active-space EA-EOMCCSDt(3p-2h) [46–48], IP-EOMCCSDt(3h-2p) [46–

48], and DEA-EOMCCSD(4p-2h){Nu} [66–69] calculations, again, with the help of auxiliary

routines used to generate the requisite lists of (N ± 1)- or (N + 2)-electron excitations [the

P -space variant of the DIP-EOMCCSD(4h-2p), which can be used to execute the analogous

DIP-EOMCCSD(4h-2p){No} [66, 67] calculations, is currently under development].

CCpy also offers a more limited selection of even higher-level particle-nonconserving

EOMCC methods that replace the the CCSD similarity-transformed Hamiltonian of the N -

electron reference system with its more accurate CCSDT counterpart in the diagonalization

step. These approaches, which include EA-EOMCCSDT(3p-2h) [44, 45], IP-EOMCCSDT(3h-

2p) [44, 53–56], and DIP-EOMCCSDT(4h-2p) [72], are useful when one needs to balance the

descriptions of the target ionized or attached electronic states and the N -electron closed-

shell ground state, as in studies where the goal is to accurately determine the IPs/DIPs or

EAs/DEAs of a molecular system [72]. If one is only interested in energy differences between

the target ground and excited electronic states, which is the case when determining excitation

153

spectra and PESs of open-shell radicals and biradicals, then replacing the underlying CCSD

similarity-transformed Hamiltonian with its CCSDT counterpart is of less importance.

As in the case of the particle-conserving EOMCC options, CCpy offers multiple initial

guess routines for performing the EA/IP/DEA/DIP-EOMCC calculations. For the EA/IP-

EOMCC options, CCpy can compute the basic 1p/1h roots for describing simple Koopmans’

states as well as more sophisticated guesses that include the valence 2p-1h/2h-1p excitations,

needed to describe the more strongly correlated doublet and quartet states in the radical

excitation spectrum. Similarly, DEA/DIP-EOMCC calculations can be initiated using 2p/2h

guess vectors, which can be augmented with valence 3p-1h/3h-1p excitations to describe more

strongly correlated excited states.

LR CC Methods for Ground and Excited States

CCpy includes a limited selection of LR CC methodologies for describing ground and

excited electronic states based on the perturbative CCn hierarchy []. In this category, CCpy

includes the CC3 approach and its higher-level CC4 counterpart, which solve an approxi-

mate form of the CCSDT and CCSDTQ equations, respectively. In order to study excited

electronic states, CCpy includes the EOM extension of the CC3 approach, abbreviated as

EOMCC3, which is equivalent to the original LR formulation of CC3 for excitation energies

(the two will differ for properties other than energy). The CC3 and EOMCC3 calculations

are compatible with RHF, UHF, and ROHF references, whereas, the CC4 implementation is

currently only available for RHF references. When using noncanonical orbitals, like ROHF,

the occupied-virtual block of the Fock matrix is treated as first order. The CCn calcula-

tions using noncanonical orbitals are not orbitally invariant unless one semicanonicalizes the

occupied and virtual orbitals prior to entering the CC3/EOMCC3 routine [346].

Perturbative CC/EOMCC Methods

Although we have cautioned against the use of perturbative CC/EOMCC methodologies

when studying systems featuring MR correlations and electronic quasi-degeneracies, CCpy

includes a selection of perturbative CC/EOMCC corrections, which can be useful for more

154

benign SR systems and benchmarking purposes.

In particular, we provide the CCSD(T)

correction to the CCSD energetics in addition to EOMCCSD(T)(a)*, IP-EOMCCSD(T)(a)*,

and EA-EOMCCSD(T)(a)* methodologies of Ref. [121], which perturbatively correct the

EOMCCSD, IP-EOMCCSD(2h-1p), and EA-EOMCCSD(2h-1p) calculations for three-body

correlation effects, respectively. Recently, we have also formulated and implemented the

DIP-EOMCCSD(T)(a) approximation [72] to the parent DIP-EOMCCSDT(4h-2p) theory,

which incorporates the T3 correlation effects in the DIP-EOMCC calculations using the

CCSD(T)(a) framework adopted in Ref. [121]. All perturbative CC/EOMCC methods can be

used with RHF, UHF, and ROHF references. As in the previous case regarding noncanonical

CCn calculations, the occupied-virtual block of the Fock matrix is treated as first order, and

orbital invariance may be lost unless semicanonicalization steps are performed [108, 347].

CR CC/EOMCC Approaches

In the majority of chemical problems, it is better to address higher–than–pair correlations

using the CR CC/EOMCC corrections based on the MMCC framework instead of their

perturbative counterparts discussed in the preceding section. CCpy offers a number of CR-

CC methodologies, including the CR-CC(2,3) triples correction to CCSD along with its

CR-CC(2,4) extension that accounts for the T4 effects missing from CCSD as well. The

higher-level CR-CC(3,4) method has also recently been included in CCpy, which corrects the

CCSDT energetics for missing four-body clusters, resulting in a very robust approximation

to the parent CCSDTQ energetics.

The CR-EOMCC methods available in CCpy include the CR-EOMCC(2,3) triples cor-

rection to the EOMCCSD total energies along with the rigorously size-intensive δ-CR-

EOMCC(2,3) correction to the EOMCCSD vertical excitation energies. Recently, we have

also extended the CR-EOMCC corrections to the EA/IP-EOMCC frameworks, resulting

in the EA-CR-EOMCC(2,3) and IP-CR-EOMCC(2,3) approaches, which correct the ener-

getics resulting from the EA-EOMCCSD(2p-1h) and IP-EOMCCSD(2h-1p) calculations for

missing 3p-2h and 3h-2p correlations, respectively. With the exception of the EA/IP-based

155

CR-EOMCC methods, which are designed to work with RHF references to maintain spin

symmetry, all CR CC/EOMCC methods can be used with RHF, UHF, and ROHF references.

CC(P;Q) Methodologies for Converging High-Level CC/EOMCC Energetics

As discussed throughout this dissertation, there are situations when MR correlations

become stronger, leading to substantial coupling between higher- and lower-rank cluster

and excitation operators that render the CR CC/EOMCC corrections to the lower-level

CC/EOMCC energetics ineffective. To address this challenge, CCpy specializes in per-

forming highly efficient and robust CC(P ;Q) computations aimed at converging high-level

CC/EOMCC energetics at tiny fractions of the computational effort. The CC(P ;Q) schemes

currently available in CCpy aimed at converging the full CCSDT/EOMCCSDT energetics

include the active-orbital-based CC(t;3) method, the semi-stochastic CC(P ;Q) and CIPSI-

driven CC(P ;Q) approaches, as well as the adaptive CC(P ;Q) methodology. Extensions

of CC(P ;Q) of the adaptive and CIPSI-driven varieties aimed at converging the CCS-

DTQ/EOMCCSDTQ and similar energetics are currently being developed. We have also re-

cently implemented the active-orbital based EA-CC(t;3) and IP-CC(t;3) approaches, which,

in analogy to the CC(t;3) method, are capable of recovering the EA-EOMCCSD(3p-2h) and

IP-EOMCCSD(3h-2p) and energetics to within a fraction of a millihartree by correcting the

results obtained from EA-EOMCCSDt(3p-2h) and IP-EOMCCSDt(3h-2p) calculations for

the missing 3p-2h and 3h-2p correlations, respectively. The development of black-box vari-

ants of these EA/IP-CC(P ;Q) approaches driven by CIPSI and adaptive moment expansions

will be subjects of future studies as well.

ACP Approaches for Describing Strongly Correlated Systems

One of the most interesting ways of describing strong correlation in many-electron systems

using CC theory is found by turning to the ACP framework (see, for example, Refs. [210,

348–357]). The existing ACP methods and their various modifications retain all doubly

excited cluster amplitudes, while using subsets of nonlinear diagrams of the CCD/CCSD

equations. This eliminates failures of conventional CC approaches, including CCSD and

156

even CCSDT or CCSDTQ, in strongly correlated situations created by the Mott metal-

insulator transitions, as modeled by linear chains, rings, or lattices of hydrogen atoms and

the π-electron networks described by the Hubbard and Pariser–Parr–Pople Hamiltonians that

model one-dimensional metallic systems with periodic boundary conditions. The basic ACP

approaches available in CCpy include the ACCD and ACCSD methods. The T3 correlations

missing from the ACCD/ACCSD calculations can be incorporated by turning to higher-level

ACCSDT approach or its more practical ACCSDt approximation, which includes the leading

contributions to T3 clusters using an active-space treatment similar to CCSDt. Triples

corrections to the ACCSD approach based on the CR-CC framework are available with the

ACC(2,3) option, while corrections to the ACCSDt energetics for missing T3 correlations are

obtained via the ACC(t;3) method.

Hybrid ec-CC Methodologies Combining CI and CC Wave Functions

The final class of CC/EOMCC approaches available in CCpy belong to the ec-CC hier-

archy, discussed earlier in this dissertation. In particular, CCpy includes the CIPSI-driven

ec-CC-II and ec-CC-II3 approaches discussed in Section 2.4 and Chapter 5 of this disserta-

tion as well as its ec-CC-II3,4 extension that corrects for T4 correlations missing from the

CIPSI wave function as well. As in the CIPSI-based CC(P ;Q) implementation in CCpy, the

determinants and associated coefficients characterizing the external CI wave function must

be stored in the format adopted by Quantum Package. Once this is done, the ec-CC options

in CCpy are capable of extracting the corresponding T3 and T4 via cluster analysis and per-

form the subsequent ec-CC-II and ec-CC-II3 calculations. In fact, other non-CC sources of

T3 and T4 clusters can also be used in conjunction with the ec-CC-II/ec-CC-II3 routines in

CCpy if the requisite CI coefficients and determinants are provided to CCpy in an external

file. In this way, the ec-CC module in CCpy is very flexible and allows one to perform a

variety of hybrid ec-CC calculations in a straightforward and unified fashion.

157

APPENDIX B

DERIVATION OF THE CC(P;Q) MOMENT EXPANSIONS

In this appendix, we provide a derivation of the key formulas defining the CC(P ;Q) en-

ergy corrections, Eqs. (2.33)–(2.37), introduced in Chapter 2. Following Refs. [126, 146,

147], the derivation starts by considering the asymmetric energy expression involving the

CC(P )/EOMCC(P ) ket state

E(FCI)
µ

= ⟨Ψ(FCI)
µ

|HR(P )

µ eT (P )|Φ⟩/⟨Ψ(FCI)

µ

|R(P )

µ eT (P )|Φ⟩.

(B.1)

Equation (B.1) is easily verified by acting on ⟨Ψ(FCI)

µ

| with H on the left via the ”turnover

rule”. Without loss of generality, the exact, FCI, bra state can be parameterized as ⟨Ψ(FCI)

µ

| =

⟨Φ|Lµe−T (P ), where the deexcitation operator

Lµ = L(P )

µ + δL (P )

µ

,

with

and

L (P )

µ = δµ,01 + X

ℓµ,K(EK)†

|ΦK ⟩∈H (P )

δL (P )

µ =

X

ℓµ,K(EK)†,

|ΦK ⟩∈(H (0)⊕H (P ))⊥

(B.2)

(B.3)

(B.4)

is formally obtained by solving the adjoint FCI eigenvalue problem in the entire many-

electron Hilbert space

⟨Φ|LµH (P ) = E(FCI)

µ

⟨Φ|Lµ.

Using this form of the FCI state, Eq. (B.1) becomes

E(FCI)
µ

= ⟨Φ|LµH (P )R(P )

µ |Φ⟩/⟨Φ|LµR(P )

µ |Φ⟩.

(B.5)

(B.6)

Expanding the quantity Lµ into many-body components according to Eq. (B.2), Eq. (B.6)

can be written as

E(FCI)
µ

= [⟨Φ|L (P )

µ R(P )

µ |Φ⟩]−1[⟨Φ|L (P )

µ H (P )R(P )

µ |Φ⟩ + ⟨Φ|δL (P )

µ H (P )R(P )

µ |Φ⟩],

(B.7)

158

where we have used the fact that ⟨Φ|δL (P )

µ R(P )

µ |Φ⟩ = 0. Next, we impose the normalization

⟨Φ|L (P )

µ R(P )

µ |Φ⟩ = 1,

(B.8)

which, while not identical to the biorthonormality relationship satisfied by L(P )

µ and R(P )

µ

[Eq.

(2.31)], is nonetheless a reasonable condition given the similarity between L (P )

µ

and L(P )
µ .

With the help of Eq. (B.8), which also allows us to identify E(P )

µ = ⟨Φ|L (P )

µ H (P )R(P )

µ |Φ⟩,

Eq. (B.7) takes on the more revealing form

E(FCI)
µ

= E(P )

µ + ⟨Φ|δL (P )

µ H (P )R(P )

µ |Φ⟩,

(B.9)

and the corrections to the CC(P )/EOMCC(P ) energetics toward FCI can be defined as

E(FCI)
µ

− E(P )

µ = ⟨Φ|δL (P )

µ H (P )R(P )

µ |Φ⟩.

(B.10)

Inserting a resolution of identity between δL (P )

µ

and H (P ) in Eq. (B.10) results in an exact

form of the moment energy expansions

E(FCI)
µ

− E(P )

µ =

X

ℓµ,K Mµ,K(P ),

(B.11)

|ΦK ⟩∈(H (0)⊕H (P ))⊥

where Mµ,K(P ) are the generalized moments of CC(P ) [Eq. (2.34)] and EOMCC(P ) [Eq.

(2.35)] equations and ℓµ,K are the amplitudes entering the operator δLµ characterizing the
exact, FCI, bra state ⟨ ˜Ψ(FCI)

|.

µ

As it stands, Eq. (B.11) provides the formulas for the corrections to the energies resulting

from CC(P )/EOMCC(P ) calculations toward the exact, FCI limit. Unfortunately, we have

not yet arrived at a computationally feasible recipe for obtaining these corrections, since the

amplitudes ℓµ,K multiplying moments Mµ,K(P ) in Eq. (B.11) are determined by solving the

exact left-eigenstate problem Eq. (B.5). Simply put, there is no free lunch in obtaining FCI.

However, the advantage of Eq. (B.11) lies in the fact that approximations to the coefficients

ℓµ,K, even relatively low-order ones, can be used to provide useful noniterative corrections.

In the rigorously size-extensive biorthogonal CC(P ;Q) framework pursued in this work, we

adopt the approximation to ℓµ,K resulting from the quasi-perturbative L¨owdin partitioning of

159

Eq. (B.5). In particular, by projecting Eq. (B.5) onto the set of determinants corresponding

to the content of δL (P )

µ

, we obtain

⟨Φ|L (P )

µ H (P )|ΦK⟩ + ⟨Φ|δL (P )

µ H (P )|ΦK⟩ = E(FCI)

µ

ℓµ,K,

|ΦK⟩ ∈ (H (0) ⊕ H (P ))⊥,

(B.12)

and assuming that H (P ) is approximately diagonal in the complementary space [meaning,
⟨ΦL|H (P )|ΦK⟩ ≈ δKL⟨ΦK|H (P )|ΦK⟩ for |ΦK⟩, |ΦL⟩ ∈ (H (0) ⊕ H (P ))⊥], we can solve for the

amplitudes ℓµ,K in Eq. (B.12) directly to obtain

ℓµ,K = ⟨Φ|L (P )

µ H (P )|ΦK⟩/(E(FCI)

µ

− ⟨ΦK|H (P )|ΦK⟩),

|ΦK⟩ ∈ (H (0) ⊕ H (P ))⊥.

(B.13)

We can then replace the exact quantities L (P )

µ

and E(FCI)

µ

in Eq. (B.14) with their respective

L(P )

µ and E(P )

µ

approximations obtained by solving the CC(P )/EOMCC(P ) problem to arrive

at a computationally tractable form of the coefficients ℓµ,K, namely,

ℓµ,K(P ) = ⟨Φ|L(P )

µ H (P )|ΦK⟩/(E(P )

µ − ⟨ΦK|H (P )|ΦK⟩),

|ΦK⟩ ∈ (H (0) ⊕ H (P ))⊥.

(B.14)

Finally, we substitute Eq. (B.14) into Eq. (B.11), limiting ourselves to correcting the CC(P )

and EOMCC(P ) energetics for correlation effects due to |ΦK⟩ in the Q-space H (Q) rather

than in the entire remaining subspace of the Hilbert space (H (0) ⊕ H (P ))⊥. This results in

the noniterative CC(P ;Q) moment corrections summarized in Eqs. (2.33)–(2.37).

160

APPENDIX C

EFFICIENT ALGORITHM FOR THE CC(P) AND EOMCC(P)
APPROACHES AIMED AT CONVERGING CCSDT AND EOMCCSDT

This appendix contains information about our novel algorithm for solving the CC(P ) and

EOMCC(P ) equations within a P space spanned by all singly and doubly excited determi-

nants and a subset of triply excited determinants, which could be obtained with the help

of active orbitals, CIQMC/CCMC wave function propagations, CIPSI calculations, or the

adaptive CC(P ;Q) moment expansions. Our focus will primarily be on the algorithm used

to solve the CC(P ) equations aimed at converging CCSDT. Our implementation for the

EOMCC(P ) approach, which solves a subset of the EOMCCSDT equations, and the com-

panion left-eigenstate CC(P )/EOMCC(P ) methods are virtually the same as the CC(P )

case after an appropriate definition of one- and two-body intermediates.

C.1 General Considerations

We begin by analyzing the structure of the CC(P )/EOMCC(P ) equations, in which

the cluster and EOM excitation operators in the P space are T (P ) = T1 + T2 + T (P )

3

and

R(P )

µ = rµ,0 + Rµ,1 + Rµ,2 + R(P )

µ,3 , respectively. Let us, for now, assume that the T (P )

3

and R(P )

µ,3 operators are defined over distinct P spaces, denoted as H (P )

µ

, where µ = 0

corresponds to the P space for the ground state and µ > 0 denote P spaces for excited states.

Focusing on the ground-state CC(P ) case, if we define the CCSD-like similarity-transformed

Hamiltonian H (2) = e−T1−T2HeT1+T2 along with the normal-product form of H (2) relative

to |Φ⟩, H (2)

N ≡ H (2) − E(P )

0 1, which naturally arise as computational intermediates in the

CCSDT system, then Eq. (2.24) becomes

M0,K(CCSD) + ⟨ΦK|[H (2)

N , T (P )

3

]|Φ⟩ = 0,

|ΦK⟩ ∈ H (P )

0

,

(C.1)

where M0,K(CCSD) = ⟨ΦK|H (2)

N |Φ⟩ are generalized moments of CCSD equations and we

have used the fact that the CC(P ) amplitude equations corresponding to projections onto

singly, doubly, and triply excited determinants eliminate terms that are nonlinear in T (P )

3

.

The programmable expressions for the one- and two-body components of H (2)

N , denoted as

161

p and ¯hrs
¯hq

pq, are given in Table C.1. Similarly, the σ-vectors [Eq. (2.16)] corresponding to

projections of the EOMCC(P ) eigenvalue problem on the relevant singly, doubly, and subset

of triply excited determinants are given by

Mµ,K(CCSD) + ⟨ΦK|[H (2)

N , R(P )

µ,3 ]|Φ⟩ + ⟨ΦK|[X (2)

µ , T (P )

3

]|Φ⟩,

|ΦK⟩ ∈ H (P )

µ

,

(C.2)

where Mµ,K(CCSD) = ⟨ΦK|X (2)
with X (2)

µ = H (2)

µ |Φ⟩ are the generalized moments of EOMCCSD equations,

N (rµ,0 + Rµ,1 + Rµ,2) denoting the intermediates used to factorize the con-

tributions to the EOMCCSDT equations due to three- and four-body components of the

CC(P ) similarity-transformed Hamiltonian. Thanks to the definition of H (2)

N and X (2)

µ , the

Eqs. (C.1) and (C.2) are linear in T (P )

3

and R(P )

µ,3 . In addition, Eqs. (C.1) and (C.2) have

similar structures, and indeed, we will exploit this similarity to implement the EOMCC(P )

algorithm by simply recycling code written for the CC(P ) case. Before we can discuss

these details further, we must first address a concern regarding the potential inclusion of

size-intensivity-violating terms in the EOMCC(P ) equations.

Equation (C.2) formally contains contributions corresponding to the ground state in the

form of rµ,0[M0,K(CCSD) + ⟨ΦK|[H (2)
spaces for the ground and excited states are identical, meaning that H (P )

]|Φ⟩] for |ΦK⟩ ∈ H (P )

N , T (P )

µ

3

. In situations where the P

0 = H (P )

µ ≡ H (P )

for all µ, the ground-state contributions vanish as a result of solving the CC(P ) amplitude

equations [Eq. (C.1)] and size-intensivity of the resulting excitation energies is automati-

cally retained. This is always the case for conventionally truncated EOMCC approximations

as well as active-orbital-based CC(P ;Q) methods based on the underlying CCSDt/EOM-

CCSDt, CCSDtq/EOMCCSDtq, etc. calculations, which de facto adopt a single P space

for all electronic states. For the semi-stochastic, CIPSI-driven, or adaptive CC(P ;Q) ap-

proaches, we have the possibility of constructing separate P spaces for each state, and a

decision for how to handle the ground-state contributions in the resulting EOMCC(P ) equa-

tions is required. In the semi-stochastic excited-state CC(P ;Q) studies of Refs. [133, 253]

and the CIPSI-based CC(P ;Q) calculations for CH+ reported in Section 3.2.3 of this dis-

sertation, the terms proportional to rµ,0 were eliminated in the EOMCC(P ) calculations

162

by adopting a single P space for all electronic states belonging to a particular symmetry.

However, in our adaptive CC(P ;Q) calculations for water discussed in Section 4.2.4, separate

P spaces were constructed for each electronic state, and unless we enforce overlap between

the P spaces for the relevant ground and excited states, the EOMCC(P ) equations formally

retain terms proportional to rµ,0. Our solution for this issue is quite simple: we drop the

size-intensive-violating terms from Eq. (C.2), which in practice, is accomplished by simply

redefining the computational intermediates X (2)

µ = [H (2)
µ,pq entering the newly defined quantity X (2)

N (Rµ,1 + Rµ,2)]C. The one- and two-

µ are provided in

body components ¯x q

µ,p and x rs

Table C.2. Although our neglect of size-intensive-violating terms may appear ad hoc, it is

worth mentioning that this idea is also adopted in other approximations to EOMCCSDT,

including EOMCCSDT-n [117, 119] and EOMCCSD(T)(a) [121]. Similar considerations also

enter the rigorously size-intensive δ-CR-EOMCC(2,3) [152] triples correction to EOMCCSD

as well as its perturbatively defined analogs [119, 328, 329].

With the matter of size-intensivity behind us, we now discuss the similarity between

Eqs. (C.1) and (C.2). In particular, the second term in Eq. (C.1) is identical to the second

term in Eq. (C.2) if one replaces replaces R(P )

µ,3 in the latter by T (P )

3

. The third term in Eq.

(C.2) is also the same as the second term in Eq. (C.1) after replacing H (2)

N by X (2)

µ . Thus, it

should not come as a surprise that we can adopt an essentially identical strategy for solving

the EOMCC(P ) equations as in the CC(P ) case. To be more specific, our EOMCC(P )

implementation is obtained by making two copies of the CC(P ) source code. In the first

copy, the quantity T (P )

3

is replaced with R(P )

µ,3 , while in the second copy, the matrix elements of

H (2)

N are replaced by X (2)

µ . With this in mind, we can restrict our attention for the remainder

of this appendix on the CC(P ) case. Since we are only considering the ground state, we will

drop the subscript µ from the P spaces H (P )

µ

and work with a single P space H (P )

0 ≡ H (P )

from here on out.

Returning to our development of the CC(P ) algorithm, the next step is to expand the

commutator in Eq. (C.1) and insert a resolution of identity between the resulting products

163

of H (2)

N and T (P )

3

to obtain the working equation used in our CC(P ) algorithm,

M0,K(CCSD)
|
{z
}
(I)

+ X
|Φdef
|

lmn⟩∈H (P )

⟨ΦK|H (2)

N |Φdef

lmn⟩tlmn
def

= 0,

|ΦK⟩ ∈ H (P ).

(C.3)

{z
(II)

}

In what follows, we will describe the strategies adopted for evaluating terms (I) and (II)

separately.

Table C.1 The one- and two-body components of the CCSD-like similarity-transformed
Hamiltonian H (2)
a and
tij
ab respectively, and matrix elements of the Fock and two-electron interaction operators,
denoted as f q

N expressed in terms of the one- and two-body cluster amplitudes, ti

pq ≡ ⟨pq|v|rs⟩ − ⟨pq|v|sr⟩.

p ≡ ⟨p|f |q⟩ and vrs

Component of H (2)
N
¯he
m
¯hi
j
¯hb
a
¯hef
mn
¯hef
am
¯hie
mn
¯hef
ab
¯hij
mn
¯hie
am
¯hij
am
¯hie
ab
χ′ie
am
χie
am
τ ij
ab

2vef
2vbf

jntin
ef
mntmn
af

Expressiona
mntn
m + vef
f e
f
j + ¯he
e + 1
jmtm
jti
f i
e + vie
a − ¯hb
e − 1
amtm
a + vbe
mtm
f b
vef
mn
mntn
am − vf e
vef
a
mnti
mn + vf e
vie
f
ab − Aabvef
amtm
mnτ mn
2vef
b
ef + A ijvje
mnτ ij
nmti
2vef
e
f − ¯hie
mntin
a + vef
nmtn
amti
af
ef + A ij(¯hjf
amtij
a + 1
mntin
2vef
¯hie
ab − Aab(χie
f + 1
amtm
mntmn
2
am + 1
vie
2vef
amti
e
¯hie
¯hie
am + 1
nmtn
a
2
ab + A ijti
tij
atj

vef
ab + 1
mn + 1
vij
am + vf e
vie
ae − ¯hij
nmtn
ab + vef
ab ti

b

am + ¯he
vij
ab − ¯he
vie

mtij
mtim

af + χ′ie
b − vef

amtj
e)
bntin
af )

a In each expression, summation is carried out over repeated upper and lower indices.

164

Table C.2 The one- and two-body components of the intermediates X (2) expressed in terms
of the one- and two-body components of the cluster operator, EOM excitation operator, and
the CCSD-like similarity-transformed Hamiltonian, entering Eq. (C.2) that are introduced
in order to evaluate the contributions to the EOMCC(P ) equations due to the three- and
four-body components of the CC(P ) similarity-transformed Hamiltonian. In this table, we
drop the subscript µ labeling the one- and two-body components of Rµ and X (2)
so that
µ,pq ≡ ¯xrs
µ,a ≡ ri
r i
pq.

p, and ¯x rs

µ,ab ≡ rij

µ,p ≡ ¯xq

a, r ij

ab, ¯x q

µ

Component of X (2)
µ
¯xe
m
¯xi
j
¯xb
a
¯xef
ab
¯xij
mn
¯xie
am
¯xij
am
¯xie
ab

Expressiona
¯hef
mnrn
f
e + ¯hif
¯he
¯hef
f + 1
jnrn
jri
jnrin
ef
2
a + ¯hbf
−¯hb
¯hbf
f − 1
anrn
mrm
mnrmn
af
2
¯hef
¯hef
−Aab
b + 1
mnrmn
amrm
ab
2
A ij¯hje
¯hef
mnrij
e + 1
nmri
2
a + ¯hef
f − ¯hie
¯hf e
nmrn
amri
e] − ¯hij
ae + ¯hie
A ij[¯hje
nmrn
amrj
mnrin
ab + A ab[¯hie
¯hie
f + 1
mtim
ab − ¯xe
mnrmn
2

¯hf e
ab ri

ef
mnrin
af
¯hef
amrij
a + 1
2
b + ¯hf e
amrm

ef
amrim
f b ]

a In each expression, summation is carried out over repeated upper and lower indices.

165

C.2 Evaluation of Term (I)

The first term in Eq. (C.3) is a generalized moment of the CCSD equations. In particular,

when |ΦK⟩ ≡ |Φa

i ⟩ or |ΦK⟩ ≡ |Φab

ij ⟩, the corresponding one- and two-body moments of the

CCSD equations, M i

0,a(CCSD) and M ij

0,ab(CCSD), represent the standard CCSD amplitude

equations. These terms are computed in a fully vectorized fashion (i.e., avoiding the use

of explicit loops) by taking advantage of efficient matrix transposition and multiplication

routines provided by BLAS. Because our CC(P ;Q) calculations aimed at converging the

full CCSDT energetics include all singly and doubly excited determinants in the P spaces,

evaluating M i

0,a(CCSD) and M ij

0,ab(CCSD) involves the conventional n2

on4

u computational

steps characterizing CCSD. When |ΦK⟩ represents a triply excited determinant in the P

space, |Φabc

ijk⟩, the generalized three-body moment of the CCSD equations is evaluated using

1
4
where A pqr ≡ Apqr = 1 − (pq) − (pr) − (qr) + (pqr) + (prq) is a three-index antisymmetrizer

0,abc(CCSD) = A ijkAabc

ec − X

amtmk
I ij

¯hie
abtjk

M ijk

(C.4)

(X

bc ),

m

e

ae [cf. Eqs. (59) and (62) of Ref. [151]]. Within every iteration of the

and I ij

am = ¯hij

am −P
CC(P ) solution procedure, the ¯hq

¯he
mtij

e

p, ¯hrs
so that Eq. (C.4) can be evaluated for each triply excited determinant |Φabc

am intermediates are precomputed and stored
ijk⟩ ∈ H (P ) via dot

pq, and I ij

products between ¯hie

ab and I ij

am and the T2 cluster amplitudes. This allows us to reduce the

number of CPU operations associated with determining the contribution of M ijk

0,abc(CCSD)

to the CC(P ) equations relative to its CCSDT counterpart by a factor of (D/d), where D is

the number of all triples and d is the number of triples included in the P space. As we will

discuss later on in Section C.3.2.3, Eq. (C.4) is also used to absorb the contributions to the

CC(P ) equations due to three-body components of H (2)

N , which formally enter in term (II),

with the help of suitable modifications to the quantities ¯hie

ab and I ij

am.

C.3 Evaluation of Term (II)

We now turn our attention to term (II) in Eq. (C.3), which represents the contributions

to the CC(P ) equations that are linear in T (P )

3

. In principle, term (II) is nothing more than

a product between the matrix representation of H (2)

N in the singles-triples (⟨Φa

i |H (2)

N |Φdef

lmn⟩),

166

doubles-triples (⟨Φab

ij |H (2)

N |Φdef

lmn⟩), and triples-triples (⟨Φabc

ijk|H (2)

N |Φdef

lmn⟩) sectors of the P space

and the vector of three-body cluster amplitudes. Precomputing the matrix H (2)

N for use in

a matrix-vector product each iteration is not practical for most problems due to the size of

the singles-triples, doubles-triples, and triples-triples blocks. The strategy we adopt instead

is to compute matrix elements of H (2)
carried out such that we skip the evaluation of ⟨ΦK|H (2)

N on the fly, but in order to be efficient, this must be

N |Φdef

lmn⟩ terms which are zero.

C.3.1 Projections onto Singly and Doubly Excited Determinants

In the case of the projections onto singly and doubly excited determinants, this can be

accomplished using an amplitude-driven strategy, where we iterate over the triply excited

determinants in the P space and evaluate all contributions to the CC(P ) equations projected

on singly and doubly excited determinants coming from a given amplitude tijk

abc correspond-
ijk⟩ ∈ H (P ). This strategy is outlined in Algorithm C.1. There are four distinct

ing to |Φabc

diagrams in the CC(P ) equations corresponding to projections onto singly and doubly ex-

cited determinants that involve T (P )

3

, referred to as Diagrams 1–4. The specific expressions

used to evaluate these diagrams in Algorithm C.1 can be derived using standard many-body

techniques. The only potentially murky point is the use of an additional summation over

index permutations operators in lines 6, 12, and 19 of Algorithm C.1, which, from a diagram-

matic perspective, exchanges the labels carried by internal lines connecting the Hamiltonian

vertex and T (P )

3 with those carried by external lines emanating from the T (P )

3

vertex going

into the projection. The presence of these additional permutations, which does not normally

enter our diagrammatic considerations, is consequence of working with a permutationally

unique set of triply excited determinants and cluster amplitudes [i.e., the summations over

permutations could removed if one includes the redundant triply excited determinants and

corresponding amplitudes in the list used in the CC(P ) procedure].

167

ij ⟩, denoted as Di
ijk⟩ ∈ H (P ) do

Get t3 ← tijk
abc

i ⟩ and |Φab
1: for |Φabc
2:
3:
4:
5:
6:
7:

Algorithm C.1 Contributions to term (II) in Eq. (C.3) corresponding to projections onto
|Φa
ab, respectively.

a and Dij

▷ Loop over triply excited determinants in the P space

▷ The three-body cluster amplitude associated with |Φabc
ijk⟩

// Permutations corresponding to (k/ij)(c/ab)
for P ∈ {(1), (ik), (jk), (ac), (bc), (ik)(ac), (ik)(bc), (jk)(ac), (jk)(bc)} do

PDi
PDij
end for

a ← PDi
ab ← PDij

a + (−1)PP¯hab
ab + (−1)PP¯hc

ij × t3
k × t3

// Permutations corresponding to (j/ik)(c/ab)
for P ∈ {(1), (ij), (jk), (ac), (ab), (ij)(ac), (ij)(ab), (jk)(ac), (jk)(ab)} do

for m = 1, no do

PDmj

ab ← PDmj

ab − (−1)PP¯hmc

ik × t3

end for

end for

▷ Diagram 1
▷ Diagram 2

▷ Diagram 3

// Permutations corresponding to (k/ij)(b/ac)
for P ∈ {(1), (ik), (jk), (ab), (bc), (ik)(ab), (ik)(bc), (jk)(ab), (jk)(bc)} do

for e = 1, nu do

PDij
end for

end for

eb ← PDij

eb + (−1)PP¯hac

ek × t3

ij ⟩ ∈ H (P ) do
ab ← A ijAabDij

ab

▷ Diagram 4

▷ Antisymmetrize the two-body residual

8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
21:
22:
23:
24: end for
25:
26: for |Φab
Dij
27:
28: end for

168

C.3.2 Projections onto Triply Excited Determinants

In the contributions to term (II) of Eq. (C.3) corresponding to projections onto triply

excited determinants in the P space, the nonzero matrix elements of the ⟨Φabc

lmn⟩,
lmn⟩ ∈ H (P ), type are most clearly represented using the diagrams shown

ijk⟩, |Φdef

with |Φabc

ijk|H (2)

N |Φdef

in Figure C.1, which involve the one- and two-body components of H (2)

N [the contributions

due to three-body components of H (2)

N can be embedded within the computation of term (I)

projected onto triply excited determinants; see Section C.3.2.3 for further details].

In Figure

Figure C.1 Hugenholtz diagrams corresponding to the contributions in Eq. (C.3) originating
from the one- and two-body components of H (2)
N . In these diagrams, the open circles are T3
vertices, and interaction vertices represent the one- and two-body components of the CCSD-
like similarity-transformed Hamiltonian H (2).

C.1, the six black external lines in each diagram specify the determinant in the bra, |Φabc

ijk⟩,

while the blue internal lines connecting the H (2)

N and T3 vertices, in addition to the black

lines emanating from the T (P )

3

and going into the projection, determine the corresponding

kets |Φdef

lmn⟩ ∈ H (P ). As a consequence of the ket and bra sharing the indices carried

by a subset of the black lines, the resulting matrix elements of H (2)

N are guaranteed to be

nonzero. To be more specific, the nonzero matrix element types corresponding to the diagram
ijk|H (2)

in Figure C.1, from left to right, are ⟨Φabc

mjk⟩, ⟨Φabc

ijk⟩, ⟨Φabc

ijk|H (2)

ijk|H (2)

N |Φef c

N |Φabc

N |Φebc

ijk ⟩,

⟨Φabc

ijk|H (2)

N |Φabc

mnk⟩, and ⟨Φabc

ijk|H (2)

N |Φebc

mjk⟩, respectively. Thus, in order to efficiently construct

the CC(P ) equations, we must find a way to selectively iterate over pairs of triply excited

determinants in the bra and ket that share some or all of their hole and particle indices.

169

C.3.2.1 Radix Sort Algorithm for the List of Triply Excited Determinants

In order to accomplish this task, we must be able to judiciously sort the list of triply

excited determinants entering the CC(P ) calculation without encountering exorbitant CPU

or memory costs. One particularly clever way of doing this is based on a technique known as

the radix sort. Radix sorting the list of triply excited determinants will allow us to organize

determinants according to a subset of their hole and particle indices in such a way that we

can directly identify connected pairs that share a particular combination of hole and particle

indices. This strategy lies at the heart of our novel CC(P ) algorithm.

To acquaint ourselves with the radix sorting procedure, suppose we want to arrange the

numbers (53, 89, 150, 36, 633, 233) in ascending order. In the radix sort, we recursively sort

the sequence of numbers according to a given set of keys. Different sorting tasks may require

different definitions of the keys, but for the simple case of sorting nonnegative integers, the

appropriate keys are the digits of each number, from least significant (rightmost) to most

significant (leftmost). In this example, we must sort according to the least significant digit

first and move toward the most significant digit in order to obtain a correctly sorted list

in the end. Sorting the array according to the leftmost digit, we obtain (150, 53, 633, 233,

36, 89). Note that the individual sorting passes in the radix sort algorithm must be stable,

meaning that objects with equivalent key values retain their original order. Sorting the list

according to the middle digit results in (633, 233, 36, 150, 53, 89), and sorting according to

the rightmost digit gives the sorted array (36, 53, 89, 150, 233, 633). It is worth making

a few comments about the time complexity of the radix sort algorithm. In practice, each

individual sorting pass is accomplished using a simple counting sort algorithm. The counting

sort algorithm uses n + k steps, where n is the number of objects being sorted according to a

given key and k is the number of values that the key can take on. Since the radix sort runs

the counting sort algorithm m times, where m is the number of keys, the time complexity

of radix sort is O(m(n + k)), assuming that each key has the same range of values (in the

case of sorting numbers according to their digits, each key takes on values from 0–9). In

170

particular, the O(m(n + k)) complexity of radix sort can be substantially less than the

O(n log n) complexity of more common state-of-the-art sorting algorithms if m, k ≪ n.

Let us now switch to the context that interests us. In our CC(P ) algorithm, we will ulti-

mately want to perform a radix sort on the list of triply excited determinants entering the P

space. In what follows, we define Φ3 as a d × 6 matrix containing the list of triply excited de-

terminants in the P space, in which each row of Φ3 corresponds to a particular triply excited
ijk⟩ ∈ H (P ) is represented as a tuple of the

determinant. Each triply excited determinant |Φabc

particle indices followed by the hole indices, (a, b, c, i, j, k), where a < b < c and i < j < k.

For example, a system with 4 electrons (i, j, k = 1, 2, 3, or 4) and 4 spatial orbitals (a, b, c =

5, 6, 7, or 8) would have all triply excited determinants arranged in Φ3 (in no particular order)

as

follows


5
5


5


5

5


5


5


5

6


6


6


6

5


5


5
5

6
6
6
6
7
7
7
7
7
7
7
7
6
6
6
6

7
7
7
7
8
8
8
8
8
8
8
8
8
8
8
8

1
1
2
1
1
1
2
1
1
1
2
1
1
1
2
1

2
3
3
2
2
3
3
2
2
3
3
2
2
3
3
2


3
4


4


4

3


4


4


4

3


4


4


4

3


4


4
4

With the above concrete representation of Φ3, we now introduce the critical notion of

uvxw-major sorting of Φ3. In particular, the array Φ3 is said to be sorted in uvxw-major

order if the rows in Φ3 characterized by the same values in columns u, v, x, and w appear

consecutively one after another. In order to bring Φ3 into uvxw-major order, we can perform

a radix sort on Φ3, where the keys are now the columns u, v, x, and w. Given the above form

of Φ3 for a 4-electron–4-orbital system, let us bring it to a 1235-major order using the radix

sort. Unlike in the previous case of sorting nonnegative integers digit-by-digit, the order in

which we sort the columns 1, 2, 3, and 5 does not matter. Therefore, we will proceed from

left to right, starting by sorting column 1 and finishing with sorting column 5.

171


5
5


5


5

5


5


5


5

6


6


6


6

5


5


5
5

6
6
6
6
7
7
7
7
7
7
7
7
6
6
6
6

7
7
7
7
8
8
8
8
8
8
8
8
8
8
8
8

1
1
2
1
1
1
2
1
1
1
2
1
1
1
2
1

2
3
3
2
2
3
3
2
2
3
3
2
2
3
3
2


3
4


4


4

3


4


4


4

3


4


4


4

3


4


4
4

→


5
5


5


5

5


5


5


5

5


5


5


5

6


6


6
6

6
6
6
6
7
7
7
7
6
6
6
6
7
7
7
7

7
7
7
7
8
8
8
8
8
8
8
8
8
8
8
8

1
1
2
1
1
1
2
1
1
1
2
1
1
1
2
1

2
3
3
2
2
3
3
2
2
3
3
2
2
3
3
2


3
4


4


4

3


4


4


4

3


4


4


4

3


4


4
4

→


5
5


5


5

5


5


5


5

5


5


5


5

6


6


6
6

6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7

7
7
7
7
8
8
8
8
8
8
8
8
8
8
8
8

1
1
2
1
1
1
2
1
1
1
2
1
1
1
2
1

2
3
3
2
2
3
3
2
2
3
3
2
2
3
3
2


3
4


4


4

3


4


4


4

3


4


4


4

3


4


4
4

→


5
5


5


5

5


5


5


5

5


5


5


5

6


6


6
6

6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7

7
7
7
7
8
8
8
8
8
8
8
8
8
8
8
8

1
1
2
1
1
1
2
1
1
1
2
1
1
1
2
1

2
3
3
2
2
3
3
2
2
3
3
2
2
3
3
2


3
4


4


4

3


4


4


4

3


4


4


4

3


4


4
4


5
5


5


5

5


5


6


6

5


5


5


5

5


5


6
6

→

6
6
6
6
7
7
7
7
6
6
6
6
7
7
7
7

7
7
8
8
8
8
8
8
7
7
8
8
8
8
8
8

1
1
1
1
1
1
1
1
1
2
1
2
1
2
1
2

2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3


3
4


3


4

3


4


3


4

4


4


4


4

4


4


4
4

In each step, we have permuted the rows in Φ3 such that the indices in the red columns

are sorted. By the end of the radix sort, we obtain the matrix Φ3 in the 1235-major order, in

which all the rows with the same values appearing in columns 1, 2, 3, and 5 are placed next

to one another. We can perform this uvxw-major sorting for any combination of (distinct)

keys defined by the integers u, v, x, and w, where u, v, x, w = 1–6. Based on the previous

analysis of the radix sort, we expect the time complexity of these sorting steps to scale as

O(4(d+nu)), where nu is the number of unoccupied orbitals in the problem, which represents

the largest range of values that can be taken on by any key in the radix sort. Given that

the number of triply excited determinants in the P space, d, is much greater than nu, the

radix sort roughly has complexity O(4d), which is typically less than O(d log d) resulting

from conventional sorting algorithms.

In the above example, we also notice that as a result of sorting Φ3 in the 1235-major

order, Φ3 can be partitioned into blocks, where each block is associated with a specific

combination of the values appearing columns 1, 2, 3, and 5. For example, the first two rows

of Φ3 above form a block characterized by the values 5, 6, 7, and 2 in columns 1, 2, 3, and

5, respectively. Similarly, rows 3 and 4 form another block with the values 5, 6, 8 and 2 in

columns 1, 2, 3, and 5, respectively. In this way, we can find 8 distinct blocks in the above

Φ3, which are associated with the respective tuples of values (5,6,7,2), (5,6,8,2), (5,7,8,2),

(6,7,8,2), (5,6,7,3), (5,6,8,3), (5,7,8,3), and (6,7,8,3) in columns 1, 2, 3, and 5. Note that the

number of rows within each block are associated with the different combinations of indices

172

that appear in the unsorted columns, which are columns 4 and 6 in this case. Given this

blocked structure of Φ3, it is useful to number each block according to a lexical index denoted

as y. In the above Φ3 for example, the block characterized by (5,6,7,2) can be assigned y = 1,

and the next block, associated with (5,6,8,2), can be y = 2, and so on. In the end we will

map each block to a particular lexical index y = 1–8. In a general uvxw-sorting, the lexical

indices y range between 1 and the total number of uvxw-blocks that can exist within the list

Φ3, which we denote as nuvxw.

It follows from this discussion that if each block in Φ3 is associated with a particular

lexical index y, and if the determinants in a given block share the values in columns u,

v, x, and w, which we arbitrarily denote as p, q, r, and s, respectively, then there must

exist a one-to-one mapping between the tuple (p, q, r, s) and the lexical index y.

In our

considerations, it will be helpful to know this mapping in order to move back and forth

between tuples (p, q, r, s) and corresponding indices y. We will denote this mapping as

y = fuvxw(p, q, r, s). In particular, the function fuvxw(p, q, r, s) returns the lexical index y

associated with the block in Φ3 containing rows in which the column u has value p, column v

has value q, column x has value r, and column w as value s. For a general list of triply excited

determinants Φ3, determining a computable formula for the mapping function fuvxw(p, q, r, s)

is not straightforward, and may even be impossible depending on the nature of Φ3 (this

amounts to finding a perfect hashing function). However, we can pre-assign the lexical

indices corresponding to all possible tuples (p, q, r, s) that can appear in list Φ3 and store

them in fuvxw(p, q, r, s), effectively treating fuvxw(p, q, r, s) as a lookup table that returns the

lexical index y for a given combination of p, q, r, and s values. Thus, in our implementation,

the function fuvxw(p, q, r, s) is represented as a four-dimensional array, where each dimension

of the array is associated with a particular argument. In this way, we can trivially retrieve

the lexical index of the block in Φ3 sorted in uvxw-major order associated with a given p,

q, r, and s. Adopting the previous example of 1235-major sorting, the array f1235 is an

nu × nu × nu × no array. Based on our assignment of the lexical indices y = 1–8, we would

173

have f1235(5, 6, 7, 2) = 1, f1235(5, 6, 8, 2) = 2, f1235(5, 7, 8, 2) = 3, and so on.

The last piece of information we need to efficiently access the sorted determinants in

Φ3 is a final auxiliary array, which we denote as Auvxw. The array Auvxw has length equal

to the number of lexical indices associated with uvxw-major sorting, nuvxw. The entry at

Auvxw(y) gives the row index in Φ3 that starts the sorted block associated with lexical index

y. The ending row of a given block characterized by y is de facto Auvxw(y + 1) − 1, where we

enforce that Auvxw(nuvxw) = d + 1. Again, in our previous example of 1235-major sorting,

A1235(1) = 1, A1235(2) = 3, A1235(3) = 5, . . ., A1235(7) = 15, and A1235(8) = 17. Note that

the arrays fuvxw(p, q, r, s) and Auvxw are meant to be used in tandem. Thus, for a given

(p, q, r, s), one can use fuvxw(p, q, r, s) to retrieve the lexical index y and then identify the

rows in Φ3 lying between Auvxw(y) and Auvxw(y + 1) − 1 (inclusively) to obtain those triply

excited determinants that possess indices p, q, r, and s in column positions u, v, x, and w.

C.3.2.2 Contributions Due to One- and Two-Body Components of H (2)
N

Armed with an understanding of how we can use the radix sort to organize the list of

triply excited determinants in any uvxw-major order and retrieve any desired block of triply

excited determinants in it, we can apply it to form and evaluate matrix elements of the

⟨Φabc

ijk|H (2)

N |Φabc

mjk⟩, ⟨Φabc

ijk|H (2)

N |Φebc

ijk⟩, ⟨Φabc

ijk|H (2)

N |Φef c

ijk ⟩, ⟨Φabc

ijk|H (2)

N |Φabc

mnk⟩, and ⟨Φabc

ijk|H (2)

N |Φebc

mjk⟩

types corresponding to the diagrams in Figure C.1. Let us focus on matrix elements of the

⟨Φabc

ijk|H (2)

N |Φef c

ijk ⟩ type corresponding to the particle-particle ladder diagram in Figure C.1,

which represents the most expensive term in CCSDT. For this matrix element, we must sort

Φ3 in a 3456-major order. However, we should be mindful of the fact that we are working

with permutationally unique lists of triply excited determinants. Therefore, in addition to

3456-major sorting, we should also consider contributions originating from 1456- and 2456-

major sorting configurations as well. Similarly, matrix elements of the ⟨Φabc

ijk|H (2)

N |Φabc

mnk⟩

type will require sorting Φ3 in 1234-, 1235-, and 1236-major order and matrix elements

of the ⟨Φabc

ijk|H (2)

N |Φebc

mjk⟩ category involve sorting Φ3 in nine different ways: 2356-, 2345-,

2346-, 1356-, 1345-, 1346-, 1256-, 1245-, and 1246-major order. The matrix elements of the

174

⟨Φabc

ijk|H (2)

N |Φabc

mjk⟩ and ⟨Φabc

ijk|H (2)

N |Φebc

ijk⟩ types technically involve sorting with respect to five

indices. This can be done, but it requires storing 5-dimensional arrays in the lookup table

f . To avoid creation of prohibitively large auxiliary arrays, we perform the computation of

⟨Φabc

ijk|H (2)

N |Φabc

mjk⟩ and ⟨Φabc

ijk|H (2)

N |Φebc

ijk⟩ while evaluating ⟨Φabc

ijk|H (2)

N |Φabc

mnk⟩ and ⟨Φabc

ijk|H (2)

N |Φef c

ijk ⟩,

respectively (see Algorithms C.2 and C.3). Our implementation of the five diagrams in

Figure C.1 defining the contributions to term (II) in Eq. C.3 corresponding to projections

onto triply excited determinants resulting from one- and two-body components of H (2)

N are

given in Algorithms C.2–C.6. As in the case of the expressions given in Algorithm C.1, the

structure of the expressions match what one would obtain using many-body diagrammatics,

except that additional loops over permutations of the indices attached to internal lines and

their external counterparts emanating from T (P )

3

are added to accommodate the use of a

permutationally unique set of triply excited determinants in Φ3.

C.3.2.3 Contributions Due to Three-Body Components of H (2)
N

Compared to the contributions due to one- and two-body components of H (2)

N in term

(II) of Eq. (C.3) corresponding to projection onto triply excited determinants, the strategy

for handling the three-body components of H (2)

N is, fortunately, much simpler. The only

three-body components of H (2)

N that we must explicitly consider arise from contributions of

the ⟨Φabc

ijk|(HN T2T (P )

3

)C|Φ⟩ type. The best approach to handling these terms results from

first contracting HN with T (P )
ab = ¯hie

defining I ie

ab + ηie

ab and I ′ij

3

¯hef
mntimn
am, where ¯hie

am are the intermediates
entering Eq. (C.4), we can embed the contributions due to three-body components of H (2)
N

ab and I ij

am = I ij

am + ηij

to form ηie

ab = − 1
2

abf and ηij

am = 1
2

¯hef
mntijn

aef . Then, after

within the evaluation of Eq. (C.4) by replacing ¯hie

ab and I ij

am by I ie

ab and I ′ij

am, respectively. All

that remains is finding an efficient strategy for computing ηie

ab and ηij

am given the spotty subset

of triply excited determinants defining T (P )

3

. Fortunately, these terms can be evaluated using

the same amplitude-driven method adopted in Algorithm C.1. The routines for evaluating

ab and ηij
ηie

am, which help us embed the contributions of three-body components of H (2)

N into

the computation of Eq. (C.4), are shown in Algorithm C.7.

175

Algorithm C.2 Particle–particle ladder diagram contributions to term (II) in Eq. (C.3)
ijk⟩ ∈ H (P ). The resulting projections are stored in the
corresponding to projections onto |Φabc
d × 1 array r3, where d is the number of triply excited determinants in the P space.
1: for I = 1, . . . , d do
2:
3:
4:
5:
6:
7:
8:
9:

// Sorting order 1456 [permutations correspond to (a/bc)]
for P ∈ {(1), (ab), (ac)} do

a = Φ3(I, 1) b = Φ3(I, 2) c = Φ3(I, 3)
i = Φ3(I, 4) j = Φ3(I, 5) k = Φ3(I, 6)

▷ Loop over projections ⟨Φabc
ijk|

a′ = Pa
y = f1456(a′, i, j, k)
for J = A1456(y), A1456(y + 1) − 1 do

e = Φ3(J, 2)
f = Φ3(J, 3)
r3(I) ← r3(I) + (−1)PP¯hef
bc × t3(J)
r3(I) ← r3(I) + (−1)PAbcAef [P (¯hf

c δbe) × t3(J)]

end for

end for

▷ Loop over kets |Φa′ef
ijk ⟩

▷ Evaluate ⟨Φabc

ijk|H (2)|Φa′ef
ijk ⟩

// Sorting order 2456 [permutations correspond to (b/ac)]
for P ∈ {(1), (ab), (bc)} do

b′ = Pb
y = f2456(b′, i, j, k)
for J = A2456(y), A2456(y + 1) − 1 do

e = Φ3(J, 1)
f = Φ3(J, 3)
r3(I) ← r3(I) + (−1)PP¯hef
ac × t3(J)
r3(I) ← r3(I) + (−1)PAacAef [P (¯hf

c δae) × t3(J)]

end for

end for

▷ Loop over kets |Φeb′f
ijk ⟩

▷ Evaluate ⟨Φabc

ijk|H (2)|Φeb′f
ijk ⟩

// Sorting order 3456 [permutations correspond to (c/ab)]
for P ∈ {(1), (ac), (bc)} do

c′ = Pc
y = f3456(c′, i, j, k)
for J = A3456(y), A3456(y + 1) − 1 do

e = Φ3(J, 1)
f = Φ3(J, 2)
r3(I) ← r3(I) + (−1)PP¯hef
ab × t3(J)
r3(I) ← r3(I) + (−1)PAabAef [P (¯hf

b δae) × t3(J)]

▷ Loop over kets |Φef c′
ijk ⟩

▷ Evaluate ⟨Φabc

ijk|H (2)|Φef c′
ijk ⟩

10:
11:
12:

13:

14:
15:
16:
17:
18:
19:
20:
21:

22:
23:
24:

25:

26:
27:
28:
29:
30:
31:
32:
33:

34:
35:
36:

37:

38:
39:
40:
41: end for

end for

end for

176

▷ Loop over projections ⟨Φabc

a = Φ3(I, 1) b = Φ3(I, 2) c = Φ3(I, 3)
i = Φ3(I, 4) j = Φ3(I, 5) k = Φ3(I, 6)

Algorithm C.3 Hole–hole ladder diagram contributions to term (II) in Eq. (C.3) corre-
ijk⟩ ∈ H (P ). The resulting projections are stored in the
sponding to projections onto |Φabc
d × 1 array r3, where d is the number of triply excited determinants in the P space.
1: for I = 1, . . . , d do
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:

// Sorting order 1234 [permutations correspond to (i/jk)]
for P ∈ {(1), (ij), (ik)} do

i′ = Pi
y = f1234(a, b, c, i′)
for J = A1234(y), A1234(y + 1) − 1 do

ijk| ∈ H (P )

▷ Loop |Φabc

i′mn⟩

m = Φ3(J, 5)
n = Φ3(J, 6)
r3(I) ← r3(I) + (−1)PP¯hjk
r3(I) ← r3(I) − (−1)PA jkA mn[P (¯hj

mn × t3(J)

▷ Evaluate ⟨Φabc

ijk|H (2)|Φabc

i′mn⟩

mδnk) × t3(J)]

end for

end for

// Sorting order 1235 [permutations correspond to (j/ik)]
for P ∈ {(1), (ij), (jk)} do

j′ = Pj
y = f1235(a, b, c, j′)
for J = A1235(y), A1235(y + 1) − 1 do

▷ Loop |Φabc

mj′n⟩

m = Φ3(J, 4)
n = Φ3(J, 6)
r3(I) ← r3(I) + (−1)PP¯hik
mn × t3(J)
r3(I) ← r3(I) − (−1)PA ikA mn[P (¯hi

▷ Evaluate ⟨Φabc

ijk|H (2)|Φabc

mj′n⟩

mδnk) × t3(J)]

end for

end for

// Sorting order 1236 [permutations correspond to (k/ij)]
for P ∈ {(1), (ik), (jk)} do

k′ = Pk
y = f1236(a, b, c, k′)
for J = A1236(y), A1236(y + 1) − 1 do

▷ Loop |Φabc

mnk′⟩

m = Φ3(J, 4)
n = Φ3(J, 5)
r3(I) ← r3(I) + (−1)PP¯hij
mn × t3(J)
r3(I) ← r3(I) − (−1)PA ijA mn[P (¯hi

▷ Evaluate ⟨Φabc

ijk|H (2)|Φabc

mnk′⟩

mδnj) × t3(J)]

13:

14:
15:
16:
17:
18:
19:
20:
21:
22:
23:
24:

25:

26:
27:
28:
29:
30:
31:
32:
33:
34:
35:
36:

37:

38:
39:
40:
41: end for

end for

end for

177

Algorithm C.4 Particle–hole ring diagram contributions to term (II) in Eq. (C.3) corre-
ijk⟩ ∈ H (P ). The resulting projections are stored in the d × 1
sponding to projections onto |Φabc
array r3, where d is the number of triply excited determinants in the P space. [Part 1 of 3]
ijk| ∈ H (P )

▷ Loop over projections ⟨Φabc

1: for I = 1, . . . , d do
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:

end for

end for

a = Φ3(I, 1) b = Φ3(I, 2) c = Φ3(I, 3)
i = Φ3(I, 4) j = Φ3(I, 5) k = Φ3(I, 6)

// Sorting order 2356 [permutations correspond to (a/bc)(i/jk)]
for P ∈ {(1), (ij), (ik), (ab), (ac), (ij)(ab), (ij)(ac), (ik)(ab), (ik)(ac)} do

j′ = Pj k′ = Pk
b′ = Pb c′ = Pc
y = f2356(b′, c′, j′, k′)
for J = A2356(y), A2356(y + 1) − 1 do

e = Φ3(J, 1)
m = Φ3(J, 4)
r3(I) ← r3(I) + (−1)PP¯hie

am × t3(J)

▷ Loop over kets |Φeb′c′

mj′k′⟩ ∈ H (P )

▷ Evaluate ⟨Φabc

ijk|H (2)|Φeb′c′

mj′k′⟩

// Sorting order 2346 [permutations correspond to (a/bc)(j/ik)]
for P ∈ {(1), (ij), (jk), (ab), (ac), (ij)(ab), (ij)(ac), (jk)(ab), (jk)(ac)} do

i′ = Pi k′ = Pk
b′ = Pb c′ = Pc
y = f2346(b′, c′, i′, k′)
for J = A2346(y), A2346(y + 1) − 1 do

e = Φ3(J, 1)
m = Φ3(J, 5)
r3(I) ← r3(I) + (−1)PP¯hje

am × t3(J)

end for

end for

▷ Loop over kets |Φeb′c′

i′mk′⟩ ∈ H (P )

▷ Evaluate ⟨Φabc

ijk|H (2)|Φeb′c′

i′mk′⟩

// Sorting order 2345 [permutations correspond to (a/bc)(k/ij)]
for P ∈ {(1), (ik), (jk), (ab), (ac), (ik)(ab), (ik)(ac), (jk)(ab), (jk)(ac)} do

i′ = Pi j′ = Pj
b′ = Pb c′ = Pc
y = f2345(b′, c′, i′, j′)
for J = A2345(y), A2345(y + 1) − 1 do

e = Φ3(J, 1)
m = Φ3(J, 6)
r3(I) ← r3(I) + (−1)PP¯hke

am × t3(J)

▷ Loop over kets |Φeb′c′

i′j′m⟩ ∈ H (P )

▷ Evaluate ⟨Φabc

ijk|H (2)|Φeb′c′
i′j′m⟩

14:
15:
16:
17:
18:
19:
20:
21:
22:
23:
24:
25:

26:
27:
28:
29:
30:
31:
32:
33:
34:
35:
36:
37:

38:
39:
40:
41: end for

end for

end for

178

Algorithm C.5 Same as Algorithm C.4. [Part 2 of 3]

▷ Loop over projections ⟨Φabc

ijk| ∈ H (P )

1: for I = 1, . . . , d do
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:

end for

end for

a = Φ3(I, 1) b = Φ3(I, 2) c = Φ3(I, 3)
i = Φ3(I, 4) j = Φ3(I, 5) k = Φ3(I, 6)

// Sorting order 1356 [permutations correspond to (b/ac)(i/jk)]
for P ∈ {(1), (ij), (ik), (ab), (bc), (ij)(ab), (ij)(bc), (ik)(ab), (ik)(bc)} do

j′ = Pj k′ = Pk
a′ = Pa c′ = Pc
y = f1356(a′, c′, j′, k′)
for J = A1356(y), A1356(y + 1) − 1 do

e = Φ3(J, 2)
m = Φ3(J, 4)
r3(I) ← r3(I) + (−1)PP¯hie

bm × t3(J)

▷ Loop over kets |Φa′ec′

mj′k′⟩ ∈ H (P )

▷ Evaluate ⟨Φabc

ijk|H (2)|Φa′ec′

mj′k′⟩

// Sorting order 1346 [permutations correspond to (b/ac)(j/ik)]
for P ∈ {(1), (ij), (jk), (ab), (bc), (ij)(ab), (ij)(bc), (jk)(ab), (jk)(bc)} do

i′ = Pi k′ = Pk
a′ = Pa c′ = Pc
y = f1346(a′, c′, i′, k′)
for J = A1346(y), A1346(y + 1) − 1 do

e = Φ3(J, 2)
m = Φ3(J, 5)
r3(I) ← r3(I) + (−1)PP¯hje

bm × t3(J)

end for

end for

▷ Loop over kets |Φa′mc′

i′mk′ ⟩ ∈ H (P )

▷ Evaluate ⟨Φabc

ijk|H (2)|Φa′ec′
i′mk′⟩

// Sorting order 1345 [permutations correspond to (b/ac)(k/ij)]
for P ∈ {(1), (ik), (jk), (ab), (ac), (ik)(ab), (ik)(ac), (jk)(ab), (jk)(ac)} do

i′ = Pi j′ = Pj
a′ = Pa c′ = Pc
y = f1345(a′, c′, i′, j′)
for J = A1345(y), A1345(y + 1) − 1 do

e = Φ3(J, 2)
m = Φ3(J, 6)
r3(I) ← r3(I) + (−1)PP¯hke

bm × t3(J)

▷ Loop over kets |Φa′ec′

i′j′m⟩ ∈ H (P )

▷ Evaluate ⟨Φabc

ijk|H (2)|Φa′ec′
i′j′m⟩

14:
15:
16:
17:
18:
19:
20:
21:
22:
23:
24:
25:

26:
27:
28:
29:
30:
31:
32:
33:
34:
35:
36:
37:

38:
39:
40:
41: end for

end for

end for

179

Algorithm C.6 Same as Algorithm C.4. [Part 3 of 3]

▷ Loop over projections ⟨Φabc

ijk| ∈ H (P )

1: for I = 1, . . . , d do
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:

end for

end for

a = Φ3(I, 1) b = Φ3(I, 2) c = Φ3(I, 3)
i = Φ3(I, 4) j = Φ3(I, 5) k = Φ3(I, 6)

// Sorting order 1256 [permutations correspond to (c/ab)(i/jk)]
for P ∈ {(1), (ij), (ik), (ac), (bc), (ij)(ac), (ij)(bc), (ik)(ac), (ik)(bc)} do

j′ = Pj k′ = Pk
a′ = Pa b′ = Pb
y = f1256(a′, b′, j′, k′)
for J = A1256(y), A1256(y + 1) − 1 do

e = Φ3(J, 3)
m = Φ3(J, 4)
r3(I) ← r3(I) + (−1)PP¯hie

cm × t3(J)

▷ Loop over kets |Φa′b′e

mj′k′⟩ ∈ H (P )

▷ Evaluate ⟨Φabc

ijk|H (2)|Φa′b′e

mj′k′⟩

// Sorting order 1246 [permutations correspond to (c/ab)(j/ik)]
for P ∈ {(1), (ij), (jk), (ac), (bc), (ij)(ac), (ij)(bc), (jk)(ac), (jk)(bc)} do

i′ = Pi k′ = Pk
a′ = Pa b′ = Pb
y = f1246(a′, b′, i′, k′)
for J = A1246(y), A1246(y + 1) − 1 do

e = Φ3(J, 3)
m = Φ3(J, 5)
r3(I) ← r3(I) + (−1)PP¯hje

cm × t3(J)

end for

end for

▷ Loop over kets |Φa′b′e

i′mk′⟩ ∈ H (P )

▷ Evaluate ⟨Φabc

ijk|H (2)|Φa′b′e
i′mk′⟩

// Sorting order 1245 [permutations correspond to (c/ab)(k/ij)]
for P ∈ {(1), (ik), (jk), (ac), (bc), (ik)(ac), (ik)(bc), (jk)(ac), (jk)(bc)} do

i′ = Pi j′ = Pj
a′ = Pa b′ = Pb
y = f1245(a′, b′, i′, j′)
for J = A1245(y), A1245(y + 1) − 1 do

e = Φ3(J, 3)
m = Φ3(J, 6)
r3(I) ← r3(I) + (−1)PP¯hke

cm × t3(J)

▷ Loop over kets |Φa′b′e

i′j′m⟩ ∈ H (P )

▷ Evaluate ⟨Φabc

ijk|H (2)|Φa′b′e
i′j′m⟩

14:
15:
16:
17:
18:
19:
20:
21:
22:
23:
24:
25:

26:
27:
28:
29:
30:
31:
32:
33:
34:
35:
36:
37:

38:
39:
40:
41: end for

end for

end for

180

ab and ηij

am intermediates resulting from factorization of

▷ Loop over triply excited determinants in the P space

▷ The three-body cluster amplitude associated with |Φabc
ijk⟩

// Permutations corresponding to (j/ik)(c/ab)
for P ∈ {(1), (ij), (jk), (ac), (ab), (ij)(ac), (ij)(ab), (jk)(ac), (jk)(ab)} do

am + (−1)PP¯hab

mj × t3

▷ Evaluate ηij

am = 1
2

¯hef
mntijn

aef

Get t3 ← tijk
abc

ijk⟩ ∈ H (P ) do

Algorithm C.7 Evaluation of ηie
three-body components of H (2)
N
1: for |Φabc
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18: end for

for m = 1, no do
am ← Pηik

for e = 1, nu do
ca ← Pηke

Pηke
end for

Pηik
end for

end for

end for

// Permutations corresponding to (k/ij)(b/ac)
for P ∈ {(1), (ik), (jk), (ab), (bc), (ik)(ab), (ik)(bc), (jk)(ab), (jk)(bc)} do

ca − (−1)PP¯heb

ij × t3

▷ Evaluate ηie

ab = − 1
2

¯hef
mntimn
abf

181

C.4 Computational Efficiency and Benefits of the Novel CC(P )/EOMCC(P )

Algorithms

Finally, we make a few remarks about the computational costs and additional benefits

characterizing our novel CC(P )/EOMCC(P ) algorithm driven by radix sorting of the list

of triply excited determinants. As stated earlier, the cost of the radix sorting operations

scale linearly with d, with a small prefactor. For the most expensive particle-particle lad-

der diagram in CCSDT, the CC(P ) algorithm based on computing matrix elements of the

⟨Φabc

ijk|H (2)

N |Φef c

ijk ⟩ type will involve D × n2

u, or n3

on5

u, floating point operations in the limit that

all triply excited determinants are included in Φ3, i.e., (d/D) → 1. In fact, the n3

on5

u opera-

tion count of the CC(P ) method using the complete manifold of triply excited determinants

in the P space exactly matches the costs of CCSDT. The same is true for the EOMCC(P )

calculations, which aim at converging EOMCCSDT. In practice, the CC(P )/EOMCC(P )

computations are carried out with (d/D) ≪ 1 (e.g., using a few percent of triply excited

determinants or less), and in this regime, the cost of the CC(P )/EOMCC(P ) calculations

is roughly d ×

(cid:17)2

(cid:16) d
D nu

. Therefore, we can expect reductions in CPU effort compared to the

canonical n3

on5

u cost of CCSDT or EOMCCSDT by a factor on the order of (d/D)2.

There are also a number of additional benefits offered by this general approach to solving

the CC(P )/EOMCC(P ) equations, which we will briefly comment on as well. First of all, the

resulting CC(P )/EOMCC(P ) routines are optimal in both CPU operation count and stor-

age, and they can work with any list of triply excited determinants. This not only includes

the spotty subsets of triply excited determinants obtained using semi-stochastic, CIPSI-

based, or adaptive CC(P ;Q) methodologies, but also the more structured lists adopted in

active-orbital-based CC(t;3) calculations. Alternative choices of lists can also be used to

perform different types of CC(P ;Q) calculations not considered in this dissertation, such

as core-valence-separated EOMCC schemes, which are useful for studying core excitation

and ionization spectra. Another advantage offered by our CC(P )/EOMCC(P ) routine is

that it works directly with permutationally unique lists of triply excited determinants. This

182

helps reduce the costs associated with storing the tijk

abc and r ijk

µ,abc arrays by only retaining

the unique elements with i < j < k and a < b < c (the use of such triangular arrays

would not be compatible with the optimized BLAS routines adopted in most conventional

CC/EOMCC codes). Finally, by including only those triply excited determinants of a partic-

ular irrep, the CC(P )/EOMCC(P ) routine can trivially take advantage of speedups offered

by molecular point group symmetry, which is usually more difficult to implement in high-level

CC/EOMCC codes.

To illustrate these benefits and highlight the efficiency of our novel CC(P ) algorithm,

we compare the single-core CPU timings characterizing the active-orbital-based CC(t;3)

calculations for the stretched F2 molecule in the aug-cc-pVQZ [10, 358] basis set obtained

using CCpy and GAMESS in Table C.3. Our calculations used an active space consisting of

the five highest-energy occupied orbitals, corresponding to the πu, πg, and σg shells, and the

lowest-energy σu orbital unoccupied in the RHF reference. To demonstrate the capability of

our CC(P ) code to take advantage of speedups offered by molecular point-group symmetry,

we performed CC(t;3) calculations using CCpy based on two different lists of triply excited

determinants. The first list included all triply excited determinants selected with the help of

active orbitals, consistent with the truncation of T3 adopted in CCSDt, while the second list

retained the subset of triply excited determinants present in the first list belonging to the

Sz = 0 Ag(D2h) symmetry. The CCSDt and CC(t;3) calculations performed with GAMESS

relies on the highly efficient and automatically generated vectorized codes, which cannot take

advantage of point-group symmetry at this time.

Upon comparing the timings reported in Table C.3, it is clear that CCpy is competitive

with GAMESS in performing the CC(t;3) calculations. Indeed, the CPU timing obtained

with GAMESS is within 5% of its counterpart resulting from the CCpy calculations that

do not advantage of the D2h Abelian symmetry of F2. More importantly, the timings per

iteration of the CCSDt calculations executed using CCpy, without using symmetry, and

GAMESS differ by just 1.4 minutes. This is impressive, especially when we consider that

183

the CCSDt code in CCpy adopts a devectorized strategy for constructing the amplitude

equations corresponding to projections onto triply excited determinants in the P space. In

contrast, the CCSDt code in GAMESS is fully vectorized, making use of vendor-optimized

BLAS routines for matrix transposition and multiplication that achieve peak performance

on modern hardware. Typically, vectorization is always more efficient, and indeed, this is

reflected in the slightly better performance of the CCSDt routines present in GAMESS,

but the fact that the CC(P ) algorithm in CCpy can remain very competitive speaks to its

efficiency. In most circumstances, a vectorized implementation will be ∼10x faster than a

devectorized one, but our CC(P ) code is able to close this gap with a combination of efficient

looping and removal of the CPU operations associated with permutationally redundant three-

body amplitudes, which must be considered in the BLAS operations.

The flexibility of the CC(P ) approach also gives us the ability to speed up our calculations

by filtering the list of triply excited determinants to only retain those of the ground-state

symmetry. The timings for our CC(t;3) calculations that exploit the D2h symmetry group of

F2 reported in Table C.3 demonstrate that our CC(P ) code can be even more efficient than

the vectorized implementation in GAMESS when taking advantage of point-group symmetry

to shorten the list of triply excited determinants. Indeed, with the help of spatial symmetry,

the total CPU timings of the calculations performed with CCpy and GAMESS, which are

177.4 and 185.7 minutes, respectively, are reduced to 67.4 minutes. One may wonder why

the speedup characterizing the calculation exploiting symmetry is only approximately a

factor of three when the number of Sz = 0 triply excited determinants entering the CCSDt

calculations is reduced by a factor of eight for the Ag(D2h)-symmetric ground state of F2

considered here. This sub-optimal speedup is, in part, because a significant fraction of the

CPU time corresponding to each iteration of the CCSDt calculation is spent constructing the

H (2)

N intermediates. Fortunately, the computation of H (2)

N is completely vectorized. Future

work may consider accelerating this step using high-performance parallel architectures, like

GPUs.

184

Table C.3 Comparison of the computational timings characterizing the CC(t;3) calculations
for the stretched F2 molecule, as described using the aug-cc-pVQZ basis set, obtained using
CCpy and GAMESS. The F–F bond length in F2 was set to 2Re, where Re = 2.66816
bohr defines the equilibrium geometry, and the core orbitals correlating with the 1s shells
of fluorine were frozen in all post-RHF steps. The active space defining the CC(t;3) and
underlying CCSDt calculations consisted of the highest-energy πu, πg, and σg shells occupied
in the RHF reference and the lowest-energy σu orbital unoccupied in the reference state.

CPU Timea
Iterative Stepsb Noniterative Stepsc Total
177.4

1.6

Software

CCpyd
CCpye
GAMESSd

175.8 (7.4f)
65.9 (2.6f)
183.7 (6.0f)

1.5

2.0

67.4

185.7

a All reported timings, in CPU minutes, correspond to single-core runs on the Precision 7920 workstation from
Dell equipped with Intel Xeon Silver 4114 2.2 GHz processor boards. The computational times associated
with the execution of the integral, RHF, and integral transformation and sorting routines preceding the CC
steps are ignored.
b In executing the iterative steps of the CC(t;3) calculations, a convergence threshold of 10−7 hartree was
assumed. The timings corresponding to the iterative steps include the times required to construct and
solve the CCSDt amplitude equations and the companion left-CCSD-like eigenstate involving the respective
similarity-transformed Hamiltonians using the two-body approximation.
c In the language of Q spaces adopted by the CC(P;Q) formalism, the computational times required to deter-
mine the noniterative triples corrections of CC(t;3) correspond to the remaining triply excited determinants
missing from the P space characterizing the CCSDt calculations.
d No advantage of the D∞h symmetry of F2 or its D2h Abelian subgroup was taken in the post-RHF steps.
e The P space defining the CCSDt and CC(t;3) calculations was filtered to retain the Sz = 0 triply excited
determinants of the Ag(D2h) symmetry.
f The average time, in CPU minutes, corresponding to a single iteration of the CCSDt calculation.

C.5 Prospects for Future Improvements

To conclude this appendix, we offer a few additional remarks about our ideas for im-

proving the current CC(P )/EOMCC(P ) routines. By and large, the CC(P )/EOMCC(P )

algorithm that we have presented thus far is highly efficient. There are no unnecessary FLOPs

used when constructing the CC/EOMCC equations and, as just discussed, the performance

of our CC(P ;Q) implementation in CCpy is competitive with the vectorized routines avail-

able in GAMESS. Key technical improvements to our current implementation will come

from fine-tuning the operations executed within the loops over higher–than–doubly excited

determinants, or perhaps reorganization of the looping structure.

First of all, the mapping given by fuvxw(p, q, r, s) should be represented using a hash

185

table or dictionary, rather than a full multidimensional array. The latter is not only unnec-

essary, since only a subset of (p, q, r, s) tuples enter the problem, but also consumes excessive

amounts of memory. A large fuvxw(p, q, r, s) object may even lead to deterioration in perfor-

mance for larger problems when cache-misses become more frequent while performing lookups

for lexical indices y. Indeed, the most natural and efficient data structure for fuvxw(p, q, r, s)

is a hash table that maps tuples of integers (keys) to lexical indices y (values). In order to

create this hash table, a given tuple (p, q, r, s) must be encoded to a key using a hash function.

Although off-the-shelf hashing functions exist, one could avoid hash conflicts altogether by

encoding each (p, q, r, s) into the linear index I = (p−1)n2n3n4+(q−1)n3n4+(r−1)n4+s+1,

where ni, i = 1–4, refer to the number of possible values (either nu or no) that each index

can take on. Modified linear indices can be defined to address triangular subarrays cor-

responding to conditions resulting from permutational symmetry, like p < q, p < q < r,

p < q, r < s, and p < q < r < s. Thus, fuvxw(p, q, r, s) could be represented as a dictionary

that uses linear indices as keys and returns lexical indices y as values. Even better, the

values in fuvxw(p, q, r, s) may directly correspond to the starting and ending positions of a

given uvxw-block in Φ3, removing the need for the addressing array Auvxw. Introducing a

dictionary would not only reduce memory requirements and potentially speed up the code,

but it would also allow us to extend our sorting schemes to sort more than four indices

without introducing large arrays. This dictionary representation would be invaluable when

implementing CC(P )/EOMCC(P ) methods aimed at converging CCSDTQ/EOMCCSDTQ,

where sorting according to six indices is necessary.

Another potentially interesting idea for future optimization involves restructuring the

loops in Algorithms C.2–C.6. Currently, we perform an outer loop over higher–than–doubly

excited determinants in the P space, and for each one, we locate the connected determinants

and loop over them, performing computations for each one. One potential drawback of this

structure is that the lexical index y is obtained using fuvxw(p, q, r, s) at least once per iteration

of the outer loop. Given that this outer loop is generally quite long (entailing potentially

186

billions of iterations), performance may suffer if the the operation y = fuvxw(p, q, r, s) is not

highly efficient. We already discussed that using a dense array for fuvxw(p, q, r, s) should be

replaced with a hash table to reduce expensive cache misses, but even a hash table will lead

to slowdowns when billions of lookups are required. Thus, we may benefit from restructuring

our current algorithm so that instead of iterating over determinants directly, we loop over

the distinct lexical indices (or sorted blocks) in our problem, and for each one, we locate the

connected blocks of determinants. This optimization has the advantage of minimizing the

number of times the map fuvxw(p, q, r, s) is used, which may be beneficial for larger problems.

Finally, with the previous two optimizations taken into account, the CC(P )/EOMCC(P )

code is likely at near-optimal performance on a single core. Thus, it is natural to consider

parallelizing routines, such as those in Algorithms C.2–C.6. To accomplish this, we may

distribute the work over higher–than–doubly excited determinants to separate processes

using message passing interface (MPI) instructions. Within each process, the work can be

subsequently subdivided using OpenMP directives. This would represent a natural parallel

scheme, similar to the one adopted in the CIPSI code in Quantum Package (see Ref. [184]

for further details).

187

APPENDIX D

CC/EOMCC AND FCI ENERGIES FOR THE GROUND- AND
EXCITED-STATE POTENTIAL CUTS OF WATER

This appendix contains the results of the CC and EOMCC calculations for the ground- and

excited-state PES cuts of the water molecule, as described by the TZ basis set of Ref. [327],

corresponding to the H2O → H + OH dissociation, discussed in Section 4.2.3 of Chapter 4 of

this dissertation, along with the associated full CI data taken from Refs. [161, 327]. Table

D.1 compares the results of the CCSDT and EOMCCSDT calculations with full CI. The re-

maining Tables D.2–D.13 compare the energies of the ground and excited states of water con-

sidered in this work obtained with the CCSD/EOMCCSD, CR-CC(2,3)/CR-EOMCC(2,3),

CCSDt/EOMCCSDt, CC(t;3), and adaptive CC(P ;Q) approaches for the selected values of

the O–H bond-breaking coordinate defining the H2O → H + OH dissociation pathway with

the parent CCSDT/EOMCCSDT data.

188

Table D.1 Total ground and excited electronic state energies of water, as described by
the TZ basis set of Ref. [327], along the O–H bond-breaking coordinate, ROH, in bohr.
All energies are reported as E + 75 hartree. The values in parentheses are errors in the
CCSDT/EOMCCSDT energies relative to FCI, in millihartree. The FCI data were taken
from Refs. [327] and [161]. The lowest-energy orbital correlating with the 1s shell of oxygen
was kept frozen in post-RHF calculations. Adapted from Ref. [138].

ROH
1.3
1.6
1.809a
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

ROH
1.3
1.6
1.809a
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

ROH
1.3
1.6
1.809a
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

ROH
1.3
1.6
1.809a
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

X 1A′

1 1A′′

Full CI

CCSDT

Full CI

EOMCCSDT

Full CI

1 3A′
EOMCCSDT

Full CI

EOMCCSDT

Full CI

EOMCCSDT

Full CI

−1.01567 −1.015602 (0.07)
−1.14894 −1.148832 (0.11)
−1.16847 −1.168316 (0.15)
−1.16417 −1.163964 (0.21)
−1.13058 −1.130217 (0.36)
−1.09255 −1.091972 (0.58)
−1.06129 −1.060470 (0.82)
−1.03889 −1.037848 (1.04)
−1.02451 −1.023323 (1.19)
−1.01967 −1.018463 (1.21)
−1.01603 −1.014832 (1.20)
1 3A′′

−0.70187 −0.702504 (−0.63)
−0.85127 −0.851860 (−0.59)
−0.88566 −0.886181 (−0.52)
−0.90098 −0.901368 (−0.39)
−0.92945 −0.929326 (0.12)
−0.95679 −0.956186 (0.60)
−0.97626 −0.975300 (0.96)
−0.98862 −0.987453 (1.17)
−0.99617 −0.994949 (1.22)
−0.99870 −0.997508 (1.19)
−1.00065 −0.999498 (1.15)
1 1A′

−0.72170 −0.722266 (−0.57)
−0.87187 −0.872393 (−0.52)
−0.90726 −0.907714 (−0.45)
−0.92363 −0.923946 (−0.32)
−0.95148 −0.951300 (0.18)
−0.97420 −0.973528 (0.67)
−0.98837 −0.987281 (1.09)
−0.99640 −0.995001 (1.40)
−1.00091 −0.999363 (1.55)
−1.00236 −1.000790 (1.57)
−1.00345 −1.001888 (1.56)
2 3A′′

−0.62821 −0.628899 (−0.69)
−0.77055 −0.771190 (−0.64)
−0.79860 −0.799146 (−0.55)
−0.80575 −0.806139 (−0.39)
−0.81062 −0.810399 (0.22)
−0.81932 −0.818515 (0.81)
−0.82860 −0.827420 (1.18)
−0.83618 −0.834787 (1.39)
−0.84172 −0.840275 (1.44)
−0.84379 −0.842366 (1.42)
−0.84546 −0.844080 (1.38)
2 1A′

−0.65020 −0.650784 (−0.58)
−0.79340 −0.793913 (−0.51)
−0.82511 −0.825513 (−0.40)
−0.84272 −0.842928 (−0.21)
−0.89158 −0.891420 (0.16)
−0.93780 −0.937554 (0.25)
−0.96789 −0.967653 (0.24)
−0.98540 −0.985183 (0.22)
−0.99516 −0.994978 (0.18)
−0.99822 −0.998050 (0.17)
−1.00046 −1.000308 (0.15)

2 3A′
EOMCCSDT

−0.55577 −0.556467 (−0.70)
−0.70873 −0.709251 (−0.53)
−0.74823 −0.748568 (−0.34)
−0.76263 −0.762831 (−0.20)
−0.78565 −0.785421 (0.23)
−0.81625 −0.815525 (0.73)
−0.83261 −0.831462 (1.15)
−0.84103 −0.839577 (1.45)
−0.84553 −0.843920 (1.61)
−0.84693 −0.845298 (1.63)
−0.84797 −0.846350 (1.62)

2 1A′′
EOMCCSDT

Full CI

EOMCCSDT

Full CI

EOMCCSDT

Full CI

−0.62356 −0.624198 (−0.64)
−0.78052 −0.781036 (−0.52)
−0.82109 −0.821469 (−0.38)
−0.83571 −0.835975 (−0.26)
−0.82785 −0.828038 (−0.19)
−0.80175 −0.801866 (−0.12)
−0.77863 −0.778512 (0.12)
−0.76634 −0.765773 (0.57)
−0.76393 −0.763214 (0.72)
−0.76335 −0.762712 (0.64)
−0.76232 −0.761785 (0.54)
3 3A′′

−0.54185 −0.542618 (−0.77)
−0.68730 −0.687943 (−0.64)
−0.71786 −0.718367 (−0.51)
−0.72455 −0.724899 (−0.35)
−0.71631 −0.716154 (0.16)
−0.72567 −0.724539 (1.13)
−0.73203 −0.730092 (1.94)
−0.73289 −0.730473 (2.42)
−0.72999 −0.727678 (2.31)
−0.72738 −0.725296 (2.08)
−0.72415 −0.722336 (1.81)
3 1A′

Full CI

EOMCCSDT

Full CI

EOMCCSDT

−0.36985 −0.370374 (−0.52)
−0.51578 −0.516222 (−0.45)
−0.55368 −0.554100 (−0.42)
−0.58208 −0.582672 (−0.59)
−0.60132 −0.602051 (−0.73)
−0.65695 −0.655011 (1.94)
−0.69947 −0.698187 (1.28)
−0.71715 −0.716719 (0.43)
−0.71837 −0.718451 (−0.08)
−0.71706 −0.717253 (−0.19)
−0.71581 −0.716073 (−0.26)

−0.39097 −0.391520 (−0.55)
−0.57861 −0.578996 (−0.39)
−0.63701 −0.637243 (−0.23)
−0.66800 −0.668055 (−0.05)
−0.69672 −0.696531 (0.19)
−0.67863 −0.678272 (0.36)
−0.66287 −0.661466 (1.40)
−0.66382 −0.660620 (3.20)
−0.66878 −0.665985 (2.80)
−0.66943 −0.665657 (3.77)
−0.66889 −0.664958 (3.93)

−0.61400 −0.614700 (−0.70)
−0.76861 −0.769204 (−0.59)
−0.80694 −0.807409 (−0.47)
−0.81992 −0.820292 (−0.37)
−0.81008 −0.810444 (−0.36)
−0.78177 −0.782217 (−0.45)
−0.75430 −0.754820 (−0.52)
−0.73276 −0.733362 (−0.60)
−0.71846 −0.719183 (−0.72)
−0.71372 −0.714539 (−0.82)
−0.71037 −0.711300 (−0.93)
3 3A′
EOMCCSDT

Full CI

−0.44370 −0.444133 (−0.43)
−0.61947 −0.619787 (−0.32)
−0.67629 −0.676486 (−0.20)
−0.71482 −0.714899 (−0.08)
−0.73737 −0.737345 (0.03)
−0.70581 −0.705526 (0.28)
−0.67660 −0.675863 (0.74)
−0.65564 −0.654127 (1.51)
−0.64297 −0.640046 (2.92)
−0.63921 −0.635295 (3.92)
−0.63687 −0.631806 (5.06)

a The equilibrium value of the O–H bond length obtained in Ref. [327] using the CCSD/cc-pVTZ method.

189

Table D.2 The total electronic energies, reported as errors relative to CCSDT in millihartree,
obtained with the CCSD, CR-CC(2,3), CCSDt, CC(t;3), and adaptive CC(P ;Q) approaches
for the X 1A′ state of the water molecule, as described by the TZ basis set of Ref. [327], along
the O–H bond-breaking coordinate, ROH, in bohr. The lowest-energy orbital correlating with
the 1s shell of oxygen was kept frozen in post-RHF steps. Adapted from Ref. [138].

%T = 1d

%T = 2e

ROH
1.3
1.6
1.809f
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

CCSD CR(2,3)a
−0.226
2.771
−0.269
3.063
−0.298
3.307
−0.325
3.562
−0.398
4.230
−0.500
5.150
−0.613
6.389
−0.724
7.929
−0.828
9.622
−0.875
10.448
−0.918
11.224

CCSDtb
2.068
2.229
2.317
2.361
2.328
2.187
2.023
1.872
1.753
1.709
1.674

CC(t;3)c CC(P ) CC(P ;Q) CC(P ) CC(P ;Q)
−0.133
−0.163
−0.182
−0.124
−0.171
−0.205
−0.144
−0.196
−0.216
−0.175
−0.230
−0.219
−0.258
−0.335
−0.206
−0.284
−0.379
−0.178
−0.275
−0.369
−0.151
−0.231
−0.350
−0.130
−0.195
−0.308
−0.116
−0.183
−0.276
−0.112
−0.162
−0.247
−0.110

1.696
1.713
1.857
2.141
2.689
2.768
2.575
2.338
2.115
2.043
1.997

1.374
1.331
1.390
1.652
2.132
2.087
1.898
1.667
1.469
1.405
1.374

a CR-CC(2,3) calculations.
b CCSDt calculations using the active space consisting of the three highest occupied and two lowest unoc-
cupied RHF orbitals.
c CC(t;3) calculations using the active space consisting of the three highest occupied and two lowest unoc-
cupied RHF orbitals.
d CC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited determinants
and 1% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
e CC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited determinants
and 2% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
f The equilibrium value of the O–H bond length in the ground electronic state of water, as obtained in Ref.
[327] using the CCSD/cc-pVTZ method.

190

Table D.3 The total electronic energies, reported as errors relative to EOMCCSDT in
millihartree, obtained with the EOMCCSD, CR-EOMCC(2,3), EOMCCSDt, CC(t;3), and
adaptive CC(P ;Q) approaches for the 1 1A′′ state of the water molecule, as described by
the TZ basis set of Ref. [327], along the O–H bond-breaking coordinate, ROH, in bohr. The
lowest-energy orbital correlating with the 1s shell of oxygen was kept frozen in post-RHF
steps. Adapted from Ref. [138].

ROH
1.3
1.6
1.809f
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

EOMCCSD CR(2,3)a

−0.081
0.049
0.301
1.015
4.319
7.904
10.917
13.263
14.933
15.537
16.012

0.917
1.125
1.145
1.005
0.505
−0.067
−0.875
−1.810
−2.700
−3.095
−3.448

EOMCCSDtb
1.601
1.661
1.710
1.794
1.972
1.873
1.705
1.594
1.542
1.531
1.527

CC(t;3)c
0.708
0.718
0.702
0.648
0.485
0.443
0.480
0.537
0.595
0.623
0.649

%T = 1d

%T = 2e

EOMCC(P ) CC(P ;Q) EOMCC(P ) CC(P ;Q)

2.404
2.458
2.573
2.943
3.802
3.944
3.973
3.637
3.130
2.889
2.995

0.615
0.612
0.656
0.700
0.620
0.587
0.696
0.869
0.934
1.021
0.999

2.049
1.983
2.124
2.480
3.093
2.802
2.569
2.261
1.935
1.897
1.789

0.549
0.595
0.587
0.660
0.534
0.527
0.574
0.681
0.686
0.725
0.725

a CR-EOMCC(2,3) calculations.
b EOMCCSDt calculations using the active space consisting of the three highest occupied and two lowest
unoccupied RHF orbitals.
c CC(t;3) calculations using the active space consisting of the three highest occupied and two lowest unoc-
cupied RHF orbitals.
d EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited deter-
minants and 1% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
e EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited determi-
nants and 2% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
f The equilibrium value of the O–H bond length in the ground electronic state of water, as obtained in Ref.
[327] using the CCSD/cc-pVTZ method.

191

Table D.4 Same as Table D.3 for the 1 3A′ state.

ROH
1.3
1.6
1.809f
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

EOMCCSD CR(2,3)a

−0.351
−0.188
0.185
1.118
3.596
4.501
4.653
4.488
4.166
3.977
3.783

0.884
1.024
1.055
1.019
1.099
1.163
1.176
1.208
1.263
1.294
1.326

EOMCCSDtb
1.550
1.603
1.646
1.758
2.001
1.915
1.801
1.740
1.716
1.713
1.714

CC(t;3)c
0.693
0.719
0.754
0.776
0.909
1.005
1.043
1.093
1.133
1.153
1.184

%T = 1d

%T = 2e

EOMCC(P ) CC(P ;Q) EOMCC(P ) CC(P ;Q)

2.150
2.191
2.192
2.435
2.461
2.330
2.061
1.801
1.710
1.681
1.660

0.495
0.483
0.474
0.514
0.712
0.873
0.959
1.031
1.046
1.067
1.066

1.827
1.721
1.638
1.865
2.058
1.711
1.469
1.272
1.131
1.104
1.053

0.457
0.458
0.465
0.482
0.610
0.705
0.738
0.727
0.708
0.712
0.708

a CR-EOMCC(2,3) calculations.
b EOMCCSDt calculations using the active space consisting of the three highest occupied and two lowest
unoccupied RHF orbitals.
c CC(t;3) calculations using the active space consisting of the three highest occupied and two lowest unoc-
cupied RHF orbitals.
d EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited deter-
minants and 1% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
e EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited determi-
nants and 2% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
f The equilibrium value of the O–H bond length in the ground electronic state of water, as obtained in Ref.
[327] using the CCSD/cc-pVTZ method.

192

Table D.5 Same as Table D.3 for the 1 3A′′ state.

ROH
1.3
1.6
1.809f
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

EOMCCSD CR(2,3)a

−0.325
−0.205
0.046
0.757
3.847
7.203
10.310
13.094
15.388
16.323
17.120

0.887
1.088
1.110
0.999
0.663
0.326
−0.213
−0.899
−1.594
−1.909
−2.192

EOMCCSDtb
1.545
1.596
1.641
1.732
1.957
1.890
1.721
1.598
1.533
1.518
1.510

CC(t;3)c
0.722
0.730
0.716
0.669
0.532
0.490
0.517
0.566
0.621
0.648
0.687

%T = 1d

%T = 2e

EOMCC(P ) CC(P ;Q) EOMCC(P ) CC(P ;Q)

2.427
2.175
2.537
2.650
3.621
4.063
4.246
3.869
3.407
3.377
3.231

0.679
0.658
0.728
0.777
0.764
0.771
0.874
1.088
0.988
1.035
1.033

2.012
1.893
2.029
2.370
3.151
3.147
2.851
2.535
2.025
1.923
1.791

0.595
0.624
0.636
0.724
0.696
0.666
0.679
0.728
0.653
0.670
0.645

a CR-EOMCC(2,3) calculations.
b EOMCCSDt calculations using the active space consisting of the three highest occupied and two lowest
unoccupied RHF orbitals.
c CC(t;3) calculations using the active space consisting of the three highest occupied and two lowest unoc-
cupied RHF orbitals.
d EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited deter-
minants and 1% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
e EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited determi-
nants and 2% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
f The equilibrium value of the O–H bond length in the ground electronic state of water, as obtained in Ref.
[327] using the CCSD/cc-pVTZ method.

193

Table D.6 Same as Table D.3 for the 1 1A′ state.

ROH
1.3
1.6
1.809f
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

EOMCCSD CR(2,3)a

−0.018
0.298
0.890
1.950
5.922
10.351
14.047
17.027
19.348
20.261
21.024

0.964
1.129
1.189
1.147
0.781
0.088
−0.790
−1.737
−2.649
−3.063
−3.448

EOMCCSDtb
1.466
1.495
1.530
1.599
1.829
1.853
1.718
1.596
1.527
1.509
1.499

CC(t;3)c
0.660
0.695
0.706
0.691
0.545
0.489
0.462
0.501
0.551
0.578
0.600

%T = 1d

%T = 2e

EOMCC(P ) CC(P ;Q) EOMCC(P ) CC(P ;Q)

2.364
2.455
2.676
3.053
4.025
4.809
5.056
4.533
3.678
3.414
3.296

0.472
0.475
0.520
0.547
0.434
0.490
0.632
0.746
0.695
0.734
0.744

1.092
1.918
1.984
2.369
3.117
3.282
3.147
2.656
2.258
2.070
1.915

0.435
0.418
0.478
0.518
0.407
0.467
0.504
0.540
0.575
0.561
0.532

a CR-EOMCC(2,3) calculations.
b EOMCCSDt calculations using the active space consisting of the three highest occupied and two lowest
unoccupied RHF orbitals.
c CC(t;3) calculations using the active space consisting of the three highest occupied and two lowest unoc-
cupied RHF orbitals.
d EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited deter-
minants and 1% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
e EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited determi-
nants and 2% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
f The equilibrium value of the O–H bond length in the ground electronic state of water, as obtained in Ref.
[327] using the CCSD/cc-pVTZ method.

194

Table D.7 Same as Table D.3 for the 2 3A′ state.

ROH
1.3
1.6
1.809f
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

EOMCCSD CR(2,3)a

−0.930
−0.057
0.958
1.669
4.621
8.219
11.561
14.620
17.204
18.274
19.197

0.877
0.950
1.006
1.122
0.995
0.352
−0.328
−1.109
−1.892
−2.253
−2.590

EOMCCSDtb
1.291
1.545
1.781
1.913
2.117
1.961
1.750
1.608
1.531
1.510
1.497

CC(t;3)c
0.705
0.708
0.733
0.781
0.722
0.545
0.524
0.551
0.591
0.612
0.632

%T = 1d

%T = 2e

EOMCC(P ) CC(P ;Q) EOMCC(P ) CC(P ;Q)

2.160
2.159
2.488
2.878
3.750
4.360
4.423
4.030
3.674
3.555
3.695

0.529
0.523
0.593
0.742
0.729
0.585
0.660
0.739
0.741
0.769
0.701

2.026
1.989
2.008
2.478
2.898
3.051
2.913
2.389
2.059
1.918
1.818

0.538
0.506
0.545
0.637
0.656
0.540
0.533
0.519
0.462
0.450
0.439

a CR-EOMCC(2,3) calculations.
b EOMCCSDt calculations using the active space consisting of the three highest occupied and two lowest
unoccupied RHF orbitals.
c CC(t;3) calculations using the active space consisting of the three highest occupied and two lowest unoc-
cupied RHF orbitals.
d EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited deter-
minants and 1% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
e EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited determi-
nants and 2% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
f The equilibrium value of the O–H bond length in the ground electronic state of water, as obtained in Ref.
[327] using the CCSD/cc-pVTZ method.

195

Table D.8 Same as Table D.3 for the 2 3A′′ state.

ROH
1.3
1.6
1.809f
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

EOMCCSD CR(2,3)a

−0.547
0.286
1.181
1.877
2.210
2.703
5.070
11.868
21.864
25.767
28.470

0.933
1.044
1.049
1.052
1.111
1.472
2.801
6.744
11.891
13.294
13.976

EOMCCSDtb
1.370
1.620
1.825
1.927
1.740
1.477
1.355
1.409
1.504
1.503
1.482

CC(t;3)c
0.736
0.711
0.679
0.647
0.596
0.543
0.505
0.484
0.481
0.483
0.625

%T = 1d

%T = 2e

EOMCC(P ) CC(P ;Q) EOMCC(P ) CC(P ;Q)

2.392
2.430
2.808
3.071
3.906
4.215
5.281
7.607
9.244
9.172
9.097

0.726
0.672
0.710
0.840
1.066
1.145
1.284
1.634
1.753
1.595
1.523

2.314
2.165
2.323
2.540
3.334
3.384
3.706
4.470
4.926
4.889
4.806

0.655
0.608
0.623
0.714
0.808
0.777
0.831
0.844
0.804
0.736
0.705

a CR-EOMCC(2,3) calculations.
b EOMCCSDt calculations using the active space consisting of the three highest occupied and two lowest
unoccupied RHF orbitals.
c CC(t;3) calculations using the active space consisting of the three highest occupied and two lowest unoc-
cupied RHF orbitals.
d EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited deter-
minants and 1% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
e EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited determi-
nants and 2% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
f The equilibrium value of the O–H bond length in the ground electronic state of water, as obtained in Ref.
[327] using the CCSD/cc-pVTZ method.

196

Table D.9 Same as Table D.3 for the 2 1A′ state.

ROH
1.3
1.6
1.809f
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

EOMCCSD CR(2,3)a

−0.720
0.014
0.920
1.886
5.132
11.838
18.528
25.315
30.370
31.801
32.540

0.865
0.918
0.908
0.872
1.267
0.215
−2.178
−4.968
−6.883
−7.063
−6.732

EOMCCSDtb
1.300
1.509
1.677
1.785
1.853
1.896
2.218
2.706
3.185
3.377
3.531

CC(t;3)c
0.679
0.663
0.643
0.627
0.772
0.892
0.868
0.743
0.743
0.646
0.621

%T = 1d

%T = 2e

EOMCC(P ) CC(P ;Q) EOMCC(P ) CC(P ;Q)

2.302
2.197
2.655
2.819
4.413
3.710
3.149
3.299
3.606
3.653
3.722

0.473
0.400
0.446
0.433
0.597
0.786
0.999
1.109
1.101
1.161
1.090

2.069
1.830
1.908
2.219
3.264
2.598
2.364
2.405
2.504
2.440
2.353

0.477
0.405
0.414
0.464
0.533
0.611
0.656
0.753
0.912
0.841
0.731

a CR-EOMCC(2,3) calculations.
b EOMCCSDt calculations using the active space consisting of the three highest occupied and two lowest
unoccupied RHF orbitals.
c CC(t;3) calculations using the active space consisting of the three highest occupied and two lowest unoc-
cupied RHF orbitals.
d EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited deter-
minants and 1% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
e EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited determi-
nants and 2% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
f The equilibrium value of the O–H bond length in the ground electronic state of water, as obtained in Ref.
[327] using the CCSD/cc-pVTZ method.

197

Table D.10 Same as Table D.3 for the 2 1A′′ state.

ROH
1.3
1.6
1.809f
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

EOMCCSD CR(2,3)a

−0.688
−0.078
0.649
1.252
1.371
1.147
1.309
2.177
4.156
5.598
7.284

0.912
0.963
0.905
0.845
0.776
0.846
1.103
1.642
2.805
3.693
4.759

EOMCCSDtb
1.340
1.582
1.776
1.875
1.717
1.483
1.348
1.289
1.274
1.277
1.283

CC(t;3)c
0.722
0.684
0.642
0.604
0.562
0.523
0.493
0.473
0.461
0.459
0.460

%T = 1d

%T = 2e

EOMCC(P ) CC(P ;Q) EOMCC(P ) CC(P ;Q)

2.153
2.202
2.565
2.685
3.247
3.039
3.586
3.757
4.680
4.871
5.282

0.702
0.607
0.652
0.777
0.890
0.805
0.790
0.783
1.140
1.217
1.208

2.209
2.049
2.171
2.465
2.886
2.577
2.588
2.493
2.823
2.820
2.800

0.624
0.583
0.593
0.686
0.701
0.578
0.588
0.544
0.734
0.637
0.098

a CR-EOMCC(2,3) calculations.
b EOMCCSDt calculations using the active space consisting of the three highest occupied and two lowest
unoccupied RHF orbitals.
c CC(t;3) calculations using the active space consisting of the three highest occupied and two lowest unoc-
cupied RHF orbitals.
d EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited deter-
minants and 1% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
e EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited determi-
nants and 2% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
f The equilibrium value of the O–H bond length in the ground electronic state of water, as obtained in Ref.
[327] using the CCSD/cc-pVTZ method.

198

Table D.11 Same as Table D.3 for the 3 3A′′ state.

ROH
1.3
1.6
1.809f
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

EOMCCSD CR(2,3)a

0.779
1.104
1.241
1.399
3.843
62.239
65.442
53.786
42.844
39.743
38.464

1.263
1.518
1.660
1.643
2.230
37.018
6.569
−0.457
−4.801
−5.151
−4.513

EOMCCSDtb
1.977
2.431
2.825
3.427
5.071
2.134
2.247
1.909
1.606
1.522
1.470

CC(t;3)c
0.604
0.662
0.663
0.557
0.322
0.133
0.531
0.511
0.741
0.737
0.496

%T = 1d

%T = 2e

EOMCC(P ) CC(P ;Q) EOMCC(P ) CC(P ;Q)

2.175
2.645
2.800
3.421
4.367
25.551
14.778
10.641
7.891
7.713
7.241

0.807
0.864
1.019
1.199
2.134
6.740
2.541
1.780
1.486
1.491
1.373

2.001
2.261
2.330
2.768
3.916
12.680
8.348
6.420
4.998
4.603
4.380

0.699
0.735
0.823
0.962
1.551
2.195
1.144
0.852
0.678
0.665
0.623

a CR-EOMCC(2,3) calculations.
b EOMCCSDt calculations using the active space consisting of the three highest occupied and two lowest
unoccupied RHF orbitals.
c CC(t;3) calculations using the active space consisting of the three highest occupied and two lowest unoc-
cupied RHF orbitals.
d EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited deter-
minants and 1% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
e EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited determi-
nants and 2% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
f The equilibrium value of the O–H bond length in the ground electronic state of water, as obtained in Ref.
[327] using the CCSD/cc-pVTZ method.

199

Table D.12 Same as Table D.3 for the 3 1A′ state.

ROH
1.3
1.6
1.809f
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

EOMCCSD CR(2,3)a

1.189
1.674
2.399
3.392
5.272
6.462
12.832
23.485
29.845
30.434
32.035

1.471
1.605
1.598
1.517
0.714
0.852
2.422
5.089
5.471
3.564
2.203

EOMCCSDtb
1.698
1.820
1.870
1.874
1.659
1.586
1.712
1.782
1.661
1.493
1.411

CC(t;3)c
0.995
1.052
1.042
1.010
0.795
0.684
0.742
0.676
0.449
0.643
0.754

%T = 1d

%T = 2e

EOMCC(P ) CC(P ;Q) EOMCC(P ) CC(P ;Q)

2.737
2.551
2.741
3.087
4.087
3.981
5.674
7.575
8.569
7.649
8.006

0.526
0.534
0.624
0.770
0.584
0.571
1.479
1.980
2.733
1.735
1.133

2.168
2.005
1.855
2.392
3.029
2.994
3.886
4.353
4.717
3.591
3.338

0.425
0.497
0.513
0.631
0.482
0.525
0.732
0.674
1.418
0.561
0.612

a CR-EOMCC(2,3) calculations.
b EOMCCSDt calculations using the active space consisting of the three highest occupied and two lowest
unoccupied RHF orbitals.
c CC(t;3) calculations using the active space consisting of the three highest occupied and two lowest unoc-
cupied RHF orbitals.
d EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited deter-
minants and 1% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
e EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited determi-
nants and 2% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
f The equilibrium value of the O–H bond length in the ground electronic state of water, as obtained in Ref.
[327] using the CCSD/cc-pVTZ method.

200

Table D.13 Same as Table D.3 for the 3 3A′ state.

ROH
1.3
1.6
1.809f
2.0
2.4
2.8
3.2
3.6
4.0
4.2
4.4

EOMCCSD CR(2,3)a

0.720
1.352
2.074
2.950
3.158
4.223
6.986
11.320
16.730
19.566
22.282

1.272
1.531
1.592
1.393
0.986
1.175
1.528
2.330
3.829
5.000
6.656

EOMCCSDtb
1.929
1.980
1.991
2.060
1.760
1.541
1.452
1.412
1.379
1.363
1.348

CC(t;3)c
0.985
1.066
1.045
0.923
0.674
0.726
0.772
0.802
0.800
0.777
0.826

%T = 1d

%T = 2e

EOMCC(P ) CC(P ;Q) EOMCC(P ) CC(P ;Q)

2.377
2.595
2.694
3.051
3.907
3.931
4.083
4.111
4.471
4.984
5.525

0.498
0.625
0.734
0.732
0.704
0.702
0.799
1.203
1.611
1.847
2.073

1.910
2.005
1.933
2.542
3.037
2.879
2.817
2.793
3.003
3.134
3.323

0.414
0.558
0.572
0.629
0.618
0.580
0.580
0.676
0.858
0.958
0.996

a CR-EOMCC(2,3) calculations.
b EOMCCSDt calculations using the active space consisting of the three highest occupied and two lowest
unoccupied RHF orbitals.
c CC(t;3) calculations using the active space consisting of the three highest occupied and two lowest unoc-
cupied RHF orbitals.
d EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited deter-
minants and 1% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
e EOMCC(P ) and CC(P ;Q) calculations using P spaces consisting of all singly and doubly excited determi-
nants and 2% of triply excited determinants identified by the adaptive CC(P ;Q) algorithm.
f The equilibrium value of the O–H bond length in the ground electronic state of water, as obtained in Ref.
[327] using the CCSD/cc-pVTZ method.

201