PHYSICAL REVIEW A 72, 062110 共2005兲

General method of solving the Schrödinger equation of atoms and molecules
Hiroshi Nakatsuji
Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering, Kyoto University, Katsura, Nishikyo-ku,
Kyoto 615-8510, Japan and Fukui Institute for Fundamental Chemistry, Kyoto University, 34-4 Takano-Nishihiraki-cho, Sakyo-ku,
Kyoto 606-8103, Japan
共Received 26 June 2005; published 13 December 2005兲
We propose a general method of solving the Schrödinger equation of atoms and molecules. We first construct
the wave function having the exact structure, using the ICI 共iterative configuration or complement interaction兲
method and then optimize the variables involved by the variational principle. Based on the scaled Schrödinger
equation and related principles, we can avoid the singularity problem of atoms and molecules and formulate ageneral method of calculating the exact wave functions in an analytical expansion form. We choose initial
function ␺0 and scaling g function, and then the ICI method automatically generates the wave function that has
the exact structure by using the Hamiltonian of the system. The Hamiltonian contains all the information of thesystem. The free ICI method provides a flexible and variationally favorable procedure of constructing the exact
wave function. We explain the computational procedure of the analytical ICI method routinely performed in
our laboratory. Simple examples are given using hydrogen atom for the nuclear singularity case, the Hooke’s
atom for the electron singularity case, and the helium atom for both cases.
DOI: 10.1103/PhysRevA.72.062110

PACS number共s兲: 03.65.Ca, 03.65.Ge, 31.10.⫹z, 31.15.⫺p

I. INTRODUCTION

As noted by Dirac in 1929 关1兴, the Schrödinger equation
共SE兲 provides a fundamental mathematical law for chemistry,though the importance of the relativistic theory is muchlarger now than then considered. The SE has quite a significant power of predicting chemical phenomena with very
high accuracy. It also connotes physical and chemical lawsthat are conceptually useful for understanding chemical phenomena like the Fukui-Woodward-Hoffman rule 关2,3兴, etc.
So establishing a general method of solving the SE as accurately as possible is one of the ultimate purposes of theoretical chemistry. The scientific and practical merits brought
about by this method are huge. First, we can make accurate
predictions of chemical and related phenomena. This is a
new evolution of quantum chemistry since it has long been aqualitative or semiquantitative science and could not do very
accurate predictions except for very simple systems. Second,
we can get chemical concepts as accurately as possible andcan get even very fine understanding of chemistry. Third, it
gives sound starting points of approximate theories. Since
one has exact wave functions, one can make up approximate
theories by neglecting some complex but minor terms. Finally, it even helps to predict a new phenomenon. Whenaccurate theory and accurate experiment contradict severely,
something must be wrong or missing in either of them andthis may become a clue for a discovery of a new science.
Such a situation could have never been expected before,
since our theories could not be sufficiently accurate.
Despite its importance, however, there was no general
method of solving the SE. Only one practical method of
calculating the “exact” wave function was the full-CI method
关4,5兴. But the number of variables involved in the full CI isastronomical: even for moderate-size molecules withmoderate-quality basis sets, the numbers of the variables arehuge and cannot be handled even with modern computers.
Therefore, applications of the full CI have been very limited.
1050-2947/2005/72共6兲/062110共12兲/$23.00

Furthermore, even if we could get the full-CI solutions, they
are actually far from the true solutions of the SE, because thebasis sets used there are usually far from complete. It isnecessary to develop a general theory for calculating the exact wave function in an analytical form.
In our series of papers 关6–13兴, we have studied the structure of the exact wave functions, aiming to develop a general
method of calculating the exact wave function. By exact
wave function, we mean the solution of the SE. We have
posed two questions: 共1兲 What mathematical structure theexact wave function has? 共2兲 Is a general theory possible that
constructs the exact wave function systematically? Recently,
we could have been able to accomplish this aim and theresults were summarized in the previous communication
关13兴. We have formulated a general systematic method ofsolving the SE of atoms and molecules using the ICI 共iterative configuration or complement interaction兲 formalism inan analytical expansion form up to a desired accuracy. A
purpose of this paper is to formulate this method in a moregeneral and detailed context.
The exact wave function ␺ is the solution of the SE,
H␺ = E␺ ,共1兲

where the Hamiltonian H is given by1
H = 兺 − ⌬i − 兺 兺 ZA/rAi + 兺 1/rij = K + V = K + Vne
2
i
i A
i⬎j
+ Vee .

共2兲In our studies for formulating a general theory of solving theSE 关6–13兴, two stages were experienced. The first stage wasto clarify the structure of the exact wave function and toformulate a general theory to construct the exact wave function 关6–11兴. In the SE, the Hamiltonian determines the exact
wave function, so that the exact wave function may be ex-

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PHYSICAL REVIEW A 72, 062110 共2005兲

HIROSHI NAKATSUJI

pressed as a functional of the Hamiltonian, f共H兲, applied tosome appropriate function ␺0, say,

␺ = f共H兲␺0 .

共3兲

We have investigated some possible forms of the function
f共H兲, based on the combined use of the variational principle
and the H-square theorem, and shown that the ICI and the
SECC 共simplest extreme coupled cluster兲 methods give theexact wave function in the sense of Eq. 共3兲 above 关6–8兴. The
general ECC method, called ECCND method 共ND stands forthe number of division of the Hamiltonian兲, was not guaranteed to be exact. A special example of ECCND was the
CCGSD 共coupled cluster general singles and doubles兲
method considered by Nooijen 关14兴 and the present author
关6兴 almost at the same time. Nooijen conjectured that it may
describe the exact wave function with only GSD number ofvariables. The present author also started his study with asimilar optimism, but reached a negative conclusion that the
CCGSD cannot be exact and, instead, he proposed the
ICIGSD method as a method to reach the exact wave function with essentially the GSD number of variables 关6兴. Later,
the present author relaxed his opinion to that the CCGSD is
not guaranteed to be exact: though the CCGSD cannot be
guaranteed to be exact, it may happen to be exact because of
its highly nonlinear character with respect to the GSD variables 关8兴. Later on, several studies have been reported about
the exactness of the CCGSD 关15–20兴: in particular, Mukherjee and Kutzelnigg 关20兴 have strengthened the present author’s proof for the GSD case 关6兴 by introducing the Leealgebraic concept, which is in the author’s language,“generalized” Thouless transformation concept used in theargument for the coupled cluster singles 共CCS兲 in Ref. 关6兴.
We have shown that the full-CI solutions can actually beobtained with essentially single to GSD number of variables
by performing the simplest ICI 共SICI兲 and the ICIGSD calculations for some molecules in closed- and open-shell electronic structures 关10,11兴. We have shown a unique merit of
the ICIGSD method that gave the full-CI solutions in a few
iterations 关11兴. However, even so, the full-CI solutions areusually far from the true solutions of the SE, because of theincompleteness of the basis sets. So we wanted to solve theSE in an analytical form.
A serious problem had occurred in the second stage inconstructing a general theory for solving the SE of atoms andmolecules in an analytical form. That was the singularity
problem 关12,13兴 caused by the singularity of the Coulomb
potential V involved in the Hamiltonian given by Eq. 共2兲.
The nuclear attraction term Vne and the electron repulsion
term Vee diverge when rAi and rij, respectively, become zero.
For these singularities, the integrals of the higher powers ofthe Hamiltonian diverge and so we cannot go further. We had
to overcome this difficulty for developing the general theoryof solving the SE.
We have proposed two ways to overcome this difficulty.
One was done by introducing the inverse Schrödinger equation 共ISE兲 关12兴 and the other by introducing the scaled
Schrödinger equation 共SSE兲 关13兴. Both the ISE and SSE areequivalent to the original SE. In the formulation based on the
ISE, it was difficult to explicitly represent the inverse Hamil-

tonian in a closed analytical form, though even the expanded
form worked well 关12兴. On the other hand, the formulation
based on the SSE was easy and rigorous and yet had nopractical difficulties.
In the previous communication 关13兴, we have also introduced the free ICI method. This method is very flexible andyet gives a solution variationally better than the ordinary
fixed ICI method 关6,7兴. We have formulated the free ICI
method based on the SSE as a general method of solving theSE and given very briefly the preliminary results of applications to hydrogen atom, helium atom, and hydrogen molecule. The results were very encouraging, showing a highpotentiality of the proposed method.
The purpose of this paper is to explain in some details thegeneral method of solving the SE in an analytical expansion
form, which is currently performed in our laboratory. We
explain the theoretical background, introduce the SSE andrelated principles, and give a general method of constructing
the wave function that has an exact structure. A merit of thefree ICI method is described in some detail. Then, themethod is applied to the hydrogen atom that has nuclear
singularity, to Hooke’s atom that has electron singularity, andto the helium atom that has both nuclear and electron singularities.
In the subsequent papers, we will show more general applications of the proposed method to the hydrogen molecule关21兴 and to a few-electron atoms: helium, lithium, beryllium,
and boron—i.e., from two- to five-electron atoms 关22兴. We
will also apply the present method to Hooke’s atom in amore general way than given in this paper. Further, an extension of the present formalism to the relativistic case for solving the relativistic Dirac and Dirac-Coulomb equations has
successfully been formulated and will appear in a separate
paper 关23兴.
II. FORMALISM OF THE METHOD OF SOLVING THE
SCHRÖDINGER EQUATION
A. Basic theory

The SE given by Eq. 共1兲 defines the exact wave function.
The variational principle given by具␺兩H − E兩␦␺典 = 0

共4兲

is equivalent to the SE when the variation of ␺ is arbitrary.
Actually, when the variation is done within some limited
functional space, it gives the best possible ␺ within thatspace. On the other hand, the H-square equation given by具␺兩共H − E兲2兩␺典 = 0

共5兲

is valid only for the exact ␺. If ␺ satisfies the H-squareequation, this ␺ is guaranteed to be exact. Comparing Eqs.
共4兲 and 共5兲, we can easily get the following theorem 关7兴:
When a ␺ including only one variable C satisfiesd␺= 共H − E兲␺ ,dC

共6兲

it has the structure of the exact wave function, because thevariationally best ␺ satisfies the H-square theorem and so isexact.

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GENERAL METHOD OF SOLVING THE…

Two simplest ␺’s that have the structure of the exact wave
function with only one variable are the simplest ICI 共SICI兲
wave function 关7兴 and the simplest extreme coupled cluster
共SECC兲 wave function 关8兴. The SICI wave function is defined by the recursion formula␺n+1 = 关1 + Cn共H − En兲兴␺n ,共7兲

where n stands for the iteration number. At convergence, the
SICI wave function is guaranteed to become exact though it
includes only one variable at each iteration. The SECC wave
function is compactly written as␺ = exp关C共H − E兲兴␺0 ,共8兲

which includes only one variable C. One can show, however,from a different argument that the optimal C of the SECC
would be minus infinite 关24兴.
Within a given limited basis set, the SICI provides amethod to calculate the “exact” full-CI solution by using
only one variable, in contrast to a huge number of variablesin the ordinary full CI. Actually, we have applied the SICI
method to many molecules in closed- and open-shell electronic structures using minimal and double-␨ basis sets andshown that it smoothly converges to the exact full-CI solution 关10兴. Because each iteration step of the SICI is variational, it converges from above to the exact solution. However, since the number of the variable is so small—i.e., only
1—the convergence speed was slow: it took 30–70 iterations
to get the energies correct to 8 decimal figures of the full-CI
energies 关10兴.
We can introduce a larger number of variables in the ICI
formalism 关6,7兴. Dividing the Hamiltonian into ND pieces
共ND stands for the number of division兲,ND

H = 兺 HI共9兲

I=1

and introducing the partial energy EI,n = 具␺n兩HI兩␺n典, we candefine the ICIND as

冋

册

ND

␺n+1 = 1 + 兺 CI,n共HI − EI,n兲 ␺n ,I

共10兲

which is also guaranteed to become exact at convergence.
When we introduce the second-quantized Hamiltonian† †H = 兺 vrpar†a p + 兺 wrs
pqar as aqa p
pr

共11兲

pqrs

and divide it into singles and doubles pieces, we can define
the ICIGSD as

冉

冊

† †
␺n+1 = 1 + 兺 crpar†a p + 兺 crs
pqar as aqa p ␺n ,
pr

pqrs

共12兲

which has the GSD number of variables 兵crp , crs
pq其. ThisICIGSD is also guaranteed to become exact 共full CI兲 at convergence 关6兴. Actually, we have formulated the ICIGSD first关6兴, showing that the exact full-CI solution can be solved byusing only the GSD number of variables that are muchsmaller than the huge number of variables in the ordinary
full CI.

We have used the ICIGSD method to calculate the full-CI
solutions of closed- and open-shell molecules using the minimal and double-␨ basis sets 关11兴. We have shown that theconvergence speed of the ICIGSD was much faster than thatof the SICI: typically only 3–9 iterations were enough incontrast to 30–70 iterations in the SICI. This was essentially
due to an increase in the number of variables from one to the
GSD number. Further, we have discussed a special merit of
the ICIGSD among the general ICIND.
Thus, with the ICI formalism, we can formulate a methodof calculating the exact full-CI solution using only one to
GSD number of variables at each iteration. However, as
stated in the Introduction, the full-CI solution is usually far
from the true solution of the SE. To get a true solution of theSE, we have to formulate a method of solving the SE in ananalytical form.
When we consider a method of analytically solving theSE, we soon encounter a big obstacle, which is the singularity problem caused by the Coulomb potential included in the
Hamiltonian. As explained in the Introduction, the second
and third terms of the Hamiltonian given by Eq. 共2兲 have theso-called nuclear and electron singularities, respectively. In
the SE given by Eq. 共1兲, the right-hand side 共RHS兲 has no
singularity and therefore the singularities caused by theseterms must be canceled out completely within the lhs of theSE. For example, in the hydrogen atom, the nuclear attractive force must be canceled out by the centrifugal force produced by the kinetic motion of an electron and this cancellation occurs only for the exact wave function. Generally,
from such considerations, we can formulate the nuclear andelectron cusp conditions of the exact wave functions, whichwere first derived by Kato 关25兴. The exact wave function
satisfies these conditions and therefore no singularities areleft in the SE of Eq. 共1兲.
However, for approximate wave functions, such complete
cancellation does not occur and some serious problems occur. For example, we show in Table I the integrals of Hn forthe hydrogen atom over the exact and approximate wave
functions 共Stater type and Gaussian type兲. The exact wave
function always gives the correct value, but the approximate
one gives divergence for n 艌 3. This divergence is due to the
nuclear singularity. Table II shows the integrals of the higher
powers of the electron-electron repulsion operator, 共1 / r12兲n,over the approximate Slater-type and Gaussian-type helium
wave functions with the optimized orbital exponents. Again,
the integrals of 共1 / r12兲n diverge for n 艌 3 for both types oforbitals. This divergence is due to the electron singularity. It
is easy to show that such divergence is general for the integrals of the higher powers of Hamiltonian over approximate
wave functions. An intuitive explanation of this divergence is
as follows. When you make n large for 共1 / r兲n, it becomes
like a well of an infinite depth, so that any integrals of thispotential over approximate wave functions of atoms andmolecules must diverge.
These results imply that the theory involving higher powers of Hamiltonian has difficulties when applied to atoms andmolecules: numerical calculations become impossible when
the basic integrals involved become infinite. The SICI secular equation involves the integrals of third power of Hamiltonian 关7兴 and the SECC wave function involves higher pow-

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HIROSHI NAKATSUJI

TABLE I. Integrals of the higher powers of the Hamiltonian and the scaled Hamiltonian of the hydrogen
atom over the Slater-type and Gaussian-type orbitals.
Slater-typea

Operator Q in
具␺兩Q兩␺典 / 具␺ 兩 ␺典
H
H2
H3
␷H␷
H4
Hg共gH兲2
HgHgH
HgH2
HgH3

Exact 共␣ = 1.0兲Approximate 共␣ = 0.8兲Gaussian-type
␣ = 0.3−0.5
0.25−0.125−⬁0.0625−0.5625−0.25−0.1875
0.09375−0.48
0.4608−⬁−⬁⬁−0.624−0.2624−0.24576⬁−0.4240387
0.4886535−⬁−⬁⬁−0.1402396−0.0711785−0.5396678⬁

H = −1 / 2共d2 / dr2兲 − 1 / r共d / dr兲 − 1 / r, ␷ = −1 / r, g = r.

aers of Hamiltonian 关8兴. Therefore, both SICI and SECC
suffer singularity problems when they are applied straightforwardly. The t-expansion method 关26兴, the connectedmoments method 关27兴, and the surplus function method 关28兴are the methods that suffer similarly the singularity problem.
We note that this singularity problem is inevitable as far as
the exact wave function is expressed like Eq. 共3兲 as a functional of the Hamiltonian applied to an approximate wave
function: the Maclaulin expansion of the function f of Eq.
共3兲 with respect to H always includes the higher powers of
H, which causes the singularity problem.
This singularity problem cannot be circumvented by introducing the finite-nucleus model 关29兴. Further, the finiteness of the nucleus is from a different physical origin and thetheory itself should be valid independent of the finiteness ofthe nucleus. We also note that the singularity problem wasrather vague in the modern quantum-chemistry technology
where everything is expanded with finite basis functions.
There, when you have a Hamiltonian matrix H definedwithin the finite basis space, you can calculate Hn essentiallyto an arbitrary n, though this is impossible if the basis iscomplete, since then 共Hn兲ij = 具i兩Hn兩j典 and the integrals on the

RHS diverge when n 艌 3 as shown above. When the basis set
is far from complete, the calculation may proceed without
much difficulty, but it is simply due to the incompleteness ofthe basis set.
Now, how do we overcome this singularity problem? Otherwise, we cannot proceed further to find a way of calculating the exact wave functions of atoms and molecules. We
have proposed two methods of solution: one was by introducing the inverse Schrödinger equation 关12兴 and the otherwas by introducing the scaled Schrödinger equation 关13兴. In
the case of the ISE, a problem was how to define the inverse
Hamiltonian explicitly, though we have shown that even the
expanded form of the inverse Hamiltonian was useful. On
the other hand, in the case of the SSE reported in the previous communication, we could formulate a general straightforward method of calculating the exact wave function withavoiding the singularity problem. We will explain thismethod in some detail in the next section.

TABLE II. Integrals of several operators over approximate helium wave functions.

where the scaling function g is a function of electron coordinates. It is a multiplicative operator and does not commute
with the Hamiltonian. It is always positive 共or always negative兲 except at the singular points. Even at the singular point
r0, the scaling function satisfies

Wave functiona

Operator

Slater-type
␣opt = Z − 5 / 16

Gaussian-type
␤opt = 0.7670

H−Z / r1 − Z / r2 + 1 / r12
1 / r12
共1 / r12兲2
共1 / r12兲3
共1 / r12兲4−729/ 256−729/ 128135/ 128
243/ 128+⬁+⬁−2.301−4.602
0.988
1.534
+⬁
+⬁

We define the scaled Schrödinger equation 共SSE兲 byg共H − E兲␺ = 0,

lim gV ⫽ 0,r→r0

␺ = ␹共1兲␹共2兲关␣共1兲␤共2兲 − ␤共1兲␣共2兲兴 / 冑2, where Slater-type means␹ = exp共−␣r兲 and Gaussian-type means ␹ = exp共−␤r2兲.
a

B. Scaled Schrödinger equation and related theorems

共13兲

共14兲

where V is the potential operator in the Hamiltonian, in order
not to eliminate the information of the Hamiltonian at the
singular points: the singularity of the potential itself is also
important information of the Hamiltonian, though it causes
the singularity problem. Equation 共14兲 means that 1 / g is amilder function than the potential itself in the limit ofr → r 0.
The SSE is equivalent to the original SE: When the SE
holds, the SSE holds. Conversely, when the SSE holds, we

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GENERAL METHOD OF SOLVING THE…

multiply 1 / g from the left of Eq. 共13兲 and obtain the SE.
This is valid even at the singular points because of Eq. 共14兲.
There are many possibilities in the choice of the g function. A choice is g = 1 / 共−Vne + Vee兲, where Vne and Vee are thenuclear-electron attraction operator and electron-electron repulsion operator, respectively. A different choice is g
= −1 / VneVee or more explicitly the inverse of the product ofeach Coulombic potential. However, this g function is very
complex and makes the ICI wave function very complex.
Further, when one of the product terms in the g function
becomes zero, the other terms can be arbitrary, so that the
restriction may not be complete. When we adopt the free ICI
method described below, the freedom of the choice of the gfunction becomes wider and one choice is g = −1 / Vne+ 1 / Vee or, more explicitly, the sum of all the inverses of the
Coulomb potentials involved in the Hamiltonian. In thiscase, however, the ICI formulation also produces the functions that are singular, so that we have to eliminate suchfunctions from our expansion bases of the free ICI wave
function: the wave function must be integrably finite by its
definition. Another choice is to take g = r / 共r + a兲 with a being
a constant for each Coulomb potential 1 / r. This applies toboth the product form and the sum form explained above.
Actually, the functional form of the scaling function is very
important in the ICI calculations. The choice of the g function affects the easiness of integral evaluations and the convergence speed of the ICI calculations.
The SSE is very flexible as seen from its definition, and so
it should be useful not only for atomic and molecular systems where the potential has Coulomb singularity, but also
for more general cases where the singularity is steeper or lesssteeper than the Coulombic one.
Table I shows the integrals of the operators including bothg and H for the case of hydrogen atom. For the potential V= −1 / r, we have chosen the scaling function g = r. We seethat the integrals of the higher powers of the scaled Hamiltonian, 共gH兲n, exist for both exact and approximate wave
functions even for n 艌 3, in contrast to the higher powers of
the original Hamiltonian H. Note further that the integrals of
the operators of the forms H共gH兲n and H2共gH兲n also exist.
These operators appear when we calculate the Hamiltonian
matrix element and the matrix of the H2 operator, respectively, over the ICI wave functions based on the SSE. So it isgenerally true that the integrals gnHm over ordinary wave
functions exist as long as 0 艋 m − n 艋 2. Similarly, the integrals of the higher powers of the electron-electron repulsion
operator, 共1 / rij兲n, exists if n 艋 2. We will see later that theseproperties are very important in constructing the free ICI
wave functions, as explained below, and calculating the exact
wave functions of atoms and molecules. Note that 具␺兩gn兩␺典
exists since ␺ is exponentially decaying function which decays much faster than the powers of r.
The variational principle given by Eq. 共4兲 and theH-square theorem given by Eq. 共5兲 play very important rolesin the formulation of our theory. We now show that the SSE
also has the variational equation and the H-square theorem as
the original SE. This makes it possible to formulate the ICI
theory based on the SSE, which leads to the ICI theory freefrom the singularity problem. Since the SE and the SSE areequivalent, this ICI theory also leads to the exact wave function.
The variational formula associated to the SSE is writtenfrom the SSE as
具␦␺兩g共H − E兲兩␺典 = 0.

共15兲

For arbitrary ␦␺, this equation is equivalent to the SSE. Thisequation may also be used to calculate the best possible
wave function within the given functional form of ␺. Theenergy E may be defined by
E = 具␺兩gH兩␺/具␺兩g兩␺典,

共16兲

which is called scaled energy and is different from the Ritz
energy,E = 具␺兩H兩␺典/具␺兩␺典.

共17兲

The scaled energy does not have the upper-bound property,
differently from the Ritz energy. For the exact wave function,
both scaled and Ritz energies become equal to the exact energy.
The H-square theorem for the SSE is written as
具␺兩共H − E兲g · g共H − E兲兩␺典 = 0,

共18兲which is valid only for the exact wave function that satisfies
the SSE or SE. The proof is very simple: this equation isrewritten as 兰兩g共H − E兲␺兩2d␶ = 0, whose integrand is always
positive or zero, and therefore, the integrand must be zero inorder that the integral of them is zero, which leads to theSSE. The converse is self-evident. Q.E.D. Another importantH-square theorem of the SSE is
具␺兩共H − E兲g共H − E兲兩␺典 = 0,

共19兲which is valid since g is always positive or zero, so that we
can define 冑g and 兰兩冑g共H − E兲␺兩2d␶ = 0 is satisfied only when冑g共H − E兲␺ = 0, which leads to the SE. Equation 共19兲 is useful for showing the exactness of the ordinary 共Ritz兲 variational solution of the SICI based on the SSE, as shown below.
We can formulate the SICI based on the SSE, just like thatbased on the SE. When a ␺ including only one variable C
satisfiesd␺= g共H − E兲␺ ,
dC

共20兲

it has the structure of the exact wave function. Putting Eq.
共20兲 into the variational formula given by Eq. 共15兲, we obtain
Eq. 共18兲: the variationally best ␺ satisfies the H-square theorem and is therefore exact. In this formulation, we used thevariational principle associated to the SSE, but we can also
use the ordinary variational principle given by Eq. 共4兲. Whenyou put Eq. 共20兲 into the ordinary variational principle given
by Eq. 共4兲, we obtain Eq. 共19兲 which guarantees that thevariational best is exact if ␺ has the structure of the exact
wave function given by Eq. 共20兲. This argument is importantbecause when we have ␺ having the structure of the exact
wave function as defined by Eq. 共20兲, we can calculate its
unknown variable C either with the ordinary variational principle given by Eq. 共4兲 or by the scaled variational principle

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HIROSHI NAKATSUJI

given by Eq. 共18兲. Of course, the ordinary variational principle is more useful, because it has the upper-bound nature
共Ritz兲 but the scaled one does not.
Now we can formulate the simplest ICI theory based on
the SSE. The SICI based on the SSE is defined by the recursion formula

The H-square theorem based on the SSE can be generalized by introducing the partitioned Hamiltonian

␺n+1 = 关1 + Cng共H − En兲兴␺n .

where ND stands for the number of division. The partitioning
may be into kinetic, nuclear-attraction, and electronrepulsion operators of the Hamiltonian given by Eq. 共2兲,
where the nuclear attraction operator may be further divided
into different nuclei or into the singles and doubles terms for
the Hamiltonian given by Eq. 共11兲. The H-square theorem forthe partitioned Hamiltonian is given by a set of the ND equations,

共21兲

It is different from the original one given by Eq. 共7兲 simply
by the existence of the g function which works to eliminate
the singularity problem. By differentiating ␺n+1 with respect
to the single variable Cn, we obtain an equation similar to
Eq. 共20兲 and this equation becomes identical to Eq. 共20兲 atconvergence where everything does not depend on the iteration number n. Thus, at convergence, the SICI based on the
SSE is guaranteed to be exact.
The unknown variable Cn of the SICI can be calculated in
two different ways. One is to use the variational principle
given by Eq. 共15兲 and the other is to use the ordinary variational principle given by Eq. 共4兲. From Eq. 共15兲 we obtain
two-dimensional secular equation
具␺n兩g共H − En+1兲兩␺n典C0,n + 具␺n兩共H − En+1兲g · g共H − En兲⫻兩␺n典C1,n = 0,
具␺n兩共H − En兲g · g共H − En+1兲兩␺n典C0,n + 具␺n兩共H − En兲g · g共H− En+1兲g共H − En兲兩␺n典C1,n = 0,

共22兲

and from Eq. 共4兲 a secular equation
具␺n兩共H − En+1兲兩␺n典C0,n + 具␺n兩共H − En+1兲g共H − En兲兩␺n典C1,n
= 0,

ND

H = 兺 HI ,具␺兩共H − E兲g · g共HI − EI兲兩␺典 = 0,

具␺兩共H − E兲g共HI − EI兲兩␺典 = 0,

␺ = exp关Cg共H − E兲兴␺0 .

共24兲

It is easy to show that this ␺ satisfies Eq. 共20兲 and is therefore exact. It gives a compact expression of the exact wave
function. Because of the existence of the g function, this
SECC is free from the singularity problem. It is known, however, that the coefficient C is infinitely large for the exact
ground state and this makes the usage of the SECC lesspopular. However, we have formulated the usage of the
SECC wave function by expanding it in finite terms and theresults of applications were encouraging 关8兴. In this paper
our main concern is the ICI theory and therefore detailed
discussions on the SECC wave functions based on the SSE
are postponed to the future papers.

共27兲

with I = 1 , . . . , ND again, because we can define 冑g from thedefinition of the g function. We can show that when either ofthese partitioned H-square theorems holds for all the ND
equations, this ␺ is exact. The proof is straightforward and so
is not given here, except we note that we use the variational
equation given by Eq. 共15兲 for the proof of Eq. 共26兲 and Eq.
共4兲 for the proof of Eq. 共27兲. You may refer to the proofs
given in Refs. 关6,7兴.
Based on the partitioned H-square theorem, we can generalize the ICI theory so as to include the ND variables,

冋

册

ND

␺n+1 = 1 + 兺 CI,ng共HI − EI,n兲 ␺n ,共23兲

where the coefficient Cn of Eq. 共21兲 is given by Cn= C1,n / C0,n: because of the normalization relation, we have
only one variable. Referring to Table I, we see that all the
integrals involved in Eqs. 共22兲 and 共23兲 exist. As describedabove, the solution of Eq. 共23兲 satisfies the Ritz variational
principle and is therefore more favorable in actual calculations than that of Eq. 共22兲.
We can also formulate the simplest ECC theory based on
the SSE as

共26兲

with I = 1 , . . . , ND. EI is defined by 具␺兩g共HI − EI兲兩␺典 = 0. Similarly to Eq. 共19兲, we also have another H-square theorem

具␺n兩共H − En兲g共H − En+1兲兩␺n典C0,n + 具␺n兩共H − En兲g共H− En+1兲g共H − En兲兩␺n典C1,n = 0,

共25兲

I=1

I

共28兲

which was called ICIND method 关7兴. When this ICIND is
calculated with either of the variational equations, Eqs. 共15兲and 共4兲, it gives the exact wave function at convergence. In
the proof, we use either combination, Eqs. 共15兲 and 共26兲 or
Eqs. 共4兲 and 共27兲.
The calculation of the unknown variables 兵CI,n其 in the
ICIND wave function given by Eq. 共28兲 is done either withthe variational principle given by Eq. 共15兲 or with the ordinary variational principle given by Eq. 共4兲. The secular equation obtained with the variational principle given by Eq. 共15兲is written as
具␺n兩g共H − En+1兲兩␺n典C0,n + 兺 具␺n兩g共H − En+1兲 · g共HI − EI,n兲I⫻兩␺n典CI,n = 0,
具␺n兩共HJ − EJ,n兲g · g共H − En+1兲兩␺n典C0,n + 兺 具␺n兩共HJ
I− EJ,n兲g · g共H − En+1兲g共HI − EI,n兲兩␺n典CI,n = 0, 共29兲and the secular equation derived from the ordinary variational principle given by Eq. 共4兲 is written as

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GENERAL METHOD OF SOLVING THE…

具␺n兩共H − En+1兲兩␺n典C0,n + 兺 具␺n兩共H − En+1兲 · g共HI − EI,n兲I⫻兩␺n典CI,n = 0,
具␺n兩共HJ − EJ,n兲g共H − En+1兲兩␺n典C0,n + 兺 具␺n兩共HJ − EJ,n兲
I⫻g共H − En+1兲g共HI − EI,n兲兩␺n典CI,n = 0,

共30兲both for all J 共J = 1 , . . . , ND兲. The coefficient CI,n of Eq. 共28兲is related to the coefficients CI,n of Eq. 共29兲 or 共30兲 by CI,n
= CI,n / C0,n. When we do not partition the Hamiltonian, Eqs.
共29兲 and 共30兲 reduce to Eqs. 共22兲 and 共23兲, respectively, of
the SICI case. In actual calculations, Eq. 共30兲 is more convenient, since the solution satisfies the upper-bound nature of
the Ritz variational principle.
Here, we explain how we perform analytical calculations
with the ICI method. We take the SICI method given by Eq.
共21兲 for example. The initial function ␺0 may be HartreeFock wave function, etc., which is anyway some analytical
function. For analytical calculations, we use the Hamiltonian
given by Eq. 共2兲, which is the sum of the kinetic operator andthe potential operators. The kinetic operator is a sum of thedifferentiation operators with respect to the electron coordinates and the differentiations are always possible for any
kinds of analytical functions. There are many mathematicalprograms available that perform such differentiations easily,
like MAPLE, MATHEMATICA, REDUCE, etc. The potential operators and the g function are simply multiplicative functions.
Thus the result of applying the SICI operator to ␺0 is ananalytical function with the unknown variable C0. Repeating
iterations, we always obtain the analytical functions withsome unknown coefficients. In some general choice of the gfunction, the resultant function may include the functions
that have singularities causing the divergence of the integrals: such functions should be eliminated since they do notsatisfy the integrability condition that the physical wave
function must satisfy. Finally, the variable coefficients involved are calculated with the variational principle.
The ICI method we routinely use in our laboratory forcalculating the analytical exact wave function is not the SICIor ICIND given above. These ICI methods are characterized
by the following two features. 共1兲 They generate the functions that converge iteratively to the exact wave function. 共2兲
The number of the variables is small and fixed to one or ND
throughout the calculations. In our routine calculations, we
relax the second property. Namely, we collect all the independent functions 兵␾k其共n兲, k = 1 , . . . , M n, included in the nthiteration function of the SICI given by Eq. 共21兲 and use themfor expanding our wave function—namely,Mn␺n+1 = 兺 ck,n␾k ,

共31兲

k=1

where M n is the number of the independent functions included in 兵␾k其共n兲. The coefficients 兵ck,n其, k = 1 , . . . , M n, arecalculated with the variational principle 共the ordinary Ritz
variational principle is easier to use兲. Since the functions ␾k
are generated with the scaled Hamiltonian, we no longer

have the singularity problem. Even if we may have somesingular functions due to a general use of the g functions, we
simply throw such functions out since they do not satisfy thephysical condition 共Q condition兲 for the wave function. We
call this method the free ICI method. Since the ICI formulation guarantees to reach the exact wave function, the free ICI
method gives the best possible wave function, in a variational sense, at each iteration. When M n is not so large, weneed not to keep it to ND. Further, in the free ICI method, noaccumulation of errors occurs, since no variables of the earlier iteration step are used in the later iteration step. We canstart nth iteration step without performing the earlier n − 1
iterations. The free ICI method is very flexible. It makes
many variants possible.
Thus, at sufficiently large n, we have the analytical function in the form of Eq. 共31兲 that has the structure of the exact
wave function: when the variables involved are determinedby the variational principle, it gives the exact wave function.
We can reach any accuracy by adjusting the iteration numbern.
We have applied the ICI method, the free ICI method inparticular, to atomic and molecular systems. In order to confirm the accuracy and the usefulness of the method, we haveapplied it to the hydrogen atom, helium atom, and hydrogenmolecule, for which the exact or very accurate wave functions are already known. We have further applied it to manyelectron atoms: lithium, beryllium, and boron 共three- to fiveelectron atoms兲 and He- H+, a two-electron molecule. We
will summarize these calculations separately in the literatures
关21,22,30兴
In the present paper, we show the applications to the hydrogen atom, to the Hooke’s atom, and to the helium atom.
They are the examples of the systems including the nuclear
singularity, the electron singularity, and the both, respectively.
III. SIMPLE APPLICATIONS
A. Hydrogen atom

The exact solution of the hydrogen atom is a classicalsubject of quantum chemistry. The Hamiltonian for the sstate of the hydrogen atom isH=−1 ⳵2 1 ⳵ Z− ,−2 ⳵r2 r ⳵r r

共32兲and the exact wave function and energy are ␺ = exp共−Zr兲 andE = −0.5 a.u. for Z = 1. As an ICI problem, the hydrogen atom
is an example of the nuclear singularity and therefore, evenfor this simplest example, the ICI approach fails without the
introduction of the SSE and ISE. The solution with introducing the ISE was given previously 关12兴, and here we give thesolution with introducing the SSE.
As a simplest choice of the g function, we choose g = r,which satisfies the condition given by Eq. 共14兲. The initial
function ␺0 is chosen to be ␺0 = exp共−␣r兲 with ␣ = 1.5. ␣= 1.0 is the exact wave function. We perform three different
ICI—namely, SICI 共ND = 1兲, ICIND 共ND = 3兲, and free ICI. In
ICIND 共ND = 3兲, we have divided the Hamiltonian given by

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HIROSHI NAKATSUJI

TABLE III. ICI calculation of hydrogen atom based on the SSE.a
SICI 共ND= 1兲nRitz
energyScaledenergy

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30

−0.375−0.375−0.465594827−0.481256589−0.488165311−0.492183091−0.494625230−0.496147453−0.497154350−0.497839922−0.498324376−0.498674837−0.498934961−0.499131605−0.499283108−0.499401536−0.499495476−0.499570866−0.499632078−0.499682257−0.499723785−0.499758430−0.499787561−0.499812222−0.499833240−0.499851257−0.499866789−0.499880247−0.499891966−0.499902215−0.499911217−0.625−0.375−0.411923413−0.434488719−0.448981506−0.458915958−0.465959942−0.471256973−0.475303857−0.478500137−0.481062407−0.483160593−0.484899446−0.486362182−0.487604403−0.488671137−0.489594209−0.490399870−0.491107471−0.491733231−0.492289504−0.492786793−0.493233307−0.493636113−0.494000854−0.494332439−0.494634860−0.494911621−0.495165616−0.495399407−0.495615134

ICIND 共ND= 3兲
n
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Ritz
energyScaledenergy−0.375−0.375−0.497053531−0.499652721−0.499974617−0.499989900−0.499998499−0.499999309−0.499999805−0.499999903−0.499999961−0.499999979−0.499999990−0.499999994−0.499999997−0.499999998−0.499999999−0.499999999−0.500000000−0.625−0.375−0.516537938−0.494662663−0.499580437−0.499133267−0.499699622−0.499755042−0.499859453−0.499897917−0.499930983−0.499948993−0.499963105−0.499971758−0.499978750−0.499983211−0.499986999−0.499989465−0.499991650−0.499993097−0.499994420

Free ICI
n

M

0
1
2
3
4
5
6
7
8

1
2
3
4
5
6
7
8
9

Ritz
energyScaledenergy−0.375−0.491025404−0.465594827−0.499954132−0.499997229−0.499999844−0.499999992−0.500000000−0.625−0.512259526−0.501470244−0.500144830−0.500011697−0.500000825−0.500000053−0.500000003−0.500000000

an: iteration number. M: number of independent functions.

Eq. 共32兲 into three terms. The number of variables is unity in
SICI, 3 in ICIND 共ND = 3兲, and increases as the iteration
proceeds in the ICI. We calculated both Ritz energy andscaled energy given by Eqs. 共16兲 and 共17兲, respectively, andthe results are given in Table III.
We expect from Table I that the singularity problem
should not occur in the present method and actually, as seenfrom Table III, we get a smooth convergence to the exact
energy in all cases, though the convergence rate is different
among the different ICI methods and between the different
definitions of the energy. For the SICI case, we have only
one variable and, therefore, the convergence rate is very
slow. When we use three variables in ICIND 共ND = 3兲, theconvergence becomes much faster. When we use the free ICI
method, the number of the variables increases as the iteration
proceeds, one by one in this case, as the iteration number

increases, and the convergence is fastest among the three ICI
methods: the Ritz energy converges after seven iterations andthe scaled energy by eight iterations to the exact energy up tonine decimal figures. Between the Ritz and scaled energies,
the Ritz energy always converges faster and satisfies theupper-bound nature. The scaled energy does not necessarily
satisfy the upper-bound nature: it may be even lower than thetrue energy.
The results of Table III indicate the usefulness of the free
ICI method with the Ritz energy. The ICIND 共ND = 3兲method may also be useful because in this case the numberof the variables is always 3 or ND.
Starting from the initial function ␺0 = exp共−1.5r兲, the ICI
method with g = r improves the wave function order by orderas the iteration proceeds. Actually, the wave function at nthncir i兲. Since the exiteration is written as ␺n = exp共−1.5r兲共兺i=0062110-8

PHYSICAL REVIEW A 72, 062110 共2005兲

GENERAL METHOD OF SOLVING THE…

act wave function is exp共−1.0r兲, the ICI correction term is
nothing else but the variational approximation to the expansion, exp共0.5r兲 = 兺i=0cir i, with ci = 0.5i / n!. The free ICI wave
function at n = 7 was 共c0 , c1 , c2 , c3 , c4 , c5 , c6 , c7兲 = 共1.000 000,
0.500 262, 0.124 126, 0.022 132, 0.001 656, 0.000 621,
−0.000 049, 0.000 008兲. The sequence given by ci = 0.5i / n! is
共1.000 000, 0.500 000, 0.125 000, 0.020 833, 0.000 260,0.000 022, 0.000 002兲. Since the energy is correct to ninedecimal figures, the wave function would be correct roughly
to four decimal figures, in accordance with the Eckert theorem. At higher iterations, we have actually confirmed that thefirst eight coefficients of the free ICI wave function agree
exactly with the above numbers of the mathematical sequence. For general atomic and molecular systems, the exact
wave functions cannot be written with single analytical functions, so that their best analytical expressions would be in ananalytical expansion form as given in this example.
In comparison with the usage of the inverse SE reported
in the previous paper 关12兴, the calculation reported here ismore straightforward. Though we could do analytical calculations based on the ISE for the hydrogen atom with the
expanded inverse Hamiltonian and the results were rather
satisfactory, it is difficult to generalize such method to manyelectron atoms and molecules, but the present method basedon the SSE is straightforwardly applicable to any atoms andmolecules.
B. Hooke’s atom

Hooke’s atom 关31兴 is not a real atom, but an imaginative
two-“electron” atom whose attractive potential is harmonic
and the inter-“electron” repulsion is Coulombic; namely, its
Hamiltonian is given by11k.
H = − 共⌬1 + ⌬2兲 + 共兩r1兩2 + 兩r2兩2兲 +22
兩r1 − r2兩2

共33兲

The harmonic potential is nonsingular, but the Coulombic
potential is singular 关32兴. We consider Hooke’s atom whose
Hooke’s harmonic constant is k = 1 / 4. A remarkable character of Hooke’s atom is that its exact solution is known. For
the above Hamiltonian, the exact solution is冋1
␺ = exp − 共r21 + r22兲
4

册冉 冊
1+

r12
2

TABLE IV. ICI energy and wave function for Hooke’s twoelectron atom.
Dimension

Energy 共a.u.兲

⌬␺a

0
1
2

1
2
2

2.064189584
2.0
2.0

1
+r12 / 20␺ = ␺0 ⌬␺.

a

C. Helium atom

The helium atom has both nuclear and electron singularities and therefore is a very interesting example. There are a
lot of calculations in the literature with various types of functions, like Hylleraas 关33兴, Kinosita 关34兴, Pekeris 关35兴,
Thakkar and Koga 关36兴, and others 关37兴. In particular, recently Schwartz 关38兴 has reported very accurate calculations,giving the energy correct up to 35 decimal figures. Thoughthese wave functions were cleverly formulated, the physics
behind was not necessarily clear, except for the first few
dominant terms. For general atomic and molecular systems,
it is difficult to estimate the functional form of the exact
wave function only by intuition as cleverly as in the heliumatom case. In the present ICI formalism, the wave function is
automatically generated in an analytical form, once wechoose the initial function ␺0 and the scaling factor g, and it
is guaranteed to approach the exact wave function. The purpose of the calculation presented here is to give a roughunderstanding of the free ICI method applied to helium
atom: we start from a rather simple initial function ␺0 using
again a simple choice of the g function. We examine what
wave function the ICI method generates and how different it
is from the existing wave functions. The state-of-the-art calculations of this atom with the ICI method will be publishedin a forthcoming paper 关22兴.
The calculations are performed by using the coordinate
defined by s = r1 + r2, t = −r1 + r2, and u = r12 with the range ofthe variables s 艌 u 艌 兩t兩. With this coordinate, the Hamiltonian of the helium atom is written as
H=−

共34兲

and the exact energy is E = 2.0 a.u.
We apply the ICI method to the above Hooke’s atom. We
take our initial guess as ␺0 = exp关− 41 共r21 + r22兲兴, an eigenfunction when the “electron” repulsion term in Eq. 共33兲 is missing, and our g factor as g = r12: the g factor is necessary only
for the electron repulsion operator. The free ICI result issummarized in Table IV. We see that the result converges tothe exact solution at second iteration. There, both the energy
and wave function become equal to the exact ones. Even if
we perform further iterations, the correction is identicallyzero for both the energy and wave function, as it must be so.
Thus the ICI method has given the exact solution of Hooke’s
atom.

Iteration

−

⳵2
⳵2
⳵2
t u2 − s2 ⳵2
s u2 − t2 ⳵2
−
2
−
−
−
2
⳵s2 ⳵t2 ⳵u2
u s2 − t2 ⳵s ⳵u
u t2 − s2 ⳵u ⳵t
4t ⳵
4sZ
1
4s ⳵ 2 ⳵− 2 2 − 2 2+ ,−2s − t ⳵s u ⳵u t − s ⳵t s − t
u
2

共35兲

where the last two terms represent the potential operator
共nuclear attraction and electron repulsion兲 and the otherterms represent the kinetic operator. The initial function wechose is a primitive one: ␺0 = exp共−␣s兲, with ␣ being a variational parameter. The scaling function is g = 共VneVee兲−1= u共s2 − t2兲 / s to prevent the nuclear and electron singularity
problem.
Table V summarizes the ground-state energy of the helium atom calculated with the free ICI method. As the iteration proceeds, the number of the independent functions generated by the free ICI method increases and the Ritz energy
approaches the best variational value, −2.903 724 377 a.u.,reported in the literature 关38兴. The chemical accuracy was

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PHYSICAL REVIEW A 72, 062110 共2005兲

HIROSHI NAKATSUJI
TABLE V. Free ICI calculation of the helium atom with ␺0 andg given in the text.
Iteration n

M na

Ritz energy

Optimal ␣

0
1
2
3
4
5
6
7
Best valueb

1
6
26
74
159
291
481
738

−2.847 656 250−2.901 577 012−2.903 708 675−2.903 723 901−2.903 724 347−2.903 724 373−2.903 724 376−2.903 724 377−2.903 724 377

1.68751.67281.88032.03302.19982.33072.4862
共2.68兲

a

Number of independent functions.
Reference 关38兴.

bobtained already at the second iteration with 26 functions
共the energy is correct to four decimal figures兲. At the fourth
iteration, the free ICI energy is correct to seven decimal figures and at the seventh iteration it is correct to nine decimal
figures. Clearly, by continuing the iteration, we should be
able to improve our wave function to any order. This is certainly the behavior we expected from the proof of the theory.
We particularly want to note that these results were obtained
automatically by the free ICI algorithm described above.
The present helium wave function generated automatically by the free ICI theory was different from those existingin the literature. It was the Hylleraas-type wave function
written as

␺=

冉兺

l,m,n

冊

clmnslt2mun exp共− ␣s兲,

共36兲

but was different from the original Hylleraas wave functionin that our l ran both positive and negative regions, while theoriginal l was only positive as well as m and n. Our wave
function was closer to the Kinoshita one though different inhigher-order terms. Actually, the functional form of the ICI
wave function depends on the choices of the zeroth-orderwave function ␺0 and the g factor. The Hylleraas expansion
actually corresponds to choosing g = u共s2 − t2兲, differently
from the present one. This implies that a different convergence would be obtained by a different choice of these functions. In this sense, the examinations of different functions in
the description of the helium wave function given by
Thakkar and Koga 关36兴 are useful. More detailed examinations of the choice of ␺0 and g in the ICI formalism of thehelium atom will be given in a forthcoming paper 关22兴.

IV. CONCLUDING REMARKSIn modern quantum chemistry, one first prepares somebasis functions by empiricism, therefrom a set of orthonormalized orbitals 共Hartree-Fock orbitals mostly兲, and then expands everything with these set of orbitals. The Hamiltonian
is rewritten in a second-quantized form and the quantumchemical calculations are transformed into matrix and vector

manipurations. Ultimately, the best theory is the full-CI
method. However, due to the incompleteness of the basis
functions first prepared, the full-CI solutions are usually far
from the true analytical solutions of the SE. When we enlarge the basis function to make it closer to be complete, thefull-CI dimension increases formidably and, therefore, thisroute of accurate quantum chemistry must be reconsidered.
We have reported previously 关6–11兴 the methodologies toreduce the number of variables that are necessary to obtainthe full-CI solutions into general singles and doubles 共GSD兲
number 关6兴 and even to unity 关7兴 and showed that their actual
performances were certainly good 关10,11兴. However, even inthese theories, we had to handle full-CI number of configurations and this made these methods still not very useful.
In the previous communication 关13兴, we proposed a general method of solving the Schrödinger equation in an analytical expansion form and the details of the proposed
method are presented in this paper. The method has alreadybeen extended to solve the relativistic Dirac and DiracCoulomb equations in an analytical form and preliminary
calculations were encouraging 关23兴. In these methodologies,
the strategy is very different from that of the modern quantum chemistry described above. We first prepare an initial
function ␺0 and choose an appropriate g function. Then, thefree ICI method automatically generates the most appropriate
basis functions 共complement functions兲 for the system byusing the Hamiltonian of the system. The Hamiltonian includes all the information of the system. Further, the theoremof the ICI guarntees that the resultant wave function has the
structure of the exact wave function as the iteration proceeds.
For efficient calculations of this step, we need a symbolic
manipulation mathematical program like MAPLE and MATHEMATICA. This point is quite different from traditional modern quantum-chemistry methodologies. Finally, the unknown
variables included in the free ICI wave function are determined by the variational principle or by some variants
thereof.
The good performance of the calculations reported here
and carried out in the author’s laboratory 关13,21–23兴 implies
a high potentiality of the proposed new methodology. The
present ICI formalism based on the SSE may become a basisof the general systematic method of calculating the exact
wave functions of atoms and molecules in an analytical expansion form. It was remarkable that the present ICI wave
functions were different from the existing ones. Hylleraas
关33兴, James-Coolidge 关39兴, Kolos 关40兴, and others wereclever enough to imagine the exact wave functions of helium
and hydrogen molecule only with intuitions. But for generalmolecules, it is difficult to estimate the analytical form of theexact wave function only by intuition, but the present theory
makes it possible to generate it automatically, since the differentiations 共involved in the Hamiltonian兲 are always possible and, furthermore, the integrals in the secular equation
step have no singularity problem. It is guaranteed that atconvergence we will get the exact wave function.
A different description of the wave function is obtained,
depending on the different choice of the initial function ␺0
and the g function. Since the exact wave function is a uniquereality, one may feel this to be strange. But this is something
like to using different complete functional sets for describing

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GENERAL METHOD OF SOLVING THE…

one entity. Many possibilities exist, but all the descriptions
are on a single reality: the exact wave function. So one description may be more efficient than the other. Actually, we
will show in the succeeding paper that different choices of ␺0
and g give finally the same result, but the convergence
speeds are different. It is impossible to write up the exact
wave function of many-electron atoms and molecules by asingle analytical function 共without including operators兲 except for one-particle cases, so that the best analytical expression would be in an analytical expansion form.
Since applying the Hamiltonian to some function means aseries of differentiations and multiplications, it is always
possible to obtain the free ICI wave function in the form of
Eq. 共31兲. Further, at convergence, this wave function is guaranteed to have the structure of the exact wave function: whenthe variables 兵ck,n其 are determined by the variational principle, it gives the exact wave function. In our theory, theprocess of forming the functions given by Eq. 共31兲 is rather
easy when a symbolic mathematical program like MAPLE isused. Usually, the most time-consuming step is a step for thevariational calculations of the coefficiens 兵ck,n其. The best procedure is to solve the secular equation by calculating analytically the Hamiltonian and overlap integrals over the functions 兵␾k其共n兲. We used this method in the present andsucceeding papers. However, different from differentiations,
integrations are not always possible for any functions. This
gives some restrictions on the choices of the g function and␺0. For general molecular systems, this problem becomes
more serious and we have to introduce different procedures.
Though Slater-type functions were used in our former calculations, Gaussian-type functions would also be useful for ␺0.
The Hamiltonian operator and the g function do not change
the functional form so that the types of functions are determined on the choice of ␺0. The integrations over theGaussian-type functions are easier than for the Slater-typefunctions. This merit may make up the well-known deficiencies of the Gaussian type functions on the cusp properties. In
principle, the cusp properties are improved even with theGaussians when many 共infinite兲 Gaussians are used. Anothergeneral procedure may be provided by the Monte Carlo–typeapproach 关41兴. A combination of the local energy procedureand the Monte Carlo sampling method may provide a usefultool. We note that our ICI wave functions should potentiallygive very good local energies, so that we can expect small
mean-square deviations 共variances兲 in the Monte Carlo integration processes. This is a key feature of the ICI wave function. Studies along these lines are now in progress in ourlaboratory.
Though we have concentrated in this paper only on the
ground state, the present theory is applicable also to calculations of the excited states. They are calculated as higher solutions of the same secular equation. For efficient calculations, some considerations of the choice of initial function ␺0
are necessary. For example, we may take a linear combination of several functions having the exponents covering both
ground and excited states under study.
We have recently extended the present methodology to
the relativistic case 关23兴. We have developed a general theoryof analytically solving the exact solutions of the Dirac andDirac-Coulomb equations. There, the ICI formalism was
similar to the present nonrelativistic case. A new problem
was variational collapse, which commonly occurs in the relativistic variational calculations. However, the exactness of
the ICI formalism has shed light on this problem and aninteresting reader may refer to Ref. 关23兴.
In separate papers 关21,22,30兴, we are going to summarize
the results of the applications of the present theory to the
many-electron atoms He, Li, Be, and B and to two-electron
molecules H2 and He- H+. We are also going to summarize
the results for Hooke’s atom in a more general context thanthat given in this paper.

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ACKNOWLEDGMENTS

The author thanks H. Nakashima and T. Kurokawa forsome helpful discussions. This research was supported by agrant for Creative Scientific Research from the Ministry of
Education, Science, Culture, and Sports of Japan.

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