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NEW COUPLED-CLUSTER METHODS FOR MOLECULAR
POTENTIAL ENERGY SURFACES

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Ian Sedrick O. Pimienta

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NEW COUPLED-CLUSTER METHODS FOR MOLECULAR
POTENTIAL ENERGY SURFACES

By

Ian Sedrick O. Pimienta

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Chemistry

2003

ABSTRACT

NEW COUPLED-CLUSTER METHODS FOR MOLECULAR
POTENTIAL ENERGY SURFACES

By

Ian Sedrick O. Pimienta

The new single-reference approach to the many-electron correlation problem,
termed the method of moments of coupled-cluster equations (MMCC), which can
be used to extend the applicability of the standard CC methods to bond breaking,
is further developed by considering the higher-than-quadruply excited moments of
the CCSD (coupled-cluster singles and doubles) equations. The main idea behind the
MMCC theory is that of the noniterative a posteriori energy corrections 6 which, when
added to the energies obtained in the standard CC calculations, recover the exact, full
configuration interaction (CI), energies. In this study, the MMCC theory is extended
by incorporating the generalized moments of the CCSD equations that correspond to
projections of these equations on pentuply and hextuply excited configurations into
the MMCC energy corrections. It is demonstrated that these higher-order moments
can be very important in studies of the most difficult cases of multiple bond breaking.
The trial wave functions that are used to calculate the MMCC energy corrections are
provided by the relatively inexpensive CI methods and by the truncated forms of
the exponential, CC-like, wave functions. The resulting CI-corrected MMCC meth-
ods and the quadratic MMCC (QMMCC) methods including higher-than-quadruply

excited moments of the CCSD equations are tested in studies of different types of

bond breaking, including the single bond breaking in the HF molecule, the simul-
taneous breaking of both O—H bonds in the water molecule, the complicated case
of triple bond breaking in the nitrogen molecule, and the extremely difficult case of

bond breaking in the C2 molecule.

for Rio

iv

ACKNOWLEDGMENTS

I would like to express my heartfelt thanks to the following peOple for their part
in this work and in my graduate studies:
my advisor, Dr. Piotr Piecuch, for his endless support, guidance, and encouragement;
Dr. Karol Kowalski, for his openness and willingness to help me in whatever capacity
he could;
my advisorial committee, including Dr. James Harrison, the second reader, Dr.
Katharine Hunt, Dr. James Jackson, and Dr. Simon Garrett, for their critical in-
sights on various issues that facilitated my understanding of my research;
Mike, Peng-Dong, Maricris, Ruth, and DJ, for their friendship and companionship,
for their willingness to help, and for listening to my practice seminars;
the MSU Filipino Club for all things “Pinoy”;
Tita Luchie, Tita Estela, and Tito Amboy, for their unselfish love and support;
Ma and Dad, Lolo Flor, and my sister Ira, for their unconditional love and encour—
agement;
my wife Ria, for her love and friendship, and for everything she is;

and finally, God Almighty, for without Him, there is nothing.

TABLE OF CONTENTS

List of Tables

List of Figures

1. Introduction

2. Project Objectives

3. The Method of Moments of Coupled-Cluster Equations: An Overview
of the General Formalism

4. The Existing MMCC(mA, m3) Approximations
4.1. The CI-corrected MMCC(2,3) and MMCC(2,4) Methods

4.2. The Renormalized and Completely Renormalized CCSD[T], CCSD(T),

and CCSD(TQ) Methods

5. The New MMCC(mA, m3) Approaches for Multiple Bond Breaking

5.1. The CI-corrected MMCC(2,5) and MMCC(2,6) Approximations
5.1.1. Theory
5.1.2. Examples of Applications
5.1.3. Conclusion

5.2. The Quasi-variational and Quadratic MMCC Approximations
5.2.1. Theory
5.2.2. Examples of Applications
5.2.3. Conclusion

6. Summary, Concluding Remarks, and Future Perspectives
Appendix A. An elementary derivation of Eq. (21)

Appendix B. The many-body diagrams for the generalized moments of
the CCSD equations

Appendix C. Algebraic expressions for the M‘1‘2‘3 (2), M31233; 4 (2),

010203

Mz‘l‘gfii‘jfasa), and /\/l‘“2'~°"“~"‘6 (2) moments of the CCSD equations

010203040506

References

vi

vii

viii

15

16

25
28

37

54
57
58
60
69
71
72
86
98 .

100

106

110

134

142

List of Tables

Table 1. A comparison of the standard CC, completely renormalized CCSD[T],
CCSD(T), and CCSD(TQ), and CI—corrected MMCC(2,3) and MMCC(2,4) ener-
gies with the corresponding full CI, CISDt, and CISth results obtained for a few
internuclear separations R of the HF molecule, as described by the DZ141 basis set.“

154

Table 2. A comparison of the standard CC, completely renormalized CCSD[T],
CCSD(T), and CCSD(TQ), and CI-corrected MMCC(2,3) and MMCC(2,4) energies
with the corresponding full CI, CISDt, and CISth results obtained for the equilib—
rium and two displaced geometries of the H20 molecule with the DZ141 basis set.“

155

Table 3. A comparison of the standard CC, completely renormalized CCSD[T],
CCSD(T), and CCSD(TQ), and CI-corrected MMCC(2,3) and MMCC(2,4) energies
with the corresponding full CI, CISDt, and CISth results for a few internuclear
separations of the N2 molecule, as described by the DZ141 basis set.“ 156

Table 4. A comparison of the CI-corrected MMCC (2,5) and MMCC(2,6) energies and
QMMCC(2,4), QMMCC(2,5), and QMMCC(2,6) energies with the results of the full
CI, standard CC, and CISthp and CISthph calculations for the equilibrium and
two displaced geometries of the H20 molecule with the DZ141 basis set.“ 157

Table 5. A comparison of the CI-corrected MMCC(2,5) and MMCC(2,6) energies
and QMMCC(2,4), QMMCC(2,5), and QMMCC(2,6) energies with the results of the
full CI, standard CC, and CISthp and CISthph calculations for a few internuclear
separations of the N2 molecule with the DZ141 basis set.“ 158

Table 6. A comparison of the QMMCC(2,4), QMMCC(2,5), and QMMCC(2,6) ener—

gies with the results of the full CI and standard CC calculations for a few internuclear
separations R of the HF molecule, as described by the DZ141 basis set.“ 159

Table 7. A comparison of the QMMCC(2,4), QMMCC(2,5), and QMMCC(2,6) en-
ergies with the results of the full CI, standard CC, and completely renormalized
CCSD(TQ) calculations for a few internuclear separations of the C2 molecule with

the DZMl basis set.“ 160

vii

List of Figures

Fig. 1. The orbital classification used in the active-space CI and the CI-corrected
MMCC(2,3), MMCC(2,4), MMCC(2,5), and MMCC(2,6) approaches. Core, active,
and virtual orbitals are represented by solid, dashed, and dotted lines, respectively.
Full and open circles represent core and active electrons of the reference configuration

lo). 161

Fig. 2. Potential energy curves for the N2 molecule, as described by the DZ basis set.
A comparison of the results obtained with the CR—CC and CI-corrected MMCC(2,3)
and MMCC(2,4) methods with the results of the standard CC and full CI calculations.

162

Fig. 3. Potential energy curves for the N2 molecule, as described by the DZ basis
set. A comparison of the results obtained with the CI-corrected MMCC(2,5) and
MMCC(2,6) methods with the results of the standard CC, CR-CCSD(TQ), and full
CI calculations. 163

Fig. 4. Potential energy curves for the N2 molecule, as described by the DZ basis set.
A comparison of the results obtained with the QMMCC methods with the results of
the standard CC and full Cl calculations. 164

Fig. 5. Potential energy curves for the C2 molecule, as described by the DZ basis set.
A comparison of the results obtained with the QMMCC methods with the results of
the standard CC, CR-CCSD(TQ), and full CI calculations. 165

viii

1. Introduction

One of the most challenging problems that quantum chemists of today face is the
accurate study of molecular potential energy surfaces (PESs) that involve breaking or
making of chemical bonds. Bond breaking and bond making are difficult to describe
with quantum mechanical methods, particularly when one is interested in high accu-
racies that are expected from modern computer simulation techniques. The early ab
initio methods developed to obtain energies for atomic and molecular systems, such
as the Hartree—Fock (HF) approach};2 do not work when bond breaking or making
is involved. The failure of the HF methods in describing bond breaking or making
can be attributed to the lack of electron correlation in these methods. Because of
the presence of the electron-electron repulsion terms in the molecular Hamiltonian,
the motion of electrons in a molecule is highly correlated and difficult to describe.
Although the bulk of the total electronic energy is obtained from the HF calculations
(~99%), the remanent energy that the HF methods do not describe is involved in
various chemical processes, including bond breaking and making. Other quantities
of interest in chemistry (i.e. ionization potentials, electron affinities, electronic ex-
citation energies, vibrational energies, etc.) are also sensitive in varying degrees to
the many-electron correlation effects. The residual energy, known as the correlation
energy obtained as the difference between the HF energy and the exact energy,3 is
the single most important piece that quantum mechanical models must describe if we
are to use them in accurate and predictive calculations for molecular systems.

There are a number of techniques that seek to improve the HF wave function. The

method of choice depends very much on the characteristics of the problem. Some of
the desirable features of a method involving electron correlation are:4 ( 1) it should be
well-defined, giving a continuous PBS and a unique energy for any nuclear configu-
ration, (2) it should be “size-consistent”, i.e. the energy of a sum of non-interacting
fragments should be exactly the sum of energies obtained in separate calculations
on the fragments, (3) it should be exact when applied to a two-electron system, (4)
it should be efficient, i.e. it should be applicable to larger basis sets, (5) it should
be accurate enough to be an adequate approximation to the exact result, and (6) it
should be variational, i.e. the energy should be an upper bound to the exact result.
Unfortunately, no current method satisfies all of the above criteria!

Nowadays, there are three main ab initio approaches that describe the many-
electron correlation effects: (1) the configuration interaction (CI) method,""9 (2)
the many-body perturbation theory (MBPT),1°_” and (3) the coupled-cluster (CC)
theory.l5—19 The conceptually simplest way of describing electron correlation is via the
CI method. CI uses a wave function which is a linear combination of the HF (or other
reference) determinant and Slater determinants obtained by promoting (exciting)
electrons from the occupied to unoccupied orbitals. The CI method is variational
and, if the expansion is complete (full CI), the exact correlation energy (and the
exact total energy) for a basis set used in the calculations is obtained. The number
of Slater determinants in full CI grows factorially with the system size, making the
method impractical for all but the smallest few-electron systems. For this reason
the CI expansion is usually truncated at some level of excitation. For example, in
the CISD (CI with singles and doubles) method, only the singly and doubly excited

2

determinants, in addition to the reference determinant, are considered while in the
CISDT (CI with singles, doubles, and triples) approach, only the singly, doubly, and
triply excited determinants are considered, and so on. The problem with all truncated
CI methods is that the convergence of the CI results towards the full CI limit with the
excitation level is often very slow. One has to use very long CI expansions to obtain
high accuracies required in modern ab initio calculations. Another problem with the
truncated CI approaches is that they are not size-extensive, i.e. the truncated CI
methods incorrectly describe the dependence of electronic energies on the size of the
system. For CISD, an approximate way to correct for the lack of size extensivity is
by introducing the Davidson corrections?!“21

Perhaps the easiest way to incorporate electron correlation into quantum chemical
calculations is by using the MBPT. In this approach, the exact Hamiltonian is decom-
posed into the unperturbed part that corresponds to the HF (or some other single-
determinantal) description and the perturbation that describes the many-electron
correlation effects. The Rayleigh-Schrbdinger perturbation theory is used to calcu-
late the corrections to the HF (or other single-determinantal reference) wave function
and energy. The MBPT energies are size—extensive, but not variational. Perturbation
theory relies on the starting wave function being close to the exact wave function.
When this is the case, i.e. when the electronic wave function is well represented by
a single Slater determinant, the convergence of the MBPT series is usually rapid.
However, when chemical bonds are stretched, the MBPT series becomes divergent.

For molecules near their equilibrium geometries, the MBPT energies and properties

are often more accurate than the corresponding limited (or truncated) CI results,

3

but unlike CI, the MBPT methods are usually useless when PESs involving bond
breaking are examined.

The failing of MBPT is that it is basically an order-by-order perturbation ap-
proach. In many cases, it is necessary to go to higher orders of MBPT. A practical
solution to this problem is provided by the CC theory. In the CC theory, an ex-
ponential form of the wave function in terms of the cluster operator T is employed.
The advantage of all CC methods is that the higher-order excitations are partially
included as products of the lower order excitations, making it possible to describe
the high-order effects at a low level of approximation. The use of the infinite-order
exponential ansatz guarantees the correct scaling of molecular properties with the
number of electrons. Like MBPT, the CC results are not variational, and energies
can go below their exact full CI counterparts, but this is never a problem when the
closed-shell systems are considered, since in this case the CC methods are remarkably
accurate. It is possible to combine some aspects of MBPT with CC theory to retain
much of the infinite-order aspect of CC while saving significantly in the computational
cost for higher cluster operators. This can be done iteratively or noniteratively (cf.
the discussion below).

Over the past 30 years, the CC theory15—19 has become the most reliable compu-
tational method of electronic structure theory for the prediction of molecular energies
and properties whenever high-accuracy results are desired. In 1966,15 and later in a
review article published in 1969,16 Ciiek laid down the foundations of the use of the
CC method in electronic structure theory in formal terms. The sophisticated math-

ematical techniques that he used, such as diagrams and second-quantization, were,

4

unfortunately, beyond the experience of most quantum chemists at that time. In the
19708, CC theory began to gain some interest in the quantum chemistry community.
Ciiek and Paldusl7 derived expressions for coupled-cluster doubles (CCD) theory (at
that time referred to as coupled-pair many—electron theory (CPMET)), using rather
elementary mathematical techniques that could be easily understood by others. This
was actually an algebraic re-derivation of the diagrammatic equations presented ear-
lier by CiZek.15 In 1972,22 Paldus, CiZek, and Shavitt performed the first ab initio
coupled cluster calculations using a variety of CC methods including singly, doubly,
and even triply excited clusters, although computer programs that they developed
were not of the general purpose type. At the end of 19708, the first general purpose
computer implementations of the CC theory began to appear. Pople et al.23 and
Bartlett and Purvis24 developed the first general purpose spin—orbital CCD codes.
In the early 19808, a particularly important achievement was that of Purvis and
Bartlett,25 who derived the vectorizable form of the CCSD (coupled-cluster with sin-
gles and doubles) equations and implemented them in a practical computer code that
could be applied to a wide range of problems. At the end of the 19808 and through-
out the 19908, the popularity of CC methods grew at a rapid pace, as more effort
was directed towards the construction of highly efficient CC codes, spin-adaptation
of open-shell CC wave functions, calculations of properties other than energy, exten-
sions of CC theory to excited states and multireference problems requiring a truly
multideterminantal description, and inclusion of higher excitations in the CC wave
function?“30 Nowadays, accurate CC calculations for closed-shell and simple open-

shell molecular systems at their equilibrium geometries involving up to 20—30 light

5

atoms or a few heavy atoms are routine.

The CC theory has grown to become one of the most popular methods in accu-
rate electronic structure calculations, but severe problems have been discovered in
applications of the CC methods to PESs involving bond breaking. Horn the time the
CCSD25—29 method was implemented, numerous CC methods have been developed
with the intention of improving the quality of PESs while maintaining the relatively
low cost of the CC calculations. The CCSD method uses only the singly and doubly
excited clusters in the CC expansion. It has been observed from many numerical
calculations that the CCSD method fails to describe bond stretching or breaking,
particularly when multiply bonded systems are considered (cf. the discussion below).
The absence of the higher-than-doubly excited clusters in the expansion is one of the
reasons why the CCSD method fails in the bond breaking region. As it turns out, the
CCSD approach is often inadequate when high accuracy results for nuclear geometries
near the equilibrium are sought. The inadequacy of the CCSD method prompted the
quantum chemists to develop approximate methods that include higher-than-doubly
excited clusters in the calculations. The most practical approximations that emerged
from this effort include the noniterative CCSD+T(CCSD) = CCSD[T]3°—32 and the
CCSD(T)33 approaches, where the information about energy contributions due to
triply excited clusters is obtained using MBPT. Later, similar methods were devel-
oped that account for the combined effect of triply and quadruply excited clusters.
One such approach is the CCSD(TQf) method,34 where one adds the relatively inex-
pensive corrections due to triply and quadruply excited clusters to the CCSD energies.

Other inexpensive approaches, termed the approximate coupled-pair of approximate

6

CCSD methods accounting for triples, quadruples, and related higher-order clusters
were developed by Piecuch et (H.3132 The addition of the triply and quadruply excited
clusters, on which the CCSD[T], CCSD(T), and CCSD(TQf) approaches are based,
to the CCSD energies has been proven time and time again in numerous molecular ap-
plications to produce methods that give the best compromise between high accuracy

and relatively low computer cost in molecular applications?“—39

Unfortunately, the CCSD[T], CCSD(T), and CCSD(TQf) methods fail in sit-
uations where bond breaking is involved, particularly when the spin-adapted re—
stricted Hartree-Fock (RHF) configuration is used as a reference (cf., for example,
Refs. 38, 40—53 and references therein). The failure of the CCSD[T], CCSD(T), and
CCSD(TQf) approaches in describing breaking of chemical bonds is due to the fol-
lowing two factors: (i) the singly and doubly excited clusters, obtained in CCSD
calculations, which are used to determine the energy corrections due to triples and
quadruples in the CCSD[T], CCSD(T), and CCSD(TQf) approaches, are significantly
different from their exact values in the bond breaking region and (ii) the noniterative
triples and quadruples corrections, defining the CCSD[T], CCSD(T), and CCSD(TQf)
methods, which are based on the standard MBPT arguments, fail due to the divergent
behavior of the MBPT series at larger internuclear separations. As a consequence, the
PESs produced by the CCSD(T), CCSD(TQf), and other noniterative CC approaches
are completely unphysical.38"’°_53 As a matter of fact, even the iterative analogs of
the CCSD[T], CCSD(T), and CCSD(TQf) methods, i.e. the CCSDT-n3l’54_57 and
CCSDTQ--158 approaches, and the higher-order noniterative CCSDT+Q(CCSDT)

= CCSDT[Q]58 and CCSDT(Q;)34 approximations that all use MBPT to evaluate

7

higher-than-doubly excited clusters, fail in the bond breaking region, particularly
when multiple bonds are broken.”49 Thus, new CC methods that can be applied to
molecular PESs involving single and multiple bond breaking must be developed.
One way to eliminate the failures of the perturbative, noniterative and iterative
CC approximations in the bond breaking region is by the inclusion of the higher-
than-doubly excited clusters, i.e. the triply, quadruply, and even the pentuply excited
clusters, completely and in a fully iterative manner. Kallay and Surjan developed a
programming technique that allows one to write efficient computer codes for iterative
CC methods of any rank.59 The complete and iterative incorporation of the higher-
order clusters in CC calculations helps to resolve the failures of the single-reference
CC approaches, especially at larger internuclear separations. However, the resulting
methods are much too expensive for routine applications. For example, the most
expensive iterative steps of the full CCSDT approach (CC method with singly, dou-
bly, and triply excited clusters)“:61 and its higher-level CCSDTQ (CC with singles,
doubles, triples, and quadruples)”.65 and CCSDTQP (CC singles, doubles, triples,

3 5 4

analogs scale as n n n n6 and 125127 respectively (no

)66
o u, o u, o u,

quadruples, and pentuples
(nu) is the number of occupied (unoccupied) orbitals in the molecular orbital (MO)
basis). This means that the computer time associated with the full CCSDT, CCS-
DTQ, and CCSDTQP calculations grows as N8, N1", and M”, respectively, with
molecular size (N is a general measure of the molecular size). On the other hand, the
CCSD(T) calculation requires iterative steps that scale as ngnfi (or JV“) and nonitera-
tive steps that scale as ngn: (or AC). The expensive steps in the CCSDT, CCSDTQ,

and CCSDTQP calculations restrict their applicability to very small systems, at best

8

consisting of 2—3 light atoms, using small basis sets. This should be contrasted with
the fact that the CCSD(T) approach can be applied to systems with up to 30 light
atoms or a few heavy atoms. In fact, Schiitz and Werner performed CCSD(T) cal-
culations for systems with ~100 atomsm—69 using the local correlation formalism of
Pulay and Saeb¢.7°"72

One can eliminate the above problems with the single-reference CC methods
by switching to the genuine multireference CC (MRCC) approaches of the state-

135"°”5"73—88 or valence-universal:”"""‘8'89_92 type, which are specifically designed

universa
to handle general open-shell and quasi-degenerate states, including, at least in prin-
ciple, all cases of bond breaking. However, the MRCC approaches are often plagued
with intruder states and multiple, singular, or unphysical solutions (cf., e.g., Refs.
76, 81-85, 93—95). The standard single—reference CC methods do not suffer from
such problems. Furthermore, the single-reference CC methods are much easier to use
than their MRCC counterparts. The state-specific MRCC approaches (cf., e.g., Refs.
96—103), which are based on the genuine multireference formalism may change this
situation, but none of the existing state-specific MRCC methods are simple or gen-
eral enough to be as widely applicable as the standard CCSD, CCSD(T), and similar
approaches.

The difficulties and problems associated with the use of the multireference ap-
proaches indicate that, in developing new methods for bond breaking, it may be more
worthwhile to focus on methods that use only a single reference formalism. In recent

years, a great deal of effort has been undertaken towards developing single-reference or

single-reference—like approaches that would, potentially, eliminate the pervasive failing

9

of the standard CC approximations at larger internuclear separations, while avoiding
the complexity of the genuine multireference theory and the astronomical costs of the
CCSDTQ, CCSDTQP, and similar calculations. These include the reduced multiref-
erence CCSD (RMRCCSD) method,38'1°“_110 the active-space CC approaches (which
can also be classified as the state-selective MRCC methods),“1"‘3’4""“8'65’1“—125 the
orbital-optimized CC methods?“127 the noniterative approaches based on the par-
titioning of the similarity-transformed Hamiltonian,1""8—131 and the renormalized (R)
and completely renormalized (CR) CC methods.“5-"’1’5:"'1"“"136 The latter approaches
are based on the more general formalism of the method of moments of CC equations
(MMCC),“5—“7'51"5237432.137 which can be applied to ground- and excited-state PESs.
All of these methods have been deve10ped with the intention of improving the descrip-
tion of bond breaking, while retaining the simplicity of the single-reference description
based on the spin- and symmetry-adapted references of the RHF type.

The RMRCCSD method of Paldus and M3330“—no and the active—space CC ap-

41,43,44,48,65,lll—122

proaches of Adamowicz, Piecuch and co—workers are interesting and

worth further development. They have been proved to accurately describe quasi-

degenerate ground states,43'65'1°4’1°7'“8 bond breaking,“,43,44,48,105,106,108—110,115,116,119,l22

ro—vibrational term values including highly excited states near dissociation,“'1°9'“°
and property functions.121 In analogy to the genuine multireference approaches, the
RMRCCSD and active-space CC approaches require choosing active orbitals, but this
is done in such a way that the formal structure of the single-reference CC theory is

largely preserved. The RMRCCSD and active-space CC methods definitely represent

the step in the right direction, but the question remains if one could completely or

10

almost completely eliminate the elements of multireference calculations, such as ac-
tive orbitals, and yet obtain a highly accurate description of PESs involving bond
breaking.

The perturbative CC approaches developed by Head-Gordon et al., which are
based on the partitioning of the similarity-transformed Hamiltonian,128_131 and the
renormalized (R) and completely-renormalized (CR) CC approaches developed by
Piecuch et al.,45_51'53 which employ the MMCCformalism,“5_“7'51""2’37'132—137 represent
an attempt to entirely eliminate active orbitals and other elements of multireference
theory from the calculations. These methods retain the simplicity and the “black-
box” character of the standard CC methods of the CCSD(T) type. The perturbative
CC approaches developed by Head-Gordon et al. improve the description of bond
breaking, but they are not as accurate as the MMCC methods and CR-CC approaches
(cf., e.g., Ref. 48). Thus, pursuing the MMCC methodology is probably the best
idea at this time, when it comes to relatively simple coupled-cluster methods that
may improve the description of bond breaking with an effort similar to the single-
reference calculations. In the MMCC formalism and the related R—CC and CR—CC
approaches, noniterative energy corrections are computed which, when added to the
energies obtained in the standard CC calculations, such as CCSD, recover the exact,
i.e. full CI, energies. For example, in the renormalized and completely renormalized
CCSD(T) and CCSD(TQ) approaches, one adds relatively inexpensive noniterative
corrections due to triples or triples and quadruples to the CCSD energies. These
methods preserve the conceptual and computational simplicity of the noniterative
CC methods, such as CCSD(T) and CCSD(TQf), while improving the results in the

11

bond breaking region. At least part of the success of the MMCC approaches is due
to the fact that in designing the MMCC approximations including the R—CC and
CR—CC methods, we directly focus on the quantity of interest which is the difference
between the full CI and CC energies.

Two types of the MMCC methods have been considered so far, namely, the CI-
corrected MMCC schemes“5'51’52”?—134 and the R-CC and CR—CC
methods.“5_51"53:13“:136 In the CI-corrected MMCC approaches, one has to perform
an a priori limited CI calculation to construct the noniterative corrections to the
standard CC energies.“"”“152432—134 The R-CC and CR-CC methods“"’"51"""'1"“'136 do
not require any a priori non-CC calculations. Thus, they are as easy to use as the
standard CC “black-boxes” of the CCSD(T) or CCSD(TQf) type. Furthermore, the
R-CC and CR—CC methods can easily be incorporated in any electronic structure
package that has the CCSD(T) and CCSD(TQf) options in it. In fact, the renormal-
ized and completely renormalized CCSD[T] and CCSD(T) methods (the R—CCSD[T],
R—CCSD(T), CR-CCSD[T], and CR—CCSD(T) methods, respectively) have recently
been incorporated138 in the popular GAMESS package.139

It has been well-established by now that the CR—CCSD(T) and CR-CCSD(TQ)
(completely renormalized CCSD(TQ)) methods provide an excellent description of en-
tire PESs involving single and double bond dissociations,“5"“3"“3’50“"2'134'136'138 highly-
excited vibrational term values near dissociation,5°'51'13“:138 and entire PESs for ex-
change chemical reactions of the general type: closed shell + closed shell —-) doublet +
doublet,"’3’13“'136 and for transition states having a diradical character.140 The results

obtained in the CR-CCSD(T) and CR-CCSD(TQ) calculations for triple bond break-

12

ing and for certain types of double bond dissociations, however, are somewhat less
impressive.“7'51“3“"136 In particular, for the double zeta (DZ)1“l model of N2, the small
~1 millihartree errors, relative to full CI, near the equilibrium geometry Re, obtained
with variant “b” of the CR—CCSD(TQ) method increase to 10-25 millihartree at
larger N—N separations (see Sect. 4.2 for the discussion on completely renormalized
CC approaches; see Table 3 for the numerical results). These are still quite signif-
icant errors if one is aiming at high accuracy. Thus, for more complicated types
of bond breaking, including cases where multiple bonds are stretched or broken, we
need better methods than CR-CCSD(T) or CR—CCSD(TQ). The main challenge is in
formulating methods that preserve the relative simplicity of the noniterative coupled-
cluster approximations of the CCSD(T) or CCSD(TQ) type. We believe that such
methods can be proposed if we use the higher-order MMCC approximations.

The main goal of this dissertation is the development, implementation, and test-
ing of the new types of noniterative MMCC methods that lead to a highly accurate
description of multiple bond breaking. We have formulated two classes of the MMCC
methods that work especially well for double and triple bond breaking. The first class
includes the CI-corrected MMCC(2,6) approach.134 In this approach, a relatively in-
expensive CI calculation is performed and the results are combined with the complete
set of moments of the CCSD equations, including the previously ignored pentuply and
hextuply excited moments (the CR-CCSD(T) and CR—CCSD(TQ) methods use only
the triply and quadruply excited moments). The second class of approaches that
we have developed, implemented, and tested includes the so-called quadratic MMCC

(QMMCC) approach,“"‘_137 which is an approximate form of the more general quasi-

13

variational MMCC (QVMMCC) formalism. The QMMCC method can be viewed
as a natural extension of the CR—CCSD(T) and CR-CCSD(TQ) methods mentioned
earlier. In particular, the QMMCC approach preserves the “black-box” character of
the CR—CCSD(T) and CR—CCSD(TQ) approximations. In the QMMCC method, as
in the case of the CR—CCSD(T) and CR—CCSD(TQ) approaches, we add relatively
straightforward noniterative corrections to the standard CCSD energies. Details of
the CI-corrected MMCC(2,5) and MMCC(2,6) approaches and the QMMCC approx-
imation are described in Sect. 5. Examples illustrating the performance of these new

approaches in calculations of molecular PESs are described in Sect. 5, too.

14

2. Project Objectives

The main objectives of this work are:

A. The development of the higher-order variants of the MMCC formalism, includ-
ing higher—than-quadruply excited moments of the CCSD equations, that are
capable of eliminating the failure or poorer performance of the existing stan-
dard and renormalized CC methods for cases involving multiple bond breaking.
This will be accomplished by formulating and implementing the CI-corrected
MMCC(2,6) method and the quadratic MMCC approximation mentioned in the

previous section.

B. Testing the proposed MMCC methods in benchmark calculations for various
types of bond breaking, including single bond breaking in the HF molecule,
simultaneous dissociation of two O—H bonds in the H20 molecule, double bond

breaking in the C2 molecule, and triple bond breaking in N2.

15

3. The Method of Moments of Coupled-Cluster Equations: An
Overview of the General Formalism

In the single-reference CC theory, the ground-state wave function I‘llo) of an N-

electron system, described by the Hamiltonian H, is written in the exponential form

We) = eTl‘I’), (1)

where T is the cluster operator and |<I>) is the independent-particle-model (IPM) ref-
erence configuration (e.g., the Hartree—Fock determinant) defining the Fermi vacuum.
In the exact CC theory, T is a sum of all many-body cluster components that can be

written for a given N -electron system. We have,

where the n-body cluster component T, is defined as

2
_ E : t1...tn Ola-an _ __ “may; alman
Tn _ talqun Eil...in _ (n' to] "can Bil-"in I (3)
i1<"’<in
al<...<an
with
n
a]...an _ a .
Eij...in _ H6 “Ch: (4)
rc=l

and till'fgn representing the corresponding excitation operators and cluster ampli-

tudes, respectively. Whenever possible, the Einstein summation convention over re-

peated upper and lower indices is employed. In our notation, c" (cp) are the usual

16

creation (annihilation) operators (c? = CI) associated with a given orthonormal spin-
orbital basis set {p}. The letters 2' and a designate the occupied and unoccupied
spin-orbitals, respectively, whereas p is a generic index that runs over all (occupied
and unoccupied) spin-orbitals. The singly and doubly excited clusters, T1 and T2,
respectively, that are used in the CCSD calculations, are given by the following ex-

pressions:

_ 2 : i1 01 = i1 01
T1 — talE'1 _t01Eil , (5)
ii
01
T _ tilt: E0102 —lti1i3 E0102 (6)
2— 010.2 iliz — 4 0102 £11.? .
i1<i2
01(02

The standard CC approximations are obtained by truncating the many-body ex-

pansion for T, Eq. (2), at a given excitation level m A < N, so that

"EA
T x TM) = 2T”. (7)

n=1

The standard CCSD method is obtained by setting mA = 2 in Eq. (7). In the
CCSDT method, m A is set at 3; in the CCSDTQ approach, mA = 4, etc. The cluster
operator T”), Eq. (7), defining the standard approximation A or the amplitudes
t“'°"" , n = 1, . . . , m A, which define it, is obtained by solving the system of nonlinear,

al ".01;

energy-independent, algebraic equations,

QnH(A)|<I>)=0, n=1,...,mA, (8)

17

where

HM) = e-T“’HeT"" .—. (HeTW)C (9)

is the similarity-transformed Hamiltonian of the CC method A (subscript C desig-
nates the connected part of the corresponding operator expression (cf. Appendix
B). (2,, is the projection operator onto the subspace spanned by all n—tuply excited

configurations relative to reference |<I>), i.e.,

Q" = Z |<I>?,‘.'.'.'.-‘f.">(@3333?|, (10)
il<"'<in
ax<~~<an
with
Witt???) = 5313?.” |<1>> (11)

representing the n-tuply excited configuration. For example, the standard CCSD

equations for the singly and doubly excited cluster amplitudes till and tail";2 defining
the operators T1 and T2, respectively, look as follows:
QnHCCSDIQ) = 0, n = 1, 2, (12)
or, simply,
a — CCSD _
(Qi: IH |¢> "‘ 0 a (13)
<¢gl1i22|HCCSD|q>> = 0, 31 < 72, 0.1 < (12 , (14)
where
HCCSD = e—(T1+T2)H6T1+T2 : (HeT1+T2)C (15)

18

is the similarity-transformed Hamiltonian of the CCSD approach. The system of
CC equations, Eq. (8), is obtained by inserting the CC wave function |\Ilo), Eq.
(1), into the electronic Schréidinger equation H |\I!o) = EOIQO), premultiplying both
sides of the Schrbdinger equation on the left by e’T and projecting the resulting
connected cluster form of the Schrddinger equation, H IQ) = EolQ), with H defined
as e'THeT (or (H eT)C), in which T = T“), onto the excited configurations IQfl‘jjit"),
n = 1, . . . , m ,4, included in TM) (represented in Eq. (8) by the projection operators
@0-

Once the system of equations, Eq. (8), is solved for T”), we calculate the CC

energy corresponding to the approximate CC method A by using the expression

E3“ = <<I>IH<A>I<I>>. (16)

obtained by projecting the connected cluster form of the Schrbdinger equation, H IQ) =
EOIQ), where T = T”), onto IQ). For many-electron systems described by the Hamil-

tonians H containing up to two-body interactions, i.e., when

H = zgcpcq + i-vgcpcqc,cr, (17)

the CC energy E3“ is calculated using only the T1 and T 2 clusters, independent of

the truncation scheme used to define TM) (assuming that mA 2 2). We have

E3“ = <<I>IH|<I>)+<<I>IlH~(T1+ T2 + #81014». (18)

19

where H N = H — (QIH IQ) is the Hamiltonian in the normal-ordered form. The 23
and 12;; coefficients defining Hamiltonian H, Eq. (17), are the usual one- and anti-
symmetrized two-electron molecular integrals, (plzlq) and (pqlvlrs)A = (pqlvlrs) -
(pqlvlsr), respectively.

In the ground-state MMCC formalism of Kowalski and Piecuch,45"47 the energy
EéA) obtained from the standard single-reference CC calculations with some approx-
imate method A, Eq. (16), is improved upon by adding the suitably designed non-
iterative correction 63A) to Eff), which in the exact limit recovers the exact full CI
ground-state energy E0. The energy formula used by all MMCC methods can be

written as

334““ = 133’“ + 63"), (19)

where EéA) is the standard CC (e.g., CCSD) energy and 63A) is the MMCC nonitera-
tive energy correction, which is described below. In the exact MMCC theory, EyMCC
becomes the exact full CI energy E0. The approximate MMCC methods are obtained
by approximating the explicit many-body expression for 63") described below.

The MMCC formula for the noniterative correction 63A) entering Eq. (19) is
expressed in terms of the generalized moments of CC equations. We obtain these
readily available quantities by projecting the CC equations for method A on the
excited configurations that are not included in the calculations with method A. Thus,
if IQfl‘Lfl) designate the k-tuply excited configurations relative to IQ), the generalized

moments Mi‘lffgk(m,4), corresponding to CC method A, are defined as

20

Mil;::.‘:.(mA) = <<I>‘-‘""°"|H""I<I>>- (20)

:1...“

Note that for k g m A, the corresponding generalized moments Mill'jngmA) vanish.
This is a consequence of the fact that cluster operator T”) satisfies Eq. (8). This
implies that the exact correction 65f), defined as the difference between the full CI and
CC energies, must be expressed in terms of the generalized moments Mfggfgk (mA)

With k > mA- We obtain“5—"‘7’134

N n
63A’EEo-E3A’= Z Z <voIan.-k<mA>M.(mA)I<I>>/<wole"“’l<1>>, (21>

nzmA +1 kzmA-l-l

where

Cn—k(mA) = (BTW )n—k (22)

represents the (n — k)-body component of the CC wave operator 6““, defining the

CC approximation A, No) is the exact ground-state wave function, and

M.(mA)I<I>>stH<A’I<I>>= Z Mi‘.~::.‘:.(mA)I<I>°*-'“*. (23)

3'1...“

i1<m<ik
al<°"<ak

An elementary derivation of Eq. (21), based on applying the resolution of identity to

an asymmetric energy expression, termed the MMCC functional, i.e.

ACCM = mm - ElA’)eT""I<I>>/<wIeT""I<I>>, (24)

21

which gives the exact value of 68’” when IQ) is the full CI state |\Ilo), is shown in
Appendix A (the original proof is found in Appendix A of Ref. 46). An alterna-
tive derivation of Eq. (21), based on the analysis of the mathematical relationships
between multiple solutions of nonlinear equations representing different CC approxi-
mations (CCSD, CCSDT, etc), has been presented in Ref. 45.

Equation (21) is the basic equation of the MMCC formalism. According to this
equation, we can obtain the exact full CI energy E0 by determining the generalized
moments Mill'jfgk(m,4) with k > mA and by adding the resulting correction 6),"), Eq.
(21), to the CC energy E84). In particular, we can use Eq. (21) to improve the results
of the CCSD calculations (the m A = 2 case). In this case, we must calculate moments
Mi‘,’j_‘f§k(2) with k = 3 — 6 (it can be shown that moments M213; (2) with k > 6
vanish) and use these moments to calculate the noniterative correction 630081)) to the
CCSD energy. According to Eq. (21), the explicit formula for the correction 630051))
is

N min(n,6)
63‘3“” 2 E0 — Elms“ = Z 2: <on62. 0.42) Mk(2)|‘1’)/(‘I’o|eT‘+T’|‘I’>, (25)
71:3 k=3

where ESCCSD) is the CCSD energy, T1 and T2 are the singly and doubly excited

cluster components obtained in the CCSD calculations,
Cn-k(2) : (eTl+T2)n—k a (26)

and

22

Mk(2)|Q) = Z M“""* (2) (oar-etc), k = 3 — 6, (27)

chuck £1...“
i1<~~<ik
al<---<ak
with
Mi.‘;::i:.(2) = <<I>?:.::'.—‘,'." (HeT‘+T2)cI<I>> (28)

representing the generalized moments of the CCSD equations.

The exact MMCC corrections 6),"), Eq. (21), or 630051)), Eq. (25), have the form
of the complete many-body expansions involving all n-tuply excited configurations
with n = mA, . . . , N, where N is the number of electrons in a system (these n-
tuply excited configurations are represented in Eqs. (21) and (25) by the projection
operators Qn). With an exception of very small few-electron systems, the complete
many-body expansions of the type of Eq. (21) are not manageable. Thus, in order
to come up with the practical MMCC schemes, based on Eqs. (21) or (25), we

or 63cc”) at some reasonably low

must truncate the many-body expansions for 63")

excitation level m3. This leads to the so—called MMCC(mA, m3) schemes, in which

we calculate the energy as follows:“5““7v5151’3””)133

ES‘MCWmA, m3) = El“ + 60m, mg). (29)

where the formula for 60(mA, m3) is

50(mA,ma)= Z Z (‘I'olQnCn-k(mA)Mk(mA)I‘I’Vf‘l’oleTWI‘I’>- (30)

n=mA+1 kzmA-f-l

Non-zero values of the correction 60(mA,mB) are obtained only when m3 > mA.

23

-VI

When m3 = N and when I‘llo) is exact, one obtains the exact MMCC theory de-
scribed by Eq. (21). The renormalized and completely renormalized CCSD(T) and
CCSD(TQ) methods mentioned in Sect. 1 are the MMCC(mA,mB) schemes with
mA = 2 and m3 = 3 (the CCSD(T) case) or m3 = 4 (the CCSD(TQ) case).

The only issue that has to be addressed before using Eqs. (29) and (30) in prac-
tice is the form of the wave function |\Ilo) that enters the MMCC(mA,mB) energy
expressions. The wave function |\II0) that enters the formula for the exact correction
66A) or 630081)), Eqs. (21) and (25), respectively, is the full CI ground state, which we
usually do not know. Thus, in order to propose computationally feasible approaches
based on the MMCC theory, one must approximate |\II0) with some inexpensive ab
initio method. Several ways of approximating |\II0) in the MMCC(mA, my) energy
expressions, Eqs. (29) and (30), are possible. The renormalized and completely renor-
malized CCSD(T) and CCSD(TQ) methods employ the low-order MBPT expres-
sions to define I‘llo) in the MMCC(2,3) and MMCC(2,4) formulas.“5-“3’50’51'53'87'13‘m3‘5

45'51’52’132fl3“ including the CI-corrected

The CI-corrected MMCC approximations,
MMCC(2,6) approach developed in this work, use the wave functions I‘llo) origi-
nating from the limited CI calculations. Finally, the QMMCC methods,13“_137 also
developed in this work, use the truncated forms of the exponential, CC-like, wave
functions IQO). In general, it has been suggested that the wave functions |\Ilo) used in

the MMCC(mA, m3) formulas, Eqs. (29) and (30), do not contain higher-than-mg-

tuply excited components relative to the reference IQ).

24

4. The Existing MMCC(mA, m3) Approximations

We restrict our discussion to the MMCC(mA, m3) schemes with mA = 2, which
can be used to correct the results of the CCSD calculations. The MMCC(2,mB)
energy expressions can be obtained by setting mA = 2 in Eqs. (29) and (30) or by
truncating the summation over n in the exact Eq. (25) at n = mg. The simplest
approximations belonging to the MMCC(2,m3) hierarchy are the MMCC(2,3) and
MMCC(2,4) approaches.45-53'132—134’136 They are obtained by setting m3 = 3 (the
MMCC(2,3) case) or m3 = 4 (the MMCC(2,4) case) in Eqs. (29) and (30). In the

MMCC(2,3) approach, one computes the energy using the following expression:

Eg‘MCCQ, 3) = Egccsm + 60(2, 3) , (31)

where ESCCSD) is the CCSD energy and the correction 60(2, 3) is defined as

60(2, 3) = (‘I'olQaM3(2)|<1’)/ (‘I’oleT‘+T“|‘I>) . (32)

As in all MMCC(2,mB) schemes, the T1 and T2 clusters entering Eq. (32) are obtained
in the standard CCSD calculations. The M3(2)|Q) quantities entering Eq. (32) are

obtained using the formula,

M3(2)|<I>> = Z Mi‘.‘:::.(2)l<1>?:.-‘;::3>, (33>
i1<i2<i3
a1<02<a3
where
25

'I

Matias): <<I>‘-"-“:“3 (He mac I<I>> (34)

11:213

represent the triply excited moments of the CCSD equations (the CCSD equations
projected on the triply excited configurations IQfl‘gjgi’».

The MMCC(2,4) approach is an extension of the MMCC(2,3) scheme in which we
use both the triply excited moments of the CCSD equations, M23342), Eq. (34),
and the quadruply excited moments,

Mi‘,‘§§i§‘a.(2)= (‘1’9‘92"”‘|(H6T‘+T’)c|‘1’) . (35)

11321334

where IQa‘aiaf‘“) are the quadruply excited configurations relative to IQ). The MMCC (2,4)

’1'2‘334

energy is computed as follows:

EMMCC(2, 4): E “059’ +60 (2, 4) (36)

where

5 0(2 4) =<‘I’0I{Q3 Ms(2 )+ Q4 [M4(2) + T1M3(2)I}|‘I’>/<‘I’0I€TI+T2I‘D)- (37)

The triexcited moments M33342) enter Eq. (37) through quantities M3(2)|Q),

defined by Eq. (33). The quadruply excited moments Milligjgi‘a‘fl) enter Eq. (37)

through quantities M4(2)IQ), which are defined as

26

01020304 i1i2i3i4

M4(2)I<I>)= Z M‘l‘2‘3": (2)|<I>“‘“’°3“‘ . (38)

i1 <f2<53<54
a) <02 <03 <04

The explicit expressions for the triply and quadruply excited moments of the

CCSD equations, M“"‘3 (2) and M‘1i253“ (2), respectively, along with their higher-

010203 01020304

level pentuply and hextuply excited Mg;g;3,;4;3,5 (2) and Mgl'ggggfggao (2) counterparts,
which will be needed to formulate the CI-corrected and quadratic MMCC(2,5) and
MMCC(2,6) schemes (cf. Sect. 5), in terms of molecular integrals defining the Hamil-
tonian and cluster amplitudes defining T1 and T2, are shown in Appendix C. These
expressions are obtained by drawing all relevant Hugenholtz diagrams for the CCSD
equations projected onto triply, quadruply, pentuply, and hextuply excited configu-
rations, as shown in Appendix B.

Different types of the MMCC(2,3) and MMCC(2,4) approximations can be pro-
posed by using different choices of I\Ilo) in Eqs. (31)—(38). So far, only the CI-corrected
MMCC(2,3) and MMCC(2,4) methods, in which the wave fuction |\Ilo) is provided
by the multireference, CI-like, CISDt and CISth calculations, respectively, and

the renormalized and completely renormalized CCSD[T], CCSD(T), and CCSD(TQ)

methods, in which the wave function I‘IIO) is obtained from the second-order MBPT ex-

pressions, have beehave been developed. The CI—corrected MMCC(2,3) and MMCC(2,4)

methods are discussed in Sect. 4.1. The renormalized and completely renormalized
CCSD[T], CCSD(T), and CCSD(TQ) methods are discussed in Sect. 4.2. In both
cases, the mathematical and computational concepts are illustrated by examples of

applications to molecular systems.

27

'I

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

The CI-corrected MMCC(2,3) and MMCC(2,4) methods use simple forms of the
wave functions I‘IIO) in which higher-than-doubly excited components are defined
through active orbitals. Thus, one uses the wave functions I\Ilo) obtained in the
active-space CISDt“"""’1'52 and CISth52 calculations, which provide the qualitatively
correct description of bond breaking, to construct the 60(2, 3) and 60(2, 4) corrections
to the CCSD energy. In order to define the CISDt and CISth wave functions |\Ilo) for
the MMCC(2,3) and MMCC(2,4) calculations, one divides the available spin-orbitals
into core spin-orbitals (i1, i2, . . . ), active spin-orbitals occupied in the reference de-
terminant IQ) (ll, 12, . . . ), active spin—orbitals unoccupied in IQ) (A1, A2, . . . ), and
virtual spin-orbitals (a1, a2, . . .) (see Fig. 1). Once the active orbitals are selected
(typically, active orbitals correspond to the valence orbitals of a molecule), we define

the CISDt and CISth wave functions I\I!o) as follows:“5'51'52

I‘I’SISD‘) = (Co + Cl + 02 + Ca)|<1>>, (39)

I‘I’SISth) = (00 + 01 + C2 + C3 + C4)I(I)), (40)

where COIQ), CIIQ), and Cng) are the reference, singly excited, and doubly excited

components of IQEISD‘) and I913: ISth) and

A A
c3IQ) = Z c‘l‘l‘ga'IQflfa’z-s 3) , (41)
I1 >i2>53
01>02>A3
28

_ E : aiaaAaA4 aiaaAsA4
C4I¢> — chlziah I‘Dlllaiah >' (42)
It>ln>i3>i4
01>02>A3>A4

As one can see, the CISDt wave function IQSISD‘), Eq. (39), which is used to
calculate the 60(2, 3) correction of the CI—corrected MMCC(2,3) method is a limited
CI wave function including all singles and doubles from IQ) and a relatively small
set of internal and semi-internal triples containing at least one active occupied and
one active unoccupied spin-orbital index (cf. Eq. (41)). The CISth wave function
IQEISDW‘), Eq. (40), which is used to construct the correction 60(2,4) of the CI-
corrected MMCC(2,4) approach, is obtained by including all singles and doubles from
IQ), a relatively small set of internal and semi-internal triples containing at least one
active occupied and one active unoccupied spin—orbital index, and a relatively small
set of quadruples containing at least two active occupied and at least two active
unoccupied spin-orbital indices (cf. Eqs. (41) and (42)). The CI coefficients that
define the CISDt and CISth wave functions are determined variationally.

There are two main factors contributing to the relatively low computer costs of the
CI-corrected MMCC(2,3) and MMCC(2,4) calculations: (1) the use of active orbitals
in the process of constructing the CISDt and CISth wave functions I\Ilo) and (2)
the noniterative character of the 60(2, 3) and 60(2, 4) corrections. The most expensive
steps in the CISDt and CISth methods scale as NoNungn: and NgNznfinz, respec-
tively (No (Nu) is the number of active orbitals occupied (unoccupied) in IQ)). Since
No and Nu are typically much smaller than no and nu, respectively, the costs of the

CISDt and CISth calculations represent a small fraction of the parent CISDT (CI

29

singles, doubles, and triples) and CISDTQ (CI singles, doubles, triples, and quadru—
ples) calculations. In fact, the CISDt and CISth calculations are often only a few
times more expensive than the CCSD or CISD calculations. Thus, the use of a small
set of triples and quadruples in the CISDt and CISth calculations greatly reduces
the cost of the CI—corrected MMCC(2,3) and MMCC(2,4) calculations. The storage
requirements for the CISDt and CISth calculations are also greatly reduced because
only small fractions of the total numbers of triples and quadruples are being used
when the number of active orbitals is small. The numbers of triples and quadruples
considered in the CISDt and CISth calculations are NoNungnfi and Nfongnfi, re-
spectively, which is much less than the total number of triples and quadruples (ngnfi
and ngnfi, respectively).

Once the wave functions IQOCISD‘) and IQEISD“) are obtained by performing the
variational CISDt and CISth calculations, one has to construct the corrections
60(2, 3) and 60(2, 4) using Eqs. (32) and (37). The costs of constructing the 60(2, 3)
and 60(2, 4) corrections, which normally are ngnfi and ngni, respectively, are reduced
further when one uses the CISDt (the MMCC(2,3) case) and CISth (the MMCC(2,4)
case) wave functions I\Ilo) since only the generalized moments of the CCSD equations
corresponding to projections of these equations on the internal and semi-internal
triples and quadruples present in the CISDt and CISth wave functions have to be
considered.

The CI-corrected MMCC(2,3) and MMCC(2,4) approaches were tested by us by

calculating the PESs for the single bond breaking in the HF molecule and the si-

multaneous dissociation of the two O—H bonds in the H20 molecule.52 We used the

30

RHF reference and a DZ141 basis set (see Tables 1 and 2). The active spaces used
to define the CISDt and CISth wave functions employed in these tests were the
30, hr, 21r, and 40 orbitals for the HF molecule and the lbl, 3a], 1b, 4a1, 2b1, and
2b2 orbitals for the H20 molecule. In other words, we used small numbers of active
orbitals resulting from the valence shells of the relevant H, F, and O atoms. In the
case of the HF molecule (see Table 1), the 1.634 millihartree error in the CCSD result
relative to full CI at the equilibrium geometry, R = R,, = 1.7328 bohr (R is the
H—F internuclear separation), which increases to 12.291 millihartree in the dissoci-
ation region (R = 5B,), is reduced to 1.195 millihartree at R = R,, and to 3.255
millihartree at R 2 5R,” when the CI-corrected MMCC(2,3) approach is employed,
and to 1.207 and 3.066 millihartree at R = Re and R = 5&, respectively, when
the CI-corrected MMCC(2,4) calculation is performed. In the R = R,2 — 2B,, region,
the CCSD[T], CCSD(T), and CCSD(TQf) methods, which represent standard ways
of correcting the CCSD energies for the effects of triples and quadruples, perform
quite well. However, once the H—F bond is significantly stretched (R 2 3B,), the
errors resulting from the CCSD[T], CCSD(T), and CCSD(TQf) calculations dramat-
ically increase. For example, the CCSD(T) and CCSD(TQf) methods produce large
negative errors of —53.183 and —35.078 millihartree, respectively, at R = ML. The
CI-corrected MMCC(2,3) and MMCC(2,4) approaches are capable of reducing these
large errors to a few millihartree.

Interestingly enough, the CISDt method itself is not capable of providing a quan-
titative description of the bond breaking in HF. The 16—34 millihartree errors in the
CISDt calculations in the R > 2R8 region are substantial. In spite of the poor perfor-

31

ll

mance of the CISDt method, the CISDt-corrected MMCC(2,3) results are excellent
when the H—F bond is broken. This indicates the robust nature of the MMCC energy
corrections. We can use a relatively poor source of the wave function I\Ilo) in con-
structing the MMCC corrections 6 and yet obtain excellent results for bond breaking
when the MMCC corrections 6 are added to the (also poor) CCSD energies.

One of the reasons for the very good performance of the MMCC approximations
in the bond breaking region is the presence of the (\IloleT1+T2IQ) denominator in the
MMCC(2,3) and MMCC(2,4) energy expressions. These denominators increase with
R, damping the unphysically large negative triples corrections of the CCSD(T) and
similar approximations. For example, the CISDt (\IIOIeT1+T’IQ) denominator ranges
from ~1.0 at R = Re to 23—24 at R = 5R6 in the case of HF.

The HF molecule represents a case where T4 clusters play a very small role so
that the CISDt-corrected MMCC(2,3) approach, which describes the effects due to
singles, doubles, and triples only, suffices. The CISth—corrected MMCC(2,4) results
are not much more accurate than the CISDt-corrected MMCC(2,3) results in the HF
case. Let us, therefore, examine the performance of the CI-corrected MMCC(2,3) and
MMCC(2,4) methods in the case of the double dissociation of H20, where both the
T3 and T4 clusters play a significant role (cf. the CCSDT — CCSD and CCSDTQ —
CCSDT differences in Table 2, which are —11.544 and 2.319 millihartree, respectively
at R = 2B,; R is the 0—H bond length).

The standard noniterative CC approximations such as the CCSD[T], CCSD(T),
and CCSD(TQf) are incapable of dealing with the large T; and T4 effects character-

izing the situation where both O—H bonds in H20 are stretched by a factor of 2 (the

32

R = 2R, case in Table 2). Indeed, the unsigned errors in the CCSD[T], CCSD(T),
and CCSD(TQf) results, relative to full C1, are 11.220, 7.699, and 5.914 millihartree,
respectively, at R = 2Re. Only the complete inclusion of the T3 and T4 clusters via the
expensive CCSDTQ method reduces the error at R = 2R9 to 0.108 millihartree. The
full CCSDT approach produces a negative, —2.211 millihartree, error at this geometry,
which is an indication of the breakdown of the CCSDT method in the R > 2Re region,
due to the lack of important T4 clusters. The CCSD[T], CCSD(T), and CCSD(TQf),
work well in the R S 1.5Re region, but they give large negative errors at R = 212,,
which is an indication of the complete breakdown of these methods when both O-—H
bonds in H20 are broken.

The CI-corrected MMCC(2,3) and MMCC(2,4) approximations provide excel-
lent results at stretched geometries of H20, when compared to the standard CC
approaches, such as CCSD[T], CCSD(T), and CCSD(TQf). In fact, the simplest
CISDt-corrected MMCC(2,3) method, which requires a very small computer effort
in the process of constructing the triples energy correction 60(2, 3), outperforms the
expensive full CCSDT approach at R = 2R? With the choice of the active orbitals
mentioned earlier (the three highest-energy occupied orbitals, lbl, 3a1, and lbz, and
the three lowest-energy unoccupied orbitals, 4a1, 2b1, and 2b2), the description of the
entire R = Re — 2B,, region by the CISDt-corrected MMCC(2,3) approximation is
excellent, in spite of the poor performance of the underlying CISDt approach. The
MMCC(2,3) method reduces the large ~19 millihartree error in the CISDt result at
R = 1.5Re and the huge ~50 millihartree error in the CISDt energy at R = 2R,” to
the very small 2.407 and 1.631 millihartree errors, respectively.

33

my.

The CI-corrected MMCC(2,4) approach offers a more balanced description of the
simultaneous breaking of both O—H bonds in H20, when compared to the CI-corrected
MMCC(2,3) method. Indeed, the errors in the CI-corrected MMCC(2,3) results ini-
tially increase, from ~0.8 millihartree at R = Re to ~25 millihartree at R = 1.5Re,
and then decrease to ~16 millihartree at R = 2R9. This might be the first sign of
the breakdown of the CI-corrected MMCC(2,3) approximation at very large distances
R due to the lack of T4 corrections in the MMCC(2,3) energy expressions. This be-
havior should be contrasted with the slow and monotonic increase of errors in the
CI-corrected MMCC(2,4) results with increasing R (from 0—5 millihartree at R = Re
to 2.416 millihartree at R = 2Re).

The excellent performance of the CISth-based MMCC(2,4) approach in the cal-
culations for double bond breaking in H20 is due to several factors. One of the
factors is the fact that the CISth method, which is used to construct the wave
function I‘IIo) entering the MMCC(2,4) energy expressions, provides a much better
description of the simultaneous breaking of both O-H bonds in H20 than the CISDt
method used in the MMCC(2,3) calculations (see Table 2). The incorporation of the
quadruply excited moments of the CCSD equations in the MMCC(2,4) calculations
provides improvements, too, particularly in the R > 1.5R,3 region. As already men-
tioned, one of the strengths of the MMCC(2,3), MMCC(2,4), and similar methods is
the ability of these approaches to produce excellent results even when the wave func-
tions |\I!o) that are used to construct the relevant energy corrections are themselves
rather poor. In the case of double dissociation of H20, this is particularly true for

the CISDt-corrected MMCC(2,3) method. Another factor that helps to improve the

34

MMCC(2,3) and MMCC(2,4) results at stretched nuclear geometries is the presence
of the (\IloleT1+T’ IQ) denominators in the MMCC(2,3) and MMCC(2,4) energy expres-
sions. In the case of double bond breaking in H20, these denominators increase their
values from ~1.0 at R = R1, to 1.5—1.7 at R = 2Re. Without these denominators, the
MMCC(2,3) and MMCC(2,4) results would be almost as bad as the CCSD(T) and
CCSD(TQf) results.

We also tested the CI-corrected MMCC(2,3) and MMCC(2,4) methods in the
calculations for N2, which is the most difficult case involving triple bond breaking
(see Table 3 and Fig. 2). The active orbitals used by us in the CI-corrected MMCC
calculations included the 309, hr“, 27ru, 11rg, 21rg, and 3a,, orbitals correlating with
the 2p shells of the N atoms. As in the case of HF and H20, we used the DZ basis
set, for which the exact full CI energies are available.47

The N2 molecule is an example of the catastrophic failure of all standard single-
reference CC methods. The standard CCSD(T), CCSD(TQf), CCSDT, and
CCSDT(Q¢) methods work well in the equilibrium, R = Re, region, providing er-
rors as low as 0.323 and 0.010 millihartree at R = Re when the CCSD(TQf) and
CCSDT(Q;) methods are employed, but at R 2 2R, or R > 2R, the results of the
standard CC calculations are disastrous. The CCSD(T) method provides a curve
that goes hundreds of millihartree below the full CI curve (similarly for the CCSD
curve). The CCSD(TQf) and CCSDT(Q;) approaches, which include triples and
quadruples, produce curves that go hundreds of millihartree above the full CI curve
in the R > 1.75Re region. The CISDt-corrected MMCC(2,3) approximation reduces

these large errors somewhat, but the MMCC(2,3) curve at larger R values is com-

35

pletely unphysical. Clearly, one needs higher-than-triply excited clusters to obtain
a better description of the N2 curve. The corrections due to T4 clusters are present
in the CISth—corrected MMCC(2,4) method, but they are insufficient to provide a
high quality curve for N2. The CISth-corrected MMCC(2,4) curve is clearly much
better than all other curves provided by the standard CC methods, but the (~16) -
(—31) millihartree negative errors in the results of the CISth-corrected MMCC(2,4)
calculations in the R Z 2R6 region indicate the need for the inclusion of the effects
due to higher-than-quadruply excited determinants in the CI-corrected MMCC cal-
culations, if we are to achieve a highly accurate description of triple bond breaking
by the MMCC theory. New classes of the MMCC methods that can handle triple
bond stretching or breaking will be discussed in Sect. 5.

In summary, the CI-corrected MMCC(2,3) and MMCC(2,4) methods work very
well for single and double bond breaking, but they are not sufficiently accurate
to provide a good description of triple bond breaking. For cases involving single
or double bond dissociations, they are capable of eliminating the failures of the
CCSD(T), CCSD(TQf), and CCSDT(Q;) methods by reducing large errors in the
standard single-reference CC calculations to a few millihartree, but they are not ca-
pable of handling more complicated cases of multiple bond breaking. The CI-corrected
MMCC(2,3) and MMCC(2,4) methods are capable of providing excellent results in
spite of the use of the rather inaccurate CI approximations to construct the relevant
corrections 60(2, 3) and 60(2, 4). This stresses the robust nature of the MMCC theory.
Although the CI—corrected MMCC(2,3) and MMCC(2,4) approximations are capable
of providing high quality results, they require performing the a priori CI calculations

36

and choosing active orbitals. The MMCC (2,3) and MMCC(2,4) methods that provide
results of similar quality for single and double bond breaking, but which do not re-
quire performing any additional calculations or choosing active orbitals, are discussed

next.

4.2. The Renormalized and Completely Renormalized CCSD[T],
CCSD(T), and CCSD(TQ) Methods

In the renormalized and completely renormalized CC methods, the low-order
MBPT expressions are used to define the wave function I\Ilo) in the MMCC(mA, m3)
energy formulas.‘5"51'53'l3“’138 Thus, in the renormalized and completely renormalized
CCSD[T], CCSD(T), and CCSD(TQ) methods, MBPT(2)-like (second-order
MBPT-like) wave functions I\II0) are combined with the MMCC(2,3) and MMCC(2,4)
approximations introduced earlier (cf. Eqs. (31) and (36)). Just as their standard
CCSD[T], CCSD(T), and CCSD(TQf) counterparts, the renormalized and completely
renormalized CCSD[T], CCSD(T), and CCSD(TQ) approaches can be used to correct
the results of the CCSD calculations through simple noniterative corrections. The
difference between the standard and renormalized or completely renormalized CC
methods is that only the latter methods can be used to study bond breaking.

The completely renormalized CCSD(T) method (the CR-CCSD(T) approach),
which is the basic renormalized CC approximation, is an example of the MMCC(2,3)
scheme, in which the wave function I‘llo) is replaced by the very simple, MBPT(2)[SDT]-

37

like, expression,

(wCCSDW) = (1 + T1 + T2 + TI” + Z3)|<I>), (43)

where T1 and T2 are the singly and doubly excited clusters obtained by solving the

CCSD equations, the TI” term,

Tlill<1>> = R33’(VNT2)CI<I>>. (44)

in Eq. (43) is a CCSD analog of the connected triples contribution to the MBPT(2)
wave function, and

23|<I>> = Rli’vme) (45)

is the disconnected triples correction that is needed to distinguish between the (T)
triples corrections and the [T] corrections due to triples discussed below. In the
above expressions, R83) designates the three-body component of the MBPT reduced
resolvent and VN represents the two-body part of the Hamiltonian in the normal-

ordered form. The energy formula defining the CR-CCSD(T) method is (cf. Eq.

(32) )45—48,50,51 ,53,134,l36,138

EER‘CCSD‘T’ = Elms") + (\I’CCSD‘T’IQs M3(2)|<I>>/<‘I'CCSD‘T’|6T‘+T’IQ), (46)

where ESCCSD) is the CCSD energy, IQCCSDm) is defined by Eq. (43), and M3(2)IQ)
is the quantity defined in terms of triexcited moments M31333 (2), as shown in Eq.

(33). One can also consider the completely renormalized CCSD[T] method (the CR-

38

CCSD[T] approach), in which I‘IIO) in the MMCC(2,3) energy expression is replaced

by another MBPT(2)ISDTI-like wave function,

IQCCSDITI) = (1 + T1 + T2 + T3[2])|<I>), (47)

where T I21 is defined by Eq. (44). The only difference between the wave functions

IQCCSDITI) and IQCCSDITI) is in the Z3IQ) term, Eq. (45), which appears in the
CR-CCSD(T) theory, but is not included in the CR—CCSDIT] approach. The CR-

CCSDIT] energy expression is (cf. Eq. (32))“"’—“8~5°»"’1v53~134»136v138

EER'CCSD‘T] = E3003”) + (wCCSDITllca Ma(2)|<I>>/<‘PCCSD[T]|8T‘+T”|<I>>. (48)

Typically, the CR—CCSD(T) method provides somewhat better and more balanced
results than the CR-CCSDIT] approach (cf. Tables 1 and 2). It is important, however,
to discuss the CR-CCSDIT] theory to understand the connection between the CR-CC
methods and their higher-order QMMCC counterparts.

In addition to the CR—CCSDIT] and CR-CCSD(T) methods, it is worth con-
sidering the renormalized CCSD[T] and CCSD(T) approaches (the R-CCSDIT] and
R-CCSD(T) approaches). The R-CCSDIT] and R-CCSD(T) methods are obtained by
replacing the Milfgjgam) moments in the CR-CCSDIT] and CR—CCSD(T) formulas,
Eqs. (48) and (46), respectively, by their lowest-order estimates, i.e.
(Q2122? (VNT2)CIQ). The R-CCSDIT] and R-CCSD(T) energies are defined as

follOWS°‘5‘48~50~5l53.134

39

ES'CCSDIT‘ = E5003”) + (‘I’CCSDITIIQa(VNT2)c|‘I>)/<‘1'CCSD[TI|6T‘+T“I‘D) . <49)

Ei'CCSD‘T’ = EéCCSD’ + <wCCSD<THa (VNT2)c|<I>>/<\I’CCSD‘T’leT‘+T’|<I>>. (50)

In practice, the calculations of PESs involving bond breaking require using the CR-
CCSDIT] and CR-CCSD(T) methods rather than their simplified R-CCSDIT] and
R—CCSD(T) versions.“5"“5*48'5°'51’53:134 However, the R—CCSDIT] and R-CCSD(T) ap-
proaches allow us to understand the relationship between the standard and completely
renormalized CC approaches.

The relationship between the CCSD(T) (or CCSD[T]) and CR-CCSD(T) (or CR-
CCSDITI) approaches can be best understood by rewriting the CR—CCSDIT], CR-

CCSD(T), R—CCSDIT], and R—CCSD(T) energies, Eqs. (48), (46), (49), and (50),

respectively, in the following formzm‘h138
EER'CCSD‘“ = Elms") + NCRlTl/Dl‘”. (51)
EEK-CCSD(T) = E(()CCSD) + NCR(T)/D(T), (52)
Egl-CCSDIT] z Eéccsp) + Nm/DIT}, (53)
EE'CCSDCT) : EéCCSD) + N(T)/D(T), (54)

where the N GRIT], N GRIT), N [T], and N (T) numerators, entering the above expressions,

40

are defined as

NOR)“ = <<I>I<T12‘)*M3(2>I<I>>. (55)
NOW) = NCRIT1+ <<I>I(Za)*Ms(2)I<I>>. (56)
NIT] = <<I>|(T:I2])*(V~T2)cl<1>), (57)
N”) = NIT] + <<I>I(Za)*(vNT2)cI<I>>. (58)

and the DIT] and D“) denominators, representing the overlaps between IQCCSDITI)

and the CCSD ground state, are calculated as

and |\I,CC80(T))

pm a (\pCCSDlTIIeTI+T2|<I>) = 1+ (QITllTllQ) + (<1>|T,l (T2 + i712) |<1>)
+<<I>|(Tl2‘)*(T1T2 + 471814). (s9)
0““) a (\IICCSDITIIeT‘+T“|Q) = DIT] +(QIZ§(T1T2 + §T3)|<1>). (60)

The N [T] and N (T) numerators defining the R—CCSDIT] and R-CCSD(T) energies,

Eqs. (53) and (54), respectively, are related to the noniterative triples corrections
Eli” = <<I>I<T121>*(v~a)cl<1>>. (61)

and

Egsf‘ = (¢I(23)I(VNT2)CI‘I’>a (62)

defining the standard CCSD[T] and CCSD(T) energiesfmrff"31

41

Egcsnm : EéCCSD) + Em, (63)

and

EECSDm = EECSDIT] + Efl ___ EéCCSD) + E13] + E36}, (64)

respectively. We can write

NIT] .-. Eff], (65)
NIT) = 131;] + ESI. (66)

One can clearly see from the above equations that the R-CCSDIT] and R-CCSD(T)
approximations, which are obtained by simplifying the CR—CCSDIT] and
CR—CCSD(T) methods, reduce to the standard CCSD[T] and CCSD(T) approaches,
when the DIT] and 0m denominators in Eqs. (49) and (50) or (53) and (54) are
replaced by l. The approximation of the DIT] and Dm denominators by 1 is a jus-
tified step from the point of view of MBPT, since both denominators equal 1 plus
terms which are at least of the second order in perturbation VN (see Eqs. (59) and
(60); the lowest-order TJTZ term (excluding 1) in Eqs. (59) and (60) is at least of
the second order in VN). The presence of the Dm and Dm denominators in Eqs.
(51)-(54) is not important at equilibrium geometries, where these denominators are
usually very close to 1. However, these denominators are essential for improving the
results of the standard CC calculations in the bond breaking region. In the bond
breaking region, the values of DITI and DIT) are significantly greater than 1, damp-

ing the excessively large negative triples corrections to the CCSD energies, which

42

cause the failures of the standard CCSD[T] and CCSD(T) approximations at larger

internuclear separations/“5'47

From the above analysis, it follows that the R-CCSDIT], R-CCSD(T),
CR—CCSDIT], and CR-CCSD(T) methods can be viewed as the MMCC-based exten-
sions of the standard CCSD[T] and CCSD(T) approaches. Very similar extensions
can be formulated for other noniterative CC approaches. For example, one can use
the MMCC formalism to renormalize the CCSD(TQf) method of Ref. 34, in which
the correction due to the combined effect of triples and quadruples is added to the
CCSD energy. The resulting completely renormalized CCSD(TQ) (CR-CCSD(TQ))
approaches are examples of the MMCC(2,4) approximation, defined by Eq. (36). As
in the case of the CR-CCSDIT] and CR-CCSD(T) methods, one uses MBPT(2)-like
expressions to define the wave function I‘IIO) in the CR-CCSD(TQ) energy formulas
(this time, however, we include the leading terms due to triples as well as quadru-
ples in I‘IIO)). Two variants of the CR—CCSD(TQ) method, labeled by extra letters
“a” and “b”, are particularly useful. The CR—CCSD(TQ),a and CR-CCSD(TQ),b

energies are defined as follows:“5—“8~5°~51J34,136

EEK-CCSD(TQ),X = Eéccsn) + <q,CCSD(TQ),xl {Q3 443(2)
+Q4 [T1M3(2) + M4(2)I}I‘I’)/

(‘I’CCSD‘TQ”"|6T‘+T’I‘I’) (x = a. b), (67)

where M3(2)|Q) and M4(2)IQ) are the quantities expressed in terms of the triply and

quadruply excited moments Mig‘gjgaa) and M3123?“ 4(2), respectively, according to

43

Eqs. (33) and (38).

IWCCSD(TQ),a> = I‘IICCSDIT)) + %T2T2(1)|¢>, (68)

and

I‘I’CCSm TQ), b>_ IWCCSDH) > + ngI‘P) (69)

are the relevant MBPT(2)[SDTQ]-like wave functions that enter the CR-CCSD(TQ),a
and CR-CCSD(TQ),b corrections to CCSD energy. The IQCCSDlTI) wave function
entering Eqs. (68) and (69) is given by Eq. (43), and the T2”) operator entering
Eq. (68) is the first-order MBPT estimate of the cluster operator T2. In analogy to
the CR-CCSDIT] and CR-CCSD(T) methods, one can rewrite the formulas for the

CR-CCSD(TQ),a and CR-CCSD(TQ),b energies, Eq. (67), in a more compact form:

EEK-CCSD(TQM : ESCCSD) + NcmTQ),x/D(TQ),X (X = a, b), (70)
where
NCRITQ) =<1NCRT>+ ,(<p|T,,*(T, 0’) )[T1M3(2)+M4(2)]|<I>), (71)
NCR<TQ>vb= NCR<T>+( ,(<I>|( (T,1) [)T(1M32)+M4(2)]IQ), (72)
D(TQ),a = (q,CCSD(TQ),a|eT1+T2|(p)
= D<T+1<<I>IT1(T“))(1+T1 1TET2+1T1>I<I>> (73)

44

TI

and

D(TQ),b ___ <wCCSD(TQ),b I 8T1 +T2 IQ)

= D”) +1<<I>I(TJ)“(1T§+1T3T2 + 1.1T?)I<I>>. (74)

with N GRIT) and 0m defined by Eqs. (56) and (60), respectively. One can also in-
troduce the renormalized CCSD(TQ) (R-CCSD(TQ)) methods by considering the
lowest-order estimates of the M17333 (2) and M31333?“ (2) moments, which enter
Eq. (67) via quantities M3(2)|Q) and M4(2)IQ), and by dropping the higher-order
T1M3(2) term in Eq. (67). Several types of the R-CCSD(TQ) methods can be
considered.45—“8‘50'51’13“:136 In the R—CCSD(TQ)-1,x (x=a,b) methods, one replaces
443123239) by ((1)313? |(V~T2)c|<1>) and M33213?“ (2) by (@31‘3’133,“ IIVN(%T3+T2I2])lc|‘P),
where T3] is defined by Eq. (44). In the R-CCSD(TQ)-2,x methods, one replaces
M11.‘:;‘1.(2>by (42.1.1123 HVN(T2+1T1)101<1>> and 442.212.1212) by (44:11:13: (1V~T1>CI<I>>.

For example, the R-CCSD(TQ)-1,x energies, x = a, b, are calculated as follows:

Eg-CCSD(TQ)-l,x = ESCCSD) + N(TQ)-1,x/D(TQ),X (X = a, b), (75)

where

Nam-Ir = N“) + <<I>IT1(T1")*lv~(1T1+ T125101»). (76)

1
2

and

N‘qu‘lv" = N”) + 1<<I>I(T1)2(vN(1T1+ T125161»). (77)

45

with NIT) and TIA defined by Eqs. (58) and (44), respectively.

In analogy to the R-CCSDIT] and R-CCSD(T) approaches and their standard
CCSD[T] and CCSD(T) counterparts, it can be shown that the
R-CCSD(TQ)-1,a scheme, obtained by simplifying the CR-CCSD(TQ),x
(x=a,b) equations according to Eqs. (75) and (76), reduces to the factorized
CCSD(TQf) approach of Kucharski and Bartlett,34 when the D(TQ).a denominator
in the R-CCSD(TQ)-1,a energy expression, Eq. (75), is replaced by 1. Indeed, the

CCSD(TQf) energy can be given the following form:

where N (TQI'I'a‘ is defined by Eq. (76).34 Clearly, one can immediately obtain Eq.
(78) from the R-CCSD(TQ)-1,a energy formula, Eq. (75), by replacing DITQM in the
latter equation by 1. This very simple relationship between the R-CCSD(TQ)-1,a
and CCSD(TQf) methods, combined with the fact that all R—CCSD(TQ) methods
are obtained by simplifying the CR-CCSD(TQ) energy expressions, implies that the
R—CCSD(TQ) and CR—CCSD(TQ) approaches represent MMCC extensions of the
standard CCSD(TQf) method of Ref. 34. As in the case of the (C)R-CCSDIT] and
(C)R—CCSD(T) approaches, the presence of the D(TQ)'x denominators, Eqs. (73) and
(74), in the (C)R—CCSD(TQ),x (x = a, b) energy expressions is essential for improving
the results of the standard CCSD(TQf) calculations at larger internuclear separations.
The overlaps of the IQCCSDITQl’X) (x = a, b) and CCSD wave functions, defining the

DITQM denominators, increase with the internuclear distance, damping the exces-

46

sively large and, thus, completely unphysical values of the noniterative triples and
quadruples corrections resulting from the standard CCSD(TQf) calculations, which
cause the failure of the CCSD(TQf) approach in the bond breaking region.“47 r
The simple relationships between the renormalized and completely
renormalized CCSD[T], CCSD(T), and CCSD(TQ) methods and their standard coun-
terparts, discussed above, imply that the computer costs of the R-CCSDIT], R-
CCSD(T), CR-CCSDIT], CR—CCSD(T), R-CCSD(TQ)-n,x, and CR—CCSD(TQ),x
(n = 1,2, x = a, b) calculations are essentially identical to the costs of the stan-
dard CCSD[T], CCSD(T), and CCSD(TQf) calculations. In analogy to the stan-
dard CCSD[T] and CCSD(T) methods, the R—CCSDIT], R—CCSD(T), CR—CCSDITI,
and CR-CCSD(T) approaches are ngn: procedures in the noniterative steps involv-
ing triples and 11312,“, procedures in the iterative CCSD steps. More specifically, the
CR-CCSDIT] and CR—CCSD(T) approaches are twice as expensive as the standard
CCSD[T] and CCSD(T) approaches in the steps involving noniterative triples cor-
rections, whereas the costs of the R-CCSD[T] and R—CCSD(T) calculations are the
same as the costs of the CCSD[T] and CCSD(T) calculations.138 The memory and disk
storage requirements characterizing the R-CCSDIT], R-CCSD(T), CR—CCSDIT], and
CR-CCSD(T) methods are essentially identical to those characterizing the standard
CCSD[T] and CCSD(T) approaches (see Ref. 138 for further details). In complete
analogy to the noniterative triples corrections, the costs of the R—CCSD(TQ)-n,x cal-
culations are identical to the costs of the CCSD(TQf) calculations (the CCSD(TQf)
method is an ngn: procedure in the triples part and an ngni procedure in the noniter-

ative steps involving quadruples). Again, the CR—CCSD(TQ),x approaches are only

47

 

twice as expensive as the CCSD(TQf) method in the steps involving the noniterative
energy corrections due to triples and quadruples.

These relatively low computer costs, combined with the ease-of-use of the com-
pletely renormalized CCSD[T], CCSD(T), and CCSD(TQ) methods that can only
be matched by the standard CCSD[T], CCSD(T), and CCSD(TQf) approaches and
with the fact that the CR—CCSDIT], CR—CCSD(T), and CR—CCSD(TQ),x approaches
remove the failing of the CCSD[T], CCSD(T), and CCSD(TQf) methods at larger in-
ternuclear separations, make the CR—CCSDITI, CR—CCSD(T), and CR—CCSD(TQ),x
approaches attractive alternatives to the existing multireference methods, such as
multireference CI (MRCI). The MRCI methods describe bond breaking in a cor-
rect manner, but the effort involved is significantly larger and one has to define
many additional elements, such as reference configurations, active orbitals, etc. to
set up multireference calculations. None of these elements has to be considered in the
CR—CCSDITI, CR-CCSD(T), and CR-CCSD(TQ),x calculations. As shown in ear-
lier works,“5—“8’5°’51“53:13“138 the CR-CCSDIT], CR-CCSD(T), and CR—CCSD(TQ),x
methods provide a highly accurate description of molecular PESs that can compete
with the results of multireference calculations. The R-CCSDIT], R-CCSD(T), and
R-CCSD(TQ)-n,x approaches are capable of improving the results of the standard
CCSD[T], CCSD(T), and CCSD(TQf) calculations for the intermediate stretches of
chemical bonds, but the overall behavior of the R-CC methods at larger distances
is not as good as the behavior of the CR-CC methods, so that the use of the R-CC
approaches for studies of entire molecular PESs is not recommended. The CR-CC

methods are more trustworthy in this regard.

48

To illustrate the performance of the completely renormalized CCSD[T], CCSD(T),
and CCSD(TQ) methods, we review the results of the CR-CCSDITI, CR-CCSD(T),
and CR-CCSD(TQ) calculations for the HF, H20, and N2 molecules. As shown in
Table 1, already the simple CR—CCSDITI and CR—CCSD(T) methods provide con-
siderable improvements over the poor description of the potential energy curve for
the HF molecule by the standard CCSD, CCSD[T], and CCSD(T) approaches. The
CR—CCSDIT], CR-CCSD(T), and CR-CCSD(TQ) methods are particularly effective
at larger internuclear separations. For example, the 11.596 millihartree error in the
CCSD result and the large —38.302, —24.480, and —18.351 millihartree errors in the
CCSD[T], CCSD(T), and CCSD(TQf) results, respectively, at R 2 3R, reduce to
2.508, 2.100, 0.425, and 0.316 millihartree when the CR-CCSDIT], CR—CCSD(T),
CR—CCSD(TQ),a, and CR—CCSD(TQ),b methods are employed (cf. Table 1). The
reduction of errors in the CCSD[T], CCSD(T), and CCSD(TQf) results at R = 5B,,
offered by the CR-CCSDIT], CR—CCSD(T), and CR—CCSD(TQ) methods, is even
more impressive. The huge —75.101, —53.183, and —35.078 millihartree errors, relative
to full CI, obtained with the CCSD[T], CCSD(T), and CCSD(TQf) calculations, re-
spectively, are reduced to 3.820, 1.650, 0.454, and 0.689 millihartree when one switches
to the CR-CCSDIT], CR-CCSD(T), CR-CCSD(TQ),a, and CR-CCSD(TQ),b meth-
ods. The CR-CCSDIT], CR-CCSD(T), CR—CCSD(TQ),a, and CR-CCSD(TQ),b re-
sults for geometries near the equilibrium (R m Re) are as good as the CCSD[T],
CCSD(T), and CCSD(TQf) results, producing very small errors not exceeding 0.5
millihartree. The excellent results obtained with the CR—CCSDIT] and CR—CCSD(T)

methods for the HF molecule imply that already the simplest CR-CC approximations

49

 

are sufficient to obtain accurate PESs for single bond breaking. The CR—CCSD(TQ)
approaches provide further improvements in this case, but their use is not essential
to obtain high quality results.

The CR—CCSDIT] and CR—CCSD(T) methods are also sufficiently accurate for
cases involving a simultaneous dissociation of two single bonds. This is illustrated
in Table 2, where we consider a simultaneous breaking of both O—H bonds of the
H20 molecule. As shown in Table 2, the CR-CCSDIT], CR—CCSD(T), and CR-
CCSD(TQ),x (x=a,b) methods reduce the large —11.220, —7.699, and —5.914 mil-
lihartree, errors in the CCSD[T], CCSD(T), and CCSD(TQf) results at R = 2R,2
to the small 1.163, 1.830, 1.461, and 2.853 millihartree errors, respectively. The <
2.5 millihartree errors, relative to full CI, obtained with the CR-CCSDIT] and CR-
CCSD(T) methods in the entire R = R, — 2R, region for the doubly dissociating
water molecule indicate that there is no apparent need to use the higher—level CR-
CCSD(TQ),x (x=a,b) methods in cases involving a simultaneous stretching of two sin-
gle bonds. However, the CR—CCSD(TQ),x (x=a,b) approaches provide a more stable
description of the double dissociation of H20, when compared with the CR-CCSDIT]
and CR-CCSD(T) methods. Indeed, the initially small, 0.560 millihartree, error in
the CR-CCSDITI result at R = Re increases to 2.053 millihartree at R = 1.5Re, to
decrease again to 1.163 millihartree at R = 2R, (see Table 2). A similar behavior
is observed for the CR-CCSD(T) approach. The non-monotonic error changes in
the CR-CCSDITI and CR—CCSD(T) results are the first signs of the possible even-
tual breakdown of these methods in the R > 2R, region. The CR-CCSD(TQ),x

(x=a,b) methods behave much better in this regard, since the small errors in the

50

 

CR—CCSD(TQ),x energies, relative to full CI, increase monotonically with the O—H
separation.

Although the CR—CCSDIT] and CR-CCSD(T) approaches seem to be accurate
enough to describe the simultaneous stretching of two single bonds, the stretching
or breaking of multiple bonds requires using the higher-order CR—CC theories, such
as CR—CCSD(TQ),b. For example, when one applies the CR—CCSD(T) method to
triple bond breaking in the N2 molecule, the resulting potential energy curve has a
hump at the N—N separation R z 1.75Re and the CR-CCSD(T) curve is located
below the full CI curve at larger N —N separations (see Table 3 and Fig. 2). The N2
molecule is characterized by large T3 as well as T4 effects, even at the equilibrium
geometry R = Re. It is quite possible that higher-than-quadruply excited clusters
play an important role when the N—N internuclear separation becomes large. The
apparent importance of the higher-order clusters in the N2 case, which cannot be
easily approximated using the conventional MBPT or CC arguments, explains why
the CR—CCSD(T) approach fails in this case.

The results of the CR-CCSD(TQ),a and CR—CCSD(TQ),b calculations for the
DZ model of N2 are shown in Table 3 and Fig. 2. We can immediately see that
these methods, particularly variant “b” of the CR—CCSD(TQ) approach, provide
significant improvements in the standard CC and CR-CCSD(T) results. The CR-
CCSD(TQ),b method, which is a relatively simple modification of the conventional
CCSD(TQf) approach and which uses elements of MBPT to estimate the effects due
to T3 and T4, provides a potential energy curve which is quite close to the exact

full CI curve. For example, at R = 2.25Re, the huge 334.985 millihartree error in

51

the CCSD(TQf) result and the 133.313 millihartree error in the CR-CCSD(T) re-
sult reduce to 14.796 millihartree, when the CR-CCSD(TQ),b method is employed
(see Table 3). The CR-CCSD(TQ),b curve is located above the full CI curve in
the entire R = 0.75Re — 2.25Re region and almost all pathologies observed in the
standard single-reference CC and CR-CCSD(T) calculations are eliminated when the
CR-CCSD(TQ),b method is employed. The huge humps on the CCSD, CCSD(T),
CR-CCSD(T), and CCSDT curves and a nearly singular behavior of the CCSD(T),
CCSD(TQf), and CCSDT(Qf) approaches at large R values are almost entirely elim-
inated by the CR—CCSD(TQ),b approach (cf. Table 3 and Fig. 2). Although there is
a bump on the CR—CCSD(TQ),b curve, the size of this hump, as measured by the dif-
ference between the CR-CCSD(TQ),b energies at the maximum corresponding to the
hump and at R = 2.25Re, is small (~4.9 millihartree). There are 10—25 millihartree
differences between the CR-CCSD(TQ),b and full CI energies at the intermediate val-
ues of R, but the fact that one can obtain a reasonably accurate potential energy curve
for the triply bonded N2 molecule, which is also located above the full CI curve in the
entire R = 0.75Re — 2.25Re region, with the ease of use characterizing the standard
noniterative CC “black-boxes” of the CCSD(TQf) type, is an encouraging finding.
However, in spite of the remarkable improvements offered by the CR-CCSD(TQ),b
method in the N2 case, the 10—25 millihartree errors in the region of the intermediate
N—N separations are not sufficiently small for high accuracy calculations. It would
also be great to be able to eliminate or, at least, further reduce the ~4.9 millihartree
hump in the CR—CCSD(TQ),b curve. As shown in Sect. 5.1, the hump on the CR-

CCSD(TQ),b curve can be completely eliminated by the new CISthph-corrected

52

 

MMCC(2,6) approach, which is an extension of the CISth-corrected MMCC(2,4)
method described in Sect. 4.1 that incorporates the corrections due to pentuply and
hextuply excited moments of the CCSD equations in addition to triply and quadru-
ply excited moments present in the MMCC (2,4) approaches. The CISthph-corrected
MMCC(2,6) method provides a virtually exact description of the N2 potential energy
curve. Similar improvements in the CR—CCSD(TQ),b description of the triple bond
breaking in N; are obtained when we use the quadratic MMCC approach, developed

in this thesis work and discussed in Sect. 5.2.

53

5. The New MMCC(mA, m3) Approaches for Multiple Bond
Breaking

The MMCC(2,3) approximation, on which the CISDt-corrected MMCC(2,3)
method and the (C)R-CCSDIT] and (C)R—CCSD(T) approaches are based, uses only
the triply excited CCSD moments, M“"‘3 (2), corresponding to projections of the

010303
CCSD equations on triply excited configurations IQfl‘SZ-g"), whereas the MMCC(2,4)
scheme, on which the CISth—corrected MMCC(2,4) approach and the R-CCSD(TQ)-
n,x and CR-CCSD(TQ),x (n=1,2, x=a,b) methods are based, uses the triply and
quadruply excited moments, Miljgggam) and M31233; 4(2), respectively, correspond-
ing to projections of the CCSD equations on triply and quadruply excited configura-
tions. The presence of the triply and quadruply excited moments in the MMCC(2,3)
and MMCC(2,4) energy expressions (cf. Eqs. (31) and (36) with 60(2, 3) and 60(2,4)
defined by Eqs. (32) and (37), respectively) is usually sufficient to obtain a highly
accurate description of single and double bond dissociation, but triple bond disso-
ciation and other complicated types of multiple bond breaking may require a more
accurate treatment. From the numerical experiments with the MMCC-based renor-
malized and completely renormalized CCSD[T], CCSD(T), and CCSD(TQ) methods
and their CI-corrected counterparts, it follows that it may be necessary to incorpo-
rate the pentuply and hextuply excited moments of the CCSD equations, M31732; (2),
k = 5 and 6, respectively, if we are to formulate accurate MMCC approximations

.....

for multiple (e.g. triple) bond breaking. The inclusion of the M""'3"'5 (2) and

0102030405

31:213141515 ' ' ' i1i2i3 i1i2i3i4
Malaaaaa‘awo (2) moments, in addition to the M010203(2) and M01020304(2) moments

54

that are already present in the MMCC(2,3) and MMCC(2,4) approaches, leads to the
MMCC(2,5) and MMCC(2,6) approximations.

The MMCC(2,5) and MMCC(2,6) approximations are obtained by fixing the value
of mA in Eqs. (29) and (30) at 2 and by setting m3 = 5 (the MMCC(2,5) case) and
m3 = 6 (the MMCC(2,6) case) in the resulting expressions. The formulas defining

the MMCC(2,5) and MMCC(2,6) approaches are as followszl34_137

El‘MCCe. 5) = E1005” + 112.5) . (79)

133414000, 6) = EICCSD’ + 60(2, 6), (80)

where

60(2, 5) = (‘I’olan 1143(2) + Q4 [M4(2) + T1M3(2)I
+Q5 [II/15(2) + T1M4(2) + (T2 + §T12)M3(2)]}I‘I’)

/<‘PoleT‘+T’ I‘P) . (81)

50(2) 6) = (‘I’olan 1143(2) + Q4 IM4(2) + T1M3(2)I
+Q5 [945(2) + T1M4(2) + (T2 + %TE)M3(2)I
+Q6 [M6(2) + T1M5(2) + (T2 + §T12)M4(2)

+(T1T2 + 171311439)”I‘I’)/<‘I’0I€T‘+T2Iq’)- (82)

The quantities M3(2)IQ) and M4(2)IQ) in Eqs. (81) and (82) are defined by Eqs. (33)

55

and (38), respectively. The M3(2)IQ) and M4(2)IQ) quantities are expressed in terms

of triply and quadruply excited moments, .M‘m‘3 (2) and M“"‘3“ (2), respectively.

010203 61020304

The new quantities M5(2)|Q) and M6(2)IQ) are defined as

Ms(2)l<I>)= 2 MW“: (2)l<I>‘-“-“"’-“‘-’“‘“”>, (88)

0102030405 11 22133415
i1<i2<i3<i4<i5
01<02<03<a4<05

and
Ms(2)I<I>) = Z 442.212.22.824»lest-21:11:28“ , (84)
i1<i2 <f3<i4 (i5 (i6
a1<az<03<04<05<06
where
442.313.222.42) = <¢ff§§i§ff§1°5 (HeT‘+T’)c|<I>), (85)
and
Mixtjzifixmz) = (811213.123; (HeT’+T’)CI‘I’), (86)

are the pentuply and hextuply excited moments of the CCSD equations, obtained
by projecting these equations on the pentuply and hextuply excited configurations,
IQfl‘,:§:§:};““) and IQflli‘ggffig‘fga“), respectively.

As in all MMCC(2,mB) energy formulas, the MMCC(2,5) and MMCC(2,6) ener-
gies are expressed in terms of the generalized moments of the CCSD equations and
the ground-state wave function IQO). The generalized moments of the CCSD equa-
tions can be easily determined using the converged CCSD cluster amplitudes, till and
ti152 respectively (see Appendix B), whereas IQO) must be replaced by some approx-

0102’

imate form of the ground-state wave function. If we want to use the MMCC(2,5)

56

and MMCC(2,6) approximations in practice, we must use a simple form of I‘Ilo)
that facilitates the calculations. In order to satisfy this “simplicity” requirement,
we have formulated two classes of the MMCC(2,5) and MMCC(2,6) methods: (i)
the CI-corrected MMCC(2,5) and MMCC(2,6) approaches, which can be viewed as
extensions of the CI-corrected MMCC(2,3) and MMCC(2,4) methods where a priori
limited CI calculations are performed to obtain I‘llo), and (ii) the quasi-variational
and quadratic MMCC approaches, which are extensions of the CR-CCSD(T) and
CR-CCSD(TQ) methods where the exponential, CC-like, form of the wave function
|\I'0) is used to construct the 60(2, 5) and 60(2, 6) corrections. These two new classes

of the MMCC approximations are discussed in Sects. 5.1 and 5.2, respectively.

5.1. The CI—corrected MMCC(2,5) and MMCC(2,6) Approximations

The CI-corrected MMCC(2,3) and MMCC(2,4) approaches, described in Sect.
4.1, provide very good results for single and double bond breaking. However, when
triple bonds (e.g. in N2) are broken, the results of the CI-corrected MMCC(2,3) and
MMCC(2,4) calculations are less impressive. As shown in Table 3 and Fig. 2, the rel-
atively small (a few millihartree) errors in the CISth-corrected MMCC(2,4) results
for the N—N separations R < 1.512., (R = Re is the equilibrium N—N bond length)
become relatively large (> 10 millihartree) for R _>_ 1.5Re. At large internuclear
separations, such as R = 2Re, the CISDt-corrected MMCC(2,3) approach and the
CISth—corrected MMCC(2,4) method suffer from non-variational collapse, similar to

57

that plaguing the standard CCSD(T) approximation, although the ,S 30 millihartree
absolute errors in the CISth-corrected MMCC(2,4) results for R S 2.25R,3 are an
order of magnitude smaller than the analogous errors characterizing the CCSD(T)
and CCSD(TQf) results (cf. Table 3). In cases like this, we have to switch to the
CI-corrected MMCC(2,6) method in which, in addition to moments Mill‘gjgaa) and

Mi‘ligjii‘mm), Eqs. (34) and (35), respectively, we consider moments Myl‘ggggfasa)

and Myl'gggggggaoa), Eqs. (85) and (86), respectively, corresponding to projections
of the CCSD equations on pentuply and hextuply excited configurations. As we will
see, the CI-corrected MMCC(2,5) approximation is not sufficiently accurate for triple
bond breaking in the region of larger internuclear separations, although the situation

dramatically changes when we consider the quadratic version of the quasi-variational

MMCC(2,5) theory described in Sect. 5.2.

5.1.1. Theory

In analogy to the CI-corrected MMCC(2,3) and MMCC(2,4) methods, the wave
functions |\II0) entering the MMCC(2,5) and MMCC(2,6) energy corrections, Eqs.
(81) and (82), respectively, are obtained by solving the limited CI equations. Thus,
in the CI-corrected MMCC(2,5) approach, the wave function |\Ilo) in Eq. (81) is
replaced by the wave function obtained in the CISthp calculation. The CISthp
wave function is calculated by solving the CI eigenvalue problem with all singles and
doubles, internal and semi-internal triples and quadruples defined by Eqs. (41) and

58

(42), and internal and semi-internal pentuply excited configurations defined by the
excitation operator

Cs|<I>> = Z c““’""“‘|<1>i‘,‘i‘:£2£;“). (87)

I1 1313 i4 i5

11 >12 >13 >i4 >i5
01>02 >As>Aq>A5

where 11, 1;, and 13 indices represent active occupied spin-orbitals and A3, A4, and
A5 indices represent active unoccupied spin-orbitals (cf. Fig. 1). In a similar manner,
the CI-corrected MMCC(2,6) method requires that we first perform the CISthph
calculation by solving the CI eigenvalue problem with all singles and doubles, internal
and semi-internal triples and quadruples defined by Eqs. (41) and (42), internal and
semi-internal pentuples defined by Eq. (87), and internal and semi—internal hextuply
excited configurations defined by the excitation operator

A A A A A A A
eels) = 2 8:81.84.“ “Isaiah‘s.“ °>, (88)

11 >12 >13)“ >i5>i6
61 >02>A3>A4>A5>Ao

where 11, lg, 13, and L. are active occupied spin-orbitals and A3, A4, A5, and A3
are active unoccupied spin-orbitals. The CISthph-corrected MMCC(2,6) energy
expression is obtained by replacing the wave function |\Ilo) in Eq. (82) by the wave
function obtained in the CISthph calculation, as described above.

As in the case of the CI-corrected MMCC(2,3) and MMCC(2,4) approaches, the
use of the active orbitals in defining the selected pentuples and hextuples entering Eqs.
(87) and (88) significantly reduces the computer costs of the CI-corrected MMCC(2,5)
and MMCC(2,6) calculations, since we do not have to determine all pentuply and hex-

tuply excited moments of the CCSD equations. It is sufficient to calculate moments

59

Milizlfiilszzl and M:§:§k?;i‘ste(2), corresponding to projections of the CCSD
equations on a relatively small set of internal and semi-internal pentuply and hex-
tuply excited configurations of the |¢§if§figx“) and l<I>flf§fi$§£°Afi types. Our
experience with the CI-corrected MMCC(2,5) and MMCC(2,6) methods indicates
that these are the only types of projections of the CCSD equations on the pentuply
and hextuply excited configurations that matter in calculations of triple bond break-

ing. This will be seen by analyzing the results of the CI-corrected MMCC(2,5) and

MMCC(2,6) calculations for the triple bond breaking in the N2 molecule.

5.1.2. Examples of Applications

The CI-corrected MMCC(2,5) and MMCC(2,6) methods developed in this thesis
project were implemented using the explicit expressions for the generalized moments
of the CCSD equations given in Appendix B. As in the case of the MMCC(2,3) and
MMCC(2,4) calculations, the T1 and T2 cluster amplitudes were obtained with the
orthogonally spin-adapted CCSD method of Ref. 29 using computer programs devel-
oped by Piecuch and Paldus. The required RHF calculations and the transformation
from the atomic to molecular basis set were performed with the GAMESS codes.139

We begin our discussion of the CI—corrected MMCC(2,5) and MMCC(2,6) results
with the DZ model of H20. In this case, the simultaneous dissociation of both O—H
bonds is reasonably well described by the CI-corrected MMCC(2,3) and MMCC(2,4)

approaches (see Table 2), but it is interesting to examine if one can further reduce the

60

1—2 millihartree errors resulting from the CI-corrected MMCC(2,3) and MMCC(2,4)
calculations at stretched nuclear geometries of H20 by performing the higher-level
MMCC(2,5) and MMCC(2,6) calculations.

The results of the CI-corrected MMCC(2,5) and MMCC(2,6) calculations for the
double dissociation of H20 are shown in Table 4. As in the case of the CI-corrected
MMCC(2,3) and MMCC(2,4) calculations discussed in Sect. 4.1, the active space used
in the CI-corrected MMCC(2,5) and MMCC(2,6) calculations consisted of the 1b;,
3a1, lbz, 4a], 2b1, and 2b2 valence orbitals. As one can see, the 0811—2407 millihartree
errors in the CISDt-corrected MMCC(2,3) results and the 0501—2416 millihartree
errors in the CISth—corrected MMCC(2,4) results in the R = Re — 2Re region reduce
to 0421—0730 millihartree when we switch to the CISthp-corrected MMCC(2,5)
approach (see Table 4). The CISthph-corrected MMCC(2,6) method reduces the
very small errors resulting from the CISthp-corrected MMCC(2,5) calculations even
further, to 0.417 millihartree at R = Re, 0.477 millihartree at R = 1.5Re, and 0.538
millihartree at R = 2R6.

These excellent results obtained with the CI-corrected MMCC(2,5) and MMCC(2,6)
approaches are, at least in part, a consequence of the relatively good description of
the double dissociation of H20 by the CISthp and CISthph methods that are used
to generate wave functions |\Ilo) for constructing the MMCC(2,5) and MMCC(2,6)
corrections, 60(2, 5) and 60(2, 6), respectively. The CISthph method is particularly
good in this case, producing the relatively small 1.922—2.600 millihartree errors in the
entire R = Re — 2R8 region. It is interesting to observe, though, that the MMCC

theory is capable of reducing the small 2.628—3.7 32 millihartree errors in the CISthp

61

results and even smaller 1.922—2.600 millihartree errors in the CISthph results by a
relatively large factor of 4—6, once the (318th and CISthph wave functions |\Ilo)
are inserted into the MMCC(2,5) and MMCC(2,6) energy expressions, Eqs. (79) and
(80), respectively. One might expect that the use of these high-quality wave functions
lilo) in the MMCC calculations would only lead to small improvements in the results
for H20. This is not the case. The reduction of errors in the CISthp and CISthph
results for the double dissociation of H20 by a factor of 4—6 is not as impressive as
the reduction of large errors in the CISDt results, when the rather poor CISDt wave
function I‘Ilo) is inserted into the MMCC(2,3) energy formula (of. Tables 2 and 5),
but the reduction of the ~ 24 millihartree errors in the CISthp and CISthph
results to as little as 04—07 millihartree is very encouraging.

We have already seen that neither the CI-corrected MMCC(2,3) theory nor its
CI—corrected MMCC(2,4) extension is accurate enough to study the most challenging
types of multiple bond breaking, including the triple bond breaking in N2 (see Table
3 and Fig. 2). The N2 molecule is characterized by large T3 and T4 effects and, for
stretched nuclear geometries, by sizable contributions due to higher-than-quadruply
excited clusters, in addition to the huge T3 and T4 effects. The effect of T3 clusters at
the equilibrium geometry (R = Re) and for the DZ basis set in N2, obtained as the
difference of the CCSDT and CCSD energies, is —6.182 millihartree (see Tables 3 and
5). The difference of the CCSDTQ and CCSDT energies, which measures the effect
of T4 clusters, is —1.912 millihartree.47 The full CCSDTQ method is virtually exact
in this case. For geometries near the equilibrium, the T3 and T4 effects are accurately

described by the perturbative CCSD(T) and CCSD(TQf) approaches. For example,

62

the difference between the CCSD(T) and CCSD energies of —6.133 millihartree at
R = Re is virtually identical to the —6.l82 millihartree difference between the CCSDT
and CCSD energies. However, the situation becomes much more complicated when
the N—N bond is stretched. The 8.289 millihartree difference between the full CI and
CCSD energies at R = Re, which measures the combined effect of all higher-than-
doubly excited clusters, rapidly increases to 33.545 millihartree at R = 1.5Re (see
Tables 3 and 5). At R z 1.75Re, the CCSD potential energy curve has an unphysical
hump, and for the N—N distances greater than 3.74 bohr, the CCSD potential energy
curve goes significantly below the exact full CI curve (see Tables 3 and 5 and Figs.
2 and 3). Similar remarks apply to the CCSDT curve, which has a well-pronounced
hump at larger N—N separations. All of these imply that we need to incorporate
T4 clusters to obtain a correct description of the N2 curve. It is quite likely that
higher-than-quadruply excited clusters play an important role too when R becomes
large, since triple bond breaking in N 2 requires at least some hextuple excitations in
a CI sense. This can be seen by comparing the CISthph and CISthp energies.
The CISthp approach, which neglects the hextuple excitations altogether, provides
a significantly worse description of bond breaking in N2 than the CISthph method
(see Table 5). It can also be shown that the difference between the CISDTQ (CI
singles, doubles, triples, and quadruples) and full CI energies at R = 2Re is almost 40
millihartree,106 indicating the possible need for higher-than-quadruply excited clusters
at larger N —N separations.

As mentioned in Sect. 4, the CCSD(T) and CCSD(TQf) approximations, which

describe the effects due to triply and quadruply excited clusters perturbatively, lead

63

to completely erroneous results at larger N —N distances (cf. Tables 3 and 5 and Figs.
2 and 3). The relatively small 2.156 millihartree difference between the CCSD(T) and
full CI energies at R = Re increases (in absolute value) to 51.869 millihartree at R =
1.75Re, 246.405 millihartree at R = 2R“ and 387.448 millihartree at R = 2.25Re. The
CCSD(TQf) approach fails, too, giving 92.981 and 334.985 millihartree errors at R =
2He and 2.25Re, respectively. As shown in Figs. 2 and 3, the potential energy curves
obtained in the CCSD(T) and CCSD(TQf) calculations are completely pathological.
At larger internuclear separations, the CCSD(T) curve is located significantly below
the full CI curve, and there is a well-pronounced bump on the CCSD(T) curve for
the intermediate values of R. The CCSD(TQf) potential energy curve is located
significantly above the full CI curve. None of the standard single-reference methods
of improving the poor CCSD results at larger N—N separations, based on adding
the noniterative corrections due to triples and quadruples to the RHF-based CCSD
energies, leads to a satisfactory description of bond breaking in N2. In fact, even
the CCSDT(Q{) method, in which the effects due to T4 clusters are added to the
CCSDT energies, fails (see Table 3). The failure of the CCSD(T), CCSD(TQf), and
CCSDT(Q;) approaches at larger N—N distances is a consequence of the divergent
nature of the MBPT series and the failure of the CCSD and CCSDT methods to
provide reasonable information about the T1, T2 and (in the case of CCSDT) T3
cluster amplitudes, which are used to construct the relevant (T) and (Qf) corrections.

The MMCC formalism provides us with two different ways of removing the failures
of the standard single-reference CC approximations at larger N —N separations in N2.

The excellent results for the triple bond breaking in N; can be obtained either by using

64

the CISthph-corrected MMCC(2,6) method or by employing the quadratic MMCC
approximation. In this section, we focus on the CISthph-corrected MMCC(2,6)
results. The quadratic MMCC calculations are discussed in Sect. 5.2.

The results of the CISthph-corrected MMCC (2,6) calculations and their CISthp-
corrected MMCC(2,5) analogs can be found in Table 5 and Fig. 3. As in the CI-
corrected MMCC(2,3) and MMCC(2,4) calculations, the active space used in the
CI-corrected MMCC(2,5) and MMCC(2,6) calculations consisted of the 309, lvru,
21ru, 11rg, 27rg, and 3a,, valence orbitals.

As one can see, the CISthph-corrected MMCC(2,6) approach reduces the huge
errors in the results of the CCSD, CCSD(T), and other standard CC calculations
at large N—N distances R to as little as 4.443 millihartree at R = 2R,3 and 4.552
millihartree at R = 2.25Re. The errors in the MMCC(2,6) calculations for N2 range
between 1.217 and 4.552 millihartree in the entire R = 0.7512,, -— 2.25R, region. This
is an excellent (and intriguing) result considering the fact that the MMCC(2,6) cal-
culations utilize the generalized moments of the failing CCSD approach. This result
clearly demonstrates that the MMCC formalism can handle all kinds of problems,
including the most difficult problem of triple bond breaking. In spite of the un-
physical shape of the CCSD PBS at the intermediate and larger N—N distances, the
MMCC(2,6) corrections to the CCSD energies lead to an excellent potential energy
curve, which is located only slightly above the exact full CI curve (see Fig. 3). The
dissociation energy De, defined here as the difference of energies at R = 2.25Re
and R = Re, resulting from the MMCC(2,6) calculations, is 6.68 eV, in excellent

agreement with the exact full CI value of 6.61 eV. This should be contrasted with

65

the fact that it is impossible to calculate De for the standard RHF—based single-
reference CC methods due to unphysical shapes of the potential energy curves re-
sulting from the single-reference CC calculations. It is, therefore, very encouraging
that the MMCC(2,6) method employing the RHF reference is capable of providing
an accurate representation of the PES of N2, even at larger N—N separations, where
all standard CC approximations fail. The 2.022 millihartree error in the MMCC(2,6)
result at R = Re is not as small as the 0.323 or ~1 millihartree errors in the stan-
dard and completely renormalized CCSD(TQ) results (cf. Table 3), but the overall
performance of the CISthph-corrected MMCC(2,6) approximation is much better
than the performance of the standard and renormalized CCSD(T) and CCSD(TQ)
methods. The only other noniterative CC method, employing the cluster amplitudes
obtained in the CCSD calculations, that can provide results that are comparable
with the results of the CI-corrected MMCC(2,6) calculations, is the aforementioned
quadratic MMCC approach discussed in Sect. 5.2.

The results of the CI—corrected MMCC(2,3), MMCC(2,4), and MMCC(2,5) calcu-
lations are not as good as the CI-corrected MMCC(2,6) results. This clearly indicates
that the MMCC(2,3), MMCC(2,4), and MMCC(2,5) levels of the MMCC theory are
not sufficient to obtain an accurate description of the triple bond breaking in N2,
when the limited CI wave functions are used as the wave functions |\Ilo) in construct-
ing the relevant energy corrections. As mentioned in Sect. 4.1, the CI-corrected
MMCC(2,3) approach completely fails for N2 (cf. Table 3 and Fig. 2). The CI-
corrected MMCC(2,4) and MMCC(2,5) results are good for R g 1.75Re, with errors

not exceeding a few millihartree (see Tables 3 and 5). However, for R 2 2R,” the

66

energies obtained in the CI-corrected MMCC(2,4) and MMCC(2,5) calculations are
significantly below the corresponding full CI energies. At larger N —N separations, the
CI-corrected MMCC(2,4) and MMCC(2,5) approaches suffer from a non-variational
collapse similar to that characterizing the CCSD and CCSD(T) approximations. On
the other hand, the CI-corrected MMCC(2,4) and MMCC(2,5) approaches performed
much better than the CCSD, CCSD(T), and other standard noniterative CC methods.
The negative errors relative to full CI in the results of the CI-corrected MMCC(2,4)
and MMCC(2,5) calculations for larger values of R, which for R = 2Re are ap-
proximately —20 millihartree, are much smaller (in absolute value) than the errors
in the CCSD, CCSD(T), CCSD(TQf), and CCSDT(Qf) results. We can, therefore,
conclude that the CI-corrected MMCC(2,4) and MMCC(2,5) methods provide sig-
nificant improvements in the results of the standard single-reference calculations for
triple bond breaking in N2. The only problem is that the improvements offered by the
CI-corrected MMCC(2,4) and MMCC(2,5) approximations at larger N—N separations
are not as great as we would like them to be. This behavior of the CI-corrected MMCC
methods is in sharp contrast to the behavior of the quadratic MMCC approximations
which will be discussed in Sect. 5.2. As we will see in the next section, already the
quadratic MMCC approaches of the MMCC(2,5) type are capable of providing very
small, at most a few millihartree, errors in the entire R = 0.75Re — 2.25Re region
of the ground-state PES of N2. As a matter of fact, the most expensive moments

...........

of the CCSD theory, i.e. Ma‘l‘gfig‘fiasfl) and Myfgggygjggaom), can be ignored in the

quadratic MMCC calculations for N2 without significant loss of accuracy at larger

N—N separations. This is in great contrast with the CI-corrected MMCC methods

67

discussed in this section, since the only level of the CI-corrected MMCC theory that
guarantees very good results for the triple bond breaking in N2 is the MMCC(2,6)
level, which includes the entire set of the generalized moments of the CCSD equations.

From the results in Tables 3 and 5 and Figs. 2 and 3, it immediately follows that
the CI-corrected MMCC methods provide the correct shape of the potential energy
curve and small errors relative to full CI only when the limited CI wave functions |\Ilo)
used to construct the MMCC energy corrections provide a qualitatively correct de-
scription of bond breaking. Clearly, the CISthph method, on which the CI-corrected
MMCC(2,6) approach is based, provides a qualitatively correct representation of the
potential energy curve of N2 (see Table 5). In consequence, the CISthph-corrected
MMCC(2,6) method provides excellent results for all N—N separations. It is inter-
esting to note that the CISthph-corrected MMCC(2,6) approximation reduces the
5390—8372 millihartree errors in the CISthph results for N2 by a factor of 2—4. As
expected, the use of the CISDt, CISth, and CISthp wave functions, which lack the
important contributions from the hextuply excited configurations, results in a poor
performance of the CISDt, CISth, and CISthp methods and their CI-corrected
MMCC(2,3), MMCC(2,4), and MMCC(2,5) analogs at larger N—N separations.

The CISthph-corrected MMCC(2,6) method is essentially the only approach
among all CI-corrected MMCC approximations that provides excellent and well-
balanced results for smaller and larger N—N distances. For simpler types of bond
breaking, including various examples of single and double bond breaking, very good
results can already be obtained with the MMCC(2,3) and MMCC(2,4) approxima-

tions (cf. Sect. 4.1). The results in Tables 4 and 5 and the earlier results in Tables

68

1—3 show that the MMCC formalism always offers considerable improvements in the
results of the limited CI calculations that are used to provide wave functions I‘Ilo) for
constructing the MMCC energy corrections, but one has to use the right form of the
CI wave function |\Ilo) for a given type of bond breaking to obtain highly accurate CI-
corrected MMCC results (CISDt for single bond breaking, CISth for double bond

breaking, and CISthph for triple bond breaking).

5.1.3. Conclusion

The results for various types of bond breaking reported in Tables 1—5 and Figs.
2 and 3 show the systematic behavior of the CI-corrected MMCC approximations.
The CI-corrected MMCC results systematically improve when we go from the basic
MMCC(2,3) approach to the intermediate MMCC(2,4) and MMCC(2,5) levels and
to the highest-level MMCC(2,6) approximation. The systematic improvements in the
results of the CI-corrected MMCC calculations in a sequence of the MMCC(2,3) —>
MMCC(2,4) -> MMCC(2,5) —> MMCC(2,6) approximations are a consequence of the
fact that along with incorporating higher and higher moments of the CCSD equations,
we are also systematically improving the quality of the CI wave functions that enter
the MMCC(2,mB), m3 = 2 — 6, energy expressions (from CISDt in the MMCC(2,3)
case to CISthph in the MMCC(2,6) case).

With a judicious choice of the CI wave function I‘IIO), the CI—corrected MMCC

methods are capable of providing very good results for all kinds of bond breaking. The

69

choice of active orbitals in the CI-corrected MMCC calculations is usually straight-
forward since we often know which valence orbitals are involved in the bond breaking
process under consideration, but clearly it would be very useful to be able to describe
PESs involving bond breaking without having to select active orbitals. Undoubtedly,
it would be desirable to have robust approaches, which combine the simplicity of the
“black-box” noniterative CC methods, such as CCSD(T), with the efficiency with
which active-space or MRCC and MRCI approaches describe quasi-degenerate elec-
tronic states and bond breaking. Noniterative single-reference CC approaches which
are flexible and powerful enough that they can handle bond breaking in spite of using
elements of MBPT, would be particularly desirable in situations where it is not easy
to define the appropriate active space. The completely renormalized CCSD(T) and
CCSD(TQ) methods that were described in Sect. 4.2 and the new quasi-variational
and quadratic MMCC approaches developed in this thesis work and described in Sect.
5.2 are good candidates for such “black-box”-type methods. The quasi-variational and
quadratic MMCC approaches can be viewed as extensions of the completely renor-
malized CCSD(T) and CCSD(TQ) methods which work for single as well as multiple

bond breaking, including triple bond breaking in N2.

70

5.2. The Quasi-variational and Quadratic MMCC Approximations

The excellent results obtained with the CISthph-corrected MMCC(2,6) approach
for the triple bond breaking in N2 indicate that in searching for the “black-box”
extensions of the completely renormalized CCSD(T) and CCSD(TQ) methods that
might provide an excellent description of triple bond breaking, one should consider
approximations involving the pentuply and hextuply excited moments of the CCSD

equations, Mi‘l‘gjgg‘jfasfl) and Myl'gggygjggaam), respectively. This implies that, in
designing the extensions of CR—CCSD(T) and CR—CCSD(TQ) methods for multiple
bond breaking, we can no longer use the MBPT(2)-like wave functions, defined by
Eqs. (68) or (69), as the wave functions I‘llo) in the relevant MMCC expressions, since
they do not contain higher-than-quadruply excited components that must be present
in the formula for |\II0), if we are to benefit from the presence of the M“"‘3“‘5 (2)

0102030405

and Mi‘ligjgg‘jfggaefi) contributions to the higher-order MMCC(2,5) and MMCC(2,6)
energy expressions. The MBPT(3) wave function is the lowest-order wave function
that has the pentuply and hextuply excited contributions, which can engage the
pentuply and hextuply excited moments of the CCSD equations in Eq. (25) to give a
nonzero MMCC(2,6) correction 60(2,6) (cf., also, Eq. (82)). Unfortunately, in spite
of several attempts, we have not succeeded in improving the CR-CCSD(TQ) results
for triple bond breaking, reported in Ref. 47 and discussed in Sect. 4.2, by using the
MBPT(3)-like expressions for I‘llo). Clearly, a different way of designing the MMCC

corrections is required if we are to improve the CR-CCSD(TQ) results for multiple

bond breaking by adding noniterative corrections to CCSD energies that are solely

71

based on the CCSD values of T1 and T2. This new way of designing “black-box”
MMCC methods for multiple bond breaking is discussed in this section.

As part of this thesis research, we have suggested a new idea of exploiting the
exponential, CC-like, forms of lilo) in Eq. (25) for the MMCC-based 630080) correc-
tion to the CCSD energy, instead of the more usual MBPT-like forms used in the
CR-CCSD(T) and CR—CCSD(TQ) approaches. As shown in this section, the CC—like
functions |\Ilo) or their truncated variants are much more effective in introducing the
higher-order terms into Eq. (25) than the finite-order MBPT expressions exploited
in the existing CR—CCSD(T) and CR—CCSD(TQ) approaches. The resulting quasi-
variational and quadratic MMCC methods provide excellent results for multiple bond

breaking, while preserving the ease of use of the standard and completely renormal-

ized CCSD(T) and CCSD(TQ) approaches.

5.2.1. Theory

Since the MBPT(3)-like wave functions |\IIo) proved to be inefficient in construct-
ing the highly accurate corrections 630031)) for multiple bond breaking, we decided to
consider an alternative approach, in which we replace |\Ilo) in Eqs. (21) or (25) by a

CC—like exponential wave function‘"’""137

I‘DSJVMMCC) = 821(1)), (89)

72

where, in practice, the excitation operator 2 is an approximation to the exact cluster
operator T. By inserting Eq. (89) into Eqs. (21) and (25) and by choosing dif-
ferent forms of E in Eq. (89), a hierarchy of the so—called quasi-variational MMCC
(QVMMCC) approximations can be proposed. In general, the QVMMCC nonitera-

tive energy correction to the standard CC energy Eff” is defined as follows (cf. Eq.

(21)):134—137

68A)(QVMMCC) E E3VMMCC __ ((JA)
N n
= Z Z (@62th Cn_k(mA) Mk(mA)|‘I’>/

n=mA+1 kzmA-f-l

(cle8*eT“’|<1>>. (90)

If the CC method A represents the CCSD theory, the corresponding QVMMCC

energy, ESWMMCC, is calculated by adding the correction
N min(n,6) t
aéCCSD’<QVMMcc> = Z Z <<I>le” 62.0.-..(2) M.(2)I<I>>/
n=3 k=3
(olez‘eTHTqr) (91)

to the CCSD energy. The name “quasi-variational” in reference to the MMCC ap-
proximations based on Eqs. (90) or (91) is related to the fact that the QVMMCC

energy
E3VMMCC = E3“ + 53A)(QVMMCC) (92)

represents an upper bound to the exact energy E0, if 2 in Eq. (90) equals TM) (or

73

2 in Eq. (91) equals T1 + T2). Indeed, by replacing |\II0) in Eqs. (21) and (24) by

IwgmMCC), Eq. (89), we immediately obtain

63A’(QVMMCC> = <<I>Ie’3*HeT“’I<I>>/<<I>Ie‘3’eT""I<I>> — 3"). (93)

so that

E3VMMCC = (<1>|e“3t HeTW |<I>)/(<I>|e£teT(A) IQ)). (94)

When 2 = T“), the QVMMCC energy E3VMMCC, Eq. (94), reduces to the expecta-
tion value of the Hamiltonian with the CC wave function eTW |<I>), which provides an
upper bound to the exact ground-state energy E0.

Clearly, the QVMMCC theory becomes exact if 2 in Eqs. (90) or (91) is the
exact cluster operator T. Also, the exponential form of I‘llo), Eq. (89), used to define
the QVMMCC corrections 63")(QVMMCC), Eq. (90), or 630051))(QVMMCC), Eq.
(91), guarantees that the resulting energies EOQVMMCC, Eq. (92), are rigorously size—
extensive. The question is if we can propose efficient computational schemes, based
on Eqs. (90) or (91), that would allow us to obtain a highly accurate description
of multiple bond breaking. In developing these practical schemes, the form of the
operator 2 entering Eqs. (90) and (91) plays an important role.

The simplest choice of E in Eq. (91) might be 23 2 T1 + T2, with T1 and T2 ob—
tained in the standard CCSD calculations. As shown above, the resulting corrections
(SSCCSD)(QVMMCC), when added to the CCSD energies, would provide upper bounds

to the exact energies. Thus, the choice of E = T1 + T2 in Eq. (91), in analogy to

74

142,143 would lead to a qualitatively correct description of

the variational CCD theory,
triple bond breaking in N2. In particular, the non-variational collapse of the standard
CCSD approach at larger N —N separations discussed in the earlier sections would be
entirely eliminated. However, we do not obtain the desired improvements in the quan-

titative description of multiple bond breaking when the wave function [Elfin/MMCC),

Eq. (89), with E 2 T1 + T 2, is used to calculate the correction décCSD)(QVMMCC),
Eq. (91), to the CCSD energy. This is a consequence of the fact that the choice
of 2 = T1 + T2 in Eq. (91) is not bringing any meaningful information about the
connected triply excited clusters T3, which are very important in all quantitative
calculations (particularly when chemical bonds are stretched or broken).

The T3 clusters can be approximated by considering the MBPT(2)-like expres-
sion, Eq. (44), used in the CCSD[T], CCSD(T), (C)R-CCSD[T], and (QR-CCSD(T)
methods. Thus, based on the fact that the T3 cluster components are essential in
studies of bond breaking, we can suggest the following form of the cluster operator 2

for the energy calculations based on Eq. (91):

2=n+n+fl3 mm

The cluster operator 2, Eq. (95), represents an approximation to the exact cluster
operator T, which is correct through the second order of the MBPT wave function
(T1 and T3 contribute, for the first time, in the second order and T2 contributes, for
the first time, in the first order; T4, T5, etc. do not contribute in the first two orders

of MBPT). We can, of course, think of some other, more elaborate, forms of 2 in Eq.

75

(95), but all of our benchmark calculations to date indicate that a simple definition
of 2 given by Eq. (95) is sufficient to provide very good results for multiple bond
breaking.

The use of the exponential wave function I‘IIEWMMCC), Eq. (89), in calculating the
noniterative corrections 680051)), has only one practical drawback, namely, all many-
body terms resulting from the presence of e” in Eq. (91), including the N -body ones,
where N is the number of electrons, contribute to the correction 630081)) (QVMMCC).
Although this does not change the fact that the generalized moments of the CCSD
equations, corresponding to projections of those equations on higher-than-hextuply

excited configurations, vanish (so that the summation over k in Eq. (91) is still limited

to the k = 3 — 6 terms), the full use of the exponential wave function IWEVMMCC)
requires that we deal with the full CI expansion of Iwgmmcc) in calculating 63cc”).

In order to alleviate this situation, we have decided to consider approximate forms of

Eq. (91), in which the power series expansion for em, i.e.,

 

N I l
(921 = Z (22!) , (96)

with 2 defined by Eq. (95), is truncated in Eq. (91) at some low power of 2". Two
approximations are particularly important here, namely, the linearized QVMMCC
(LMMCC) model, in which est in Eq. (91) is truncated at the linear terms in 2*, so

that the wave function |\P3VMMCC), Eq. (89), is approximated by

I‘I’t‘MMCC) = (1 + 2)|‘I>>, (97)

76

and the quadratic QVMMCC (QMMCC) model, in which est in Eq. (91) is truncated

at the (2")2 terms, so that the wave function |W3VMMCC

) is approximated by
IwfiMMCC) = (1+ 2: + §22)|<1>). (98)

Cubic, quartic, and other QVMMCC models based on truncating the power series

expansion for |\II(?VMMCC)

at higher powers of E can be proposed, too, although it
seems to us that the LMMCC and QMMCC approximations are usually sufficiently
accurate.

It can be easily shown that the LMMCC model with 2 defined by Eq. (95) is
equivalent to the CR—CCSD[T] method, introduced in Refs. 45 and 46 and overviewed
in Sect. 4.2. As such, the LMMCC method cannot describe multiple bond breaking
due to the absence of the quadratic §T§ terms and other nonlinear terms in the wave
function [\IIO) defining the relevant energy correction 65mm) (cf., e.g., Ref. 47). The
LMMCC E CR-CCSD[T] method and its CR-CCSD(T) analog work well for the
PESs involving single bond breaking. The QMMCC model can be thought of as the
extension of the CR-CCSD[T] and CR—CCSD(T) methods that includes terms that
are quadratic in 23*. In particular, the QMMCC method contains the quadratic $432
terms in the wave function Illlo) that can also be found in variant “b” of the CR-
CCSD(TQ) theory, introduced in Ref. 47 and discussed in Sect. 4.2. In addition to
the £71} terms in I‘llo) (the %(T2l)2 terms in Eq. (91)), the QMMCC energy expression
contains other terms that are bilinear in cluster amplitudes, such as T2" (szlf‘, which

are not present in the CR-CCSD(TQ) method. These additional bilinear terms are

77

essential for obtaining a highly accurate description of multiple bond breaking.

We begin our description of the QMMCC energy expressions by considering the
complete QMMCC theory within the TM) = Tl + T2 and 2: = T1 + T2 + 742] approxi-
mations, referred to as the QMMCC(2,6) method. By replacing E in Eq. (91) by Eq.
(95) and by truncating the power series expansion for e“3t in Eq. (91) at the (202

term, we obtain the following expression for the QMMCC(2,6) energy:
EOQMMCC(2,6) = EéCCSD) + JSCCSD)(QMMCC(2,6)), (99)

where

(SéCCSD)(QMMCC(2, 6)) = NQMMCC(2,6)/DQMMCC(2,6), (100)

with the numerator

NQMMCC‘“) = 26: EnjwfiMMCCIQn C.-t(2) Mt(2)I<I>>
= 23%;; + (T§2’)*1M3(2)
+9992 + TI(T§2])*1[M4(2) + T1M3(2)1
+TJ<T§21>WM5<2> + Time) + (T2 + 9T3>Me<2n
+%[(T§2‘)*12[Mo(2) + TiMs(2) + (T2 + iTE)M.(2)

+(T1T2+ Haitians»). (101)

78

and the denominator

DQMMCC(2,6) : (ngMccleT1+T2lq))

= 1+ <<I>ITIT1|<I>> + <<I>IITJ + in": >21(T2 + %T3)l<1>>
+<<I>HTJTJ + (Tflmmn + tram
+<<I>|[%(TJ)2 + THTE‘msT: + 97127; + amid»
+<<I>ITJ(T§2‘)*(%T1T§+ mire + 93mm

+<<I>l%[(T3[2')*]2(%T§ + iTET: + time + straw). (102)

As we can see, the QMMCC(2,6) method introduces the T; (T3[2])l and %[(T;,E2])l]2 com-
ponents into the energy expression. These components originate from the (2")2 terms
in Eq. (91). Their presence in Eq. (101) leads to the appearance of the M5(2)|<IJ)
and M6(2)|<I>) quantities and, in consequence, the pentuply and hextuply excited

......

moments, Milligiig‘jfasa) and M;‘,‘g;3;‘;f;§ae(2), respectively, in Eq. (99). The linear
terms in 2*, including (T§2])l, that are already present in the low-order LMMCC or
CR-CCSD[T] and CR—CCSD(T) models and that engage the triply excited moments
Milfgjgfi), and the quadratic %(T§)2 terms that are present in the CR-CCSD(TQ),b
energy and that engage the quadruply excited moments M31123?“ 4 (2), are contained in
the QMMCC(2,6) energy expression, Eq. (99), too. As a result, the QMMCC(2,6) en-
ergy formula involves the complete set of the generalized moments of the CCSD equa-
tions, including the desired pentuply and hextuply excited moments, Milligjgi‘jfas (2)
and M“‘2‘3“‘5‘° (2), respectively, and, at the same time, it contains all lower-order

010203040506

terms of the CR-CCSD(T) and CR—CCSD(TQ) methods. The QMMCC(2,6) method

79

should give us an accurate description of triple bond breaking due to the presence

of the Mill‘gmfifasw) and Mgfgggggggafiw) moments and the T; (T§2])I and %[(T§2])l]2
terms in Eq. (100).

We have also tested the approximate QMMCC schemes in which we ignore the
higher-order terms involving the most expensive hextuply or pentuply and hextuply
excited CCSD moments. For example, the summation over n in Eq. (101) reduces
to 25 when we neglect the %[(T?E2])I]2 terms. The resulting QMMCC approxima-

n=3

tion, termed the QMMCC(2,5) method, does not require the calculation of the most

expensive hextuply excited moments ./\/f""“"“""6 (2). The QMMCC(2,5) energy ex-

010203040506

pression can be given the following form:
EOQMMCC(2,5) = E30030) + 680031)) (QMMCC(2, 5)), (103)

where

63CCSD)(QMMCC(2, 5)) : NQMMCC(2,5)/DQMMCC(2,5), (104)

with

NQMMCW’ = Z EanMCCIQ. 0..-..(2)Mt(2)l<1>>
n=3 k=3
= <<I>IITITJ +(Ti2')*1M3(2)
+[%(T2*)2 + TMTWnMic) + Timon

+T;(T§2])th5(2) + T1M4(2) + (T2 + §T12)M3(2)]|<I>), (105)

80

and

DQMMCC‘2~5’ = 1+ <<I>|TlTil<I>> + <0HTJ + i(T:)21(T2 + %T3)|<I>>
+<0IiTJTJ +(T3‘21)*)(T1T2 + trim
+<<I>I[%(TJ)2 + TI(T§2‘)*](%T§ + in"); + am»)

+<<I>|TJ(T§2])*(%T1T§ + iris + Mm). (100)

120

As we will see in Sect. 5.2.2, the absence of the Mi‘l‘gjgi‘jfigaJm moments and the
%[(T§2])I]2 components in the QMMCC(2,5) energy expression has no detrimental
effect on the results of the QMMCC calculations for multiple bond breaking. If we
further ignore the T; (T421)r terms in the QMMCC(2,6) energy formula, Eq. (99), and
the %[(T;£2])"]2 terms that have already been neglected in the QMMCC(2,5) approach,

we obtain the QMMCC(2,4) method. The QMMCC(2,4) energy is calculated as

follows:
EOQMMCCQA) = EéCCSD) + 63CCSD)(QMMCC(2,4)), (107)
where
déCCSD)(QMMCC(2, 4)) : NQMMCC(2,4)/DQMMCC(2,4), (108)
with

NQMMCW’ = Ziwgmmlencaia)Mt(2)I<I>>
n=3 k=3
= <<I>I[TITJ + (Téz‘mMeo)
+9092 + T: (T§21)‘1[M4(2) + T1M3(2)]|<I>), (109)

81

and

DQMMW‘” = 1+ (Wm) + (0)0; + %(Tf)2l(T2 + 971%)
+(0HTJTJ + (T420000. + who

+<<I>|[%(TJ)2 + T3(T§2])*](%T§ + i-Tfrre + 21—4050). (110)

As we can see, the QMMCC(2,4) method requires that we only consider the triply
and quadruply excited moments of the CCSD equations, M31333 (2) and M11333?“ 4 (2),
respectively. In consequence, the QMMCC(2,4) energy formulas, Eqs. (107)—(110),
resemble the expressions defining the CR-CCSD(TQ),b approach described in Sect.
4.2 (cf. Eqs. (70), (72), and (74)). The lowest level of the QMMCC approximations,
i.e. the QMMCC(2,3) approach, is obtained by neglecting the [%(TJ)2 + TI(T§2])I]
terms in Eqs. (109) and (110), so that the only moments of the CCSD equations

that are included in the QMMCC(2,3) calculations are the triply excited Millfgjz3(2)

moments. We calculate the QMMCC(2,3) energy by using the formula

EOQMMCC(2,3) : Eéccsn) + ééCCSD)(QMMCC(2,3)), (111)

where

JSCCSD)(QMMCC(2, 3)) = NQMMCC(2,3)/DQMMCC(2,3), (112)

with

NQMMCC(2,3) = (‘PgMMCCIQ3 00(2) M3(2)|(p>

82

= (0)1111"; +(T121)*)M3(2)I0>. (113)
and

DQMMCC‘2'3’ = 1+ <<I>ITIT1|<I>> + <<I>IITJ + %(T{‘)21(T2 + %T3)I<I>>

+<0IiTJTJ + (T§2])"](T1Te + $510) (114)

The QMMCC(2,3) energy formulas, Eqs. (111)—(114), are practically identical to
the CR—CCSD[T] or CR—CCSD(T) expressions, Eqs. (51) and (52). Thus, we can
expect that the QMMCC(2,3) results are virtually identical to the CR—CCSD[T] or
CR—CCSD(T) results.

The formal similarity of the QMMCC(2,4) and CR-CCSD(TQ),b approximations
immediately implies that the computer costs of the QMMCC(2,4) calculations re-
semble those of the CR—CCSD(TQ),b calculations. Thus, in analogy to the CR-
CCSD(TQ),b method and its standard CCSD(TQf) counterpart, the cost of calcu-
lating the QMMCC(2,4) energy correction, Eq. (108), is ngnfl, where no and nu are
the numbers of occupied and unoccupied orbitals, respectively. The QMMCC(2,5)
and QMMCC(2,6) methods are somewhat more expensive, although, as shown in
Section 5.2.2, we can ignore the most expensive pentuply and hextuply excited mo-

ments, Mgl‘gmgfasm) and M:;g;g;4;3;g%(2), respectively, in the QMMCC(2,5) and

QMMCC(2,6) energy formulas, Eqs. (103) and (99), respectively, without affecting

the excellent QMMCC(2,5) and QMMCC(2,6) results.

The most expensive steps of the full QMMCC(2,6) approximation, on which all

83

other QMMCC methods are based, scale as 112123. The relatively low, nS-like, cost of

computing the noniterative QMMCC(2,6) energy correction may be somewhat sur-

000000

. . . - 311233": 1 '
prismg, Since the hextuply excrted moments Moio2o3ofo§o5(2) are twelve-index quan—

tities, but one has to keep in mind the highly factorized character of the QMMCC

energy corrections. For example, the most expensive %(<I>|[(T3[2])l]2 M6(2)|<I>) term of

the QMMCC(2,6) method has the following form:

i<<1>I[(CI‘3""‘)*]2 Me(2)l<1>> = z%<<1>|[(T§2‘)‘l2(V~T;‘)c|<I>>, (115)
since
Mi‘.‘:;'::‘.f:z:a.(2) = 90:22:32,290 (VNTE‘M‘D). (110)

where, as usual, VN represents the two-body part of the Hamiltonian in the normal-
ordered form. The highly factorized character of Eq. (115) allows us to form the
intermediates from the (TRY deexcitation and T2 excitation cluster amplitudes by
connecting the (3)21)" and T2 diagrams with at least two fermion lines. In consequence,
we do not have to construct and store the twelve-index hextuply excited moments

......

Mg’ggggmaoa) to calculate terms, such as Eq. (115). This leads to a reduction of
the operation count to the 112713 or less expensive noniterative steps. Similar remarks
apply to the QMMCC(2,5) theory, which formally uses the moments Millfgizi‘jfasfl).
Again, all terms entering the QMMCC(2,5) energy formula are highly factorized and

we can completely eliminate the need for calculating and storing ten-index quantities,

such as Milligjgg‘jfas (2), by connecting the T; and (THYf vertices entering the pentuply

84

excited part of Eq. (105) with the T1 and T2 vertices entering

M'1'2‘3“'5 (2) = (@91910091°0|[VN(%T§' + fiTfT§)]C|<I>) (117)

0102030405 11121314“

to form the relevant intermediates.

Let us summarize our discussion of the QVMMCC, LMMCC, and QMMCC meth-
ods. The QMMCC(2,4) theory is similar in content and computer cost to the exist—
ing CR-CCSD(TQ),b method, described in Sect. 4.2. The CR-CCSD(TQ),b method
can be viewed as a slightly modified version of the QMMCC(2,4) theory, in which
the only bilinear term of the %()3")2 type, multiplying [M4(2) + T1M3(2)], is the
lowest-order %(T§)2 term. As demonstrated in Sect. 5.2.2, the QMMCC(2,4) and
CR—CCSD(TQ),b results are virtually identical. The LMMCC approach is equiva-
lent to the CR-CCSD[T] method, which is, in turn, only slightly less accurate than
the CR—CCSD(T) and QMMCC(2,3) approaches, and considerably less accurate than
the CR-CCSD(TQ),b method in applications involving multiple bond breaking (see
Sect. 5.2.2). Finally, the QMMCC(2,4) approach represents an approximation to the
more complete QMMCC(2,5) and QMMCC(2,6) models. This means that the CR-
CCSD(TQ),b or QMMCC(2,4) approaches can be regarded as intermediate steps be-
tween the less accurate LMMCC = CR—CCSD[T], CR—CCSD(T), and QMMCC(2,3)
methods, which work very well for the ground-state PESs involving single bond break-
ing (see Sects. 4.2 and 5.2.2), and the more accurate QMMCC(2,5) and QMMCC(2,6)
approaches, which, as shown in Sect. 5.2.2, are capable of providing an accurate de-

scription of the PESs involving multiple bond breaking. We expect, therefore, to

85

observe the following accuracy patterns in the actual calculations for bond breaking:
LMMCC E CR—CCSD[T] ,S CR-CCSD(T) z QMMCC(2, 3)
< CR-CCSD(TQ), b m QMMCC(2, 4)

< QMMCC(2, 5) ,5 QMMCC(2, 6) a QMMCC 5 Full CI. (118)

5.2.2. Examples of Applications

In order to test the performance of the QMMCC methods, we implemented the
entire family of the QMMCC approximations described in the previous section. As
in the case of the CI-corrected MMCC(2,5) and MMCC(2,6) methods, we used the
spin-adapted CCSD code of Piecuch and Paldus to generate the required T1 and T2
cluster amplitudes. The RHF calculations needed to generate molecular orbitals and
the integral transformation from the atomic to molecular orbital basis were performed
with GAMESS. In programming the QMMCC methods, we relied on the expressions
for the generalized moments of the CCSD equations given in Appendices B and C
(obtained using the diagrammatic method of MBPT; see Appendix B for the details).
We tested the performance of the QMMCC methods by performing the calculations
for single bond breaking in HF, double bond breaking in H20, triple bond breaking
in N2, and the very challenging type of bond breaking in C2. In all cases, we used
the relatively small DZ basis set, which allowed us to compare the QMMCC results

with those obtained with the exact full CI approach.

86

We already know that all types of single bond breaking (such as bond break-
ing in HF) are accurately described by the lowest-order MMCC(2,3) approxima-
tions, including the CISDt-corrected MMCC(2,3) approach discussed in Sect. 4.1
and the completely renormalized CCSD[T] and CCSD(T) approaches discussed in
Sect. 4.2. According to Eq. (118), we expect the results from the QMMCC(2,3)
and CR-CCSD[T] or CR-CCSD(T) calculations for single bond breaking in HF to be
almost identical. Our test calculations for HF using the QMMCC(2,3) method con-
firmed this expectation: The QMMCC (2,3) results turned out to be virtually identical
(to within less than a millihartree) to the CR—CCSD[T] and CR-CCSD(T) results.
Thus, in the following, we focus on the higher-level QMMCC(2,4), QMMCC (2,5), and
QMMCC(2,6) approximations. In the case of HF, the small ~l millihartree unsigned
errors in the QMMCC(2,4) results in the entire R, _<_ R 3 5R, region are practically
identical to the CR—CCSD(TQ) results (see Tables 1 and 6). This is a clear reflection
of the accuracy patterns described by Eq. (118). The higher level QMMCC(2,5)
and QMMCC(2,6) calculations provide results that are very similar to their lower
level QMMCC(2,4) counterparts (see Table 6). This indicates that the inclusion
of the more expensive pentuply and hextuply excited moments of the CCSD equa-

ooooooooooo

tions, Mgl'ggyggasm and Myl'gggygjggaem), is not providing us with any additional
improvements in the QMMCC(2,4) or CR-CCSD(TQ) results for processes involving
a dissociation of a single chemical bond. In these cases, the very simple MMCC(2,3)
approximations, such as CR-CCSD(T), are sufficient for an accurate description of
single bond breaking and the QMMCC(2,4) or CR-CCSD(TQ) approaches provide

additional small improvements.

87

The QMMCC approximations have been deve10ped primarily to improve the
poor description of multiple bond dissociation by the standard CC methods, such
as CCSD, CCSD[T], CCSD(T), or CCSD(TQf). In the context of the simultane-
ous breaking of both O—H bonds in the H20 molecule, we would like to reduce,
if not completely eliminate, the large negative errors resulting from the CCSD[T],
CCSD(T), and CCSD(TQf) calculations in the R = 2R6 region (cf. Table 2). We
have already discussed the considerable improvements in the results of the standard
CC calculations for the double dissociation of water by the completely renormal-
ized CC approaches, such as CR—CCSD(T) or CR-CCSD(TQ) (see Sect. 4.2). In
particular, the CR—CCSD(T) and CR-CCSD(TQ) methods reduce the large negative
((—6)—(—11) millihartree) errors, relative to full CI, obtained in the CCSD[T], CCSD(T),
and CCSD(TQf) calculations at R = 2Re to small (< 2—3 millihartree) positive er-
rors (cf. Table 2). We have also mentioned that we can obtain similar improve-
ments in the results of the standard CC calculations by switching to the CI-corrected
MMCC(2,3) and MMCC(2,4) methods (see Sect. 4.1; particularly Table 2). The
CISDt-corrected MMCC(2,3) method or the CISth-corrected MMCC(2,4) approach
reduces the larger negative errors in the CCSD[T], CCSD(T), and CCSD(TQf) results
for R = 2R,z to relatively small ~2 millihartree positive errors. As shown in Sect. 5.1,
the CISthp-corrected MMCC(2,5) scheme and the CISthph-corrected MMCC(2,6)
approach reduce these errors further, to 0.730 and 0.538 millihartree, respectively (see
Table 4). It is interesting to examine whether the QMMCC methods, which can be
viewed as the natural extensions of the “black-box” CR-CCSD[T], CR—CCSD(T), and

CR—CCSD(TQ) approaches, are capable of providing similar accuracies.

88

The QMMCC(2,4), QMMCC(2,5), and QMMCC(2,6) results for the H20 molecule
are listed in Table 4. As expected (cf. Eq. (118)), the QMMCC(2,4) approximation,
defined by Eqs. (107)-—(110), provides results that are virtually identical to the CR-
CCSD(TQ) results shown in Table 2. In analogy to the CR—CCSD(TQ) case, the
small ~0.3 millihartree error in the QMMCC(2,4) energy at R = Re monotonically
increases with R to reach ~2 millihartree in the R = 2R,Z region (cf. Table 4).
The QMMCC(2,4) results are also similar to the results of the CISth—corrected
MMCC(2,4) calculations (cf. Tables 2 and 4). This is quite promising, since, unlike
the CISth-corrected MMCC(2,4) theory, the QMMCC(2,4) method does not require
selecting active orbitals. The QMMCC(2,5) and QMMCC(2,6) methods provide fur-
ther improvements in the description of the double dissociation of water, particularly
in the R = 2Re region. The 1.163—2.853 millihartree errors in the CR-CCSD[T],
CR-CCSD(T), CR-CCSD(TQ), and QMMCC(2,4) results at R = 2Re reduce to <
0.6 millihartree when the QMMCC(2,5) and QMMCC(2,6) methods are employed.
The description of the double dissociation of the H20 molecule by the QMMCC(2,6)
method is as good as the excellent description of this process by the CISthph-
corrected MMCC(2,6) approach, which requires that we first perform multireference-
like CISthph calculations (see Table 4). It is remarkable to observe that the 1.790,
5.590, and 9.333 millihartree errors in the CCSD results at R = Re, 1.5Re, and 2R6,
respectively, can be reduced to < 0.7 millihartree in the entire R = Re — 2Re region
by adding to the CCSD energies, the a posteriori noniterative QMMCC energy cor-
rections, employing only T1 and T2 components obtained in the CCSD calculations.

The results of the QMMCC(2,5) calculations are somewhat surprising (in a pos-

89

itive sense). The QMMCC(2,5) energies are essentially identical to the excellent

QMMCC(2,6) energies at all values of R. This implies that we can almost cer-

tainly ignore the most expensive hextuply excited Mgll'ggggfggasu) moments and

[2]

the corresponding %[(T3 )WI‘IIO) wave function components that are present in the

QMMCC(2,6) energy expression without sacrificing the high quality of the QMMCC

results. In fact, we can neglect the Milligigi‘jfasa) and Myfgmggggasm) moments in

the complete QMMCC or QMMCC(2,6) energy formula, Eq. (99), without com-

promising the excellent QMMCC(2,6) results. The 0.2—0.7 millihartree errors in the

QMMCC results for H20, obtained by zeroing the Mi‘fgm‘jfas (2) and M‘1'9‘3“'5‘° (2)

010203040506

moments in the QMMCC(2,6) energy formulas, are as small as the errors in the re-
sults of the complete QMMCC(2,6) calculations, in which all generalized moments

of the CCSD equations are included (see Table 4). By ignoring the Milligjgi‘jfasfl)

and M"”'3“‘5‘° (2) moments of the CCSD equations in the QMMCC(2,6) energy

alaga3a4a5ag
expression, we are essentially preserving the simplicity and the relatively low cost of
the CR—CCSD(TQ) calculations. This means that we may be able to obtain < 1 mil-
lihartree errors for cases involving a simultaneous breaking of two single bonds with
an effort comparable to the CR—CCSD(TQ) calculations. Based on the results of the
QMMCC calculations for H20, shown in Table 4, the QMMCC method, in which the

nnnnnn

Mi‘l‘gggg‘jfasfl) and M;‘l'g;fi;‘;f;§%(2) moments are ignored, is definitely worth further
exploration. We can also use the QMMCC(2,5) method, defined by Eq. (103), and
obtain results for the double dissociation of water that can only be matched by the ex-

cellent results of the QMMCC(2,6) or CISthph-corrected MMCC(2,6) calculations

(see Table 4). Our benchmark calculations for the water molecule clearly reflect the

90

accuracy pattern described by Eq. (118).

The most critical test of the QMMCC theory is provided by triple bond breaking in
N2. As mentioned in Sect. 4.2, the ground-state PESs obtained in the CR—CCSD(T)
and CR-CCSD(TQ) calculations for dissociations of single and double bonds are
usually very good but, when triple bond breaking is involved, the CR—CCSD(T)
and CR-CCSD(TQ) results become less accurate. For example, the 25.069 milli-
hartree error in the CR-CCSD(TQ),b results for N2 at R = 2R,“ although much
smaller than the 246.405 and 92.981 millihartree errors obtained with the standard
CCSD(T) and CCSD(TQf) approaches, is far too big for the highly accurate de-
scription of the potential energy curve for N2. The CR-CCSD(T) results for N2 at
larger values of R are even worse than the CR-CCSD(TQ) results (see Table 3).
Thus, the CR—CCSD(T) and CR—CCSD(TQ) methods are not sufficient to obtain
highly accurate PESs involving multiple bond breaking, although, undoubtedly, the
CR—CCSD(TQ),b results for N2 are much better than the results of the standard
CC calculations (including the relatively expensive full CCSDT and CCSDT(Q¢) cal-
culations; cf. Table 3 and Fig. 2). The unphysical humps and other unphysical
features of PESs obtained in the standard CCSD, CCSD(T), CCSD(TQf), CCSDT,
and CCSDT(Qf) calculations are almost entirely eliminated by the CR—CCSD(TQ),b
approach (see Table 3 and Fig. 2), but the 10—25 millihartree differences between
the CR-CCSD(TQ),b and full CI energies at intermediate and larger values of the
N—N separation R are too large for the majority of applications including the dy-
namics describing the N2 molecule. We already know that the CISthph-corrected

MMCC(2,6) approach provides the desired improvements in the description of triple

91

bond breaking in N2, reducing the 10—25 millihartree differences between the CR-
CCSD(TQ),b and full CI energies at intermediate and larger internuclear separations
R to 4.0—4.5 millihartree (see Sect. 5.1, particularly Table 5 and Fig. 3). We must
keep in mind, however, that the CISthph-corrected MMCC(2,6) approach requires
that we select active orbitals for the CISthph calculations that are needed to con-
struct wave function |\Po) entering the MMCC(2,6) energy expression (82). In other
words, although the CISthph-corrected MMCC(2,6) can be viewed as a nonitera-
tive CC approximation that provides an excellent description of triple bond breaking,
this method is not a pure “black-box” of the CCSD(T) or CCSD(TQf) type, since
we must make some arbitrary decisions about active orbitals in order to carry out
the related CISthph calculations (and these calculations also increase the computer
effort). Clearly, it is interesting to examine if we can reduce the 10—25 millihartree
errors resulting from the CR-CCSD(TQ),b calculations at larger values of R to a few
millihartree by using the QMMCC(2,6) method, which is a “black-box” equivalent of
the highly successful CISthph-corrected MMCC(2,6) approximation.

The QMMCC results for the N2 molecule obtained by adding the QMMCC(2,4),
QMMCC(2,5), and QMMCC(2,6) corrections to the CCSD energies according to Eqs.
(99)—(110), are shown in Table 5 and Fig. 4. As we can see, the QMMCC(2,5) and
QMMCC(2,6) methods are capable of providing an excellent description of the entire
R = 0.75Re - 2.25Re region of the N2 potential energy curve. The very large negative
errors in the CCSD results for N2 in the R > 1.75Re region and the 13.517, 25.069, and
14.796 millihartree errors in the CR—CCSD(TQ),b results at R = 1.75Re, 2B,, and

2.25Re, are reduced by the complete QMMCC(2,6) method to 1.380, 6.230, and -3.440

92

millihartree, respectively. This is quite remarkable, considering the single-reference
and noniterative character of the QMMCC energy corrections and the complete failure
of the CCSD approximation, on which the QMMCC(2,6) and other QMMCC methods
are based, at larger internuclear separations. As in the case of H20, the QMMCC(2,5)
results for N2 are almost identical to the excellent QMMCC(2,6) results.

As shown in Fig. 4, the QMMCC(2,6) potential energy curve for N2 is located
above the full CI curve in the entire R < 2.25Re region, in spite of the catastrophic
failure and the apparently non-variational behavior of the CCSD method at larger
N—N separations. The QMMCC(2,6) potential energy curve and the full CI curve
almost overlap in the entire 0.75Re S R S 2.25Re region. The QMMCC(2,6) poten-
tial is a monotonically increasing function in the entire 2.068 bohr g R _<_ 4.35 bohr
region (recall that R = 2.068 bohr is the equilibrium bond length in N2). It is only
when R z 2.25Re that the QMMCC(2,6) energies slightly decrease, but even in this
case the errors in the QMMCC(2,6) results, relative to full C1, are less (in absolute
value) than 3.5 millihartree (see Table 5). The approximate dissociation energy De,
calculated by forming the difference between the QMMCC(2,6) energies at R = 4.35
bohr and R = Re, is 6.59 eV, in excellent agreement with the full CI value of De of
6.61 eV. Thus, the QMMCC theory represents an approximately variational formal-
ism, which is capable of providing an excellent description of the large part of the
potential energy curve of N2.

Our experience with the QMMCC results for the simultaneous dissociation of
both O—H bonds in H20 tells us that it might be worth examining the effect of ne-
glecting the most expensive moments of the CCSD equations, i.e. Mgfa’fig‘jfas (2) and

93

)\/l""'3‘“5‘6 (2), in the QMMCC calculations for N2. We have already noticed that

010203040506

the QMMCC(2,5) method, in which the M'm’3m5‘5 (2) moments are ignored, pro-

010203040506
vides results of the full QMMCC(2,6) quality (cf. Table 5 and Fig. 4). The large,
30—120 millihartree, unsigned errors in the CCSD results in the R _>_ 1.5Re region
are reduced to 3.756 millihartree at R = 1.5Re, 1.415 millihartree at R = 1.75Re,
6.672 millihartree at R = 2Re, and 2.638 millihartree at R = 2.25Re when the
QMMCC(2,5) approach is employed. When the pentuply and hextuply excited mo-

ments, 1142;333:042) and Mgggggggfggafim), respectively, are neglected, the errors in

the QMMCC(2,6) calculations increase slightly, but the overall performance of the

nnnnnnnnnnn

QMMCC(2,6) approximation, in which .M'm'i‘m5 (2) and M"”'3‘"5‘° (2) are ze-

0102030405 010203040506

roed, remains very good (see Table 5). The potential energy curve of N2, obtained by
performing the QMMCC(2,6) calculations in which the pentuply and hextuply excited
moments are zeroed, is monotonically increasing in the entire Re 3 R 5 2R, region.

The approximate dissociation energy De, obtained by forming the difference between

nnnnnnnnnnn

the QMMCC(2,6)(M“”'3“‘5 (2) = Minimum (2) = 0) energies at R = 2He and

0102030405 010203040506

R = Re, is 6.54 eV, in very good agreement with the full CI De value of 6.61 eV.

The somewhat more complete QMMCC(2,6) calculations, in which only the hextuply

excited M""‘3“'5’° (2) moments are zeroed, and the QMMCC(2,5) calculations pro-

010203040506

duce D... values, which are only slightly more accurate than the value of De resulting

oooooo

from the QMMCC(2,6)(Milligjgi‘jfas(2) = Myl'gggggggaaa) = 0) calculations (6.59 and
6.61 eV, respectively). Based on the results of the QMMCC calculations for the N2

molecule, we can conclude that it is not absolutely necessary to include the pentuply

and hextuply excited moments of the CCSD equations in order to obtain an excel-

94

lent description of triple bond breaking as long as we maintain the specific, highly
factorized, many-body structure of the noniterative energy corrections defining the
QMMCC(2,5) and QMMCC(2,6) approaches (cf. Eqs. (100)—(102) and (104)—(106)).

As in the case of H20 (and as predicted by Eq. (118)), the QMMCC(2,4) results
for N2 are essentially identical to those obtained with the CR-CCSD(TQ),b approach
(cf. Tables 3 and 5 or Figs. 2 and 4). This implies that we cannot ignore the T1} (T31)t
terms in the QMMCC(2,6) or QMMCC(2,5) energy expressions, Eqs. (99)-(106),
although, as shown above, we can certainly ignore the corresponding pentuply excited
Milligigi‘jfasfl) moments. Our QMMCC(2,5) calculations imply that we can safely

neglect the %[(T:§2])l]2 terms and the corresponding hextuply excited Mg‘ggg‘gggaom)
moments in the QMMCC(2,6) energy expressions. The accuracy patterns described
by Eq. (118) are clearly reflected in our calculations for the N2 molecule. This
means that the noniterative QMMCC corrections provide us with a systematic way
of improving the standard and renormalized CC results for multiple bond breaking.
We also applied the QMMCC methods to the very complicated case of bond break-
ing in the C2 molecule. The C2 molecule is characterized by an unusual ordering of
the 0(2p) and 1r(2p) shells and the proximity of the highest occupied 1r and lowest un-
occupied a orbitals, which cause a quasi-degeneracy of the ground-state wave function
even at the equilibrium geometry. In analogy to N2, the C2 molecule is characterized
by large Tn effects, where n > 2, even at the equilibrium geometry. The effect of T3
clusters in C2 for the DZ basis set at R = Re, as measured by the diflerence between

the CCSDT and CCSD energies, is 18.593 millihartree. The difference between the

CCSDT and full CI energies at R = Re is 2.091 millihartree, which indicates that

95

T4 clusters are important too. As in the N2 case, all standard single-reference CC
approaches, including the iterative CCSD and CCSDT methods and their perturba-
tive CCSD(T) and CCSD(TQf) extensions, fail to provide a correct description of the
potential energy curve of C2 (see Fig. 5).

As shown in Table 7, the error in the CCSD result at R = R3 is 20.684 millihartree.
For R 2 1.75Re, the CCSD results seem to improve, providing unsigned errors of
36.280, 15.772, and 3.422 millihartree at R = 1.75Re, 2R9, and 2.5Re, respectively.
However, at R z 2.5Rc, the CCSD potential energy curve goes below the full CI curve
(see Table 7 and Fig. 5). This is a clear indication of the imminent breakdown of
the CCSD theory at larger internuclear C—C separations. The standard noniterative
CC methods, such as CCSD(T) and CCSD(TQf), in which the effects due to triples
and quadruples are obtained from MBPT, provide reasonable results for R < 1.75R,.,,
but in the R > 17512,. region, the CCSD(T) and CCSD(TQf) approaches completely
fail. The unsigned errors in the CCSD(T) and CCSD(TQf) calculations, which are
80.231 and 29.196 millihartree, respectively, at R = 2.5Re, increase to 96.055 and
67.237 millihartree at R 2 3R... The CCSD, CCSD(T), and full CCSDT curves have
big humps in the region of intermediate R values (see Fig. 5). The CR-CCSD(TQ),b
approach is capable of reducing the large errors in the standard CC results in the
R 2 2R. region to ~20 millihartree, and the shape of the CR—CCSD(TQ),b curve is
qualitatively correct, but it is interesting to examine if we can get further improve-
ments in the results for C2 by switching to the QMMCC theory. As shown in Table
7, the QMMCC(2,4), QMMCC(2,5), and QMMCC(2,6) methods reduce the 20.684

millihartree error in the CCSD result at R = Re to 2875—4991 millihartree. For

96

stretched geometries, the improvements offered by the QMMCC approximations are
even more dramatic. In particular, the huge errors in the CCSD(T) and CCSD(TQf)
results in the R = 2.5Re — 3Re region reduce to relatively small ~11—13 millihartree
errors when the QMMCC(2,5) and QMMCC(2,6) calculations are performed. The
QMMCC(2,5) and QMMCC(2,6) potential energy curves are located near the full
CI curve and all QMMCC energies are invariably above their exact full CI counter-
parts (see Fig. 5 and Table 7). In analogy to the N2 molecule, the QMMCC(2,4)
results are not as good as the QMMCC(2,5) and QMMCC(2,6) results. As one might
expect based on Eq. (118), the QMMCC(2,4) results are almost identical to the
CR—CCSD(TQ),b results.

It is again interesting to note that the QMMCC(2,5) and QMMCC(2,6) results
are almost identical (see Table 7). Indeed, the energy differences of the QMMCC(2,5)
and QMMCC(2,6) results in the entire R = 0.75Re — 3R8 region do not exceed 0.012
millihartree. This means that we can ignore the most expensive hextuply excited

moments, 11423331333042), in the calculations for C2 without affecting the results.

In analogy to the N2 molecule, the simplified QMMCC(2,6) method, obtained by

zeroing the Mi‘l‘gjgi‘jfasfl) and Mgl'ggggfggafia) moments, produces results that are
similar to the results of the complete QMMCC(2,6) calculations. The fact that we
still have 11—23 millihartree errors in the R Z 1.5Re region with the QMMCC(2,5)
and QMMCC(2,6) methods needs to be addressed, but we must keep in mind the chal-
lenging nature of the bond breaking in C2 on the one hand and the single-reference

character of the noniterative QMMCC corrections on the other hand. We think that

one of the reasons why we find these relatively large deviations from the full CI re-

97

sults with the QMMCC methods is the fact that the CCSD approach, on which the
QMMCC approximations are based, provides cluster amplitudes that are much too
unphysical to yield a highly accurate description of the C2 curve by means of nonit-
erative QMMCC corrections. In Sect. 6, we suggest a method that might improve

the QMMCC results for C2 in the region of intermediate and large C—C separations.

5.2.3. Conclusion

In this section, we introduced the new classes of the MMCC “black-box” ap-
proximations, referred to as the QVMMCC and QMMCC models. As in all MMCC
methods, the main idea of the QVMMCC and QMMCC approximations is that of
the noniterative energy corrections which, when added to the energies obtained in
the standard CC (e.g., CCSD) calculations, provide highly accurate description of
quasi-degenerate states and bond breaking.

We demonstrated that the QMMCC corrections to the standard CCSD energies
lead to an excellent description of double dissociation of water and triple bond break-
ing in N2. The QMMCC corrections are capable of restoring the virtually exact
description of the large part of the N2 potential energy curve, in spite of the apparent
failure of the CCSD theory at larger internuclear separations and in spite of the fact
that we only use the CCSD values of the T1 and T2 cluster amplitudes in constructing
the QMMCC corrections obtained with the RHF reference.

The QMMCC methods can be regarded as the higher-order analogs of the previ-

98

ously pr0posed (also MMCC-based) CR—CCSD[T], CR-CCSD(T), and CR-CCSD(TQ)
approximations, which work well for single and double bond breaking, but which fail
to provide a highly accurate description of triple bond breaking. The QVMMCC and
QMMCC methods provide a systematic way of improving the CR-CCSD[T], CR-
CCSD(T), and CR-CCSD(TQ) results, particularly when multiple bond breaking is
considered.

The main reason for the success of the QMMCC methods is a specific, highly fac-
torized, many-body structure of the corresponding energy corrections 63am”, which
would have never been discovered if we had only relied on the conventional MBPT
arguments in the design of these corrections. The presence of the higher-order pen-
tuply and hextuply excited moments of the CCSD equations in the QMMCC energy
expressions seems to be of secondary significance. The presence of various bilinear
terms, such as %(T2")2 and T§(T§2])l, seems to be much more important in reducing
the huge errors in the CCSD results for multiple bond breaking to a few millihartree,

rather than the presence of the pentuply and hextuply excited CCSD moments.

99

6. Summary, Concluding Remarks, and Future Perspectives

In this thesis, we have described several new ideas in the area of highly accurate
coupled-cluster (CC) calculations for molecular potential energy surfaces involving
bond breaking. The main focus of this work has been the extension of the applicability
of the method of moments of coupled-cluster equations (MMCC) and renormalized
coupled-cluster approaches of Kowalski and Piecuch to multiple bond breaking. We
have accomplished this goal by developing new classes of the CI-corrected MMCC
methods and quasi-variational (QV) and quadratic (Q) MMCC approaches.

The main idea of all MMCC methods, including new methods developed in this
work, is that of noniterative energy corrections which, when added to the energies
obtained in the standard CC calculations, such as CCSD, recover a virtually exact de-
scription of many-electron systems. The MMCC formalism provides us with rigorous
relationships between the standard CC energies and their exact full CI counterparts.
By estimating the differences of the full CI and CC energies in a given basis set using
the MMCC expressions, we can obtain results that are significantly better than the
results of standard CC calculations. This is particularly true in quasi-degenerate sit-
uations, such as bond breaking, where the conventional arguments originating from
MBPT, on which the standard noniterative CC approximations, such as CCSD(T),
are based, fail due to the divergent behavior of the MBPT expansions at larger inter-
nuclear separations.

The MMCC formalism leads to various approximations, including the renormal-

ized (R) and completely renormalized (CR) CCSD(T) and CCSD(TQ) methods. The

100

CR—CCSD(T) and CR-CCSD(TQ) approaches remove the pervasive failing of the
standard CCSD(T) and CCSD(TQf) approximations for single bond breaking and
some cases of multiple bond dissociation, but in severe cases of multiple bond break-
ing, they do not work well. We can use the CI-corrected MMCC(2,3) and MMCC(2,4)
methods instead of the CR—CCSD(T) and CR—CCSD(TQ) approaches, but their per-
formance in cases of multiple bond breaking is almost identical to the performance
of the CR—CCSD(T) and CR-CCSD(TQ) methods.

In order to change this situation, we have proposed two new classes of the MMCC
approximations: (i) the CI-corrected MMCC(2,5) and MMCC(2,6) methods and (ii)
the QVMMCC and QMMCC approaches. In the former methods, we design the
MMCC corrections to the CCSD energies by utilizing all or almost all generalized
moments of the CC equations, and by performing the a pgrz'ori limited CI calculations
with a small set of pentuple and hextuple excitations defined through active orbitals
that provide the wave function I‘IIo) entering the MMCC energy expressions. Our
test calculations for the double dissociation of water and triple bond breaking in N2
clearly indicate that the CI-corrected MMCC(2,6) approach is capable of providing
~1 millihartree or better accuracies in the description of multiple bond breaking.
This is an important finding, demonstrating the usefulness of the MMCC theory in
designing noniterative CC methods for molecular potential energy surfaces.

Two factors contribute to the success of the CI-corrected MMCC(2,6) method.
First of all, the MMCC(2,6) approach is the lowest-order MMCC approximation that
utilizes the complete set of moments of the CCSD equations, including the pentuply

and hextuply excited moments ignored in the earlier studies. Second, the CISthph

101

wave function |\Ilo) used in the design of the MMCC(2,6) energy correction provides
a reasonable description of multiple bond breaking. By inserting the CISthph wave
function |\Ilo) into the MMCC(2,6) energy expression, we obtain a virtually exact
description of multiple bond dissociation.

The only practical problem with the CI-corrected MMCC(2,6) approach is that
this method requires defining active orbitals. In other words, the CI corrected
MMCC(2,6) method is not as easy to use as the standard and renormalized CCSD(T)
and CCSD(TQ) approaches. This issue has been addressed in this thesis research by
developing the QVMMCC and QMMCC methods. The QMMCC methods can be re-
garded as the higher-order extensions of the existing “black-box” CR-CCSD(T) and
CR—CCSD(TQ) approaches that work for single as well as multiple bond breaking.
We confirmed this by performing calculations for the double dissociation of water and
triple bond breaking in N2, obtaining ~1 millihartree accuracies in both cases. For
single bond breaking (we used HF as a test case), the QMMCC methods work as well
as the CR-CCSD(T) and CR—CCSD(TQ) approximations, which provide excellent
results in this case.

The only case where the QMMCC methods did not work as well as expected was
the bond breaking in C2. The QMMCC methods provide significant improvements
over the standard and renormalized CCSD(T) and CCSD(TQ) results in this case, but
the potential energy curves obtained in the QMMCC calculations are characterized
by 10—20 millihartree errors at larger C—C separations. The main reason for this
poorer performance of the QMMCC methods is the extremely low quality of the T1

and T2 cluster amplitudes used to construct the QMMCC corrections.

102

The poorer performance of the QMMCC approximations for C2 calls for further
studies. We believe that the best course of action in this area will be to improve the
quality of the T1 and T2 cluster operators that are used in the design of the QMMCC
and other MMCC energy corrections. We might, for example, try to minimize the

expectation value of the Hamiltonian with the CCSD wave function, i.e.,
EXCC(T1. T2) = (0IeT1’ ”‘2’ Hew‘iI<I>>/(<1>Ie"I ”’2’ eT1+Til<I>> . (119)

with respect to cluster amplitudes till and till";2 defining T1 and T2 instead of calculat-
ing T1 and T2 by the standard CCSD method.”""’143 This would enable us to obtain
the variationally best T1 and T2 operators, which might be better for the MMCC
calculations. Unfortunately, it is very diflicult to propose an efficient algorithm for
calculating the energy expressions of the type of Eq. (119), since the expectation
value of the Hamiltonian with a CC wave function represents a nonterminating series
in cluster components.16 There are, however, other ways of improving the quality of T1
and T2 clusters. For example, Piecuch et. al.135'136'14“ have recently demonstrated that
one can obtain reasonably good T1 and T2 cluster amplitudes, in a computationally
tractable fashion, without resorting to the nonterminating series in cluster compo-
nents resulting from Eq. (119), by using the extended CCSD (ECCSD) method or
one of its approximate variants based on the ECC theory.1“‘""155 Just like the standard
CCSD theory, the ECCSD approach uses only one and two-body cluster components.

The only difference between the standard CCSD method and the ECCSD approach

is the use of two independent sets of singly and doubly excited cluster amplitudes in

103

the ECCSD calculations. As shown by Piecuch et al.,135'136'1“ the T1 and T2 clusters
resulting from the ECCSD calculations are of much higher quality than the T1 and
T2 clusters obtained with the standard CCSD method. In particular, the ECCSD
method applied to N2 does not suffer from the non-variational collapse observed in
the CCSD calculations at larger N —N separations. Thus, it will be very interesting to
examine the effect on the QMMCC and other MMCC results due to using the better
T1 and T2 clusters resulting from the ECCSD calculations. The preliminary results
for the minimum basis set model of N2 are most encouraging.144 There is, therefore,
some hope that we will have new ways of improving the results for the most chal-
lenging types of bond breaking (such as that in C2) by combining the ECCSD and

MMCC ideas.

104

Appendices

105

Appendix A. An elementary derivation of Eq. (21)

The MMCC theory and all MMCC approximations, including the renormalized
and completely renormalized CCSD[T], CCSD(T), and CCSD(TQ) approaches, the
CI-corrected MMCC methods, and the quasi-variational and quadratic MMCC meth-
ods, are based on Eq. (21). An elementary derivation of Eq. (21), based on applying
the resolution of identity to the asymmetric energy expression, Eq. (24), is presented
in this Appendix (see Ref. 46 for the original proof; cf., also, Ref. 45 for the alternative
derivation).

First, we insert the resolution of identity,

N
P+ZQn=1, (120)
n=l

where

P = I<I>)(<I>| (121)

is the projection operator onto the one-dimensional subspace spanned by the refer-
ence configuration I<I>) and
Qn = Z l¢?,‘.'.2'.-‘f.">(9?,{13-1"l (122)

‘1 <...<in
01 <...<0n

is the projection operator onto the subspace spanned by the n-tuply excited configu-

rations relative to I<I>), into Eq. (24) defining the MMCC functional ACCI‘II]. We get

ACCIW) = (01PM—Eé"))eT“’I0>/(0Ie"‘"l0>

N
+Z<0IQ.(H— Es“)eT""I0)/<0IeT“’I0). (123)

n=1

106

The cluster operator T“) is an excitation operator so that (T(A))I and its positive

powers annihilate I<I>). Thus, the formula for the CC energy can be rewritten as

follows (cf. Eq. (16)):
Eff” = (<I>|e-T""HeT“’|<I>) = (cleeT“’|<I>). (124)
This implies that the first term on the right-hand side of Eq. (123) vanishes. Indeed,

<0|P(H—EéA’)eT""I0>/(0IeT“’I0) ——- (0)0)((0IHeT""I0>- é")/
(WIeT""I0>

= 0, (125)

so that

N
000111) = [(0)021 (H — EéA’)eT""I0)/(0Ie"“’l<1>>. (120)

Next, we apply the well-known property of the CC exponential ansatz,15'16'35'381“5

HeT“’|<i>) = eT“’(HeT“’)CI<1>) = eT‘“ H<A>|<1>), (127)

where H”) is the similarity-transformed Hamiltonian of the CC theory, Eq. (9), to

obtain the expression

Q. HeT“’I<I>) = Q. J” HWI<I>> = 20.. (eT"")..-e H£A’I<I>>, (128)
k=0

107

where, in general, 0), represents the k-body component of operator 0. We can obtain
Eq. (127) by multiplying HeTWIQ) on the left by eTWe‘TW = 1 and by realizing
that the similarity-transformed Hamiltonian H”) can be identified with the connected
product of H and CT“) (see, e.g., Refs. 35, 38, 45 for more information). The I: = 0
term in Eq. (128) gives the unlinked, energy-dependent, part of Q" H 8TH) IQ), namely,
EgMQneTWIQ) (see Eq. (16)). This immediately implies that Eq. (128) can be

rewritten in the following form:“5146

Q. (H — EéA’)eT""I<I>> = ZQ. (eT"")._r 111%) = 262.. 0..-).(mn) Mr(m.t)I<I>>.
k=l k=l

(129)

where Cn_k(m,4) is the (n — k)-body component of ET”), Eq. (22), and Mk(mA)IQ)

is defined by Eq. (23). In deriving Eq. (129), we used the identity
1114’10) = 91.110410) 2 M.(m,.)10>. (130)

Before we proceed further, let us comment on the many-body structure of Eq.
(129). Since the zero-body contribution Co(mA) = (eT(A))o equals 1, the k = n
term in Eq. (129) corresponds to the connected component of Qn HeTmIQ), i.e.,
Qn(HeTW)CIQ). Thus, all remaining terms in Eq. (129) with 0 < k < 12 represent
the linked but disconnected components of Q" HeT(A)|Q) (see Ref. 45 for further
details).

In order to complete the derivation of Eq. (21), we substitute Eq. (129) into Eq.

108

(126) to obtain the following result:

N n
4000111 = 2 201102.. 0..-).(m1) Mt(mn)l0>/<0IeT“’I0) . (131)

n=l 1:21

From Eq. (24), it immediately follows that we can replace ACCIQ] by the energy
difference E0 — E84) E 68"), where E0 is the exact, full CI, energy if IQ) is the full CI

state |\Ilo). Thus,

71

N
03’” = Z (0010.. 0..-,(m1)Mt(m1)l0)/<0oIeT“’)0) . (132)

n=l k=l

Eq. (132) is valid for any values of the cluster amplitudes defining T”). In practical
applications of the MMCC theory, we assume, of course, that T“) is obtained by
solving the standard CC equations, Eq. (8), such that the generalized moments
M““"* (171A) with k = 1,...,m,4 vanish. This immediately implies that (cf. Eqs.

01 ODIak

(8), (20), and (23))
Mk(mA)IQ) =0, fork =1,...,mA. (133)

The substitution of Eq. (133) into Eq. (132) reduces the summations over n and k

N n
in Eq. (132) to Z Z , giving us the desired result, Eq. (21).

n=mA+l k=mA+1

109

Appendix B. The many-body diagrams for the generalized moments
of the CCSD equations
The explicit expressions for the generalized moments of the CCSD equations that
appear in the MMCC energy formulas can be most conveniently obtained using di-
agrammatic methods of MBPT. The basic elements of the diagrammatic language
used by us in deriving the expressions for the complete set of the CCSD moments,

i.e- Max:220), M31332... (2). Mr,‘:;::.::i..(2). and Mia‘giifiizinw). and the diagrams
representing these moments are described in this Appendix. The resulting algebraic
expressions in terms of molecular integrals and cluster amplitudes are given in Ap-
pendix C.

Basically there are two closely related types of many—body diagrams: the Hugen-
holtz diagrams and the Goldstone diagrams. Both are inspired by the Feynman
diagrams used in quantum field theory and time-dependent perturbation theory, al-
though in the development of quantum chemical methods, it is sufficient to use the
time-independent formulation of diagrammatic techniques introduced to chemistry
by Citewvm

The diagrammatic theory starts from selecting a Fermi vacuum state IQ), which
is a single-determinantal state that is typically chosen to provide a reasonable ap-
proximation to the ground electronic state of a given many-fermion system under

consideration. When we promote an electron from a spin-orbital occupied in IQ) to

a virtual spin-orbital unoccupied in IQ), we create a hole in the Fermi vacuum and a

0102

particle in the virtual spin-orbital. For example, the Slater determinant IQ“: ) is a

2-hole/2-particle state (two electrons occupying hole or occupied spin-orbitals 2'1 and

110

2'2 are promoted to virtual or unoccupied or particle spin-orbitals a1 and a2). As a
rule of thumb in drawing diagrams, one says that holes “run backwards in time” and
particles “run forward” (although time plays here only a formal role; there is no time
in the time-independent description used by us).

There are two conventions for drawing the Goldstone and Hugenholtz diagrams:
(1) the formal time for determining the hole-particle character of single-particle states
flows from the right to the left, or (2) the formal time flows from the bottom to the t0p.
We will use the former convention, which is to say that the hole lines (representing
occupied single-particle states) run from the left to the right and the particle lines
representing the unoccupied single-particle states run from the right to the left. We
will use the Hugenholtz diagrams, which are less numerous and easier to draw, when
compared to the Goldstone diagrams.

Essentially, every diagrammatic calculation reduces to the following operations:

1. First, we represent the second quantized operators A“, u = 1,. . . , M, entering
M

a given operator product “A”, by basic diagrams. In the design of these
11:1

diagrams, we use vertices with incoming lines representing annihilation opera-
tors and outgoing lines representing creation operators. Each basic vertex A),
contains information about matrix elements in a spin-orbital basis defining op-
erator A”. For example, the basic diagrams relevant to our MMCC calculations
(in a Hugenholtz representation), representing the one- and two-body parts of

the Hamiltonian H N in a normal-ordered form,

FN = Z ngicch} e Mich} (134)

P00

111

and

VN z: i Z vgN{cpcqc,c,} = l1)”"N{c“’c"c,c,} , (135)

- 4 129
p,q,r,a
respectively (f is the Fock operator and N { . . .} is the normal product) and the

cluster operators T1 and T2 are

 

respectively.

. Next, we align the basic diagrams A), along the invisible horizontal line using

the same ordering of these diagrams as in the corresponding operator product

M
11 A1»
p=l

 

M
I—IAu ——> [Afi [AZJW IE

u=1

. In the third step, we connect (contract) the lines of diagrams A), in all possible

ways using the rules

112

 

 

._£,_. ...... ""——>——oq : c p > q o

o—Ee— ...... '———<9—. I o p F %
o—Be— ...... q : 0
._p>_, ...... q : O ,

which reflect the fact that the only nonvanishing contractions of creation and an-
nihilation operators are c” 0,, = 6WH (q) and cp c" = JWII — H (q)] , where H (q) =
l for q being a hole and 0 for q representing a particle (5109 is the usual Kro-
necker delta). Connecting lines of diagrams is equivalent to using the Wick’s
theorem.156 Only nonequivalent resulting diagrams should be considered in the
final analysis (some resulting diagrams can be obtained in several ways, but we
eliminate these repetitions). Specific expressions (such as those in CC theory)

may lead to the elimination of certain types of resulting diagrams.

. Once the allowed resulting diagrams are drawn, we assign the algebraic expres-

sion to each and every one of them using the rules discussed below. The final
M

expression for the operator product of interest, H A,“ is a sum of the alge-
u=l

braic expressions corresponding to nonequivalent resulting diagrams allowed by

a given many-body theory.

Every resulting diagram consists of one or more basic (e.g., VN or T2) vertices that

are connected by oriented (i.e. fermion) lines and that may have some external (i.e.

uncontracted) lines. In general, we interpret such a diagram by forming a product of

(a) the weight factor, (b) the sign factor, (c) the scalar factor, and (for example, in

the wave function expressions) ((1) the operator part, and by summing the resulting

113

expressions over all relevant hole and particle indices, if such summations exist in

operators A”. For the Hugenholtz diagrams used here, the weight factor is 2"‘,

where k is the number of pairs of topologically equivalent (e.g., identically oriented)

fermion lines. We must remember that the identically oriented lines carrying fixed

(i.e. not summed) spin-orbital indices can never be regarded as equivalent lines. The

scalar factor is a product of matrix elements associated with the individual vertices
M

A), from the product H A” entering the resulting diagram. The operator part (if
:1

it appears in the expression) is a product of the creation and annihilation operators

associated with the uncontracted external lines.

The Hugenholtz diagrams do not specify the overall sign of the contribution of
the diagram. This is due to the fact that the basic Hugenholtz diagrams, such as
VN or T2, use antisymmetrized matrix elements 12;, till‘ga, etc. In order to determine
the sign, it is necessary to draw one Goldstone representative, called the Brandow
diagram,157 for each Hugenholtz diagram. This can be done by “expanding” the

Hugenholtz vertices. For example, the Hugenholtz VN vertex in a resulting diagram

must be replaced (only for the purpose of sign determination) by

 

(Hugenholtz) (Brandow)
In general, in order to obtain a Brandow representative of a given resulting Hugen-
holtz diagram, we replace all basic Hugenholtz vertices by Goldstone vertices, as
shown above, while keeping the directions of the lines and the connectivity intact.

Usually more than one possibility exists, but this is not a problem: we choose any

114

Goldstone representative of a given Hugenholtz diagram as a Brandow diagram. Once
the Brandow diagram is drawn, we determine the sign factor for it by counting the
number of loops (1) and the number of internal hole lines (h) and by using the sign
formula (—1)‘+".
We used the Hugenholtz (and Brandow) diagrams to derive the explicit many-body
expressions for all the terms that correspond to the generalized moments M3323. (m A),
= 3 — 6, Eq. (28). There are several methods of obtaining all resulting Hugen-
holtz diagrams in a highly efficient manner.156 We follow the recipe, described in Ref.
156, in which we first draw the nonoriented Hugenholtz skeletons from nonoriented
basic vertices representing FN, VN, T1, and T2. The orientations of fermion lines are
introduced in a subsequent step, once the nonoriented skeletons are determined. In

the case of expressions for the generalized moments of CC equations, where we have

01 .Oiak

to project (H NeT‘+T’)CIQ) on the excited determinants IQihik ), we do not draw

the diagram representing the state (Qfl‘fak . Instead, we draw all permissible dia-

..i,.
grams for (H NeT1+T’)C|Q) with k incoming and k outgoing external lines labeled by
fixed indices 2'1, . . . , 2),, a1, . . . , ak. This facilitates the process of drawing the resulting
diagrams and makes the resulting diagrams less complicated.

We begin our discussion with the Mill‘gjgsm) moments obtained by projecting the
CCSD equations onto triply excited configurations. From Eq. (34), it immediately
follows that the following expression must be examined in this case:

MffifiJZ) = (‘1’?1‘1‘2123lliFNT22 + VN(T2 + 7172 + 571272 + 2154113712 + %T2?

+iTiT§)]oI<I>). (130)

115

where the subscript C designates the connected part of the operator expression. The
connected part of an operator expression, such as that in Eq. (136), can be represented
graphically by drawing all nonequivalent connected diagrams formed by “connecting”
the FN or VN vertices with zero, one, two, three, or four T (i.e., T1 or T2) vertices, as

shown symbolically below.

0.

)1

ENE!

0. 0

At most four T cluster components can be attached to the Hamiltonian operator
vertex, since H N = FN + VN contains up to two—body interactions.

The seven terms in Eq. (136) lead to the following eight nonoriented Hugenholtz

diagrams (or skeletons):

(III)

(IV)
; (V) (VD (VII) (VIII)

 

 

v><>o>c
FN WV 7'1 T2

 

 

All eight skeletons have six external lines extending to the left that correspond to the

116

projection of (H MET” T’)CIQ) in Eq. (34) on triply excited configurations. The final
oriented Hugenholtz diagrams corresponding to skeletons (1)—(VIII) are obtained by
inserting line orientations in all possible ways. The resulting oriented Hugenholtz

diagrams representing moment Mi‘l‘gjzsfl), Eqs. (34) and (136), are given below.

Oriented Hugenholtz diagrams for M“‘"3 (2) (skeletons I—VIII)

010203

E Sir/iissaraz/a, i
a?

(I) ‘3 (I)

 

117

il a?
— 2 '3 + s. . s (I:
= Sin‘z/‘ss‘I/“zas a; '1/‘133 ‘1‘2/‘3 0,
I3 I;

(II) (IIA) (IIB)

 

   

(In) (IIIA) (IIIB)

    

(IIIC)

(IV)

   

(VB)

118

E Sis/1.3111121. °

 

(VI)
(VIA) (VIB)

+ Silt/i3 Sal/02123 a

 

(VII)

+ 85/3135‘1/‘1/‘3

 

 

(VIII)

119

The S operators in front of the Hugenholtz diagrams above (and, also, in the ori-

ented Hugenholtz diagrams corresponding to the remaining moments M31123?“ (2),

......

2142;333:1304», and Mgl'gggggggaom) considered below) are the diagram symmetrizers
that are used to permute the indices labeling external lines for both the occupied and
unoccupied spin-orbitals, to obtain the nonequivalent diagrams resulting from the
same arrangement of oriented fermion lines in a given skeleton. The actual form of
these symmetrizers depends on the topology of a particular diagram and the equiva-
lencies among external lines. For example, the symmetrizers 851),”, and 8;, /1‘2/1‘a seen
in front of several of the diagrams corresponding to the triply excited moments are

defined as follows:

Sir/i212: : liiizis + (2.17:2) + (2.13.3) 2 (137)
Sir/iz/ia = 1110213 + (3112) + (iris) + (his) + (212233) + (ilisizli (133)

where limit is a three-index identity operator, (i122), (2123), and (i223) are the two-
index permutations (e.g., (2122) means interchange labels 2'1 and 2'2), and (211'22'3) and
(£12322) designate the three-index cyclic permutations (e.g., (2'12'22'3) indicates that
label 2'1 is replaced by 2'2, in is replaced by i3, and 23 is replaced by 2'1). Notice that in
the slashed quantities Sir/1213 and 851/12)“, the only permutations allowed are those
which interchange the indices belonging to different groups of symbols separated by
the “/” sign. For example, we do not permute 1'2 and is in Shh-2,3. The analogous
formulas can be given for the unoccupied labels al, a2, etc. and one can easily extend
the definitions (137) and (138) to cases involving permutations of more than three

indices (needed in the case of higher-than-triply excited moments M23232) with

120

k>m.

We can use the above methodology to draw the Hugenholtz diagrams representing

the remaining moments M‘1‘9‘3“ (2), /\/f""""“5 (2), and M'1'9’3““'° (2). The for-

01020304 0102030405 010203040506

nnnnnn

i1i2i3i4 i1i2i3i4i5 £11213141535
mulas for the moments Malagaaa‘(2), Malagasamm), and Mmaaasamaofl) that must

be represented diagrammatically are

Mi‘.‘::::‘..(2) = <<I>?:.-‘;1;‘:.“*I[vr(%T§ + 111,132+ 1,2: + iTETEHch. (139)
Mil,‘:;'::‘::..(2) = («1311;33:331qung + tTiT:)1oI<I>>. (140)
Mi‘,‘:§:§‘.ffiia,(2) = (23.2233312223“ (;-,V~T2“)cl<1>>- (141)

In the next few pages, the diagrams are written in the following sequence: (i) the
nonoriented Hugenholtz skeletons representing Miljgjggmz), Millfgjgi‘jfasa), and
Mi‘l‘gjig‘mgaafl), followed by (ii) all oriented Hugenholtz diagrams that result from

the nonoriented diagrams for these moments. The Brandow diagrams representing

the Hugenholtz diagrams for all CCSD moments M“""* (2), k = 3 — 6, are listed at

01.n0k

the end of this Appendix.

121

Nonoriented Hugenholtz diagrams for M31123?“ (2)

 

 

 

 

 

 

 

 

 

 

IX
( ) (X)
5
(X1) (x11)

122

Nonoriented Hugenholtz diagrams for Milligjgi‘jfasw) ((XIII) and (XIV)) and

Muzzmuzms (2) (XV)

010203040506

 

 

 

 

 

(XIII)
(XIV)

 

 

 

 

(XV)

123

Nonoriented Hugenholtz diagrams for M‘m'3m5 (2) ((XIII) and (XIV)) and

...... “102030405

Mtitztsutsts (2) (xv)

010203040506

 

 

 

 

 

(XIII)
(XIV)

 

 

 

 

 

(XV)

123

 

Oriented Hugenholtz diagrams for skeletons (IX)—(XV) representing M“""* (2),

   

01mm;
with k = 4 - 6
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124

 

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(XIV) (XIVA)
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E swig/insura/azae/asat

or
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(XV)

126

 

The Brandow diagrams for the complete set of generalized moments of the CCSD
equations (corresponding to Hugenholtz diagrams I—XV)

 

 

Sin/ii} Sana/“3

 

 

 

(I)

Silt/i3 Sal/“2‘3
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132

SiIiA/iz/‘js Salas/“243”“

 

 

 

 

 

Siliyizi‘fij‘sa‘as/apdaaa4

 

 

 

 

133

Appendix C. Algebraic expressions for the Milfgjgza), M31323; 4(2),
M‘l'2'3‘4‘5 (2), and ./\/i'”2'3'“~"'6 (2) moments of the CCSD equations

(1102030405 010203040506

In this Appendix, we present the final algebraic expressions for the Hugenholtz and
Brandow diagrams (1)—(XV) representing the complete set of moments of the CCSD
equations used in the MMCC calculations. Let us recall that the weight factor for
each diagram is read from the Hugenholtz form of the diagram. The sign factors and
the scalar factors, expressed in terms of matrix elements fg E (pl f |q), where f is the
Fock operator, and 2);; E (pqlvlrsh = (pqlvlrs) — (pqlvlsr), where v is the interaction,
and cluster amplitudes t?! E (alltllil) and ‘33, E <0102|t2li1i2>A are read from the
Brandow diagrams. To prOperly account for the diagrams resulting from the action of
the symmetrizer S on a specific oriented Hugenholtz diagram, an analogous operator
A, called the antisymmetrizer, must be introduced in the corresponding algebraic
expression. For example, the antisymmetrizer Ail/i225 corresponding to symmetrizer

Shh-2,3 is defined as

Ail/i253 : liiizia _ (hi2) — (213.3) - (142)

Similarly, the four-index antisymmetrizer Aim/Mi, corresponding to the diagram

symmetrizer 8,1,, /,-3 /i4 can be defined as

Ania/ism = 1i1i2i3i4 — (212's) — (M4) — (his) — (i2i4) — (1.3331) + (i1i3)(i2i4)

+(i1i31'4) + (i2i3i4) + (2.11.425) + (i2i4i3) —' (i1i3i2i4) » (143)

where (i1i3z'22'4) is a four-index cyclic permutation. In some cases, there are dia-
grammatic contributions to the quadruply-excited moments M21233?“ (2) that do not

134

require the genuine four-index antisymmetrizers and use the three-index Operators
instead (see the algebraic expressions below).
The antisymmetrizers, signs, and weights corresponding to the Hugenholtz/Brandow

diagrams (1)—(XV) listed in Appendix B are shown in the following table:

 

 

Diagram Antisymmetrizer Weight I h Signs (—1)‘+"

I Ai1/i2i3 Amen/03 71' 0 1 “
HA Aim /i3 A01 /0203 i 0 1 —
HB Ail hm A0102 /03 i 0 0 +
IIIA Afih/iaAO, Mm % 0 2 +
IIIB A, /,-,,-3Ama,/a3 % O 0 +
IIIC Ar, #5534401 /ag/03 % O 1 —
HID Ail /i2/i3-Aa1/a2a3 % 0 1 -
IVA «4n /i2i3-A~olag/a3 % 0 2 +
IVB Ail /i2/i3-A01 /a2a3 % 0 2 +
IVC Ail/igi3A01/ag/a3 % 0 1 —
IVD Aim /i3 A01 /a203 i 0 1 —
VA 442'1/1'21'3 A0102 /03 i O 2 +
VB Aim/53A), /aza3 % O 2 +
VIA A41 /i2 /i3 A0102 /03 § 1 2 —
VIB A4. /i2i3 A0102 /03 § 0 2 +
VIC Am, /i3 A01 /aga3 % 0 l —
VID Ail ig/ia Aa1/aa/a3 :1; 1 1 +

 

135

 

 

Diagram Antisymmetrizer Weight I h Signs (-1)’+"
VIIA Ailiz/ionla3/a2 % 0 2 +
VIIB 441-, fig /,-3.Ama3/a2 % l 2 —
VIIC A, /,,,,A,,, m,” § 0 2 +
VIID Ag, /52i3A0102/a3 % 0 2 +
VIII «41', /i2i3Aalaz/a3 41 1 2 —
IXA Alia/ism A0, mag 11?, 0 2 +
IXB Ail /i2i4 /i3 A0103 /a2 /a.. i 0 1 —
IXC Ail /i2i3 Am a; /a3 /04 fi 0 0 +

XA Ail /i2i3/i4 A0, /aza4 /03 i 0 2 +
XB A11 /i2 /i3 /i4 A01/0203 31; 0 2 +
XC A4, /i2i3A01 /a2 /a3/a4 11; 0 1 —
XD Aim /i2 /i3 A0, /0203/a4 % 0 1 —
XIA Am; /i3 /i4 A01 /a2a3 31—2 0 2 +
X113 A, Mi. /.'3 4401/0203 /a. i 1 2 -
XIC Ail/521’. A0102 /03/a4 315 0 2 +
XIIA AwgiaAmn/aa/a, ,1—6 0 2 +
XIIB Aim, /i2 M An; ma, /03 i 0 2 +
XIIC A,,,/.-3/,-,A.,,/a,a, ,1—6 0 2 +
XIIIA Ail /i2i3/i4i5 Aala5/aza4 /a3 1173 0 2 +
XIIIB Ail i5 /i2i3/i4 A01 /a2 a4 /a3a5 117; 0 1 —
XIVA Am. /i2i3/i5 A01/0205/a3a4 1173 0 2 +
XIVB A.“ m /i2 /i3i5 A0105 #1203 /04 11—6 0 2 +
XV Ania /i2i4 /i5i5 A0105 macs/0304 61—4 0 2 +

 

The final algebraic expressions for all Hugenholtz/Brandow diagrams (1)—(XV) rep-

resenting moments Mi‘l'figk (2), k = 3 — 6, are as follows:

136

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References

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153

__..A_.._-f'~ 04"

Table 1. A comparison of the standard CC, completely renormalized CCSD[T],
CCSD(T), and CCSD(TQ), and CI-corrected MMCC(2,3) and MMCC(2,4) ener—
gies with the corresponding full CI, CISDt, and CISth results obtained for a few
internuclear separations R of the HF molecule, as described by the DZ141 basis set.“

 

 

Method R = R.” R = 2R. R = 3R. R = 5Re
Full CI" -O.160300 0.021733 0.014719 0.016707
CCSD 1.634 6.047 11.596 12.291
CCSDT‘ 0.173 0.855 0.957 0.431
CCSD[T]“ -0070 -2725 -38.302 -75.101
CCSD(T)“ 0.325 0.038 -24.480 -53.183
CCSD(TQ,)€ 0.218 -0.081 -18.351 -35.078
CR-CCSD[T]“ 0.163 0.700 2.508 3.820
CR—CCSD(T)"' 0.500 2.031 2.100 1.650
CR-CCSD(TQ),af 0.053 0.396 0.425 0.454
CR-CCSD(TQ),bf 0.060 0.299 0.316 0.689
CISDt“ 5.783 16.000 29.238 33.627
CISthe~9 5.466 6.730 7.456 7.468
MMCC(2,3)/CI“ 1.195 2.708 3.669 3.255
MMCC(2,4)/CF19 1.207 2.225 3.015 3.066

 

° The full CI total energies are reported as (E + 100) hartree. The standard CC,
CI, CR—CC, and CI-corrected MMCC energies are in millihartree relative to the
corresponding full CI energy values.

b The equilibrium H—F bond length, Re, equals 1.7328 bohr.

c From Ref. 41.

d From Refs. 45 and 46.

c From Ref. 52.
’ From Ref. 134.

9 The active space consisted of the 30, 177, 217, and 40 orbitals.

154

Table 2. A comparison of the standard CC, completely renormalized CCSD[T],
CCSD(T), and CCSD(TQ), and CI-corrected MMCC(2,3) and MMCC(2,4) energies
with the corresponding full CI, CISDt, and CISth results obtained for the equilib-
rium and two displaced geometries of the H20 molecule with the DZ141 basis set.“

 

 

Method R = Re" R = 1.5R: R = 2R;
Full CI 0157866" 00145210 0.094753c
CCSD 1.790 5.590 9.333
CCSDT" 0.434 1.473 -2211
CCSDTQ” 0.015 0.141 0.108
CCSD[T]! 0.362 0.751 41.220
CCSD(T)! 0.574 1.465 -7.699
CCSD(TQm 0.166 0.094 -5914
CR-CCSD[T]’ 0.560 2.053 1.163
CR-CCSD(T)f 0.738 2.534 1.830
CR-CCSD(TQ),af 0.195 0.905 1.461
CR—CCSD(TQ),bf 0.195 0.836 2.853
CISDt-‘1'” 6.922 18.884 49.948
CISth-‘b” 2.702 2.919 5.638
MMCC(2,3)/01M 0.811 2.407 1.631
MMCC(2,4) K3191" 0.501 0.942 2.416

 

° The full CI total energies are reported as (E + 76) hartree. The standard CG,
CI, CR-CC, and CI-corrected MMCC energies are in millihartree relative to the

corresponding full CI energy values.

b The equilibrium geometry and full CI result from Ref. 158.

c The geometry and full CI result from Ref. 159.

d From Ref. 60.

c From Ref. 63.

I From Ref. 46.

9 From Ref. 52.

h The active space consisted of the 101, 3a1, 1b2, 4a1, 2b1, and 202 orbitals.

155

Table 3. A comparison of the standard CC, completely renormalized CCSD[T],
CCSD(T), and CCSD(TQ), and CI-corrected MMCC(2,3) and MMCC(2,4) energies
with the corresponding full CI, CISDt, and CISth results for a few internuclear
separations of the N2 molecule, as described by the DZ141 basis set.“

 

 

Method 0.7512,, R,” 1.2512,, 1.5R. 1.75R, 2R. 2.25R.
Full CI“ 0.549027 1.105115 1.054626 0.950728 0.889906 0.868239 0.862125
CCSD 3.132 8.289 19.061 33.545 17.714 -69.917 -120.836
CCSDTc 0.580 2.107 6.064 10.158 -22.468 409.767 455.656
CCSD(T)“ 0.742 2.156 4.971 4.880 -51.869 -246.405 -387.448

CCSD(TQf)“ 0.226 0.323 0.221 -2.279 -14.243 92.981 334.985
CCSDT(Q;)“ 0.047 -0.010 -0.715 -4.584 3.612 177.641 426.175

CR—CCSD(T)° 1.078 3.452 9.230 17.509 -2.347 -86.184 -133.313
CR-CCSD(TQ),a“ 0.448 1.106 2.474 5.341 1.498 40.784 -69.259
CR—CCSD(TQ),b° 0.451 1.302 3.617 8.011 13.517 25.069 14.796

CISDt” 8.380 19.303 40.247 75.379 126.472 182.996 216.760
CISthe" 5.101 7.233 10.651 18.003 30.226 41.978 51.126

MMCC(2,3)/01"" 1.589 4.076 10.117 18.926 -O.564 -85.380 433.437
MMCC(2,4)/(318»! 1.242 2.354 5.363 11.639 10.831 46.086 30.720

 

“ The full CI total energies are reported as —-(E + 108) hartree. The standard CC,
CI, CR—CC, and CI-corrected MMCC energies are in millihartree relative to the
corresponding full CI energy values. The lowest two occupied and the highest two
unoccupied orbitals were frozen in correlated calculations.

" The equilibrium N—N bond length, Re, equals 2.068 bohr.

“ From Ref. 47.

“ From Ref. 49.

c From Ref. 134.

I The active space consisted of the 309, 11r,,, 2n,“ 1179, 27rg, and 3a,, orbitals.

156

 

Table 4. A comparison of the CI-corrected MMCC(2,5) and MMCC(2,6) energies and
QMMCC(2,4), QMMCC(2,5), and QMMCC(2,6) energies with the results of the full
CI, standard CC, and CISthp and CISthph calculations for the equilibrium and
two displaced geometries of the H20 molecule with the DZ141 basis set.“

 

 

Method R = Re“ R = 1.5R,c R = 2R;
Full CI 0157866" -0.014521° 0.094753°
CCSD 1.790 5.590 9.333
CCSDT“ 0.434 1.473 -2.211
CCSDTQe 0.015 0.141 0.108
CCSD[T]! 0.362 0.751 41.220
CCSD(T)! 0.574 1.465 -7.699
CCSD(TQf)’ 0.166 0.094 -5914
CISthp-‘l’h 2.628 2.578 3.732
CISthph-‘l'h 2.600 2.187 1.922
MMCC(2,5)/019'“ 0.421 0.584 0.730
MMCC(2,6)/019'” 0.417 0.477 0.538
QMMCC(2,4)i 0.271 0.959 2.005
QMMCC(2,5)i 0.202 0.688 0.549
QMMCC(2,6)‘ 0.202 0.688 0.546
QMMCC(2,6) 0.202 0.688 0.546
<M::::33:3“°<2> = 0)i

QMMCC(2,6) 0.206 0.708 0.657

(11491030394059) = 0,

“32‘33435

M91.a?°99495“6(2) _—. 0)i

313233143515

 

“ The full CI total energies are reported as (E + 76) hartree. The standard CC,
CI, QMMCC, and CI-corrected MMCC energies are in millihartree relative to the
corresponding full CI energy values.

“ The equilibrium geometry and full CI result from Ref. 158.

c The geometry and full CI result from Ref. 159.

d From Ref. 60.

c From Ref. 63.

f From Ref. 46.

9 From Ref. 134.

h The active space consisted of the 1b1, 301, 102, 4a1, 2b1, and 202 orbitals.
‘ From Ref. 137

157

Table 5. A comparison of the CI-corrected MMCC(2,5) and MMCC(2,6) energies
and QMMCC(2,4), QMMCC(2,5), and QMMCC(2,6) energies with the results of the
full CI, standard CC, and CISthp and CISthph calculations for a few internuclear
separations of the N2 molecule with the DZ“u basis set.“

 

 

Method 0.75R, R.” 1.25R. 1.5Re 1.7512,, 2R. 2.25R.
R111 016 0.549027 1.105115 1.054626 0.950728 0.889906 0.868239 0.862125
CCSD 3.132 8.289 19.061 33.545 17.714 -69.917 420.836
CCSDT“ 0.580 2.107 6.064 10.158 -22.468-109.767-155.656
CCSD(T)° 0.742 2.156 4.971 4.880 -51.869 -246.405 -387.448
CCSD(TQf)“ 0.226 0.323 0.221 -2279 44.243 92.981 334.985
CCSDT(Qf)“ 0.047 -0010 -0715 -4.584 3.612 177.641 426.175
CISthpeJ 5.401 6.969 8.880 12.086 20.037 28.161 34.276
CISthphe" 5.390 6.799 7.558 6.707 7.189 7.777 8.372
MMCC(2,5)/019! 1.220 2.089 3.527 5.493 1.631 .24.410 39.124
MMCC(2,6)/CI” 1.217 2.022 2.909 3.186 4.048 4.443 4.552
QMMCC(2,4)9 0.458 1.384 3.916 8.362 13.074 22.091 10.749
QMMCC(2,5)9 0.384 1.012 2.365 3.756 1.415 6.672 -2.638
QMMCC(2,6)9 0.384 1.012 2.373 3.784 1.380 6.230 -3440
QMMCC(2,6) 0.384 1.013 2.397 3.782 1.378 6.240 -3.418
(Mttittittmw) = 0)9

QMMCC(2,6) 0.387 1.040 2.533 4.317 2.062 4.674 -6.499

( 11491930994059) = 0,

3132331435

M91.°?a.39‘95°6(2) = 0)::

313233143535

 

“ The full CI total energies are reported as —(E + 108) hartree. The standard CG,

CI, QMMCC, and CI-corrected MMCC energies are in millihartree relative to the
corresponding full Cl energy values. The lowest two occupied and the highest two
unoccupied orbitals were frozen in correlated calculations.

9 The equilibrium N—N bond length, Re, equals 2.068 bohr.

c From Ref. 47.

d From Ref. 49.

‘3 From Ref. 134.

f The active space consisted of the 309, 1n“, 217", 1179, 2179, and 30,, orbitals.
9 From Ref. 137.

158

Table 6. A comparison of the QMMCC(2,4), QMMCC(2,5), and QMMCC(2,6) ener—
gies with the results of the full CI and standard CC calculations for a few internuclear
separations R of the HF molecule, as described by the DZ“fl basis set.“

 

 

Method R = R.” R = 2R. R = 5R.
111m CI“ -0.160300 3.021733 0.016707
CCSD 1.634 6.047 12.291
CCSDT” 0.173 0.855 0.431
CCSD[T]“ -0070 -2725 45.101
CCSD(T)“ 0.325 0.038 33.183
CCSD(TQf)“ 0.218 -0.081 35.078
QMMCC(2,4)f 0.198 0.074 0.277
QMMCC(2,5)f 0.091 -0312 -0797
QMMCC(2,6)’ 0.092 -0.308 -0.798

 

“ The full CI total energies are reported as (E + 100) hartree. The standard CC and

QMMCC energies are in millihartree relative to the corresponding full CI energy
values.

9 The equilibrium H—F bond length, Re, equals 1.7328 bohr.
c From Ref. 41.

d From Refs. 45 and 46.

9 From Ref. 52.

I From Ref. 160.

159

Table 7. A comparison of the QMMCC(2,4), QMMCC(2,5), and QMMCC(2,6) en-
ergies with the results of the full CI, standard CC, and completely renormalized

CCSD(TQ) calculations for a few internuclear separations of the Cg molecule with
the DZ141 basis set.“

 

 

Method 0.75R. R.” 1.5R. 1.75R. 2Re 2.5R. 3R.
Full 016 0.293301 0.641867 0.519881 0.484166 0.462769 0.449248 0.446985
CCSD 16.795 20.684 37.640 36.280 15.772 3.422 -7.438
CCSDT“ 0.698 2.091 14.393 17.704 -4.508 32.502 -26.161
CCSD(T)C -2791 0.389 11.926 1.877 35.176 30.231 -96.055
CCSD(TQ;)" 3.594 -0735 11.347 12.308 5.800 29.196 67.237
CR-CCSD(TQ),b‘-‘ 4.096 4.993 17.830 26.812 24.344 21.096 20.282
QMMCC(2,4)“ 3.379 4.991 19.325 27.897 24.370 20.237 19.192
QMMCC(2,5)“ 2.554 2.875 15.451 23.468 18.696 13.180 11.646
QMMCC(2,6)“ 2.555 2.875 15.458 23.470 18.689 13.169 11.634
QMMCC(2,6) 2.555 2.875 15.458 23.469 18.689 13.169 11.634
(943223333sz = 0)“

QMMCC(2,6) 2.609 3.064 15.879 23.777 18.892 13.243 11.665

(11491032394059) = 0,

3192333495

M91.°?”.39495°6(2) = 0)4

Ingzatqtsto

 

“ The full CI total energies E [reported as —(E + 75)] are given in hartree. The
standard CC, CR—CC, and QMMCC energies are reported in millihartree relative
to the corresponding full CI energy values. The lowest two occupied and the highest
two unoccupied orbitals were frozen in correlated calculations.

9 The equilibrium C-C bond length, Re, equals 2.348 bohr.
c From Ref. 51.
d From Ref. 160.

160

1FIIIIII

unoccupied(al, 02,

virtual(al, 82 , ...)

active(Al, A2, ...)

 

occupied(i1, l2, . .

----.G_-®___--
----.G__e____-
——o—o——
——o—o—

' core(il,i2, ...)

—o—o—
———o—o—

active(ll, 12, ...)

-)

Fig. 1. The orbital classification used in the active-space CI and the CI-corrected
MMCC(2,3), MMCC(2,4), MMCC(2,5), and MMCC(2,6) approaches. Core, active,
and virtual orbitals are represented by solid, dashed, and dotted lines, respectively.
Full and open circles represent core and active electrons of the reference configuration

14>).

161

 

 

 

 

 

 

 

 

 

 

-108.5 ~ ' ’ T‘ ' ’ ‘ -
9
f 0080 /*
-108.6 - . CCSD(T) , .
» o CCSD(TQ,) ,' .
—108.7 . *CCSDT ’9 -
a; v CR-CCSD(T) I,
9 A CR—CCSD(TQ),b ,5
t -108.8 — - MMCC(2,3)/Cl ,1 5
g > MMCC(2,4)/CI 21.-4-3,.
v _ - ‘ ' 9*>--1>_
> 108.9
c»
213
c -109.0 _ -‘
LU
-109.1 7 \- -
409.2 ~ 'K‘ -
\‘4
_109.3 ...LL111.1....1...111...1.1111..
1.5 2.0 2.5 3. 4.0 4.5

0 3.5
RN-N (bohr)

Fig. 2. Potential energy curves for the N 2 molecule, as described by the DZ basis set.
A comparison of the results obtained with the CR—CC and CI-corrected MMCC(2,3)
and MMCC (2,4) methods with the results of the standard CC and full CI calculations.

162

 

T I V I T Y T 7 I I I I I I V
.

 

 

 

 

 

 

 

P.
1 T CCSD J
' 08°“ ” - CCSD(T) ,1
-108.7 7 A CR-CCSD(TQ),b II 2
g <1 MMCC(2,5)/CI
a _ o MMCC(2,6)/Cl yo q
t ”108'8 Full Cl
m -
5 ;
> -108.9 —
c»
E
C -109.0 '—
LIJ
-109.1 .
—109.2 —
_109.3 41..1....1....1....1....1...11..
1.5 2.0 2.5 4.0 4.5

3.0 3.5
RN-N (bohr)

Fig. 3. Potential energy curves for the N2 molecule, as described by the DZ basis
set. A comparison of the results obtained with the CI-corrected MMCC(2,5) and

MMCC(2,6) methods with the results of the standard CC, CR—CCSD(TQ), and full
CI calculations.

163

 

—108.5

I
l

 

—108.6 1- CCSD ,1 —
. CCSD(T) I,
o CCSD(TQ,) ,6
:1 QMMCC(2,4) 1’
<>QMMCC(2,5) 1
-108.8 ” ° QMMCC(2,6) I *
FU" CI ."1:-.. 5‘ --1

—108.7

1

 

 

Energy (hartree)

 

 

 

—108.9 - -

—109.0 - —

409.1 — _ -

409.2 — K -

\‘1

_109.3 1 1 L11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 111 1 1 1 1 14 111 1
1.5 2.0 2.5 3. 4.0 4.5

O 3.5
RM, (bohr)

Fig. 4. Potential energy curves for the N2 molecule, as described by the DZ basis set.
A comparison of the results obtained with the QMMCC methods with the results of
the standard CC and full CI calculations.

164

 

-75.3 T V V T T T I I I I Y I' Y T T T V Y T I V V V I' r T

-75.4 ,a’ .

l

 

 

 

     
 

g ease/£7813
h _
g 1* 11
5 —75.5 - -
53 “1.1.
(T) CCSD “fin-n,
C 30CSDT
UJ ACCSD(T)
-755 - n 0030011,) -
' oCR—CCSD(TQ),b
<>QMMCC(2,5)
. QMMCC(2,6)

 

 

 

[Full Cl

1 l l l l L I l

 

_757 111.111

1.7 2.7 5:3 5.7 6.7

7
C_C (bohr)

Fig. 5. Potential energy curves for the Ca molecule, as described by the DZ basis set.
A comparison of the results obtained with the QMMCC methods with the results of
the standard CC, CR-CCSD(TQ), and full CI calculations.

165

 

   

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