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NEW ALTERNATIVES FOR ELECTRONIC STRUCTURE
THEORY: THE APPLICATION OF TWO-BODY CLUSTER
EXPANSIONS IN HIGH ACCURACY AB INITIO
CALCULATIONS

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Peng-Dong Fan

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NEW ALTERNATIVES FOR ELECTRONIC STRUCTURE
THEORY: THE APPLICATION OF TWO-BODY CLUSTER
EXPAN SION S IN HIGH ACCURACY AB IN ITIO CALCULATIONS

By

Peng-Dong Fan

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Chemistry

2005

ABSTRACT

NEW ALTERNATIVES FOR ELECTRONIC STRUCTURE
THEORY: THE APPLICATION OF TWO-BODY CLUSTER
EXPANSIONS IN HIGH ACCURACY AB INITIO CALCULATIONS

By
Peng-Dong Fan

In this thesis, the applicability of exponential cluster expansions involving one-
and two-electron operators in high accuracy ab initio calculations is discussed. First,
the extended coupled-cluster method with singles and doubles (ECCSD) is tested
in the most demanding studies of potential energy surfaces involving multiple bond
breaking. The numerical results for a few cases of multiple bond breaking show that
the single-reference ECCSD method is capable of providing a qualitatively correct
description of entire potential energy surfaces, eliminating, in particular, the fail-
ures and the unphysical behavior of all standard coupled-cluster methods in similar
cases. It is also demonstrated that one can obtain entire potential energy surfaces
with millihartree accuracies by combining the ECCSD theory with the noniterative
a posteriori corrections obtained by using the generalized variant of the method of
moments of coupled-cluster equations. This is the first time when the relatively sim-
ple single-reference formalism, employing one— and two-body clusters only, provides
a highly accurate description of the dynamic and significant nondynamic correlation
effects characterizing multiply bonded systems. Second, an evidence is presented that
one may be able to represent the exact or virtually exact ground- and excited-state
wave functions of many-electron systems by exponential cluster expansions employing

general two-body operators. Calculations for small many-electron systems indicate

the existence of finite two-body parameters that produce the numerically exact wave
functions. This finding may have a significant impact on future quantum calcula-
tions for many-electron systems, since normally one needs triply excited, quadruply
excited, and other higher—than—doubly excited Slater determinants, in addition to all
singly and doubly excited determinants, to obtain the exact or virtually exact wave

functions.

ACKNOWLEDGMENT

I feel very privileged to have worked with my supervisor, Dr. Piotr Piecuch. To him I
owe a great debt of gratitude for his patience, inspiration, and the warmest support.
Piotr has taught me a great deal about electronic structure theory and guided me to

explore new alternatives in this field.

I also owe a great debt of gratitude to my Guidance Committee, including Dr.
Katharine C. Hunt, Dr. Paul F. Mantica, and Dr. James K. McCusker, for advice,

support, and setting up and overseeing my graduate study program.

I would also like to thank Dr. Karol Kowalski, a former postdoctoral research asso-
ciate in our group, who shared the results of his work when I started working on this
project. I would like to thank Ian, Mike, Ruth, DJ, and Maricris for their friendship

and help in practicing my seminar presentations.

Finally, I would like to acknowledge the Department of Chemistry for providing the
teaching assistantship and the Dissertation Completion Fellowship, and the US De-

partment of Energy for the research assistantship.

iv

Table of Contents

List of Figures vi
List of Tables vii
1 Introduction 1

2 Practical Ways of Improving Standard Coupled-Cluster Methods
Employing Singly and Doubly Excited Clusters via Extended Coupled-

Cluster Theory 5
2.1 Extended Coupled-Cluster Theory: General Formalism ........ 5
2.2 Extended Coupled-Cluster Methods with Singles and Doubles . . . . 9
2.2.1 The Piecuch-Bartlett ECCSD Approach ............ 9
2.2.2 The Arponen—Bishop ECCSD Approach ............ 12
2.3 Numerical Results for Multiple Bond Breaking ............. 17
2.3.1 The Piecuch-Bartlett ECCSD Approach ............ 17
2.3.2 The Arponen-Bishop ECCSD Approach ............ 34
2.4 Conclusion ................................. 41

3 Noniterative Corrections to Extended Coupled-Cluster Energies for
High Accuracy Electronic Structure Calculations: Generalized Method

of Moments of Coupled-Cluster Equations 43
3.1 The Standard Method of Moments of Coupled-Cluster Equations . . 44
3.2 The Generalized MMCC Formalism ................... 50
3.3 The ECCSD(T), ECCSD(TQ), QECCSD(T), and QECCSD(TQ) Meth-

3.4

ods and their Performance in Calculations for 'Il'iple Bond Breaking in

4 Exactness of Two-Body Cluster Expansions in Many-Body Quantum

Theory 70
4.1 Theory ................................... 71
4.1.1 The exp(X) Conjecture ..................... 72

4.1.2 Formal Arguments in Favor of the exp(X) Conjecture (Ground
States) ............................... 75
4.1.3 Extension of the exp(X) Conjecture to Excited States ..... 81
4.2 Numerical Results ............................. 82
4.3 Summary ................................. 89
Bibliography 90

2.1

2.2

2.3

2.4

2.5

3.1

3.2

List of Figures

Potential energy curves for the MBS S4 system. ECCD represents
the Piecuch-Bartlett variant of the ECCD approach (in this case, the
ECCD and ECCSD approaches are equivalent). ............

Potential energy curves for the N2 molecule, as described by the STO-

3G basis set. The LECCSD, BECCSD, QECCSD, and ECCSD meth-
ods are based on the Piecuch-Bartlett variant of the ECC theory.

(a) The overlaps of the normalized CCSD, BECCSD, QECCSD, and
ECCSD wave functions, Eq. (2.62), with the normalized full CI wave
function for the STO-3G N2 molecule, as functions of the N—N separa-
tion R (in bohr). (b) The difference between the CCSD and ECCSD
cluster operators T and the difference between the ECCSD cluster op-
erators T and E, as defined by the quantity d(Y, Z), Eq. (2.63), for the

STO-3G N2 molecule, as functions of the N—N separation R (in bohr).

Ground-state potential energy curves of the N2 molecule, as described
by the STO-3G basis sets. The QECCSD and ECCSD methods are
based on the Arponen-Bishop ECC theory. ..............
Ground-state potential energy curves of the N2 molecule, as described
by the DZ basis sets. QECCSD represents the quadratic version of the
ECCSD theory of Arponen and Bishop ..................

Ground-state potential energy curves of the N2 molecule as described
by the STO-3G basis set. The QECCSD and ECCSD calculations were
performed using the ECC theory of Arponen and Bishop ........

Ground-state potential energy curves of the N2 molecule as described
by the DZ basis set. The QECCSD calculations were performed using
the ECC theory of Arponen and Bishop. ................

vi

19

26

32

38

40

68

69

2.1

2.2

2.3

2.4

2.5

3.1

3.2

4.1

4.2

List of Tables

Ground-state energies of the MBS S4 model system as functions of the
parameter a. ...............................

Ground-state energies of the N 2 molecule, as described by the STO-3G
basis set ...................................

Ground-state energies of the N2 molecule, as described by the STO-3G
basis set. .................................

The ground-state energies of the N2 molecule obtained for several in-
ternuclear separations R with the STO-3G basis set. .........
The ground-state energies of the N2 molecule obtained for several in-
ternuclear separations R with the DZ basis set. ............

The ground-state energies of the N2 molecule obtained for several in-
ternuclear separations R with the STO-3G basis set. All ECC-related
calculations were performed with the Arponen-Bishop ECC theory.

The ground-state energies of the N2 molecule obtained for several in-
ternuclear separations R with the DZ basis set. ............

A comparison of the ground-state energies of the MBS H8 system ob-
tained with the exp(X)-like wave functions, where X = X2 (a purely
two-body operator) or X1 + X2 (a sum of one- and two-body opera-
tors), with the exact, full CI, energies and energies obtained in various
CI and CC calculations. Full CI energies are total energies in hartree
and all other energies are errors relative to full CI (also in hartree). We
also give the overlaps of the normalized ground-state wave functions
obtained with the exp(X) ansatz and~the full CI approach. The over-
laps of the reference states Km) and |<I>o) with the full CI ground-state
wave function |\IIO) are given for comparison ...............

Same as Table 4.1 for the first excited state of the 1Ag symmetry.

vii

25

87
88

Chapter 1 Introduction

Great advances have been made in ab initio quantum chemistry. Highly accurate
calculations for closed-shell and simple open-shell molecular systems involving a
few atoms are nowadays routine. This, in particular, applies to coupled-cluster
(CC) theory1—5, which has become the de facto standard for high accuracy calcu-
lations for atomic and molecular systems‘3—13. The basic single-reference CC meth-
ods, such as CCSD (CC approach with singles and doubles)”, and the noniterative
CCSD + T(CCSD) = CCSD[T]l5 and CCSD(T)16 approaches that account for the
effect of triexcited clusters by using arguments based on the many-body perturbation
theory (MBPT), in either the spin-orbitall4"16 and spin-freer"-19 or spin-adapted'm—22
forms, are nowadays routinely used in accurate electronic structure calculations. The
idea of adding the a posteriori corrections due to higher-than—doubly excited clusters
to CCSD energies, on which the CCSD[T] and CCSD(T) approaches and their more
recent CCSD(TQf) extension23 are based, is particularly attractive, since it leads to
methods that offer an excellent compromise between high accuracy and relatively
low computer cost, as has been demonstrated over and over in numerous atomic and
molecular applications7—10'12’13.

There are, however, Open problems in CC theory. First and foremost is the per-
vasive failing of the standard single-reference CC methods, such as CCSD, CCSD[T],
CCSD(T), and CCSD(TQf), at larger internuclear separations, when the spin-adapted

restricted Hartree-Fock (RHF) configuration is used as a reference, which limits the

applicability of the standard CC approaches to molecules near their equilibrium ge-
ometries. Second is the large computer effort associated with the need for using
higher-than—doubly excited clusters in calculations involving quasi-degenerate elec-
tronic states and bond breaking when larger many-electron systems are examined.
Undoubtedly, it would be very useful to extend the applicability of the standard
single-reference CC methods to entire molecular potential energy surfaces (PESs)
involving bond breaking, and quasi-degenerate electronic states in general, without
invoking complicated and time-consuming steps associated with the more traditional
multi-reference approaches, in which one has to choose active orbitals and multi-
dimensional reference spaces on an ad hoc molecule—by—molecule basis. Ideally, one
would like to develop a straightforward theory which could provide a virtually exact
description of many-electron wave functions with the exponential, CC-like, expansions
involving one- and two-body operators only, since the molecular electronic Hamilto-
nian does not include higher-than—two-body interactions.

There are several specific challenges in all those areas. First of all, the RHF-based
CCSD method, on which the noniterative CCSD[T], CCSD(T), and CCSD(TQf)
approaches are based, is inadequate for the description of bond breaking and quasi-
degenerate states, since it neglects all higher—than—doubly excited clusters, including
the important triply and quadruply excited T3 and T4 components. Second, the non-
iterative triples and quadruples corrections defining the CCSD[T], CCSD(T), and
CCSD(TQf) methods aggravate the situation even further, since the usual arguments
originating from MBPT, on which these standard noniterative CC approximations
are based, fail due to the divergent behavior of the MBPT series at larger inter-
nuclear separations (or when the strong configurational quasi-degeneracy and large

nondynamic correlation effects set in). In consequence, the ground-state PESs ob-

tained with the CCSD[T], CCSD(T), CCSD(TQf), and other noniterative CC ap-
proaches are completely pathological when the RHF configuration is used as a ref-
erence (cf., e.g., Refs. 9, 11—13, 24—37 and references therein). The iterative ex-
tensions of the CCSD[T], CCSD(T), and CCSD(TQf) methods, including, among
many others, the CCSDT-112135“41 and CCSDTQ-l42 approaches, and the nonitera-
tive CCSDT + Q(CCSDT) = CCSDT[Q]42 and CCSDT(Q;)23 methods, in which the
a posteriori corrections due to T4 cluster components are added to the full CCSDT

43"“ energies, improve the description of PBS in the

(CC singles, doubles, and triples)
bond breaking region, but ultimately all of these approaches break down at larger
internuclear distances (see, e.g., Refs. 28, 29, 34). One might try to resolve the fail-
ures of the standard single-reference CC approaches in the bond breaking region and
for quasi-degenerate electronic states in a brute-force manner by including the triply
excited, quadruply excited, pentuply excited, etc. clusters fully and in a completely
iterative fashion, but, unfortunately, the resulting CCSDTQ (CC singles, doubles,
triples, and quadruples)45_48, CCSDTQP (CC singles, doubles, triples, quadruples,
and pentuples)”, etc. approaches are far too expensive for routine molecular appli-
cations. For example, the full CCSDTQ method requires iterative steps that scale
as ngnfi (no (nu) is the number of occupied (unoccupied) orbitals in the molecular
orbital basis). This N10 scaling with the system size (N) restricts the applicability of
the CCSDTQ approach to very small systems, consisting of ~ 2 - 3 light atoms de-
scribed by small basis sets. For comparison, CCSD(T) is an ngnfi (or JV“) procedure
in the iterative CCSD steps and an 723123 (or M) procedure in the noniterative part
related to the calculation of the triples (T) correction. In consequence, it is nowadays

possible to perform the CCSD(T) calculations for systems with 10—20 light atoms

or a few heavier (transition metal) atoms. This indicates that in searching for new

methods that would help to overcome the failures of the standard CC approaches in
the bond breaking region, one should focus on the idea of improving the results of
the low-order CC calculations, such as CCSD, with the noniterative corrections of
the CCSD(T) type, since only such methods have a chance to be applied to larger
molecular systems in the not—too—distant future.

In view of the above discussion, the question: Can one improve the quality of
standard CC wave functions in the bond breaking region at the basic CCSD level of
the single-reference CC theory? seems to be particularly important. In this thesis,
we show that this can be accomplished by exploring the so-called extended coupled-
cluster (ECC) theory. The basic ECCSD results, particularly when multiple bonds are
stretched or broken, are qualitatively much better than the corresponding standard
CCSD results. However, they are not yet fully quantitative. This prompts another
question: Can one improve the quality of the ECCSD results by adding a simple a
posteriori correction to the ECCSD energy which is obtained by using the singly and
doubly excited cluster amplitudes obtained with the ECCSD approach? In this thesis,
'we show that the answer to this question is affirmative if we use the generalized
method of moments of coupled-cluster equations (GMMCC). Eventually, of course,
one would prefer to use only one- and two-body clusters to obtain an exact or virtually
exact description of many-electron systems, since, as we have already mentioned,
the Hamiltonians used in quantum chemistry do not contain higher-than—pairwise
interactions. This prompts the third and the final question of this thesis research:
Can one obtain the exact or virtually exact many-electron wave functions by using

two-body exponential cluster expansions?

Chapter 2 Practical Ways of Improving Standard
Coupled-Cluster Methods Employing Singly and
Doubly Excited Clusters via Extended
Coupled-Cluster Theory

2.1 Extended Coupled-Cluster Theory: General Formalism

The extended coupled-cluster (ECC) theory is based on the asymmetric, doubly con-

nected energy functional50—60,

Elm) =<<I>Ifil<1>>, (2.1)

where |<I>) is the independent particle model reference configuration (e.g., the Hartree-

Fock determinant) and

H = e2t(e'THeT)e’2t = eZtHe’El = (Eff-[)6 = [eEt(HeT)C]C (2.2)
is the doubly transformed Hamiltonian, obtained by transforming the similarity trans-
‘ formed Hamiltonian H used in the standard CC theory,

H = e-THeT = (HeT)C, (2.3)

where H is the Hamiltonian and C stands for the connected part of the corresponding
operator expression, with the exponential operator e'Et. T is the usual cluster opera-
tor, which is a particle-hole excitation operator generating the connected components

of the many-electron ground-state wave function
I‘I/o) = 871(1)). (24)

5

and BI is the auxiliary hole-particle deexcitation operator. In the exact theory, T is

a sum of all many-body components Tn with n = 1,. . . , N,
N
T = Z Tn, (2 5)
n=1

where N is the number of electrons and T... is defined as

_ i1...ifl a1...an
Tn" Z ta1°"anEi1---in , (2.6)
i1<~-<i,,,al<---<a,,
with
n .
aloooan_ an .
Eiwd" _ H c cm (2.7)

K. = 1
representing the excitation operators and $1123" designating the corresponding
cluster amplitudes. We use a notation in which c” (cp) are the usual creation (anni-
hilation) operators (c? = cl) associated with a given orthonormal spin-orbital basis
set {p}. Letters i1, . . . , in represent the occupied spin-orbitals in |<I>) and a1, . . . ,an

designate the unoccupied spin-orbitals. The auxiliary operator 21 is defined as

N
2* = Z 231,, (2.8)
n=1

where

:31, = . Z 0511.1".an13311'33fi'n. (2.9)

0 Zn
The operators

E31221?" = II Cancun (2.10)
K, = 1

. a . . . a . . .
and the coefficrents Ui 1. . n are the corresponding deexc1tatlon operators and am-

1 o 0 Zn
plitudes, respectively.

The operators T and 2* (or the corresponding amplitudes till 1211.3" and 02911.1... ia")
78
defining the wave function I‘llo) through Eq. (2.4) and the energy EéECC) through Eq.

(2.1) can be determined in various ways. In the ECC theory of Piecuch and Bartlett“,
which can be applied to both ground and excited states, the T and El operators are
determined by considering the doubly transformed form of the electronic Schrodinger
equation, i.e.,

line) = E0|<I>), (2.11)

and its left-hand analog,

(<1)le Eo(<I>|, (2.12)

where (<I>| is the left eigenstate of II corresponding to the right reference eigenstate
|<I>). We obtain Eq. (2.11) by inserting the formula for the CC wave function, Eq.

(2.4), into the Schrbdinger equation,
Hl‘I’o> = Eol‘I’ola (2-13)

and by premultiplying the resulting equation by eve—T, while utilizing the fact that
|\Ilo) = eT|<I>) = eTe‘EIIQ) (23t is a deexcitation operator, so that (El)"|<1>) = 0 for
k > 0). In general, the (<I>| dual state entering Eq. (2.12) depends on the values of
T and El defining R. In the ECC theory of Piecuch and Bartlett, we simply require
that T and El are such that (<I>| = (<I>|. Thus, the final system of equations used to

determine the two different cluster operators T and El consists of Eq. (2.11) and
(<12le = <<I>|Eo. (2.14)

which is the left-hand counterpart of Eq. (2.11). It is worth mentioning that Eq.
(2.14) can also be obtained by considering the bra counterpart of the connected

cluster form of the electronic Schréidinger equation,

chp) = E0|<I>), (2.15)

where H is defined by Eq. (2.3), i.e.
(<I>|(1+ ME = E0(<I>|(1+ A), (2.16)

where A is the well-known “lambda” deexcitation operator of the analytic gradient
CC theorym'”, and by identifying the left-hand ground eigenstate of H, (<I>|(l + A),
with (<I>|ezl. An alternative reasoning that leads to Eq. (2.14) is based on considering

the adjoint form of the electronic Schrodinger equation, i.e.
(‘i’olH = Eo<‘i’o|, (2.17)

where the dual state (\Ilol, satisfying the condition (\Ilolqlo) = 1, is the CC bra ground
statemm’“,

(\iIOI = (<1>|(1 + A)e"T. (2.18)

Clearly, in the exact, full C1 or full CC, limit, there exists a deexcitation operator
U, such that (1 + A) = e”, so that one can always give the dual CC state (Ilol a

completely bi-exponential form,
(to) = (slave-T. (2.19)

By inserting Eq. (2.19) into Eq. (2.17) and by multiplying the resulting equation on
the right by eTe‘Sl, we obtain the desired Eq. (2.14).

In the original work by Arponen and Bishopm—59, the operators T and 21 of the
ECC formalism are determined by imposing the stationary conditions on the energy

functional ESECC), Eq. (2.1), with respect to operators T and 21,

6E3ECC)
6T

6 EéECC)
62"

 

 

=0, =0. (2%)

The bi-variational character of the ECC theory of Arponen and Bishop is particularly

useful in calculations of molecular properties other than energy, since one can apply

8

the Hellmann-Feynmann theorem in such calculations‘55‘73. The question addressed
in this thesis is how the bi-variational ECC theory of Arponen and Bishop and the
ECC theory of Piecuch and Bartlett, which uses Eqs. (2.11) and (2.14) rather than
Eq. (2.20) to determine T and 2", work when molecular PESs along bond breaking

coordinates are examined.

2.2 Extended Coupled-Cluster Methods with Singles and Dou-
bles

2.2.1 The Piecuch-Bartlett ECCSD Approach

The approximate ECC methods, such as the ECCSD approaches tested in this work
and developed in Refs. 74—77, are obtained as follows: First, as in all standard CC
approximations, we truncate the many-body expansions of T and El, Eqs. (2.5) and

(2.8), respectively, at some excitation level mA < N, so that T is replaced by T“),

mA
T“) = 2T", (2.21)
n=1
and 2" is replaced by
mA
2”” = Z 21,. (2.22)
n=l

Next, we use either the Piecuch-Bartlett approach (Eqs. (2.11) and (2.14)) or the

Arponen-Bishop approach (Eq. (2.20)) to obtain a system of equations for the un-

”in

’ Z. a a! o o o a
known cluster amplitudes ta1 (1 and 0-1 . ",
l ’ ' ’ n 21 ,

_ 2.1.0.2.”
”Zn n — 1,...,mA. Once tal"'an

and 0?". ° 1“ n are determined, we use the approximate form of the energy functional

1 .02"

(2.1),
35,") = (cplfiwlcp), (2.23)

where

HM) ___ 82( ) (8_T(A)H6T(A))e-E(A)

to calculate the ground-state energy.

(2.24)

In the specific case of the ECCSD approach, T is approximated by the sum of

one- and two-body components, T1 and T2, respectively,

where

and

with

and

T g T(ECCSD) = T1 + T2,
i a
= 2228
i,a

__ ab
T2 — Z tajbEij ’
i<fia<b

a__ a,
Ei—ccz

Eab— — c acicbcj'

(2.25)

(2.26)

(2.27)

(2.28)

(2.29)

representing the elementary single and double excitation operators, and ti, and tab

designating the corresponding singly and doubly excited cluster amplitudes. A similar

truncation scheme is applied to operator El, i.e.,

where

and

Wm

_ a i
_Zai Ea
i,a

2;: Z abEij ab’

(zlECCSW = 2: + 2;,

(2.30)

(2.31)

(2.32)

with E3 and E22 designating the one- and two—body elementary deexcitation opera-
tors, Ei = (Bi-W and E3) = (E3351, respectively, and a? and 0%? representing the

corresponding deexcitation amplitudes.

In the ECCSD method of Piecuch and Bartlett“, we obtain the equations for the

a
i)

and 03;), defining T1, T2, 21, and 23;, respectively, by

cluster amplitudes t5, t3, 0
left- and right-projecting Eqs. (2.11) and (2.14), where T is given by Eq. (2.25) and 2*
is given by Eq. (2.30), on the singly and doubly excited determinants IQ?) = Eglé)

and W???) = EgybICP). We obtain,

(<13 |§<ECCSD>|<1>) = 0, (2.33)
<¢gjblfiaccsml¢> = 0, i < j, a < b, (2.34)
(olfimsmlcg‘) = 0, (2.35)
(¢|RIECCSD)|<I>§‘JI-’) = 0, 2' < j, a < b, (2.36)
where
111230050) = 621”); (e—Tl—T2 HeT1+T2) 6—21—23
= e£l+2lH(CCSD>e-2l-El (2.37)

is the doubly transformed Hamiltonian of the ECCSD method, with
H(CCSD) = €_T1_T2H€TI+T2 = (HeT1+T2)C (238)

representing the similarly transformed Hamiltonian of the CCSD approximation.

Once Eqs. (2.33)-(2.36) are solved for operators T1, T2, 21 and 2;, the ground-state

energy ESECCSD) is calculated as follows:

ESECCSD) = ((1,, fiscasmlé)
: ((plezl-I-Eg (e—Tl —T2 H8T1+T2)€_EI_E;|¢)
= (q)|e’3l+zi(e‘Tl‘T2HeT1+T2)|<I>), (2.39)

11

where we used the fact that e‘zllQ) = IQ).

Since the full ECCSD formalism defined by Eqs. (2.33)—(2.39) is rather complex,
in this thesis, in addition to the full ECCSD method, we consider the systematic
sequence of the linear (LECCSD), bi-linear (BECCSD), and quadratic (QECCSD)
approaches, which represent approximations to full ECCSD. The LECCSD, BECCSD,
and QECCSD methods are obtained by replacing the doubly transformed Hamiltonian
1330080), Eq. (2.37), in Eqs. (2.33)—(2.36) and (2.39) by

ITLECCSD) = (1 + 21 + EDI-{(CCSDKI — El — 2;), (2.40)
in the LECCSD case,

13000009 = (1 + 2) + 293000041 _ 21 _ 2;)

+%(21 + 2;)2H(CCSD) + %H(CCSD)(EI + 2;)2’ (2.41)

in the BECCSD case, and
= 1 _
91000009 = [1 + 2; + 1:; + 5(2:) + 2.).)0] 140001»
1
x [1 — )3} — 2; + 5(2)} + 2})2] , (2.42)

in the QECCSD case.

2.2.2 The Arponen-Bishop ECCSD Approach

In the case of the ECCSD method of Arponen and Bishop50—59, we obtain the system
of equations for the singly and doubly excited cluster amplitudes t2 and thjb defining

T1 and T2 and the deexcitation amplitudes 09 and of}? defining El and 2* from the
z z] I 2

stationary condition represented by Eq. (2.20). The resulting equations can be given

in the following form:

BEéECCSD)

80“ = (leezl‘nglL-I‘CCSDHQ) = 0, (2.43)
i

12

aE(ECCSD) b t f _
_0__ = ((1)3. [621+22H(CCSD)|<I>) = 0, i < j, a < b, (2.44)
3029‘]? ‘7
(ECCSD)
6E0 . : <¢|ezl+zg[g(ccsn)’ qu»
3th
I I —
= (QIeElHSZ (H(CCSD) E?)C|q)) : 0, (2.45)
(ECCSD)
6E0 .. = (@l €21.02; [ H(CCSD) EQbHQ)
at” ’ 1.?
ab

1' _
= (0|e2l+52 (H‘CCSDlEng’)C|<I>) = 0, i< j, a < b, (2.46)

where EéECCSD) is the ECCSD energy functional, Eq. (2.39), and H<CCSD) is the
similarity-transformed Hamiltonian defined by Eq. (2.38).

It can be shown that the full ECCSD method, defined by Eqs. (2.43)—(2.46), leads
to computational steps that scale as N10 with the system size. This is a lot more
than the JV'6 steps of the standard CCSD approach. In order to reduce the computer
costs of the full ECCSD calculations, we must truncate the power series expansion
for e21”; in the ECCSD energy functional EéECCSD), Eq. (2.39), at the low power
of (2‘; + 23;). The simplest approximation of this type that one might suggest is the
linearized ECCSD (LECCSD) formalism, in which we replace the energy functional

EéECCSD)’ Eq. (2.39), by the expression linear in El and 2;,

EéLECCSD’ = <<I>|(1+ 21 + 292003210)
= (012003000) + (0211110039009 + <0I21H‘CCSD’I0>

: <¢|H(CCSD)|¢) + Z 0.76:1. <¢?|H(CCSD)I¢>
i, a
ab ab ‘ (CCSD)
+ Z a” (<1,le |Q). (2.47)
i < j,a < b
The most expensive steps of the LECCSD method, as defined by Eq. (2.47), scale as

ngnfi (or N6 with the system size), but, unfortunately, the LECCSD method based

13

on the Arponen-Bishop theory does not improve the CCSD results at all, since once

we impose the stationary conditions on the energy functional ESLECCSD) with respect

to operators T1, T2, 2}, and 2;, we obtain the following equations:

aE(Lsccsn) _
334 = <0ilH<CCSD>I<I>> = o. (2.48)
z
E(LECCSD) __
L3)?»— = <0ijblH‘CCSD’l0> = 0. .-< j, a < 0, (2-49)
which are the usual CCSD equations for T1 and T2, and
aEmsccsn) _
JEF— = <<I>l(1 + 21 + £1) (H‘CCSD’ 13,9980) = 0. (2.50)
a
aE(LECCSD) _
05%,, = <<I>|(1 + >31 + 2;) (H‘CCSD’ E#)c|<1>> = 0.
ab

i < j, a < b, (2.51)

which are the El equations that are completely decoupled from the CCSD system,
Eqs. (2.48) and (2.49), and solved only after the T1 and T2 cluster are determined. In
other words, the LECCSD approximation does not provide for the coupling between
the T and El equations, which is necessary for improving the quality of T1 and T 2
clusters in the bond breaking region.

The above analysis indicates that in order to obtain the T1 and T2 clusters, which
are better than those provided by the LECCSD = CCSD approximation, we must
introduce nonlinear terms in 21 and 23; into the LECCSD energy functional that
couple the T and 2" equations. The lowest-order approximation of this type is ob-
tained by truncating the ECCSD energy functional, Eq. (2.39), at terms quadratic
in (2} + 5.3;). We call it the QECCSD approximation. The formula for the QECCSD
energy functional used in the Arponen-Bishop version of the QECCSD theory is

1 _
123000000) = (010 + (21 + >31) + 5(21 + 292111208210)

14

= <0IH<CCSD>I0> + <¢|ElH‘eeSD’l<I>>
1 _
+<<I>I12£+ 5(212111‘0052109

+<<I>Izizlfilccsml0> + §<<I>I(>:;)2H<CCSD>I<I>> (2.52)
or, somewhat more explicitly,

E(QECCSD)_ (q, I H(ccsn)|¢) + Z 030%” 1:1(ccsn) I‘D)

i, a

+ Z (aijb + Aijaa 0?)(<I)ajblH(CCSD)|<P)

i < j

a < b
+ Z Ai/jk'Aa/bcOilajk<¢qbcleCSml¢>

i<j<k

a < b < c

CCSD

+ Z ij/klAb/cd Uijb 01.1 edl‘l’mz lH‘ WP) (2 53)

i<j<k<l

a<b<c<d

where Aij = 1— (ij), Ai/jk = 1— (ij) — (ik), Aa/bc = 1- (ab) — (ac), and
A, j / kl = l— (ik) - (il) - (jk) - (jl) + (ik)(jl) are the suitable index antisymmetrizers
((i j) and (ab) are index interchanges). The stationary conditions of the type of Eqs.

(2.43)—(2.46), written for the QECCSD energy functional EéQECCSD), Eq. (2.52) or

(2.53), are
aEtlQECCSD) a '(CCSD) b ab - CCSD)
T = (‘1’: |H |Q)+Zaj(Qz-j|H( |Q)
’ Lb
+ Z J k<<1> W|H(CCSD>|¢)= (2.54)
j < k, b < c
0E0 E(QECCSD)
ab = (Qajb|H(CCSD)Iq)> + 20k<¢§3 bC|H(CCSD)|(p>
60 Uij k, c
+ Z ag‘l’(<1>al’cd|H<CCSD>|<p)= (2.55)
k<Lc<d

15

8 (QECCSD)
=(<I>|[1+(EI+E§ )+— %(2{+2;) 2](H CCSD)EC‘)C |<I>>

ati
= (¢I(H(CCSD)E?)C|<I>) + 20?. ((D2|(H(CCSD)EZQ)CI(I)>
1311
+ Z (a ek+Aka jag)(<1>gg|(H<CCSD>E3)C|¢)
j<hb<c
+ Z 4“]°/1cz*4z;/a1"$201651i (‘1’ d“ H(eeSD’Ee)c I‘D)
j<k<l
b<c<d
+ Z Mbcade ((1,de |(H(CCSD)Ea)C I‘D)
Ajk/lm Ac/de Ojk lm jklm
j<k<l<m
b<c<d<e
= 0, (2.56)
aE(QECCSD) 1 _ b
—°—---—,, = <<I>|[1 + (21 + 2;) + E(>31 + 21.)?) (H<CCSD>Egj )C|q>)
atab
= <<I>|(H‘CCSD’E,‘-‘}’)cld>) + Zag(<1>il(H‘CCSD)E3})C|q>)
k,c
+ Z (agghAklogaquWduH (CCSD)Eab)C (<5)
k < l, c < d
0031) ab
+ Z Ak/lm'A c/de‘7k0'zm(‘I’ic‘zi ml(H( )Ei )0 VP)
k < l < m
c<d<e

+ Z Akl/mnAd/ef akl amn f<<pkl€fn I(H(CCSD)Eab)C |<I>)
k<l<m<n
c<d<e<f

= 0. (2.57)

As one can see, the QECCSD equations for the t3, and t3) amplitudes, Eqs. (2.54)

and (2.55), respectively, are strongly and nontrivially coupled with the equations for

ab
U

standard CCSD approach, the T1 and T2 clusters are no longer calculated independent

the a? and az- amplitudes, Eqs. (2.56) and (2.57), respectively, so that unlike in the

of the El and 2; clusters. Similarly strong coupling is present in the QECCSD,

m

BECCSD, and ECCSD methods based on the Piecuch-Bartlett ECC theory described
in Section 2.2.1. As shown below, a strong coupling between T and 2* equations has
a positive effect on improving the quality of the T1 and T2 clusters resulting from the
QECCSD and similar calculations. At the same time, the most expensive steps of
the QECCSD approach based on Eqs. (2.54)—(2.57) scale as n3 (JV6 with the system
size). Similar remarks apply to the QECCSD and BECCSD methods based on the

Piecuch-Bartlett approach discussed in Section 2.2.1.

2.3 Numerical Results for Multiple Bond Breaking

2.3.1 The Piecuch-Bartlett ECCSD Approach

In this section, we discuss the results of the full and approximate ECCSD calculations
for the minimum basis set (MBS) S4 model“;-80 and the STD-3G81 model of N2
employing the ECC theory of Piecuch and Bartlett. All ECCSD, LECCSD, BECCSD,
and QECCSD calculations reported here were performed using the original computer

codes developed in this work, in which the relevant cluster amplitudes t3, t5] , a“ and

i :
9b
3.7
the ECCSD system, Eqs. (2.33)—(2.36), with the downhill simplex method”. This

a are determined by minimizing the sum of the squares of the equations constituting
algorithm is based on an obvious fact that the global minimum being sought is zero.
For smaller values of the parameter a describing the S4 model (a z 2.0 bohr) and in
the vicinity of the equilibrium geometry of N2 (the N—N separation R z 2.0 bohr),
where the standard CCSD approach provides a reasonable description of the ground-

state wave function, the initial guesses for cluster amplitudes ti , t2] , 09, and age were
a ab 2

3.7
obtained by using the CCSD values for ti and t2] and by assuming that the initial
(1 ab

17

ab
3.7

the first-order MBPT estimates of the cluster operator T and the auxiliary operator 2

ab=tzj

values of a? and 02- satisfy the conditions a? =ta and a ab (one can show that

are identicals3'84). For larger values of a and R, where T and 23 significantly differ, we

ab amplitudes, obtained for smaller

used the previously converged tinte ajb’a “,and 0i]
values of a and R, to initiate the numerical procedures used to determine T and )3.

We begin the discussion of our ECCSD test calculations with the MBS S4 model,
which consists of four hydrogen atoms arranged in a square configuration, described by
a small basis set consisting of only one s orbital centered on each hydrogen atomn‘w.
The geometry of the S4 model is determined by a single parameter a, which is de-
fined as a distance between the nearest-neighbor hydrogen atoms. The small number
of electrons and the fact that the molecular orbital basis set consists in this case of
only four orbitals that are fully determined by the high spatial symmetry of the S4
model cause that there are only eight spin- and symmetry-adapted configurations
that are relevant to the ground-state full CI problem. In addition to the RHF deter-
minant, which we use as a reference, we only need six doubly excited configurations
and one quadruply excited configuration to describe the exact, full CI, ground state
of the MBS S4 model. There are no single and triple excitations in the full CI ex-
pansion of the ground-state wave function, so that T1 = T3 = O and the CCD, CCSD
and CCSDT approximations become completely equivalent. Similarly, the one- and
three-body components of the deexcitation operator 23" vanish, so that the ECCD
and ECCSD methods give identical results. Because of the presence of only four elec-
trons in the MBS S4 model, there is no difference between the BECCSD = BECCD,
QECCSD = QECCD, and full ECCSD = ECCD approximations in this case. Thus,

we only report the results of the full ECCD calculations, which we compare with the

CCD and exact full CI results.

18

 

 

—1.96

 

 

 

 

—1.98

93> —2.00

'5

5 —2.02

> .

9 -2.04

3:: ’ ACCD
-2-°6 _' VECCD -
-2.08 _ OFullCl
_2.1O . 1 . n . n . 1 .

2.0 3.0 4.0 5.0 6.0 7.0
0c(bohr)

Figure 2.1: Potential energy curves for the MBS S4 system. ECCD represents the
Piecuch-Bartlett variant of the ECCD approach (in this case, the ECCD and ECCSD

approaches are equivalent).

19

The MBS S4 models with larger values of a create a serious challenge for the
standard single-reference CC methods (even the genuine MRCC approaches may have
a difficulty in describing these models-’8'“). This is related to the fact that larger
a values correspond to a dissociation of the 84 model into four hydrogen atoms.
This process is very difficult to describe by the RHF-based single-reference methods,
particularly in the region of the intermediate a values where the ground-state wave
function of the S4 system undergoes a significant rearrangement of its structure. As
shown in Table 2.1 and Figure 2.1, the CCD = CCSD approximation breaks down at
larger values of a. For a S 2.0 bohr, the unsigned errors in the CCD results, relative
to full CI, do not exceed 2.834 millihartree. However, for 3.5 bohr 5 a g 4.5 bohr,
the unsigned errors in the CCD results increase to 13—14 millihartree and the CCD
potential energy curve corresponding to the dissociation of the S4 model into four
hydrogen atoms goes significantly below the exact, full CI, curve (see Figure 2.1).
For a g 2.0 bohr, the ground-state wave function of the MBS S4 model is dominated
by two configurations: the RHF determinant and the doubly excited determinant
corresponding to the HOMO —+ LUMO biexcitation78-80. For larger values of a,
essentially all electron configurations present in the full CI expansion become very
important, creating a strongly quasi-degenerate situation, and the role of the T4
cluster component becomes significant in the intermediate 2.5 bohr S a S 6.0 bohr
region. This leads to the failure of the CCD approximation observed in Table 2.1 and

Figure 2.1.

20

Table 2.1.: Ground-state energies of the MBS S4 model system as functions of the

parameter a.“

 

 

 

a Full CI CCD b ECCD c (CCD)d (ECCD)d
0.5 3.952114 —0.093 0.015 0.032 0.015
1.0 —0.668783 —0.424 0.110 0.236 0.110
1.5 -1.694327 —1.235 0.376 0.862 0.376
2.0 —1.975862 —2.834 0.871 2.188 0.871
2.5 —2.043797 —5.674 1.609 4.447 1.609
3.0 —2.044850 —9.568 2.373 6.895 2.373
3.5 —2.028186 —13.083 2.754 7.698 2.754
4.0 —2.011713 —14.385 2.496 6.218 2.496
4.5 —2.000438 -13.094 1.812 3.896 1.812
5.0 —1.994021 —10.292 1.107 2.054 1.107
6.0 -1.989199 -4.748 0.297 0.431 0.296
7.0 —1.988164 —1.743 0.062 0.076 0.061

 

 

“The full CI total energies are in hartree. The CCD and ECCD energies are in millihartree
relative to the corresponding full CI energy values. The parameter a is in bohr.

”For the MBS 84 model, CCD = CCSD.

cThe ECCD method of Piecuch and Bartlett. For the MBS S4 model,
ECCD = ECCSD = BECCSD = QECCSD.

d(X) (X = CCD, ECCD) is the expectation value of the Hamiltonian with the 6T2 |<I>) wave

function, where T2 is obtained with method X (see Eq. (2.58)).

21

 

The results in Table 2.1 and Figure 2.1 show that the ECCD theory provides
substantial improvements in the poor description of the dissociation of the S4 model
into four hydrogen atoms by the CCD method. The 13-14 millihartree unsigned errors
in the CCD energies in the 3.5 bohr g a g 4.5 bohr region reduce to 2-3 millihartree,
when the ECCD approach is employed. The considerable reduction of errors in the
CCD results is observed at all values of (1, even in the a S 2.0 bohr region, where
the maximum unsigned error in the ECCD results is 0.871 millihartree, as opposed to
2.834 millihartree obtained with the CCD approach (see Table 2.1). Unlike the CCD
potential energy curve shown in Figure 2.1, which is located significantly below the
full CI curve, the ECCD potential energy curve describing the dissociation of the S4
system into four H atoms is located slightly above the full CI curve. Thus, in spite of
its formally nonvariational character, the ECCD approach based on the ECC theory
of Piecuch and Bartlett60 provides a highly accurate and variational description of
the breaking of all four H—H bonds in the S4 system.

The substantial improvement in the description of the S4 model system offered by
the ECCD approach suggests that the T2 clusters resulting from the ECCD calcula-
tions are considerably better than the CCD T2 values. This can be seen by calculating
the expectation values of the Hamiltonian, designated by (X), where X = CCD or

ECCD, with the normalized CCD-like wave functions

5113’”) = N<X>eTéX’|<1>), (X = CCD, ECCD), (2.58)
where Téx) (X = CCD, ECCD) are the T2 cluster components obtained with the
CCD and ECCD methods, respectively, and N (X) = (@Ie‘TigX))teT2(X)|<I>)‘1/2 are the
appropriate normalization factors. Clearly, the (CCD) and (ECCD) values provide

the upper bounds to the exact, full CI, energies. However, as demonstrated in Table

2.1, the (CCD) energies remain poor in the 3.5 bohr S a g 4.5 bohr region, whereas

22

the expectation values of the Hamiltonian calculated with the ECCD wave function
IWSECCD)» Eq. (2.58), are very close to the corresponding full CI energies.

The MBS S4 model is so simple that we can clearly understand the reasons of
the excellent performance of the ECCSD = ECCD theory at larger 0 values. The
MBS S4 system has only four electrons and the T1 and T3 components vanish due to
the high symmetry of the Hamiltonian. Thus, in order to obtain a highly accurate
description of the ground electronic state of the MBS S4 system, we must use a
method which is capable of providing an accurate description of the effects due to
connected quadruply excited (T4) cluster components, whiCh are missing in CCSD.
It turns out that at least some of the T4 effects are brought into the ECC formalism
as products of the low-order many-body components of 23‘ and T. Indeed, as shown,
for example, in Ref. 23, the leading, fifth-order, contribution to the energy due to T4

clusters can be estimated by adding the E5]Q term, defined as

1
E818 = Z<2I<T§)?(V~T§)CI<I>>, (2.59)
to
1
E814 = 5<9I<T§>Z<VNT§2UCI95 (2.60)
where
TI” = R83’(V~T2)c. (2.61)

In the above expression, R83) designates the three-body part of the MBPT reduced re-
solvent and VN represents the two-body part of the Hamiltonian in the normal-ordered
form, H N = H — (<I>|H|<I>). For the MBS S4 model, the E8.) energy component, Eq.
(2.60), vanishes, since Ty] = 0. Thus, the entire fifth-order effect due to T, can be
estimated in this case by the E5]Q contribution, Eq. (2.59), which appears in the

ECCD energy as the %(<I>|(E;)2(VNT22)C|<I>) term, since T 2 and 232 are similar when

23

MBPT converges. This means that the ECCD energy for the MBS S4 model contains
a great deal of information about the leading effects due to T4 and these are sufficient
to provide the excellent results at all values of the parameter a observed in Figure
2.1 and Table 2.1.

The MBS S4 model allows us to obtain useful insights into the performance of the
ECC approximations in the calculations for quasi-degenerate electronic states, but
we cannot use it to test all important aspects of the ECC theory. For example, the
MBS S4 system is too simple to analyze the importance of the T1 and 23} components
and it does not allow us to understand the significance of terms that distinguish the
LECCSD, BECCSD, QECCSD, and full ECCSD approximations defined in Section
2.2.1. Moreover, multiple bond breaking in real molecules can be considerably more
complicated than the dissociation of the H4 cluster represented by the S4 system into
four hydrogen atoms. A good example of the very challenging situation, which is
considerably more complex than the situation created by the S4 model, is provided
by the triple bond breaking in N2, where the standard CCSD approach displays a
colossal failure (see Table 2.2 and Figure 2.2). We tested the ECCSD, LECCSD,
BECCSD, and QECCSD methods, based on Eqs. (2.33)—(2.42), using the minimum
basis set STD-3G81 model of N2. In all correlated calculations for N2 discussed below,

the lowest two core orbitals were kept frozen.

24

Table 2.2.: Ground—state energies of the N2 molecule, as described by the STD-3G

 

 

 

basis set.“

Rb Full CI CCSD BECCSDc QECCSDc ECCSDc
1.0 -101.791600 0.319 0.298 0.298 0.298
1.5 — 106.720117 1.102 0.885 0.885 0.885
2.0 -107.623240 3.295 1.897 1.897 1.897
2.5 — 107.651880 9.220 3.442 3.442 3.428
3.0 - 107.546614 13.176 3.919 3.908 3.758
3.5 — 107.473442 —38.645 5.280 5.322 4.746
4.0 -107.447822 — 140.376 15.580 15.968 14.122
4.5 —107.441504 — 184.984 26.795 27.769 24.039
5.0 -107.439549 -200.958 34.134 35.732 30.390
5.5 —107.438665 —206.974 38.368 40.491 33.867
6.0 —107.438265 —209.538 40.730 43.227 35.746
7.0 — 107.438054 —21 1.915 42.754 45.595 37.306
8.0 —107.438029 -—213.431 43.405 46.320 37.799

 

 

“The full CI total energies are in hartree. The CC and ECC energies are in millihartree

relative to the corresponding full CI energy values. The lowest two occupied orbitals were

frozen in the correlated calculations.

bThe N—N separation in bohr. The equilibrium value of R is 2.068 bohr.

“The BECCSD, QECCSD, and full ECCSD methods are based on the Piecuch-Bartlett

variant of the ECC theory.

25

 

-107035 ' l v u v I v

    

     

 

 

 

 

 

—107.40 - "figi
: c 3 c )
-107.45 1' UCCSD ‘
Ia" <LECCSD
a; * " 0350630 1
e QECCSD
2 -107.50 . 4\ ECCSD -
g 1! <1 OFuIICI
>, E \
9 . <1~ .
0:, —107.55 (1‘46“
LIJ “Gm-43:]
‘ 11
—107.60 - ' -
L‘
—107.65 7 = = = J
r
-107. 70
2.0 3.0 4.0 50 60 7.0 8.0
RN_N(bohr)

Figure 2.2: Potential energy curves for the N2 molecule, as described by the STD-3G
basis set. The LECCSD, BECCSD, QECCSD, and ECCSD methods are based on

the Piecuch-Bartlett variant of the ECC theory.

26

The results in Table 2.2 and Figure 2.2 clearly demonstrate that the complete
ECCSD formalism of Piecuch and Bartlett“, in which all nonlinear terms in (21 +
23;) and (T 1 + T2) are included, and its bilinear and quadratic variants, BECCSD
and QECCSD, respectively, defined by the truncated Hamiltonians 1718300313) and
H‘QECCSD), Eqs. (2.41) and (2.42), respectively, provide remarkable improvements in
the very poor description of the potential energy curve of N2 by the standard CCSD
method. The huge negative errors in the CCSD results at larger N—-N separations,
R 2 4.5 bohr, of about -—200 millihartree, reduce to much smaller positive errors when
the ECC methods are employed (24—38 millihartree in the full ECCSD case, 27—43
millihartree in the BECCSD case, and 28—46 millihartree in the QECCSD case). We
also observe a considerable reduction of errors for smaller values of R, including the
equilibrium, R z 2.0 bohr, region (see Table 2.2). As shown in Table 2.2 and Figure
2.2, the BECCSD, QECCSD, and full ECCSD approaches eliminate the pathological
behavior of the CCSD method at larger N—N distances. As in the MBS S4 case, the
BECCSD, QECCSD, and full ECCSD approaches restore the variational description
of the potential energy curve of N2 at all internuclear separations.

Our results for N2 show that it is not necessary to insist on the bi-variational
determination of the 2" and T operators, exploited in the Arponen-Bishop ECC
formalism, in order to obtain great improvements in the description of multiple bond
breaking by the ECC theory. The fact that the ECC theory of Piecuch and Bartlett
is not rigorously bi—variational seems to be of little significance, since our BECCSD,
QECCSD, and ECCSD results obtained with this theory are of the same quality as
the strictly bi-variational QECCSD results for N2 reported in subsection 2.3.2. The
presence of quadratic terms in 2* in QECCSD), H‘BECCSD), and ITQECCSD), Eqs. (2.37),

(2.41), and (2.42), respectively, and the use of two independent cluster operators T

27

and E in the ECC formalism, which are optimized by solving a coupled system of
equations, are more important for improving the results in the bond breaking region

than the particular way of obtaining the t3, tbjb’ of, and 09b amplitudes that defines

3.7
the ECCSD formalism of Piecuch and Bartlett.

The importance of the quadratic terms in (21 + 23;), such as %(E{ + 2;)2 H(CCSD)
and %H(CCSD) (21 + 2;)2, in the ECCSD equations becomes apparent when we com-
pare the BECCSD or QECCSD results with the results of the LECCSD calculations.
These quadratic terms are ignored in the LECCSD method (see Eq. (2.40)) and,
in consequence, the LECCSD potential energy curve for N2 has the same type of
hump for the intermediate values of R as the CCSD curve (see Figure 2.2). It is
interesting to observe, though, a substantial reduction of errors in the CCSD re-
sults at larger N—N separations, when the LECCSD approach is employed. This
corroborates our earlier statements that the use of two independent cluster oper-
ators, T and 2", in the ECC theory is more important for improving the poor
CCSD results at larger R values than the specific procedure used to determine T

and 2". It is also worth noticing that we can safely neglect the higher-order nonlinear

terms, such as $031 + 23;)? ECCSD) (231 + 2;), $031 + 2;)?!(0051’) (2’; + 2;)”, and
1
4
methods (cf. Eqs. (2.42) and (2.37), respectively) and absent in the BECCSD ap-

(21 + 2;)2 H(CCSD) (2‘; + 2;)2, which are present in the QECCSD and full ECCSD

proach (see Eq. (2.41)). The BECCSD results are of the same quality as the QECCSD
and full ECCSD results (see Figure 2.2 and Table 2.2). The BECCSD theory repre-
sents the lowest-order ECC approach among various ECC methods examined in this

work capable of providing the qualitatively correct description of triple bond breaking

in N2.

28

Table 2.3.: Ground-state energies of the N2 molecule, as described by the STD-3G

 

 

 

basis set.3

Rb Full CI (CCSD)c (BECCSD)c (QECCSD)c (ECCSD)c
1.0 —101.791600 0.298 0.298 0.298 0.298
1.5 — 106.720117 0.890 0.888 0.888 0.888
2.0 — 107.623240 2.004 1.946 1.946 1.946
2.5 —107.651880 4.316 3.775 3.775 3.775
3.0 — 107.546614 5.288 4.160 4.160 4.161
3.5 — 107.473442 16.755 3.378 3.387 3.388
4.0 -107.447822 80.696 6.922 7.206 7.145
4.5 — 107.441504 95.003 10.603 11.506 11.277
5.0 — 107.439549 91.561 12.391 13.877 13.327
5.5 —107.438665 86.652 13.056 14.931 14.224
6.0 — 107.438265 83.037 13.276 15.356 14.553
7.0 —107.438054 79.607 13.393 15.571 14.724
8.0 -107.438029 78.563 13.451 15.578 14.758

 

 

aThe full CI total energies are in hartree. The remaining energies are in millihartree relative
to the corresponding full CI energy values. The lowest two occupied orbitals were frozen in
the correlated calculations.

bThe N—N separation in bohr. The equilibrium value of R is 2.068 bohr.

c(X) (X = CCSD, BECCSD, QECCSD, ECCSD) is the expectation value of the Hamilto-
nian with the eT1+T2|<I>) wave function, where T1 and T2 are obtained with method X (cf.

Eq. (2.62)).

29

Finally, before describing the calculations employing the Arponen-Bishop ECC
theory, let us discuss the quality of the T1 and T2 clusters resulting from various types
of the ECCSD calculations for N2. The remarkable improvements in the description
of triple bond breaking in N2, offered by the BECCSD, QECCSD, and full ECCSD
methods, imply that the T1 and T2 clusters resulting from the bilinear, quadratic,
and full ECCSD calculations are much more accurate than the T1 and T 2 operators
obtained with the standard CCSD approach. As in the case of the MBS S4 model,
we examined the quality of the T1 and T2 clusters obtained in the CCSD and various
ECC calculations for N2 by computing the expectation values of the Hamiltonian,
designated by (X), where X = CCSD, BECCSD, QECCSD, and ECCSD, with the

normalized CCSD-like wave functions
103’”) = N“) efo’+Téx’|<l>), (X = CCSD, BECCSD, QECCSD, ECCSD), (2.62)

where 7",“) and 7.)") (X = CCSD, BECCSD, QECCSD, ECCSD) are the T1 and
T2 cluster components obtained with the CCSD, BECCSD, QECCSD, and complete
ECCSD methods, respectively, and N (X l = (<1>|e(T{X))t+(TéX))teTlxb'Tén|<I>)"1/2 are the
corresponding normalization factors (see Table 2.3). As demonstrated in Table 2.3,
the expectation values of the Hamiltonian obtained with the CCSD wave function

are extremely poor at larger N—N separations, whereas the expectation values of

the Hamiltonian calculated with the BECCSD, QECCSD, and full ECCSD wave

SBECCSD)), I‘I’SQECCSD)), and I‘IIEECCSD)), respectively, are very close to

functions, |\II
the corresponding full CI energies at all values of R. The fact that we observe a fairly
substantial (2—3-fold) reduction of unsigned errors, when the standard CCSD energy

EéCCSD) = (®|H(CCSD)|<D), is replaced by the expectation value of the

expression,
Hamiltonian with the CCSD wave function, does not help the CCSD theory too much,

since the (CCSD) energies are characterized by the very large, 79—95 millihartree,

30

errors in the R 2 4.0 bohr region. When the BECCSD, QECCSD, and ECCSD
energy expressions, based on Eq. (2.39), are replaced by the expectation values of the
Hamiltonian with the BECCSD, QECCSD, and ECCSD wave functions, the 24—46
millihartree errors in the BECCSD, QECCSD, and ECCSD energies in the R 2 4.5
bohr region reduce to 11—13 millihartree in the BECCSD case, 12—16 millihartree
in the QECCSD case, and 11—15 millihartree in the full ECCSD case (see Table
2.3). The fact that we can improve the description of the potential energy curve of
N2 by simply replacing the BECCSD, QECCSD, and ECCSD energy expressions,
based on Eq. (2.39), by the expectation values of the Hamiltonian calculated with
the BECCSD, QECCSD, and ECCSD wave functions, IqlgBECCSD)), IWSQECCSD», and
I‘IISECCSDU, respectively, is interesting and worth further exploration.

The high quality of the T1 and T2 clusters resulting from the BECCSD, QECCSD,
and full ECCSD calculations and the poor quality of the T1 and T2 clusters obtained
with the standard CCSD approach can also be seen by examining the overlaps of
the normalized CCSD, BECCSD, QECCSD, and ECCSD wave functions, lwgccsm),
IWSBECCSDB, IWéQECCSD)), and IQéECCSD)), respectively, Eq. (2.62), with the nor-
malized full CI wave function IWSWICI)» as functions of the N—N separation R (see
Figure 2.3 (a)). As demonstrated in Figure 2.3 (a), the overlaps between the normal—
ized BECCSD, QECCSD, or ECCSD wave functions and [WSW] CI)) vary between 0.98
and 1.0 in the entire R region. For comparison, the overlap of the normalized CCSD
and full CI wave functions, which is close to 1.0 in the vicinity of the equilibrium

geometry (R z 2.0 bohr), decreases to ~ 0.93 for larger N—N separations.

31

 

 

.055 as m comuwumaom zlz 2: .8 328:8 ma 638—2: aZ 0mr08m 25 .80 A33 .60 AN .03 055:6 2: .3 @258 an .N
van .h 93.230 .8630 Dmovm on» 5253 8:8on 2: was E 93330 .8339 Qmoom van QmOO 2: 8253 8:8me
2:. Q: 650: as m 838.8% 212 on... 00 283an mm @3838 «Z Umr09m one .80 .85an 953 HO :8 Bum—«8.8: 2:
fits .393 .dm .maosoaé 963 Dmoom v.8 .QmOOm—O .Qmoomm .QmOO Buzmgo: 2: .3 83.83 23. A3 “mum 223m

 

 

         
  

     

 

9.68 2-2m 38: ii
0.0 OK 0.0 0.0 0.? 0.0 0.N 0._. . 0.0 0.5 0.0 0.0 0.? 0.0 0.N 0.—. .
- . r . - . - . t . - 0 0 1 i t . - . - . - e - . 1 N0 0
. no 03
. . I . N6 1 vmd
LawUUmVNHN «Boom. I? O lxrzl
r EmoomwhflN .BmroPHV D .. 0.0 QWUUNHX 4 rm
.AomuumovhflN .Ewdghm> D . QWUUm—OHN D 0 Wm
LomoomehflN 5802—3? 4‘ . #0 \m r QWUUMQNX 0 g 00.0 W
a ...A omooux n. N
. md IZx <m
m 1h". lhnleMl I
. od . i i- “a. mad
A ’0
. N0 .
t ”.0 1 cos—v

 

 

 

 

 

32

The substantial differences between the T1 and T 2 cluster components resulting
from the standard CCSD calculations and their analogs obtained in the BECCSD,
QECCSD, and ECCSD calculations at larger N—N distances that lead to the big
differences between the behavior of the CCSD and BECCSD/QECCSD/ECCSD ap-
proaches in the bond breaking region are shown in Figure 2.3 (b). Suppose Y and

Z are two excitation operators, defined by the amplitudes y; and 2.], respectively (tf,

C}

2 ab for the cluster operator

2'3

2). The following quantity provides us with an accurate measure of how different or

and t3, for the cluster operator T and 03 = a and 02],) = a

how similar the two operators Y and Z are:

 

d(Y, 2) = \/):(y., — 2m. (2.63)
J

When Y = Z, we obtain d(Y, Z) = 0. When the cluster amplitudes y; and ZJ defining
operators Y and Z, respectively, have similar values, d(Y, Z) is close to 0. Otherwise,
the value of d(Y, Z) is significantly greater than 0. As shown in Figure 2.3 (b), the
cluster Operators T resulting from the CCSD and BECCSD/QECCSD/ECCSD calcu-
lations are almost identical when R is small or when R is close to the equilibrium bond
length in N2 (R z 2.0 bohr). This is why the CCSD and BECCSD/QECCSD/ECCSD
wave functions and the corresponding energies (particularly, the expectation values of
the Hamiltonian calculated with the CCSD and BECCSD/QECCSD/ECCSD wave
functions) are virtually identical for the equilibrium and smaller values of R. The sit-
uation drastically changes when the triple bond in N2 is stretched or broken. For R >
3.0 bohr, the differences between the CCSD and BECCSD/QECCSD/ECCSD oper-
ators T increase so much (cf. Figure 2.3 (b)) that the behavior of the CCSD method
and the behavior of the BECCSD/QECCSD/ECCSD approaches for stretched nuclear
geometries are totally different. As we have already discussed, the standard CCSD

method completely fails at larger R values, whereas the BECCSD/QECCSD/ECCSD

33

x1

approaches provide very good results. Figure 2.3 (b) illustrates another important
feature of the ECCSD theory, namely, the similarity of the T and 23 operators in the
equilibrium region and the significant difference between the T and )3 operators ob—
tained in the ECCSD calculations in the region of larger N -N distances. As mentioned
earlier, the lowest-order MBPT estimates of the operators 2 and T are identical (cf.
Refs. 60—62, 83, 84). In consequence, in the equilibrium region, where the MBPT
series rapidly converges, we. have 2 z T (see Figure 2.3 (b)). The situation changes,

when the convergence of the MBPT series is slow or when the MBPT series diverges,

 

as is the case when the N —N bond is stretched or broken. For larger N—N separations,
the operators 2 and T become completely different. Figure 2.3 (b) provides us with

a direct evidence that this is indeed what happens at larger R values.

2.3.2 The Arponen-Bishop ECCSD Approach

So far, we have tested the ECCSD methods based on the ECC formalism of Piecuch
and Bartlett. We have demonstrated considerable improvements offered by the ECCSD
approximations when multiple bonds are broken. The question is if similar improve-
ments can be obtained when the alternative formulation of the ECC theory, proposed
by Arponen and Bishop, is exploited. This question is addressed in this subsection.
The usefulness of the bi—variational ECCSD theory of Arponen and Bishop based
on Eqs. (2.39) and (2.43)—(2.46) in improving the results for multiple bond breaking
becomes apparent when we examine the results for the STD-3G model of N2 shown in
Table 2.4 and Figure 2.4. As one can see, the ECCSD method of Arponen and Bishop
employing the ground-state RHF determinant as a reference provides remarkable im-
provements in the very poor description of the potential energy curve of N2 by the

standard CCSD method. The results are as good as those obtained with the ECCSD

34

approach of Piecuch and Bartlett. Indeed, the huge negative errors in the CCSD re-
sults at larger N—N separations, which exceed —200 millihartree in the R > 4.5 bohr
region, reduce to much smaller positive errors (on the order of 31—38 millihartree
when R > 4.5 bohr) in the ECCSD/Arponen-Bishop case. As shown in Figure 2.4,
the ECCSD approach of Arponen and Bishop completely eliminates the pathologi-
cal behavior of the standard CCSD method at larger N—N distances, restoring the
variational description of the potential energy curve of N2 at all internuclear separa-
tions. As in the case of the Piecuch-Bartlett theory, the ECCSD approach of Arponen
and Bishop is capable of capturing the most essential nondynamic correlation effects
(which the small STD-3G basis set used in these calculations already describes) in
spite of the use of the single, spin— and symmetry-adapted, RHF determinant as a
reference. The comparison of the CCSD and ECCSD potential energy curves for N2
shown in Table 2.4 and Figure 2.4 confirms once again that the T1 and T2 cluster
components obtained in the ECCSD calculations are of much higher quality in the
bond breaking region than the T1 and T2 clusters resulting from the standard CCSD
calculations.

As shown in Table 2.4 and Figure 2.4, the QECCSD/Arponen-Bishop results for
the STD-3G model of the N2 molecule, obtained by truncating the ECCSD energy
functional at terms quadratic in (2} + 2;) (see Eqs. (2.52)-(2.57)), are much better
than the corresponding CCSD results and almost as good as the results of the full
ECCSD calculations. The huge negative errors in the CCSD results in the R > 4.5
bohr region, which exceed -200 millihartree, reduce to much smaller positive errors
on the order of 35—46 millihartree, when the QECCSD method is employed. As in
the case of the full ECCSD approach, the QECCSD approximation employing the

Arponen-Bishop ECC formalism eliminates the pathological behavior of the CCSD

35

 

method at larger N —N distances, restoring the variational description of the potential
energy curve of N2 at all internuclear separations. The variational and qualitatively
correct behavior of the QECCSD method based on the Arponen-Bishop ECC theory
is independent of the basis set. Indeed, as shown in Table 2.5 and Figure 2.5, the
QECCSD potential energy curve for the double zeta (DZ)85 model of N2 is much
better than the corresponding CCSD curve. The QECCSD potential energy curve
for the N2 molecule obtained with the DZ basis set, shown in Figure 2.5, is located
above the full CI curve. The large negative errors in the CCSD results in the R 2 2Re
region of (—70) — (—121) millihartree (RC = 2.068 bohr is the equilibrium value of R)
are replaced by the considerably smaller positive errors of 40—50 millihartree when the
QECCSD method is employed. As in the case of the STD-3G basis set, the QECCSD
approach based on the Arponen-Bishop ECC theory eliminates the well pronounced
hump on the CCSD potential energy curve, when the DZ basis set is employed.

Thus, the QECCSD approach exploiting the Arponen-BishOp ECC theory pro-
vides a practical method of capturing the large nondynamic correlation effects in N2,
in spite of the single-reference nature of the ECC formalism, in spite of the use of
the RHF determinant in the QECCSD calculations, and, what is probably most re-
markable, in spite of the two-body character of the QECCSD (or ECCSD) theory,
which uses one- and two-body cluster T1, T2, 21, and 22 only. As a matter of fact,
the QECCSD results at larger internuclear separations of N2 are much better than
those obtained with the standard CC methods with singly, doubly, triply, and even
quadruply excited clusters (cf. the QECCSD results in Table 2.5 and Figure 2.5 with
the corresponding CCSD(T), CCSDT, CCSD(TQf), and CCSDT(Qf) results).

36

Table 2.4.: The ground-state energies of the N2 molecule obtained for several inter-

nuclear separations R with the STO-3G basis set.“

 

 

 

Rb Full CI CCSD QECCSDc ECCSDc
1.5 -106.720117 1.102 0.886 0.885
2.0 —107.623240 3.295 1.898 1.897
2.5 —107.651880 9.220 3.443 3.427
3.0 — 107.546614 13.176 3.909 3.757
3.5 - 107.473442 —38.645 5.294 4.746
4.0 —107.447822 — 140.376 15.815 14.148
4.5 —107.441504 - 184.984 27.792 24.198
5.0 —107.439549 —200.857 35.335 30.590
5.5 —107.438665 —206.974 39.983 34.158
6.0 —107.438265 -209.538 42.609 36.082
7.0 — 107.438054 -211.915 44.839 37.671
8.0 —107.438029 —213.431 45.508 38.161

 

 

a"The full CI energies are in hartree. The CCSD, QECCSD, and ECCSD energies are in
millihartree relative to the corresponding full CI energy values. The lowest two occupied
orbitals were kept frozen.

bThe N—N separation in bohr. The equilibrium value of R is 2.068 bohr.

cThe QECCSD and full ECCSD methods are based on the Arponen-Bishop ECC theory.

37

 

-107.35 - .

 

 

 

 

—107.40 -
-107.45 - f
3 . .
9
“C -107.50 P
2 {2 .FullCl
v \ 00050
g; \ Aoecoso
5 —107.55 \ veccso -
c |
Lu \
0
-107.60 \ 4
\\
\
—107.65 Q‘Qom—o---
—107.7o**1‘4-'-‘*-‘
2.0 3.0 4.0 5.0 6.0 7.0 8.0
RN_N(bohr)

Figure 2.4: Ground-state potential energy curves of the N2 molecule, as described
by the STD-3G basis sets. The QECCSD and ECCSD methods are based on the

Arponen-Bishop ECC theory.

38

Table 2.5.: The ground-state energies of the N2 molecule obtained for several inter-

nuclear separations R with the DZ basis set.“

 

 

 

Method 0.7512. 12.b 1.2512. 1.512. 1.7512. 212. 2.2512.
CCSD 3.132 8.289 19.061 33.545 17.714 —69.917 —120.836
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 —2.279 —14.243 92.981 334.985
CCSDT(Q;)“ 0.047 -0.010 —0.715 —4.584 3.612 177.641 426.175
QECCSD“ 2.506 6.236 13.609 23.485 31.060 40.085 49.741

 

 

“All energies are in millihartree relative to the corresponding full CI energy values, which are

—108.549027, —109.105115, -109.054626, —108.950728, —108.889906, —108.868239, and

—108.862125 hartree at R = 0.75R., Rm 1.25R., 1.5R., 1.75R., 2R., and 2.25R., respec-

tively. The lowest two occupied and the highest two unoccupied orbitals were frozen in

correlated calculations.

bThe equilibrium value of R, R. = 2.068 bohr.

cFrom Ref. 31.
dFrom Ref. 34.

eThe quadratic approximation to the full ECCSD theory of Arponen and BishOp.

39

 

-108.5 . .

Full Cl

 

-108.6 - -—- ccso ,’ -
aleccsor I
' Dccsom I

_ . OCCSDG'Q.) I .
1 08'7 AQECCSD ,

 

-108.8 2

—108.9 -

Energy (hartree)

—109.0 -

 

 

 

 

—109.1 ~

-109.2 .

_109.3 --.1....1....1...L11...1L..L14
1.5 2.0 2.5 3.0 3.5 4.0 4.5

RM, (bohr)

Figure 2.5: Ground-state potential energy curves of the N2 molecule, as described by
the DZ basis sets. QECCSD represents the quadratic version of the ECCSD theory

of Arponen and Bishop.

40

2.4 Conclusion

We can summarize this chapter by stating that the ECCSD approach of Piecuch and
Bartlett, and its approximate BECCSD and QECCSD variants—”“76, and the ECCSD
method of Arponen and Bishop and its QECCSD {variant77 represent interesting new
alternatives for accurate electronic structure calculations of quasi-degenerate elec-
tronic states and bond breaking. The BECCSD, QECCSD, and full ECCSD methods
remove the pervasive failing of the standard CCSD approach at larger internuclear
separations and provide very good values of the T1 and T2 cluster amplitudes in the
bond breaking region, in spite of using the RHF configuration as a reference and
in spite of the two-body character of all ECCSD approximations. The BECCSD,
QECCSD, and full ECCSD approaches improve the quality of the T1 and T2 cluster
components so much that we can start thinking about using the BECCSD, QECCSD,
or full ECCSD theories to design new single-reference ab initio methods for quantita-
tive, high-accuracy calculations for bond breaking. For example, by having an access
to very good T1 and T2 cluster amplitudes, resulting from the BECCSD, QECCSD,
or ECCSD calculations, which are much better than the T1 and T2 clusters resulting
from the standard CCSD calculations, we should be able to propose simple nonitera-
tive corrections to the BECCSD, QECCSD, or ECCSD energies, which may provide
further improvements in the ECC results in the bond breaking region (see Chapter
3).

Based on a comparison of the ECCSD and QECCSD results for N2 obtained with
the Piecuch-Bartlett and Arponen-Bishop theories, we do not expect the differences
between the ECC methods of Arponen and Bishop‘w—59 and Piecuch and Bartlett60
to be large in the context of bond breaking. It seems to us that the rigorously bi-

variational character of the ECC formalism of Arponen and Bishop is of the secondary

41

importance in the calculations of PESs involving bond breaking (cf. Section 2.3).
Our experiences to date indicate that the most important factor that contributes
to significant improvements in the quality of the T1 and T 2 cluster components in
the bond breaking region is the flexibility of the ECC theories, which rely on two
independent sets of cluster amplitudes that are optimized by solving coupled systems
of equations. The standard CC theory uses only one set of cluster amplitudes and
this is not sufficient to obtain a correct description of multiple bond breaking by the
standard CCSD method.

On the other hand, costs of the ECCSD calculations employing the ECC theory
of Arponen and Bishop are somewhat smaller than those charactering the ECCSD
approach of Piecuch and Bartlett. This, in particular, applies to the QECCSD approx-
imation, which is simpler when the Arponen-Bishop ECC theory is employed. The
QECCSD method based on Arponen’s and Bishop’s formulation of the ECC theory is
an .N’6 procedure. The relatively low cost of the QECCSD approximation, the single
reference (“black-box”) character of the QECCSD calculations, and the qualitatively
correct description of multiple bond breaking by the QECCSD approach make the
QECCSD method an attractive theory for the design of noniterative corrections to

CC energies. These corrections are discussed next.

42

Chapter 3 Noniterative Corrections to Extended
Coupled-Cluster Energies for High Accuracy
Electronic Structure Calculations: Generalized
Method of Moments of Coupled-Cluster Equations

In Chapter 2, we showed that we can provide substantial improvements in the quality
of the calculated potential energy curves and electronic quasi-degeneracies if we switch
from the standard CCSD theory to its extended ECCSD counterpart. The question
arises if we can improve the ECCSD (or QECCSD) results even further and obtain
the quantitative description of bond breaking by adding the a posteriori corrections
to ECCSD energies that would be reminiscent of the popular triples and quadruples
corrections of the CCSD(T) and CCSD(TQf) methods and that would eliminate fail-
ures of these methods at larger internuclear separations. In this chapter, we show
that such corrections to ECCSD energies can be developed if we use the generalized
version of the method of moments of CC equations (the MMCC theory) of Piecuch
and Kowalskill-13’3k32’74’75. Before describing the generalized MMCC approach, we

discuss the standard MMCC formalism.

43

 

3.1 The Standard Method of Moments of Coupled-Cluster
Equations

The main idea of the standard MMCC theory‘1‘13’30—9’257‘L75 is that of the noniterative
energy correction

53") E E0 — E3“, (3.1)

which, when added to the energy ESA) obtained in some standard approximate CC
calculation A, such as CCSD, recovers the exact, full CI, energy E0. The objective of
the approximate MMCC methods is to approximate corrections 6),"), such that the

resulting MMCC energies, defined as
EgMMCC) = 133") + 63"), (3.2)

are close to the corresponding full CI energies E0. The ground-state MMCC for-

12’13’3‘H38 and genuine multi-reference CC

malism can be extended to excited states
theories32’89'90. In this thesis, we focus on the ground-state problem and merging the
single-reference MMCC formalism with the non-standard CC theories, such as the
ECC method of Arponen and Bishop50—59 (see Refs. 75 and 77).

In the standard formulation of the ground-state MMCC theory, we use the non-
iterative corrections 66’4) to improve the results of the standard CC calculations. By
the standard CC calculation, we mean any single-reference CC calculation in which
the many-body expansion for the cluster operator T, defining the CC ground state
|\Ilo), Eq. (2.4), is truncated at some excitation level mA < N (recall that N is the
number of electrons in a system). The general form of the truncated cluster operator

T”), defining the standard CC approximation A characterized by the excitation level

mA, is given by Eq. (2.20). An example of the standard CC approximation is the

44

CCSD method. In this case, m A = 2 and the cluster operator T is approximated by
T z T(CCSD) = Tl + T2, (3.3)

where T1 and T2 are defined by Eqs. (2.25) and (2.26), respectively. Other examples

of the standard CC approximations are the full CCSDT, CCSDTQ, and CCSDTQP

approaches mentioned in the Introduction, in which mA 2 3, 4, and 5, respectively.
The standard CC system of equations for the cluster amplitudes tall 1112.3" defining

the Tn components of T (A) has the following general form:

(@2111 °"ia"|H(A)|<I>) = 0, i1 < < in, a1 < < an, (3.4)
o o o n
where n = 1,...,mA,

H“) = e"T(A)HeTW = (HeTW)C (3.5)

is the similarity-transformed Hamiltonian of the CC theory, and

Id);l ' ‘ '19") = E2911 ° ’ 'ia"|<I>) are the n-tuply excited determinants. In particular, the
n n

standard CCSD equations for the singly and doubly excited cluster amplitudes t3

and 4sz defining operators T1 and T2, respectively, are

(‘I’flH‘CCSD’l‘M = 0, (3-6)

(¢g]b|H<CCSD>|<1>) = 0, 2‘ < j, a < b, (3.7)

where H‘CCSD) is the similarity-transformed Hamiltonian of the CCSD approach de-
fined by Eq. (2.37). Once the system of equations, Eq. (3.4), is solved for T”) (or
Eqs. (3.6) and (3.7) are solved for T1 and T 2), the CC energy corresponding to the

standard approximation A is calculated as

E5” = (<1>|H(A)|<I>). (3.8)

45

The fundamental formula of the ground-state MMCC formalism introduced in
Refs. 11 and 30 (cf., also, Refs. 12, 30—32, 74, 75), which expresses the energy differ-
ence 6“), Eq. (3.1), in terms of the generalized moments of the single-reference CC
equations of method A, has the following form:

6W=E— E”): ZN: Z (-eonC..(mA)M.(mA)I<I>>/<\II0IT""I«I>>. (3.9)

n=m4+lk=m4+l

Iiere,

Cn—k(mA) = (6TH) )n—k (3-10)

is the (n — k)-body component of the wave operator 8TH) defining the CC method A,

|\Ilo) is the full CI ground state, and

M.(mA)I<I>> = Z Max:113: (m mg“ ,, an. (3.11)
i] < ' ° ' < ik
0.1 < ' ' ' < ak
The coefficients Mal iffk(m,() entering Eq. (3.11) represent the general moments
of the CC equations of method A,
Maizzzé‘Jm/a) = ((pgllxjiikIHMnQ). (3.12)
By comparing Eqs. (3.12) and (3. 4), we can state that moments .Mallz .i" k(mA)

entering 63") represent the projections of the CC equations of method A on excited
determinants Willi?) with k > m ,4.

Equation (3.9) states that by calculating quantities Cn_k(m,4), Eq. (3.10), and
moments M311'_'.'.ic’fk (mA), Eq. (3.12), with k > mA, we can determine the nonit-
erative energy correction 66A) to the CC energy BSA) that we recovers the full CI
energy E0. The determination of moments Mill'ffjt’z‘JmA) for the low-order CC
methods, such as CCSD, is relatively straightforward (cf. Eqs. (3.15)-(3.18) below).

The Cn_k(mA) terms entering Eq. (3.9) are easy to calculate too. The zero-body

46

term, Co(m,4), equals 1; the one-body term, Cl(m,4), equals T1; the two-body term,
1

02(mA), equals T2 +%T12 if mA 2 2; the three-body term, C3(mA), equals TIT; + 6T13

if 172); = 2 and T3 + T 1T2 + -1-T ,3 if mA 2 3, etc. Thus, the above formula for 63"),

6
Eq. (3.9), is an excellent starting point for developing noniterative CC approaches, in
which the ground-state energies are calculated by adding corrections 68A) to the stan-
dard CC energy EéA). For example, we can develop a hierarchy of approximations, in
which the energy corrections based on Eq. (3.9) are added to the CCSD energies. In

this case, the formula for the correction 66cc”), which must be added to the CCSD

energy ESCCSD) to recover the full CI energy E0, is

N min(n,6)
68‘3“”) 2 E0 — E3003”) = Z Z (wolc.-.(2)M.(2)I<I>>/<%IeT1+T2|<I>>, (3.13)
n=3 k=3
where
Mk(2)I<I>> = Z M23112... (2) I331 2'. a). (3.14)
i1 < ' ' ' < ik

0.1 < - ° - < a),
with moments M211 11125,: (2), k = 3 — 6, defined as projections of the CCSD equations

on triply, quadruply, pentuply, and hextuply excited determinants,

1 1 1 1
MWQ): (cpgjk|[HN(T2+TlT2+ —T22+ +—T2T2+—T1T§+—T?T2)]c|<1>), (3-15)

Mabc 2 2 1 2 6
1 1
M23“(2)=(<1>ab,g;1|(H~(-;- Tg+ 41T2+——T:+1T3T2)]C|q>), (3.16)
3] 2 6 4
kl
Mijbcd’g‘e): g<<1> 3,”,§§‘,$,IlH~(T§ +T1T3>JCI<I>> (3.17)
ijklmn __ abcdef
Mam, (2) — 24<¢,,k,mnI(H~13)cl<I>>. (3.18)

The exact MMCC corrections 63"), Eq. (3.9), or 630030), Eq. (3.13), have the
form of the complete many-body expansions involving all n-tuply excited configura-

tions with n = mA + 1, . . . , N, where N is the number of electrons in a system (cf.

47

the summations over 17. in Eqs. (3.9) and (3.13)). Thus, in order to develop prac-
tical methods based on the MMCC theory, we must first truncate the many-body

or 6(ccsn)

expansions for corrections (SW at some, preferably low, excitation level m3

satisfying mA < 7113 < N. This leads to the MMCC(mA, m3) schemes, in which we

calculate the energy as followsl1‘13’3°_32'74’75:

EéMMCC)(mAa m3) : E(()A) + 60(mAa m3), (3'19)
where E3” is the energy obtained with the CC method A and

60<mmma)= Z Z (‘I’olCn .(mi)Mk<mA)I<I>>/<oneT“’I<I> <I>> (3.20)

n=mA+l k: mA-l-l
is the relevant MMCC correction. Examples of the MMCC(mA, m3) schemes are the
basic MMCC(2,3) and MMCC(2,4) approximations, in which energies are calculated

as followsl“1330—3523435.

E‘MMCC’e 3): E‘CCSD’ + (‘I’olM3(2)|‘I’>/(‘I’o IeTI+T2I<I>> (3.21)

E<MMCC)(2 4): ESCCSD) + (‘I'olfM3(2) + [M4(2) + T1M3(2)l}|‘p)/<‘I’OIBTI+T2I‘D),
(3.22)
where Ewcsm is the CCSD ground-state energy. The MMCC(2, 3) approach re-
quires that we only determine the triply excited moments M2532), Eq. (3.15). The
MMCC(2,4) method requires that we determine the triply excited moments Mijk(2),

szkl d,(2) Eq. (3.16). The second

Eq. (3.15), and the quadruply excited momentsM
issue that has to be addressed before the MMCC(mjc, m3) methods can be used in
practice is the fact that in the exact MMCC theory the wave function lilo) that
enters Eqs. (3.9) and (3.13) is the exact, full CI, ground state. Thus, in order
to propose the computationally tractable approaches based on the MMCC theory,

we must approximate |\Ilo) in some, preferably inexpensive, manner. Various ways

48

of approximating |\Ilo) in Eqs. (3.20)—(3.22), leading to the completely renormal-
ized (CR) CC methods"_‘3*3°_35'37’74’75, the quasi-variational and quadratic MMCC
methods-”776'”, the CI-corrected MMCC approachesl1'13’75'86’87'92, and the multiref-r
erence MBPT-corrected MMCC approach93 — all employing the CCSD values of T1

ééCCSD)

and T2 to construct the relevant corrections — have been suggested. For exam-

ple, the CR—CCSD(T) method‘1’13’30’31"3""35'37'74’75'9“_96 is obtained by replacing the
wave function |\Ilo) in the MMCC(2,3) formula, Eq. (3.21), by the MBPT(2)[SDT]-
like expression

|¢§CSD(T’) = (1 + T1 + T2 + T421 + Z3)|<I>), (3.23)
where T1 and T2 are the singly and doubly excited clusters obtained in the CCSD
calculations, T91 is an approximation of the connected triply excited clusters T3,

defined by Eq. (2.61), and
Zsl<1>) = Rés’VNTIId» (3.24)

is the disconnected triples correction, which is responsible for the difference between
the [T] and (T) triples corrections of the standard CCSD[T] and CCSD(T) ap-
proaches. The higher-order CR-CCSD(T Q) methodsl1‘13’30’31’33735174'75 are obtained
in a similar manner, by inserting the MBPT(2)[SDTQ]-like expressions for |\Ilo) into
the MMCC(2,4) formula, Eq. (3.22). For example, the wave function |\Ilo) defin-
ing variant “b” of the CR-CCSD(TQ) approach (the CR-CCSD(TQ),b method) is
defined as follows:
IwECSD‘TQ”) = Wm) + 223231». (3.25)
where |W€CSD(T)) is given by Eq. (3.23).
As shown in Refs. 11—13, 30—35, 37, 74, 75, and 94—96, the CR—CCSD(T) and

CR-CCSD(TQ) approaches eliminate or considerably reduce the failures of the stan-

dard CCSD(T) and CCSD(TQf) methods at larger internuclear separations and for

49

diradicals without making the calculations substantially more difficult or expensive.
In particular, the CR—CCSD(T) and CR-CCSD(TQ) methods provide a very good
description of single bond breaking. Unfortunately, performance of these methods for
multiply bonded systems is often much less impressive, partly due to the very poor
quality of the singly and doubly excited cluster amplitudes resulting from the CCSD
calculations, on which the CR—CCSD(T) and CR-CCSD(TQ) methods are based, in
calculations involving multiple bond breaking. The purpose of this work is to ex-
amine an alternative approach, in which the MMCC corrections 63A) or 65,0030) are
constructed using the cluster components obtained in the ECC or ECCSD calcula-
tions. As shown in Chapter 2, the ECCSD method provides much better values of
T1 and T2 clusters than the CCSD approach when multiple bonds are stretched or
broken, so that we may be able to improve the CR-CCSD(T) and CR-CCSD(TQ)
results in cases where the standard CCSD values of T1 and T2 are of very poor quality.
The use of the ECC or ECCSD values of cluster amplitudes in MMCC calculations
requires the generalization of the MMCC theory to non-standard CC methods, which

is described in the next section.

3.2 The Generalized MMCC Formalism

Interestingly, Eqs. (3.9) and (3.13) can be generalized to a situation where the trun-
cated cluster operator T“) is not determined by solving the standard CC equa-
tions, Eq. (3.4). Here is how this works: When the cluster operator T“) is ob-
tained in a non-standard way, we can no longer assume that the generalized moments
Mall'jfffik (mA) with k = 1,. . . , mA vanish; they only vanish in the standard CC case

(cf. Eqs. (3.4) and (3.12)). It has been shown in Refs. 13 and 30 that when mo-

50

ments M211°.°.'.z(’1‘k(m,4) with k = 1, . . . m A do not vanish, we have to use the following

expression for the exact, full Cl energy E0 instead of Eq. (3.9):

N

E. = ZZ<WOIC.-.(m.)M.(mA)I<I>>/<\IIoIeT“’I<I>>

n=0 k=0
N n

= Moon.)+ZZ<wo|c._.(mA)M.(m.)I<I>>/<%IeT“’I<I>>. (3.26)
n=l k=1

Here, Mo(mA) designates the zero-body moment, which is calculated in exactly the

same way as the CC energy Eé"), i.e. (cf. Eq. (3.8))
Mum.) = <<I>IFI""I<I>>. (3.27)

Although the formulas for Mo(m,4) and E84) are identical, there is a fundamental
difference between Mo(mA) and Eff). The energy E5,“ is determined using the cluster
amplitudes originating from the standard CC equations, Eq. (3.4), and E84) is the
energy expression used in the CC theory. The zero—body moment Mo(mA) can be
computed with any cluster operator T“), obtained, for example, by performing some
nonstandard calculations, such as the ECC calculations, and Mo(mA) does not have
to represent the energy expression of the nonstandard CC theory used to generate
T(A) (e.g., Mo(m,4) does not represent the energy expression of the ECC formalism).

As in the case of the standard MMCC formalism defined by Eq. (3.9), the
Cn-k(m,4) and Mk(m,4)|<l>) quantities entering Eq. (3.26) are defined by Eqs. (3.10)
and (3.11), respectively, although we must consider now all quantities Mk(mA)|<I>)
with k 2 1, not just those with k > mA. In other words, since we no longer assume
that the generalized moments M311 'o'.'.igk(m,4) with k = 1,. . . , mA vanish, since Eq.
(3.4) is no longer satisfied, we must consider all generalized moments Mill '. ’. 11.3,: (m A),

which a given cluster operator T“) produces. In particular, if we want to use the T1

and T2 clusters to construct the full CI energy E0, which are no longer determined

51

by solving the standard CCSD equations, we must use the following formula for the

exact energy E0:

N min(n,6)
E0 = 140(2) +2 2 (10(0,_,.(2) Mk(2)|<I>)/(\Ilo|eT1+T2|<I>), (3.23)
=1 k=l

where Mk(2)|<I>) is defined by Eq. (3.14). As one can see, Eq. (3.28) is very similar
to Eq. (3.13). In particular, we do not have to consider momentsMé‘lzng) with
k > 6, since for Hamiltonians containing up to two-body interactions the generalized
moments Mall‘jji’fka) with k > 6 vanish, independent of the source of T1 and T2
clusters. There is, however, a difference between Eqs. (3.13) and (3.28): in Eq. (3.28)
we consider the singly and doubly excited moments, 1143(2) = ((bgll—Iwcsmkb) and
MZJAQ) = (¢?Jb|H(CCSD)|<I>), respectively, which are no longer zeroed, along with
moments M31, 11115,,(2) with k = 3 — 6 considered in the standard CCSD case; Eq.
(3.13) uses moments M31115), (2) with k = 3 — 6 only, since Mg(2) = Mijb(2) = 0
in the standard CCSD case. The formulas for the M32) and M329) moments,

entering Eq. (3.28), in terms of the T1 and T2 clusters, are identical to the left-hand

sides of the standard CCSD equations. Thus, we obtain (cf. Eqs. (3.6) and (3.7)),

- 1 1
M242) = (¢?|[H~(1 + T, + T2 + 5T1? + T171, + 6T3)]C|<I>), (3.29)
M11 (2) = («31.11.1|[H~(1+ T1+ T2 +1T‘1 + T1T2 +1T11 +12"1 + 171173 + —1-T4)]C|<I>).
ab 2] 2 1 6 1 2 1 2 1 24 1

(3.30)

Equations (3.26) and (3.28) define the generalized version of the ground-state
MMCC theory, designated as GMMCC. Clearly, the GMMCC theory reduces to
the standard MMCC formalism if cluster components Tfl defining operator T“) are
determined by solving the standard CC equations, Eq. (3.4). In this case, the
M311 '.'.'.ic’{k(m,4) moments with k = 1, . . . , m A vanish and the summations 25:1 22:,

entering Eq. (3.26) reduce to ELM +1 22:,“ +1, giving Eq. (3.9). The obvious ad-

52

 

vantage of Eqs. (3.26) and (3.28) is that they are much more general than Eqs. (3.9)
and (3.13), enabling us to use the non-standard values of T“) clusters. Otherwise,
we use Eqs. (3.26) and (3.28) in exactly the same way as Eqs. (3.9) and (3.13).
Thus, once the cluster components defining the truncated cluster operator T”) are
determined, we calculate the relevant moments Mo(m,4) and M311'_'.'.i§k (mA) and
use these moments to determine the ground-state energy E0.

In analogy to the standard MMCC theory, a few issues have to be addressed be-
fore using the GMMCC formalism in practical calculations. First of all, the exact
GMMCC expressions, Eqs. (3.26) or (3.28), represent the complete many-body ex-
pansions involving all n-tuply excited configurations with n = 1,. ..,N, where N
is the number of electrons in a system (see the summations over n in Eqs. (3.26)
and (3.28)). Thus, in order to develop the computationally tractable GMMCC
methods, we must first truncate the many-body energy expansions, Eqs. (3.26) or
(3.28), at some, preferably low, excitation level m3, where mA < 1713 < N. This
leads to the GMMCC(mA, m3) schemes, which are the non-standard analogs of the
MMCC(mA,mB) approximations discussed in Section 3.1. By limiting ourselves to
the wave functions |\Ilo) that do not contain higher—than—mB-tuply excited compo-
nents relative to reference |<I>) and by restricting the summation over n in Eq. (3.26)

accordingly, we obtain the following energy expression for the GMMCC(mA,m3)

methods:
mg n
EéGMMCC1(mA, m3) = Mum.) + 2 2(2010.-.(m..) Mk(mA)|‘I’)/(‘I’0|€Tw (<2).
n=1 k=l

(3.31)
where Mo(mA) is defined by Eq. (3.27). The exact GMMCC formalism, equivalent
to calculating the full CI energies, is obtained when |\Ilo) in Eq. (3.31) is the full CI

ground-state wave function and m3 = N. In this case, it is irrelevant what is the

53

value of mA and where is the cluster operator T“) taken from. When I‘Ilo) is exact
and m3 = N, Eq. (3.31) produces the exact energy, independent of the excitation
level m A and the source of cluster amplitudes defining T”).

In this thesis, we focus on the GMMCC(mA,m3) schemes with mA = 2, which
can be used to correct the results of the non-standard CCSD-like (e.g., ECCSD or
QECCSD) calculations. In analogy to the standard MMCC theory, two
GMMCC(mA, m3) approximations are expected to be particularly useful: the
GMMCC(2,3) method and the GMMCC(2,4) approach. According to Eq. (3.31),
the GMMCC(2,3) and GMMCC(2,4) energies are calculated as follows:

EéGMMCC’a. 3) = M0(2) + (on{M1(2) + (142(2) + T1M1(2)] + [M3(2)

+T.M2(2) + (T: + §T3)M1(2)1}I<I>>/<%Ie11+11I<I>> . (3.32)

ESGMMCC’Q. 4) = Mo(2) + <%I{M1(2) + (142(2) + T1M1(2)1
+[M3(2) + T1M2(2) + (T2 + %T3)M1(2)]

+[M.,(2) + T1M3(2) + (T2 + éTE)M2(2)
1

+(T1T2 + 6

T13)M1 (2)1}|<I>)/(1I’oleT1+T2 I‘P) . (333)

As one can see, in the case of the GMMCC(2,3) approximation, we must calculate
moments M3(2), MZ)(2), and M3£(2), using Eqs. (3.29), (3.30), and (3.15), re-
spectively, in addition to moment Mo(2), Eq. (3.27). In the case of the GMMCC(2,4)
approach, we must determine moments M3132) and M2520), Eqs. (3.15) and
(3.16), respectively, along with moments Mo(2), M2(2) and M320), Eqs. (3.27),
(3.29) and (3.30), respectively. Clearly, when the singly and doubly excited moments
M39) and MZJAQ) vanish, the GMMCC(2,3) and GMMCC(2,4) energy expressions,

Eqs. (3.32) and (3.33), respectively, reduce to the MMCC(2,3) and MMCC(2,4) for-

54

mulas given by Eqs. (3.21) and (3.22). This can only happen when the T1 and T 2
clusters are obtained by solving the standard CCSD equations, Eqs. (3.6) and (3.7).

The second issue that needs to be addressed before the GMMCC(2,3) and
GMMCC(2,4) methods and other GMMCC(mA,m3) approximations are used in
practice is the issue of the wave function I‘llo) that enters Eqs. (3.31)-(3.33), which in
the exact theory is a full CI ground state. Clearly, in order to make the GMMCC(2,3),
GMMCC(2,4), and other GMMCC (m ,4, m3) schemes usable in practical applications,
we must suggest some approximate forms of |\Ilo) that can be easily generated with
one of the inexpensive ab initio approaches. Finally, the third issue is the source of
the T1 and T2 cluster amplitudes which are needed to construct the GMMCC(2,3) and
GMMCC(2,4) energy expressions, Eqs. (3.32) and (3.33), respectively. As mentioned
earlier, in calculating the GMMCC(2,3) and GMMCC(2,4) energies we would like
to use the T1 and T2 cluster components which are more accurate in cases involving
multiple bond breaking than those obtained in the standard CCSD calculations. As
shown in Chapter 2, the T1 and T2 cluster components resulting from various types
of ECCSD calculations are much better than their standard CCSD counterparts,
when multiply bonded systems are examined. Thus, it is worth examining the pos-
sibility of combining the GMMCC(2,3) and GMMCC(2,4) schemes with the ECCSD
methods. The resulting ECCSD(T), ECCSD(TQ), QECCSD(T) and QECCSD(TQ)

approaches are discussed next.

55

3.3 The ECCSD(T), ECCSD(TQ), QECCSD(T), and
QECCSD(TQ) Methods and their Performance in Cal-
culations for Triple Bond Breaking in N2

The ECCSD results for triple bond breaking in N2 are so much better than their
standard CCSD analogs that it is very important to analyze the effect of replacing
the CCSD values of the T1 and T2 cluster components in the MMCC calculations by
the ECCSD values of these clusters. Since the ECCSD and QECCSD methods are no
longer the standard CC theories, so that we can no longer assume that the singly and
doubly excited moments, MZ(2) and M2,.)(2), Eqs. (3.29) and (3.30), respectively,
vanish, we must use the GMMCC formalism discussed in Section 3.2 rather than the
standard MMCC formalism discussed in Section 3.1 in such considerations. In the
following, we test an idea of using the QECCSD and ECCSD values of T1 and T2,
resulting from the application of the Arponen-Bishop variant of the ECC theory, in
the GMMCC calculations.

As explained in Section 3.2, in practice we are interested in a truncated form of
the GMMCC theory that leads to relatively low costs of calculating the final energy.
All of our tests to date, including the calculations for N2 discussed below, indicate
that the lowest-order GMMCC scheme, employing the ECCSD or QECCSD values of
T1 and T2, which provides substantial improvements in the results for multiple bond
breaking, is the GMMCC(2,4) approach defined by Eq. (3.33). In this approach,
we only consider the generalized moments Millijzigkfl) with k = 1 — 4, i.e. mo-
ments corresponding to the projections of H(CCSD)|<1>) on singly, doubly, triply, and
quadruply excited determinants. The lower-order GMMCC(2,3) approach, defined
by Eq. (3.32), employing the ECCSD or QECCSD values of T1 and T2 and ignoring

the quadruply excited moments szbgla), provides improvements too, but the ne-

56

glect of the quadruply excited moments “3331(2) in the GMMCC calculations has
a negative impact on the results for multiply bonded systems, such as N2.

As in all approximate MMCC calculations, we must decide what to do with the
wave function |\Ilo) that enters the GMMCC(2,3) and GMMCC(2,4) energy formulas,
Eqs. (3.32) and (3.33). Since we are mainly interested in the “black-box” GMMCC
approaches of the CCSD(T) or CCSD(TQ) type, in this study of the performance of
the GMMCC theory employing the ECCSD and QECCSD values of T1 and T2, we use
the same types of the wave functions I‘llo) in Eqs. (3.32) and (3.33) as the wave func-
tions (1130511111) and (113051111‘1111’) used in the CR—CCSD(T) and CR-CCSD(TQ),b
calculations (cf. Eqs. (3.23) and (3.25)). Depending on the choice of lilo) and the pre-
cise source of T1 and T2 clusters for the GMMCC(2,3) and GMMCC(2,4) calculations,

we introduce the following four approximations:

o ECCSD(T): The ECCSD(T) approach is defined as the GMMCC(2,3) method,
in which I‘IIO) is defined by Eq. (3.23) and in which T1 and T2 clusters originate

from the full ECCSD calculations of the Arponen-Bishop type.

0 QECCSD(T): The QECCSD(T) approach is defined as the GMMCC(2,3) method,
in which [\110) is defined by Eq. (3.23) and in which T1 and T2 clusters originate

from the QECCSD calculations of the Arponen-Bishop type.

0 ECCSD(TQ): The ECCSD(TQ) approach is defined as the GMMCC(2,4) method,
in which |\Ilo) is defined by Eq. (3.25) and in which T1 and T2 clusters originate

from the full ECCSD calculations of the Arponen-Bishop type.

0 QECCSD(TQ): The QECCSD(TQ) approach is defined as the GMMCC(2,4)
method, in which I‘IIO) is defined by Eq. (3.25) and in which T1 and T2 clusters

originate from the QECCSD calculations of the Arponen-Bishop type.

57

Based on the above definitions, it is easy to verify that the ECCSD(T)/QECCSD(T)

energies can be calculated as follows:

E31Q1ECCSD‘T11 = 1140(2) + N“) mm, (3.34)
where
N111 = <<I>IT.1 M1(2)I<I>> + <4>|TJ (142(2) + T1M1(2)]|<I>>
+<<I>|(T3[2’ + 232121432) + T. 1142(2) + (T: + 3T3)M1(2)]I<I>) (3.35)
and

011" = 1+<<I>|TIT1I<I>>+<<I>|TJ (T. +17?) |<I>>+<<I>I(T§2]+Za)1(T1T2+éTf)I<I>), (3.36)

with T3] and 23 defined by Eqs. (2.61) and (3.24), respectively, and T1 and T2
obtained in the ECCSD/QECCSD calculations. The ECCSD(TQ)/QECCSD(TQ)

energies are calculated as

23319110051111”) = 1140(2) + N<TQ>/D<TQ>, (3.37)
where
N111” = N1“ + é<<I>I(T2‘ )1 [M.(2) + T1M3(2) + (T2 + 1T3)M2(2)
+(T1T2 + éT?)M1(2)]I<I>> (3.33)
and
011‘” = D11" + %<<I>|(T2* H17“: + H.171 + #312), (3.39)

with N (T) and Dm defined by Eqs. (3.35) and (3.36), respectively, and T1 and T2
obtained in the ECCSD/QECCSD calculations.
The ECCSD(T)/QECCSD(T) methods reduce to the CR-CCSD(T) approach of

Refs. 11 and 30 (cf. Section 3.1) if the T1 and T2 clusters originating from the

58

ECCSD/QECCSD calculations are replaced in Eqs. (3.34)—(3.36) by their stan-
dard CCSD values. Similarly, the ECCSD(TQ)/QECCSD(TQ) methods reduce to
the CR—CCSD(T),b approach of Ref. 31 (cf., also, Section 3.1) if the T1 and T2
clusters originating from the ECCSD/QECCSD calculations are replaced in Eqs.
(3.37)—(3.39) by their CCSD analogs. These straightforward relationships between
the ECCSD(T)/QECCSD(T) and ECCSD(TQ)/QECCSD(TQ) methods on the one
hand and the CR—CCSD(T) and CR-CCSD(T),b approaches on the other hand im-
mediately imply that once the T1 and T2 clusters are determined by solving the
ECCSD/QECCSD equations, the costs of calculating the ECCSD(T)/QECCSD(T)
and

ECCSD(TQ)/QECCSD(TQ) energies are essentially the same as the costs of the cor-
responding CR-CCSD(T) and CR-CCSD(T),b calculations or their standard CCSD(T)
and CCSD(TQf) counterparts. In particular, the most expensive steps of the
ECCSD/QECCSD-based ECCSD(T)/QECCSD(T) calculations (if we ignore the costs
of the ECCSD/QECCSD calculations) scale as 11311:. The most expensive steps of the
ECCSD(TQ)/QECCSD(TQ) calculations (again, ignoring the costs of the
ECCSD/QECCSD calculations) scale as either nfinf’, or 113. The ECCSD(T) and
ECCSD(TQ) methods are much less practical, since the underlying ECCSD calcula-
tions that provide T1 and T2 clusters have steps that scale as N10 with the system
size. However, the QECCSD(T) and QECCSD(TQ) methods are very promising in
this regard, since both the underlying QECCSD calculations that provide T1 and T2
clusters and the calculations of the final QECCSD(T) and QECCSD(TQ) energies,
Eqs. (3.34) and (3.37), respectively, have steps that scale, at worst, as ngnf’, or n2 (M
or .N'6 with the system size). We must keep in mind, however, that the QECCSD(T)

and QECCSD(TQ) methods are approximations to the more complete ECCSD(T)

59

and ECCSD(TQ) approaches. The questions, therefore, are:

(i) Do the QECCSD(T) and QECCSD(TQ) methods provide the results of the full
ECCSD(T) and ECCSD(TQ) quality?

(ii) Are the QECCSD(T) and QECCSD(TQ) methods sufficiently accurate to elimi-
nate the problems observed in the standard and completely renormalized CCSD(T)

and CCSD(TQ) calculations for triple bond breaking in N2?

The answers to both questions can be provided if we examine the results of the
benchmark ECCSD(T), ECCSD(TQ), QECCSD(T), and QECCSD(TQ) calculations
for N2 shown in Tables 3.1, 3.2 and Figures 3.1, 3.2, which we performed with the
computer codes developed in this thesis work".

As shown in Table 3.1 and Figure 3.1, the ECCSD(TQ) approach employing the T1
and T2 clusters obtained in the full ECCSD (Arponen-Bishop) calculations is capable
of providing spectacular improvements in the description of triple bond breaking in
the N2 molecule, as described by the STO-3G basis set, reducing the large unsigned
errors in the CCSD and CR—CCSD(TQ),b results in the R 2 5.0 bohr region, on
the order of 201—213 and 39—54 millihartree, respectively, and the 31—38 millihartree
errors in the ECCSD results to less than 4 millihartree. Remarkably enough, the
ECCSD(TQ) results for the N2 molecule, as described by the STO-3G basis set, in
the entire R _<_ 8.0 bohr (2: 4H,) region do not exceed ~ 4 millihartree, being much
smaller in the R m Re region. As shown in Table 3.1 and Figure 3.1, the ECCSD(TQ)
potential energy curve is located only slightly above the full CI curve and there is
no unphysical bump on it. Interestingly enough, the zero-body moment Mo(2), Eq.
(3.27), calculated with the ECCSD values of T1 and T2, which is corrected in the

ECCSD(TQ) energy formula, Eq. (3.37), by adding terms expressed via moments

60

M311°.'.'.igk(2) with k = 1 — 4, is a poor approximation to the exact, full CI, energy in
the region of large N —N separations (see Table 3.1). This demonstrates the remarkable
ability of the MMCC (or GMMCC) formalism to restore high accuracies in the bond
breaking region even when the CC energy that we are trying to correct (in this case,
the Mo(2)/ECCSD energy) is itself very poor.

The ECCSD(TQ) results for the STO-3G model of N2 shown in Table 3.1 and
Figure 3.1 are very encouraging, but, as mentioned earlier, the ECCSD(TQ) ap-
proach is not too practical due to the expensive N10 steps of the underlying ECCSD
calculations. It is, therefore, important to examine if the much more manageable
QECCSD(TQ) approximation, which relies on the T1 and T2 clusters obtained in the
QECCSD calculations and which is the N“ -.N‘7 procedure, provides the results of the
ECCSD(TQ) quality. The results in Table 3.1 indicate that the QECCSD(TQ) and
ECCSD(TQ) energies are virtually indistinguishable when the internuclear separation
R does not exceed 3.5 bohr (z 1.75Re). Only when the N—N separations exceed 3.5
bohr, the differences between the QECCSD(TQ) and ECCSD(TQ) results become
larger. Although the errors in the QECCSD(TQ) results in the R > 3.5 bohr are
greater than the corresponding errors obtained with the ECCSD(TQ) method, the
QECCSD(TQ) method provides an excellent description of the potential energy curve
of the STO-3G N2 molecule, reducing the 35—46 millihartree errors in the QECCSD
results and much larger errors in the CCSD and CR—CCSD(TQ),b results in the
R > 4.5 bohr region to 5—6 millihartree (see Table 3.1). As in the ECCSD(TQ) case,
the QECCSD(TQ) potential energy curve of the STO-3G N2 molecule is located very
close to and above the exact, full CI curve (see Figure 3.1). Thus, the QECCSD(TQ)
approach provides a highly accurate description of the large nondynamic correlation

effects characterizing the N2 molecule at larger N—N separations.

61

The question is if the above observations obtained for the very small STO-3G
basis apply to larger basis sets. In order to answer this question, we performed the
QECCSD(TQ) calculations for the DZ model of N2 (see Table 3.2 and Figure 3.2). As
shown in Table 3.2, the errors in the QECCSD(TQ) results are somewhat greater than
in the case of the STO-3G basis set, but the overall patterns are the same. Thus,
the QECCSD(TQ) method provides considerable improvements in the CCSD and
QECCSD results, reducing the large negative, (—70) — (—121) millihartree, errors in
the CCSD results in the R = 2R9 —2.25Re region and the relatively large positive, 40—
50 millihartree, errors in the QECCSD results in the same region to 16—20 millihartree.
For smaller values of R, the improvements offered by the QECCSD(TQ) approach are
even greater. For example, the QECCSD(TQ) method reduces the 6.236 and 8.289
millihartree errors in the QECCSD and CCSD energies at R = Re and the 23.485
and 33.545 millihartree errors in the QECCSD and CCSD energies at R = 1.5Re to
1.002 and 6.011 millihartree, respectively. As shown in Table 3.2 and Figure 3.2, the
QECCSD(TQ) potential energy curve for the DZ N2 molecule is located above the
full CI curve and the QECCSD(TQ) approach completely eliminates the unphysical
humps on the CR-CCSD(T) and CR—CCSD(TQ) curves, obtained with the CCSD
values of T1 and T2. This shows that the use of the QECCSD rather than CCSD
values of the T1 and T2 clusters improves the results of the MMCC calculations. All
of this is very encouraging, since the QECCSD(TQ) method is a relatively inexpensive
single-reference approach employing the spin-adapted RHF reference. As shown in
Table 3.2 and Figure 3.2, all standard CC methods using the RHF determinant as a
reference, including the very expensive CCSDT and CCSDT(Q;) approaches, which
require iterative steps that scale as JV8 with the system size, completely break down

at larger N—N separations. The QECCSD(TQ) method provides a much smoother

62

and more accurate description of triple bond breaking in N 2, with the relatively small
errors which monotonically increase with R, while eliminating all of the pathologies
observed in the standard CC calculations.

Interestingly enough, even the simplest QECCSD(T) method works reasonably
well, when the DZ model of N2 is examined (see Table 3.2), although one has to
keep in mind that the errors in the QECCSD(T) energies in the R S 1.5Rf region
are 2—3 times larger than in the QECCSD(TQ) case (see Table 3.2). Moreover, the
QECCSD(T) energies do not vary with R as smoothly as the QECCSD(TQ) energies
(cf. the nonmonotonic changes in errors in the QECCSD(T) results shown in Table
3.2). The same behavior is observed when the STO-3G basis set is employed. The
calculations for the STO-3G basis set show that the QECCSD(T) and ECCSD(T)
methods are incapable of providing significant improvements in the corresponding
QECCSD and ECCSD results when the internuclear separation R becomes large (see
213(2)
in the QECCSD(T) and ECCSD(T) energy expressions. It is interesting to observe,

Table 3.1). This is related to the absence of the quadruply excited moments M

though, that the QECCSD(T) results for the DZ model of N2 are much better than
the results of the CR-CCSD(T) calculations (not to mention the CCSD(T) results), in
which the quadruply excited moments M23219) are neglected too. This shows once
again that the T1 and T2 clusters resulting from QECCSD (and ECCSD) calculations
are of much higher quality than the T1 and T2 clusters obtained with the standard

CCSD approach, improving the results of the MMCC calculations.

63

3.4 Summary

We have demonstrated that the noniterative ECCSD(TQ) and QECCSD(TQ) meth-
ods, obtained by merging the ECC formalism of Arponen and Bishop with the gen-
eralized version of the MMCC theory that enables one to use the non-standard clus-
ter components to design the noniterative corrections to CC energies, provide an
accurate and variational description of potential energy surfaces involving multiple
bond breaking with the ease of a single-reference “black-box” calculation. In partic-
ular, the GMMCC-based ECCSD(TQ) and QECCSD(TQ) approximations employ-
ing T1 and T2 clusters obtained in ECCSD and QECCSD calculations do not suffer
from the non-variational collapse or unphysical behavior observed in the standard
CCSD, CCSD(T), CCSD(TQf), CCSDT, and CCSDT(Qf) calculations. The use of
the ECCSD and QECCSD values of the singly and doubly excited clusters has a
very positive impact on improving the results of the MMCC calculations in the bond
breaking region. In particular, the ECCSD(TQ) and QECCSD(TQ) methods em-
ploying the ECCSD and QECCSD values of T1 and T2 clusters improve the results
of the CR-CCSD(T) and CR-CCSD(TQ) calculations for triple bond breaking in N2,
which also use the MMCC theory but rely on the T1 and T2 clusters obtained with
the standard CCSD approach.

The results obtained in this thesis show that the new GMMCC theory is a flex-
ible formalism, which enables one to design the relatively inexpensive noniterative
CC methods employing the non-standard values of cluster components. The gener-
alized version of the MMCC theory provides us with precise information about the
many-body structure of the exact energy and the corrections that must be added to
standard or non-standard CC energies to recover full CI results. This is particularly

valuable in situations involving multiple bond stretching or breaking, where the stan-

64

dard arguments based on MBPT fail due to the divergence of the MBPT series at
larger internuclear separations. The fact that one can use the standard as well as
non-standard cluster amplitudes in the GMMCC calculations is a very useful feature,
which gives us an opportunity to improve the results by using non-traditional sources
of cluster amplitudes that are more suitable for the applications of interest. The
results obtained in this thesis, in which we used the ECCSD and QECCSD methods
to generate the T1 and T2 clusters for the GMMCC calculations, are a clear demon-
stration of how useful the idea of using the non-standard values of cluster amplitudes
might be in the most challenging cases involving multiple bond breaking where all
standard single-reference CC methods (including high level methods such as CCSDT
or CCSDT(Qf)) fail.

We tested a few ECCSD- and QECCSD-based GMMCC approximations, in-
cluding ECCSD(T), QECCSD(T), ECCSD(TQ), and QECCSD(TQ). Although the
best results are obtained with the ECCSD(TQ) method, we cannot recommend this
method at this time for practical calculations due to the large costs of the under-
lying ECCSD calculations. The QECCSD(TQ) approach employing the QECCSD
values of T1 and T2 clusters is more promising in this regard, offering an accurate
description of large nondynamic and substantial dynamic correlation effects with the

computational steps that scale as N6 — .IV7 with the system size.

65

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66

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Table 3.2.: The ground-state energies of the N 2 molecule obtained for several inter-

nuclear separations R with the DZ basis set.“

 

 

 

Method 0.7512. 12..b 1.2512. 1.512. 1.7511. 2R. 2.2512.
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 —109.767 —155.656
CCSD(T)“ 0.742 2.156 4.971 4.880 —51.869 —246.405 —387.448
CCSD(TQ.)c 0.226 0.323 0.221 —2.279 —14.243 92.981 334.985
CCSDT(Q;)d 0.047 —0.010 —0.715 —4.584 3.612 177.641 426.175
CR—CCSD(T)C 1.078 3.452 9.230 17.509 -2.347 —86.184 -133.313

CR-CCSD(TQ),ac 0.448 1.106 2.474 5.341 1.498 -40.784 -69.259
CR—CCSD(TQ),bc 0.451 1.302 3.617 8.011 13.517 25.069 14.796

QECCSD 2.506 6.236 13.609 23.485 31.060 40.085 49.741
Mo(2)/QECCSD° 1.908 4.164 7.935 13.623 33.109 70.120 106.399

QECCSD(T)‘ 1.000 2.941 7.121 11.661 8.454 10.330 17.977
QECCSD(TQ)S 0.412 1.002 2.443 6.011 11.393 ' 16.103 19.958

 

 

“All energies are in millihartree relative to the corresponding full CI energy values, which are
—108.549027, -109.105115, -109.054626, —108.950728, —108.889906, —108.868239, and
—108.862125 hartree at R = 0.75Re, Re, 1.25Re, 1.5Re, 1.75Re, 2B.” and 2.25Re, respec-
tively. The lowast two occupied and the highest two unoccupied orbitals were frozen in
correlated calculations.

bThe equilibrium value of R, 1?.: = 2.068 bohr.

cFrom Ref. 31.

dFrom Ref. 34.

eThe zero-body moment or the CCSD-like energy expression, Eq. (3.27), calculated using
the QECCSD values of T1 and T2 (obtained with the Arponen-Bishop ECC theory).

fThe GMMCC(2,3) result obtained using the QECCSD values of T1 and T2 and |\Ilo) defined
by Eq. (3.23) (obtained with the Arponen-Bishop ECC theory).

“The GMMCC(2,4) result obtained using the QECCSD values of T1 and T2 and |\Ilo) defined

by Eq. (3.25) (obtained with the Arponen-Bishop ECC theory).

67

 

 

 

 

 

 

-107.35 r I ' I ' I V v f r 7
-107.40 - :9
. fl”
4; '_‘—:;:m‘M-‘
-107.45 - f
A m
E *~)|$__*__
t —107.50 - -
2 Q .FulICl
V 4 \ 00080 <
5‘: \ .'*'CR-CCSD(TQ).b
5 -107.55 - '\ Aoeccso -
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u.| . uosccsocro)
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\
§>\ *
l ‘- eke»
-107.65 9---<>---
-107.7o-‘A--‘-‘-‘A
2.0 3.0 4.0 5.0 6.0 7.0 8.0

RN_N (bohr)

Figure 3.1: Ground-state potential energy curves of the N2 molecule as described by

the STD-3G basis set. The QECCSD and ECCSD calculations were performed using

the ECC theory of Arponen and BishOp.

68

 

7"er 'tvr vvvv 'i" rii" fit" 7
U I Y I T I

-108.5 q

Full Cl

 

-108.6 r ——— ccso I -
alt-CCSDT I
’ ECCSD(T) /
_ . CCSD(TQ.) I .

1 08'7 AQECCSD ,'
Voeccscxm) ¢ ‘

 

-108.8 L

-108.9 -

Energy (hartree)

—109.0 -

-109.1 -

-109.2 -

 

 

 

 

_109.3 ..LLL.;L.1..-.1....I....1...;1..
1.5 2.0 2.5 3.0 3.5 4.0 4.5
RN_N(bohr)

Figure 3.2: Ground-state potential energy curves of the N2 molecule as described by
the DZ basis set. The QECCSD calculations were performed using the ECC theory

of Arponen and BishOp.

69

Chapter 4 Exactness of Two-Body Cluster
Expansions in Many-Body Quantum Theory

As demonstrated in Chapters 2 and 3, we can considerably improve the description of
chemical bond breaking and quasi-degenerate electronic states by performing ECCSD
calculations and by adding new types of noniterative energy corrections, based on
the GMMCC formalism, to the ECCSD energies. The ECCSD approaches and the
ECCSD-based GMMCC approaches are capable of eliminating the failures of the
standard CCSD, CCSD(T), and CCSD(TQf) methods, in spite of the fact that we only
use one— and two—body cluster operators in the ECCSD and ECCSD-based GMMCC
calculations. Our positive experiences with the ECCSD-based methods imply that
there is a lot of unexplored flexibility in the CC theories using only one- and two-body
clusters. The natural question arises if one can improve the quality of many-electron
wave functions based on the cluster expansions involving one— and two—body (or only
two-body) operators to a degree where the results become exact or virtually exact.
After all, the many-electron Hamiltonians used in quantum chemistry and atomic
and molecular physics do not contain higher—than-two—body terms. Thus, one may
wonder if there is a way to obtain the exact or virtually exact description of many-
electron systems with the CC—like theories that does not use higher—than—two—body
operators to construct the corresponding many-particle wave functions. An issue of

the exactness of the exponential cluster expansions employing two-body operators is

70

discussed in this chapter.

4.1 Theory

It is well known that one can always obtain the exact solution of the electronic
Schrodinger equation within a given basis set by performing the full CI calculations.
Unfortunately, the dimension of the full CI eigenvalue problem can easily run into
astronomical figures, even for small many-electron systems. This can be easily veri-
fied by applying the well-known Weyl’s formula97 for the number f (n, N, S) of spin-
adapted electron configurations that enter the full CI expansion of an eigenstate of

the electronic Hamiltonian H, i.e.,

2S+1 n+1 n+1
f(n’N’S)=m(N/2—S)(N/2+S+1)’ (4'1)

where n represent the number of correlated orbitals, N is the number of correlated
electrons, S is the total spin of an eigenstate under consideration, and (’1?) =
m!/[k!(m — k)!] is the binomial coefficient. For example, a full CI calculation of a sin-
glet electronic state for a system consisting of 10 electrons (e.g., the HF molecule or the
Ne atom) and described by only 20 orbitals requires using f (20, 10, O) = 52, 581, 816
configurations. A modest increase of the number of orbitals from 20 to 30 results
in a steep increase in the number of configurations defining the full CI problem to
4.04 x 109. Those numbers should be compared with the much smaller numbers of
22,155 and 108,345 two-electron integrals defining the electronic Hamiltonians for the
n = 20 and 30 cases, respectively. The point-group symmetry and other symmetries
of the Hamiltonian can, on occasion, reduce the dimension of the full CI problem, but
savings resulting from the use of symmetry are minimal compared to the rapidly in-

creasing numbers of configurations defining the full CI wave functions with n and N.

71

Indeed, the realistic calculation for a 10 electron system would require using ~ 100
orbitals. According to Eq. (4.1), the n = 100, N = 10, and S = 0 case leads to an
astronomical number of 9.94 x 1014 configurations in the corresponding full CI expan-
sion. This should be compared to a much smaller number of 12,753,775 two-electron
integrals defining the electronic Hamiltonian for the n = 100 case. Clearly, the num-
bers of full CI configurations would be significantly larger for N > 10 and for larger
n values due to the factorial scaling of the dimension of the full CI problem with the
system size. This should be contrasted with the relatively slow, n4-like, increase of
the number of two-electron integrals defining the Hamiltonian H with n. Thus, there
seems to be a conflict between the huge dimensionality of the full CI problem, which
prevents us from performing exact ab initio calculations for larger systems, and the
fact that the Hamiltonians for many-electron systems involve only one- and two-body
integrals, whose numbers are much smaller than the numbers of full CI coefficients
defining the exact wave functions. In spite of the tremendous progress in the area of
full CI calculations and computer technology (cf., e.g., Ref. 98 and references therein),
the exact ab initio calculations employing the full CI method remain limited to a few

electron systems described by small (usually, n ~ 20 — 30) basis sets.

4.1.1 The exp(X) Conjecture

It has recently been suggested that it may be possible to represent the exact or
virtually exact ground-state wave function of an arbitrary many-fermion pairwise in-
teracting system by an exponential cluster expansion involving a general two-body

99-105. If these statements were true, completely new ways of performing

Operator
ab initio quantum calculations for many-fermion (e.g., many-electron) systems might

be suggested, which could provide enormous reductions in computational require-

72

ments for accurate quantum calculations for pairwise-interacting many-fermion sys-
tems, eliminating the astronomical costs of generating the exact many-particle wave
functions by solving the full CI eigenvalue problem. Specifically, it has been proposed
that the exact ground-state wave function I‘IIO) of a given many-fermion system de-

scribed by the Hamiltonian,
1
H = zgcch + ivggopcqcscr, (4.2)

containing up to two-body terms, obtained in a finite spin-orbital basis set, has the
following simple form”:

No) 5 I‘I’0(X)) = exlq’o), (43)

where X is a general two-body operator and |<I>o) is a normalized reference state,
which in principle is an arbitrary wave function that has a nonzero overlap with
|\Ilo), but in practice should provide us with a reasonable approximation of I‘IIO). In
Eq. (4.2) and equations presented below, we use, whenever possible, the Einstein
summation convention over repeated upper and lower indices. As in the earlier parts
of this thesis, the op and op operators represent the usual creation and annihilation
operators, respectively, associated with the one particle basis {p}. The 25% = (plilq)
and 2253 = (pqlfilrs) represent the usual one- and two-particle integrals defining the

Hamiltonian. In the language of second quantization,

1

X=X2=2

£3019 cqcscr, (4.4)

where 23% are some coefficients. According to Nooijen99 (cf., also, Refs. 104, 105),
the number of independent coefficients 2:53 should be identical to the number of two-
particle integrals v53 entering the Hamiltonian H, Eq. (4.2). One could redefine the

operator X by considering the one- and two—body components in Eq. (4.4)99’101'102

73

and write

1
X = X1 + X2 = xgopql + Exacpcqcscr, (4.5)

but this is not really necessary, since for a fixed number of particles (N), one can
always rewrite the Hamiltonian H, Eq. (4.2), in terms of two-body terms only. A

straightforward manipulation shows that

l

H 2 Ehggcpcqcscr, (4.6)
where
fig}; 2 v53 + (2;,63 + 6523)/(N — 1), (4.7)

with 6;], representing the usual Kronecker delta. On the other hand, it may be bene-
ficial to use Eq. (4.5) rather than Eq. (4.4) in actual calculations, since the presence
of one-body term X1 = xgcpccl in the operator X may accelerate the convergence of
the resulting wave functions and energies.

The above representation of the exact ground-state wave function, Eq. (4.3), is
reminiscent of the exponential ansatz of the single-reference CC theory, Eq. (2.4).
There is, however, a fundamental difference between Eqs. (4.3) and (2.4). The
cluster operator T entering Eq. (2.4) is defined in terms of the particle-hole excitation
operators Exist-2", Eq. (2.7), where i1, . . . , in (a1, . . . , an) are the spin-orbitals that
are occupied (unoccupied) in the reference configuration |<I>o) (which is then a single
Slater determinant), and it contains all many-body terms Tn with n = 1,. . . , N in the
exact case. The operator X, Eq. (4.4), or its analog defined by Eq. (4.5), entering Eq.
(4.3), has at most two-body terms, but of the general type (excitations, deexcitations,
and all other combinations of indices p, q, r, 3). Moreover, the reference configuration

|<I>o) can be a multi-determinantal state. One could, of course, truncate the many-

body expansion for the cluster operator T in Eq. (2.4) at a two-body component

74

T2, Eq. (2.27), but then the resulting wave functions eT2|<I>o) and eT1+T2|<I>o), which
are used in the CCD and CCSD methods, respectively, are only approximate wave
functions. Thus, as we can see, there are significant differences between Eqs. (4.3) and
(2.4). However, because of the formal similarity of the CCD wave function, eT2|<I>o),
and Eq. (4.3), Nooijen and Lotrich106 and Van Voorhis and Head-Gordon104 call the
wave function ansatz defined by Eqs. (4.3) and (4.4) generalized CCD (GCCD) (the
wave function ansatz using Eqs. (4.3) and (4.5) is then called generalized CCSD or

GCCSD). A similar terminology has been used in Refs. 100—103.

4.1.2 Formal Arguments in Favor of the exp(X) Conjecture (Ground
States)

There are several facts that speak in favor of the correctness of Eq. (4.3). Nooijen
based his reasoning99 on the fact that the number of two-body coefficients $53 is iden-
tical to the number of components of the Nakatsuji two-particle density equation107,
which is, in turn, equivalent to the time—independent Schriidinger equation for Hamil-
tonians containing up to two-body terms. The problem that was left unsolved by
Nooijen is the solubility of a rather complicated exponential variant of the Nakatsuji
density equation, which forms an essential part of Nooijen’s analysis (cf. Eq. (11)
in Ref. 99). Van Voorhis and Head-Gordon based their reasoning104 on the fact (ex-

ploited in Quantum Monte Carlo techniques) that one can always obtain the exact

wave function by considering the expression

I‘I’o) = £13330 eztléo), (48)
where the two-body operator Z, is defined as follows:

Z, = —(H - Eo)t, (4.9)

75

with E0 representing the exact energy. They used Eq. (4.8) to write the two-body

operator X defining the exact wave function |\Ilo) via Eq. (4.3) as

t—mo

Similar arguments and equations have been presented by N akatsujimr‘“103

, who also
considered the conditions for the wave functions parameterized by two-body oper-
ators to be exactloo. The problem with using Eq. (4.8) in this fashion is that the
operator 2,, Eq. (4.9), provides the exact wave function only in the t —+ 00 limit,
whereas the Operator X, Eq. (4.4) or (4.5), entering Eq. (4.3), is a finite operator.
This immediately implies that the operator X defining the exact wave function |\Ilo)
through Eq. (4.3) cannot be constrained to be of the Hamiltonian form, Eq. (4.9)
(cf. below for additional remarks). Eq. (4.8) does not Open up the possibility of the
existence of a finite two-body operator X, which is not necessarily defined through
infinite coefficients 3;; (which Eq. (4.10) produces). There also seem to exist some
contradictions between statements made in Refs. 100 and 102, 103. The exactness
of the GCCD or GCCSD wave functions was questioned in Ref. 100 and supported
in Refs. 102, 103 (see Refs. 108—112 for further debate; see the discussion below for
additional comments).

We have recently provided an evidence105 that the exact ground state of a many-
fermion system, described by the Hamiltonian containing one- and two-body terms,
may indeed be represented by the exponential cluster expansion employing a general

two-body operator, Eq. (4.3), by connecting the problem with the Horn-Weinstein

formula for the exact energym,

E0 = tl_i+m ((pole—tHHI‘IHD/(cPMC-‘tflI‘po)

= 11m Em; =31“) E(Xt), (4.11)

t—mo

76

where

130.2 E(X.) = (eolexi‘ HextI<I>O>/<<I>olexi‘extl<1>o> (4.12)
and
1
X1: —§tH, (4.13)

and by determining the Operator X entering Eq. (4.3) through a direct minimization

of the expectation value expression
E00?) = <<I>ole*’He*I<I>o>/<<I>ole*‘e*I<I>o> (4.14)
over general two-body Operators
X = -~ggcpcqcscr (4.15)

This analysis (taken from Ref. 105) is described below.

Let us consider the family M of all two-body operators X , Eq. (4.15), that are
defined by finite coefficients 5;}; and that have a general structure of the Hamiltonian
H, Eq. (4.6). This means that M consists of all two-body operators that are, for
example, Hermitian, since H is Hermitian; that satisfy relations, such as 5:53 = 53;),
since has] = hag, etc. Obviously, the number of independent parameters 553 is
identical to the number of coefficients hgg or 1253 defining the Hamiltonian. It should
be noticed that all operators Xi, Eq. (4.13), and Z,, Eq. (4.9), belong to M, although
M is a much larger operator family, which contains infinitely many Operators that are
not multiples of H. This remark is important for the considerations discussed in this
section, since one can always Obtain the exact wave function and energy by applying
Eqs. (4.8) and (4.11), with Z, and X; defined by Eqs. (4.9) and (4.13), respectively.
As pointed out above, neither the operator Z, nor its X, analog can provide the

exact description of a many-fermion system for a finite value of t. We should search

77

for the operator X defining |\Ilo) via Eq. (4.3) by minimizing the expectation value
expression, E0(X), Eq. (4.14), over all Operators in M.

Let us, therefore, examine what the direct minimization of E0(X) in M leads
to. According to the Ritz variational principle, E0(X) is bounded from below by the
exact, full CI, energy, so that

&S%d) mm

for all operators X E M. This implies that there should exist a two-body operator

X E M that minimizes E0(X). We can write

E0(X) = min E0(X). (4.17)
XEM
Obviously,
E0 3 E0(X). (4.18)

Let us now consider the energy expression E0,“ Eq. (4.12), for an arbitrary (fixed)

value of t. We can immediately write,
E0(X) < Eat, (4.19)

since E0(X) is a minimum value of E0(X), Eq. (4.14), in a space of all two-body
Operators X, whereas E0; = Eo(X¢) is the value Of E0(X) at X = X, (cf. Eqs.
(4.14) and (4.12)). As a matter of fact, for a given value of t, one can always find a
two-body Operator Y from M such that E0(Y) < Em. An example of such operator
might be provided by Xt: with t’ > t, since, as shown in Ref. 113, E0,“ Eq. (4.12), is a
monotonically decreasing function Of t. However, since the Operator family M is much
larger than the “one-dimensional” manifold of Operators X t, which are multiples of H,
there is a chance that there exist two-body Operators Y E M which satisfy E0(Y) <

Em: and which are not given by Eq. (4.13). This indicates that the Operator X

78

minimizing E0(X ) may very well be a finite Operator (i.e., defined by finite coefficients
x53 and not Obtained by considering the limiting case of the t —+ 00 operators Xt),
although we cannot provide a rigorous mathematical proof that this is indeed the
case and the existence of finite Operator X may depend on the actual form of the
reference state |<I>o), which does not have to be a single Slater determinant (see, e.g.
Refs. 108-110). The existence of a finite Operator X E M that minimizes E0(X)
according to Eq. (4.17) and that is not of the Hamiltonian form is supported by the
numerical calculations for a few many-electron systems105 (see the discussion below).

The inequalities (4.18) and (4.19) can be combined into the following result:
E0 S 13000 < Eat, (4-20)

true for any value of t. In view of the Horn-Weinstein energy expression, Eq. (4.11),

by considering the t —> oo limit in Eq. (4.20), we Obtain the identity
E0 = E0(X)- (4.21)

This means that the two-body operator X, obtained by minimizing the expectation
value expression E0(X), Eq. (4.14), gives the exact energy E0 and, by the virtue of
the variational principle, the exact ground state |\Ilo), as stated in Eq. (4.3).

The above analysis makes the exactness of Eq. (4.3) a real possibility, but one
should not treat it as a complete mathematical proof of Eq. (4.3), since we cannot rig-
orously prove the existence of the finite coefficients 1:53 that would define the optimum
Operator X corresponding to a global minimum of E0(X) for an arbitrary reference
|<I>o). We can make, however, several useful Observations. First, the reasoning pre-
sented above, which is based on combining the Horn-Weinstein energy formula, Eq.
(4.11), with the minimization of E0(X), has an advantage over the arguments given

in Refs. 102—104 in that it frees us from necessarily assuming that the operator X can

79

only be obtained by studying the t —+ 00 Operators Xt, Eq. (4.13), or Z,, Eq. (4.9).
By minimizing E0(X), Eq. (4.14), in a space of all two-body operators (or, equiva-
lently, by minimizing E0(X) in a finite-dimensional space of variables :2”), which is
exactly how we obtained Operator X in Ref. 105 (cf. the examples below), we may
be able to find finite parameters 113%, defining the exact wave function |‘Ilo), precisely
because the operator X is not constrained to be a multiple of the Hamiltonian. If the
finite operator X, determined by some numerical procedure for minimizing E0 (X) in
M, is a local rather than a global minimum on the E0(X) multi-parameter surface,
then the resulting energy E0(X), calculated by substituting X = X into Eq. (4.14),
and the resulting wave function |\I!o(X)), Eq. (4.3), do not have to be exact. How-
ever, even in this case, the operator X may provide excellent results, opening up a
possibility of using the exponential wave functions (4.3), with X defined by Eq. (4.4)
(or (4.5)), in high-accuracy ab initio calculations. Second, the mathematical analysis
described above does not tell us anything about the specific form of the reference
state |<I>o) that should be used in the calculations exploiting the exp(X) ansatz. It is
possible, for example, that the finite parameters 2:53 that give the exact state |\IIo(X))
via Eq. (4.3) exist only for certain types of references |<I>o) that do not have to be
represented by a single Slater determinant (based on the analyses presented in Refs.
108—110, it is rather unlikely that |<I>o) is a single determinant although there is no
proof of this statement). It may happen, however, that highly accurate results are
already obtained with the ordinary Hartree—Fock reference Km) and the additional
improvements are obtained by using a multi-determinantal reference state |<I>o) in the
definition of |\II0(X)), Eq. (4.3). All Of these issues are explored here by preforming

numerical calculations for a few small many-electron systems.

80

4.1.3 Extension Of the exp(X) Conjecture to Excited States

The exp(X) conjecture was originally proposed for the ground states, but we can
easily extend it to the excited states using the standard variational approach. For

example, we can construct the trial wave function for the first excited state |\I!1) as
WW» = |\I’1(X“’)) — (wolw1(X<”)>Iwo>. (422)

Here,

In» =Noexw)|‘i’o) (4.23)

is the previously obtained normalized ground state (.No is the normalization factor,
X (0) is the optimum two-body Operator representing |\Ilo), and I‘io) is the single- or

multi-determinantal reference state used in the calculation of |\I'o)) and
l‘pl(X(l))):N1€X(l)|(i’l> (4.24)

is the normalized exp(X)-like form of the first excited state, where |<I>1) represents
the reference state for |\Ill). By minimizing the energy functional

:__ (@1(X‘1’)IHI@1(X‘”)>

~ (1)
EM ) <@.(X<1>)I®.(X<l>)>

 

(4.25)

over all two-body Operators X (I) 6 M, we Obtain the energy of the first excited state
and the corresponding wave function |\II1) E |\P1(X(1))), where X (1) is the Optimum
value of X (1) obtained by minimizing E1 (X (1)), Eq. (4.25).

For the k-th excited state, we can define the trial wave function for variational

calculations as

k—l
likd‘k’» = I‘l'k(X"")> — Z<wmlwk(x‘*>»lwm>. (4.26)
m=0
where |\Ilm), m = 1, . . . , k — 1, are the previously computed states and
WW) =N.e*‘*’|<i>.> (4.27)

81

is the normalized exp(X)-like form of the k-th excited state I‘IIk), with |<I>k) rep-
resenting the reference state for the k-th excited state. By minimizing the energy

functional
_ (‘i'k(X(k))|Hl‘i’k(X(k)))
(‘1’ka "")l\i'k(X (*0)

 

BAX“) (4.28)

over all two-body Operators X“) E M, we obtain the energy of the k-th excited state
and the corresponding wave function l‘IIk) E |\II,,(X("))), where X (k) is the optimum
value of X (k).

Since excited states of many-electron systems are almost always very
multi-determinantal and since they often have significant singly excited components,
we may improve accuracies and accelerate the convergence of the numerical calcula-
tions for excited states using Eqs. (4.22)—(4.28) by using Eq. (4.5) (rather than Eq.
(4.4)) to represent operator X“), while Obtaining the reference states ling) by diag-
onalizing the Hamiltonian H in a small space spanned by a few Slater determinants
that we believe are important to describe the ground and excited states of interest.

This is what we did in our excited-state calculations described below.

4.2 Numerical Results

In our calculations for a few many-electron systems, we first concentrated on the fol-
lowing two aspects of theory: (i) the existence of finite coefficients x53 defining the
two-body operator X, Eq. (4.4), and (ii) the non-Hamiltonian nature of the Operator

X. Initially, we focused on the ground-state theory. We tested the exact ground-

state theory, in which we used the unexpanded form Of the exponential operator ex

to define the E0(X) energy rather than the truncated power series expansion in X.

This was made possible by representing the operators H and X as matrices in the

82

finite-dimensional N-electron Hilbert spaces relevant to a molecular system under
consideration (using all symmetry-adapted Slater determinants |<I>o) and @313“),
n = 1,. . . ,N, defining the full CI problem, as basis states). In order to calculate
the exact value of E0(X), Eq. (4.14), in a given iteration of the numerical proce-
dure used to minimize E0(X), we first diagonalized the matrix representing X with
some unitary matrix (7 to obtain the diagonal matrix D = 0X0“. Next, we con-
structed eb by taking the exponentials of the diagonal elements of D. Finally, after
constructing eb, we calculated the matrix representing ex as U‘leDU and applied
it to a column vector representing |<I>o) to Obtain |\IIO(X)) according to Eq. (4.3).
The value of 13.02) was obtained by calculating (\IIO(X)|H|\II0(X))/(\IIO(X)|\II0(X)),
using the matrices representing H and |\IIO(X)), as described above. In order to
determine the Optimum Operator X and the corresponding energy E0(X), we used
the downhill simplex method82 to minimize the energy E0(X) in M (recall that M
is a family of all two-body operators). Typically, our calculations required ~ 100
iterations to Obtain a reasonably converged result, although in some cases we had to
iterate much longer, particularly if we wanted to determine high decimal places. We
realize that the numerical procedures used to obtain the Optimum Operator X are not
suitable for routine calculations or for larger many-electron systems. The main point
Of this study is addressing the question if the exact or virtually exact many-fermion
wave functions can be represented by Eq. (4.3). In this context, the efficiency of the
numerical procedures used to determine X is of the secondary importance.

Our test calculations were performed for a few many-electron systems, including,
among others, the 8-electron/16-spin-orbital H8 modell”, which consists Of eight
hydrogen atoms arranged in a distorted octagonal configuration and described by

the minimum basis set (MBS) obtained by placing one 3 function on each hydrogen

83

atom. An example of the H8 model is very important for the discussion of the
exactness of Eq. (4.3). This model provides us with a highly nontrivial situation,
where the number of independent two-body parameters x1733, defining operator X,
or the number of independent two-body integrals hf’g’ defining the Hamiltonian,
is considerably smaller than the dimension Of the corresponding N-electron Hilbert
space. Indeed, the number of two—body parameters 253 defining operator X is in
this case 186, whereas the number of all spin- and symmetry-adapted configurations
defining the corresponding full CI problem is 468. The one-body parameter 3;], that
enter Eq. (4.5) do not change this situation. The total number of one- and two-body
parameters defining X, Eq. (4.5), is 198, which is still a lot less than the number of
full CI configurations describing the exact wave function. The H8 model is described
by a single parameter a (in bohr), which describes the deviation of the geometry of
the Dgh-symmetric H8 system from the regular octagonm. The following values of
a were particularly important for testing: a = 1.0 and a = 0.0001. The a = 1.0
H8 system is somewhat less demanding, since in this case the exact ground-state
wave function is dominated by the RHF configuration |<I>o). The more demanding
a = 0.0001 H8 model is characterized by a strong configurational quasi-degeneracy of
the ground electronic state involving the RHF reference MM) and the doubly excited
configuration |<I>1) of the HOMO-LUMO typel”.

The results of our calculations for the ground-sate of the MBS H8 system are
shown in Table 4.1. As we can see, the exp(X) ansatz gives remarkably accurate
results for the ground-state of H8, even when the overlap of the reference state is as
small as 0.668268, as is the case when a = 0.0001 and we use the RHF determinant
as a reference. Independent of the value of a, we obtain microhartree accuracies,

which are a lot better than the accuracies resulting from various traditional CI and

84

CC calculations. Our exp(X) calculations for the ground state employing two-body
or one- plus two-body Operators X are order of magnitude more accurate than the
CISD and CCSD calculations, which also use one- and two-body operators only. In
fact, the exp(X) calculations provide a significantly better description Of the ground
state than the CISDTQ calculations employing one-, two-, three-, and even four-
body excitations. The number of parameters used to describe the CISDTQ wave
function is much bigger than the number Of parameters used to define our exp(X)
wave functions. The results of our exp(X) calculations employing at most two-body
operators are as good as the CCSDTQ results. In fact, by using one- and two-body
parameters in X and the two-determinantal reference I430) (see Table 4.1), we Obtain
the results which are considerably better than those obtained with CCSDTQ, when
the challenging case of the quasi-degenerate a = 0.0001 H8 model is examined.

As we can see, the presence of one-body operator X1 = xgcpql in X improves
the convergence toward the full CI results. The use of a two-determinantal refer-
ence state I450), which reflects the predominantly two-determinantal character of the
ground-state wave function at a = 0.0001, has a positive impact on the accuracy
of exp(X) calculations. Even if our numerical procedures do not produce the exact
(in a mathematical sense) energies, the microhartree accuracies, obtained with the
exponential cluster expansions involving up to two—body operators only, are truly in-
triguing. The parameters 2:53 or 3% and 2:53 defining the Optimum Operators X are
finite. For example, if we Optimize X assuming that X = X2 = é-mggcpcqcsor, the
largest values if 1:53 obtained in our ground-state calculations are 0.316845 at a = 1.0
and 0.596623 at a = 0.0001. The corresponding operators X do not commute with
the Hamiltonian, so that the Optimum X operators producing the highly accurate

results in Table 4.1 are not of the Hamiltonian form.

85

In Table 4.2 we show the results of the exp(X) calculations for the first excited
state of the 1A9 symmetry. This state is dominated by the RHF configuration |<I>o)
and the doubly excited determinant of the HOMO-LUMO type |<I>1) in the quasi-
degenerate, a = 0.0001, region, so that it is natural to choose a two-determinantal
state |<I>1) = c01|<I>o) + c11|<I>1) as a reference in our exp(X) calculations (see footnote
“a” in Table 4.2). The exp(X) calculations for the first excited 1Ag state were per—
formed using the numerical procedure described in Section 4.1.3. To facilitate our
numerical effort, we considered the truncated form of the exp(X) expansion, where
the exp(X ) series is truncated at the X 5° term. Since the Optimum coefficients 22% and
$53 are rather small and X " enters the exp(X) expansion as iX", the truncation of
the exp(X) expansion at the 5%X50 term produces essentially no errors (errors that
cannot be detected in numerical calculations with the double precision Fortran). As
shown in Table 4.2, the exp(X) calculations for the first excited 1A9 state of H8 at
a = 0.0001 produce microhartree accuracies. For oz = 1.0 we Obtained a somewhat
larger error on the order of 0.4 millihartree, but this can be understood if we real-
ize that the reference state |<I>1) is not a very good representation of the exact first
excited-state wave function in this region of a. We must realize, however, that this
error is on the same order as the error resulting from the equation-of-motion (EOM)
CCSDTQ calculations (for basic information about EOMCC methods, see, e.g., Refs.
9, 60, 63), which use a much larger number Of parameters to define the corresponding
wave function than the exp(X) expansion employing one- and two-body parameters
only. For a = 0.0001, the exp(X) calculations give a microhartree-level accuracy

for the first excited 1A9 state of H8. None Of the conventional C1 or CC methods,

including CISDTQ or even EOMCCSDTQ can produce the results of this quality.

86

Table 4.1.: A comparison of the ground-state energies of the MBS H8 system obtained
with the exp(X)-like wave functions, where X = X2 (a purely two-body operator) or
X1 + X2 (a sum of one- and two-body operators), with the exact, full CI, energies
and energies obtained in various CI and CC calculations. Full CI energies are total
energies in hartree and all other energies are errors relative to full CI (also in hartree).
We also give the overlaps of the normalized ground-state wave functions Obtained with
the exp(X) ansatz and the full CI approach. The overlaps Of the reference states |<I>o)

and I60) with the full CI ground-state wave function |\Ilo) are given for comparison.

 

 

Number of parameters

in the wave function a = 1'0 a = 0000]

Method (wave function)

 

 

Energ'es

Full CI 467 —4.352990 —4.204803
Noexp(X2)|<I>o)a 186 0.000008 0.000052
No exp(X1 + )6,.)|<1>0)a 198 0.000002 0.000020
1% exp(X1 + X2)|<I>o)b 198 0.000005 0.000007
CISD 46 0.008396 0.022779
CISDT 146 0.007984 0.016064
CISDTQ 320 0.000254 0.000449
CCSD 46 0.000546 0.005034
CCSDT 146 0.000026 —0.008362
CCSDTQ 320 -0.000001 —0.000035
Overlaps with a full CI wave function:

Noexp(X2)I(I’o)a 0.999998 0.999987
No exp(Xl + X2)|<1>o)a 0.999999 0.999995
No exp(Xl + x2)|<i>o)b 0.999998 0.999998
|<1>o)a 0.939657 0.668268
|<i>o)b 0.942804 0.909461

 

 

Who) is the ground—state RHF determinant.

bléo) = cooléo) +c10|¢I>1), where |<I>o) is the RHF determinant and Mn) is the doubly excited
determinant of the (HOMO)2 —+ (LUMO)2 type. The coefficients coo and 010 defining the
reference I‘io) were obtained by diagonalizing the Hamiltonian in a space spanned by |<I>o)
and |<I>1) and by selecting the lower energy eigenstate of H in this two-dimensional subspace

87

Table 4.2.: Same as Table 4.1 for the first excited state of the 1A9 symmetry.

 

 

Number of parameters

in the wave function a = 1'0 a = 00001

Method (wave function)

 

 

Energ'es

Full CI 467 -3.998978 —4.144027
M exp(X1 + X2)|<i>1)a 198 0.000405 0.000020
CISD 46 0.059314 0.042374
CISDT 146 0.031697 0.009726
CISDTQ 320 0.001608 0.000435
EOMCCSD 46 0.019605 0.015011
EOMCCSDT 146 —0.003174 —0.010505
EOMCCSDTQ 320 —0.000140 -0.000384
Overlaps with a full CI wave function:

JV. exp(X1 + X2)I<i>1)a 0.999998 0.999992
|<i>1)a 0.791762 0.902255

 

 

3|<I>1) = c01|<I>o) +c11|<I>1), where |<I>o) is the RHF determinant and |<I>1) is the doubly excited
determinant of the (HOMO)2 —+ (LUMO)2 type. The coefficients cm and cu defining the
reference |<I>1) were obtained by diagonalizing the Hamiltonian in a space spanned by Km)
and Mn) and by selecting the higher energy eigenstate of H in this two—dimensional subspace

83 l‘i’il-

88

 

4.3 Summary

Let us summarize the results discussed in this chapter. By combining the theoretical
arguments based on the Horn-Weinstein energy expression with variational princi-
ple and numerical calculations, we demonstrated that one can obtain the virtually
exact description of pairwise interacting many-fermion systems, including molecular
systems, by representing their ground- and exited-state wave functions by exponen-
tial cluster expansions employing general two-body or one- and two-body Operators.
Based on the evidence reported in this chapter, we can conclude that the “Optimum
two-body or one- and two-body operators defining these cluster expansions are fi—
nite and not of the Hamiltonian form. The results discussed in this chapter confirm
that one can tremendously improve the description of many-electron wave functions
without using higher—than—two-body cluster operators. All of this implies that there
is a lot of flexibility in the exponential cluster expansions, which was not utilized
in the past. Based on the formal arguments and the extraordinarily high accuracies
obtained in the calculations based on the exp(X) expansions, where X is a sum of
one- and two-body components or a purely two-body operator, we can conclude that
it is quite likely that one can formulate the exact or virtually exact many-electron

theories based on these kinds of expansions.

89

 

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90

 

9999”!"

10.

11.

12.

13.

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