Many-Center Coulomb Sturmians andShibuya–Wulfman Integrals

JOHN AVERYH. C. Ørsted Institute, University of Copenhagen, Universitetsparken 5,Copenhagen, DK-2100, Denmark

Received 29 August 2003; accepted 1 October 2003Published online 2 December 2003 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.10820

ABSTRACT: When momentum space is projected onto the surface of a unit 4-Dhypersphere by means of Fock’s mapping, Coulomb Sturmian basis functions can besimply represented in terms of hyperspherical harmonics. The properties of theseharmonics can be used to evaluate Shibuya–Wulfman integrals and other integrals thatarise when the Sturmian basis functions are used in molecular calculations. © 2003Wiley Periodicals, Inc. Int J Quantum Chem 100: 121–130, 2004

Key words: Sturmians; hyperspherical harmonics; Shibuya–Wulfman integrals;exponential-type orbitals

Introduction

T he solution of the Schrodinger equation for thehydrogen atom was a landmark in the early

history of quantum theory, and it was natural tothink of using the set of solutions as a basis forbuilding up solutions to more complicated prob-lems. However, it soon became clear that sets ofhydrogen-like orbitals are awkward to use as basissets because they are not complete unless the con-tinuum is included. To remedy this situation, Hyl-eraas, Shull, Lowdin, and others [1] introduced setsof exponential-type radial basis functions that con-sisted of polynomials in r multiplied by a factore�kr, the constant k being the same for all the mem-bers of the set. These authors were able to show that

such sets were complete without the inclusion ofthe continuum in the sense that any well-behavedfunction of r obeying the same boundary conditionscould be expressed in terms of them. Later, Roten-berg named these functions “Sturmians” to empha-size their connection with Sturm–Liouville theory[2, 3]. Goscinski generalized the Sturmian conceptby interpreting Sturmian basis sets as solutions toan approximate Schrodinger equation with aweighted potential, the weighting factors beingchosen in such a way as to make all the functions inthe set isoenergetic [4–7]. The polynomials that ap-pear in a Sturmian basis set can be chosen in manyways, but in this article we will consider CoulombSturmians, i.e., the particular choice where the Stur-mian basis set is identical with the familiar set ofhydrogen-like orbitals except that Z/n is replacedby the constant k. In other words, when we couplethe radial Sturmian basis set with the appropriatespherical harmonics, we consider the set of func-tions

This article is dedicated to the memory of Prof. Herbert W.Jones, one of the most important pioneers of the use of expo-nential-type orbitals in quantum chemistry.

International Journal of Quantum Chemistry, Vol 100, 121–130 (2004)© 2003 Wiley Periodicals, Inc.

�nlm�x� � Rnl�r�Ylm��, ��, (1)

where

Rnl�r� � �nl�2kr�le�krF�l � 1 � n�2l � 2�2kr�

�nl �2k3/ 2

�2l � 1�! � �l � n�!n�n � l � 1�! , (2)

F(a�b�x) being a confluent hypergeometric function.

F�a�b�x� � 1 �axb �

a�a � 1� x2

b�b � 1�2! � . . . . (3)

It is easy to verify that these functions are just thefamiliar hydrogen-like orbitals, where we havemade the replacement Z/n 3 k. Making the samereplacement in the Schrodinger equation for thehydrogen atom, we can see that our Coulomb Stur-mian basis set obeys the one-electron wave equa-tion:

��12 �2 �

12 k2��nlm�x� �

nkr �nlm�x�. (4)

This equation can be used to show that the set ofCoulomb Sturmians obeys potential-weighted or-thonormality relations because another (complexconjugated) member of the set obeys a similar equa-tion,

��12 �2 �

12 k2��*n�l�m��x� �

n�kr �*n�l�m��x�. (5)

Multiplying (4) and (5) on the left, respectively, by�*n�l�m�(x) and �nlm(x), subtracting (5) from (4), inte-grating, and making use of the Hermeticity of theoperator in square brackets, we obtain the relation

� d3x�*n�l�m��x���12 �2 �

12 k2��nlm�x�

� � d3x�nlm�x���12 �2 �

12 k2��*n�l�m��x�

� 0 � �n � n��k � d3x�*n�l�m��x�1r �nlm�x�, (6)

from which it follows that

� d3x�*n�l�m��x�1r �nlm�x� � 0 if n � n�. (7)

When the Coulomb Sturmians are normalized as isshown in Eq. (2), the potential-weighted resultingorthonormality relations can be written in the form

� d3x�*n�l�m��x�1r �nlm�x� �

kn �n�n�l�l�m�m, (8)

where the factor �l�l�m�m results from the orthonor-mality of the spherical harmonics. Because the Cou-lomb Sturmians are identical with the familiar hy-drogen-like orbitals, except the Z/n has beenreplaced by k, they also obey the relation

� d3x��nlm�x��2 � 1. (9)

Fock’s Mapping: HypersphericalHarmonics

In an early and remarkably brilliant article [8, 9],Fock was able to show that when momentum spaceis mapped onto the surface of a 4-D unit hyper-sphere by the transformation

u1 �2kp1

k2 � p2 � sin � sin �p cos �p

u2 �2kp2

k2 � p2 � sin � sin �p sin �p

u3 �2kp3

k2 � p2 � sin � cos �p

u4 �k2 � p2

k2 � p2 � cos � (10)

the Fourier transformed Coulomb Sturmians can beexpressed simply in terms of 4-D hypersphericalharmonics [8–24]. Fock showed that

�n,l,mt �p� � M�p�Yn�1,l,m�u�, (11)

where

M� p� �4k5/ 2

�k2 � p2�2 (12)

AVERY

122 VOL. 100, NO. 2

and where the Fourier transform and its inverse aregiven by

�nlm�x� �1

�2�3/ 2 � d3peip�x�nlmt �p�

�nlmt �p� �

1�2�3/ 2 � d3xe�ip�x�nlm�x�. (13)

The 4-D hyperspherical harmonics in Eq. (12) aredefined by the relationships

Y,l,m�u� � �,lC�ll�1�u4�sinl�Yl,m��p, �p� (14)

and

�,l � ��1�il�2l �!!�2� � 1�� � l �!� � l � 1�! , (15)

where Yl,m(�p, �p) is a spherical harmonic and C� is

a Gegenbauer polynomial:

C��u4� � �

t�0

�/ 2 ��1�t� � � � t�t!� � 2t�!���

�2u4��2t.

(16)

The first few 4-D hyperspherical harmonics definedby Eqs. (10) and (14)–(16) are shown in Table I.Aquilanti and coworkers at the University of Peru-

gia studied alternative orthonormal sets of 4-D hy-perspherical harmonics and the transformationslinking them [25–32]. One such alternative set (Ta-ble II) corresponds, through Fock’s mapping, to theCoulomb Sturmians that result when the hydrogen-like Schrodinger equation is separated in paraboliccoordinates [19].

Shibuya–Wulfman Integrals

If we let

���x� � �nlm�x � Xa�, (17)

where � stands for the set of indices {n, l, m, a}, thenwe can define a set of integrals ���,� by the relation-ship

� d3x�*���x���12 �2 �

12 k2����x� � k2���,�.

(18)

These integrals were first introduced and studiedby Shibuya and Wulfman, who extended Fock’smomentum-space treatment of the hydrogen atomto the problem of an electron moving in a many-center Coulomb potential [17–63]. In Shibuya andWulfman’s pioneering article [33], the integrals ���,�are expressed, through Fock’s mapping, as integralsover hyperspherical harmonics:

TABLE I ______________________________________4-D hyperspherical harmonics.

l m �2 Y,l,m(u)

0 0 0 11 1 1 i�2 (u1 � iu2)1 1 0 � i2u3

1 1 �1 � i�2 (u1 � iu2)1 0 0 � 2u4

2 2 2 � �3 (u1 � iu2)2

2 2 1 2�3 u3(u1 � iu2)2 2 0 � �2 (2u3

2 � u12 � u2

2)2 2 �1 � 2�3 u3(u1 � iu2)2 2 �2 � �3 (u1 � iu2)2

2 1 1 � i2�3 u4(u1 � iu2)2 1 0 2i�6 u4u3

2 1 �1 2i�3 u4(u1 � iu2)2 0 0 4u4

2 � 1

TABLE II ______________________________________Alternative hyperspherical harmonics.

n1 n2 m �2 Yn1,n2,m(u)

0 0 0 10 0 1 � i�2 (u1 � iu2)0 0 �1 � i�2 (u1 � iu2)1 0 0 �2 (u4 � iu3)0 1 0 �2 (u4 � iu3)0 0 2 � �3 (u1 � iu2)2

0 0 �2 � �3 (u1 � iu2)2

2 0 0 �3 (u4 � iu3)2

0 2 �0 �3 (u4 � iu3)2

1 1 0 �3 (u12 � u2

2 � u32 � u4

2)1 0 1 �6 (u4 � iu3)(u1 � iu2)1 0 �1 �6 (u4 � iu3)(u1 � iu2)0 1 1 �6 (u4 � iu3)(u1 � iu2)0 1 �1 �6 (u4 � iu3)(u1 � iu2)

COULOMB STURMIANS AND SHIBUYA–WULFMAN INTEGRALS

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 123

���,� � � d�eip��Xa��Xa�Y*n��1,l�,m��u�Yn�1,l,m�u�, (19)

where d� is the solid angle element on the hyper-sphere defined by Eq. (12):

d� � sin2� sin �pd�d�pd�p � � 2kk2 � p23

d3p. (20)

To see the relationship between Eqs. (18) and (19),we first let ��

t(p) be defined by

��t�p� � e�ip�Xa�nlm

t �p� (21)

so that

���x� � �nlm�x � Xa� �1

�2�3/ 2 � d3peip�x��t�p�. (22)

Then,

��12 �2 �

12 k2����x� �

1�2�3/ 2 � d3peip�x�k2 � p2

2 ��t�p�

(23)

and

k2���,� � � d3x�*���x���12 �2 �

12 k2����x�

� � d3p�k2 � p2

2 � 1

�2�3/ 2 � d3xe�ip�x���t �x�*��

t�p�

� � d3p�k2 � p2

2 ���t*�p���

t�p�

� � d3p�k2 � p2

2 eip��Xa��Xa����t*�p���

t�p�. (24)

Substituting (12), (13), and (20) into (24), we obtain(19). Similarly, one can show that

� d3x�*���x����x�

� � d��1 � u4�eip��Xa��Xa�Y*n��1,l�,m��u�Yn�1,l,m�u� (25)

and

� d3x�*���x���12 �2����x�

�12 � d��1 � u4�eip��Xa��Xa�Y*n��1,l�,m��u�Yn�1,l,m�u�. (26)

From Eq. (19) and from the orthonormality of thehyperspherical harmonics, it follows that for thespecial case where a� � a

���,� � � d�Y*n��1,l�,m��u�Yn�1,l,m�u�

� �n�n�l�l�m�m, if a� � a. (27)

Combining this with (24), we obtain the momentumspace orthonormality relations for the Fourier-transformed Coulomb Sturmians:

� d3p�k2 � p2

2k2 �n�,l�m�t* �p��n,l,m

t �p� � �n�n�l�l�m�m.

(28)

Weniger [63] pointed out that because the CoulombSturmians obey a momentum-space orthonormalityrelation of this type, and because their direct-spaceorthonormality relation (8) can be expressed in theform

� d3x�*n�l�m��x����2 � k2

2k2 ��nlm�x� � �n�n�l�l�m�m,

(29)

the functions form the basis of a Sobolev space.

Shibuya–Wulfman Integrals andTranslations

The momentum-space orthonormality conditions(28) allow us to expand a plane wave in terms of theFourier-transformed Coulomb Sturmians. If we let

AVERY

124 VOL. 100, NO. 2

eip�x � �k2 � p2

2k2 �n�l�m�

�n�l�m�t* �p�an�l�m�, (30)

then the expansion coefficients an�l�m� can be deter-mined by multiplying (30) from the left by �nlm

t (p)and integrating over momentum space. Then, withthe help of (28) we obtain

� d3p�nlmt �p�eip�x

� �n�l�m�

an�l�m� � d3p�k2 � p2

2k2 �nlmt �p��n�l�m�

t* �p�

� �n�l�m�

an�l�m��n�n�l�l�m�m � anlm. (31)

Comparing this result with (13) we can see that

anlm � �2�3/ 2�nlm�x� (32)

so that

eip�x � �2�3/ 2�k2 � p2

2k2 �nlm

�nlmt* �p��nlm�x�. (33)

The expansion shown in (33) was first obtained byShibuya and Wulfman, who used it to evaluate theintegrals ���,�. Aquilanti demonstrated that theplane-wave expansion (33) can also be used to de-rive a representation of translations based on theCoulomb Sturmians. The derivation is as follows:From (13) we have

�nlm�x � Xa� �1

�2�3/ 2 � d3peip��x�Xa��nlmt �p�. (34)

But, from (33) it follows that

eip�x � �2�3/ 2�k2 � p2

2k2 eip�Xa� �n�l�m�

�n�l�m�t* �p��n�l�m��x�.

(35)

Replacing eip�x in (34) by this expansion, we obtain

�n,l,m�x � Xa� � �n�l�m�

�n�l�m��x � Xa��

� d3p�k2 � p2

2k2 eip��Xa��Xa��n�l�m�t* �p��nlm

t �p�, (36)

and comparing this result with (24) we can see theShibuya–Wulfman integrals form a representationof translations based on the Coulomb Sturmians.Thus, a Coulomb Sturmian located on the center Xa

can be expanded in terms of Coulomb Sturmianscentered at another point, Xa�, the expansion coef-ficients being the Shibuya–Wulfman integrals:

�nlm�x � Xa� � �n�l�m�

�n�l�m��x � Xa�����,�. (37)

Because two translations performed in successionproduce an effect equivalent to a single translationfrom the initial point to the final one, it follows fromEq. (37) that

�n�l�m�

�� ,�����,� � �� ,�. (38)

Further, from Eq. (24), it can be seen that

�*��,� � ��,��. (39)

Also, combining (24) and (4), we can obtain therelationship

���,� �nk � d3x�*���x�

1�x � Xa�

���x�

�nk � d3x�*n�l�m��x � Xa��

1�x � Xa�

�nlm�x � Xa�.

(40)

Equations (37) and (40) allow us to express thematrix element of a many-center nuclear attractionpotential in terms of Shibuya–Wulfman integrals:

�a� d3x�*n�l�m��x � Xa��

Qa

�x � Xa��n l m �x � Xa �

� �nlma

��,� � d3x�*n�l�m��x � Xa��Qa

�x � Xa��nlm�x � Xa�

� ��

���,�

Qakn ��,� . (41)

COULOMB STURMIANS AND SHIBUYA–WULFMAN INTEGRALS

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 125

Thus, if V(x) is the attractive potential of a set ofnuclei with effective charges Qa located at thepoints Xa,

V�x� � ��a

Qa

�x � Xa�, (42)

then the matrix elements of V(x) based on themany-center Coulomb Sturmians of Eq. (17) aregiven by

���,� � �1k � d3x�*���x�V�x��� �x� � �

�

���,�

Qa

n ��,� .

(43)

Essentially what is happening in Eq. (43) is that thefunctions �*n�l�m�(x � Xa�) and �nlm(x � Xa) are trans-lated, respectively, from the points Xa� and Xa to thepoint Xa. Here, the matrix element of the attractiveCoulomb potential of nucleus a is evaluated bymeans of Eq. (8). Note that in making this interpre-tation we must make use of Eq. (39) to derive thetranslation properties of the complex conjugatedfunction �*n�l�m�(x � Xa�). In the case of a more gen-eral many-center potential of the form

U�x� � �a

ua�x�, (44)

the rule for translations can be used to derive asimilar formula:

U��,� � � d3x�*���x�ua�x��� �x� � ��,��

���,��u��,���,� ,

(45)

where

u��,� � �a�,a � d3x�*n�l�m��x�ua�x��nlm�x�. (46)

Now, suppose that we wish to solve the Schrod-inger equation

��12 �2 � V�x� � Ej��j�x� � 0 (47)

for an electron moving in the field of a many-centerCoulomb potential. We can represent the solutionas a superposition of the basis functions ��(x):

�j�x� � ��

���x�C�, j. (48)

If we now impose the constraint

kj � ��2Ej (49)

and substitute the expansion (48) into (47), we ob-tain

��

��12 �2 �

12 kj

2 � V�x�����x�C�, j � 0. (50)

The choice of kj shown in Eq. (49) ensures that allthe members of the isoenergetic basis set have anenergy that is equal to the energy of the eigenstatethat they are meant to represent. All of the basisfunctions thus have the correct asymptotic behaviorin the region where penetration is classically for-bidden. Numerical experience shows that with atruncated basis set optimum accuracy is obtainedwhen this constraint is imposed. The special fea-tures of the Sturmian secular equation that we willderive below follow from choosing kj in accordancewith Eq. (49). Multiplying (50) from the left by aconjugate function in our basis set, integrating overthe coordinates, and making use of (24), we obtainthe set of secular equations

��

����,� � kj���,�C�, j � 0. (51)

The Sturmian secular equations, (51), have a differ-ent form from the equations that one would obtainby diagonalizing the Hamiltonian of the system.The kinetic energy term has vanished, and the eig-envalues are not energies but a spectrum of valuesof the constant k, a constant that functions as ascaling parameter in the Sturmian basis set and isrelated to the energy through Eq. (49). The matrixelements ���,� do not depend explicitly on k butrather (as we shall see) on a dimensionless param-eter s � k(Xa� � Xa). One can pick values of s tospecify the shape of a molecule, its size being un-known. Inserting these into (51), one solves thesecular equations and obtains a spectrum of k val-ues, kj, j � 1, 2, 3, 4, . . . . The largest of these valuescorresponds, through Eq. (49), to the ground state,

AVERY

126 VOL. 100, NO. 2

and for this state the basis functions fall off rapidlybecause of the rapidly decaying exponential factore�kr. Smaller values of kj correspond to excitedstates of the system, and for these states the basisset is more diffuse, as is appropriate for the descrip-tion of a less tightly bound state. Thus, one obtainsat one stroke a spectrum of energy values and alsothe most appropriate basis functions for each state.Having found the kj values, one then knows the sizeof the nuclear configuration. Repeating the calcula-tion for other values of s, one obtains curves repre-senting the energies as functions of the systemssize, as illustrated in Figure 1.

The secular Eq. (51) can be written in a differentform by introducing the matrix

K��,� � �Qa�Qa

n�n ���,�. (52)

Combining (43) and (52) we have

���,� � � n�nQa�Qa

��

K��,� K� ,� (53)

so that (51) becomes

��� �

�

K��,� K� ,� � kjK��,��C�, j � 0. (54)

Now, suppose that we have found a set of coeffi-cients Ctau, j that satisfy the secular equations

��

�K��,� � kj���,�C�, j � 0. (55)

Then, by substituting (55) into (54), we can see thatthese coefficients will also be solutions of (54). Thus,if the basis set is complete (55), (54), and (51) con-tain the same information. However, when the ba-sis set is truncated, as it always is in practice, theinformation contained in these equations is not pre-cisely the same, and experience has shown that fora given size of basis (51) gives the most accurateresults. This is especially true when direct methodsare available for evaluating the nuclear attractionintegrals, as is the case for diatomic molecules.

Evaluation of Shibuya–WulfmanIntegrals

Substituting (11) and (12) into (43), we can re-write the plane wave expansion in the form

eip�R � �2

k 3/ 2

�1 � u4� �nlm

�nlm�R�Y*n�1,l,m�u�. (56)

If we let R � Xa� � Xa, then with the help of (56) Eq.(19) can be rewritten in the form

���,� � �2

k 3/ 2 �n l m

�n l m �Xa� � Xa� � d��1 � u4�

Y*n �1,l ,m �u�Y*n��1,l�,m��u�Yn�1,l,m�u�. (57)

The Shibuya–Wulfman integrals, written in thisform, can be evaluated directly by using the 4-Dangular integration formula [18, 19]

� d� j�1

4

ujnj �

42

�n1 � n2 � n3 � n4 � 2�!!

j�1

4

�nj � 1�!! (58)

or using the alternative 4-D angular integrationformula [19]

FIGURE 1. Excited states of an electron moving inthe field of two nuclei with charges Q1 � 1 and Q2 �2. The energies, which were calculated using Eq. (51),are shown as functions of the internuclear distance R.Both the energies and distances are expressed in a.u.In the united-atom limit, the energies approach those ofthe Li2� ion.

COULOMB STURMIANS AND SHIBUYA–WULFMAN INTEGRALS

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 127

� d�F�u� � 42 ���0

� 1�2��!!�2� � 2�!! ���F�u�u�0,

(59)

where

� ��2

�u12 �

�2

�u22 �

�2

�u32 �

�2

�u42 . (60)

Equation (59) holds if F(u) is any polynomial inu1, . . . , u4. One can also use the generalized Cleb-sch–Gordan coefficients for 4-D hyperspherical har-monics. To do this we begin by evaluating theintegral

� d�eip�RYn�1,l,m�u� � �2

k 3/ 2 �n�l�m�

�n�l�m��R�

� d��1 � u4�Y*n��1,l�,m��u�Yn�1,l,m�u�. (61)

It can be shown that

� d��1 � u4�Y*n��1,l�,m��u�Yn�1,l,m�u�

� �l�l�m�m�n�n � �l�l�m�m�n�,n�1

12 ��n � l ��n � l � 1�

n�n � 1�

� �l�l�m�m�n�,n�1

12 ��n � l ��n � l � 1�

n�n � 1�(62)

and therefore

� k2

3/ 2 � d�eip�RYn�1,l,m�u�

� �nlm�R� �12 ��n � l ��n � l � 1�

n�n � 1��n�1,l,m�R�

�12 ��n � l ��n � l � 1�

n�n � 1��n�1,l,m�R�. (63)

Factoring out the spherical harmonic that is com-mon to all the functions on the right side of (63), wecan rewrite it in the form

� d�eip�RYn�1,l,m�u� � �2

k 3/ 2

fnl�s�Ylm�s�, (64)

where

s � �sx, sy, sz� � kR (65)

and where

fnl�s� � Rnl�R� �12 ��n � l ��n � l � 1�

n�n � 1�Rn�1,l�R�

�12 ��n � l ��n � l � 1�

n�n � 1�Rn�1,l�R�. (66)

The radial functions Rnl in Eq. (66) are defined byEq. (2), and it is understood that

Rn�1,l�R� � 0 if l � n � 1. (67)

The first few functions fnl(s) are shown in Table III.Equation (64) can be used to generate Shibuya–Wulfman integrals by resolving products of two4-D hyperspherical harmonics into sums of singleharmonics. For example, from Table I we can seethat

Y*1,1,�1�u�Y1,1,1�u� �1 �2

3 Y2,2,2�u� (68)

TABLE III _____________________________________fnl(s), Eq. (66), where s � k�Xa� � Xa�.

n l k � 3/2fnl(s)

1 0 e � s(1 � s)

2 0 �23 e�ss2

2 12

3�3e�ss�1 � s�

3 013 e�ss2��2 � s�

3 11

3�2e�ss�1 � s � s2�

3 21

3�10e�ss2�1 � s�

4 0 �2

15 e�ss2�5 � 5s � s2�

4 12

15�15e�ss�5 � 5s � 12s2 � 3s3�

4 22

15�5e�ss2�2 � 2s � s2�

AVERY

128 VOL. 100, NO. 2

and thus from Eqs. (19) and (68) we have

�a�,2,1,�1;a,2,1,1 �1 �2

3 � d�eip�RY2,2,2�u�

�1 �2

3 �2

k 3/ 2

f3,2�s�Y2,2�s�. (69)

Inserting the function f3,2(s) from Table III, we ob-tain

�a�,2,1,�1;a,2,1,1 �16 e�s�1 � s��sx � isy�

2. (70)

As a second example, we can also see from Table Ithat for all n, l, and m

Y*0,0,0�u�Yn�1,l,m�u� �1

�2Yn�1,l,m�u� (71)

so that Eq. (64) yields

� d�eip�RY*0,0,0�u�Yn�1,l,m�u� �1

�2�2

k 3/ 2

fnl�s�Ylm�s�.

(72)

Then from Eq. (19) we obtain the Shibuya–Wulf-man integrals involving 1s orbitals in the form

�a�,1,0,0;a,n,l,m �2�

k3/ 2 fnl�s�Ylm�s�. (73)

As a third example, note that from Eq. (62) andTable I it follows that

Y*1,0,0�u�Yn�1,l,m�u�

�1

�2���n � l ��n � l � 1�

n�n � 1�Yn,l,m�u�

� ��n � l ��n � l � 1�

n�n � 1�Yn�2,l,m�u��, (74)

where it is understood that

Ylm � 0 if � l (75)

and therefore

�a�,2,0,0;a,n,l,m �2�

k3/ 2 � ��n � l ��n � l � 1�

n�n � 1�fn�1,l�s�

� ��n � l ��n � l � 1�

n�n � 1�fn�1,l�s��Ylm�s�, (76)

where it is understood that

fn�1,l�s� � 0 if n � 1 � l � 1. (77)

The generalized Clebsch–Gordan coefficients of 4-Dhyperspherical harmonics have been extensivelystudied by Dunlap [36] and by Aquilanti and co-workers at the University of Perugia [25–29, 31, 32].Aquilanti and Caligiana derived closed-form ex-pressions for the Shibuya–Wulfman integrals and(in the two-center case) for the Coulomb potentialmatrix elements ���,� that appear in Eqs. (43) and(51) [31]. They have done this both for the sphericalpolar case (Table I) and for the case of parabolicSturmians (Table II). Aquilanti and Caligiani havealso written Fortran programs for evaluation ofthese integrals that are available on the Internet[32]. In the case of spherical polar coordinates, thegeneral expression of Aquilanti and Caligiana [31]for the Shibuya–Wulfman integrals can be writtenin terms of the functions fnl(s) defined by Eq. (66)and illustrated in Table III. Their expression is es-sentially as follows:

S��,� � ��1�l��l�M2�

k3/ 2 �N,L

�n�nN�2l� � 1��2l � 1�

�l�, �m�; l, m�L, M��n� � 1

2n � 1

2N � 1

2n� � 1

2n � 1

2N � 1

2l� l L

fNL�s�YLM�s�, (78)

where M � m � m� and where �..; ..�..� is a Clebsch–Gordan coefficient while the expression in curlybrackets is a 9j symbol. In Eq. (78) allowance hasbeen made for a slight difference in the definition ofthe Shibuya–Wulfman integrals between this articleand that of Aquilanti and Caligiana. Professor C.Weatherford of the University of Florida has inde-pendently written a Fortran program for the calcu-lation of Shibuya–Wulfman integrals. Thus the ap-plication of many-center Coulomb Sturmians to

COULOMB STURMIANS AND SHIBUYA–WULFMAN INTEGRALS

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 129

molecular calculations is a promising and rapidlydeveloping field.

References

1. Shull, H.; Lowdin, P.- O. J Chem Phys 1959, 30, 617.

2. Rotenberg, M. Ann Phys NY 1962, 19, 62.

3. Rotenberg, M. Adv Atom Mol Phys 1970, 6, 233–268.

4. Goscinski, O. Preliminary Research Report 217; QuantumChemistry Group, Uppsala University: Uppsala, Sweden,1968.

5. Gazeau, J. P.; Maquet, A. J Chem Phys 1980, 73, 5147–5154.

6. Bang, J. M.; Vaagen, J. S. Z Phys A 1980, 297, 223–236.

7. Bang, J. M.; Gareev, F. G.; Pinkston, W. T.; Vaagen, J. S. PhysRep 1985, 125, 253–399.

8. Fock, V. A. Z Phys 1935, 98, 145.

9. Fock, V. A. Kgl Norske Videnskab Forh 1958, 31, 138.

10. Bandar, M.; Itzyksen, C. Rev Mod Phys 1966, 38, 330, 346.

11. Podolski, B.; Pauling, L. Phys Rev 1929, 34, 109.

12. Coulson, C. A. Proc Cambr Phys Soc 1941, 37, 55, 74.

13. Coulson, C. A.; Duncanson, W. E. Proc Cambr Phys Soc 1941,37, 67.

14. Duncanson, W. E. Proc Cambr Phil Soc 1941, 37, 47.

15. McWeeny, R.; Coulson, C. A. Proc Phys Soc Lond A 1949, 62,509.

16. McWeeny, R. Proc Phys Soc Lond A 1949, 62, 509.

17. Wen, Z.- Y.; Avery, J. J Math Phys 1985, 26, 396.

18. Avery, J. Hyperspherical Harmonics; Applications in Quan-tum Theory; Kluwer Academic: Dordrecht, The Nether-lands, 1989; p 255.

19. Avery, J. Hyperspherical Harmonics and Generalized Stur-mians; Kluwer Academic: Dordrecht, The Netherlands, 2000,p 196.

20. Avery, J. In: Problemes Non Lineaires Appliques, 1990–1991;Calcul Scientifique et Chimie Quantique; Ecoles CEA-EDF-INRIA: Rocquencourt, France, 1991, pp 1–28.

21. Avery, J. In: Herschbach, D. R.; Avery, J.; Goscinski, O., eds.Dimensional Scaling in Chemical Physics; Kluwer Aca-demic: Dordricht, The Netherlands, 1992, pp 139–164.

22. Avery, J. In: Kryachko, E. S.; Calais, J. L., eds. ConceptualTrends in Quantum Chemistry, vol. 1; Kluwer Academic:Dordrecht, The Netherlands, 1994; pp 135–169.

23. Avery, J. In: Tsipis, C. A.; Popov, V.; Herschbach, D. R.;Avery, J., eds. New Methods in Quantum Theory; KluwerAcademic: Dordrecht, The Netherlands, 1995; pp 281–296.

24. Avery, J.; Herschbach, D. R. Int J Quantum Chem 1992, 41,673.

25. Aquilanti, V.; Cavalli, S.; De Fazio, D.; Grossi, G. In: Tsipis,C. A.; Popov, V. S.; Herschbach, D. R.; Avery, J. S., eds. NewMethods in Quantum Theory; Kluwer: Dordrecht, The Neth-erlands, 1996; pp 233–250.

26. Aquilanti, V.; Cavalli, S.; Coletti, C.; Grossi, G. Chem Phys1996, 209, 405.

27. Aquilanti, V.; Cavalli, S.; Coletti, C. Chem Phys 1997, 214, 1.28. Aquilanti, V.; Cavalli, S.; Coletti, C. Phys Rev Lett 1998, 80,

3209.29. Aquilanti, V.; Cavalli, S.; Coletti, C.; Di Domenico, D.; Grossi,

G. Int Rev Phys Chem 2001, 20, 673.30. Aquilanti, V.; Avery, J. Adv Quantum Chem 2001, 399, 71.31. Aquilanti, V.; Caligiana, A. Preprint 2002.32. http://bong.chm.unipg.it/downloads/download.html.33. Shibuya, T.; Wulfman, C. E. Proc Roy Soc A 1965, 286, 376.34. Wulfman, C. E. In: Loebel, E. M., ed. Group Theory and Its

Applications; Academic Press: New York, 1971.35. Judd, B. R. Angular Momentum Theory for Diatomic Mole-

cules; Academic Press: New York, 1975.36. Dunlap, B. I. Chem Phys Lett 1975, 30, 39.37. Monkhorst, H. J.; Jeziorski, B. J Chem Phys 1979, 71, 5268–

5269.38. Navasa, J.; Tsoucaris, G. Phys Rev A 1981, 24, 683.39. Koga, T.; Murai, T. Theor Chim Acta 1984, 65, 311–316.40. Koga, T. J Chem Phys 1985, 82, 2022.41. Koga, T.; Matsumoto, S. J Chem Phys 1985, 82, 5127.42. Koga, T.; Kawaai, R. J Chem Phys 1986, 84, 5651–5654.43. Koga, T.; Matsuhashi, T. J Chem Phys 1987, 87, 1677–1680,

4696–4699.44. Koga, T.; Matsuhashi, T. J Chem Phys 1988, 89, 983.45. Koga, T.; Yamamoto, Y.; Matsuhashi, T. J Chem Phys 1988,

88, 6675–6676.46. Koga, T.; Ougihara, T. J Chem Phys 1989, 91, 1092–1095.47. Koga, T.; Horiguchi, T.; Ishikawa, Y. J Chem Phys 1991, 95,

1086–1089.48. Duchon, C.; Dumont-Lepage, M. C.; Gazeau, J. P. J Chem

Phys 1982, 76, 445–447.49. Duchon, C.; Dumont-Lepage, M. C.; Gazeau, J. P. J Phys A

Math Gen 1982, 15, 1227–1241.50. Avery, J. J Phys Chem 1993, 97, 2406.51. Avery, J.; Antonsen, F. Theor Chim Acta 1993, 85, 33.52. Avery, J. J Math Chem 1994, 15, 233.53. Avery, J.; Børsen Hansen, T.; Wang, M.; Antonsen, F. Int J

Quantum Chem 1996, 57, 401–411.54. Avery, J.; Børsen Hansen, T. Int J Quantum Chem 1996, 60,

201–211.55. Aquilanti, V.; Avery, J. Chem Phys Lett 1997, 267, 1–8.56. Avery, J. J Math Chem 1997, 21, 285–304.57. Avery, J.; Antonsen, F. J Math Chem 1998, 24, 175–185.58. Avery, J. Adv Quantum Chem 1999, 31, 201–229.59. Avery, J. J Mol Struct Theochem 1999, 458, 1–9.60. Avery, J. J Math Chem 1998, 24, 169–174.61. Avery, J.; Sauer, S. In: Hernandez-Laguna, A.; Maruani, J.;

McWeeney, R.; Wilson, S., eds. Quantum Systems in Chem-istry and Physics, vol. 1; Kluwer Academic: Dordrecht, TheNetherlands, 2000; pp 19–39.

62. Avery, J.; Shim, R. Int J Quantum Chem 2000, 79, 1–7.63. Weniger, E. J. J Math Phys 1985, 26, 276.

AVERY

130 VOL. 100, NO. 2