VOLUME 93, N UMBER 3

week ending
16 JULY 2004

PHYSICA L R EVIEW LET T ERS

Scaled Schrödinger Equation and the Exact Wave Function
Hiroshi Nakatsuji
Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering, Kyoto University,
Kyoto-Daigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan
Fukui Institute for Fundamental Chemistry, Kyoto University, 34-4 Takano-Nishihiraki-cho, Sakyo-ku, Kyoto 606-8103, Japan(Received 27 September 2003; published 16 July 2004)
We propose the scaled Schrödinger equation and the related principles, and construct a general
method of calculating the exact wave functions of atoms and molecules in analytical forms. The nuclear
and electron singularity problems no longer occur. Test applications to hydrogen atom, helium atom,and hydrogen molecule are satisfactory, implying a high potentiality of the proposed method.
DOI: 10.1103/PhysRevLett.93.030403

PACS numbers: 03.65.Ca, 03.65.Ge, 31.10.+z, 31.15.–p

To develop a general systematic method of solving the
Schrödinger equation (SE) and the corresponding relativistic equation is of central importance in theoreticalphysics and chemistry. So far, there was no general
method of solving the SE in an analytical form. Only
full configuration interaction (CI) is a practical method of
calculating the ‘‘exact’’ wave function, but the number ofthe variables involved is astronomical and therefore theapplication is limited. Furthermore, the full CI solutions
are actually far from the solutions of the SE, because thebasis sets are usually far from complete. In our series of
papers [1–6], we have studied a general method of solving
the SE by investigating the structure of the exact wave
function. Here, we introduce a variant of the SE andpropose a general method of calculating the exact wave
functions of atoms and molecules in analytical forms.
Test applications are given to some elementary systems.
In the nonrelativistic Born-Oppenheimer approximation, the atomic and molecular system is defined by the
Hamiltonian
X 1
XX
X
H
 i 
ZA =rAi  1=rij
(1)
2
i
i A
i>j

wave function and the H-square equation is valid only forthe exact wave function. From this criterion, we haveproposed the iterative CI (ICI) method and the simplest
extreme (or extended) coupled-cluster (SECC) method as
methods of constructing the exact wave function.
that we divide our H into ND parts, H 
PNSupposeDH,
anddefine the variable operator S by S 
I
PI1
ND
C
H
with
ND variables fCI g. Then, the ICI wave
I1 I I
function is defined by the recurrencen1  1  Sn  n :(4)

The variables are optimized by the variational principle,
and, at convergence, n is proved to be exact. Since eachstep of the ICI is variational, its energy converges monotonically to the exact value. When ND  1, it is thesimplest ICI (SICI). The ECC wave function is defined by expS 0 :(5)where H 1 H  HH 1  1. Note that in the ISE, H is
actually the positive Hamiltonian [5].
We summarize briefly the past results [1–6] to introduce our background. A condition for the wave function
to have the structure of the exact wave function is expressed as follows. When the unknown variables included
in are optimized by the variational principle, and when
the resultant satisfies the H-square equation [1–3], thenthis has the structure of the exact wave function. This isbecause the variational principle gives the best possible

The simplest one (SECC) for ND  1 is proved to be
exact, but the optimal C would be minus infinite from adifferent argument. On the other hand, the general ECC(ND  1) is not guaranteed to be exact [1,3], though it
may happen to become exact because of its highly nonlinear nature [3]. This was in contrast to the conjecture
given by Nooijen [7] and Piecuch et al. [8] (CCGSD is a
special case of ECCND for second-quantized Hamiltonian). Descriptions of the excited states with the ICI
and ECC methods were given in some detais in Refs. [2,4].
The above arguments are also valid for the inverse
Hamiltonian case [5].
Singularities.—The atomic and molecular Hamiltonians have nuclear and electron singularities in thesecond and third terms, respectively, of Eq. (1). In the
SE of Eq. (2), the right hand side has no singularities, and,
therefore, these singularities caused by the Hamiltonian
must be canceled out within the left hand side of the SE.
(For example, in a hydrogen atom, the attractive Coulomb
force is canceled by the centrifugal force.) This consideration leads easily to the nuclear and electron cusp conditions first derived by Kato [9]. The exact wave functions

030403-1

 2004 The American Physical Society

and is described by the exact wave function that is asolution of the SE
H E ;(2)or the inverse Schrödinger equation (ISE) [5]H 1  E1 ;(3)0031-9007=04=93(3)=030403(4)$22.50

030403-1

PHYSICA L R EVIEW LET T ERSVOLUME 93, N UMBER 3

satisfy these conditions, and, therefore, no singularities
exist in the SE. However, for approximate wave functions,this complete cancellation of the singularities in H does
not occur and some problems are caused. For example, inTable I, the integrals of the nth power of the Hamiltonianof the hydrogen atom give correct values for the exact
wave function, but diverge for the approximate wave
functions for n larger than 3.
In the ICI and ECC formalisms, there appear the
integrals over H n with n  3, so that these theories
have difficulties for atoms and molecules. This is true forany theories that involve such higher powers of Hamiltonian [10 –12]. But, when we use H 1 instead of H, sucha difficulty does not occur [5]: a problem there was how towrite the inverse Hamiltonian in a closed form, thougheven the expanded form worked well [5]. We show in thisLetter an approach that is free from such a difficulty.
Scaled Schrödinger equation.—We introduce the scaled
Schrödinger equation (SSE) bygH  E  0;

(6)

which is equivalent to the SE. The function g, called thescaling function, is a function of electron coordinates. It
is a multiplicative operator and generally does not commute with the Hamiltonian. It is always positive (ornegative) and nonzero except at the singular points.
Even at the singular point r0 ,(7)

lim gV  0;

r!r0

where V is the potential operator in the Hamiltonian, inorder not to eliminate the information of the Hamiltonianat the singular points: the singularity itself is also important information of the Hamiltonian. In other words,
if we introduce 1=g, its singularity at r0 is less steep thanor equally steep as that of the potential in the
Hamiltonian.
The proof of the equivalence between the SE and SSE is
almost self-evident. If SE is valid, SSE is valid.
Contrarily, if SSE is valid, by multiplying 1=g from theweek ending
16 JULY 2004

left of the SSE, we obtain the SE. This is valid even at the
singular points because of Eq. (7).
A choice of g is g  1=vne  vee , where vne andvee are the nuclear-electron attraction operator andelectron-electron repulsion operator, respectively. A different choice is g  1=vne vee .
Variational principle and square theorem for theSSE.—The variational principle and the square theorem
are important for developing the theory of the exact wave
function. From Eq. (6), the variational formula for the
SSE is written ash jgH  Ej i  0(8)

for arbitrary  . This principle may also be used to
calculate the best possible wave function within the givenfunctional form of . The energy E may be defined by
E  h jgHj i=h jgj i, which is called scaled energy
and is different from the Ritz energy, E  h jHj i=
h j i. The scaled energy does not necessarily have thestationary property, different from the Ritz energy. For
the exact wave function, both scaled and Ritz energies
become equal to the exact energy.
The square theorem for the SSE is written as
h jH  Eg gH  Ej i  0;(9)which is valid only for the exact wave function thatsatisfies the SSE or SE. The R
proof is very simple.
This equation is rewritten as jgH  E j2 d  0,
whose integrand is always positive or zero, and, therefore,
the integrand must be zero in order that the sum of themis zero, which leads to the SSE. The converse is selfevident. (Q.E.D.)
If is chosen such that its variation put into the variational principle, Eq. (8), leads to the square theorem,
Eq. (9), then this has the structure of the exact wave
function, since this is guaranteed to become exact after
variational calculation.
Simplest ICI and simplest ECC theories based on the
SSE.—We can formulate the ICI and ECC theories
associated with the SSE. From the above argument, theTABLE I. Integrals of the higher powers of the Hamiltonian and the scaled Hamiltonian of the hydrogen atom over the Slatertype and Gaussian-type orbitals.
Operator Q ina,b,ch jQj i=h j i

Exact (  1:0)

Approximate (  0:8)

Gaussian-type
  0:3

H
H2
H3
H
Hg gH2
HgHgH
HgH 2
HgH 3

0:5
0.25
0:125
1
0:5625
0:25
0:1875
0.093 75

0:48
0.4608
1
1
0:624
0:2624
0:245 76
1

0:424 038 7
0.488 653 5
1
1
0:140 239 6
0:071 178 5
0:539 667 81aSlater-type

2

d
1 d
1 b
1 c
H   12 dr
2  r dr  r .    r . g  r.

030403-2

030403-2

week ending
16 JULY 2004

PHYSICA L R EVIEW LET T ERSVOLUME 93, N UMBER 3
simplest ECC (SECC)

 expCgH  E 0 ;(10)

and the simplest ICI (SICI) defined by the recursion
formulan1  1  Cn gH  En  n

(11)

are exact when their variables are determined by thevariational principle.
Square theorem for the partitioned Hamiltonian andthe ICIND method.—The square theorem
P Dis generalizedwith the partitioned Hamiltonian H  N
I1 HI . Namely,when the partitioned square theorem,h jH  Eg gHI  EI j i  0;

(12)

h n jgH  En1 j n iC0;n Xholds for all I (I  1; . . . ; ND ), then this is exact. Here,EI is defined by h jgHI  EI j i  0. The proof isstraightforward and so not given here.
Based on the partitioned square theorem, the ICI
theory that involves ND variables (ICIND) is formulated.
We introduce the S operator by
Sn 

ND
X

CI;n gHI  EI;n 

(13)

I

and define the ICI wave function by using Eq. (4) again(note that the definition of the S operator is different).
When this ICIND is calculated with Eq. (8), it gives theexact wave function at convergence.
The secular equation at nth iteration is written as

h n jgH  En1  gHI  EI;n j n iCI;n  0(14)

h n jHJ  EJ;n g gH  En1 gHI  EI;n j n iCI;n  0

(15)

I

and
h n jHJ  EJ;n g gH  En1 j n iC0;n XI

for all J (J  1; . . . ; ND ). When we do not partition the
Hamiltonian, it is SICI, and we see from Table I that all
the integrals involved exist.
Free ICI Method.—The ICI method for calculating theexact wave function is characterized by two features. (1) It
generates the functions that converge iteratively to theexact wave function. (2) The number of the variables issmall and fixed to ND throughout the calculations.
Let us now relax the second property. Namely, we use
all the independent functions fk gn included in the nthiteration function of the SICI given by Eq. (11) for expanding our wave function, namely,
n1 

Mn
X

ck;n k ;

(16)

k1

where Mn is the number of the independent functions. The
coefficient ck;n is calculated by the ordinary Ritz variational principle. Since the functions k are generated withthe scaled Hamiltonian, we no longer have the singularity
problem. We call this method the free ICI method. Since
the ICI formulation is guaranteed to reach the exact wave
function, this free ICI gives the best possible wave function, in a variational sense, at each iteration. When Mn is
not large, we need not keep it to ND . We further note thatin the free ICI, no accumulation of errors occurs, since novariables of the earlier iteration step is used in the later
iteration step. We can start nth iteration step without
performing the earlier n  1 iterations.
We apply the present ICI method based on the SSE tothe analytical calculations of the exact wave functions ofhydrogen atom, helium atom, and hydrogen molecule.
Hydrogen atom.—The Hamiltonian for the s state of@2
1 @Z
the hydrogen atom is H   12 @r
2  r @r  r . The scaling
function is chosen as g  r. We performed the free ICI
030403-3

using 0  expr with   1:5.   1:0 is the exact
wave function. We see from Table I that the singularity
problem does not occur in the present method. Table IIgives the results. As the iteration proceeds, the number ofthe independent functions increases one by one and theRitz energy converges after seven iterations and thescaled energy by eight iterations to the exact energy up
to nine decimal figures.
Helium atom.—The helium atom has both nuclear andelectron singularities and therefore is very interesting.
There are a lot of calculations in the literature withvarious types of functions [13–16]. Though these wave
functions were cleverly formulated, the physics behindthem was not necessarily clear, except for the first few
dominant terms. In the present ICI formalism, the wave
function is automatically generated in an analytical form,
once we choose the initial function 0 and the scaling
factor g, and guaranteed to approach the exact wave
TABLE II. Free ICI calculation based on the sSE for thehydrogen atom starting from 0  exp1:5r.
Iteration

Mn a

Ritz energy

Scaled energy

0
1
2
3
4
5
6
7
8

1
2
3
4
5
6
7
8
9

0:375
0:491 025 404
0:499 316 143
0:499 954 132
0:499 997 229
0:499 999 844
0:499 999 992
0:500 000 000

0:625
0:512 259 526
0:501 470 244
0:500 144 830
0:500 011 697
0:500 000 825
0:500 000 053
0:500 000 003
0:500 000 000aNumber of independent functions.

030403-3

VOLUME 93, N UMBER 3

PHYSICA L R EVIEW LET T ERSTABLE III. Free ICI calculation based on the sSE for the
helium atom with 0 and g given in the text.
Iteration

Mn a

Ritz energy

Optimal 

0
1
2
3
4
5
6

1
6
26
74
159
291
481

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

1.68751.67281.88032.03302.19982.33072.4862

Best valueba2:903 724 377

Number of independent functions. bReference [16].

function. We want to test how good this automatic method
is. 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 helium
atom case.
We introduce the coordinate defined by s  r1  r2 ,t  r1  r2 , u  r12 with s  u  jtj. We choose theinitial function by 0  exps with  as a variational
2
2parameter and the scaling function by g  u s ttosprevent nuclear and electron singularity problem.
Table III summarizes the result of the free ICI based on
the SSE. As the iteration proceeds, the Ritz energy approaches the best variational value, 2:903 724 377 a:u:,reported in the literature [16]. At the fifth iteration,
the free ICI energy was 2:903 724 373 a:u: with 291
independent functions, and at n  6, it was 
2:903 724 376 a:u: with 481 functions. These energies
were higher by only 4 and 1 nhartrees, respectively,from the best value, which guarantees the validity ofthe present method. We particularly note that these results
were obtained automatically by the ICI algorithm as
described above. The present ICI wave function included
the terms of both positive and negative powers of thevariable s. It is different from the Hylleraas expansionthat includes only positive powers of s, t, and u, but closer
to the Kinoshita one, though different in higher-orderterms. The functional form of the ICI wave function
depends on the choices of the zero-order wave function
0 and the g factor, and, in this sense, it supports thegeneral Thakkar-Koga expansions [16]. The Hylleraas
expansion actually corresponds to choosing g  us2 
t2  different from the present one. Examinations of thedependence on 0 and g will be given in a forthcoming
paper.
Hydrogen molecule.—The free ICI method has also
been applied to a hydrogen molecule with RH-H 
1:4 a:u: Using elliptic coordinate [17], the second iterationof the free ICI with the initial function
0 exp'1  '2 =21  (  (2  with   2:2 and g ('21  )21 '22  )22 ='1 '2 gave the Ritz variational en-

030403-4

week ending
16 JULY 2004

ergy of 1:174 475 703 a:u: with 609 independent functions. This energy is lower than the (nonrelativistic) valueof 1:174 475 686 a:u: obtained by Kolos in 1994 [18].
More details will be published in the near future.
Conclusion.—Good performance of the calculations
particularly for the helium atom and the hydrogen molecule implies a high potentiality of the proposed method.
It is noteworthy that their ICI wave functions are different
from the existing ones. Thus, the present ICI formalism
based on the SSE provides a general systematic method
for calculating the exact wave functions of atoms andmolecules in an analytical form. For He and H2 ,Hylleraas, James-Coolidge, Kolos, and others were cleverenough to imagine the exact wave functions only withintuition. But for general molecules, it is difficult toestimate the analytical form of the exact wave function
only by intuition, but the present method generates it
automatically, since the differentiations (involved in the
Hamiltonian) are always possible, and, furthermore, theintegrals in the secular equation step have no singularity
problem. It is guaranteed that at convergence we will get
the exact wave function. When (analytical) integrations
are difficult, the Monte Carlo method [19] may provide a
useful tool.
The author thanks Mr. H. Nakashima and Y. Kurokawa
for some help in the calculations of helium atom andhydrogen molecule, respectively. This research was supported by the Grant for Creative Scientific Research from
the Ministry of Education, Science, Culture, and Sportsof Japan.

[1] H. Nakatsuji, J. Chem. Phys. 113, 2949 (2000).
[2] H. Nakatsuji and E. R. Davidson, J. Chem. Phys. 115,
2000 (2001).
[3] H. Nakatsuji, J. Chem. Phys. 115, 2465 (2001).
[4] H. Nakatsuji, J. Chem. Phys. 116, 1811 (2002).
[5] H. Nakatsuji, Phys. Rev. A 65, 052122 (2002).
[6] H. Nakatsuji and M. Ehara, J. Chem. Phys. 117, 9 (2002).
[7] M. Nooijen, Phys. Rev. Lett. 84, 2108 (2000).
[8] P. Piecuch et al., Phys. Rev. Lett. 90, 113001 (2003).
[9] T. Kato, Commun. Pure Appl. Math. 10, 151 (1957).
[10] D. Horn and M. Weinstein, Phys. Rev. D 30, 1256 (1984).
[11] J. Cioslowski, Phys. Rev. Lett. 58, 83 (1987).
[12] H. Huang et al., J. Chem. Phys. 110, 3703 (1999).
[13] E. A. Hylleraas, Z. Phys. 54, 347 (1929).
[14] T. Kinoshita, Phys. Rev. 105, 1490 (1957).
[15] C. L. Pekeris, Phys. Rev. 112, 1649 (1958).
[16] A. J. Thakkar and T. Koga, Phys. Rev. A 50, 854 (1994).
[17] H. M. James and A. S. Coolidge, J. Chem. Phys. 1, 825(1933).
[18] W. Kolos, J. Chem. Phys. 101, 1330 (1994).
[19] B. L. Hammond, W. A. Lester, Jr., and P. J. Reynolds,
Monte Carlo Methods in Ab Initio Quantum Chemistry(World Scientific, Singapore, 1994).

030403-4