JOURNAL OF CHEMICAL PHYSICSVOLUME 115, NUMBER 6

8 AUGUST 2001

Structure of the exact wave function. III. Exponential ansatz
Hiroshi Nakatsujia)
Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering,
Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan

共Received 22 January 2001; accepted 22 May 2001兲
We continue to study exponential ansatz as a candidate of the structure of the exact wave function.
We divide the Hamiltonian into N D 共number of divisions兲 parts and extend the concept of thecoupled cluster 共CC兲 theory such that the cluster operator is made of the divided Hamiltonian. Thisis called extended coupled cluster 共ECC兲 including N D variables 共ECCND兲. It is shown that the
S共simplest兲ECC, including only one variable (N D ⫽1), is exact in the sense that it gives an explicit
solution of the Schrödinger equation when its single variable is optimized by the variational or
H-nijou method. This fact further implies that the ECCND wave function with N D ⭓2 should also
have a freedom of the exact wave function. Therefore, by applying either the variational equation or
the H-nijou equation, ECCND would give the exact wave function. Though these two methods givedifferent expressions, the difference between them should vanish for the exact wave function. This
fact solves the noncommuting problem raised in Paper I 关H. Nakatsuji, J. Chem. Phys. 113, 2949
共2000兲兴. Further, ECCND may give more rapidly converging solution than SECC because of its
non-linear character, ECCND may give the exact wave function at the sets of variables different
from SECC. Thus, ECCND is exact not only for N D ⫽1, but also for N D ⭓2. The operator of the
ECC, exp(S), is an explicit expression of the wave operator that transforms a reference function into
the exact wave function. The coupled cluster including general singles and doubles 共CCGSD兲
proposed in Paper I is an important special case of the ECCND. We have summarized the methodof solution for the SECC and ECCND truncated at order n. The performance of SECC and ECC2
is examined for a simple example of harmonic oscillator and the convergence to the exact wave
function is confirmed for both cases. Quite a rapid convergence of ECC2 encourages an applicationof the ECCND to more general realistic cases. © 2001 American Institute of Physics.
关DOI: 10.1063/1.1385371兴

I. INTRODUCTION

in a coordinate representation, orA purpose of this series of studies1,2 is to clarify the
structure of the exact wave function. Since the Hamiltonianin the Schrödinger equation is simple, involving the operators only up to two-particle interaction terms, the exact wave
function, its eigenfunction, should also have some simple
structure, reflecting this simplicity of the Hamiltonian operator. We have already shown in the previous two papers1,2 thatcertainly the exact wave function can be described with asmaller number of variables than ever thought. After clarifying the structure of the exact wave function, we study theefficient method to calculate it. The Schrödinger equation
governs essentially most of physics and all of chemistry,3 sothat if we can solve this equation with a realistic cost, we canmake very precise predictions and its scientific and practical
merits are huge.
The Schrödinger equation is given by
共 H⫺E 兲 ␺ ⫽0,

H⫽

H⫽

1
M full-Cl⫽
m⫹1

M GSD⫽m 2 ⫹

N

m⫹1

1
N
2

1
,
N⫹1
2

共1.3兲

冋

册

2
m
共 m⫺1 兲 .
2

共1.4兲This is the number of the general singles and doubles 共GSD兲
substitution operators and is much smaller than M full-CI . We
have proposed the iterative configuration interaction 共ICI兲
method involving GSD number of variables 共ICIGSD兲 as a

共1.2a兲

a兲Electronic mail: hiroshi@sbchem.kyoto-u.ac.jp

0021-9606/2001/115(6)/2465/11/$18.00

冉 冊冉 冊m⫹1

easily runs into astronomical figures even for small molecules. In Eq. 共1.3兲 m is the number of active orbitals, N thenumber of electrons, and 共 兲 denotes binomial coefficient.
In the first paper of this series,1 hereafter called Paper I,
we have shown that it is possible to solve the Schrödinger
equation with M GSD variables

共1.1兲

w 共 i, j 兲 ,兺i v共 i 兲 ⫹ 兺i⬎ j

共1.2b兲

in a second-quantized representation. The most straightforward method of solving the Schrödinger equation within agiven basis set is the full CI method, but the number of thevariables, M full-CI involved in this method

where the Hamiltonian involves only one- and two-particle
operators asN⫹ ⫹v rp a r⫹ a p ⫹ 兺 w rs
兺
pq a r a s a q a p ,
pr
pqrs

2465

© 2001 American Institute of Physics

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2466

J. Chem. Phys., Vol. 115, No. 6, 8 August 2001

Hiroshi Nakatsuji

method to calculate the exact wave function. In the second
paper of this series,2 called Paper II, we have shown that aneven smaller number of variables is possible to solve the
Schrödinger equation. The number of variables depends on
how we divide the Hamiltonian operator into pieces. This
number of division, N D is, for example, from unity to M GSD :unity is for no division and M GSD for dividing H into all theterms in Eq. 共1.2b兲. This generalized ICI method was called
the ICIND method. Further, we have described how to calculate the excited states from the ICIND method.
In Paper I, we have obtained the theorem that suggests
the possibility that the exponential anzatz may represent thestructure of the exact wave function when it is combinedwith the variational principle. The coupled cluster singles
共CCS兲 is certainly exact when the Hamiltonian involves onlya single particle operator. However, the conventional CCSD
was shown to be not exact because it is not variational andbecause the operators involved are only excitation-operator
type ( At type兲 and do not include other substitution operators( Xt type兲.1 We then proposed the CCGSD method that was
the CC including general singles and doubles 共GSD兲 substitution operators, but the problem was that it included thenoncommuting operators. Nooijen8 recently considered a
combination of CCGSD with the density equation.5
In this paper, we will extend the exponential ansatz inthe light of the general theorem given in Paper II and present
extended coupled cluster 共ECC兲 theory. We will show that
the ECC in its simplest form 共SECC: simplest extended
coupled cluster兲 certainly represents the structure of the exact
wave function as expected originally in Paper I. This fact
further sheds light on the structure of the ECC theory ingeneral. We analyze the structure of the ECC wave function.
A method of solving the ECC is described and some simple
application is given.
The wave operator W is an operator that transforms anappropriate approximate wave function ␺ 0 into the exact
wave function ␺ that is a solution of the Schrödinger equation␺ ⫽W ␺ 0 .

共1.5兲

When ␺ 0 is Hartree–Fock, W includes all the correlation
effects. We will give in this paper a simple explicit expression of the wave operator based on the exponential ansatz.
II. BACKGROUND

We first summarize several equations that are equivalent
to the Schrödinger equation in the necessary and sufficient
sense. They provide valuable information for the study of thestructure of the exact wave function, because they have thesame determinative power as the Schrödinger equation when
they are solved appropriately.
The most well-known and useful one is the variational
principle

具 ␺ 兩 H⫺E 兩 ␦ ␺ 典 ⫽0,

共2.1兲

where the energy of the system, E, is defined by

具 ␺ 兩 H⫺E 兩 ␺ 典 ⫽0.
Second, the equation

共2.2兲

具 ␺ 兩 共 H⫺E 兲 2 兩 ␺ 典 ⫽0

共2.3兲is equivalent to the Schrödinger equation, and each of thefollowing two equations,具 ␺ 兩 共 H⫺E 兲 H 兩 ␺ 典 ⫽0,

共2.4兲

具 ␺ 兩 H 2 ⫺E 2 兩 ␺ 典 ⫽0,

共2.5兲

and
together with Eq. 共2.2兲 is also equivalent to the Schrödinger
equation.5
Based on these equations, we have given the following
theorem in Paper II. Namely, when we divide the Hamiltonian into the N D parts as
ND

H⫽兺 HI ,

I⫽1

共2.6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2.7兲

and Eq. 共2.2兲 is exact in a 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2.8兲

is also exact in the necessary and sufficient sense. In Eq.
共2.8兲, the partial energy E I corresponding to H I is defined by

具 ␺ 兩 共 H I ⫺E I 兲 兩 ␺ 典 ⫽0,

共2.9兲

and satisfies
E⫽兺I E I .

共2.10兲

It is convenient to call this group of equations H-square or
H-nijou equations. 共‘‘Nijou’’ is Japanese meaning square.兲
Based on the above theorem, we have introduced in Paper II the iterative CI method including N D variables
共ICIND兲 as a method that gives the exact wave function.
Actually, we can obtain the exact wave function, corresponding to any division of the Hamiltonian. For example, we canobtain the exact wave function using only one variable. We
can define the exact wave function including only the singles
number 共number of singles substitution operators兲 of variables: this is an interesting special case that may give a basisof the density functional theory 共DFT兲.6 When we divide the
Hamiltonian into the GSD parts, as in the expression of Eq.
共1.2b兲, we have the ICIGSD method proposed in Paper I.
Since each iteration process of the ICI method is variational,
the solution converges from above to the exact wave function. We have explained further in Paper II how to calculate
the excited states within the framework of the ICI method.

III. EXTENDED COUPLED CLUSTERIn Paper I, we have examined the exponential ansatz as apromising candidate of the structure of the exact wave function. This expectation was based on Theorem II-2 of Paper I
that suggested that the exponential ansatz might represent thestructure of the exact wave function. We study this problem
here in the light of the theorem given in Paper II.

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J. Chem. Phys., Vol. 115, No. 6, 8 August 2001

Structure of exact wave function. III.

We describe here an extension of the coupled cluster
共CC兲 theory. Corresponding to the division of the Hamiltonian

兺 HI ,

共3.1兲

I⫽1

ND

兺 C IH I ,

共3.2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!

N⫽Tr兵 n rp 其 ,共3.3b兲

兺 c I共 H I ⫺E I 兲 ,

共3.4兲

I⫽1

where E I is given by Eq. 共2.9兲 and satisfies Eq. 共2.10兲. The
wave function is commonly written by Eq. 共3.3兲.
Note that the coupled cluster general singles and doubles
共CCGSD兲 defined by1

␺ ⫽exp共 T 兲 ␺ 0 ,␺ ⫽exp共 T ⬘ 兲 ␺ 0 ,T ⬘⫽⫽共3.12兲rs⫹ ⫹D rp 共 1⫺n rp 兲 a r⫹ a p ⫹ 兺 D rs
兺
pq 共 1⫺n pq 兲 a r a s a q a p
p,r
pqrs

共3.13a兲
⫽

rs
rs
⫹ ⫹
d rp 共 v rp ⫺E rp 兲 a r⫹ a p ⫹ 兺 d rs
兺
pq 共 w pq ⫺E rq 兲 a r a s a q a p ,prpqrs

共3.13b兲r
rs
where D rp ,D rs
pq and d p , d pq represent the variables.

We study below the ECCND, classifying it into the twocases, namely the simplest ECC 共SECC兲 with N D ⫽1 and thegeneral ECCND with N D ⭓2.

共3.5兲

We first consider the simplest extended coupled cluster
共SECC兲, where N D ⫽1 in Eq. 共3.1兲; namely, we do not divide
the Hamiltonian operator. Then, SECC may also be referred
to as ECC1 as an extreme of the ECCND. The S operator of
the type of Eq. 共3.2兲 is written as S⫽CH, and the SECC
wave function is written as␺ ⫽exp共 CH 兲 ␺ 0

冉

with

兺
pr

共3.11兲

A. SECCND

T⫽

N 共 N⫺1 兲 ⫽Tr兵 n rs
pq 其 ,

共3.3a兲

where ␺ 0 is some reference function that is usually Hartree–
Fock, 兩0典. This is a small extension of the coupled cluster
method, so we call it extended coupled cluster 共ECC兲
method. Following the conventional notation, we call the
ECC given by Eq. 共3.3兲 ECCND 共ECC with N D variables兲
method. Another variation of ECCND is introduced by defining the variable operator S byS⫽

共3.10兲

where N is the number of electrons. Using these quantities,
we can define the CCGSD in the form of

we define the variable operator
S⫽

E rp ⫹ 兺 E rs
兺
pq ,
p,r
pq,rs

and the density matrices to

ND

H⫽

E⫽

2467

C rp a r⫹ a p ⫹

兺

pqrs

共3.6兲

is a special case of the ECCND: we use the Hamiltonian
共1.2b兲 in defining the S operator 共here, T operator兲 having
GSD number of variables.
The CCGSD in the form of Eq. 共3.4兲 is written as follows. We define the partial energies E rp and E rs
pq by

具 ␺ 兩 v rp a r⫹ a p ⫺E rp 兩 ␺ 典 ⫽0,⫹ ⫹rs
具 ␺ 兩 w rs
pq a r a s a q a p ⫺E pq 兩 ␺ 典 ⫽0,

共3.7兲

and the density matrices n rp and n rs
pq by

具 ␺ 兩 a r⫹ a p ⫺n rp 兩 ␺ 典 ⫽0, 具 ␺ 兩 a r⫹ a s⫹ a q a p ⫺n rs
pq 兩 ␺ 典 ⫽0,

共3.8兲

where
E rp ⫽ v rp n rp ,

rs rs
E rs
pq ⫽w pq n pq .

The partial energies sum up to E as

冊

1
1
⫽ 1⫹CH⫹ C 2 H 2 ⫹ C 3 H 3 ⫹¯ ␺ 0 .
2
3!

⫹ ⫹
C rs
pq a r a s a q a p

rs ⫹ ⫹c rp v rp a r⫹ a p ⫹ 兺 c rs
兺
pq w pq a r a s a q a p ,prpqrs

共3.14a兲

共3.9兲

共3.14b兲

We calculate the energy E by 具 ␺ 兩 H⫺E 兩 ␺ 典 ⫽0 and the variable C by the variational principle. Differentiating ␺ withrespect to C, we obtain⳵␺⫽H ␺ .
⳵C

共3.15兲

Inserting this relation into the variational equation, we have具 ␺ 兩 共 H⫺E 兲 H 兩 ␺ 典 ⫽0,

共3.16兲

which is Eq. 共2.4兲. Therefore, the SECC given by Eq. 共3.14兲
is exact when it is solved variationally.
Using the S operator of the type of Eq. 共3.4兲 as S⫽c(H⫺E), the SECC can be written in a different form␺ ⫽exp关 c 共 H⫺E 兲兴 ␺ 0

冋

共3.17a兲册11⫽ 1⫹c 共 H⫺E 兲 ⫹ c 2 共 H⫺E 兲 2 ⫹ c 3 共 H⫺E 兲 3 ⫹¯ ␺ 0 ,2
3!
共3.17b兲which satisfies

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2468

J. Chem. Phys., Vol. 115, No. 6, 8 August 2001

⳵␺⫽ 共 H⫺E 兲 ␺ .
⳵cHiroshi Nakatsuji

共3.18兲

Therefore, from the variational principle, we have具 ␺ 兩 共 H⫺E 兲 2 兩 ␺ 典 ⫽0,

共3.19兲

which is Eq. 共2.3兲. Therefore, the variational SECC in theform of Eq. 共3.17兲 is also exact.
Thus, we have the theorem.
Theorem III: Variational SECC is an explicit expressionof the solution of the Schrödinger equation.
Theorem III is rather surprising. It means that the variational SECC is exact, though it has only one variable. It alsomeans that the operator of the SECC, exp(CH) or exp关c(H⫺E)兴, is an explicit expression of the wave operator definedby Eq. 共1.5兲. It transforms an approximate wave function ␺ 0
into the exact one ␺. Since the operator, exp(CH) or
exp关c(H⫺E)兴, is totally symmetric, the wave function of
SECC reflects the symmetry of the reference function ␺ 0 .
This implies that the SECC is applicable to the ground stateof each spin–space symmetry.
The convergence of SECC, when it is written in an expansion form as in Eq. 共3.14b兲 or 共3.17b兲, is clear, since a
merit of the exponential ansatz is that it always converges,
though this does not necessarily mean that the convergenceis fast. The convergence rate is dependent on the quality of
the reference function ␺ 0 . It is very much desirable to calculate the SECC wave function, exp(CH)␺0 or exp关c(H⫺E)兴␺0 , directly as it is in Eq. 共3.14a兲 or 共3.17a兲 rather than
expanding it as in Eq. 共3.14b兲 or 共3.17b兲.
There is no iteration in the structure of the SECC, different from the ICI method. In comparison with the simplest
ICI 共SICI兲 given by Eq. 共3.18兲 or 共3.19兲 of Paper II, the
SECC has uniquely only one variable, while in the SICI
method, the single variable is reoptimized in each iteration
step.
The SECC given by Eq. 共3.17兲 is somewhat more transparent than that of Eq. 共3.14兲. When ␺ 0 is an eigenfunctionof H, then exp关c(H⫺E)兴⫽1, and therefore, ␺ ⫽ ␺ 0 as itshould be. When ␺ 0 is close to ␺, (H⫺E) should be small
and therefore the convergence should be fast. Actually, however, the operators, exp(CH) and exp关c(H⫺E)兴, are essentially the same: the difference lies only in the norm of the
wave function.
Solution of the Schrödinger equation in the form of
SECC has long been considered in the field of quantum molecular dynamics dealing with the time-dependent Schrödinger equation.7,8 Kozloff and Tal-Ezer,8 for example, considered the solution of the time-dependent Schrödinger
equation on the imaginary time axis. Then, the solution of
the time-dependent Schrödinger equation becomes␺ 共 ␶ 兲 ⫽exp共 ⫺H ␶ 兲 ␺ 共 0 兲 ,共3.20兲

where ␶ ⫽it with t being 共real兲 time. In comparison with Eq.
共3.14a兲, the correspondence is C⫽⫺ ␶ , ␺ ⫽ ␺ ( ␶ ), and ␺ 0⫽ ␺ (0). This ␺共␶兲 approaches the exact wave function of theground state of H after sufficiently long time interval startingfrom an initial function ␺共0兲. Expanding the initial wave
function ␺共0兲 in terms of the eigenfunctions 兵 ⌿ i 其 of H 共theordering of the states is E 0 ⬍E 1 ⬍E 2 ⬍...兲, we obtain8

␺ 共 ␶ 兲 ⫽exp共 ⫺H ␶ 兲 兺 d i ⌿ i 共 0 兲 ⫽ 兺 d i exp共 ⫺E i ␶ 兲 ⌿ i 共 0 兲 .
ii

共3.21兲
After a time ␶, the eigenfunction ⌿ i is reduced relative to ⌿ 0
by the ratio, exp关⫺(Ei⫺E0)␶兴, so that after the time ␶ larger
than m ln 10/(E 1 ⫺E 0 ), ␺共␶兲 approaches ⌿ 0 in m digits ofaccuracy. At ␶ infinity, ␺共␶兲 becomes equal to ⌿ 0 . As 兵 ⌿ i 其
are all linearly independent, this is a unique way of getting
the exact solution from Eq. 共3.21兲.
This argument means that the variable C or c in the
SECC is a large negative 共infinite兲 value to represent theexact wave function. Of course, the wave function itself is
finite when it is divided by the normalization factor. The
actual optimal value is dependent on the quality of the reference wave function ␺ 0 . We will see in the applications of
SECC that the actual optimal value of the variable is not solarge negative, as far as we solve it approximately.
The most important conclusion of this section is that thesimplest ECC, SECC, has a structure of the exact wave function and can certainly become exact with its structure of the
exponential ansatz.

B. ECCND

We next consider the general case of the ECCND whereN D ⫽1. Now, we know that even the simplest form of the
exponential ansatz, SECC, has the structure of the exact
wave function, so that we can assume that a more generalexponential ansatz, ECCND, does have freedom to represent
the structure of the exact wave function. Therefore, when we
calculate the variables 兵 C I 其 in Eq. 共3.2兲 either by the variational method or by the H-nijou method, both give the exact
wave function, because both methods are equivalent to the
Schrödinger equation in the necessary and sufficient sense.
We first apply the variational principle to the ECCND
wave function given by Eq. 共3.3兲. Since the operators 兵 H I 其 in
Eq. 共3.2兲 do not generally commute with each other关 H I ,H J 兴 ⫽0

共 I⫽J 兲 ,共3.22兲

the variation of ␺ with respect to C I is written as

冋

⳵␺
1
1
⫽ H I ⫹ 共 H I S⫹SH I 兲 ⫹ 共 H I S 2 ⫹SH I S⫹S 2 H I 兲
⳵CI
2
3!

册

⫹¯ ␺ 0 ,共3.23兲

where the S operator is given by Eq. 共3.2兲, and we have used
the relation

⳵S⫽H I .
⳵CI

共3.24兲

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J. Chem. Phys., Vol. 115, No. 6, 8 August 2001

Structure of exact wave function. III.

Equation 共3.23兲 means that ⳵ ␺ / ⳵ C I ⫽H I ␺ in general. Combining Eq. 共3.23兲 with the variational principle given by Eq.
共2.1兲, we obtain the variational working equation for the ECCND method as冋

1
G I 共 兵 C I 其 兲 ⫽ 具 ␺ 兩 共 H⫺E 兲 H I ⫹ 共 H I S⫹SH I 兲
2
⫹

册

1
共 H S 2 ⫹SH I S⫹S 2 H I 兲 ⫹¯ 兩 ␺ 0 典 ⫽0,
3! I
共3.25兲

for all I. Note that this equation is highly nonlinear for thevariable 兵 C I 其 . Multiplying C I in Eq. 共3.25兲 and summing up
for I, we obtain

具 ␺ 兩 共 H⫺E 兲 S 兩 ␺ 典 ⫽0.

共3.26兲

A simple sum of Eq. 共3.25兲 for all I gives冋1
2

具 ␺ 兩 共 H⫺E 兲 H⫹ 共 HS⫹SH 兲⫹

册1
共 HS 2 ⫹SHS⫹S 2 H 兲 ⫹¯ 兩 ␺ 0 典 ⫽0.
3!

共3.27兲

Next, we apply the H-nijou method to the ECCND wave
function. The H-nijou method is defined by Eqs. 共2.7兲 or
共2.8兲. Inserting the ECCND wave function into Eq. 共2.7兲, wehave the working equation of the H-nijou method as
F I 共 兵 C I 其 兲 ⫽ 具 ␺ 兩 共 H⫺E 兲 H I 兩 ␺ 典⫽ 具 ␺ 兩 共 H⫺E 兲关 H I ⫹H I S⫹ 21 H I S 2 ⫹¯ 兴 兩 ␺ 0 典 ⫽0,
共3.28兲
for all I, where E is defined by Eq. 共2.2兲. Equation 共3.28兲 isagain nonlinear for the variable 兵 C I 其 .
We see that Eq. 共3.25兲 of the variational method and Eq.
共3.28兲 of the H-nijou method are different. But, since theexact wave function should satisfy both variational andH-nijou equations, it must also satisfy the equation⌬ I 共 兵 C I 其 兲 ⫽ 具 ␺ 兩 共 H⫺E 兲

冋1
1
共 H I S⫺SH I 兲 ⫹ 共 2H I S 223!

册⫺SH I S⫺S 2 H I 兲 ⫹¯ 兩 ␺ 0 典 ⫽0,

共3.29兲

which is derived by subtracting Eq. 共3.25兲 from Eq. 共3.28兲.
This equation can also be written as⌬ I 共 兵 C I 其 兲 ⫽ 具 ␺ 兩 共 H⫺E 兲

再1
1
关 H I ,S 兴 ⫹ 共 2 关 H I ,S 兴 S
23!

冎⫹S 关 H I ,S 兴 兲 ⫹¯ 兩 ␺ 0 典 ⫽0,

共3.30兲

and a simple sum of this equation for all I gives⌬ 共 兵 C I 其 兲 ⫽ 具 ␺ 兩 共 H⫺E 兲

再11关 H,S 兴 ⫹ 共 2 关 H,S 兴 S
23!

冎⫹S 关 H,S 兴 兲 ⫹¯ 兩 ␺ 0 典 ⫽0.

共3.31兲

2469

Conversely, when ⌬ I ( 兵 C I 其 )⫽0 together with G I ( 兵 C I 其 )⫽0
or F I ( 兵 C I 其 )⫽0 for all I, the variational equation 共3.25兲 becomes equal to the H-nijou equation 共3.28兲 for all I, andtherefore, this ␺ must be exact from the theorem given inPaper II.
⌬ I ( 兵 C I 其 )⫽0, Eq. 共3.30兲, is a highly nonlinear equation
for the variables 兵 C I 其 , likewise the variational equation
共3.25兲 and the H-nijou equation 共3.28兲, and may be satisfied
in two ways. Case A is when all H I and S commute,关 H I ,S 兴 ⫽0. Equation 共3.31兲 is satisfied if H and S commute,关 H,S 兴 ⫽0. For Eq. 共2.4兲 being held, only 关 H,S 兴 ⫽0 suffices.
However, we feel that the commutation relation 关 H,S 兴 ⫽0 is
a rather too strong condition in the case of N D ⭓2. A milder
condition is case B, in which Eq. 共3.30兲 is satisfied as it is as
an integral relation, even though the above commutation relation does not hold.
It is true that the relation 关 H,S 兴 ⫽0 holds if all C I ⫽C:
then, this ECCND reduces to the SECC and therefore its
variational solution should be exact. If H I is truly a part of H,
not like d I H with d I being a proportionality constant, then
H I never commutes with H, and therefore 关 H,S 兴 ⫽0 holds
only when all C I ⫽C. Thus, the solution in case A is identical with that of SECC. On the other hand, case B may besatisfied even if H and S do not commute, and therefore even
if all C I are not equal. Since the ␺ at case B satisfies ⌬ I⫽0 together with G I ⫽0 or F I ⫽0, this ␺ must be exact.
When case A and case B occur simultaneously at different
sets of 兵 C I 其 , giving the same energy, the solution of ECCND
is not unique. Since ⌬ I ⫽0 is a nonlinear equation for 兵 C I 其 ,this is possible. Note, however, that since the exact wave
function is unique, the solutions corresponding to case A and
B must be identical except for a normalization factor.
Since case A is identical to SECC, it is realized only
when all terms in the expansion of Eq. 共3.3b兲 are included: it
must be strictly exp(S) as the argument of Kosloff and
Tal-Ezar8 given below Eq. 共3.21兲 implies. A merit of case B
over case A is that case B can be realized within shorter
terms of expansion of the exponential operator, the contribution of the latter’s rest of terms being smaller than a given
threshold of accuracy: this is realized since ECCND has alarger number of variables than SECC, which has only one
variable. For example, let us image CCGSD. There, the variables associated with the X t type operators can become
smaller than those of the A t-type ones as a result of thevariational principle. 共Within GSD operators, A t type operators are ordinary excitation operators that appear in CCSD,
while the X t type operators are the rest of the GSD operators
including de-excitation operators, etc.1兲 This is natural since
we know that CCSD is already a good approximation of theexact wave function. However, in the SECC only one variable exists, so that such a fine tuning is impossible and alarge number of terms, i.e., true exponential operator, is necessary to represent the exact wave function.
In an approximate method of ECCND in which we terminate the expansion of the exponential operator at someorder, case B would be realized within some accuracy andgive a lower solution than case A. When we have a larger
number of variables in ECCND, the convergence rate would

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2470

J. Chem. Phys., Vol. 115, No. 6, 8 August 2001

Hiroshi Nakatsuji

be faster. This would be especially so when we perform
CCGSD, a special case of ECC given in Paper I.9 We will
see in Sec. V that in a simple example of application, the
ECCND gives much faster convergence than SECC becauseof the realization of case B. In SECC, only case A exists.
Now, it would be interesting if we can introduce theexponential ansatz including only commutable operators.
Such formulation is certainly possible if we introduce anormal-ordered exponential ansatz10,11

␺ ⫽ 兵 exp共 T 兲 其 ␺ 0 ,共3.32兲

where the brace 兵 其 means that we include only the normalordered products in the expansion of the exponential operator. T is the GSD operator given by Eq. 共3.6兲. This expansion
does not include the noncommuting operators by its definition. This formulation has been considered first by Lindgren
to circumvent noncommuting problems in the formulation of
open-shell coupled cluster theory.10 Mukherjee also gavesome interesting formulations.11 When we apply the variational principle to the ␺ given by Eq. 共3.32兲, we obtain

具 ␺ 兩 共 H⫺E 兲 兵 a r⫹ a p exp共 T 兲 其 兩 ␺ 0 典 ⫽0,

共3.34兲

but these equations are different from Eqs. 共2.1兲 and 共2.2兲 ofPaper I. Therefore, the normal-ordered exponential anzatz
given by Eq. 共3.32兲 is not guaranteed to represent the structure of the exact wave function.
Finally, it may be useful to give the formulas for thenonvariational solution of the ECCND. They are given by

具 ␺ 0 兩 H⫺E 兩 ␺ 典 ⫽0,

共3.35兲

and具 ␺ 0 兩 H I* 共 H⫺E 兲 兩 ␺ 典 ⫽0

共 I⫽1,...,N D 兲 .

共3.36兲

When ␺ 0 is Hartree–Fock, H I ␺ 0 includes only up to twoparticle excitations, so that the calculations of Eq. 共3.36兲 iseasier than that of Eq. 共3.25兲. Note that when H I includes
only X t-type operators, most of their coefficients C I may
vanish identically as a result of Tables III and IV of Paper I.
In contrast to the variational method, this method of solution
does not guarantee the solution to be exact. This nonvariational formulation is also valid to the SECC with N D ⫽1.
The formulations given here is based on the S operator
given by Eq. 共3.2兲. Those based on the S operator given by
Eq. 共3.4兲 are easy, therefore, we do not give them for thesake of brevity. The arguments about the ECCND are valid
also for this case.

IV. SOLVING ECC
A. Solving SECC

We now formulate the basic equations for solving the
ECC. Again, we describe the case of SECC first and then thecase of ECCND with N D ⭓2. The main difference lies in thecommutation relation.

g 共 C 兲 ⫽ 具 ␺ 兩 共 H⫺E 兲 H 兩 ␺ 典 ⫽0,

共4.1兲

f 共 C 兲 ⫽ 具 ␺ 兩 共 H⫺E 兲 2 兩 ␺ 典 ⫽0,

共4.2兲

or

with E defined by Eq. 共2.2兲. Equations 共4.1兲 and 共4.2兲 areequivalent when the energy E is defined by Eq. 共2.2兲. C in
Eqs. 共4.1兲 and 共4.2兲 stands for either C or c in Eq. 共3.14a兲 or
共3.17a兲, respectively. For SECC, the variational and H-nijou
equations are the same. If we define␭⫽ 共 H⫺E 兲 exp共 CH 兲 ␺ 0 ,共4.3兲

␭⫽ 共 H⫺E 兲 exp关 c 共 H⫺E 兲兴 ␺ 0 ,

共4.4兲

or

then Eq. 共4.2兲 is written as
f 共 C 兲 ⫽ 具 ␭ 兩 ␭ 典 ⫽0.

共3.33兲

and具 ␺ 兩 共 H⫺E 兲 兵 a r⫹ a s⫹ a q a p exp共 T 兲 其 兩 ␺ 0 典 ⫽0,

The working equation for the SECC given by Eq.
共3.14a兲 or 共3.17a兲 is written as

共4.5兲

Note that always f (C)⭓0 and the solution is at the extreme,f (C)⫽0. It is very useful if we can calculate the integral
f (C) directly from Eq. 共3.14a兲 or 共3.17a兲 without expanding
the exponential operator as in Eq. 共3.14b兲 or 共3.17b兲.
Another method of calculating the SECC may be performed asymptotically by increasing term by term along the
expansion of the exponential operator as in Eq. 共3.14b兲 or
共3.17b兲. We truncate the expansion of the exponential operator at the term (1/n!)C n H n or (1/n!)c n (H⫺E n ) n : such truncated wave function is denoted as ␺ n

冉

冊

1
1
1
␺ n ⫽ 1⫹CH⫹ C 2 H 2 ⫹ C 3 H 3 ⫹¯⫹ C n H n ␺ 0 ,
2
3!
n!
共4.6兲
or

冋

1
1
␺ n ⫽ 1⫹c 共 H⫺E n 兲 ⫹ c 2 共 H⫺E n 兲 2 ⫹ c 3 共 H⫺E n 兲 3 ⫹¯
2
3!
⫹

册

1 n
c 共 H⫺E n 兲 n ␺ 0 ,
n!

共4.7兲

where E n is defined by

具 ␺ n 兩 H⫺E n 兩 ␺ n 典 ⫽0.

共4.8兲

The optimal value of the variable C or c is calculated for thetruncated ␺ n and the energy and other properties are calculated. Then, we increase n, repeat the calculation, and checkthe convergence. Though the expansion should always converge, the convergence rate would be dependent on the quality of the reference function.
The optimum value of C or c is calculated by the following methods. We define f n (C) from the truncated wave
functions given by Eq. 共4.6兲 or 共4.7兲 as
f n 共 C 兲 ⫽ 具 ␺ n 兩 共 H⫺E n 兲 2 兩 ␺ n 典 ,共4.9兲

and impose the condition

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J. Chem. Phys., Vol. 115, No. 6, 8 August 2001

Min关 f n 共 C 兲兴 ,or

Structure of exact wave function. III.

df 共 C 兲 ⫽0.
dC n

共4.10兲

For the truncated ␺ n , f n (C) is positive nonzero. By adding
term by term, the function f n (C) approaches zero fromabove and when it becomes zero, i.e., at the convergence, thecorresponding ␺ n is exact. This method is called thef (C)-minimization method and is different from the variational method described below.
We also calculate the truncated wave functions ␺ n by
applying the variational principle. Differentiating ␺ n by thevariable C or c, we obtaind␺n⫽H ␺ n⫺1 ,dC

共4.11兲

d␺n⫽ 共 H⫺E n 兲 ␺ n⫺1 ,dc

共4.12兲

orfor the truncated SECC given by Eq. 共4.6兲 or 共4.7兲, respectively. Inserting these expressions into the variational principle given by Eq. 共2.1兲, we obtain

具 ␺ n 兩 共 H⫺E n 兲 H 兩 ␺ n⫺1 典 ⫽0,
具 ␺ n 兩 共 H⫺E n 兲 2 兩 ␺ n⫺1 典 ⫽0,

共4.14兲

for the truncated SECC. E n is defined by Eq. 共4.8兲. The
optimal value of C or c is calculated from Eq. 共4.13兲 or
共4.14兲.
Equations 共4.9兲 and 共4.14兲 show the difference between
the f (C)-minimization method and the variational method,
but we note that they become close as n increases and finallybecome identical. Before then, the energy calculated by thevariational method is lower than that obtained by thef (C)-minimization method.
The SECC has some similarity to the solution of thetime-dependent Schrödinger equation on the imaginary timeaxis. Therefore, the method developed for solving the timedependent Schrödinger equation7,8 might also be useful for
the SECC.

⫹

1
共 H S 2 ⫹SH I S⫹S 2 H I 兲 ⫹¯
3! I

册

n

1
S m⫺1 H I S n⫺m 兩 ␺ 0 典 ⫽0,⫹n! m⫽1兺For ECCND with N D ⭓2, the variational equation andthe H-nijou equation are different. The method of solution
for the exact case is presented in the previous section. We
consider here the method of solution for the ECCND truncated at order n

冊

1
1
1
␺ n ⫽ 1⫹S⫹ S 2 ⫹ S 3 ⫹¯⫹ S n ␺ 0 .
2
3!
n!

共4.15兲

Since this ␺ n is an approximation to the ECCND, the variational and the H-nijou methods give different solutions ingeneral, but the difference should be small for good ␺ n .
The variational working equation is written as

共4.16兲

with E n defined by

具 ␺ n 兩 H⫺E n 兩 ␺ n 典 ⫽0.

共4.17兲

The energy E n becomes minimum for the solution of theground state.
The working equation for the H-nijou method is given
by
F In 共 兵 C I 其 兲 ⫽ 具 ␺ n 兩 共 H⫺E n 兲 H I 兩 ␺ n 典

冉1⫽ 具 ␺ n 兩 共 H⫺E n 兲 H I ⫹H I S⫹ H I S 2 ⫹¯
2
⫹

冊

1
H S n 兩 ␺ 0 典 ⫽0,n! I

共4.18兲

where E n is defined by Eq. 共4.17兲.
The difference between the variational and H-nijou
equations is given by ⌬ n⌬ In 共 兵 C I 其 兲 ⫽ 具 ␺ n 兩 共 H⫺E n 兲

再1
1
共 H I S⫺SH I 兲 ⫹ 共 2H I S 223!

⫺SH I S⫺S 2 H I 兲 ⫹¯⫹n⫺兺 S m⫺1 H I S n⫺mm⫽2⫽ 具 ␺ n 兩 共 H⫺E n 兲

再

册冎

冋

1
共 n⫺1 兲 H I S n⫺1n!
兩 ␺ 0典1
1
关 H I ,S 兴 ⫹ 共 2 关 H I ,S 兴 S
2
3!
n⫺1

1
⫹S 关 H I ,S 兴 兲 ⫹¯⫹
共 n⫺m 兲 S m⫺1n! m⫽1冎兺⫻ 关 H I ,S 兴 S n⫺m⫺1 兩 ␺ 0 典 .

B. Solving ECCND冉冋1
G In 共 兵 C I 其 兲 ⫽ 具 ␺ n 兩 共 H⫺E n 兲 H I ⫹ 共 H I S⫹SH I 兲
2

共4.13兲

or

2471

共4.19兲

Since ␺ n is approximate, ⌬ In is not necessarily zero, but itshould be small for good ␺ n : It gives a measure how good is
the quality of the calculated ␺ n . The smallness of ⌬ In would
be due to case B of the previous section: The S and H I would
not commute and all C I would not be equal. When ⌬ In ⫽0,
the difference between F In and G In becomes
F In ⫺G In ⫽ 具 ␺ n 兩1
H S n兩 ␺ 0典 .
n! I

共4.20兲

The energy calculated by the variational method is always
lower than that calculated by the H-nijou method.

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2472

J. Chem. Phys., Vol. 115, No. 6, 8 August 2001

Hiroshi Nakatsuji
TABLE I. Ground state of the harmonic oscillator by the variational method
for the truncated SECC.

V. PERFORMANCE OF SECC AND ECC2
FOR HARMONIC OSCILLATOR

We examine here the performance of the SECC by applying it to a simple example, one-dimensional harmonic oscillator.
The harmonic oscillator with the force constant k and the
reduced mass ␮ is defined by the Schrödinger equation

冉

⫺

冊

h2 d2
1 2
␺ ⫽E ␺ .
2
2 ⫹ ky8 ␲ ␮ dy2

共5.1兲

Replacing the constants by

␤ 4⫽

h2
,
4 ␲ 2k ␮␧⫽

8 ␲ 2␤ 2␮ E
,
h2

共5.2兲

and the variable byy⫽ ␤ x,共5.3兲

␣ ⫽0.55

␣ ⫽0.60

n

C opt

E opt

C opt

E opt

0
1
2
3
4
5
6
7
8
9
10
15
20
30

¯
⫺0.1989⫺0.1790⫺0.3091⫺0.2941⫺0.3808⫺0.3810⫺0.4232⫺0.4378⫺0.4607⫺0.4763⫺0.5383⫺0.5771⫺0.6237

1.004 545
1.000 015
1.001 664
1.000 077
1.000 639
1.000 175
1.000 288
1.000 173
1.000 167
1.000 133
1.000 118
1.000 069
1.000 050
1.000 033¯
⫺0.1961⫺0.1760⫺0.2887⫺0.2781⫺0.3344⫺0.3427⫺0.3681⫺0.3817⫺0.3965⫺0.4082⫺0.4513⫺0.4779⫺0.5093

1.016 667
1.000 203
1.006 160
1.000 840
1.002 515
1.001 197
1.001 334
1.001 007
1.000 920
1.000 801
1.000 726
1.000 497
1.000 394
1.000 300

the Hamiltonian is written as
H⫽⫺d2⫹x 2 ,
dx 2

共5.4兲

and the Schrödinger equation 共5.1兲 is rewritten as

冉

2

⫺

冊

d
⫹x 2 ␺ ⫽␧ ␺ .
dx 2

A. SECC

共5.5兲

The two lowest solutions of this equation are␺ g ⫽ ␲ ⫺1/4 exp共 ⫺ 21 x 2 兲 ,␧ g ⫽1,

共5.6兲

and

␺ e ⫽& ␲ ⫺1/4x exp共 ⫺ 21 x 2 兲 ,␧ e ⫽3.

共5.7兲

We take␺ 0 ⫽exp共 ⫺ ␣ x 2 兲 ,共5.8兲

with ␣ ⫽0.55 and 0.60 as our reference functions. ␣ ⫽0.5 isexact solution. The energy expectation value of the referencefunction ␺ 0 is
␧ 0 ⫽ 共 1⫹4 ␣ 2 兲 /4␣ ,which has the minimum value 1.0 at ␣ ⫽0.5, a consequenceof the variational principle.

共5.9兲

We now apply SECC first to the ground state. We use the
SECC given by Eq. 共3.14兲. A single variable C is included in
the S operator as

冉

S⫽C ⫺

冊

d2
⫹x 2 .
dx 2

共5.10兲

We have applied the variational method and thef (C)-minimization method to the truncated ␺ n for n⫽1 – 30. The calculations have been performed using MAPLE
126 and the results are given in Tables I and II. The absolutevalue of C, C opt of ␺ n increases, as expected, as the numberof terms, n increases, and the corresponding energy E opt approaches the exact value 1.0. The magnitude of C opt is, however, far from the negative infinity. Initially for small n, theresults oscillate to some extent, but become monotonic forn⬎6. The convergence rate is faster for ␣ ⫽0.55 than for

TABLE II. Ground state of the harmonic oscillator by the f (C)-minimization method for the truncated SECC.

␣ ⫽0.55

␣ ⫽0.60

n

C opt

E opt

f (C)

C opt

E opt

f (C)

0
1
2
3
4
5
6
7
8
9
10
15
20
30

¯
⫺0.1981⫺0.1969⫺0.3021⫺0.3007⫺0.3638⫺0.3736⫺0.4062⫺0.4228⫺0.4427⫺0.4580⫺0.5178⫺0.5566⫺0.60461.004 545
1.000 016
1.001 691
1.000 081
1.000 641
1.000 188
1.000 290
1.000 180
1.000 172
1.000 139
1.000 124
1.000 074
1.000 054
1.000 036¯
1.3⫻10⫺4
7.8⫻10⫺3
5.1⫻10⫺4
2.5⫻10⫺3
7.9⫻10⫺4
1.0⫻10⫺3
6.3⫻10⫺4
5.7⫻10⫺4
4.5⫻10⫺4
3.8⫻10⫺4
2.0⫻10⫺4
1.3⫻10⫺4
8.0⫻10⫺5¯⫺0.1931⫺0.1896⫺0.2737⫺0.2749⫺0.3155⫺0.3281⫺0.3490⫺0.3627⫺0.3764⫺0.3878⫺0.4305⫺0.4579⫺0.4915

1.016 667
1.000 209
1.006 221
1.000 921
1.002 517
1.001 270
1.001 370
1.001 062
1.000 968
1.000 851
1.000 774
1.000 539
1.000 429
1.000 326¯
1.7⫻10⫺3
2.8⫻10⫺2
5.0⫻10⫺3
1.0⫻10⫺2
5.2⫻10⫺3
5.2⫻10⫺3
3.9⫻10⫺3
3.5⫻10⫺3
2.9⫻10⫺3
2.6⫻10⫺3
1.6⫻10⫺3
1.2⫻10⫺3
8.4⫻10⫺4

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J. Chem. Phys., Vol. 115, No. 6, 8 August 2001

Structure of exact wave function. III.

TABLE III. First excited state of the harmonic oscillator by the variational
method for the truncated SECC.

␣ ⫽0.55

␣ ⫽0.60

n

C opt

E opt

C opt

E opt

0
1
2
3
4
5
6
7
8
9
10
15
20
30

¯
⫺0.1419⫺0.1130⫺0.2230⫺0.1983⫺0.2885⫺0.2754⫺0.3321⫺0.3365⫺0.3676⫺0.3808⫺0.4528⫺0.4998⫺0.5594

3.013 636
3.000 077
3.007 207
3.000 288
3.003 646
3.000 703
3.001 915
3.000 904
3.001 109
3.000 783
3.000 742
3.000 402
3.000 272
3.000 166¯
⫺0.1396⫺0.1115⫺0.2121⫺0.1925⫺0.2584⫺0.2573⫺0.2914⫺0.3020⫺0.3209⫺0.3333⫺0.3854⫺0.4193⫺0.4615

3.05
3.001 000
3.026 552
3.003 280
3.013 703
3.005 412
3.007 716
3.005 168
3.005 098
3.004 223
3.003 860
3.002 481
3.001 865
3.001 307

We next apply ECC2 to the ground state of the harmonic
oscillator. Two variables C K and C P are associated to thekinetic and potential operators as

共5.11兲

with ␣ ⫽0.55 and 0.60. This reference function is ungerade
for the inversion symmetry as the exact wave function: thefirst excited state is the ground state of the ungerade symmetry. The energy of the reference function is␧ 1 ⫽3 共 1⫹4 ␣ 2 兲 /4␣ ,MAPLE 6 and the results are shown in Table III. The convergence behavior of the excited state is similar to that of theground state. It initially oscillates but becomes monotonic forn⬎8. The value of C opt increases as n increases and E opt
approaches the exact value 3.0, though the convergence rateis slow.
In Tables I to III, the result for ␺ 1 is remarkably good. It
is essentially the result of the first iteration of the SICI
method.2 Since every step is variational in the ICI method
and since the variable is reoptimized in each iteration step,
the convergence of the SICI is expected to be faster than thatof the SECC, if the SECC is solved in the expansion method
as given here.

B. ECC2␣ ⫽0.60, because the quality of the reference function is better for ␣ ⫽0.55. The variational results given in Table I and
the f (C)-minimization results given in Table II show similar
behaviors. The energy of the former is always lower than thatof the latter, but the difference becomes smaller as n increases. The value of f (C), which must be positive, gradually approaches zero from above. The convergence rate isslow in both methods: we do not yet reach the convergenceespecially for ␣ ⫽0.60. But, we can conclude that the SECC
converges to the exact value for both the results given in
Tables I and II.
The SECC has also been applied to the first excited stateof the harmonic oscillator for which the exact solution is
given by Eq. 共5.7兲. Similarly to the ground state, we use the
SECC given by Eq. 共3.14兲 and we take the reference function
␺ 0 ⫽x exp共 ⫺ ␣ x 2 兲 ,2473

共5.12兲

which has a minimum again at ␣ ⫽0.5. We have applied thevariational method to the truncated ␺ n of the SECC using

S⫽⫺C K

d2⫹C P x 2 .
dx 2

共5.13兲

Note that S and H do not commute as far as C K is not equal
to C P . The variational method and the H-nijou method are
different in ECC2.
We first apply the variational method. We calculate the
ECC2 wave function truncated at n⫽2 and 3 and summarize
the results in Table IV. C K and C P in Table IV are accurate
only to five decimal figures. The result for n⫽1 is rather
special. At n⫽1, the ECC2 is identical with the first iterationof ICI2, and in this example of harmonic oscillator, the first
iteration of ICI2 is also just identical with the first iterationof SICI or ICI1 because ␺ 0 , ⫺(d 2 /dx 2 ) ␺ 0 , and x 2 ␺ 0 are
not linearly independent. Thus, the ECC2 result at n⫽1 isidentical with the SECC result at n⫽1 given in Table I.
From Table IV, we are surprised at the good convergence of the ECC2 wave function. Even at n⫽2, the energy
converges very nicely to 1.0, and at n⫽3, the results are
already very accurate. This is in sharp contrast to the resultof SECC given in Table I, where the convergence is very
slow and the energy is still considerably different from 1.0
even at n⫽30 for ␣ ⫽0.60. The optimal C K and C P of ECC2
are not equal but differ even in sign. Therefore, this ECC2
solution is different from the SECC solution. Further, theTABLE IV. Truncated ECC2 for harmonic oscillator calculated by the variational method. At n⫽1, C K ⫽C P⫽⫺0.1989 and E 1 ⫽1.000 015 for ␣ ⫽0.55, and C K ⫽C P ⫽⫺0.1961 and E 1 ⫽1.000 203 for ␣ ⫽0.60.
n⫽2

Optimal
En
G In
⌬ In

⳵En /⳵CI

CK
CP
I⫽K
I⫽ P
I⫽K
I⫽ P
I⫽K
I⫽ P

n⫽3

␣ ⫽0.55

␣ ⫽0.60␣ ⫽0.55

␣ ⫽0.60⫺0.020 10⫹0.027 92
1.000 000 044 00.227 0⫻10⫺50.264 7⫻10⫺50.042 7⫻10⫺50.030 7⫻10⫺5⫺0.267 2⫻10⫺50.311 5⫻10⫺5⫺0.036 41⫹0.056 50
1.000 002 106 4⫺1.160 5⫻10⫺50.395 9⫻10⫺5⫺0.229 0⫻10⫺5⫺0.147 5⫻10⫺5
1.417 9⫻10⫺50.483 7⫻10⫺5⫺0.020 10⫹0.027 85
1.000 000 000 21
1.229 8⫻10⫺5⫺1.116 6⫻10⫺5⫺0.064 8⫻10⫺5⫺0.059 4⫻10⫺5
1.447 4⫻10⫺5⫺1.314 2⫻10⫺5⫺0.036 62⫹0.055 80
1.000 000 031 27
1.251 7⫻10⫺5
1.038 1⫻10⫺50.125 3⫻10⫺50.209 4⫻10⫺51.530 6⫻10⫺5
1.269 4⫻10⫺5

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2474

J. Chem. Phys., Vol. 115, No. 6, 8 August 2001

Hiroshi Nakatsuji

TABLE V. Truncated ECC2 for harmonic oscillator calculated by the H-nijou method. At n⫽1, C K ⫽C P⫽⫺0.1989 and E 1 ⫽1.000 015 for ␣ ⫽0.55, and C K ⫽C P ⫽⫺0.1961 and E 1 ⫽1.000 203 for ␣ ⫽0.60.
n⫽2

OptimalEn
F In
nth-order
terma
a

CK
CP
I⫽K
I⫽ P
I⫽K
I⫽ P

n⫽3

␣ ⫽0.55

␣ ⫽0.60␣ ⫽0.55

␣ ⫽0.60⫺0.020 10⫹0.027 92
1.000 000 044 00.262 4⫻10⫺5⫺0.173 8⫻10⫺50.446 8⫻10⫺5⫺0.469 2⫻10⫺5⫺0.036 41⫹0.056 53
1.000 002 108 1
3.959 1⫻10⫺50.204 6⫻10⫺51.420 3⫻10⫺4⫺0.885 2⫻10⫺4⫺0.020 10⫹0.027 85
1.000 000 000 21
1.166 9⫻10⫺5⫺1.166 5⫻10⫺50.202 0⫻10⫺7⫺0.310 0⫻10⫺7⫺0.036 62⫹0.055 80
1.000 000 031 27⫺0.866 1⫻10⫺50.932 8⫻10⫺50.260 2⫻10⫺5⫺0.187 6⫻10⫺5

This is the term defined by Eq. 共4.20兲.

values of ⌬ Kn and ⌬ nP are remarkably very small, showingthat the variational solution is very close to the H-nijou solution. This is confirmed from Table V, in which the resultsof the H-nijou method are summarized. This smallness of ⌬ ln
is attained though S and H do not commute. This is case B,
explained in the previous section. The optimal character ofthe ECC2 results shown in Table IV is confirmed from thesmallness of the values of G In (I⫽K, P) and ⳵ E n / ⳵ C I at the
optimal values of C K and C P .
Table V shows the result of the H-nijou method applied
to n⫽2 and 3 of ECC2. The optimal values of C K and C P
are very close to those of the variational method. Actually,
the H-nijou results are the same as the variational results
except for the case of ␣ ⫽0.60 and n⫽2, where the secondorder term in the H-nijou integral F In , which is the difference
between F In and G In as seen from Eq. 共4.20兲, is the little bit
large so that the variational result and the H-nijou result become different. Note that this difference is not due to a large⌬ In value. It is also small even in this case. The optimal
character of C K and C P is seen from the smallness of thevalue F In .
The results of ECC2 given in this section very muchencourage the ECCND approach for more realistic systems.
The key is a smallness of the ⌬ In values that guarantees thehigh quality of the calculated results. This high quality of thecalculated results is certainly due to the exactness of thestructure of the ECCND wave function.
VI. CONCLUSION

In this series of papers,1,2 we have investigated the structure of the exact wave function and obtained several expressions of the exact wave function that includes much smaller
number of variables than M full-CI given by Eq. 共1.3兲. The
number of variables is from unity to M GSD , for example,
depending on the method of division of the Hamiltonian.
Since the exact wave function is unique, these different expressions represent the different aspects of a single reality,
the exact wave function. These expressions give the bases forthe further studies of the exact wave function from both theoretical and numerical aspects. They will also give a hint in
the study of the different frameworks of the quantum mechanics, like DFT and DMT 共density matrix theory兲.13
In this paper, we have examined the exponential ansatz
in the light of the general theorem given in Paper II. As

motivated in Paper I, we could prove that the simplest extended coupled cluster 共SECC兲 has the structure of the exact
wave function: it becomes exact when solved variationally orby the H-nijou method. This fact sheds light on the possibility of the ECCND wave function in general. Since even thesimplest ECC is exact, the more general ECCND should also
be exact. Then, both the variational equation and the H-nijou
equation give the exact wave function when applied to ECCND. For the exact solution, the difference between thevariational equation and the H-nijou equation should vanish
and is satisfied as an integral relation given by Eq. 共3.29兲.
Thus, the noncommuting problem pointed out in Paper I
disappears.9 Further, ECCND (N D ⭓2) may give a more rapidly converging solution than SECC. Because of its nonlinear character, ECCND may give the exact solution at thesets of variables different from SECC. We thus conclude that
the ECCND is exact for all N D . ECCND gives an explicit
expression of the wave operator W defined by Eq. 共1.5兲. As a
special case, CCGSD is also exact: a statement in Paper I
must be corrected.9 The ECCND and CCGSD must be
solved by the variational method or by the H-nijou method.
Nooijen4 considered solving the CCGSD using the density
equation,5 which belongs to the H-nijou method: this is aclever idea to give the exact solution.
We have summarized the method of solving the SECC
and ECCND for the nth-order truncated form. The variational equation and the H-nijou equation are different in thiscase, but the difference should be small: it gives a measure of
the quality of the calculated result.
The SECC and the ECCND are applied in their truncatedform to the simple example of harmonic oscillator. The
SECC certainly gave the exact wave function starting fromapproximate reference functions for both the ground- andfirst-excited states, though the convergence rates were rather
slow. On the other hand, ECC2, having different variables
for kinetic and potential operators, gave quite a rapid convergence. The variational method and the H-nijou method
gave essentially the same results: ⌬ In ⫽0 were essentially satisfied. This good performance of the ECCND method certainly reflects its exact structure as a wave function. CCGSD
is also important as a special case of ECCND. Head-Gordon
and his co-workers recently studied variational coupled cluster in some detail and reported several nice results.14 Basedon the present result, we believe that the study of ECCND

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J. Chem. Phys., Vol. 115, No. 6, 8 August 2001

would certainly open up a new wave in the study of the exact
wave function.
ACKNOWLEDGMENTS

The author thanks Professor E. R. Davidson for somee-mail discussions and Dr. M. Hada and Dr. M. Ehara forseveral useful conversations and help. The discussions withProfessor D. Mukherjee were also helpful in realizing thenormal-ordered exponential ansatz. This study has been supported financially by the special fund from the Ministry of
Education, Science, Culture, and Sports of Japan.
H. Nakatsuji, J. Chem. Phys. 113, 2949 共2000兲.
H. Nakatsuji and E. R. Davidson, J. Chem. Phys. 5, 2000 共2001兲.
3
P. A. M. Dirac, Proc. R. Soc. London, Ser. A 123, 714 共1929兲.
4
M. Nooijen, Phys. Rev. Lett. 84, 2108 共2000兲.
5H. Nakatsuji, Phys. Rev. A 14, 41 共1976兲.
1
2

Structure of exact wave function. III.

2475

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R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules 共Springer, New York, 1989兲.
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T. J. Park and J. C. Light, J. Chem. Phys. 85, 5870 共1986兲.
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R. Kosloff and H. Tal-Ezer, Chem. Phys. Lett. 127, 223 共1986兲; R. Kosloff, J. Phys. Chem. 92, 2087 共1988兲.
9
In the previous paper 共Ref. 1兲, we wrote, ‘‘Therefore, the variational
CCGSD is not exact.’’ This statement should be read, ‘‘Therefore, the trial
form of the CCGSD is not guaranteed to be exact.’’ But, the fact that the
SECC has a structure of the exact wave function implies that the CCGSD
is also exact when it is solved by the variational or H-nijou method.
10
I. Lindgren, Int. J. Quantum Chem. S12, 33 共1978兲.
11
I. Lindgren and D. Mukherjee, Phys. Rep. 151, 93 共1987兲; U. S. Mahapatra, B. Datta, B. Bandyopadhyay, and D. Mukherjee, Adv. Quantum
Chem. 30, 163 共1998兲.
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M. B. Monagan, K. O. Geddes, K. M. Heal, G. Labahn, S. M. Vorkoetter,
and J. McCarron, MAPLE 6, Waterloo Maple, Inc., Waterloo, Canada.
13
Many-Electron Densities and Reduced Density Matrices, edited by J.
Cioslowski 共Kluwer Academic, New York, 2000兲.
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T. V. Voorhis and M. Head-Gordon, J. Chem. Phys. 113, 8873 共2000兲.

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