JOURNAL OF CHEMICAL PHYSICSVOLUME 115, NUMBER 5

1 AUGUST 2001

Structure of the exact wave function. II. Iterative configurationinteraction method
Hiroshi Nakatsujia)
Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering,
Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan

Ernest R. Davidson
Department of Chemistry, Indiana University, Bloomington, Indiana 47405-6040

共Received 28 November 2000; accepted 10 May 2001兲
This is the second progress report on the study of the structure of the exact wave function. First,
Theorem II of Paper I 共H. Nakatsuji, J. Chem. Phys. 113, 2949 共2000兲兲 is generalized: when we
divide the Hamiltonian of our system into N D 共number of division兲 parts, we correspondingly havea set of N D equations that is equivalent to the Schrödinger equation in the necessary and sufficient
sense. Based on this theorem, the iterative configuration interaction 共ICI兲 method is generalized sothat it gives the exact wave function with the N D number of variables in each iteration step. We callthis the ICIND method. The ICIGSD 共general singles and doubles兲 method is an important special
case in which the GSD number of variables is involved. The ICI methods involving only one
variable 关ICION共one兲 or S共simplest兲ICI兴 and only general singles 共GS兲 number of variables 共ICIGS兲
are also interesting. ICIGS may be related to the basis of the density functional theory. The
convergence rate of the ICI calculations would be faster when N D is larger and when the quality of
the initial guess function is better. We then study the structure of the ICI method by expanding its
variable space. We also consider how to calculate the excited state by the ICIGSD method. One
method is an ICI method aiming at only one exact excited state. The other is to use the higher
solutions of the ICIGSD eigenvalues and vectors to compute approximate excited states. The lattermethod can be improved by extending the variable space outside of GSD. The underlying concept
is similar to that of the symmetry-adapted-cluster configuration-interaction 共SAC-CI兲 theory. A
similar method of calculating the excited state is also described based on the ICIND method.
© 2001 American Institute of Physics. 关DOI: 10.1063/1.1383032兴
second-order density matrix ⌫ (2) (1 ⬘ 2 ⬘ 兩 12) is enough to calculate these properties.1 Among other operators, the Hamiltonian, a key operator in quantum mechanics, is written in asecond-quantized form as2

I. INTRODUCTION

The Schrödinger nonrelativistic equation describes muchof the world of chemistry. If we can solve this equation witha realistic cost, we can make very precise predictions and its
scientific and practical merits are huge. The full CI method
gives the exact wave function within a given basis set, but
the number of variables involved in this method, M full-CI ,easily runs into astronomical figures for basis sets capable ofgiving accurate results. For singlet molecules with evennumber of electrons, it is given by1
M full-CI⫽m⫹1

冉 冊冉 冊
m⫹1

m⫹1

1
N
2

1
N⫹1
2

,

H⫽

共1.2兲

where the first term is a one-particle operator and the second
term is a two-particle operator. In atomic and molecular systems, the electronic Hamiltonian is written in a coordinate
representation as

共1.1兲

H⫽

1

兺v ⫺ 2 ⌬ v ⫺ 兺v 兺A Z A /r A v ⫹ ␮兺⬎ v 1/r ␮ v ,共1.3兲

where the first kinetic and second nuclear attraction terms are
one-electron operators and the third electron repulsion term
is a two-electron operator. We therefore expect that the exact
wave function ␺ that satisfies the Schrödinger equation

where m is the number of active orbitals, N the number ofelectrons, and 共 兲 denotes a binomial coefficient. Table I
shows M full-CI for some typical small molecules assuming adouble-zeta basis set. Even for such small molecules, thisnumber is truly astronomical, yet the energy from a doublezeta basis is not accurate enough for chemical predictions.
All the basic physical operators may be written using
only one- and two-particle operators. For this reason, the共 H⫺E 兲 ␺ ⫽0,

共1.4兲

i.e., an eigenfunction of such a simple operator, should also
have a simple structure: for example, it may be written witha number of variables that is substantially smaller thanM full-CI . In this paper, we mainly use the Hamiltonian 共1.2兲
given in a second-quantized form.

a兲

Author to whom correspondence should be addressed. Electronic mail:hiroshi@sbchem.kyoto-u.ac.jp

0021-9606/2001/115(5)/2000/7/$18.00⫹ ⫹v rp a r⫹ a p ⫹ 兺 w rs
兺
pq a r a s a q a p ,
pr
pqrs

2000

© 2001 American Institute of Physics

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J. Chem. Phys., Vol. 115, No. 5, 1 August 2001

Structure of the exact wave function. II

TABLE I. Number of variables for double-zeta basis set.
Moleculem

M full-CI

Water
Ethylene
Benzene1428
72

1 002 001
88 385 227 425⬃3.8⫻1034

M GSD
8 477
143 668
6 538 320

In a previous paper of this series,2 which is called Paper
I hereafter, one of the authors examined the structure of theexact wave function and showed that it is actually possible tocalculate the exact wave function with the number of variables that is equal to the number of general singles anddoubles 共GSD兲 substitution operators, M GSD ,

冋

册

2
m
M GSD⫽m ⫹ 共 m⫺1 兲 .
2
2

共1.5兲

We proposed the iterative configuration interaction 共ICI兲
method including M GSD variables in each iteration step to
calculate the exact wave function: it is called the ICIGSD
method. The total number of variables in ICIGSD is nM GSD ,where n is an iteration number until convergence. Table I
shows the number M GSD for the same molecules. M GSD iscertainly much smaller than M full-CI . In this paper, we continue to study the structure of the exact wave function. We
generalize the ICI method from a more general point of viewand study the structure of the ICI wave function.
To investigate the structure of the exact wave function,
we need the equations that are equivalent to the Schrödinger
equation in a necessary and sufficient sense.2 Such equations
have the same determinative power as the Schrödinger when
they are solved appropriately. First, the variational principle

具 ␺ 兩 H⫺E 兩 ␦ ␺ 典 ⫽0,

共1.6兲

is equivalent to the Schrödinger equation. The energy of thesystem E is defined by

具 ␺ 兩 H⫺E 兩 ␺ 典 ⫽0,

共1.7兲

throughout this paper. Second, the equation

具 ␺ 兩 共 H⫺E 兲 2 兩 ␺ 典 ⫽0,

共1.8兲

is equivalent to the Schrödinger equation, and the following
equation:具 ␺ 兩 共 H⫺E 兲 H 兩 ␺ 典 ⫽0,

共1.9兲

together with Eq. 共1.7兲, is also equivalent to the Schrödinger
equation. Though we believe that the equivalence of theseequations to the Schrödinger equation is well known, proof
is given in the Appendix. The density equation,1,3 which isequivalent to the Schrödinger equation in the necessary andsufficient sense in the space of the density matrix, was derived based on this theorem. In Paper I,2 a theorem is given
stating that the following set of equations:具 ␺ 兩 共 H⫺E 兲 a r⫹ a p 兩 ␺ 典 ⫽0,

共1.10兲

具 ␺ 兩 共 H⫺E 兲 a r⫹ a s⫹ a q a p 兩 ␺ 典 ⫽0,

共1.11兲

and

2001

where E is given by Eq. 共1.7兲 and the indices p,q,r,s run
through all occupied and unoccupied orbitals, is equivalent
to the Schrödinger equation. This theorem has given a basis
for constructing a method of calculating the exact wave function with the M GSD number of variables.2 Note that a r⫹ anda p in the above equations are the creation and annihilation
operators, respectively, defined by using some appropriate
orthonormal set of orbitals like Hartree–Fock.
II. THEOREM

It is shown in this section that Theorem II-1 of Paper I isa special case of a more general theorem given below.
A. Theorem II-I

We define a division of the Hamiltonian into N D parts as
ND

H⫽兺 HI .

I⫽1

共2.1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2.2兲with E given by Eq. 共1.7兲 is exact in a necessary and sufficient sense.
Proof

The necessity is trivial because if ␺ satisfies the Schrödinger equation given by Eq. 共1.4兲, it automatically satisfies
Eq. 共2.2兲. The sufficiency is also simple. If ␺ satisfies Eq.
共2.2兲 for all I, we sum them up for all I, and using Eq. 共2.1兲
we obtain Eq. 共1.9兲, which implies together with Eq. 共1.7兲
that ␺ is exact 共QED兲.
We define the partial energy E I , corresponding to H I ,by具 ␺ 兩 H I ⫺E I 兩 ␺ 典 ⫽0.

共2.3兲

Then, summing up Eq. 共2.3兲 for all I, we obtain

具 ␺ 兩 H⫺ 兺 E I 兩 ␺ 典 ⫽0,
I

共2.4兲

which implies, comparing to Eq. 共1.7兲
E⫽

兺I E I .

共2.5兲

Using the partial energy E I defined by Eq. 共2.3兲, we obtainthe following theorem.
B. Theorem II-2

The wave function ␺ that satisfies Eqs. 共2.3兲, 共2.5兲, and具 ␺ 兩 共 H⫺E 兲共 H I ⫺E I 兲 兩 ␺ 典 ⫽0,

共2.6兲

for all I (I⫽1,...,N D), is exact in a necessary and sufficient
sense.
Proof

The proof is very similar to that for Theorem II-1. The
necessity is trivial. The sufficiency is also simple. When wesum up Eq. 共2.6兲 for all I, we obtain Eq. 共1.8兲, which implies
␺ is exact 共QED兲.

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2002

J. Chem. Phys., Vol. 115, No. 5, 1 August 2001

H. Nakatsuji and E. R. Davidson

The two theorems given above have significant andbroad utility depending on how we divide the Hamiltonian.
We may divide the Hamiltonian into one- and two-electron
parts and the one-electron part further into kinetic and different nuclear attraction terms, using the definition of the
Hamiltonian given by Eq. 共1.3兲. We may divide H into theHartree–Fock part and the correlation part. We may divide H
into all p,r and p,q,r,s parts, namely into M GSD parts, using
the definition of the Hamiltonian given by Eq. 共1.2兲. In thelast case, Eq. 共2.2兲 of Theorem II-1 gives Eqs. 共1.10兲 and共1.11兲. Then, Theorem II-1 of Paper I is a special case of
Theorem II-1 of this section. We note here that how to divide
the Hamiltonian is dependent on what expression of the
Hamiltonian we use. In this paper, we mainly use the Hamiltonian 共1.2兲 written in a second quantized form.
III. ICI METHOD—GENERALIZATION

It is shown in this section that the ICIGSD method given
in Paper I is a special case from a group of variational methods that gives the exact wave function with a number ofvariables from 1 to M GSD in each iteration step.
We start from a brief explanation of the ICIGSD
method.2 We define the iterative CI method by the recurrence␺ n ⫽ 共 1⫹T n 兲 ␺ n⫺1 ,共3.1兲

where T n is defined by
T n⫽

⫹ ⫹
n r ⫹
C p a r a p ⫹ 兺 n C rs
兺
pq a r a s a q a p ,
pr
pqrs

共3.2兲

using the general singles and doubles substitution operators.
The variables in T n , n C rp , and n C rs
pq , are determined by the
secular equations

具 ␺ n 兩 H⫺E n 兩 ␺ n⫺1 典 ⫽0,

共3.3兲

具 ␺ n 兩 共 H⫺E n 兲 a r⫹ a p 兩 ␺ n⫺1 典 ⫽0,

共3.4兲

具 ␺ n 兩 共 H⫺E n 兲 a r⫹ a s⫹ a q a p 兩 ␺ n⫺1 典 ⫽0,

共3.5兲

which are obtained by applying the variational principle
Eq. 共1.6兲 to ␺ n given by Eq. 共3.1兲. This procedure is iterated
until convergence. When converged, ␺ n becomes identicalwith ␺ n⫺1␺ ⫽ ␺ n ⫽ ␺ n⫺1 ,共3.6兲

and the energy isE⫽E n ⫽E n⫺1 ,共3.7兲

and therefore, Eqs. 共3.3兲, 共3.4兲, and 共3.5兲 become identical toEqs. 共1.7兲, 共1.10兲, and 共1.11兲, respectively. This means thatthe converged solution ␺ is exact. The number of variables
in each iteration step is M GSD , as seen from Eq. 共3.2兲. Since
each iteration process is variational, the solution converges
from above to the exact solution.
The above ICI method can be generalized based on thetheorem given in the preceding section. First, we define thevariable operator S
NDS⫽兺 C IH I ,

I⫽1

共3.8兲

corresponding to the division of the Hamiltonian given by
Eq. 共2.1兲. C I (I⫽1,...,N D) in Eq. 共3.8兲 are variables to be
calculated. We now assume a recurrence␺ n ⫽ 共 1⫹S n 兲 ␺ n⫺1 ,共3.9兲

and determine the variables n C I variationally at each step.
The label n on n C I and S n denotes the iteration number.
Applying the variational principle to ␺ n , we get the secularequation

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

共3.10兲
共 I⫽1,...,N D 兲 .

共3.11兲

Note that the energy of the nth iteration E n satisfies

具 ␺ n 兩 H⫺E n 兩 ␺ n 典 ⫽0,

共3.12兲

as well as Eq. 共3.10兲. It is easily derived from Eqs. 共3.8兲–
共3.11兲. When converged, ␺ n ⫽ ␺ n⫺1 ⫽ ␺ and E n ⫽E n⫺1 ⫽E,
and we have Eq. 共1.7兲 and具 ␺ 兩 共 H⫺E 兲 H I 兩 ␺ 典 ⫽0.

共3.13兲

Summing up Eq. 共3.13兲 for all I, we obtain

具 ␺ 兩 共 H⫺E 兲 H 兩 ␺ 典 ⫽0,

共3.14兲

which combined with Eq. 共1.7兲 shows that the converged
solution is exact. We note that this procedure is valid not
only to the ground state, but also to the excited state, as
discussed in more detail later in this paper.
We can formulate the above iterative method in aslightly different way. We define the variable operator S asND

S⫽兺 c I共 H I ⫺E I 兲 ,I⫽1

共3.15兲

using E I defined by Eq. 共2.3兲. The ICI recurrence formula forthis variable operator S is also given by Eq. 共3.9兲, and applying the variational principle, we get after convergence具 ␺ 兩 共 H⫺E 兲共 H I ⫺E I 兲 兩 ␺ 典 ⫽0,

共3.16兲

instead of Eq. 共3.14兲. Summing up Eq. 共3.16兲 for all I, we
obtain具 ␺ 兩 共 H⫺E 兲 2 兩 ␺ 典 ⫽0,

共3.17兲

which again shows that the converged solution is exact.
Since the ICI method defined by the recurrence given by Eq.
共3.9兲 with the S operator given by Eq. 共3.8兲 or 共3.15兲 includes
N D variables in each iteration step, we call this the ICIND
method and it is summarized as follows.
ICIND method: when the Hamiltonian is divided into
N D parts as Eq. 共2.1兲, we define the variable operator S by
Eq. 共3.8兲 or 共3.15兲. Then, we can formulate the ICI methodby Eq. 共3.9兲, where the number of variables in each iteration
step is N D , and when converged, this method gives the exact
wave function. In the converging process, the ICI solution
approaches from above the exact solution, since each iteration step is variational.
We discuss two extreme cases of the ICI method. First,
when we do not divide the Hamiltonian, the ICI recurrence
formula Eq. 共3.9兲 is written as

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J. Chem. Phys., Vol. 115, No. 5, 1 August 2001

Structure of the exact wave function. II␺ n ⫽ 共 1⫹ n CH 兲 ␺ n⫺1 ,共3.18兲

␺ n ⫽ 关 1⫹ n c 共 H⫺E n 兲 兴 ␺ n⫺1 ,共3.19兲

ordepending on whether we take Eq. 共3.8兲 or 共3.15兲 as a S
operator. The number of variables of this method in each
iteration step is only one, so this is referred to as the ICION
共one兲 method or simplest ICI 共SICI兲 method. A similar
method has been frequently advocated and the convergenceproperty has been discussed.4
Second, when we divide the Hamiltonian into each p,rone-electron part and each p,q,r,s two-electron part, N D isequal to M GSD and the operator S is equivalent to the operator T asS⫽

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

⫽

⫹ ⫹
C rp a r⫹ a p ⫹ 兺 C rs
兺
pq a r a s a q a p ⫽T.
pqpqrs

共3.20兲

Therefore, this is the ICIGSD method proposed in Paper I of
this series.2
Another interesting division of the Hamiltonian is the
singles division that gives the ICIGS 共general singles兲
method. For example, we may divide H into M s number of‘‘singles’’ operators H rp as
H rp ,
兺pr

共3.21兲

H rp ⫽a r⫹ h rp a p ,共3.22兲

H⫽
whereh rp ⫽ v rp ⫹a s⫹ w rs
兺
pq a q .
qs

共3.23兲

Then, correspondingly, we have the operator S as
S⫽

C rp H rp .
兺pr共3.24兲

The secular equation for this method is written from Eqs.
共3.10兲 and 共3.11兲 as

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

E n ⭓E,

共3.25兲

where E is the exact energy. Since the wave function of thismethod given by Eq. 共3.9兲 with S defined by Eq. 共3.24兲 includes only M s variables, we may regard Eq. 共3.25兲 as being
related to the basic existence theorem5 of the density functional theory 共DFT兲,6 like Hohenberg–Kohn theorem4 andothers.7–9 Since there are many different ways of dividing
the Hamiltonian into M s operators, it is interesting to speculate whether such a singles method could be rewritten using
electron density or the first-order density matrix alone.
In actual calculations, the convergence property of the
ICIND method is very important. We believe that in principle, this method should converge. As easily seen from Eqs.
共3.18兲 and 共3.19兲, the SICI method is related to the power

2003

method and the Lanczos method in the eigenvalue
problem.10 It converges to largest absolute eigenvalue, but by
modification, it is made to converge to the ground state.
Since

␦

具 ␺ n兩 H 兩 ␺ n典
⫽ 具 ␺ n 兩 H⫺E n 兩 ␺ n 典 ,具 ␺ n兩 ␺ n典

共3.26兲

when 具 ␺ n 兩 ␺ n 典 ⫽1, Eq. 共3.19兲 is equivalent to a variational
step along the gradient to the average energy. The convergence rate would be accelerated when we use larger numberof variational parameters, and this is realized by using the
ICIND and the ICIGSD method, where the number of variables is N D and M GSD , respectively. Thus, since SICI converges, in principle, the ICIND and ICIGSD methods should
also converge, in principle.
In actual calculations, the convergence property is dependent on other technical matters. Since the convergence
rate would be faster when N D is larger, it is desirable to use
the largest possible N D for realizing fast convergence. Notefurther that the number of the divisions, N D , is not necessarily kept constant during the iteration process. By changing
N D , we can adjust the labor necessary for diagonalizing thematrices. Allowing such a flexibility in the computationalalgorithm would be useful in applications of the ICIND
method and some discussions are given in Sec. IV.
Another factor affecting the convergence rate is the quality of the initial function ␺ 0 . An appropriate choice of ␺ 0 insymmetry and in the nature is an important prerequisite: thebetter the quality of the initial function ␺ 0 , the faster theconvergence. Note that if we start from a single determinant
initial function, the coefficients for excitations higher than
2n-fold will be kept zero until iteration n. The energy may
converge rapidly, but the convergence rate may be dependent
on the size of the molecule. It would be useful to prepare theinitial function by the method of a high-performance algorithm along the line of the ICIND method, or by the method
now available. For example, we may adopt a nonvariational
procedure for preparing the initial guess, or we may adopt,for example, the conventional CCSD 共coupled cluster singles
and doubles兲 for ␺ 0 . In the latter case, all levels of excitations are included in ␺ 0 from the beginning in an approximate way, so that the convergence rate would be faster thanusing the Hartree–Fock initial function.
IV. EXPANSION OF THE VARIABLE SPACE OF THE
ICI METHOD

We consider in this section an expansion of the variable
space of the ICI method and see what happens by such an
expansion. We consider two different expansions: one is the
expansion from the ICIGSD method 共N D⫽M GSD) to include
triple and higher operators, and the other is from the intermediate case 共ICIND兲 where N D is between 1 and M GSD .
First, we consider the expansion from the ICIGSD
method. We define the general operator G by
G⫽H⫹L

共4.1兲

where H is the Hamiltonian of the system and L is some
operator defined later. Here, note that we are interested in the
eigenfunction of H, not of G. So, L is not a perturbation. We
rewrite the Hamiltonian operator given by Eq. 共1.2兲 as

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2004

J. Chem. Phys., Vol. 115, No. 5, 1 August 2001

H⫽

兺
pr

v rp a r⫹ a p ⫹

兺

pqrs

⫹ ⫹
w rs
pq a r a s a q a p ⫽H. Nakatsuji and E. R. Davidson

M GSD

兺i

x i t i , 共4.2兲

where x i represents the integrals, v rp and w rs
pq , and t i represents the substitution operators, a r⫹ a p and a r⫹ a s⫹ a q a p . Similarly, the operator L is also expressed asL⫽兺j y j r j ,共4.3兲

where y j is some number and r j the substitution operator.
Corresponding to H of Eq. 共4.2兲, we define the GSD operator
T as
T⫽

兺
pr

C rp a r⫹ a p ⫹

兺

pqrs

⫹ ⫹
C rs
pq a r a s a q a p ⫽M GSD

兺i c i t i , 共4.4兲

and similarly, corresponding to L we define the variable operator R asR⫽

兺j d j r j ,共4.5兲

where c i and d j represent unknown variables. Thus, corresponding to G we have defined the variable operator
U⫽T⫹R.

共4.6兲

We now consider the recurrence␺ n ⫽ 共 1⫹U n 兲 ␺ n⫺1 .

共4.7兲

Applying the variational principle given by Eq. 共1.6兲 to ␺ n ,we get具 ␺ n 兩 H⫺E n 兩 ␺ n⫺1 典 ⫽0,

共4.8兲

具 ␺ n 兩 共 H⫺E n 兲 t i 兩 ␺ n⫺1 典 ⫽0,

共4.9兲

具 ␺ n 兩 共 H⫺E n 兲 r j 兩 ␺ n⫺1 典 ⫽0,

共4.10兲

and when converged, ␺ n ⫽ ␺ n⫺1 ⫽ ␺ and E n ⫽E n⫺1 ⫽E, weobtain具 ␺ 兩 H⫺E 兩 ␺ 典 ⫽0,

共4.11兲

具 ␺ 兩 共 H⫺E 兲 t i 兩 ␺ 典 ⫽0.

共4.12兲

具 ␺ 兩 共 H⫺E 兲 r j 兩 ␺ 典 ⫽0,

共4.13兲

for all i (1⭐i⭐M GSD) and j. Equations 共4.11兲 and 共4.12兲
guarantee that the solution ␺ is exact, but Eq. 共4.13兲 has
nothing to do with the proof that guarantees the exactness of
␺ . Namely, the operators L and R do not affect the final
solution.
We classify the operator R into two cases: one is the casewhere the operator R is expressed within singles and doubles
and the other is the case where the operator R belongs to theoutside of the singles and doubles, namely to triples andhigher operators.
When we add some physical operator L to the Hamiltonian as in Eq. 共4.1兲, the corresponding R operator belongs
to the general singles and doubles space, since any physicaloperators can be expanded by the GSD operators
兵 a r⫹ a p ,a r⫹ a s⫹ a q a p 其 . This means that the addition of the variable operator R as Eq. 共4.6兲 is redundant to the already existing GSD variable space represented by the T operator

given by Eq. 共4.4兲. Then, in this case the addition of the L
and R operators has no effect at all on either the iteration
process or the final result.
On the other hand, when R includes the operators from
triple to N-ple excitation or substitution operators, it enlarges
the variable space to be larger than M GSD and certainly affects the variational process, though it does not affect theconverged solution. Actually, when R includes all triple toN-ple excitation operators, the above method gives full CI
and the iteration converges at once.2
We next consider the expansion of the variable spacefrom the intermediate ICIND method in which the number ofvariables is N D which is between 1 and M GSD . In this casethe definition of the operator G is the same, but that of U isdifferent from Eq. 共4.6兲 and is given by
U⫽S⫹R,

共4.14兲

using the S operator given by Eq. 共3.8兲. Correspondingly,
Eqs. 共4.9兲 and 共4.12兲 are rewritten as

具 ␺ n 兩 共 H⫺E n 兲 H I 兩 ␺ n⫺1 典 ⫽0,

共4.15兲

具 ␺ 兩 共 H⫺E 兲 H I 兩 ␺ 典 ⫽0,

共4.16兲

andrespectively. Since the operator space 兵 H I 其 is incomplete
within the GSD space, the operator L that is physical or thatbelongs to the GSD space cannot generally be expanded by
兵 H I 其 . Therefore, the addition of the R operator may expand
the variable space and may affect the iteration process,though of course, the final converged solution is unaffected.
Similarly, when the R operator includes the operators from
triple to N-ple excitation and substitution operators, it would
also affect the variational process, though again, it does notaffect the final solution. When the R operator affects thevariational process.
In summary, as only the Hamiltonian H defines the system, the additional operator L has nothing to do with thesolution of the ICI method. Only its variable counterpart R
plays a role for controlling the convergence of the iteration
process. Furthermore, by expanding the variable operator
space by R, we can calculate higher excited states by themethod described below.
V. EXCITED STATES BY THE ICIGSD AND ICINDMETHODSAs briefly described in Paper I, the ICIGSD method isapplicable not only to the ground state but also to the excitedstate. We describe in this section how to calculate the excitedstates by the ICIND method. Since the ICIGSD method is an
important extreme, we first discuss the excited-state calculations based on the ICIGSD method. We give three methods
called method A, B, and C.
First, since the ICIGSD method is a one-state theory, as
would be clear from the Appendix, we calculate the excitedstate iteratively as follows. We refer to this method as
method A. Suppose that we are interested in the second state
共first excited state兲. Then, we take the second state in theinitial guess, and then continue to choose only the ‘‘second’’
state in the iteration process. This would lead to convergence

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J. Chem. Phys., Vol. 115, No. 5, 1 August 2001

Structure of the exact wave function. II

to the second state, which should be exact from the theorem.
Note, however, that second in this procedure is only the
number in the initial guess and does not mean that the stateis the true second state. The ‘‘third’’ state in the initial
guess may become lower than the second state as often occurs in an ordinary iterative diagonalization process likeDavidson’s.11
In method B, we utilize the higher-energy solutions simultaneously obtained in the ICIGSD calculation of theground state as the excited states. At the convergence of the
ICI calculation, we obtain the exact ground state ␺ g and at
the same time, the higher-energy solutions ␺ K of the same
secular equation that are written as
M GSD

␺ K ⫽ 兺 c Ki t i 兩 ␺ g 典 ,

共5.1兲

i⫽1

where t i represents the singles and doubles substitution operators as defined by Eq. 共4.4兲. Since ␺ K are the solutions of
the same secular equation as for ␺ g , they satisfy

具 ␺ K 兩 ␺ g 典 ⫽0,

具 ␺ K 兩 H 兩 ␺ g 典 ⫽0

共5.2兲

and

具 ␺ K 兩 ␺ K ⬘ 典 ⫽ ␦ KK ⬘ ,具 ␺ K 兩 H 兩 ␺ K ⬘ 典 ⫽E K ␦ KK ⬘ .

共5.3兲

Since Eqs. 共5.2兲 and 共5.3兲 are important necessary conditionsof the excited states, we may take ␺ K as representing theexcited states of ␺ g .
Now, what is the quality of the excited states calculated
by method B? The excited functions ␺ K of method B are
generated by applying the GSD operators to the exact groundstate ␺ g as expressed by Eq. 共5.1兲. It is also interesting toconsider a set of functions ␸ i defined by

␸ i⫽

兺j d ji u j 兩 ␺ g 典 ,共5.4兲

where 兵 u j 其 represents the operators from triple to N-ple excitation operators. Since ␺ g is exact, it satisfies

具 ␸ i 兩 H⫺E 兩 ␺ g 典 ⫽0.

共5.5兲

However, 兵 ␺ 其 obtained simultaneously with ␺ do not necessarily satisfy the similar relation to Eq. 共5.5兲, namely
Kg

具 ␸ i 兩 H⫺E 兩 ␺ K 典 ⫽0,

共5.6兲

because ␸ i , which is linearly independent from ␺ , are notincluded in the secular equation which gives ␺ g . This meansthat the excited-state functions 兵 ␺ K 其 of method B obtained
simultaneously with ␺ g are not exact, though they satisfy
Eqs. 共5.2兲 and 共5.3兲.
The quality of the excited states by method B is therefore improved by extending the variable space asK冉冊␺ L ⫽ 兺 c Li t i ⫹ 兺 d Lj u j 兩 ␺ g 典
i

j

共5.7兲

where 兵 t i 其 represents GSD operators and 兵 u j 其 the operatorsfrom triple to N-ple excitation operators. By further diagonalizing 共only once兲 the secular equation corresponding to
Eq. 共5.7兲, we can improve the quality of the excited state, up
to the exact limit. Furthermore, by this method, we can cal-2005

culate higher multiple excitations which were rather poor orwere not calculated in method B. We refer to this method as
method C.
The methods of calculating the excited state described inthis section have some similarity to already existing
methods.12 The iterative method A is similar to the symmetry
adaptedcluster
共SAC兲method13,14
and
the
15,16
for excited and openMR共multireference兲-SAC method
shell electronic states. Method B is similar to the SAC-CI
SD-R method,12,17 and method C to the SAC-CI general-R
method.18 In the SAC-CI SD-R method, the SAC-CI excitation operators R, which are similar to the operator t i in Eq.
共5.1兲, are limited within singles and doubles, while in thegeneral-R method, they are taken not only from the SD space
but also from triple to higher excitation space. The SAC-CI
SD-R method is accurate for describing ordinary singleelectron excitation processes, but insufficient for describing
the multiple-electron excitation processes,18 for which the
SAC-CI general-R method is accurate.19
It is very interesting which of the three methods given inthis section is most efficiently accurate for the study of excited states of atoms and molecules. Probably method B,
which is easiest among the three, would be most useful for
ordinary excitation processes. Needless to say, the expressionof the excited state by Eqs. 共5.1兲 and 共5.7兲 represents thetransferability of electron correlations between ground- andexcited states. Further, by including the operators belonging
to different symmetries and the ionization and electron attachment operators in the operators t i and u j , we can also
describe different excited states 共e.g., triplet excited states兲
and ionized and electron attached states as well as the excitedstates having the same symmetry as the ground state, just as
in the SAC-CI method.12,17
It is also possible to calculate the excited states in the
ICIND formalism. Method A is essentially the same as in the
ICIGSD case. We use an initial guess function ␺ 0 having thesymmetry and the electronic structure desired for the excitedstate to be calculated, and choose to keep only such functions
until convergence. Again, the converged solution should be
exact. In the case of method B, the quality of the excitedstates obtained simultaneously with the ground state would
be worse than that for the ICIGSD method, when the variational space for the excited states of the ICIND method issmaller than that of the ICIGSD method. The quality of theexcited states can be improved and the range of the excitations calculated are expanded by extending the variable
space from ND to GSD or even to include triple to N-ple
excitation operators. This method corresponds to method Cof the above paragraphs. We believe from the experiences of
the SAC/SAC-CI calculations that the ICIGSD method, followed by method B in particular, is a useful method for
calculating the ground- and excited states of molecules andmolecular systems.
VI. CONCLUSIONThe Schrödinger equation itself should be correct even if
the basic operator involves up to N-particle interaction terms,
but in reality, the Hamiltonian operator involves only up to
two-particle interaction terms. Since the exact wave function

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2006

J. Chem. Phys., Vol. 115, No. 5, 1 August 2001

is an eigenfunction of the Hamiltonian that has such a simple
structure, the exact wave function itself should also have asimple structure. This is a basic philosophy underlying thepresent series of studies. We study the structure of the exact
wave function under the expectation that it should be simple
and hopefully beautiful.
In Paper I,2 we presented a theorem that implies an existence of the general singles and doubles description of theexact wave function. Namely, the number of variables necessary to describe the exact wave function would be reduced
to M GSD given by Eq. 共1.5兲 instead of M full-CI given by Eq.
共1.1兲. We examined both exponential and linear expansion
ansätz and proposed the ICIGSD method as a method of
calculating the exact wave function with the GSD number ofvariables.
In this paper, we have generalized Theorem II-1 of Paper
I: when we divide the Hamiltonian into N D parts, we have a
set of N D equations that is equivalent to the Schrödinger
equation in the necessary and sufficient sense. Based on thistheorem, the ICI method was generalized to calculate theexact wave function with N D number of variables where N D
ranges from 1 to M GSD where M GSD is the case of the
ICIGSD method. The simplest case with N D⫽1 共ICION or
SICI兲 is very interesting, and the general singles case whereN D is the number of general single substitution operators
共ICIGS兲 is also interesting. In actual ICI calculations, theconvergence rate would be faster when N D is larger andwhen the quality of the initial function is better. Further, N D
is not necessarily kept constant throughout the ICI calculations.
We studied the structure of the ICI method by first expanding its variable space. GSD is a good special case because the GSD operator space is complete in the sense that it
can expand any physical operators. If we extend the variable
space outside of GSD by including some triple to N-ple excitation operators, the variational process would be accelerated though the final solution should be the same.
We considered how to calculate the excited state by the
ICIGSD method. We proposed three methods. One is an iterative method aiming at only one excited state, and theother is the method utilizing the higher-energy solutions obtained simultaneously with the ground state. The first method
gives the exact solution, but the second one is not exact,
though the solutions satisfy the orthogonality and Hamiltonian orthogonality with the calculated exact ground state.
The latter method has some similarity to the SAC-CI SD-R
method. By extending the variable space after obtaining theexact ground state, we can improve the excited state: thismethod is similar to the SAC-CI general-R method for calculating the excited states, in particular, the multiple-electron
excited states. Similar methods are also described based on
the ICIND method.
ACKNOWLEDGMENT

This study has been supported financially by the special
fund from the Ministry of Education, Science, Culture, andSports of Japan.

H. Nakatsuji and E. R. Davidson

APPENDIX

We here prove that each of Eqs. 共1.8兲 and 共1.9兲 with E
defined by Eq. 共1.7兲 is equivalent to the Schrödinger equation in a necessary and sufficient sense.
The necessity is trivial: if ␺ satisfies Eq. 共1.4兲, it satisfies
Eqs. 共1.8兲 and 共1.9兲. We prove the sufficiency. We assume
that we have the solutions of the Schrödinger equation asH⌿ n ⫽E n ⌿ n .

共A1兲

Since 兵 ⌿ n 其 forms a complete set of eigenfunctions of the
Hamiltonian, we can expand our ␺ as

␺ ⫽ 兺 C n⌿ n .

共A2兲

n

Inserting Eq. 共A2兲 into our expression given by Eq. 共1.8兲, wehave
0⫽ 具 ␺ 兩 共 H⫺E 兲 2 兩 ␺ 典 ⫽
⫽

2
C*
兺
n C m 具 ⌿ n 兩 共 H⫺E 兲 兩 ⌿ m 典n,m兺n 兩 C n兩 2共 E n ⫺E 兲 2 .

共A3兲Since all terms in the last sum are non-negative, Eq. 共A3兲
stands only when all C n are zero except for one case n⫽i,for which E⫽E i and C i ⫽0. From the normalization condition, C i ⫽1, and therefore, from Eq. 共A2兲, ␺ ⫽⌿ i . Namely,␺ is an eigenfunction of the Schrödinger equation. It can beeither ground state or excited state. When E i is degenerate, ␺
becomes a linear combination of one set of the degenerate
eigenfunctions of E i of the Schrödinger equation. Thus, thesufficiency of Eq. 共1.8兲 is proved. Equation 共1.9兲 is easily
obtained from Eq. 共1.8兲 by using Eq. 共1.7兲 共QED兲.
1

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