JOURNAL OF CHEMICAL PHYSICSVOLUME 116, NUMBER 5

1 FEBRUARY 2002

Structure of the exact wave function. IV. Excited states from exponential
ansatz and comparative calculations by the iterative configurationinteraction and extended coupled cluster theories
Hiroshi Nakatsujia)
Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering,
Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan

共Received 23 July 2001; accepted 6 November 2001兲
In a previous paper of this series 关Paper III: Nakatsuji, J. Chem. Phys. 105, 2465 共2001兲兴, the author
showed a high potentiality of the extended coupled cluster 共ECC兲 method to calculate the exact
wave function of the ground state. In this paper, we propose ECC-configuration interaction 共CI兲method, which is an accurate useful method to calculate the excited states from the ECC wave
function of the ground state. In contrast to the ECC method, the standard ECC-CI method isapproximate, but we can make it exact by generalizing its excitation operator 共ECC-CI general兲. The
ECC-CI method is applicable not only to the excited states having the same spin-space symmetry as
the ground state, but also to those having different spin-space symmetries and to the ionized andelectron-attached states. The theoretical framework of the ECC-CI method is similar to that of thesymmetry-adapted-cluster 共SAC兲-CI method proposed in 1978 by the present author. Next in thispaper, we examine the performance of the methods proposed in this series of papers for a simple
one-dimensional harmonic oscillator. The iterative configuration interaction 共ICI兲 and ECC methods
are examined for the ground state and the ICI-CI and ECC-CI methods for the excited states. The
ICI method converges well to the exact ground state and the excited states are calculated nicely by
the ICI-CI method in both the standard and general active spaces. In contrast to the simplest 共S兲ECC
examined in Paper III, the ECC2 method shows quite a rapid convergence to the exact ground state,
which enables us to calculate the true exact wave function in the ECC form. The ECC-CI methods
in both the standard and general active spaces also work well to calculate the excited states. Thus,
we conclude that the ICI and ECC approaches have a potentiality to provide useful method to
calculate accurate wave functions of the ground and excited states. A merit of ECC is that it provides
the exact wave function in a simple explicit form. © 2002 American Institute of Physics.
关DOI: 10.1063/1.1430741兴

I. INTRODUCTION

N

H⫽

Toward precise predictions and gigantic-scale systems
are the two ways modern quantum chemistry must expand in
this decade. The first aim is essentially realized if we cansolve the Schrödinger equation with a realistic cost. The
full-CI method, only one general method of solving the
Schrödinger equation within a given basis set, is howeververy time consuming and highly demanding computationallyand therefore cannot be applied to chemically interesting
systems. Since the Hamiltonian operator has a very simple
structure, involving only one and two particle operators, its
eigenfunctions must also have some simple structure, reflecting the simplicity of the Hamiltonian. Though modern algorithms of the full-CI method1,2 utilize this fact in the evaluation of the matrix elements, it does not presume such asimplicity in the structure of the wave function.
The Schrödinger equation is given by
共 H⫺E 兲 ␺ ⫽0,

w 共 i, j 兲 ,兺i v共 i 兲 ⫹ 兺
i⬎ j

共1.2兲

in a coordinate representation, orH⫽⫹ ⫹v rp a r⫹ a p ⫹ 兺 w rs
兺
pq a r a s a q a p ,
pr
pqrs

共1.3兲

in a second-quantized form. In Eq. 共1.3兲 the indices p,q,r,s
run over all m reference orbitals 共both occupied and unoccupied兲. The number of terms in Eq. 共1.3兲 is

冋

册

2
m
M GSD⫽m ⫹ 共 m⫺1 兲 ,
2
2

共1.1兲

共1.4兲

where we did not assume the Hermiticity of the operators v
and w, and GSD stands for general singles and doubles.
We have studied in this series of studies3–5 the structureof the exact wave function using the equations that areequivalent to the Schrödinger equation in a necessary andwhere the Hamiltonian involves only one- and two-particle
operators asa兲Electronic mail: hiroshi@sbchem.kyoto-u.ac.jp

0021-9606/2002/116(5)/1811/14/$19.00

N

1811

© 2002 American Institute of Physics

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1812

J. Chem. Phys., Vol. 116, No. 5, 1 February 2002

Hiroshi Nakatsuji

sufficient sense. This equivalence is important, since theseequations have the same determinative power as the Schrödinger equation. The variational principle,

具 ␺ 兩 H⫺E 兩 ␦ ␺ 典 ⫽0,

共1.5兲

following criterion. When we apply the variational principle
to some candidate function ␺ , and if the variational equation
actually gives one set of the H-nijou equations, then that ␺
has the structure of the exact wave function. This is realized,
for example, whenis well-known to be equivalent to the Schrödinger equation.
The energy of the system, E, is defined by

具 ␺ 兩 H⫺E 兩 ␺ 典 ⫽0.

共1.6兲

The variational principle is used to search for the best possible wave function within the functional form of the given
␺ . If ␺ can become exact in its functional form, this principle gives the exact wave function as an extreme.
Another useful set of equations that is equivalent to the
Schrödinger equation is H-square or H-nijou 共‘‘nijou’’ isJapanese meaning square兲 equations. The equation,

具 ␺ 兩 共 H⫺E 兲 2 兩 ␺ 典 ⫽0,

共1.7兲

and each of the following equations

具 ␺ 兩 共 H⫺E 兲 H 兩 ␺ 典 ⫽0,

共1.8兲

具 ␺ 兩 H 2 ⫺E 2 兩 ␺ 典 ⫽0,

共1.9兲

and
together with Eq. 共1.6兲 are equivalent to the Schrödinger
equation. Further, based on these equations, we have shownin Paper II a useful set of equations that is equivalent to the
Schrödinger equation. We define a division of the Hamiltonian operator into N D parts by
ND

H⫽兺 HI .

I⫽1

共1.10兲

Then, the wave function ␺ that satisfies

具 ␺ 兩 共 H⫺E 兲 H I 兩 ␺ 典 ⫽0, 共 I⫽1, . . . ,N D 兲 ,共1.11兲

and Eq. 共1.6兲 is exact in the necessary and sufficient sense.
Similarly, the wave function ␺ that satisfies

具 ␺ 兩 共 H⫺E 兲共 H I ⫺E I 兲 兩 ␺ 典 ⫽0, 共 I⫽1, . . . ,N D 兲 ,共1.12兲

is also exact in the necessary and sufficient sense. In Eq.
共1.12兲, the partial energy E I corresponding to H I is definedby具 ␺ 兩 H I ⫺E I 兩 ␺ 典 ⫽0,

共1.13兲

and satisfies
ND

E⫽

兺 EI .

I⫽1

共1.14兲

We refer to Eqs. 共1.11兲 and 共1.12兲 also as H-square or
H-nijou equations. Equation 共1.8兲 is a special case of Eq.
共1.11兲 for N D ⫽1. Likewise, Eq. 共1.7兲 is a special case of Eq.
共1.12兲. It is easily seen from Eq. 共1.7兲 that the H-nijou equation is valid only to the exact wave function: when we definef 共 ␺ 兲 ⫽ 具 ␺ 兩 共 H⫺E 兲 2 兩 ␺ 典 ,共1.15兲

this quantity is always positive or zero, and f ( ␺ )⫽0 is realized only when ␺ is exact.
An underlying principle in this series of papers3–5 is that
we can find the structure of the exact wave function by the␦ I ␺ ⫽H I ␺ ,共1.16兲

␦ I ␺ ⫽ 共 H I ⫺E I 兲 ␺ ,共1.17兲

orwhere ␦ I represents a variation with respect to the variableassociated to H I included in ␺ , since then we have Eq. 共1.11兲
or 共1.12兲 as a result of the variational equation 共1.5兲. In otherwords, when this holds, the variational best is the exact wave
function: that ␺ has a freedom to be able to become the exact
wave function. Note however that the condition given by Eq.
共1.16兲 or 共1.17兲 is a bit too strong: it is enough if we could
obtain Eq. 共1.11兲 or 共1.12兲 from Eq. 共1.5兲, as we argued inPaper III. In other words, Eq. 共1.16兲 or 共1.17兲 is a sufficientcondition for ␺ to have an exact structure, but not necessary.
Theorem II-2 of Paper I3 is a special case of the above theorem.
This fact further implies that the exact wave function canbe described with only N D variables, N D being defined by
Eq. 共1.10兲. The equivalence of a set of the N D equations
共1.11兲 or 共1.12兲 to the Schrödinger equation means that we
can describe the exact wave function with only N D variables,
because these equations have the same determinative power
as the Schrödinger equation. This actually implies that theexact wave function can be described with the number ofvariables from unity to, say, M GSD . The latter corresponds tothe division of the Hamiltonian defined by Eq. 共1.3兲. These
numbers are much smaller than the number of variables infull-CI, so that this line of research may lead us to a new
simpler method of solving the Schrödinger equation.
In Paper I of this series,3 we have investigated the possibility of the general singles and doubles 共GSD兲 descriptionof the exact wave function based on Theorem II-1 of Paper I,
which is also a kind of H-nijou equation, and proposed iterative CI 共ICI兲 method including GSD number of variables. In
Paper II,4 we have presented the H-nijou equations, Eqs.
共1.11兲 and 共1.12兲, and shown that the number of variables
necessary to solve the Schrödinger equation is defined by the
number of division of the Hamiltonian operator, and therefore ranges from unity to the GSD number. Based on thisfinding, we have extended the concept of the ICI method andproposed the ICIND method, which is the ICI including N D
variables. We have also considered the methods of calculating the excited states from the ICI theory. The simplest
共S兲ICI method that includes only one variable is similar tothe surplus function approach proposed recently by Huang
et al.6 It is also related to the Lanczos method7 in eigenvalue
problems.8 In Paper III,5 we have examined the exponential
ansatz and shown that it can also be exact if the operator isdefined using the divided Hamiltonian. We have proposed
extended coupled cluster 共ECC兲 method, and two interestingcases were considered: one is the S共simplest兲ECC that includes only one variable and the other is the general ECCND
共N D ⭓2兲. SECC is exact in spite of its simple structure, but is

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J. Chem. Phys., Vol. 116, No. 5, 1 February 2002

Structure of the exact wave function. IV

rather inflexible. ECCND is also able to describe the exact
wave function rather efficiently through its flexible nonlinear
structure. Test calculations were given for harmonic oscillator, showing different behaviors of the SECC and ECCND
methods: when truncated, the SECC was rather slowly converging, but the ECCND converged quite nicely.5
One purpose of this paper is to formulate a method to
calculate the excited states from the ECCND method. We
refer to the proposed method as the ECC-CI method, since
the theoretical framework of the theory is similar to that of
the SAC共symmetry adapted cluster兲-CI theory.9,10 Though
the ECC method is exact, the ECC-CI method is not, but it
can be improved up to the exact limit by expanding the operator space. Another purpose of this paper is to examine the
ICI and ECC methods in the calculations of the ground andexcited states of harmonic oscillator. We examine the convergence behaviors of both methods and the qualities of thecalculated ground and excited states. The harmonic oscillator
is probably the simplest possible system, but the analyticalapplication of the ICI and ECC methods may clarify their
behaviors, some of which would be common to those formore general many-electron systems, which is a goal of thepresent theory.
II. ECC

We summarize here briefly the extended coupled cluster
共ECC兲 theory.5 We first define the division of the Hamiltonian into N D parts
ND

H⫽

兺 HI ,

共2.1兲

I⫽1

and correspondingly, we define the variable operator,
NDS⫽兺 C IH I ,

共2.2兲

I⫽1

which includes N D variables 兵 C I 其 . Using this variable operator, we define the coupled cluster expansion by␺ ⫽exp共 S 兲 ␺ 0

冋

册

1
1
⫽ 1⫹S⫹ S 2 ⫹ S 3 ⫹ ••• ␺ 0 ,2
3!

共2.3a兲
共2.3b兲

where ␺ 0 is some reference function, which may be
Hartree–Fock 兩 0 典 , or some other function. Since this is asmall extension of the coupled cluster theory,11–13 we called
it extended coupled cluster 共ECC兲 theory:14 since this ECC
involves N D variables, we refer to the ECC given by Eq.
共2.3兲 as ECCND method.
When we divide the Hamiltonian into all singles anddoubles parts as given by Eq. 共1.3兲, we have ECCGSD,

␺ ECCGSD⫽exp共 T 兲 ␺ 0 ,T⫽⫹ ⫹C rp a r⫹ a p ⫹ 兺 C rs
兺
pq a r a s a q a p .
pr
pqrs

共2.4兲
共2.5兲

On the other hand, the conventional coupled cluster withinsingles and doubles 共CCSD兲 is written as

␺ CCSD⫽exp共 T c 兲 ␺ 0 ,

共2.6兲

T c⫽

1813

ab ⫹ ⫹
C ai a ⫹兺
a a i⫹ 兺 C i j a a a b a ja i ,
ia
i jab

共2.7兲

where the indices i, j run over occupied orbitals and a, b overunoccupied orbitals. Since the operator T c does not correspond to a division of the Hamiltonian operator, the conventional CCSD does not have a freedom of the exact wave
function, while since the ECCGSD operator T corresponds toa division of the Hamiltonian as defined by Eq. 共1.3兲, the
ECCGSD is capable of representing the exact wave function,
as discussed in Paper III.5 Nooijen15 also proposed independently that the generalized CCSD that is equivalent to theabove ECCGSD might be exact. Van Voorhis andHead-Gordon16 recently confirmed numerically that the
ECCGSD certainly gives the exact wave function for neonand N2 .
It was also shown in Paper III that the SECC including
only one variable,

␺ ⫽exp共 CH 兲 ␺ 0 ,共2.8兲

is exact in the necessary and sufficient sense: the variational
determination of the variable C gives the H-nijou equation,
Eq. 共1.8兲. As shown by Kozloff and Tal-Ezer,17 this SECC
becomes exact with a large negative value 共negative infinite兲
of C. Therefore, a truncation of the expansion of the exponential operator at some order is not a good approximation: it
is desirable to treat it in the exponential form, not in its
approximate finite-order truncated form. This implies that theconvergence of the SECC may not be fast, though actually
its rate is dependent on the quality of the reference function
␺ 0 . A simple example was given in Paper III. The SECC isalso related to the t expansion of Horn and Weinstein18 andconnected-moments expansion of Cioslowski.19
Since even the simplest ECC 共SECC兲 is exact, the
ECCND should also be exact: it includes SECC as a specialcase. Therefore, we impose variational and H-nijou equations to the ECCND and obtain,冋1
1
G I 共 兵 C I 其 兲 ⫽ 具 ␺ 兩 共 H⫺E 兲 H I ⫹ 共 H I S⫹SH I 兲 ⫹ 共 H I S 223!

册⫹SH I S⫹S 2 H I 兲 ⫹••• 兩 ␺ 0 典 ⫽0,

共2.9兲

and
F I 共 兵 C I 其 兲 ⫽ 具 ␺ 兩 共 H⫺E 兲 H I 兩 ␺ 典

冋⫽ 具 ␺ 兩 共 H⫺E 兲 H I ⫹H I S

册1⫹ H I S 2 ⫹••• 兩 ␺ 0 典 ⫽0,2

共2.10兲

respectively, for all I(1⭐I⭐N D ) The exact wave function
should satisfy both of them. Subtracting Eq. 共2.9兲 from Eq.
共2.10兲, we obtain

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1814

J. Chem. Phys., Vol. 116, No. 5, 1 February 2002

⌬ I 共 兵 C I 其 兲 ⫽ 具 ␺ 兩 共 H⫺E 兲冋Hiroshi Nakatsuji

1
1
共 H S⫺SH I 兲 ⫹ 共 2H I S 2 ⫺SH I S
2 I
3!

册

再

冎

共2.11兲

for all I. If all H I and S commute, 关 H I ,S 兴 ⫽0, Eq. 共2.11兲
holds and Eq. 共2.9兲 reduces to Eq. 共2.10兲, but in general, such
commutation relation can not be assumed. If two of the threeequations, Eqs. 共2.9兲–共2.11兲, hold, this ␺ satisfies both variational and H-nijou equations, and therefore, it is exact. Since
Eqs. 共2.9兲–共2.11兲 are all non-linear for the variables 兵 C I 其 ,they can be satisfied not only in the SECC case, where all C I
are equal 共Case A兲, but also in a more general case where C I
takes different optimal values 共Case B兲.5 We have shown inPaper III that a quick convergence of the ECC expansion isrealized in Case B. This is natural since the standard CCSD,
an approximation of ECCGSD as explained above, is alreadya good approximation of the exact wave function for many
molecular systems.5
The ECC method describes the ground state of eachsymmetry. Since the S operator is totally symmetric, the ECC
exponential operator is also totally symmetric, and therefore,
the symmetry of the calculated state is that of the referencefunction ␺ 0 . In the next section, we consider how to calculate the excited states from the ECC theory of the groundstate.
III. ECC-CI

We now formulate the theory for the excited states fromthe ECC theory of the ground state. Though the ECC theory
for the ground state is exact, the theory presented in thissection for the excited state is not exact. We start from the
H-nijou equation for the ground state,

具 ␺ g 兩 共 H⫺E 兲 H K 兩 ␺ g 典 ⫽0 共 K⫽1, . . . ,N D 兲 ,共3.1兲

where the subscript g on ␺ g denotes the ground state. Now,
we define the 共excited兲 functions ⌽ K by
⌽ K ⫽H K ␺ g ,共3.2兲

then this set of functions 兵⌽ K 其 , (K⫽1, . . . ,N D ) satisfies theBrillouin-orthogonality with the exact ground state ␺ g ,具 ␺ g 兩 共 H⫺E 兲 兩 ⌽ K 典 ⫽0,

共3.3兲

as easily seen from Eq. 共3.1兲. This relation implies that thefunctions 兵 ⌽ K 其 are good basis for the excited states of ␺ g ,so that we approximate our excited states by a linear combination of these excited functions as
ND␺ e⫽ 兺 d K⌽ K⫽
K⫽0

冉兺 冊
ND

K⫽0

d KH K ␺ g ,共3.5a兲

␺ e ⫽R ␺ g .

1
1
关 H I ,S 兴 ⫹ 共 2 关 H I ,S 兴 S
2
3!

⫹S 关 H I ,S 兴 兲 ⫹••• 兩 ␺ 0 典 ⫽0

兺 d KH K,

K⫽0then this operator is a generator of the excited state from theground state as⫺S 2 H I 兲 ⫹••• 兩 ␺ 0 典⫽ 具 ␺ 兩 共 H⫺E 兲

ND

R⫽

共3.4兲

where we have included ␺ g (⫽⌽ 0 , H 0 ⫽1) to ensure theorthogonality of ␺ e to ␺ g . If we introduce the operator R by

共3.6兲

Note a similarity of the R operator defined by Eq. 共3.5a兲 to
the S operator defined by Eq. 共2.2兲. For convenience, wedenote the set of operators 兵 H K 其 included in Eq. 共3.5a兲 asRS ,RS ⫽ 兵 1, H K 其 ⫽ 兵 R S 其 ,共3.7兲

where the subscript S implies ‘‘standard,’’ in contrast to‘‘general’’ introduced in the next section. With the use of Eq.
共3.7兲, the excitation operator R is written as
ND

R⫽兺 d SR S .

共3.5b兲S⫽0

We refer to this method of calculating the excited states as
the ECC-CI method, since Eq. 共3.4兲 is CI-like. This namingis also due to the similarity of the present theory to the
SAC 共symmetry adapted cluster兲-CI theory for excitedstates.9,10 The SAC theory20 is a kind of coupled cluster
theory11,13 for the ground state, like ECC is for the groundstate, and the SAC-CI theory is for the excited state and wasformulated similarly to Eqs. 共3.4兲–共3.6兲 using the SAC
instead of the ECC ground state ␺ g . The
ground state ␺ SAC
g
CCLRT 共coupled cluster linear response theory兲21–23 and the
EOMCC 共equation-of-motion coupled cluster兲 theory24 –26
developed later are equivalent to the SAC-CI theory.27
The formulation for the ECCGSD case is interesting,where the division of the Hamiltonian is defined by Eq. 共1.3兲.
In this case, we start from the H-nijou equations,具 ␺ g 兩 共 H⫺E 兲 a r⫹ a p 兩 ␺ g 典 ⫽0,

共3.8兲

具 ␺ g 兩 共 H⫺E 兲 a r⫹ a s⫹ a q a p 兩 ␺ g 典 ⫽0,which were given in Paper I as Theorem II-1. Note, in theseequations, we have omitted the constant integral factors, v rp
and w rs
pq of Eq. 共1.3兲. Now, we define the excited functions
兵 ⌽ K 其 of the ECCGSD-CI method by⌽ p,r ⫽a r⫹ a p ␺ g ,⌽ pq,rs ⫽a r⫹ a s⫹ a q a p ␺ g ,共3.9兲

and approximate our excited state by a linear combination ofthese functions as,冉冊␺ e ⫽R ␺ g ⫽ d 0 ⫹ 兺 d p,r a r⫹ a p ⫹ 兺 d pq,rs a r⫹ a s⫹ a q a p ␺ g ,
p,r

pq,rs

共3.10兲

where the ground state is included to ensure the orthogonality of ␺ e to ␺ g . This is ECCGSD-CI, and this expression
would be familiar since it is similar to the standard singles
and doubles SAC-CI.
Now, we come back again to the ECCND-CI case. The
expansion coefficients d K are determined variationally by the
secular equation,

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J. Chem. Phys., Vol. 116, No. 5, 1 February 2002

具 ⌽ K 兩 H⫺E 兩 ␺ e 典 ⫽0 共 K⫽0,1, . . . ,N D 兲 ,Structure of the exact wave function. IV

共3.11a兲which is equivalent toND兺 具 ⌽ K兩 共 H⫺E 兲 兩 ⌽ L 典 d L ⫽0 共 K⫽0,1, . . . ,N D 兲 .

L⫽0

共3.11b兲
The solutions of this secular equation satisfy orthogonality
and Hamiltonian-orthogonality to the ground-state ␺ g and toeach other, namely,

具 ␺ e 兩 ␺ g 典 ⫽0, 具 ␺ e 兩 H 兩 ␺ g 典 ⫽0,

共3.12兲

具 ␺ e ⬘ 兩 ␺ e 典 ⫽ ␦ ee ⬘ ,共3.13兲

and具 ␺ e ⬘ 兩 H 兩 ␺ e 典 ⫽E e ␦ ee ⬘ ,respectively.
Though the ECC wave function for the ground state isexact, the ECC-CI wave function defined in this section for
the excited state is approximate, since the space spanned by
兵 ⌽ K 其 is not complete. The nature of the excited states described by this method is characterized and restricted by thenature of the 兵 ⌽ K 其 space. To explicitly express the quality of
the calculated excited states, we may refer to the above
ECC-CI method as ECCND-CIND method, where the first
ND stands for the size of the ECC calculations and the second ND stands for the size of the 兵 ⌽ K 其 space considered in
the ECC-CI calculations. This is in some sense similar to the
SAC-CI SD-R method in which the dimensions of both SAC
and SAC-CI are singles and doubles 共R stands for the
SAC-CI excitation operator兲.
We note that when ␺ g strictly satisfies the ECC equation,
Eq. 共3.1兲, then it is exact and not modified by the ECC-CI
method, but when ␺ g is an approximate solution of the ECC
equation, it may be improved at the ECC-CI step: the groundstate solution obtained from the ECC-CI secular equation,
Eq. 共3.11兲, may become lower than that of the input approximate ␺ g . The improved ␺ g satisfies the relation similar to
Eq. 共3.12兲.
Like in the SAC-CI method, we can also consider thenonvariational method of solution. Though the quality of thesolution would become worse, the computational labor isdiminished. In the nonvariational method, the bra-functionsof Eq. 共3.11a兲 are replaced by the set 兵 ␺ 0 , H K ␺ 0 其 , where␺ 0 is the reference function and we obtain

具 ␺ 0 兩 H⫺E 兩 ␺ e 典 ⫽0,

共3.14a兲具 ␺ 0 兩 H K* 共 H⫺E 兲 兩 ␺ e 典 ⫽0 共 K⫽1, . . . ,N D 兲 .

共3.14b兲

The nonvariational method is acceptable when the set of brasin Eq. 共3.14兲, 兵 ␺ 0 , H K ␺ 0 其 is a good approximation of theset 兵 ⌽ K 其 . We note that when ␺ 0 is Hartree–Fock and whensome H K operators are composed only of purely X t type
operators,28 which include either a ⫹i or a a , where the subscripts i and a denote occupied and unoccupied orbitals,
respectively,3 then Eq. 共3.14b兲 would vanish identically andtherefore, this nonvariational method would be inappropriate, because the number of the equations would be smaller
than that of the variables.
The above ECC-CI method produces the excited states
belonging to the same symmetry as the ground state. How-

1815

ever, like in the SAC-CI method, we can extend the ECC-CI
method so that it also gives the excited states of different
spin-space symmetries, ionized states, and electron-attachedstates. Since the operators of the ECC-CIGSD are composedof a set of the products of the creation and annihilation operators a r⫹ and a p where the suffices r and p denote spinspace orbitals, it is easy to introduce similar set of substitution operators that belong to different spin-space symmetry.
Examples are seen from the SAC-CI operators given in Refs.
9 and 10. Denoting such set of operators again by RS⫽ 兵 R S 其 , and using Eqs. 共3.5b兲 and 共3.6兲, we can calculate theexcited states belonging to the different symmetry.
Similarly, we can calculate the ionized and electronattached states. In the above case, the substitution operators
were made of the products of the same number of creation
and annihilation operators. But, if the number of the creation
operators is smaller by n than the number of the annihilation
operators, it is the n-electron ionization operator, and if thenumber of the creation operators is larger by n than the number of the annihilation operators, it is the n-electronattachment operator. If we denote such set of operators again
by RS ⫽ 兵 R S 其 , we can calculate n-electron ionized andn-electron-attached states by applying the operator R definedby Eq. 共3.5b兲 to the ground state as in Eq. 共3.6兲.
IV. ECC-CI general

Since the ECC-CI method in the standard active spacedescribed above may be insufficient to obtain sufficient accuracy, we need a theory that can reach up to the exact limit
along the same line of the formalism. We give here as suchtheory ECC-CI general theory. In this theory, we can describe the excited, ionized, and electron-attached states in a
sufficient accuracy even up to the exact limit by expanding
the size of the R operator. This method is similar to the
SAC-CI general-R method.27,29–31
We expand the ECC-CI space defined by RS of Eq. 共3.7兲
by including the operators that are linearly independent fromthem. Such linearly independent operators may be made of
the products of the lower operators as
RA ⫽ 兵 H I H J , H I H J H K , H I H J H K H L , . . . 其 ⫽ 兵 R A 其 ,共4.1兲
where the subscript A implies ‘‘additional.’’ Here, we note ingeneral, H I H J ⫽H J H I , etc. Using these higher operators, we
expand the ECC-CI operator as
RG⫽RS ⫹RA ,共4.2兲

and calculate the excited state by␺ e ⫽R ␺ g ,

共4.3兲

where
ND

R⫽

兺 d S R S ⫹ 兺A d A R A .

S⫽0

共4.4兲

We refer to this expansion of ECC-CI as ECC-CI general,from the similarity to the SAC-CI general-R theory.29 Notethat for the SECC (N D ⫽1), the space RG defined by Eq.
共4.2兲 is just the same as the Krilov space.32

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1816

J. Chem. Phys., Vol. 116, No. 5, 1 February 2002

When our ECC is ECCGSD, the standard operator
space RS is defined by singles and doubles
兵 1, a r⫹ a p , a r⫹ a s⫹ a q a p 其 , and the additional operators for the
ECC-CI general are composed of the product operators like
triples, quadruples, and highers as,

Hiroshi Nakatsuji
TABLE I. SICI for the ground states of harmonic oscillator.

⫹⫹ ⫹ ⫹ ⫹RA ⫽ 兵 a s⫹ a ⫹t a u a ra qa p , a t a u a v a w a sa ra qa p , . . . 其,共4.5兲and therefore, it is easily understood that this ECC-CI general can formally become up to exact.
By this method, the quality of the calculated excitedstates is improved, and furthermore, we can calculate higher
excitation processes that may be rather poorly described by
the standard ECC-CI method. For example, three electron
excitation processes from the ground state would not be described within the ECCGSD-CIGSD method, but would be
described when we include higher operators defined by Eq.
共4.5兲.
The formation of the higher operators by Eq. 共4.1兲 is an
example of the exponential generation 共EG兲 idea presented
in 1985.33 When lower operators R I and R J are of some
importance, the product R I R J should also be important in thedescription of the states. If it is difficult to include all ofthese higher operators, because the size of the product operators easily becomes formidably large, we may include only
such products R I R J in which both R I and R J are importantby some given threshold. By such a criterion, we can produce only the important higher excitation operators.
The modifications of the ECC-CI general method to include the excited states of different symmetries, ionizedstates and the electron-attached states are rather simple. We
can improve the quality and expand the nature of these statesby expanding the active space by the ECC-CI general
method. Many examples have actually been reported by the
SAC-CI general-R methods27,30,31 for the calculations of theionization spectra including shake-up satellite peaks.

V. PERFORMANCE OF ICI AND ECC METHODS FORHARMONIC OSCILLATOR

We here examine the performance of the ICI and ECC
methods by applying them to the ground and excited statesof a one-dimensional harmonic oscillator. This is probably
the simplest possible system. Some behavior would be specific to this simple system, but some would be common tomore complex electronic systems. We want to see the qualityof the results calculated by the ICI and ICI-CI methods andthe ECC and ECC-CI methods for the ground and excitedstates. In Paper III, we have applied the S共simplest兲ECC,
which is ECC1, and ECC2 to this model system and obtainedmuch insight on the performance of these methods. Thoughboth are exact, the performance of the truncated ECC ismuch different: the SECC is rather slowly converging, but
ECC2 shows quite a rapid convergence. This shows a different behavior of ECCND for N D ⫽1 and N D ⭓2, and is due toa nonlinear nature of the ECCND method.
The Schrödinger equation for the one-dimensional harmonic oscillator is

n

␧

C

Gerade
0
1
2
3
4
5
6
7
8
9

1.01671.000 203
1.000 004 46
1.000 001 04
1.000 000 318
1.000 000 109
1.000 000 0459
1.000 000 0206
1.000 000 0103
1.000 000 005 41⫺0.1961⫺0.1093⫺0.091 58⫺0.1142⫺0.075 76⫺0.0895⫺0.0654⫺0.0739⫺0.0571

Ungerade
0
1
2
3
4
5
6
7
8
9

3.05
3.001 00
3.000 0340
3.000 008 41
3.000 002 69
3.000 000 975
3.000 000 419
3.000 000 193
3.000 000 0984
3.000 000 0526

⫺0.139 55⫺0.0894⫺0.0790⫺0.0887⫺0.0666⫺0.0731⫺0.0581⫺0.0621⫺0.0514

冉

⫺

d2
dx

2

冊

⫹x 2 ␺ ⫽␧ ␺ ,共5.1兲

and the Hamiltonian is the sum of the kinetic and potential
operators as
H⫽⫺

d2
dx 2

⫹x 2 .

共5.2兲

Two lowest solutions of this equation are冉

冊1␺ g ⫽ ␲ ⫺1/4exp ⫺ x 2 ,2
␧ g ⫽1,
and

共5.3兲

冉

冊

1
␺ e ⫽ 冑2 ␲ ⫺1/4xexp ⫺ x 2 ,2␧ e ⫽3,

共5.4兲

which are the ground states of the gerade and ungerade symmetries, respectively. The eigenvalues of the gerade statesare 1.0, 5.0, 9.0, 13.0, 17.0, 21.0,..., and the ungerade eigenvalues are 3.0, 7.0, 11.0, 15.0, 19.0, 23.0, ... . We take

␺ 0 ⫽exp共 ⫺ ␣ x 2 兲 ,共5.5兲

␺ 0 ⫽xexp共 ⫺ ␣ x 2 兲 ,

共5.6兲

and

with ␣ ⫽0.60 as our reference functions ␺ 0 for the gerade
and ungerade states, respectively. ␣ ⫽0.5 is the exact solution.
The S operator for SICI and SECC is

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J. Chem. Phys., Vol. 116, No. 5, 1 February 2002

Structure of the exact wave function. IV

1817

TABLE II. ICI2 and ICI2-CI2 for gerade ground and excited states of harmonic oscillator.
␧

C0

CK

CP

Ground state
0
1a
2
3
4
5
6

1.01671.000 203
1.000 002 10
1.000 000 0406
1.000 000 004 85
1.000 000 000 804
1.000 000 000 160

•••
1.0
1.0
1.0
1.0
1.0
1.0

•••
⫺.196 086 246⫺.108 505 5618⫺.075 653 4009⫺.068 844 6171⫺.072 965 4691⫺.058 936 1691••••••⫺.107 506 5766⫺.075 632 6339⫺.068 844 1948⫺.072 965 4621⫺.058 936 1679

Excited state
0
1a
2
3
4
5
6

•••
5.09985.002 94
5.000 0663
5.000 001 97
5.000 000 0705
5.000 000 0226•••
1.0⫺.232 577⫺.197 249⫺.212 814⫺.363 227⫺.413 158•••⫺.999 799 67⫺.531 423 50⫺.605 446 65⫺.574 3707⫺.273 564⫺.173 683

•••
•••
1.0
1.0
1.0
1.0
1.0

n

a

For n⫽1, the rank of the matrix is 2 and therefore identical with SICI.

冉

S⫽C ⫺

冊

d2
dx

⫹x 2 ,
2states by Methods B and C of Paper II. In the terminology ofthis paper, method B is called ICIND-CIND method, andmethod C is called ICIND-CI general. The latter naming isuseful because it makes the active space involved in eachmethod clear.
Table I gives the result of SICI for the gerade and ungerade ground states. The SICI shows a good convergencefor both states. The energy converges nicely to the exact
energy, 1.0 and 3.0, both from above. The coefficients initially vibrate to some extent and would converge to zero
finally. But, it would take some more iterations before convergence. The energy converges faster than the wave function itself. Tables II and III are the results of ICI2 for thegerade and ungerade ground states, respectively. The excitedstate calculated from the same secular equation 共namely by
ICI2-CI2兲 is also given in these tables. Note for n⫽1 共first
iteration兲, the rank of the matrix is 2, and therefore the result
becomes identical with that of SICI. For higher iterations, theICI2 gives faster convergence than SICI because of an increase in the number of the variational parameters. It is in-

共5.7兲

which includes only one variable, and that for ICI2 andECC2 isS⫽⫺C Kd2
dx 2⫹C P x 2 ,共5.8兲

which includes two variables C K and C P that are associatedto the kinetic and potential operators, respectively.
A. ICI method

The ICI method is defined by the recursion,␺ n ⫽ 共 1⫹S n 兲 ␺ n⫺1 ,共5.9兲

and the variables in each iteration are solved by the variational principle. As an initial function ␺ 0 , we take the functions given by Eqs. 共5.5兲 and 共5.6兲 for the gerade and ungerade ground states, respectively. We also calculate the excited

TABLE III. ICI2 and ICI2-CI2 for ungerade ground and excited states of harmonic oscillator.
n

␧

C0

CK

CP

Ground state
0
1a
2
3
4
5
6

3.05
3.001 00
3.000 0145
3.000 000 371
3.000 000 0463
3.000 000 008 02
3.000 000 001 67

•••
1.0
1.0
1.0
1.0
1.0
1.0

•••
⫺.139 55⫺.088 643 9874⫺.065 561 1301⫺.060 767 7653⫺.062 131 7043⫺.055 270 6686••••••⫺.087 285 2785⫺.065 519 9553⫺.060 766 6567⫺.062 131 6816⫺.052 706 6821

Excited state
2
3
4
5
6

7.006 69
7.000 196
7.000 007 51
7.000 000 294
7.000 000 0938⫺.826 3355⫺.672 6282⫺.740 2273
1.0
1.0⫺.443 8106⫺.551 4566⫺.506 5120
.175 0141.102 194 12

1.0
1.0
1.0
⫺.841 6808⫺.768 860 799

a

For n⫽1, the rank of the matrix is 2 and therefore identical with SICI.

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1818

J. Chem. Phys., Vol. 116, No. 5, 1 February 2002

Hiroshi Nakatsuji

TABLE IV. ICI2-CI general for the gerade and ungerade states of harmonic oscillator.
n⫽2

n⫽4

n⫽6

Variational
E 0 共g兲
E 1 共u兲E 2 共g兲
E 3 共u兲E 4 共g兲
E 5 共u兲E 6 共g兲
E 7 共u兲E 8 共g兲
E 9 共u兲E 10 共g兲
E 11 共u兲

1.000 000 000 001 59
3.000 000 000 0238
5.000 000 004 07
7.000 000 0208
9.000 000 756
11.000 002 37
13.000 007 75
15.000 0205
17.000 693
19.001 13
21.0246
23.03211.000 000 000 000 172
3.000 000 000 002 50
5.000 000 000 750
7.000 000 003 55
9.000 000 0573
11.000 000 180
13.000 002 83
15.000 005 38
17.000 0484
19.000 0817
21.000 428
23.000 669

1.000 000 000 000 157
3.000 000 000 002 08
5.000 000 000 432
7.000 000 001 79
9.000 000 0107
11.000 000 0257
13.000 000 161
15.000 000 314
17.000 001 52
19.000 002 38
21.000 0105
23.000 0139

Nonvariational
E 0 共g兲
E 1 共u兲E 2 共g兲
E 3 共u兲E 4 共g兲
E 5 共u兲E 6 共g兲
E 7 共u兲E 8 共g兲
E 9 共u兲E 10 共g兲
E 11 共u兲

0.999 999 999 9393
2.999 999 999 435
5.000 662
7.000 006 88
8.999 641
10.999 895
13.000 853
14.997 20
17.0332
19.41421.12423.2241.000 000 000 178
2.999 999 999 791
4.999 992 01
7.000 0487
8.999 996 29
10.999 9281
13.000 220
15.001 97
17.002 46
19.003 59
21.005 1022.9789

0.999 999 999 9751
2.999 999 999 775
5.000 000 0284
7.000 000 0587
9.000 004 36
11.000 005 64
12.999 669
14.999 777
17.004 98
19.003 18
20.9883
22.9861teresting to note that the variational parameters C K and C P
become closer as the iteration proceeds for both the gerade
and ungerade ground states: they would converge to thesame value and would converge to zero. This behavior showsthat the ICI2 converges to the same solution as the SICI
given in Table I. This is due to the linear nature of the ICI
method. Even though the variables C K and C P become
closer as the iteration proceeds, using two independent variables makes the convergence faster than just using one variable. Already at 5th iteration, the energy is correct by 9 zeros
and 8 zeros after the decimal point for the gerade and ungerade ground states, respectively. Nevertheless, though the energy essentially converges, it would take some more iterations before true convergence of the wave function, whereC K and C P become zero.
The excited states calculated by method B of Paper II,
ICI2-CI2 in the present notation, also show a good convergence to the exact solution again from above, though theconvergence is slower than that for the ground state. The
results for the gerade excited states given in Table II and theungerade excited state given in Table III are similar.
Table IV shows the result of method C, ICI2-CI generalin the present notation. We have extended the variable spaceof the ICI-CI method, following Eqs. 共4.1兲 and 共4.2兲, byincluding not only the kinetic 共k兲 and potential 共p兲 operators,
but also the higher products of the potential operators, pp,ppp, pppp, ppppp in the active space, giving independent
variable to each of the operators, following the spirit of the
EGCI method.33 Namely, the operator space used for defining the S operator in this calculation is written as
ICI2-CI2 共 method B兲 :1, k, p,共5.10兲ICI2 - CI general 共 method C兲 :1, k, p, p p, p p p, p p p p, p p p p p.

共5.11兲

The dimension of the secular equation is 7 in the ICI2-CI
general method in comparison with 3 of the original ICI2CI2 method. As the 共input兲 ICI2 solution for the groundstate, we used the results of the 2nd, 4th, and 6th iterations
given in Tables II and III. We have performed both variational and nonvariational calculations.
The results for the ground and excited states obtained by
the variational ICI2-CI general method are certainly much
improved, as expected, in comparison with those obtained by
the ICI2-CI2 method. The numbers of zeros for the lower
four states of Table IV are much larger than those of thecorresponding states shown in Tables II and III. Note that the
ground state is also much improved, since the results of the
2nd, 4th, and 6th iterations are not yet the true ICI solution.
Further, by the ICI2-CI general method, we could calculate
higher excited states not calculated by the ICI2-CI2 method.
This merit of the ICI-CI general method is very important forinvestigations of higher excited and ionized states in atomic
and molecular systems.
The nonvariational results are almost always worse thanthe variational results. Many results show overshooting of
the exact energy. The accuracy of the nonvariational result
depends rather critically on the accuracy of the ICI2 groundstate. As the number of iterations n of the ICI2 calculation
increases, the accuracy of the nonvariational calculations is
improved. The higher excited states calculated by the nonvariational method are rather miserable particularly for n⫽2.

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J. Chem. Phys., Vol. 116, No. 5, 1 February 2002

Structure of the exact wave function. IV

TABLE V. ECC2-CI2 and ECC2-CI general from the exact ground states of
the gerade and ungerade states of harmonic oscillator.

B. ECC2 method

We have already applied the SECC to the present harmonic oscillator in Paper III. The convergence of the truncated SECC was slow: even at n⫽30, we had only three andtwo zeros after the decimal point for the gerade and ungerade
ground states, respectively, in contrast to the SICI result
given in Table I. Though the truncated SECC certainly converges to the exact solution, the rate was much slower thanthe SICI given in this paper. The difference is that though the
SICI has one fresh variable at each iteration step, the SECC
has only one variable at any truncation level.
We apply here the ECC2 to the same harmonic oscillator. Some results were already given in Paper III. The S
operator is given by Eq. 共5.8兲 with two variables C K and C P .
It is easily seen that the present ECC2 has at least threedifferent solutions that give the exact wave function: one is
the solution which is the same as the SECC with C K and C P
both equal to C of SECC, the second is C K ⫽0.0 and C P⫽0.1, a self-evident solution, and the third is the nontrivial
solution that was already given in Paper III for the truncated
ECC2 in 2nd and 3rd orders. This fact means that the solution of the ECC method is generally not unique, though all
gives, of course, the same unique exact wave function. Since
the first solution is already given in Paper III as the result of
SECC, we discuss here the second self-evident solution andthe third nontrivial solution of ECC2.
First, we consider the self-evident solution of ECC2:C K ⫽0.0 and C P ⫽0.1. In this case, the ECC2 becomes␺⫽exp共 0.0k⫹0.1p 兲 exp共 ⫺0.6x 兲

ECC2

2

⫽exp共 0.1x 2 兲 exp共 ⫺0.6x 2 兲⫽exp共 ⫺0.5x 2 兲⫽ ␺ exact ,共5.12兲

Variational
E 0 共g兲
E 1 共u兲E 2 共g兲
E 3 共u兲E 4 共g兲
E 5 共u兲E 6 共g兲
E 7 共u兲E 8 共g兲
E 9 共u兲E 10 共g兲
E 11 共u兲Nonvariational
E 0 共g兲
E 1 共u兲E 2 共g兲
E 3 共u兲E 4 共g兲
E 5 共u兲E 6 共g兲
E 7 共u兲E 8 共g兲
E 9 共u兲E 10 共g兲
E 11 共u兲␺ ECC2⫽exp共 0.0k⫹0.1p 兲 x exp共 ⫺0.6x 2 兲⫽exp共 0.1x 2 兲 x exp共 ⫺0.6x 2 兲⫽x exp共 ⫺0.5x 2 兲
共5.13兲

for the ungerade ground state, so that they are exact. The
values of C K and C P are common to both gerade and ungerade states. Since the values of C K and C P are different, thissolution is different from the solution of SECC given in Paper III. Note that this self-evident exact solution is possible
only for ECC2: other SICI, ICI2, and SECC do not havesuch a straightforward solution.
We apply ECC2-CI2 using this exact solution: the active
space is given by Eq. 共5.10兲. The result is given in Table V.
The solutions for the excited states are also exact in bothvariational and nonvariational methods. We further applied
ECC2-CI general using the exact solution for the groundstate: the active space is given by Eq. 共5.11兲. The result issummarized in Table V. Again, we get the exact solution forthe lowest 12 states, not only by the variational method but
also by the nonvariational method.

ECC2-CI2

ECC2-CI general

1.0
3.0
5.0
7.0

1.0
3.0
5.0
7.0
9.0
11.0
13.0
15.0
17.0
19.0
21.0
23.0

1.0
3.0
5.0
7.0

1.0
3.0
5.0
7.0
9.0
11.0
13.0
15.0
17.0
19.0
21.0
23.0

Next, we discuss the nontrivial solution of ECC2, which
is more important than the self-evident solution, since suchself-evident solution would never occur in actual complex
electronic systems. Tables VI and VII show the nontrivial
solutions for the gerade and ungerade ground states, respectively, of the harmonic oscillator calculated by the ECC2
truncated at n⫽2, 3, 4, and 5, where n denotes the order of
the truncation of the exponential operator, namely,冉

for the gerade ground state, and⫽ ␺ exact,

1819

冊

1
1
1
␺ n ⫽ 1⫹S⫹ S 2 ⫹ S 3 ⫹•••⫹ S n ␺ 0 .
共5.14兲
2
3!
n!
The results of these tables were calculated by the variational
method given in Paper III. Note that the first-order (n⫽1)
truncated ECC is identical to the first iteration of the ICI
method and therefore omitted here. The values of the variables C K and C P were optimized only up to five decimal
figures.
First, we are rather surprised by a marvelous converging
behavior of the ECC2 method. Only with the two variables,
the n⫽5 results are accurate by ten zeros and nine zeros forthe gerade and ungerade ground states, respectively. In comparison with the truncated SECC result given in Paper III,the convergence of the truncated ECC2 is much faster. In
comparison with the ICI2 result, the ECC2 energy is betterby one order of magnitude than the ICI2 one, if we compare
the order n result of ECC2 to the nth iteration result of ICI2.
This is marvelous considering the fact that in the ECC2
method, the two variables are optimized only once, but in the
ICI2 method, the two variables are reoptimized at each iteration step. Within the present accuracy of C K and C P , thesmallness of G In and ⳵ E n / ⳵ C I implies that the solution isessentially at the variational minimum, and the smallness of

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1820

J. Chem. Phys., Vol. 116, No. 5, 1 February 2002

Hiroshi Nakatsuji

TABLE VI. Truncated ECC2 for the gerade ground state of harmonic oscillator.

Optimal

CK
CP

En

n⫽2

n⫽3

n⫽4

n⫽5

⫺0.036 41⫹0.056 50⫺0.036 62⫹0.055 80⫺0.036 62⫹0.055 79⫺0.036 62⫹0.055 78

1.000 002 10

1.000 000 0313

1.000 000 000 438

1.000 000 000 0271

G In

I⫽K
I⫽ P

⫺1.1605⫻10⫺5
1.3959⫻10⫺51.2517⫻10⫺5
1.0381⫻10⫺5⫺1.4802⫻10⫺51.2320⫻10⫺5

1.1979⫻10⫺50.9969⫻10⫺5

⌬ In

I⫽K
I⫽ P

⫺0.2290⫻10⫺5⫺0.1475⫻10⫺50.1253⫻10⫺50.2094⫻10⫺50.1500⫻10⫺5⫺0.0277⫻10⫺5⫺0.1213⫻10⫺5⫺0.1000⫻10⫺5

⳵En /⳵CI

I⫽K
I⫽ P

1.4179⫻10⫺50.4837⫻10⫺51.5306⫻10⫺5
1.2694⫻10⫺5⫺1.8099⫻10⫺5
1.5065⫻10⫺5

1.4648⫻10⫺5⫺1.2190⫻10⫺5

1.999 871

2.000 011

2.000 016

1.999 987

Virial coefficient⌬ In implies that both of the variational and H-nijou equations
are essentially satisfied. The virial relation is also wellsatisfied.
Another important fact is that the optimal values of C K
and C P are different, showing that the present ECC2 result isdifferent from the SECC results given in Tables I–III of Paper III. The value of C P is larger 共in absolute magnitude兲
than that of C K . This is in sharp contrast from the SICI and
ICI2 results given in Tables II and III, where the values of
C K and C P become close as the iteration proceeds and finallyconverge to the same solution as the SICI one. This behavior
of ECC2 is due to the nonlinear nature of the ECC method.
Furthermore, at n⫽4 and 5, the values of the variables of thegerade and ungerade ground states are the same, within thepresent accuracy of calculations. This means that the qualityof the initial function ␺ 0 is the same for both states.
Probably the most impressive and important result from
Tables VI and VII is that the convergence of the variables C K
and C P is quite rapid in the truncated ECC2 method. 共It may
be noted that the converged values of C K and C P of thepresent solution of ECC2 are not zero, in contrast to the ICI
case.兲 The values of these variables converge up to five decimal figures already at the truncation of order 5. This fact

implies that we can easily calculate the exact wave function
simply by using the converged coefficients in the ECC2
wave function as␺ exact⬵Nexp共 ⫺0.036 62k⫹0.055 78p 兲 ␺ 0 ,共5.15兲

which is valid for both the gerade and ungerade groundstates. Here, k and p represent kinetic and potential operators,
respectively, in the Hamiltonian given by Eq. 共5.2兲. The energy calculated from the right-hand side of Eq. 共5.15兲 isabove the exact energy only by 0.208⫻10⫺10 and 0.623⫻10⫺10 for the gerade and ungerade states, respectively, being essentially in the same order of accuracy as the n⫽5result given in Tables VI and VII, reflecting the accuracy ofthe variables C K and C P of the present optimization. This
reflects also the Eckert theorem. The correctness of Eq.
共5.15兲 for the wave function is more clearly checked by comparing numerically the both sides. ␺ exact is given by Eqs.
共5.3兲 and 共5.4兲 for the gerade and ungerade states, respectively, and ␺ 0 is given by Eqs. 共5.5兲 and 共5.6兲, respectively,
with ␣ ⫽0.60. Therefore, we can directly compare both sidesof Eq. 共5.15兲 numerically by expanding them into

TABLE VII. Truncated ECC2 for the ungerade ground state of harmonic oscillator.

Optimal

CK
CP

En

n⫽2

n⫽3

n⫽4

n⫽5

⫺0.036 39⫹0.056 91⫺0.036 62⫹0.055 81⫺0.036 62⫹0.055 79⫺0.036 62⫹0.055 78

3.000 0146

3.000 000 307

3.000 000 004 94

3.000 000 000 152

G In

I⫽K
I⫽ P

⫺2.2053⫻10⫺5⫺1.3409⫻10⫺51.9039⫻10⫺5⫺1.6156⫻10⫺5⫺0.3047⫻10⫺50.2544⫻10⫺5

1.6060⫻10⫺5⫺1.3365⫻10⫺5

⌬ In

I⫽K
I⫽ P

⫺1.2299⫻10⫺5⫺0.7864⫻10⫺5⫺0.2076⫻10⫺50.5616⫻10⫺50.0310⫻10⫺5⫺0.5860⫻10⫺5⫺0.1626⫻10⫺5⫺0.1519⫻10⫺5

⳵En /⳵CI

I⫽K
I⫽ P

6.3107⫻10⫺5⫺3.8371⫻10⫺55.4681⫻10⫺5⫺4.6400⫻10⫺5⫺0.8750⫻10⫺50.7308⫻10⫺54.6127⫻10⫺5
3.8386⫻10⫺5

1.999 683

2.000 048

2.000 002

1.999 986

Virial coefficient

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J. Chem. Phys., Vol. 116, No. 5, 1 February 2002

Structure of the exact wave function. IV

1821

TABLE VIII. Exact wave function estimated from the ECC2 wave function at n⫽5. ai

ECC2 at n⫽5

Estimated

Exact

Gerade ground
0
1
2
3
4
5
6
7
8
9

0.751 13⫻10⫺00.375 56⫻10⫺00.938 86⫻10⫺1
0.156 44⫻10⫺1
0.195 37⫻10⫺2
0.194 82⫻10⫺30.161 13⫻10⫺4
0.112 97⫻10⫺50.676 22⫻10⫺70.340 81⫻10⫺80.751 13⫻10⫺00.375 57⫻10⫺00.938 92⫻10⫺1
0.156 49⫻10⫺1
0.195 61⫻10⫺2
0.195 61⫻10⫺30.163 01⫻10⫺40.116 44⫻10⫺50.727 74⫻10⫺70.404 30⫻10⫺80.751 13⫻10⫺00.375 56⫻10⫺00.938 91⫻10⫺1
0.156 48⫻10⫺1
0.195 61⫻10⫺2
0.195 61⫻10⫺30.163 00⫻10⫺40.116 43⫻10⫺50.727 70⫻10⫺70.404 28⫻10⫺8

Ungerade ground
0
1
2
3
4
5
6
7
8
9

1.062 25⫻10⫺00.531 11⫻10⫺00.132 76⫻10⫺00.221 13⫻10⫺10.275 89⫻10⫺2
0.274 47⫻10⫺30.225 77⫻10⫺40.156 45⫻10⫺50.913 44⫻10⫺60.435 00⫻10⫺8

1.062 26⫻10⫺00.531 13⫻10⫺00.132 78⫻10⫺00.221 31⫻10⫺1
0.276 64⫻10⫺2
0.276 64⫻10⫺30.230 53⫻10⫺40.164 67⫻10⫺50.102 92⫻10⫺60.571 77⫻10⫺8

1.062 25⫻10⫺00.531 13⫻10⫺00.132 78⫻10⫺00.221 30⫻10⫺1
0.276 63⫻10⫺2
0.276 63⫻10⫺30.230 52⫻10⫺40.164 66⫻10⫺50.102 91⫻10⫺60.571 73⫻10⫺8a

The coefficients c i of ␺ ⫽ 兺 i (⫺) i c i x 2i for gerade ground state and of ␺ ⫽ 兺 i (⫺) i c i x 2i⫹1 for ungerade groundstate are given.

␺ ⫽ 兺 共 ⫺ 兲 i c i x 2ii

共5.16兲

for the gerade state, and␺ ⫽ 兺 共 ⫺ 兲 i c i x 2i⫹1i

共5.17兲

for the ungerade state. We compare in Table VIII the coefficients c i of both sides of Eq. 共5.15兲. The column below
ECC2 at n⫽5 shows c i for the truncated wave function, thecolumn below ‘‘estimated’’ shows c i for the right-hand sideof Eq. 共5.15兲 and the column below ‘‘exact’’ shows c i for theexact wave function. We again note that the present optimization is valid only up to five decimal figures for the variables. In Table VIII, c i for the truncated ECC2 becomesdepart from the exact one for higher i, but c i of the righthand side of Eq. 共5.15兲 is always close to the exact one to
five decimal figures: such closeness of c i actually continues
to higher i than shown here. Therefore, we can conclude thatthe estimation of the exact wave function by Eq. 共5.15兲 isvery accurate. This is a very encouraging result: we can calculate the exact wave function simply by using the explicit
ECC function, when the convergence of the variables is rapid
with respect to the truncation. We can do this since we know
the explicit structure of the exact wave function as ECCND
given by Eq. 共2.3a兲. This is in sharp contrast to the ICI case.
This is valuable especially when we calculate accurate properties of molecules. In comparison with the ICI result, therapid convergence of the ECC2 seems to reflect the fact that
the ECC gives a compact accurate expression of the structureof the exact wave function.
We have already shown in Paper III that this rapid converging behavior of the truncated ECC2 is in sharp contrast

to the slow convergence of the truncated SECC. We havewritten the reasons in some details in Paper III.
Using the solutions of the truncated ECC2 given in
Tables VI and VII for the ground states of gerade and ungerade symmetries, respectively, we have performed ECC2-CI2
calculations for the excited states by both of the variationaland nonvariational methods, and the results are given in
Table IX. Though the ECC-CI method is essentially themethod to calculate excited states, it also gives the groundstate as a lowest solution, since we include ␺ g in Eq. 共3.4兲 toensure the orthogonality between the ground and excitedstates. When the ground state is approximate, as in thepresent truncated approximation, the ground state is improved also at the 共variational兲 ECC-CI level. Comparing thegerade and ungerade ground states of Table IX obtained by
the variational method to those of Tables VI and VII, we seethat the ground state is improved by one to two orders ofmagnitude by the ECC-CI method. We could calculate by the
ECC2-CI2 method one excited state for each symmetry, andTable IX shows the four lower states of the harmonic oscillator. The quality of the calculated result becomes worse as
the energy of the excited state becomes higher, but all of thestates are much improved as the order of the truncation for
the ground state ECC2 increases. At n⫽5, even the second
ungerade state, the worst state in this calculation, is correctby eight zeros to the exact energy. Further, we can use theestimated exact ground state given in Table VIII and theresult is shown in the last column of Table IX. The quality of
the calculated result is much improved.
We have also carried out the nonvariational calculations
at the ECC2-CI2 level, and the result is given in Table IX. As
expected, the nonvariational result is almost always worse
than the corresponding variational result. Further, though the

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1822

J. Chem. Phys., Vol. 116, No. 5, 1 February 2002

Hiroshi Nakatsuji

TABLE IX. Truncated ECC2-CI2 for the gerade and ungerade states of harmonic oscillator.a
n⫽2

n⫽3

n⫽4

n⫽5

Estimated exact

Variational
E 0 共g兲
E 1 共u兲E 2 共g兲
E 3 共u兲1.000 000 0360
3.000 000 313
5.000 0664
7.000 197

1.000 000 003 11
3.000 000 0348
5.000 001 69
7.000 006 73

1.000 000 000 0343
3.000 000 000 448
5.000 000 0331
7.000 000 162

1.000 000 000 000 460
3.000 000 000 007 69
5.000 000 000 815
7.000 000 004 06

1.0
3.0
5.000 000 000 125
7.000 000 000 208

Nonvariational
E 0 共g兲
E 1 共u兲E 2 共g兲
E 3 共u兲1.000 002 05
3.000 000 797
5.001 01
7.002 391.000 006 38
3.000 004 47
5.000 0698
6.996 811.000 000 0305
2.999 999 147
5.000 0141
7.000 0358

1.000 000 0150
3.000 000 395
5.000 335
6.999 992 86

1.000 000 000 000 257
3.000 000 000 001 29
5.000 003 51
7.000 005 86a

1.0 and 3.0 means that the number has more than 15 zeros after the decimal point.

variational energy is always above the true exact energy, thenonvariational one sometimes overshoots the exact energy:
this is seen for E 3 of n⫽3, E 1 of n⫽4, and E 3 of n⫽5.
When we use the estimated exact ground state given in Table
VIII, the nonvariational solution is also much improved.
Table X shows the result obtained by the ECC2-CI general calculations, extending the variable operator space asgiven by Eq. 共5.11兲. By this calculation, we can not only
improve the accuracies of all the states, but also calculate alarger number of excited states. Table X shows the twelve
lower states of the harmonic oscillator in comparison withfour of the ECC2-CI2 method shown in Table IX. Furthermore, in comparison with the ECC2-CI2 results given in
Table IX, the results of ECC2-CI general are very much
improved in quality. The number of zeros below the decimal

point increases much in this table. The accuracy of the results is improved very much as the order of the truncation nfor the ground state ECC2 increases. This is especially sowhen we use the estimated exact ground state given in Table
VIII. The solution obtained at this level is essentially exact,
like those shown in Table V. Thus, the ECC2-CI general
method is very promising for calculating accurate groundand excited states.
Table X also gives the nonvariational results of ECC2-CI
general. The accuracy of the nonvariational method is rather
capricious, and is almost always worse than that of the variational method. But, even so, the accuracy becomes steadily
better as the order n increases. In contrast to the variationalcase, we see a lot of overshooting in the nonvariational case.
However, the nonvariational solution obtained from the esti-

TABLE X. Truncated ECC2-CI general for the gerade and ungerade states of harmonic oscillator.a

a

n⫽2

n⫽3

n⫽4

n⫽5

Estimated exact

Variational
E 0 共g兲
E 1 共u兲E 2 共g兲
E 3 共u兲E 4 共g兲
E 5 共u兲E 6 共g兲
E 7 共u兲E 8 共g兲
E 9 共u兲E 10 共g兲
E 11 共u兲

1.000 000 000 001 69
3.000 000 000 0267
5.000 000 004 25
7.000 000 0226
9.000 000 781
11.000 002 52
13.000 008 13
15.000 0224
17.000 686
19.001 11
21.024423.03171.000 000 000 000 221
3.000 000 000 002 94
5.000 000 000 954
7.000 000 004 31
9.000 000 153
11.000 000 444
13.000 002 96
15.000 007 19
17.000 420
19.000 666
21.005 64
23.007 82

1.000 000 000 000 0673
3.000 000 000 000 951
5.000 000 000 418
7.000 000 001 91
9.000 000 0167
11.000 000 0507
13.000 003 16
15.000 006 23
17.000 091 38
19.000 154
21.000 84
23.001 26

1.000 000 000 000 0107
3.000 000 000 000 153
5.000 000 000 0764
7.000 000 000 316
9.000 000 0116
11.000 000 0314
13.000 000 567
15.000 001 25
17.000 0103
19.000 0193
21.000 0833
23.000 138

1.0
3.0
5.0
7.0
9.0
11.0
13.0
15.0
17.0
19.0
21.000 000 001 37
23.000 000 001 62

Nonvariational
E 0 共g兲
E 1 共u兲E 2 共g兲
E 3 共u兲E 4 共g兲
E 5 共u兲E 6 共g兲
E 7 共u兲E 8 共g兲
E 9 共u兲E 10 共g兲
E 11 共u兲

0.999 999 999 9605
2.999 999 999 509
5.000 000 180
6.999 9659
8.999 995 36
11.000 475
13.000 393
15.000 708
17.0269
19.0106
21.15423.4410.999 999 999 9438
3.000 000 003 37
4.999 999 9391
7.000 002 09
9.000 000 219
11.000 0487
13.000 199
15.000 411
17.012119.004 42
21.0280
23.1430.999 999 999 9710
3.000 000 000 125
5.000 001 01
6.999 9895
8.999 993 43
11.000 0297
13.000 192
15.000 413
17.001 89
18.996 81
21.0155
23.0698

1.000 000 000 000 262
2.999 999 999 9485
4.999 999 99549
6.999 999 9534
9.000 000 403
11.000 002 78
13.000 0231
14.999 999 9719
17.000 517
19.001 48
21.002 78
22.999 101

1.0
3.0
5.0
7.0
9.0
11.0
13.0
15.0
17.000 000 000 127
19.000 000 000 184
21.000 0387
23.000 0457

1.0, 3.0, etc., means that the number has more than 15 zeros after the decimal point.

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J. Chem. Phys., Vol. 116, No. 5, 1 February 2002

Structure of the exact wave function. IV

1823

mated exact ground state is again almost exact.
It is true that the one-dimensional harmonic oscillator isprobably the simplest possible system, so that we cannot
generalize the present result to more complex electronic systems. Nevertheless, it is thought that the present results
would already show some of the general aspects of the ICI
and ECC methods and their solutions. We may safely saythat the ICI and ECC methods are worth studying in moredetails as methods of giving exact solution of the Schrödinger equation.

Though one-dimensional harmonic oscillator is the simplest possible system, some of the present results should bein common to those of the more general complex electronicsystems.
The results of the present series of papers may lead to a
new wave in the studies of exact and very accurate wave
functions of the ground and excited states of molecular systems, making precise predictions possible in the theoreticalstudies of chemistry and physics.

VI. CONCLUSIONACKNOWLEDGMENTS

In this paper, we have first presented the method of describing the excited states, ionized states, and the electronattached states from the ECCND method. Since the ECCND
method is very promising as a method of calculating the
ground state, it is important to prepare the method of calculating the excited states of the same system. The proposed
method is called the ECC-CI method, because the theoreticalframework is very similar to that of the SAC-CI method
proposed in 1978 by the author, which was also formulatedby the variational principle. Standard ECC-CI method that is
ECCND-CIND is approximate, but by expanding the CI
space, we obtain the ECC-CI general method that can evenbecome exact. We are able not only to improve the quality of
the calculated states, but also to expand the nature of theexcited states up to higher multiple-excitation processes.
Second, we have applied the ICI and ECC methods tothe calculations of the ground and excited states of the harmonic oscillator. The ICI method gave good convergence:
the ICI2 gave a faster convergence than the SICI, and converged to the same solution as the SICI. The convergence of
the wave function takes more iterations than the convergenceof energy. The excited states were successfully calculated by
the standard ICI-CI method. The ICI-CI general method
gives better quality and a larger number of excited states than
the standard ICIND-CIND method.
The ECC2 method gave at least three different solutionsof the unique exact ground state of the harmonic oscillator:
the solution that is the same as the SECC solution, the selfevident and nontrivial solutions. The nontrivial solution,
which is most important as a general solution of ECC2,shows quite a rapid convergence to the exact wave function.
This is in sharp contrast to the slow convergence of the
SECC shown in Paper III. Using this fact, we could estimate
the exact wave function simply by using the explicit ECC
form. Further, in contrast to the ICI case, the self-evident andnontrivial solutions of ECC2 are different from that of the
SECC. Because of the flexibility of the ECC2 due to its
nonlinear nature, the truncated ECC2 gives much better result than the SECC. This is a very valuable feature of the
ECC method. The ECC-CI method gave also a very promising result. Similarly to the ICI-CI results, the ECC-CI
method gave very good results for both the ground and excited states. In particular, very good results for the groundand excited states were obtained by the ECC-CI general
method. When the ground state is exact, the ECC-CI methods, both CI2 and CI general, gave the exact excited statesby both variational and non-variational methods.

The author thanks Dr. M. Ehara and Dr. M. Hada forvaluable conversations. This study has been supported financially by the special fund from the Ministry of Education,
Science, Culture, and Sports.
P. J. Knowles and N. C. Handy, Chem. Phys. Lett. 111, 315 共1984兲.
G. L. Bendazzoli and S. Evangelisti, J. Chem. Phys. 98, 3141
共1993兲.
3
H. Nakatsuji, J. Chem. Phys. 113, 2949 共2000兲.
4H. Nakatsuji and E. R. Davidson, J. Chem. Phys. 115, 2000
共2001兲.
5
H. Nakatsuji, J. Chem. Phys. 115, 2465 共2001兲.
6
H. Huang, Q. Xie, Z. Cao, Z. Li, Z. Yue, and L. Ming, J. Chem. Phys. 110,
3703 共1999兲; H. Huang, X. Zeng, and L. Ming, ibid. 112, 5257 共2000兲; M.
G. Marmorino, ibid. 114, 2513 共2001兲.
7
C. Lanczos, J. Res. Natl. Bur. Stand. 45, 255 共1950兲.
8
E. R. Davidson, Comput. Phys. Commun. 53, 1 共1989兲; W. A. Berger, H.
G. Miller, and R. M. Dreizler, J. Phys. A 10, 1089 共1977兲; J. H. van Lenthe
and P. Pulay, J. Comput. Chem. 11, 1164 共1990兲.
9
H. Nakatsuji, Chem. Phys. Lett. 59, 362 共1978兲; ibid. 67, 329,
共1979兲.
10
H. Nakatsuji, in Computational Chemistry, Reviews of Current Trends,
Vol. 2 edited by J. Leszczynski 共World Scientific, Singapore, 1997兲, p.
62–124 and the references cited therein.
11
F. Coster, Nucl. Phys. 7, 421 共1958兲; F. Coster and H. Kümmel, ibid. 17,477 共1960兲.
12
O. Sinanoglu, J. Chem. Phys. 36, 706 共1962兲; Adv. Chem. Phys. 6, 315
共1964兲.
13
J. Cı́zek, J. Chem. Phys. 45, 4256 共1966兲; Adv. Chem. Phys. 14, 35
共1969兲.
14
The naming ‘‘Extended Coupled Cluster 共ECC兲’’ happened to be identicalto that already used by Arponen 关Ann. Phys. 共N.Y.兲 151, 311 共1983兲兴, but
the present ECC is different from that of Arponen.
15
M. Nooijen, Phys. Rev. Lett. 84, 2108 共2000兲.
16
T. Van Voorhis and M. Head-Gordon, J. Chem. Phys., 115, 5033 共2001兲.
The author thanks the authors of this paper for giving him this preprint
prior to publication. See also, T. Van Voorhis and M. Head-Gordon, J.
Chem. Phys. 113, 8873 共2000兲.
17
R. Kosloff and H. Tal-Ezer, Chem. Phys. Lett. 127, 223 共1986兲; see also T.
J. Park and J. C. Light, J. Chem. Phys. 85, 5870 共1986兲.
18
D. Horn and M. Weinsten, Phys. Rev. D 30, 1256 共1984兲.
19
J. Cioslowski, Phys. Rev. Lett. 58, 83 共1987兲.
20
H. Nakatsuji and K. Hirao, J. Chem. Phys. 68, 2053 共1978兲.
21
D. Mukherjee and P. K. Mukherjee, Chem. Phys. 39, 325 共1979兲.
22
E. Dalgaard and H. Monkhorst, Phys. Rev. A 28, 1217 共1983兲; See also, H.
Monkhorst, Int. J. Quantum Chem., Quantum Chem. Symp. 11, 421
共1977兲.
23
H. Koch and P. Jorgensen, J. Chem. Phys. 93, 3333 共1990兲; H. Koch, H. J.
Aa. Jensen, T. Helgaker, and P. Jorgensen, J. Chem. Phys. 93, 3345
共1990兲.
24
J. Geertsen, M. Rittby, and R. J. Bartlett, Chem. Phys. Lett. 164, 57
共1989兲.
25
J. F. Stanton and R. J. Bartlett, J. Chem. Phys. 98, 7029 共1993兲; D. J. Wadt
and R. J. Bartlett, ibid. 101, 3073 共1994兲.
26
J. F. Stanton and J. Gauss, J. Chem. Phys. 101, 8938 共1994兲; J. F. Stantonand R. J. Bartlett, ibid. 102, 3629 共1995兲.
27
M. Ehara, M. Ishida, K. Toyota, and H. Nakatsuji, Reviews in Modern
1
2

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1824

J. Chem. Phys., Vol. 116, No. 5, 1 February 2002

Quantum Chemistry, A Tribute to Professor Robert G. Parr, edited by K.
D. Sen 共World Scientific, Singapore, in press兲.
28
See Tables I and II of Paper I.
29
H. Nakatsuji, Chem. Phys. Lett. 177, 331 共1991兲.
30
M. Ehara, P. Tomasello, J. Hasegawa, and H. Nakatsuji, Theor. Chem.
Acc. 102, 161 共1999兲.

Hiroshi Nakatsuji
M. Ehara, M. Ishida, and H. Nakatsuji, J. Chem. Phys. 114, 8990 共2001兲;and the references cited therein.
32
J. H. Wilkinson, The Algebraic Eigenvalue Problem 共Oxford University
Press, Oxford, 1965兲.
33
H. Nakatsuji, J. Chem. Phys. 83, 5743 共1985兲; ibid. 94, 6716
共1991兲.
31

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