PHYSICAL REVIEW A, VOLUME 65, 052122

Inverse Schrödinger equation and the exact wave function
Hiroshi Nakatsuji
Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering, Kyoto University,
Sakyo-ku, Kyoto 606-8501, Japan
共Received 17 December 2001; published 10 May 2002兲
Using the inverse of the Hamiltonian, we introduce the inverse Schrödinger equation 共ISE兲 that is equivalent
to the ordinary Schrödinger equation 共SE兲. The ISE has the variational principle and the H-square group ofequations as the SE has. When we use a positive Hamiltonian, shifting the energy origin, the inverse energy
becomes monotonic and we further have the inverse Ritz variational principle and cross-H-square equations.
The concepts of the SE and the ISE are combined to generalize the theory for calculating the exact wave
function that is a common eigenfunction of the SE and ISE. The Krylov sequence is extended to include theinverse Hamiltonian, and the complete Krylov sequence is introduced. The iterative configuration interaction
共ICI兲 theory is generalized to cover both the SE and ISE concepts and four different computational methods ofcalculating the exact wave function are presented in both analytical and matrix representations. The exact
wave-function theory based on the inverse Hamiltonian can be applied to systems that have singularities in the
Hamiltonian. The generalized ICI theory is applied to the hydrogen atom, giving the exact solution without any
singularity problem.
DOI: 10.1103/PhysRevA.65.052122

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

I. INTRODUCTION

The basic principles of chemistry are described by the
Schrödinger equation 共SE兲 and the corresponding relativistic
equation 关1兴. Solving these equations as accurately as possible is, therefore, the central theme of theoretical chemistry.
In this series of papers 关2– 6兴, our aim is to formulate asystematic method of solving the SE, and for this purpose,
we have studied the structure of the exact wave function.
Since the exact wave function is an eigenfunction of the
Hamiltonian whose structure is very simple—a sum of only
one- and two-particle operators, such asH⫽

1/r i j兺i ⫺ ⌬ i ⫺ 兺i 兺A Z A /r Ai ⫹ 兺
i⬎ j
1
2

共1.1兲

for atomic and molecular systems, we expect that the exact
wave function must also have a simple structure, reflecting
the simplicity of the structure of the Hamiltonian. When the
structure of the exact wave function is clarified to be simple,
we would be able to find an easier way of solving the exact
wave function than before. We have proposed the iterative
configuration-interaction 共ICI兲 关2,3兴 and extended coupledcluster 共ECC兲 关4兴 methods to calculate the exact wave function. The methods of calculating excited states based onthese theories are formulated 关3,5兴, and the applications ofthese theories to some simple system 关4,5兴 and to the atomic
and molecular system 关6兴 have been given. Van Voorhis andHead-Gordon 关7兴 have reported that the coupled cluster general singles and doubles 共CCGSD兲 method, which is a special case of the ECC method, has reproduced the full-CI
result for Ne and N2 . We have shown more recently that for
many atoms and molecules, the ICI method reproduces thefull-CI results with only a few 共one to three兲 variables 关6兴, incontrast to a very large number of variables in the conventional full-CI algorithm 关8兴.
1050-2947/2002/65共5兲/052122共15兲/$20.00

Another approach for solving the SE is due to the densitymatrix theory 共DMT兲 关9兴. Since all the elementary physicalproperties can be calculated from the second-order density
matrix ⌫ (2) , we may utilize it as a basic variable of quantum
mechanics instead of the wave function ␺. Calculating⌫ (2) through the variational principle with some
N-representability conditions 关10,11兴 has long been a subjectof discussion 关12,13兴 since a birth of the DMT. Recently, ageneral computational method of this density-matrix variational theory 共DMVT兲 has been developed 关14兴 by using thepositive-semidefinite programming algorithm, and a high potentiality of the DMVT has been demonstrated 关15兴.
Another method in the DMT is to first derive the equationthat is equivalent to the SE but contains only density matrices as variables. The density equation 共DE兲 关16兴 is such an
equation and, therefore, we call this approach the density
equation theory 共DET兲. Later, this equation was alternatively
called the contracted Schrödinger equation 关17兴. To solve thisequation, we need relations between ⌫ (3) , ⌫ (4) , and ⌫ (1) ,⌫ (2) . Valdemoro and co-workers 关18兴 proposed approximate
relations, we proposed the improved relations using theGreen’s function method 关19兴, and Mazziotti 关20兴 gave therelations using the cumulant expansion method. We have calculated the ground state of many atoms and molecules togood accuracy by the DET 关19兴. Recently, many studies havebeen presented for the further development of the sciencelying here 关8,21兴. In particular, the cumulant expansion
method 关22兴 has been shown to be useful to systematize therelations among the density matrices in different orders andto develop further this theory 关23兴. Nooijen 关24兴 consideredsolving the CCGSD using the DE, which circumvents the
N-representability problem.
Certainly, the DMVT and the DET are very promising. It
is, however, true that an obstacle in these approaches still lies
in our limited knowledge on the N-representability condition
关10兴.

65 052122-1©2002 The American Physical Society

HIROSHI NAKATSUJI

PHYSICAL REVIEW A 65 052122

In this paper, we take the wave-function approach. We
first introduce the inverse Schrödinger equation 共ISE兲 that iscompletely equivalent to the SE. The ISE has the same determinative power as the ordinary SE. Further, similar to theordinary SE, the ISE has the variational principle and theH-square 关25兴 group of equations, which are equivalent to
the ISE. The cross-H-square equation couples the two worldsof the SE and the ISE. Based on these observations, we continuously develop the method of calculating the exact wave
functions 关2– 6兴 for general systems such as atomic and molecular systems. The ICI and the ECC theories are equally
defined for the ISE and are generalized to cover both the SE
and ISE concepts.
The ICI and the ECC theories utilize the products of the
Hamiltonian or the divided Hamiltonian applied to an appropriate reference function to construct the exact wave function. However, when the Hamiltonian includes singularities
such as those by the nuclear attraction operator and the electron repulsion operator in the atomic and molecular Hamiltonian given by Eq. 共1.1兲, their higher-order products applied
to the reference function do not satisfy the quantummechanical condition that the physical wave function mustsatisfy, i.e., integrable finiteness. The singularity of thenuclear attraction operator is too large to be solved by thefinite size of the nucleus 关26,27兴. This problem is similar to
those encountered in the analytical formulations of thet-expansion method 关28兴 and the connected-moments method关29兴. This difficulty was also pointed out 关30兴 to occur in theapplication of the surplus function method 关31兴, which issimilar to the simplest case of the ICI theory 关3兴.
However, in such a case, an introduction of the inverse
Hamiltonian of the ISE resolves the problem. This is shownby taking the Krylov sequence 关32兴 as an example: by extending the Krylov sequence into the inverse side of the
Hamiltonian, we have a sequence that naturally satisfies thequantum-mechanical condition. The ICI theory is generalized based on both the SE and ISE concepts and by theH-square and the cross-H-square group of equations. Fourdifferent formulas of the generalized ICI 共GICI兲 theory arepresented, both in the analytical form and in matrix representation. The generalized ICI theory is useful even for systems
that have singularities in the Hamiltonian. Numerical examples are given by applying the present theory to the hydrogen atom, which has a Coulombic singularity at the position of the nucleus. The concluding remarks are given in Sec.
VII.

A⫺1 C⫽CE⫺1 .

Namely, the eigenvalue of the inverse matrix is the inverse of
the eigenvalue of the original matrix, and the eigenvector iscommon to A and A⫺1 .
We apply this relation to the Schrödinger equation 共SE兲,H ␺ ⫽E ␺ .

H ⫺1 H⫽HH ⫺1 ⫽1,then it is easy to show that H ⫺1 satisfies
H ⫺1 ␺ ⫽E ⫺1 ␺ ,共2.6兲

which is called the inverse Schrödinger equation 共ISE兲. Conversely, we can obtain SE from ISE by using Eq. 共2.5兲. Thus,on the assumption that the inverse Hamiltonian exists, the SE
and the ISE are equivalent.
Note that H and H ⫺1 commute, as easily seen from Eq.
共2.5兲. As the Hamiltonian is an operator, including differential operators, etc., it is not necessarily straightforward towrite down the analytical form of the inverse Hamiltonian.
However, quantum mechanically, we can always define thematrix form of the Hamiltonian H, and then we can always
define its inverse H⫺1 . H ⫺1 may be defined as the analyticalcorrespondence of the inverse matrix H⫺1 . In the matrix
formulation of quantum mechanics 关33兴 H and H⫺1 are entirely equivalent.
Hereafter, we call H as regular Hamiltonian or simply
ordinary Hamiltonian to distinguish it from the inverse
Hamiltonian.
Here, a remark is necessary on the inverse energy. Theenergy E in Eq. 共2.4兲 may be defined by

具 ␺ 兩 H⫺E 兩 ␺ 典 ⫽0,

共2.7兲

E⫽ 具 ␺ 兩 H 兩 ␺ 典 / 具 ␺ 兩 ␺ 典 .

共2.8兲

namely, by

The inverse of the energy E ⫺1 is, therefore,

共2.1兲

where A is a regular Hermitian matrix and E and C are its
eigenvalue and eigenvector. When we introduce the inverseof A, A⫺1 as
A⫺1 A⫽AA⫺1 ⫽1,

共2.5兲

共2.9兲

On the other hand, from Eq. 共2.6兲, we obtain 具 ␺ 兩 H ⫺1
⫺E ⫺1 兩 ␺ 典 ⫽0, which gives

Suppose that we have an eigenvalue problem
AC⫽CE,

共2.4兲

Based on Eq. 共2.2兲, we define the inverse of the HamiltonianH ⫺1 asE ⫺1 ⬅1/E⫽ 具 ␺ 兩 ␺ 典 / 具 ␺ 兩 H 兩 ␺ 典 .
II. INVERSE SCHRÖDINGER EQUATION

共2.3兲

共2.2兲

E ⫺1 ⫽ 具 ␺ 兩 H ⫺1 兩 ␺ 典 / 具 ␺ 兩 ␺ 典 .

共2.10兲

For the exact wave function ␺, two expressions, Eqs. 共2.9兲
and 共2.10兲, are equivalent, but for approximate ones, they are
different. For this reason, we introduce the inverse energy i E
associated with a wave function ␾ asithen it is easy to show that the inverse of A satisfies
052122-2

E 关 ␾ 兴 ⬅ 具 ␾ 兩 H ⫺1 兩 ␾ 典 / 具 ␾ 兩 ␾ 典 .

共2.11兲

INVERSE SCHRÖDINGER EQUATION AND THE EXACT . . .

PHYSICAL REVIEW A 65 052122

If ␾ is not exact, the inverse energy i E 关 ␾ 兴 is different fromthe inverse of the energy E 关 ␾ 兴 , which is defined as 具 ␾ 兩 H⫺E 关 ␾ 兴 兩 ␾ 典 ⫽0 similarly to Eq. 共2.7兲. For the exact wave
function ␺, we havethe exact wave function. Namely, in this case, the variational
best is the exact wave function. We used this criterion toinvestigate the structure of the exact wave function 关2–5兴.
We formulate here the equations that are equivalent to the
ISE and therefore to the SE. They are useful not only as they
are, but also for calculating the exact wave function. We firstprove the inverse variational principle.
Theorem. When we define the inverse energy i E 关 ␺ 兴 byi

E 关 ␺ 兴 ⫽E 关 ␺ 兴 ⫺1 .

共2.12兲

III. EQUATIONS EQUIVALENT TO SE AND ISE

The equation that is equivalent to the SE is very important, since it has the same determinative power as the SE.
The variational equation and the H-square group of equations 关25兴 are two important equations that are equivalent to
the SE. The variational equation is written as

具 ␺ 兩 H⫺E 兩 ␦ ␺ 典 ⫽0

共3.1兲

具 ␺ 兩 H ⫺1 ⫺ i E 关 ␺ 兴 兩 ␺ 典 ⫽0,then the wave function ␺ that satisfies

␦ i E 关 ␺ 兴 ⫽0

共3.2兲

where E 关 ␺ 兴 is defined by

具 ␺ 兩 H⫺E 关 ␺ 兴 兩 ␺ 典 ⫽0.

具 ␦ ␺ 兩 H ⫺1 ⫺ i E 兩 ␺ 典 ⫹ 具 ␺ 兩 H ⫺1 ⫺ i E 兩 ␦ ␺ 典 ⫺ 具 ␺ 兩 ␺ 典 ␦ i E⫽0,

共3.12兲

where i E represents i E 关 ␺ 兴 . From 具 ␺ 兩 ␺ 典 ⫽0 and Eq. 共3.11兲,
we obtain

具 ␦ ␺ 兩 H ⫺1 ⫺ i E 兩 ␺ 典 ⫹ 具 ␺ 兩 H ⫺1 ⫺ i E 兩 ␦ ␺ 典 ⫽0,

共3.3兲

具 ␺ 兩 共 H⫺E 兲 2 兩 ␺ 典 ⫽0,

共3.4兲

具 ␺ 兩 共 H⫺E 兲 H 兩 ␺ 典 ⫽0,

共3.5兲

具 ␺ 兩 H ⫺1 ⫺ i E 兩 ␦ ␺ 典 ⫽0.
共 H ⫺1 ⫺ i E 兲 兩 ␺ 典 ⫽0,

共3.6兲

where E of Eqs. 共3.5兲 and 共3.6兲 is defined by Eq. 共3.3兲. For
Eq. 共3.4兲, such a definition of E is unnecessary. More general
H-square equations were introduced in Paper II 共Ref. 关3兴兲 asfollows. We define a division of the Hamiltonian into N D
parts asND

兺I H I ,共3.7兲

H p ⫽H⫹␧ c ,
共3.8兲

共3.9兲

for all I (I⫽1,...,N D ). In Eq. 共3.8兲, E I is defined by 具 ␺ 兩 H I⫺E I 兩 ␺ 典 ⫽0.
The variational equation is a flexible equation and is used
to calculate the best possible wave function within a givenfunctional form of ␺. The H-square equation is, on the other
hand, a rather strict equation that is valid only for the exact
wave function. Therefore, if a variation of a given ␺, substituted into the variational equation, leads to the H-squareequation, then, that ␺ is guaranteed to have the structure of

共3.16兲

where ␧ c is some positive constant with the dimension ofenergy. In the SE of H p ,or

具 ␺ 兩 共 H⫺E 兲 H I 兩 ␺ 典 ⫽0

共3.15兲

where i E⫽E ⫺1 for the uniqueness of the ISE. So, the sufficiency is proved. Conversely, when we have the ISE, i E⫽E ⫺1 from Eq. 共2.10兲 and we have Eq. 共3.11兲 from Eq.
共3.12兲. Q.E.D.
The Ritz variational principle can be inverted when the
Hamiltonian has only positive eigenvalues. Though the ordinary Hamiltonian may have positive and negative eigenvalues, it is easily made a positive operator, H p , by shifting theorigin of the Hamiltonian, i.e., by adding some positive constant to the original Hamiltonian asthen the H-square equation is given, in general, by

具 ␺ 兩 共 H⫺E 兲共 H I ⫺E I 兲 兩 ␺ 典 ⫽0

共3.14兲

For arbitrary ␦␺, Eq. 共3.14兲 gives

and

H⫽

共3.13兲

which leads to 关34兴 the inverse variational equation

The simplest H-square equations 关2兴 are given by具 ␺ 兩 H 2 ⫺E 2 兩 ␺ 典 ⫽0,

共3.11兲

is exact in the necessary and sufficient sense.
Proof. Taking the derivative of Eq. 共3.10兲, we get

and the Ritz variational principle for the ground state isgiven byE 关 ␺ 兴 ⭓E g ,共3.10兲

H p ␺ ⫽E p ␺ ,共3.17兲

⫺1
H ⫺1
p ␺ ⫽E p ␺ ,共3.18兲

E p ⫽E⫹␧ c ,共3.19兲

and in the ISE,

the energy E p is

which is positive by definition. Then, it is easy to show thatfor the ground state, the inverse positive energy i E p 关 ␾ 兴 associated with an approximate ␾,

052122-3

HIROSHI NAKATSUJI

PHYSICAL REVIEW A 65 052122i
具 ␾ 兩 H ⫺1
p ⫺ E p 关 ␾ 兴 兩 ␾ 典 ⫽0,gives a lower bound of the inverse of the positive energy
E pg ⫺1 of the true ground statei

E p 关 ␾ 兴 ⭐E ⫺1
pg .

共3.21兲

We call this relation the inverse Ritz variational principle fora positive Hamiltonian. Note that the subscript g means theground state.
Proof. We assume that we have the exact wave functions
for the positive Hamiltonian as
H p ␺ K ⫽E pK ⌿ K ,共3.22兲

⫺1
H ⫺1
p ⌿ K ⫽E pK ⌿ K .

共3.23兲

Since 兵 ⌿ K 其 is a complete set, we expand ␾ of Eq. 共3.20兲 bythis set as ␾ ⫽ 兺 K C K ⌿ K and obtaini⫺1
E p 关 ␾ 兴 ⫺E ⫺1
pg ⫽ 共 具 ␾ 兩 ␾ 典 兲再兺K具 ␺ 兩 共 H ⫺1 ⫺ i E 兲 2 兩 ␺ 典 ⫽0,

共3.20兲冎

⫺1
兩 C K 兩 2 共 E ⫺1
pK ⫺E pg 兲 ⭐0,then this equation holds if and only if ␺ satisfies the ISE.
Proof. When the ISE holds, ␺ is exact and Eq. 共3.25兲 isvalid when i E⫽E ⫺1 . Conversely, when Eq. 共3.25兲 holds, wedefine⌬⫽ 共 H ⫺1 ⫺ i E 兲 ␺ ,共3.26兲

and then Eq. 共3.25兲 implies 具 ⌬ 兩 ⌬ 典 ⫽0, which is valid only
when ⌬⫽0, namely,
共 H ⫺1 ⫺ i E 兲 ␺ ⫽0.

共3.27兲

From the uniqueness of the ISE, i E⫽E ⫺1 , namely, Eq.
共3.27兲 is the ISE. Q.E.D.
With the definition of i E⫽ i E 关 ␺ 兴 given by Eq. 共3.10兲, it iseasy to show that the following equations:具 ␺ 兩 共 H ⫺1 ⫺ i E 兲 H ⫺1 兩 ␺ 典 ⫽0

共3.28兲

具 ␺ 兩 共 H ⫺1 兲 2 ⫺ 共 i E 兲 2 兩 ␺ 典 ⫽0

共3.29兲

and共3.24兲

⫺1is always
where the last inequality arises because E pK
smaller than or equal to E pg ⫺1 . The equality occurs when all
C K ⫽0 except for K⫽0⫽g, namely, when ␾ ⫽C 0 ⌿ 0 .
Q.E.D.
Thus, in iterative variational calculations of the inverse
positive energy, the inverse energy approaches from below
the exact inverse energy. We further note that if the Hamiltonian is not positive, we do not have the inverse Ritz variational principle given above and, therefore, we cannot expect
the above behavior in an iterative process.
The above argument has clarified the necessity of introducing the positive Hamiltonian when we introduce the inverse Hamiltonian. The reason is simple. Consider a hydrogen atom, then the eigenvalues of the regular Hamiltonian
range from ⫺0.5 to 0 for the bound states and 0 to ⬁ for theunbound continuum states. Then, the eigenvalues of the corresponding inverse Hamiltonian range from ⫺0.2 to ⫺⬁ forthe bound states and from ⫹⬁ to 0 for the unbound continuum states. Thus, there exists a large discontinuous gap
from ⫺⬁ to ⫹⬁ in the inverse energy spectrum. This is
undesirable and can be easily avoided by shifting the originof the energy of the regular Hamiltonian into the positive
region. By shifting the regular Hamiltonian by unity to thepositive side, i.e., ␧ c ⫽1, the eigenvalues of the bound statesof the positive Hamiltonian range from 0.5 to 1.0 and thoseof the unbound states range from 1.0 to ⫹⬁. Then, the eigenvalue spectrum of the inverse positive Hamiltonian
ranges from 2.0 to 1.0 for the bound states and from 1.0 to
0.0 for the unbound states: they are monotonic and there isno infinite gap. Thus, when we shift the regular Hamiltonian
into the positive region, we have monotonic descriptions of
the energy eigenvalue spectra for both the regular and inverse Hamiltonians.
Next, we prove the inverse H-square theorem.
Theorem. We define the inverse H-square equation orH ⫺1 -square equation by

共3.25兲

are also equivalent to the ISE. Note that for the originalH ⫺1 -square theorem given by Eq. 共3.25兲, i E⫽E ⫺1 is automatic and, therefore, Eq. 共3.10兲 is unnecessary.
Similarly to Theorem II-1 of Paper II, we can prove thefollowing H ⫺1 -square theorem for the partitioned inverse
Hamiltonian. Namely, when the inverse Hamiltonian is divided into N D 共number of divisions兲 parts,
ND

H ⫺1 ⫽

兺I d I i H I ,共3.30兲

where d I is a constant, then the wave function ␺ that satisfies
the following partitioned inverse H-square equation:

具 ␺ 兩 共 H ⫺1 ⫺ i E 兲共 i H I ⫺ i E I 兲 兩 ␺ 典 ⫽0

共3.31兲

具 ␺ 兩 共 H ⫺1 ⫺ i E 兲 i H I 兩 ␺ 典 ⫽0

共3.32兲

orfor all I (I⫽1,...,N D ) with i E defined by Eq. 共3.10兲 and i E I
defined by

具 ␺ 兩 i H I ⫺ i E I 兩 ␺ 典 ⫽0

共3.33兲

is exact in the necessary and sufficient sense. The partitioned
inverse energy i E I satisfies

兺I d I i E I ⫽ i E.

共3.34兲

Proof. When ISE holds, both Eqs. 共3.31兲 and 共3.32兲 hold,where i E is given by Eq. 共3.10兲. Conversely, when Eq.
共3.31兲 or Eq. 共3.32兲 holds, we multiply it by d I , sum up for
all I, and using Eqs. 共3.30兲 and 共3.34兲, we obtain Eq. 共3.25兲
or Eq. 共3.28兲, so that this ␺ is exact. Q.E.D.

052122-4

INVERSE SCHRÖDINGER EQUATION AND THE EXACT . . .

Note that in the above division of the inverse Hamiltonian
given by Eq. 共3.30兲, we have given the coefficients d I oneach partitioned Hamiltonian i H I . Throughout the proof,
however, we need not to know the actual value of d I . Thismeans that all we must know for the division of the Hamiltonian is the individual functional forms of the operatorsi
H I , and not their coefficients. The same is true for the
H-square theorem of the divided Hamiltonian given in Papers I and II. This relaxes the necessary knowledge on theregular and inverse Hamiltonians and would be useful when
the detailed form of the Hamiltonian is difficult to know.
We now show an important H-square equation that combines the regular and inverse worlds of the Hamiltonian: wecall it the cross-H-square theorem, which is valid only forthe positive Hamiltonian. Namely, the following cross-Hsquare equation:⫺1
具 ␺ 兩 共 H p ⫺E p 兲共 H ⫺1
p ⫺E p 兲 兩 ␺ 典 ⫽0

共3.35兲

is equivalent to the SE and ISE in the necessary and sufficient sense.
Proof. When the SE and the ISE of the positive Hamiltonian are satisfied by ␺, Eq. 共3.35兲 holds automatically. So,
the necessity is evident. The sufficiency is as follows. We
expand ␺ by the exact eigenfunctions, ␺ ⫽ 兺 K C K ⌿ K , andinsert it into Eq. 共3.35兲 and obtain

PHYSICAL REVIEW A 65 052122

When we divide the positive Hamiltonian and its inverse
as
ND

H p⫽

H ⫺1
p ⫽

兺K 兩 C K兩 2共 E pK ⫺E p 兲 2 /E pK E p ⫽0.

i
具 ␺ 兩 共 H ⫺1
p ⫺ E p 兲 H pI 兩 ␺ 典 ⫽0

共3.37兲

共3.38兲

共3.39兲

with the definition of i E p asi
具 ␺ 兩 H ⫺1
p ⫺ E p 兩 ␺ 典 ⫽0

共 I⫽1,...,N D 兲

共3.43兲

is equivalent to Eq. 共3.39兲. Second, with Eq. 共3.38兲, the following set of equations,具 ␺ 兩 共 H p ⫺E p 兲 i H pI 兩 ␺ 典 ⫽0

共 I⫽1,...,N D 兲

共3.44兲

is equivalent to Eq. 共3.37兲. Further, each of the following two
sets of equations is equivalent to Eq. 共3.35兲,⫺1
具 ␺ 兩 共 H ⫺1
p ⫺E p 兲共 H pI ⫺E pI 兲 兩 ␺ 典 ⫽0

共 I⫽1,...,N D 兲 ,共3.45兲

具 ␺ 兩 共 H p ⫺E p 兲共 i H pI ⫺ i E pI 兲 兩 ␺ 典 ⫽0

共 I⫽1,...,N D 兲 ,共3.46兲

具 ␺ 兩 i H pI ⫺ i E pI 兩 ␺ 典 ⫽0 ,共3.47兲

and they satisfy

is necessary and sufficient for the equivalence to the crossH-square equation. Also, the following equation,i
具 ␺ 兩 共 H ⫺1
p ⫺ E p 兲 H p 兩 ␺ 典 ⫽0

共3.42兲

共3.36兲

with the definition of E p as

具 ␺ 兩 H p ⫺E p 兩 ␺ 典 ⫽0

兺I i e I i H pI ,具 ␺ 兩 H pI ⫺E pI 兩 ␺ 典 ⫽0 ,Since all elements of the summation are positive, it is valid
only when E p ⫽E pK for some K for which C K is nonzero,
and C K ⫽0 for all other K, which means ␺ ⫽C K ⌿ K . So, thesufficiency is also proved. Q.E.D.
The cross-H-square equation also has its family. Namely,
the following equation:

具 ␺ 兩 共 H p ⫺E p 兲 H ⫺1
p 兩 ␺ 典 ⫽0

ND

where the partitioned energies E pI and i E pI are defined by⫺1
兺K 兩 C K兩 2共 E pK ⫺E p 兲共 E ⫺1
pK ⫺E p 兲⫽⫺

共3.41兲

then we have cross-H-square theorems for the divided
Hamiltonians. First, with Eq. 共3.40兲, the following set of
equations:⫺1
具 ␺ 兩 共 H p ⫺E p 兲共 H ⫺1
p ⫺E p 兲 兩 ␺ 典⫽

兺I e I H pI ,共3.40兲

is again necessary and sufficient for the equivalence to thecross-H-square equation. The proof of these equations isstraightforward and so omitted for brevity. Note that i E p in
Eq. 共3.39兲 is not E ⫺1
p .

ND

E p⫽

兺I e I E pI ,

ND

i

E p⫽

兺I i e I i E pI .

共3.48兲

The proof of these equations is very similar to that for Eqs.
共3.31兲 and 共3.32兲 and is, therefore, omitted here.
We will see later in the formulation of the generalized ICI
theory that these cross-H-square equations are important,connecting the two worlds of the regular and inverse Hamiltonians. In this sense, it is natural that this theorem is valid
only for the positively shifted Hamiltonian.
The inverse Hamiltonian has the ISE, the inverse variational principle, and the H ⫺1 -square group of equations, justas the ordinary SE. Therefore, there exists entirely the same
theoretical framework as that of the SE on the inverse side of
the SE. Further, these two sides are connected by the crossH-square equation. The knowledge on the existence of theseequivalent sets of equations on both sides of the Hamiltonian
would make the physical and chemical imagination more
fertile and would help us to understand Nature more clearly.
On the regular side, we usually have an analytical expression of the Hamiltonian, but on the inverse side, we do notnecessarily have a closed analytical expression of the Hamiltonian, though we really require it. However, in the matrix
form, we can always define the inverse Hamiltonian matrix,

052122-5

HIROSHI NAKATSUJI

PHYSICAL REVIEW A 65 052122

Similarly, we define the inverse Hamiltonian Krylov sequence as

quantum-mechanical 共QM兲 condition a physical wave function must satisfy, i.e., integrable finiteness. For example,
when the Hamiltonian includes the Coulombic nuclear attraction operator, it has a singularity at the position of thenucleus, and, therefore, the function H n ␺ 0 with n⭓1 does
not satisfy the QM condition: ␺ 0 is finite but H n becomes
infinite at the nuclear position 关see Eq. 共6.2兲, for example兴
and this nuclear singularity is too large to be solved with theaid of the fact that the nucleus is a finite-size entity 关26,27兴.
This is the difficulty also encountered in the t-expansion
method 关28兴 and the connected-moments method 关29兴. Marmorino 关30兴 discussed that this difficulty should also occur inthe calculation of the surplus function method 关31兴. However, we must also note that when ␺ 0 is exact, H n ␺ 0 is
simply E n ␺ 0 , which implies that a very good reference function is necessary for the convergence of this procedure.
In the present formalism of the Krylov sequence, however, we can use in such a case the inverse Hamiltonian
Krylov sequence, where the Coulombic singularity problem
does not occur since the elements of the sequence are composed of the inverse of the Hamiltonian, H ⫺n ␺ 0 . At thenuclear position, for instance, H ⫺n becomes zero, and at a
distance apart from the nuclei, H ⫺n ␺ 0 behaves like r n ␺ 0 ,which is integrable finite because ␺ 0 decays exponentiallylike ⬃exp(⫺␣r) or ⬃exp(⫺␤r2). In comparison with theordinary Krylov sequence, the dependence of this sequenceon the quality of the reference function ␺ 0 would be dull
and, therefore, the convergence would be faster. In general, it
is convenient to use the complete Hamiltonian Krylov sequence given by Eq. 共4.6兲 and use only such parts that satisfy
the QM condition. In this case, we can calculate the groundstate starting from an approximate excited-state referencefunction ␺ 0 . In conclusion, introducing the inverse of the
Hamiltonian, the proposed method is free from the singularity problems originating from the nuclear attraction operator
and the electron repulsion operator in the atomic and molecular Hamiltonian given by Eq. 共1.1兲.

兵 ␺ 0 ,H ⫺1 ␺ 0 ,H ⫺2 ␺ 0 ,H ⫺3 ␺ 0 ,... 其 .

V. GENERALIZED ICI THEORY

when we have the regular Hamiltonian matrix. Thus, theconcept combining the SE and ISE may have a larger practical merit when we formulate our problem in a matrix representation.
We study in this series of papers 关2– 6兴 the structure of theexact wave function. Now that the exact wave function ␺ isan eigenfunction of not only H but also H ⫺1 , the structure of
the exact wave function must depend not only on H but alsoon H ⫺1 entirely on the same footing. This fact would lead us
to a deeper understanding of the structure of the exact wave
function. In the following section, we utilize the concept of
the regular and inverse Hamiltonians to generalize the Krylov sequence, and in Sec. V, the ICI theory is generalized
into four different ones by connecting the regular and inverse
sides of the Hamiltonian.
IV. COMPLETE KRYLOV SEQUENCEThe Krylov sequence,

兵 c,Ac,A2 c,A3 c,... 其 ,共4.1兲

plays an important role in the eigenvalue problem given by
Eq. 共2.1兲 关32,35兴. The Arnoldi method 关36兴 and the Lanczos
method 关37兴 are related to the Krylov sequence. The Hamiltonian Krylov space

兵 ␺ 0 ,H ␺ 0 ,H 2 ␺ 0 ,H 3 ␺ 0 ,... 其共4.2兲

is important for solving the SE: these functions are used as
basis functions to expand the wave function. As we now
know that A⫺1 plays an equivalent role to A in the eigenvalue problem, we expect a similar important role of theinverse Krylov sequence defined by

兵 c,A⫺1 c,A⫺2 c,A⫺3 c,... 其 .

共4.3兲

共4.4兲

Further by combining the two sets of Krylov sequences, we
obtain the complete Krylov sequence as

兵 ...,A⫺3 c,A⫺2 c,A⫺1 c,c,Ac,A2 c,A3 c,... 其共4.5兲

and the complete Hamiltonian Krylov sequence as

兵 ...,H ⫺3 ␺ 0 ,H ⫺2 ␺ 0 ,H ⫺1 ␺ 0 , ␺ 0 ,H ␺ 0 ,H 2 ␺ 0 ,H 3 ␺ 0 ,... 其 .

共4.6兲

The Hamiltonian Krylov sequence gives a basis of the
expansion of the exact wave function. Since the ISE is
equivalent to the SE, the inverse Krylov sequence should
also be important in the expansion of the exact wave function. The complete Krylov sequence given by Eq. 共4.6兲 provides a wider functional space that is useful for calculating
the exact wave function.
When we use the Hamiltonian Krylov sequence as a basisof the expansion of a wave function, it sometimes happensthat the element of the Krylov subspace does not satisfy the

As shown in Sec. III, the ISE has the variational principle
and the H-square group of equations just as the ordinary SE.
Therefore, the theoretical framework developed for the ordinary SE should also be formulated on the inverse side of theSE. Further, the regular and inverse sides are connected by
the cross-H-square equation, which turns out to be very important and useful in this section. The ICI and ECC theories
developed previously for calculating the exact wave function
are, therefore, generalized to cover both sides of the Hamiltonian. We describe here the GICI theory. The generalized
ECC theory will be given in a forthcoming paper 关38兴. We
show the operator formalism first and then the matrix representation. These formalisms are used in the applications
given in Sec. VI.
A. Generalized ICI theory in analytical form

In Sec. III, we have shown that in order to have a continuous correspondence between the eigenvalues of the regu-

052122-6

INVERSE SCHRÖDINGER EQUATION AND THE EXACT . . .

PHYSICAL REVIEW A 65 052122

lar and the inverse Hamiltonian, we have to introduce thepositive Hamiltonian, which is easily obtained by shifting theorigin of the Hamiltonian as in Eq. 共3.16兲. For the positive
Hamiltonian, we have the inverse Ritz variational principle
and the cross-H-square equation, but for nonpositive Hamiltonian, these useful relations do not hold. Therefore, in thissection, we limit ourselves to use only the positive Hamiltonian. Though we do not use the subscript p like in H p , forsimplicity, all the quantities given in this section are those of
the positive Hamiltonian.
The generalized ICI theory will be grouped into four, depending on how the S operator is defined and what variational principle is used in calculating the unknown variablesin the S operator. Depending on whether the regular Hamiltonian or the inverse Hamiltonian is used, we denote R or I,
respectively, so that R-R, R-I, I-R, and I-I cases occur. We
explain these four cases below.
First are the R-R and R-I cases, where we start from thedivision of the regular Hamiltonian

to the H-square equation given by Eq. 共3.9兲, which guarantees that the solution is exact. This R-R case is the formulation given in Papers I and II.
Next is the R-I case, where we use the inverse variational
principle given by Eq. 共3.14兲 to calculate the unknown variables 兵 C I 其 of the S operator and obtain

ND

H⫽兺I d I H I .

共5.1兲

We note again that we have assigned the coefficients 兵 d I 其 in
front of the divided Hamiltonian H I , differently from thepartitioning of the Hamiltonian given in Paper II. These coefficients are introduced just for making it clear that all weneed to know is the functional form of H I and we need notknow the coefficients d I . In other words, this coefficient d I
was assumed to be unity in the previous formalism 关2– 4兴.
Corresponding to Eq. 共5.1兲, we define the variable operator S
by
ND

S⫽

兺 C IH I .

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

具 ␺ n 兩 H ⫺1 ⫺ i E n 兩 ␺ n 典 ⫽0.

␺ n ⫽ 共 1⫹S n 兲 ␺ n⫺1 ,

共5.3兲

共5.9兲

At convergence, ␺ n ⫽ ␺ n⫺1 and the variational equation
given by Eq. 共5.8兲 becomes equal to the cross-H-squareequation given by Eq. 共3.43兲, which guarantees that the solution is exact. Since the present Hamiltonian is positive, theenergy converges from below to the exact inverse energy
In the above formulation, we defined our variable operator S in terms of the regular Hamiltonian. Next, in the I-R
and I-I cases, we define our variable operator in terms of theinverse Hamiltonian. For doing so, we first define the division of the inverse Hamiltonian as
ND

H

⫺1

⫽

兺I d I i H I .

共5.10兲

Again, we have introduced the unimportant coefficients 兵 d l 其 ,which we need not know throughout the calculations. For
this division of the inverse Hamiltonian, we introduce thevariable operator i S as
ND

i

S⫽

兺 iC I iH I

共5.11兲

I⫽1

We here note that in the definition of the S operator, we do
not use the coefficients 兵 d i 其 in Eq. 共5.1兲: we need not know
the coefficients 兵 d I 其 . The ICI method is defined by the recurrence

共5.8兲

E n also satisfies

共5.2兲

I⫽1

共 I⫽1,...,N D 兲 .

共5.7兲

and define the ICI wave function as␺ n ⫽ 共 1⫹ i S n 兲 ␺ n⫺1 .

共5.12兲

Now, we calculate the unknown coefficients 兵 i C l 其 in the
S operator again either by the regular or the inverse variational principle. The former is the I-R case and the latter isthe I-I case. When we use the regular variational principle,
we obtain the secular equation for the I-R case,iwhere n denotes iteration number.
Now, to calculate the unknown coefficients 兵 C I 其 in the S
operator, we can use either the regular or the inverse variational principle. When we use the regular variational principle, it is the R-R case, and we obtain the secular equation

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 I⫽1,...,N D 兲 .

共5.4兲
共5.5兲

Note that the energy of the nth iteration E n satisfies

具 ␺ n 兩 H⫺E n 兩 ␺ n 典 ⫽0

具 ␺ n 兩 H⫺E n 兩 ␺ n⫺1 典 ⫽0,

共5.6兲

as well as Eq. 共5.4兲. Because of the variational nature, theenergy E n converges to the exact energy from above 关6兴. At
convergence, ␺ n ⫽ ␺ n⫺1 , and therefore, Eq. 共5.5兲 is identical

具 ␺ n 兩 共 H⫺E n 兲 i H I 兩 ␺ n⫺1 典 ⫽0

共 I⫽1,...,N D 兲 .

共5.13兲
共5.14兲

At convergence, this equation is identical to the cross-Hsquare equation given by Eq. 共3.44兲, which guarantees thatthe solution is exact. In this case, the energy E n converges tothe exact energy from above. On the other hand, when we
use the inverse variational principle given by Eq. 共3.14兲, we
obtain the secular equation for the I-I case,052122-7

具 ␺ n 兩 H ⫺1 ⫺ i E n 兩 ␺ n⫺1 典 ⫽0,

共5.15兲

HIROSHI NAKATSUJI

PHYSICAL REVIEW A 65 052122

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

共 I⫽1,...,N D 兲 .

共5.16兲

At convergence, this is identical to the H ⫺1 -square equation
given by Eq. 共3.32兲, so that the converged solution is exact.
In the I-I case, the energy converges from below to the exact
inverse energy.
Thus, from the SE and ISE, we have four different ICI
methods for calculating the exact wave function. One would
converge more quickly than others and one would be moreeasily formulated than others. A criterion for the choice is theexistence of singularities in the Hamiltonian or in the inverse
Hamiltonian. For atomic and molecular systems, the largest
singularity is due to the nuclear attraction operator. In thiscase, the integrals involving higher products of the nuclear
attraction operator may diverge and do not satisfy the QM
condition necessary for physical wave functions, so that thecalculation is problematic. Such a situation may occur when
we use the S operator defined by using the divided regular
Hamiltonian, Eq. 共5.1兲. Namely, the R-R and R-I cases areproblematic. Therefore, we recommend using the i S operator
defined by Eq. 共5.11兲: the variational principle may be either
regular or inverse, which is the I-R or I-I case.
It is convenient if we can combine two different definitions of the recurrence formula of the ICI theory, Eqs. 共5.3兲
and 共5.12兲, into one. This is possible by introducing the combined recurrence formula asH⫽ 兵 H i j 其 ⫽ 兵 具 ␹ i 兩 H 兩 ␹ j 典 其 .

共5.18兲

For convenience, we assume in this section that we haveshifted our Hamiltonian H to be positive as defined by Eq.
共3.16兲, though we do not use the subscript p for simplicity.
The matrix H is then positive definite. The eigenvalue problem of this Hamiltonian is given by
共5.19兲

H␭⫽E␭,and that of the inverse Hamiltonian H⫺1 is given byH⫺1 ␭⫽E ⫺1 ␭,

共5.20兲

where ␭ is an eigenvector and E and E ⫺1 , which are positive, are the eigenvalue and its inverse. The corresponding
wave function is written as␺ ⫽ ␹␭⫽ 兺 ␭ i ␹ i ,i

共5.21兲

where ␹ is a row vector and ␭ is a column vector. When ␹
involves full configurations, as we assume below, this isnothing else but the full CI, and when ␹ involves only SDTQ
functions, it is SDTQ-CI.
In the matrix representation of ICI, we have four different
cases, R-R, R-I, I-R, and I-I, as in the operator formalism. In
the R-R and R-I cases, we first divide the H matrix as in Eq.
共5.1兲,ND␺ n ⫽ 共 1⫹S n ⫹ S n 兲 ␺ n⫺1 .
i

共5.17兲

In using this formula, a remark is necessary. When the QM
condition is not satisfied by the elements of S n and/or i S n ,we must omit them from the calculations. When we startfrom an approximate higher excited state as an initial guess
to calculate the ground state, some deexcitations are done by
S n or i S n and, therefore, we need the terms belonging to both
S n and i S n . For the ordinary case, however, using Eq. 共5.17兲
is essentially the same as using Eq. 共5.3兲 or Eq. 共5.12兲 separately.
B. Generalized ICI theory in matrix representation

In the above operator formalism of the ICI theory, weneed an explicit form of the inverse Hamiltonian in the R-I,
I-R, and I-I cases, but it is not necessarily given in a closed
analytical form. However, in the matrix formulation of quantum mechanics, we can always define the inverse of the
Hamiltonian, so that the matrix representation of the ICI
theory has certainly a merit in actual calculations.
Suppose that we have a set of orthonormal configuration
functions 兵 ␹i 其 for the system under consideration. It may be
a complete configuration space of full CI made of some orthonormal set of orbitals such as Hartree-Fock orbitals, orsome subspace of this set, such as the singles to quadruples
共SDTQ兲 configuration space. We define the Hamiltonian matrix defined in these configuration functions as

兺I d I HI ,H⫽

共5.22兲

and define the S operator as
NDS⫽兺 C I HI .

I⫽1

共5.23兲

Then, the ICI is defined by the recurrence␭n ⫽ 共 1⫹Sn 兲 ␭n⫺1 ,共5.24兲

where n is the iteration number. For the formulations below,
it is convenient to introduce the coefficient C 0 for the unit
matrix 1 of the above ICI equation, and redefine the ICI as

冉兺
ND

␭n ⫽

l⫽0

冊

C I,n HI ␭n⫺1 ,共5.25兲

where H0 ⫽1.
In the R-R case, we apply the regular variational principle
to this expression and obtain the secular equation of the dimension N D ⫹1,
共 hn ⫺E n sn 兲 Cn ⫽0,

共5.26兲

where Cn is a column vector composed of the coefficients
C I,n and the matrix elements of hn and sn are given by

052122-8* HI* HHJ ␭n⫺1
共 hn 兲 IJ ⫽␭n⫺1

共5.27兲

INVERSE SCHRÖDINGER EQUATION AND THE EXACT . . .

PHYSICAL REVIEW A 65 052122* i HI* i HJ ␭n⫺1 .
共 i sn 兲 IJ ⫽␭n⫺1

and* HI* HJ ␭n⫺1 .
共 sn 兲 IJ ⫽␭n⫺1

共5.28兲

As we perform iterative calculations, the energy decreases,
and at convergence the eigenvalue E n and the eigenvector ␭nbecome identical with the solution of the full CI given by Eq.
共5.19兲.
In the R-I case, we use the inverse variational principle.
By applying it to the ICI recurrence formula given by Eq.
共5.25兲, we obtain the secular equationi
共 h⫺1n ⫺ E n sn 兲 Cn ⫽0,given by* HI* H⫺1 HJ ␭n⫺1 ,共 h⫺1n 兲 IJ ⫽␭n⫺1

共5.30兲

where H⫺1 is the inverse of the Hamiltonian matrix. We
perform iterative calculations of the small eigenvalue problem given by Eq. 共5.29兲. The energy approaches from below
the inverse of the full-CI energy and at convergence, we
obtain the inverse energy and the eigenvector that are identical to the full-CI values.
In the above formulation, the variable operator S of the
ICI was defined by using the regular Hamiltonian. In the I-R
and I-I cases, we define it using the inverse Hamiltonian.
First, we define a partition of the inverse Hamiltonian,
ND

H⫺1 ⫽

兺I d I i HI ,共5.31兲

and define the i S operator as
ND

i

S⫽

兺 i C I i HI .

I⫽1

In iterations, the energy approaches from above the full-CI
energy.
Next, in the I-I case, we apply the inverse variational
principle to the ICI recurrence given by Eq. 共5.34兲 and obtainthe secular equation of the dimension N D ⫹1 asii
共 i h⫺1n ⫺ E n sn 兲 Cn ⫽0,

共5.32兲

* i HI* H⫺1i HJ ␭n⫺1 ,共 i h⫺1n 兲 IJ ⫽␭n⫺1

␭n ⫽ 共 1⫹ Sn 兲 ␭n⫺1 .

冉兺
ND

␭n ⫽

I⫽0

i

冊

C I,n i HI ␭n⫺1 ,共5.34兲

where i H0 ⫽1.
We apply first the regular variational principle to the ICI
wave function given by Eq. 共5.34兲 and obtain the secularequation for the I-R case as
共 i hn ⫺E n i sn 兲 Cn ⫽0,and

The hydrogen atom is the simplest basic system thatshows the importance of the ISE concept, because the regular
Hamiltonian has a singularity due to the nuclear attraction
operator. The ordinary ICI theory 共GICI in the R-R case兲 isproblematic for this singularity, but we show that the generalized ICI theory in the I-R and I-I cases has no problem. The
ECC study will be given in the forthcoming paper 关38兴.
The Hamiltonian for the radial part with zero angular momentum is written as
H⫽k⫹ ␯ ⫽⫺共5.35兲

where the matrix elements of i hn and i sn are given by* i HI* Hi HJ ␭n⫺1
共 i hn 兲 IJ ⫽␭n⫺1

VI. GENERALIZED ICI THEORY APPLIEDTO HYDROGEN ATOM

共5.33兲

For convenience, we assign the coefficient i C 0 for the unit
matrix 1 of the above ICI equation, and redefine the ICI as

共5.36兲

共5.39兲

and the overlap matrix i sn is common to that given by Eq.
共5.37兲. After iteration, the solution of Eq. 共5.38兲 should approach from below the inverse of the full-CI energy.
In the above formulation, we used the matrices H, HI , etc.
of the full-CI dimension, since we adopted the complete basis functions 兵 ␹ i 其 for our matrix representation. In actual ICI
calculations, it is unnecessary to keep such huge matrices inmemory. On the other hand, the matrices that appear in the
ICI secular equations, such as hn and sn of Eq. 共5.26兲, have
only N D ⫹1 dimension. Actually, the present matrix representation is equivalent to calculating the necessary integrals
by the resolution of identity method using the complete basis
兵 ␹ i 其 关6兴.
In the matrix formulation, the singularity problem is not
so explicit, because all the individual matrices are well defined. For a small basis function space, no problem would
occur except that the convergence would be slower in the
problematic case. But, as the basis function space approaches
completeness, the product of the matrices approaches thematrix of the operator product, so that the same problem asin the analytical procedure would occur in the problematiccase even in the matrix formulation. We see below such an
example in the application to hydrogen atom.

Then, the ICI is given by the recurrencei

共5.38兲

where the matrix i h⫺1
n is defined by

共5.29兲

where sn is the same as that given by Eq. 共5.28兲 and h⫺1
n is

共5.37兲

1 d2 1 d 1⫺ ,⫺2 dr 2 r dr r

共6.1兲

where the first two terms represent the kinetic operator k andthe last term represents the nuclear attraction potential ␯.
This Hamiltonian has a singularity at the origin in the last
two terms, so that the higher elements of the ordinary Krylov
sequence have a strong singularity at the origin and do not

052122-9

HIROSHI NAKATSUJI

PHYSICAL REVIEW A 65 052122

satisfy the QM condition. For example, the following integral diverges even for the exact wave function ␺ ⫽exp(⫺r)
关30兴:

具 ␺ 兩 共 ⫺1/r 兲 H 共 ⫺1/r 兲 兩 ␺ 典 ⫽⫺⬁,TABLE I. Energy of the hydrogen atom calculated by the I-R
type ICI method with increasing accuracy of the i S operator.
i

共6.2兲

S of Eq. 共6.7兲
␺0
i
S(1)
i
S(2)
i
S(3)
i
S(4)
i
S(5)
i
S(6)
i
S of Eq. 共6.10兲
␺0
i
S(1)
i
S(2)
i
S(3)
i
S(4)
i
S(5)
i
S(6)

共6.3兲

We define the positive Hamiltonian by shifting the originasH p ⫽H⫹1.

共6.4兲

We show here the application of the generalized ICI theory
to the hydrogen atom first in the analytical form and then in
the matrix representation.
A. Analytical application of the generalized ICI theory

The ordinary ICI method, which is the generalized ICI in
the R-R case, is problematic, because of the Coulombic singularity due to the nuclear attraction operator. Some integrals
involved are divergent, just like the integral shown in Eq.
共6.2兲. Therefore, the ICI in the R-R case is not applicable, in
its analytical form, even to the hydrogen atom. This problem
is circumvented by using the generalized ICI theory based on
the ISE concept, in particular, the cross-H-square theorem.
We show below the use of the generalized ICI theory in the
I-R case.
To utilize the ISE concept, we need an explicit expressionof the inverse Hamiltonian. Different from the matrix formulation, this is not necessarily straightforward in the analytic
treatment. But, we may write the inverse of the Hamiltonian
given by Eq. 共6.1兲 as
H

⫺1

冉

冊

1
1
1 ⫺1
⫽
⫽
1⫹ k
k⫹ ␯ ␯
␯⫽冉 冊 冉 冊 冎

再共6.5b兲

In Eq. 共6.5b兲, the kinetic operator is not in the inverse form.
Putting 1/␯ ⫽⫺r, we see no singularity in the inverse Hamiltonian. The inverse of the positive Hamiltonian given by Eq.
共6.4兲 is written similarly as冉冊1k⫹1 ⫺1
1
1⫹
⫽
H ⫺1
p ⫽
k⫹ ␯ ⫹1 ␯
␯
⫽

1

冉

冊

k⫹1 I⫺1

兺 ⫺ ␯
I⫽1 ␯

.

共6.6a兲

共6.6b兲

The generalized ICI theory is defined by using the positive Hamiltonian given by Eqs. 共6.4兲 and 共6.6兲. Based on theexpression of the inverse positive Hamiltonian given by Eq.
共6.6b兲, we define the variable operator i S as in Eq. 共5.11兲,

Ec

4
3
3
3
2
1

0.625
0.507 054 40.500 597 40.500 042 6
0.500 002 6
0.500 000 1
0.500 000 0

4
3
2
2
2
1

0.625
0.507 054 40.500 413 20.500 020 7
0.500 001 1
0.500 000 1
0.500 000 0a

Truncated i S operator.
Number of iterations at which the energy converges by seven decimal figures.
c
Positively shifted by ⫹1.0. The exact energy is 0.5 a.u. for the
positive Hamiltonian and ⫺0.5 a.u. for the nonpositive Hamiltonian.
b

⬁

i

S⫽

1

冉

冊

k⫹1 I⫺1

兺 CI ␯ ⫺ ␯
I⫽1

.

共6.7兲

In this case, the number of divisions N D is infinity. Using thisoperator, the generalized ICI wave function is defined by therecurrence␺ n ⫽ 共 1⫹ i S n 兲 ␺ n⫺1 ,共6.5a兲

1 2
1 3
1
1
1⫺ k⫹ k ⫺ k ⫹¯ .
␯
␯
␯
␯

nb

i

but the next one does not,具 ␺ 兩 共 ⫺r 兲 H 共 ⫺r 兲 兩 ␺ 典 ⫽⫺1.0.

Sa

共6.8兲

where n is the iteration number.
We calculate the ICI wave function by successively truncating the terms of Eq. 共6.7兲: first i S(1)⫽C 1 (1/␯ ), secondi
S(2)⫽C 1 (1/␯ )⫺C 2 (1/␯ 2 )(k⫹1), third i S(3)⫽C 1 (1/␯ )⫺C 2 (1/␯ ) 关 (k⫹1)/ ␯ 兴 ⫹C 3 (1/␯ ) 关 (k⫹1)/ ␯ 兴关 (k⫹1)/ ␯ 兴 , andso on. The initial guess is the Slater-type function

␺ 0 ⫽exp共 ⫺ ␣ r 兲 ,共6.9兲

with ␣ ⫽1.5 for which the energy expectation value is 0.625
a.u. with the positive Hamiltonian 共⫺0.375 a.u. with theregular Hamiltonian兲. ␣ ⫽1.0 is the exact wave function with
E p ⫽0.5 a.u. for H p 共E⫽⫺0.5 a.u. for the regular H兲. We
optimize the unknown variables C i by the regular variational
method for the shifted positive Hamiltonian. This is the GICIin the I-R case and the basic secular equation, which is given
by Eqs. 共5.13兲 and 共5.14兲, becomes the cross-H-square equation at convergence.
Table I gives the overall result and the convergence process is shown in Table II for i S(1), i S(3), and i S(5). When
we use only i S(1), the ICI converges by four iterations: the052122-10

INVERSE SCHRÖDINGER EQUATION AND THE EXACT . . .

PHYSICAL REVIEW A 65 052122

TABLE II. Convergence process for each i S(I) (I⫽1,3,5) of
Eq. 共6.7兲.

We emphasize here that no problem of the Coulombic
singularity has occurred in the above calculations of the I-R
case of the GICI theory, in contrast to the case of using theordinary Hamiltonian operator in the ordinary ICI method,which is the R-R case of the GICI theory. This would also
have been the case in the t-expansion method 关28兴 and theconnected-moments method 关29兴. Thus, the use of the ISE
concept given in this paper would overcome a well-known
long-continued difficulty of singularities in the study of accurate wave functions.

S

Iteration

Energya

i

S(1)

i

S(3)

0
1
2
3
4
0
1
2
3
0
1
2

0.625
0.508 974 6
0.507 112 6
0.507 056 3
0.507 054 4
0.625
0.500 045 90.500 042 7
0.500 042 6
0.625
0.500 000 2
0.500 000 1

i

i

a

S(5)

B. Application of the generalized ICI theory
in matrix representation

The variable coefficient C I converges toward zero.

coefficient C 1 converges toward 0.0 and the converged energy is 0.507 054 a.u. which is higher than the exact value,
0.5 a.u. This is because i S(1) is only a part of the inverse
Hamiltonian. For obtaining the exact solution, the i S operator must include all parts of the divided inverse Hamiltonian.
When we use i S(2), the converged energy is 0.500 597, becoming closer to 0.5. For i S(3), the convergence is very
quick as shown in Table II and the converged energy
0.500 043 is closer to 0.5. With i S(5), the converged energy
becomes much closer to 0.5, and with i S(6), the energy
converges to the exact value within seven decimal figures.
The convergence is good not only for the individual level ofi
S(I), but also for the overall level. Though the i S operator
is composed of infinite number of terms in the expression
given by Eq. 共6.7兲, only the first six terms are enough to get
the energy correct to seven decimal figures.
In the expansion of the i S operator given by Eq. 共6.7兲, theoperator (k⫹1) can be omitted, and we can expand the i S
operator solely by the inverse potential ⫺r,
⬁

冉冊

1 I
i
S⫽
CI
.
␯
l⫽1

兺

We show here the usage of the present concept of the SE
and the ISE by applying the matrix representation of thegeneralized ICI theory to the hydrogen atom. This application is particularly interesting because we can show all fourcases of the generalized ICI theory. Different from the analytical case given above, we have an explicit form of theinverse Hamiltonian matrix without ambiguity.
The basis functions of the matrix representation are the
STO-NG sets 共N⫽4 and 8兲 of O-ohata, Taketa, and Huzinaga 关39兴. For the STO-8G orbital, we further added two
GTO’s of very large exponents, 2000 and 10 000, to reinforce the region close to the nucleus. The orthonormal basis
orbitals 兵 ␹ i 其 are made by diagonalizing the overlap matrix
and transforming the basis orbitals in terms of the Löwdin
orthogonalization 关40兴. By diagonalizing the positive Hamiltonian matrix defined by Eq. 共6.4兲, we obtain the energy ofthe ground state as 0.501 009 a.u. and 0.500 004 a.u., respectively, by the STO-4G and STO-(8⫹2)G sets. The corresponding inverse energy is 1.995 970 a.u. and 1.999 985 a.u.,
respectively. Thus, these energies and inverse energies are
our goal in the matrix representation GICI calculations.
In doing the GICI, the Hamiltonian and the inverse
Hamiltonian are divided asH⫽v⫹ 共 k⫹1兲共6.11兲

H⫺1 ⫽v⫺1 ⫹r,

共6.12兲

and共6.10兲

This is possible because the operator (k⫹1) in Eq. 共6.7兲
gives the terms in lower orders of r, which already exist in
the individual i S(I) term. In other words, each i S(I) term of
Eq. 共6.7兲 can be rearranged in the form of the i S(I) term of
Eq. 共6.10兲: no new order terms appear in Eq. 共6.7兲 in comparison with Eq. 共6.10兲. Further, we can look at Eq. 共6.10兲 as
an expansion of the i S operator by the basic variable r of thissystem: for the S state of the hydrogen atom, the basic variable is only one that is r and, therefore, the i S operator
should be able to be expanded by this variable as given by
Eq. 共6.10兲.
In Table I we also give the converged energy for eachchoice of i S(I) of Eq. 共6.10兲 from I⫽1 to I⫽6. The initialenergies by the i S(I) of Eqs. 共6.7兲 and 共6.10兲 were completely the same and the converged energies were also close
to each other.

respectively, where v is the matrix of the potential operator,
k the matrix of the kinetic operator, and r is the residual
matrix. The S and i S operators are defined, corresponding to
the above equations, asS⫽C ␯ v⫹C k 共 k⫹1兲共6.13兲

S⫽ i C ␯ v⫺1 ⫹ i C r r,

共6.14兲

andirespectively.
Table III shows the converging process of the four typesof the generalized ICI calculations of the hydrogen atom
based on the STO-4G basis set. The initial reference function ␺ 0 is the second outermost Löwdin orthogonalized orbital: the first one was too good as the initial guess to show

052122-11

HIROSHI NAKATSUJI

PHYSICAL REVIEW A 65 052122

TABLE III. Generalized ICI calculations in matrix form for thehydrogen atom using a positive Hamiltonian for the STO-4G basis
set. The correct energy for the positive Hamiltonian of the STO-4G⫺1basis is E p ⫽0.501 009 a.u. and E ⫺1
p ⫽1.995 970 a.u. .
EnergyInteration

R-R case

I-R case

R-I case

I-I case

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

0.554 112
0.522 819
0.507 856
0.503 1710.501 761
0.501 103
0.501 043
0.501 022
0.501 014
0.501 011
0.501 010
0.501 010
0.501 010
0.501 009

0.827 107
0.502 693
0.501 013
0.501 009

1.909 055
1.949 066
1.975 771
1.987 305
1.991 964
1.994 120
1.995 084
1.995 546
1.995 763
1.995 869
1.995 921
1.995 946
1.995 958
1.995 964
1.995 967
1.995 969
1.995 969
1.995 970

1.758 407
1.995 881
1.995 969
1.995 970

TABLE IV. Generalized ICI calculations in matrix form for thehydrogen atom using the positive Hamiltonian for the STO-(8⫹2)G basis set. The correct energy for the positive Hamiltonianof the STO-(8⫹2)G basis is E p ⫽0.500 004 a.u. and E ⫺1p⫽1.999 985 a.u.⫺1 . For the R-R and R-I cases, the convergence
was too slow to be realistic.
I-R case

I-I case

Iteration

Guess 1aGuess 2b

Guess 1aGuess 2b

1
2
3
4
5
6

0.505 034
0.500 1610.500 013
0.500 005
0.500 004

0.973 440
0.502 669
0.500 287
0.500 024
0.500 005
0.500 004

1.960 160
1.999 8021.999 972
1.999 984
1.999 985

1.930 675
1.998 416
1.999 825
1.999 971
1.999 984
1.999 985a

The outermost orthogonalized orbital is the initial guess function
␺0 .
bThe second outermost orthogonalized orbital is the initial guess
function ␺ 0 .

the converging process. With the rather crude STO-4G basis,
all four types of the generalized ICI calculations have
converged. The R-R and I-R cases converge to E p⫽0.501 009 a.u. and the R-I and I-I cases to E ⫺1p⫽1.995 970 a.u. In the analytical case, the R-R and R-I cases
involve the calculations of the diverging integrals such asthat expressed by Eq. 共6.2兲. However, in the matrix representation with the crude STO-4G basis set, even these problematic cases showed no difficulty in the convergence process.
On the other hand, the I-R and I-I cases that have no intrinsic
theoretical problem show remarkably good convergence:
they converge with only four iterations, in contrast to 14 and18 for the R-R and R-I cases. In the R-R and I-R cases, which
use the normal variational principle, the energy converges
from above, and in the R-I and I-I cases, which use the inverse variational principle, the energy converges monotonically from below to the correct value.
We next show the GICI calculations for the STO-(8⫹2)G basis set. The STO-8G basis is already a much better
basis than the STO-4G basis. The energy of the former is
0.500 009 a.u. in comparison with that of the latter, 0.501 009
a.u. We further added two large exponent bases in order toreinforce the inner region of the AO where the divergence of
the integrals such as that given by Eq. 共6.2兲 is dictated. By
this addition, the energy decreases slightly to 0.500 004 a.u.
The initial reference function ␺ 0 is the outermost Löwdin
orthogonalized orbital.
For this better basis, the convergences of the R-R and R-Icases become much worse than for the STO-4G basis: actually, the convergence was too slow to be realistic. This is

expected since these cases are very problematic in the analytical case because of the existence of the diverging integrals. As the basis set is improved, the behavior of the matrix
formulation would become closer to the analytical one. Thus,
Table IV shows only the I-R and I-I cases. For these cases,
however, the convergence was quite fantastic: the energy
converges only with five iterations in both cases. This may
be expected, since there is no intrinsic theoretical problem inthese cases, but the performance is much better than expected. Since the convergence was so beautiful for the I-R
and I-I cases, we tried the calculations using a worse initial
guess, which is the second outermost orthogonalized orbital.
Again, both of the I-R and I-I cases showed quite a niceconvergence with only six iterations. For the I-R case, it
converges monotonically from above to the correct value,and for the I-I case, it converges from below to the correct
value.
It is interesting to see the effect of using the positive
Hamiltonian. It has been shown in the previous section thatwithout introducing the positive Hamiltonian, a natural connection between the regular and the inverse worlds is difficult and we cannot obtain the cross-H-square equations, sothat the formulation of the I-R and R-I cases becomes impossible.
Table V shows the generalized ICI calculations for the
STO-4G basis set using a nonpositive Hamiltonian, whoseenergy eigenvalue is ⫺0.498 990 a.u. and ⫺2.004 046 a.u.
for the inverse case. The problematic R-R and R-I cases show
a behavior similar to that of Table III. The I-R case, which
has no theoretical problem with the positive Hamiltonian,shows some problem with the nonpositive Hamiltonian, because the cross-H-square theorem does not hold in this case.
Namely, the convergence becomes much worse when we use
the nonpositive Hamiltonian 共13 times in comparison with
only five times兲. The I-I case shows a good convergence
共only four iterations兲 and is insensitive to whether the Hamiltonian is positive or not, as seen from the fact that the052122-12

INVERSE SCHRÖDINGER EQUATION AND THE EXACT . . .

PHYSICAL REVIEW A 65 052122

TABLE V. Generalized ICI calculations in matrix form with the
nonpositive Hamiltonian for the hydrogen atom with the
STO-4G basis set. The correct energy for the STO-4G basis isE⫽⫺0.498 990 a.u. and E ⫺1 ⫽⫺2.004 046 a.u.⫺1 .

parison with Table IV, the effect of taking a positive Hamiltonian is remarkable for the I-R case. In the nonpositive
Hamiltonian case, which is theoretically bad in the I-R case,the convergence to the correct value, ⫺0.499 996 a.u., takes
50 iterations in contrast to only five 共guess 1兲 shown in Table
IV for the positive Hamiltonian case. This is a very naturalconsequence, since the calculation must be done as thetheory requires. Even in the I-I case, the convergence for the
positive Hamiltonian is better than the nonpositive case.
With the nonpositive Hamiltonian, the I-I case gives the convergence from above, different than ‘‘from below’’ seen forthe positive Hamiltonian given in Table IV.

EnergyIteration

R-R case

I-R case

R-I case

I-I case

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

⫺0.445 888⫺0.477 181⫺0.492 144⫺0.496 829⫺0.498 239⫺0.498 729⫺0.498 897⫺0.498 957⫺0.498 978⫺0.498 986⫺0.498 989⫺0.498 990⫺0.436 302⫺0.496 044⫺0.498 222⫺0.498 678⫺0.498 842⫺0.498 920⫺0.498 956⫺0.498 973⫺0.498 982⫺0.498 986⫺0.498 988⫺0.498 989⫺0.498 990⫺1.588 918⫺1.834 904⫺1.940 275⫺1.980 685⫺1.995 473⫺2.000 914⫺2.002 900⫺2.003 627⫺2.003 893⫺2.003 990⫺2.004 025⫺2.004 039⫺2.004 043⫺2.004 045⫺2.004 046⫺1.965 856⫺2.003 510⫺2.004 033⫺2.004 046

VII. CONCLUSION

H ⫺1 -square equation is valid for both nonpositive and positive Hamiltonians. It is noteworthy that both the R-I and I-I
cases of the nonpositive Hamiltonian converge from above tothe correct value, as the ordinary variational case, contrary to
the behavior from below in Table III, which is theoreticallycorrect for the positive Hamiltonian.
Table VI shows a similar convergence process for the
nonpositive Hamiltonian with STO-(8⫹2)G basis. In com-

Introducing the inverse of the Hamiltonian, we obtain the
inverse Schrödinger equation 共ISE兲 that is entirely equivalent
to the ordinary Schrödinger equation 共SE兲. The ISE has thesame determinative power as the regular SE. The inverse
forms of the variational principle and the H-square group ofequations that are equivalent to the ISE are derived, and by
further shifting our Hamiltonian to be positive, the inverse
Ritz variational principle and the cross-H-square equations
are derived. For the positive Hamiltonian, we obtain a monotonic correspondence between the energy eigenvalues of theregular and inverse Hamiltonians. In the variational process,the inverse energy approaches ‘‘from below’’ the exact energy, in contrast to ‘‘from above’’ in the ordinary case. The
knowledge on the equivalent sets of equations on the regularand inverse sides of the Hamiltonian, including the SE and
ISE, not only makes the imaginations on physical and chemical phenomena fertile and gives a deep insight on the natureof the quantum-mechanical entities, but also provides an ef-

TABLE VI. Generalized ICI calculations in matrix form with the nonpositive Hamiltonian for thehydrogen atom with the STO-(8⫹2)G basis set. The correct energy for the STO-(8⫹2)G basis isE⫽⫺0.499 996 a.u. and E ⫺1 ⫽⫺2.000 015 a.u.⫺1 In the R-R case, the convergence was too slow to berealistic, and in the R-I case, the calculations were diverging.
I-R case

I-I case

Iteration

EnergyIteration

EnergyIteration

EnergyIteration

Energy

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

⫺0.449 651⫺0.491 420⫺0.497 023⫺0.497 824⫺0.498 260⫺0.498 542⫺0.498 784⫺0.498 981⫺0.499 147⫺0.499 285⫺0.499 400⫺0.499 497⫺0.499 578⫺0.499 646⫺0.499 703

16
17
18
19
20
21
22
23
24
25
26
27
28
29
30

⫺0.499 751⫺0.499 791⫺0.499 824⫺0.499 852⫺0.499 875⫺0.499 895⫺0.499 912⫺0.499 925⫺0.499 937⫺0.499 946⫺0.499 955⫺0.499 961⫺0.499 967⫺0.499 972⫺0.499 976

31
32
33
34
35
36
37
38
39
40,41
42– 44
45– 4950⫺0.499 979⫺0.499 982⫺0.499 984⫺0.499 986⫺0.499 988⫺0.499 989⫺0.499 990⫺0.499 991⫺0.499 992⫺0.499 993⫺0.499 994⫺0.499 995⫺0.499 996

1
2
3
4
5
6
7
8
9
10
11
12
13
14

⫺1.931 726⫺1.943 310⫺1.989 179⫺1.996 062⫺1.998 332⫺1.999 379⫺1.999 758⫺1.999 914⫺1.999 975⫺1.999 999⫺2.000 009⫺2.000 013⫺2.000 014⫺2.000 015

052122-13

HIROSHI NAKATSUJI

PHYSICAL REVIEW A 65 052122

particular, when the Hamiltonian of the system has a singularity, the integrals involving higher-order products of theregular Hamiltonian may diverge and, therefore, the method
may be problematic, but even in such a case, when we use
the inverse Hamiltonian, we can calculate the exact wave
function without having such a difficulty.
The generalized ICI theory has been applied to the hydrogen atom. Since the regular Hamiltonian has a Coulombic
singularity at the origin, the ordinary ICI theory in the R-Rcase is problematic: it involves the integrals that do not converge in the analytical formulation. Thus, even the hydrogen
atom is a very good system to test the utility of the ISEbased concept. We have performed generalized ICI calculations in both analytical and matrix formulations and we
could calculate the ground state of the hydrogen atom without the problem of singularity. The matrix formulation
showed some interesting behaviors that are clearly attributable to the SE and SE concepts given in this paper.
In conclusion, the concept of the regular and inverse
Hamiltonians and the various formulas originating fromthem seem to give an important insight for further studies
aiming at the exact understanding of atomic and molecularsystems.

fective and powerful method for formulating and calculating
the exact wave function. This is the purpose of the present
series of studies 关2– 6,38兴.
The Hamiltonian Krylov sequence is extended to include
the inverse Hamiltonian, giving the inverse Krylov sequence.
Since the Hamiltonian often involves Coulombic singularities, the element of the ordinary Krylov sequence, H n ␺ 0 withn⭓1, strongly breaks the quantum-mechanical condition
共i.e., integrable finiteness兲, while the inverse Krylov sequence does not have such a singularity problem. Combining
the ordinary and inverse Krylov sequences, we obtain thecomplete Krylov sequence, which provides a natural basis
for describing the exact wave function.
The SE and ISE concepts are combined to generalize the
ICI method for calculating the exact wave function. Basedon the variational principles in the regular and inverse forms,
and on the H-square, inverse H-square, and cross-H-squareequations, we formulated the four different cases of generalized ICI theory. They correspond to the two different formations of the variable operator S, depending on the use of H or
H ⫺1 , and to the use of the two different variational principles, either regular or inverse. They have clear internal relations, and all four ICI methods give, in principle, the exact
wave function. In these formulations, we used a positively
shifted Hamiltonian for consistency. Further, we formulated
the generalized ICI theory in a matrix representation, where a
merit is that the definition of the inverse Hamiltonian isstraightforward in comparison with the analytical case. The
four different cases of ICI theory would have merits anddemerits depending on the system, and the convergence ratewould also be different, again depending on the system. In

The author thanks Professor J. Cioslowski for some valuable conversations at the Girona Symposium and Dr. M.
Hada, Dr. M. Ehara, and Dr. J. Hasegawa for some suggestions. This study has been supported financially by the special fund from the Ministry of Education, Science, Culture,and Sports.

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