Sturmian basis sets in momentum space

Sturmian Basis Sets in Momentum Space

JOHN AVERY, TOM BIZ)RSEN HANSEN, AND MINCHANG WANG H . C. Orsted Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark

FRANK ANTONSEN Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark

Received June 20, 1994; revised manuscript received October 10, 1994; accepted December 7, 1994

ABSTRACT rn The properties of Sturmian basis sets in d-dimensional direct space and d-dimensional momentum space are reviewed, as well as the relationship between hydrogenlike Sturmians and hyperspherical harmonics. The kernel of the reciprocal-space Schrodinger equation is expanded in terms of Strumian basis sets. This expansion allows Shibuya and Wulfman’s treatment of many-center Coulomb potentials to be extended to many-center potentials of a general form, and the method is also extended to the calculation of crystal orbitals and band structures. 0 1996 John Wiley & Sons, Inc.

Introduction

t is well known in quantum theory that the I bound-state hydrogenlike wave functions do not form a complete set: For completeness, the continuum hydrogenlike wave functions must be included, and it is awkward to use the continuum states as basis functions. For this reason, Shull and Lowdin [l] introduced basis sets of a type which has come to be called Sturmian. (The name ”Sturmian” was introduced by Rotenberg [2] to emphasize the connection with Sturm-Liouville theory.) The Sturmian basis sets of Shull and Lowdin consisted of hydrogenlike orbitals, with

International Journal of Quantum Chemistry, Vol. 57, 401 -41 1 (1996) 0 1996 John Wiley & Sons, Inc.

the nuclear charge adjusted so that all the functions in the set corresponded to the same value of the energy, for all values of the quantum numbers, and the two authors were able to show that such a set is complete. All the functions in a Sturmian basis set have the correct asymptotic behavior, and in this respect, the functions are similar to the basis sets used by Hylleraas [31 and Perkeris [4].

Later, the Sturmian concept was shown to be capable of generalization [5-111. In general, a Sturmian basis set is a set of solutions of an unpertubed Schrodinger equation, with the potential, v,(x), scaled so that all the functions in the set correspond to a constant value of the energy E. Such a set of functions obeys a potential-weighted orthornomality relation, as can be seen by the

CCC 0020-7608 I96 I030401 -1 1

AVERY ET AL.

following argument: Let us suppose that we know a set of functions +,(x) which obey the equation

where

so that when d = 3N, - $ A represents the kinetic energy operator in mass-weighted coordinates for an N-particle system. Here, P, is a scaling param- eter which is introduced to ensure that all the functions in the set correspond to the same value of E , although their quantum numbers differ. An- other member of the set obeys a similar (complex conjugate) equation:

( A + 2E)+,*.(x) = 2 / 3 n r V o ( x ) + ~ r ( x ) . (3)

Multiplying Eqs. (1) and (3), respectively, by +:,(x) and +,(x), integrating over the coordinates, and subtracting the two equations, we obtain

where we have made use of the Hermiticity of the operator A + 2E. From Eq. (4), it follows that

This potential-weighted orthogonality relation does not tell us how to normalize the Stunnian basis functions, but, as we shall see below, it is con- venient to choose the normalization in such a way that

2E 1 d ~ + , * ~ ( x ) V o ( x ) + n ~ x ) = - an,n.

Now suppose that we wish to solve the SchrG dinger equation

(7)

where V ( x ) is some other potential. We can do this by expanding the wave function in terms of our

(6) P n

[ A - ~ V ( X ) + 2 E j ] t,hj(x) = 0,

Sturmian basis set:

+j(X) = C 4,(X)Cnj . (8)

Inserting this expansion into the Schrodinger equation, we obtain

n

C [ A - ~ V ( X ) + 2Ej] +,(x)C,j = 0 . (9) n

Then, making use of Eq. (1) with E = Ej, we have

(10)

If we multiply on the left by +,'*(XI, integrate over the spaces coordinates, and make use of our potential-weighted orthonormality relation, we obtain a set of secular equations:

(11)

C [ P,VO(X) - V(X)l+,(X)C,j = 0. n

C TnPn - k, an,, IC, j = 0, n

where

k z G - 2 E (12)

and

Solutions in Reciprocal Space

The Fourier transforms of the Sturmian basis functions +,(XI are given by [71

,

where

d x = dx, dx , . . . d x , (15)

and

k - x 3 klxl + k , x 2 + ... k , ~ , . (16)

The inverse relationship is

VOL. 57, NO. 3 402

STURMIAN BASIS SETS IN MOMENTUM SPACE

Inserting this last relationship in Eq. (l), and remembering that k; = - 2 E , we have

j dke - I k . ' ( k i + k2>4i(k>

= -2P, /dke-'k'xVo(x)4d(k). (18)

If we now multiply on the left by eik'x, integrate over dx, and make use of the relationship

1 dxei(k- k') .x - - ( 2 7 ~ ) ~ 6 ( k - k'), (19)

we obtain the reciprocal-space version of Eq. (1):

where

Similarly, the reciprocal-space version of the Schrodinger equation [71 is

3 L

(ki -t kr2>+/(k') = - -- ( 7 r ) d / 2

From the generalized Fourier convolution theorem [4], we have

It follows from the last result, combined with Eq. (19), that if the direct-space Sturmian basis functions are normalized as in Eq. (6) then their Fourier transforms in reciprocal space obey the orthonormality relation

Since the matrix Cnj is unitary, the Fourier transformed solutions to the Schrodinger equation obey a similar orthonormality relation:

= c c,*jj~cn,sn~n = (25)

The orthonormality relations (61, (241, and (25) are, in fact, those which are appropriate for basis sets in a d-dimensional Sobolev space. The relationship between Sturmian basis functions and basis sets in Sobolev spaces has been treated by Weniger 1121. The reader who wishes to go more deeply into the question of orthonormality and completeness of Sturmian basis sets is referred to Weniger's excel- lent article, to the books dealing with Sobolev spaces by Adams, Blanchard, and Briining, Maz'ja, Sobolev, Wiedmann, and Ziemer [13-181 and to the articles by Klahn and Binge1 [19, 201.

The kernel of the reciprocal-space Schrodinger Eq. [(22)] can be expanded in terms of the solutions, +/(k). We can see this by means of the following argument: Since these solutions form a complete set, we can expand any function of k in terms of them, and we can also expand any function of k' in terms of +?'(k'). Thus, we can write

n'n

2 2k;V'(k - k') -~

(27T)d/2 ( k ; + k 2 ) ( k ; + k r 2 )

= c +,!(k)rjtj+Tt(k'), (26) i ' i

where rj.j is a matrix which we shall try to determine. Inserting (26) into (221, and making use of the orthonormality relation, (251, we obtain

+/(k') = +,!(k')rjtj, (27) i'

so that

r. r'i = s.,. I 1' (28)

and therefore

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 403

AVERY ET AL.

and from (10,

(31)

so that, finally, making use of the unitarity of the matrix C, j , we obtain

2 (k; + k2)(ki + kr2) V’(k - k’) = --

(27r)d/2 2ki

X c 4~4k)Tfltfl4~’(k’). (32)

If we substitute (29) into (22) and make use of the orthonormality relation (251, we can see that the two equations are consistent. Indeed, (29) can be derived directly from (22) and (25) by writing

(ki + kf2)$/(k‘) = (ki + k ” ) C 6jvj$j!(k’)

n’n

i’ ( k i + k2)(ki + kr2)

X $j(i!(k’)$;f(k)$/(k)

= I d k 2ki

i’

x/dkVf(kf - k)$/(k). (33)

Equation (29) then follows from the fact that the functions $/(k) form a complete set and from the requirement that (33) must hold for each member of the set. We will see below that Eq. (32) wiU be important in treating of many-center potentials.

Hydrogenlike Sturmian Basis Functions

As a simple example to illustrate the discussion given above, we can consider the case where d = 3 and where

1 r VO(X) = - - / (34)

so that our Sturmian basis set consists of hydrogenlike wave functions, with the potential weighted so that they all correspond to the same value of energy, regardless of their quantum numbers:

+,,,(X) = R,,(t)Y,,(O, 4), (35)

where

t = kor

and

X(2t)’ePfF(Z + 1 - n ( 2 2 + 2)2t), (37)

where H a lbl s) is a confluent hypergeometric function defined by

as a(a + l )s2 F(a(b( S ) 1 + - +

b b(b + 1)2!

The hydrogenlike Sturmian basis functions obey the differential equation [2-41

If we compare this with Eq. (l), remembering that k i = -2E, we can see that for this particular Sturmian basis set

When combined with the orthonormality relations for the spherical harmonics, this implies that the radical part of the Sturmian basis set fulfills the orthonormality relation:

To illustrate the solution of the Sturmian secular equations: [(11)--(13)1/ we can consider the case where V(x) is a screened Coulomb potential

(43)

where , y ( r ) is a screening function. Since this potential is spherically symmetric, its matrix ele-

404 VOL. 57, NO. 3


ments [Eq. (1311 will be diagonal with respect to the angular quantum numbers:

(44)

where

Thus, if we expand our solutions to the Schra dinger equation in series of the form

$,;I,(X) = Y1,(@, 4) C R,,(t)C,j, (46) n

then the coefficients C n j will be solutions to the secular equation

C [f,,,, - k06,,,]C,, = 0. (47) n

To illustrate some of the interesting features of the Sturmian secular equation, we can consider the simple case where I = 0 and where

X ( r ) = e- f ’ = e - y t , (48)

where y = t/k,. For I = 0, the radial part of the Sturmian basis functions reduces to

so that

Thus, in., is independent of k,. By constructing this matrix and diagonalizing it, we obtain, as eigenfunctions, a spectrum of values of k,, from which we can find the energies through the rela- tionshp k: = - 2 E. Since the functions in our basis set are all multiplied by a factor e P f = e-’O‘, our basis set has the correct asymptotic behavior at large values of Y and this is the advantage of Sturmian basis functions. The disadvantage is that if we are interested in a particular value of 5 we do not know the value of y which corresponds to it until we have diagonalized f,,, and found k,. Thus, we have to solve the secular .equations for a range of y values, finding, in this way, solutions corresponding to a range of 5 values, or, alterna- tively, we can guess an initial value of k,, use it to evaluate the matrix in,,/ find k, by diagonalizing the matrix, and continue the process until self- consistency is reached. Solutions for the screened Coulomb potential using hydrogenlike Sturmian basis functions are shown in Table I. The table illustrates the rapid convergence which is a characteristic of Sturmian expansions.

Hydrogenlike Sturmian Basis Functions in Reciprocal Space

Fock [21, 221 was able to show that if reciprocal space is mapped onto the surface of a 4- dimensional hypersphere by the transformation

= sin x sin 8 cos 4 u1 = - k; + k 2 2kOkI

TABLE I Sturmian ground-state solutions for a screened Coulomb potential [Eqs. (46) and (4711; the table illustrates the rapid convergence which is a characteristic of Sturmian expansions.

n=l n = 2 n = 3 n = 4 n = 5 n = 6

ko - E 5

y = 0.00

1 .oooooo 0.000000 0.000000 0.000000 0.000000 0.000000 1 .oooooo 0.500000 0.000000

y = 0.04

0.999286 0.037755 0.001 586 0.000074 0.000004 0.000000 0.961 881 0.462608 0.038475

y = 0.08

0.99742 1 0.071 547 0.005650 0.000489 0.000046 0.000005 0.9271 15 0.42977 1 0.0741 69

y = 0.12

0.99471 6 0.1 0201 6 0.01 1424 0.001391 0.0001 83 0.000026 0.895196 0.400688 0.1 07423

y = 0.16

0.991 385 0.1 29650 0.018384 0.00281 5 0.000463 0.000080 0.865732 0.374746 0.13851 7


AVERY ET AL.

2knkz

k2 - k; u4 = - = cos x, k; + k2 (51)

then the Fourier transformed hydrogenlike functions can be written in the form

+;, 1 , m(k) = M(k)Yn- I,!, m ( u ) / (52)

where Y, - I , ,(u) is a +dimensional hyperspherical harmonic [7,23-251 and where M(k) is a universal function which is independent of the quantum numbers:

(53)

Fock’s derivation of this result, expressed briefly, is as follows: The Fourier-transformed Schrodinger equation for a hydrogen atom is an integral equation which can be written in the form

I 1

(54)

The solid angle in the +dimensional hyperspace defined by the unit vector u z (ul,. . . , u 4 ) is given by

dR = sin2Xsin8dXded+, (55)

from which it follows that

3 k; + k2 d 3 k = (T) dR. (56)

Furthermore (with a little work), one can show that

so that the reciprocal-space integral equation for the hydrogen atom can be written in the form

1 (k; + kr2)’+‘(kr) = -

2k0.rr2

If we let

+‘(k) = M(k)f(u), (59)

then the reciprocal-space Schrodinger equation for a hydrogenlike atom takes on a still more simple form:

From the theory of hyperspherical harmonics [7], it can be shown that

1 m

27T2

A = O A + ’ lm = c - YAIm(u)YA~m(u’), (61)

where Ci(u - u’) is a Gegenbauer polynomial and where YAIm(u u‘) is a 4-dimensional hyperspherical harmonic. If we substitute (61) into (60), we can see that functions of the form

will satisfy the reciprocal-space Schrodinger equation for the hydrogen atom, provided that ko =

l /n . This leads to the familiar energy spectrum (expressed in atomic units):

(63)

Alliluev [26, 271 was able to extend Fock’s method to d-dimensional hydrogenlike orbitals, and d-dimensional hydrogenlike Shumian basis functions can be applied to the study of correlation in the quantum mechanical many-body problem. The properties of d-dimensional Sturmian basis sets of this type have been discussed in detail by Avery and Herschbach [8].

Reciprocal-space Solutions in Many-center Potentials

Fock‘s remarkable reciprocal-space approach to the hydrogen atom problem has been generalized

406 VOL. 57, NO. 3


by Shibuya and Wulfman [281, Judd [251, Monkhorst and Jeriorski [29], Koga and Matsuhashi [301, Duchon et al. [311, Gazeau and Maquet [32], and Novosadov [331. These authors were able to apply a generalization of Fock’s method to the problem of a charged particle moving in a many-center Coulomb potential:

7

The equation, analogous to (60), then becomes

(64)

i(k‘ - k).R,

/ d f l c f (u) . 1

f(u‘) = - 25r2ko &, 1u - U1l2

If we let

65)

,

(66)

where T stands for the set of indices Y, n, I, rn, then we can rewrite Eq. (65) in the form

If we expand f(u) in a series of the form

f(U) = c V,(U)B,, (68) r

then (67) becomes

k o x r],(u)B, = c V~.(U)K,.,B,, (69) T T ‘ T

where K T , , is the overlap integral:

Equation (69) will be fulfilled if the coefficients B, satisfy the secular equations:

C[ K T , , - ko6,.,]B, = 0. (71)

In other words, solutions to the problem of a single particle moving in a many-center Coulomb potential can be found by diagonalizing the overlap matrix [(70)] between the nonorthogonal

T

Sturmian basis functions defined by Eq. (66). The eigenvalues of this matrix, k,, determine the energies of the system through Eq. (121, as well as the asymptotic behavior of the wave function at large distances from the attractive centers. The problem of calculating the wave function of a charged particle moving in a many-center Coulomb potential thus reduces to the problem of calculating the overlap matrix, K T r T [(70)1, after which it may be diagonalized using standard programs. A method for calculating the overlap matrix elements by means of the Clebsch-Gordan coefficients of the 4-dimensional hyperspherical harmonics has been discussed by Shibuya and Wulfman [28]. Alterna- tive methods have been discussed by Avery [7, 101. When many basis functi. .IS are used, the reciprocal-space Sturmian approach to the many- center Coulomb problem is able to produce very high accuracy. For instance, Koga and Matsuhashi [30] were able to obtain 10-figure accuracy when they applied this method of calculation to the hydrogen molecule ion.

~~~~ ~

Many-center Non-Coulomb Potentials

Equation (32) allows us to extend the approach discussed in the preceding paragraph to many- center non-Coulomb potentials. For example, if we let

then

With the substitutions

and

(75)

and with the help of Eq. (32), the integral equation


AVERY ET AL.

can be rewritten in the form where

where

and where the functions (bnrm(x) are the hydrogenlike Sturmian basis functions shown in Eqs. (35)-(42). If we let

F(u) = c l,(u)B, (79) 7

and

*., = c T,," S,", (80)

[ STnT being defined by Eq. (70)], then Eq. (77) will be satisfied provided the k, and B, fulfill the secular equations

7"

C [ q t , - k,S,,,] B, = 0. (81)

To illustrate Eqs. (72)-(81), we can consider a homonuclear diatomic molecule with atoms at PO- sitions R, and R, and with a screening function X ( r ) = Ze-*'. We can calculate the ground state of the molecule in an approximation where we use only two Sturmian basis functions:

7

1

1 12(u) = e'k~R~Yo,o,o(u) = - . (82) w e ik .Rz

In this very simple approximation, the matrix ST,,, , is given by

(84)

The roots of the secular equation [(80)1 are then

k, = w[l f (1 + t)e-f 1 , (86)

where w is defined in Eq. (85). In this approximation, the lowest orbital energy of the homonuclear diatomic molecule is then given by

(87)

In the united atom limited ( t = 01, this corresponds to

i.e., the energy of an electron in a screened Coulomb potential with nuclear charge 2 2 . In the separated atom limit [ ( t = m)], both the roots of Eq. (86) correspond to the energy of an electron in a screened Coulomb potential with nuclear charge Z:

7 2

(89)

For intermediate values of the internuclear distance, R, the approximate energy is given by the curve shown in Figure 1. We can repeat this calculation for a heteronuclear diatomic molecule, where the effective one-electron potential is given ap- proximately by

Then, the equations corresponding to (84) and (85) are

and

408 VOL. 57, NO. 3


0 1 I vectors aj are basis vectors of the direct lattice and

vj = 0, L- 1, -t 2, f 3 ,... . (96)

For such a periodic system, the order of the secular equation [(8l>l can be reduced by making use of the translational symmetry of the crystal. In Eq. (80, T stands for the set of indices v,n, I , m. Writ- ing these indices explicitly, we express (81) in the form

C [ q ’ n ‘ l ’ m ’ ; v n l m - k, a u f u a n ‘ n 81’1 a m ’ m I B v n l m = 0. vnlm

(97)

Let us try to see whether it is possible to find a

, ZR , B , ~ ~ ~ = eiq‘RpB nlm (9). (98)

Inserting this trial solution into the secular equations and introducing the definition

solution of the form

0 1 2 3 4 5

FIGURE 1. Ground-state orbital energy E I Z 2 as a function of internuclear distance R for an electron moving in a two-center screened Coulomb potential. Only two Sturmian basis functions [Eq. (8211 were used in the

q ’ l ‘ m ‘ ; n l m ( q ) C x ’ n ’ l ’ m ’ ; u n l m eiq.(R,-R ..) , (99) Y

calculation. From bottom to top, the curves correspond to [/Z = 0, 0.1 ,0.2,0.3. For 5 = 0, the energies are exact

we obtain a reduced set of secular

in the unites-atom and separated-atom limits.

while the equation corresponding to (86) is (100)

where, as before, t = ko R, R being the internuclear distance. In this illustrative calculation, we used only the two Sturmian basis functions shown in Eq. (82), but the accuracy could, of course, be increased by using a larger Sturmian basis set.

Periodic Potentials

Let us next consider an electron moving in a periodic potential of the form

(94)

It might be objected that .Wn.lfm,;nrm(q) ought to depend on v’, since we have summed over v but not over v’. However, because of the translational symmetry of the crystal, Z,rn,f,m,; vn,m depends only on the relative position R, - R u t and not the absc- lute position of the two unit cells. Thus, if edge effects are neglected, the lattice sum (99) is independent of its starting point.

q is a symmetry index labeling the different irreducible representations of the translational symmetry group of the crystal. In going from Eq. (97) to Eq. (1001, we are making use of the fact that basis functions corresponding to different symmetries are not hybridized in solutions to the Schrodinger equation. If cyclic boundary condi- tions are imposed at the faces of the crystal, then the symmetry index q can take on the values

where where

R, = vlal + v2a2 + v3a3 (95) I 1 = O , f 1 , k 2 , f 3 ,... j = 1 , 2 , 3 (102)

is a set of direct crystal lattice vectors. Here, the and where the vectors bi are basis of the reciprocal


AVERY ET AL.

lattice of the crystal and are related to the direct lattice basis vectors by

ai - b, = aij. (103)

In Eq. (1011, N is the number of unit cells in the crystal.

To illustrate Eq. (loo), let us consider the case where

in the very simple approximation where only one Sturmian basis function per unit cell is used, namely:

1 L &,(u) = - m eik'R"

(105)

In that case, the indices n, 2, m are 1,0,0 for all the basis functions and they need not be written. The equations analogous to (83) and (84) then become

S V N , = (1 + koIR,. - Rvl)e-kOIR~"-R~l (106)

and

Zt,#! = SV',"W, (107)

with w defined by Eq. (85). Thus, we have

Trv = ~ ~ # , " S , . , = ws,.,. (108) V -*

Then, (99) becomes

-Hq) = w c (1 + koIR, - R,,I) V

x e-kOIR "- R y 4 &.(R y - R ..) , (109)

so that, from Eq. (1001,

ko =-Hq) E(q) = -;[-Hq)l2. (110)

The higher bands, as well as an increase in accuracy, can be obtained by using a larger Sturmian basis set, and the method can be generalized in a straightforward way to crystals with many atoms per unit cell.

Discussion

We can see from the examples discussed above that Shull and Lowdin's concept of Sturmian basis

functions [l] is capable of generalization and that the concept is closely related to Fock's elegant reciprocal-space treatment of the hydrogen atom [21, 221. For a Sturmian basis set, the potential is scaled in such a way that all functions in the basis set correspond to the same value of the energy. The orthonormality and completeness properties of Shull and Lowdin's hydrogenlike Sturmian basis functions correspond, through Fock's mapping [Eqs. (51)-(53)], to the orthonormality and completeness properties of the 4-dimensional hyperspherical harmonics. The constancy of the energy within a Sturmian basis set corresponds to the constancy of the radius of the hypersphere onto which reciprocal space is mapped.

Shibuya and Wulfman [281, Monkhorst and Jeriorski [29], and also Koga and Matsuhashi [30] have already generalized Fock's momentum-space treatment of hydrogenlike atoms to many-centered Coulomb potentials. In the present article, we extended the method still further, to many-center non-Coulomb potentials. The possibility of making this generalization was foreseen by Monkhorst and Jeriorski in their 1979 article [29]. The method by which we have implemented it depends on an expansion [Eq. (32)] of the kernel of the reciprocal- space Schrodinger equation.

'Recently, an extremely promising generalization of the concept of Sturmian basis function was made by Museth 1111, who used a Sturmian-like basis set in the R-matrix method for reactive scattering. Like Koga, Museth was able to achieve remarkably high accuracy (13 figures!) using Sturmian basis functions.

We hope to discuss in another article the appli- cation of Minkowski-space hyperspherical harmonics to Sturmian basis sets in scattering prob- lems.

References

1. H. Shull and P. 0. Lowdin, J. Chem. Phys. 30, 617 (1959).

2. M. Rotenberg, Adv. At. Mol. Phys. 6, 233 (1970).

3. E. A. Hylleraas, Z. Phys. 54, 347 (1929).

4. C. L. Pekeris, Phys. Rev. 112, 1649 (1958). 5. 0. Goscinski, Preliminary Research Report No. 217

(Quantum Chemistry Group, Uppsala University, 1968).

6. J. G. Loeser and D. R. Herschbach, J. Chem. Phys. 84, 3882 (1986).

7. J. Avery, Hyperspherical Harmonics; Applications in Quantum Theory (Kluwer, Dordrecht, Netherlands, 1989).

41 0 VOL. 57, NO. 3


8. J. Avery and D. R. Herschbach, Int. J. Quantum Chem. 41,

9. J. Avery, J. Phys. Chem. 97,2406 (1993). 673 (1992).

10. J. Avery, in Conceptual Trends in Quantum Chemistry, E. S. Kryachko and J. L. Cakiis, Ed. (Kluwer, Dordrecht, Netherlands, 1994).

11. K. Museth, Thesis (Department of Physical Chemistry, Uni- versity of Copenhagen, 1994).

12. E. J. Weniger, J. Math. Phys 26, 276 (1985). 13. R. A. Adams, Sobola, Spaces (Academic Press, New York,

14. P. Blanchard and E. Bruning, Variational Methods in Mathe-

15. V. G. Maz’ja, Sobola, Spaces (Springer-Verlag, Berlin, 1985). 16. S. L. Sobolev, Applications of Functional Analysis in

Mathematical Physics (American Mathematical Society, Providence, RI, 1963).

17. J. Weidmann, Linear Operators in Hilbert Space (Springer- Verlag, New York, 1980).

18. W. P. Ziemer, Weakly Differentiable Functions (Springer- Verlag, New York, 1989).

19. B. Klahn, Adv. Quantum Chem. 13, 155 (1981); Ibid,. J. Chem. Phys. 83, 5749, 5754 (1985).

20. B. Klahn and W. A. Bingel, Theor. Chim Acta 44, 9, 27 (1977); Ibzd., Int. J. Quantum Chem. 11, 943 (1977).

1975).

matical Physics (Springer-Verlag, Berlin, 1985).

21. V. A. Fock, A. Phys. 98, 145 (1935); Ibid., Kgl. Norske Videnskab. Forh. 31, 138 (1958).

22. V. Bargman, Z. Phys. 99, 576 (1936). 23. N. K. Vilenken Special Functions and the Theory of Group

Representations (American Mathematical Society, Provi- dence, RI, 1968).

24. L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains (American Mathematical Society, Providence, RI, 1963).

25. 8. R. Judd, Angular Momentum Theory for Diatomic Molecules (Academic Press, New York 1975).

26. S. P. Alliluev, Sov. Phys. JETP 6, 156 (1958). 27. M. Bander and C. Itzykson, Rev. Mod. Phys. 38, 330, 346

28. T. Shibuya and C. E. Wulfman, Proc. Roy. SOC. A 286 (1965). 29. H. Monkhorst and B. Jeriorski, J. Chem. Phys. 71, 5268

30. T. Koga and T. Matsuhashi, J. Chem. Phys. 89, 983 (1988). 31. C. Duchon, M. C. Dumont-Lepage, and J. P. Gazeau, J.

Chem. Phys. 76, 445 (1982); [bid., J. Phys. A 15, 1227 (1982). 32. J. P. Gazeau and A. Maquet, Phys. Rev. A 20, 727 (1979). 33. B. K. Novosadov, Opt. Spectrosc. 41, 490 (1976); Ibid., J.

(1966).

(1979).

Mol. Struct. 52, 119 (1979).

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Sturmian basis sets in momentum space