N-Dimensional Isotropic

Transcription

1 Bessel Basis with Applications: N-Dimensional Isotropic Polynomial Oscillators H TAŞELI, A ZAFER Department of Mathematics, Middle East Technical University, 65 Ankara, Turkey Received July 996; accepted December 996 ABSTRACT: The efficient technique of expanding the wave function into a FourierBessel series to solve the radial Schrodinger equation with polynomial potentials, K i V r Ý vir, in two dimensions is extended to N-dimensional space It is shown that the spectra of two- and three-dimensional oscillators cover the spectra of the corresponding N-dimensional problems for all N Extremely accurate numerical results are presented for illustrative purposes The connection between the eigenvalues of the general anharmonic oscillators and the confinement potentials of the form VŽ r Zr Ý K cr i i is also discussed 997 John Wiley & Sons, Inc Int J Quant Chem 6: 95947, 997 Key words: Radial Schrodinger equation; polynomial oscillators; Bessel basis; integrals containing Bessel functions; perturbed Coulomb potentials; accurate eigenvalue calculations Introduction T he dimensionless form of the Schrodinger equation written in N-dimensional Cartesian coordinates N Ý N VŽ x, x,, x E x i Ž x, x,, x, Ž N Correspondence to: H Taşeli where lim Ž x, x,, x Ž xi N has attracted the attention of scientists for a long time due to its application in quantum field theory and molecular physics The corresponding equation in one dimension has been studied very extensively However, there is a limited number of works on multidimensional systems Before entering the complexity of the general N-dimensional oscillators it is perhaps more reasonable to deal with the spherically symmetric states for which the 997 John Wiley & Sons, Inc CCC -768 / 97 / 595-

2 TAŞELI AND ZAFER potential function is of the form V x, x,, xn Vž ' x x x N/ Ž Therefore, introducing the N-dimensional spherical coordinates N Ý r x Ž 4 the radial part of the Schrodinger equation Ž becomes d N d lž ln VŽ r dr r dr r where Ž r EŽ r, r,, Ž 5 i lim r 6 r Additionally, the wave function must behave like r l as r to have a regular solution We study here the generalized anharmonic oscillator in N dimensions described by the potentials of the form K i Ý i K V r v r, v 7 with K,,, where the v i are the coupling constants As is well known, the trivial case of the harmonic oscillator when K, ie, V r vr, admits exact solutions in terms of the associated Laguerre polynomials, ž ' / Ž N l Ž ' v r ŽNl n, l n Ž r re L v r Ž 8 for the eigenfunctions while the eigenvalues are specified by ' Ž N E v Ž N4nl, n, l,, n,l 9 in N-dimensional space Moreover, it can be deduced that, if K K, K,,, explicit solutions in closed form may be constructed whenever certain algebraic relations between the coupling constants hold, For instance, the ground-state wave function of the sextic oscillator, 4 6 V r v r vr vr, is expressible as 4 6 ŽN Ž4ar 4 Žbr r e, a,, which satisfies the boundary condition Ž 6 Substituting into Ž 5, we see that Ž is indeed an exact eigenfunction with the corresponding eigenvalue, subject to a constraint on v where Ž N E Nb,, that v b N a, v 4 a' v 6, b Ž v ' 6 The other states are characterized exactly by exponentially weighted polynomial wave functions This class of exact solutions is enumerably infinite but not complete However, if K K, such a construction of exact eigensolutions is not possible The quartic anharmonic oscillator, V r vr 4 vr is the simplest example to this case 5 4 An exponential part in the solution is necessary owing to the essential singularity at infinity, which is controlled mainly by the dominant coupling r K It is a simple matter to find out the correct asymptotic form of the wave function as r, but the introduction of an orthonormal basis set reflecting the required asymptotic behavior is the problem In a recent study 6 by the same authors, it is shown that the modification of the asymptotic boundary condition Ž 6 such that Ž L, where L is finite, enables to use Bessel functions of the first kind as a basis set in the RayleighRitz variational method for solving very accurately the two-dimensional radial Schrodinger equation with any polynomial potential in Ž 7 In this article we truncate again the semiinfinite interval of r and consider the Dirichlettype boundary condition Ž r rl Ž 4 over a spherical surface of radius L The unperturbed Schrodinger equation is defined as d N d lž ln Ž r dr r dr r r 5 in which a zero potential energy, ie, V r, is assumed, where is a constant When N the unperturbed equation so defined is directly equiv- 96 VOL 6, NO 5

3 BESSEL BASIS WITH APPLICATIONS alent to Bessel s differential equation whose normalized eigenfunctions form a complete orthonormal set over the interval r L with respect to the weighting function r 6 In the general case of N dimensions we introduce a new dependent variable Ž r such that where r with ŽŽ N r r r 6 satisfies the differential equation ž r / d 4 Ž r Ž 7 dr l N, 8 which can be solved analytically as well It is clear that if N, then Ž r is identical to Ž r As a result of Ž 6, the solution of the transformed full Schrodinger equation d 4 dr E V r r, r r, L 9 may be postulated to be an expansion in terms of the exact eigenfunctions of Ž 7 Such an expansion is presented in Section Note that the accompanying conditions with Ž 9 are now Ž r r as r, L, and that Ž 5 and Ž 9 have completely the same eigenvalues As a matter of fact, truncating the interval of r we consider an enclosed quantum mechanical system of multidimensional isotropic oscillators which also has received great interest in different fields of physics and astronomy 7 Our strategy in this work, however, is to determine the spectrum of the corresponding unbounded system, where L, making use of the finite boundary as a nonlinear optimization parameter This approach is justified analytically in Section Another remark is that the transformed equation Ž 9 is concerned explicitly with the parameter rather than the angular quantum number l and the dimension of the space N, separately In other words, Eq Ž 8 implies that any eigensolu- ŽN ŽN tion of 9 denoted by, E 4 n, l n, l remains un- changed for a prescribed value of l N For instance, n, Ž n, Ž4 and En, Ž En, Ž4 However, the situation with the eigenfunctions n, Ž N l of the original problem Ž 5 is different More specifi- Ž Ž4 cally, we see from 6 that n, rn, and hence n, Ž n, Ž4, although the corresponding eigenvalues are equal to each other The last section is, therefore, devoted to the discussion of these interesting degeneracies of the spectrum of N-dimensional polynomial potentials, and the numerical applications Furthermore, we establish functional relationships between the anharmonic oscillators considered in this work and the perturbed Coulomb problem, which was studied especially from the view-point of the perturbation theory and Lie algebra technology 8, 9 Eigenfunction Expansion As is readily seen the Bessel functions of the first kind multiplied by r r cr J r stands for the exact solution of Ž 7 which behaves like r Ž at the origin Then the second condition in Ž is fulfilled if L is a positive root of the equation J having a countable infinite set of distinct zeros, L,,,, n Furthermore, when c is properly chosen, the functions in Ž may be normalized so that where H L ², : Ž r Ž r dr, Ž n m n m ' r Ž r J Ž rl n n, n,, LJ Ž n Ž 4 and is the Kronecker delta Now according to the theory of orthogonal functions, an eigenfunction expansion of the form Ý n n n Ž r a Ž r Ž 5 is proposed to characterize the solution of 9 satisfying the conditions The only thing which remains is the determination of the coefficients a n INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 97

4 TAŞELI AND ZAFER To this end, the standard RayleighRitz method leads to the algebraic equations Ý n n H E a, m,, 6 where the matrix elements H K Ý are defined by j Ž, j H v L I Ž 7 n j j for the polynomial potentials given in Ž 7 Here the I Ž, j denote the integrals of the type H Ž, j Ž j I c J Ž J Ž d, n m Ž c Ž 8 J Ž J Ž n m to be evaluated, provided J Ž n for any fixed In the exceptional case of j, I Ž, because of the orthonormality relation Ž If the dimension of the space is even, N, 4,, then becomes an integer from Ž 8, and we encounter integrals containing Bessel functions of integer order which have been treated in 6 Actually, it has been shown that such integrals may be evaluated analytically by means of recursive relations On the other hand, whenever N is odd, N, 5,, Bessel functions of fractional order related to spherical Bessel functions appear in the integrand of Ž 8 These functions can simply be expressed in terms of circular functions, the first three of which are / sin / ž / ž / ž / J Ž sin, Ž 9 ž J Ž cos, Ž ž J Ž 5 sin cos Ž In general, for m,,, we have ž / J Ž f Ž sin g Ž cos m Ž m m, Ž where f Ž and g Ž m m are polynomials in inverse powers of As specific examples, let us first consider, ie, N and l, so that, from Ž 8, H Ž, j Žj I c J Ž J Ž d, n m where J Ž From Ž 9 n we have n n, for n,,, and therefore I Ž, j reduces to Ž, j I c ' H Ž j sinž n sinž m d Ž 4 Ž Using to calculate c it follows that j n m j n m Ž, j I Ž S Ž S Ž, wherein the integral denoted by S, j j j j k 5 k S H cos d, S j 6 can be evaluated both recursively and explicitly The recursion relation S Ž jž j S Ž sin j cos, j j j,, 7 with the initial condition S Ž sin is the computationally more useful one For,ie,Nand l orn5 and l, we face the integral H Ž, j Žj I c J Ž J Ž d, n m 8 where the n s are now the positive roots of J Ž Equation Ž implies that the n s should satisfy the equation tan 9 98 VOL 6, NO 5

5 BESSEL BASIS WITH APPLICATIONS In connection with Ž and Ž, the integral in Ž 8 may be written as I Ž, j sin nsin m ž / n sin m ž m / m sin n j H cos n cos d Ž resulting again an expression in terms of S, j I Ž, j sin nsin m ½ S Ž S Ž j n m j n m j S Ž S Ž n m j n m j n m 5, Ž n m with j The integral I Ž, j with a larger involves more labor in the calculations but can be evaluated in a similar fashion However, a straightforward recursive evaluation of I is preferable Therefore, Ž, j using the general recurrence relations for the Bessel functions, we have derived that if n m, then c Ž n n m Ž, j I J Ž J Ž jc Ž m m n Ž n m J J Ž n m n m J J J J n m n m 4j Ž n m Ž, j n m Ž Ž j I Ž, j n m j I with c Ž n n m Ž, I J Ž J Ž and if n m, then cnn Ž Ž, j nn I with j n J J, m m n nj j J n nj n jž j J Ž J Ž 4 n n n Ž j j Žv, j I Ž 4 nn Ž j n Ž c Ž, nn nn n n I J Ž J Ž, Ž 5 where It should be noted that evaluating such integrals even numerically is not an easy task, and the above formulas being quite general in the sense that no assumption is made about n can be effectively used As in our case, if n s are zeros of J Ž x with a specific, then J Ž n can be written in terms of J Ž for,,,kj and j,,,k, where k is a fixed positive integer, and therefore the integrals I Ž, k are expressible in terms of the zeros n of J Thus, no evaluation of any Bessel function is required, which is of course computationally very efficient At the numerical side of the present study, Ž 6 is truncated to a homogeneous system of M equations Therefore the roots of the determinantal equation det H E 6 MM then give approximately the eigenvalues En, Ž N l of N-dimensional oscillators as a function of the boundary parameter L The numerical algorithm requires only the zeros n, which are calculated on Mathematica, of the Bessel functions depending on and the potential coefficients v i as its input data n INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 99

6 TAŞELI AND ZAFER Variation of Eigenvalues with Respect to L In this section, we examine briefly the qualitative behavior of the energy eigenvalues with respect to the boundary parameter L Regarded as EEL and Ž r, L,Eq Ž 9, satisfied by such a pair of eigensolution, may be reconsidered in the form subject to L, 4 L V r E L r r Ž r, L r constant, Ž r, L rl Ž In fact, the constant in Ž is zero from Ž unless the parameter is equal to If we differentiate both sides of Ž with respect to L, there follows de L, Ž L dl where L The result of multiplying Ž L by and integrating over the interval r, L takes the form de ² L,:, Ž 4 L dl where we have assumed that the wave function is normalized to have ², : On integrating by parts and using Ž the last equation may be written as L ; de, L Ž 5 dl L r L r in which the inner product on the right-hand side is zero from Ž The conditions in Ž imply that d at r and r L Thus we have / d r ž dr dl r L r dl Ž 6 L r and / ž / d rl ž dr dl r L rl to obtain the relations dl Ž 7 r L rl, Ž 8 L L r r rl rl which make it possible to express 5 in a more neat form ž / de Ž 9 dl r rl showing that dedl is always negative On the other hand, the limit energy EL as L approaches zero may be determined analytically Indeed, making use of the linear transformation, rl,eq Ž 9 becomes d 4 LE L LVŽ L d,, When L, we revert to the unperturbed Schrodinger equation Ž 7 for which lim LE L, L where is any root of Ž Thus EL goes to infinity like L as L tends to zero and decreases monotonically, since dedl from Ž 9, as L increases However, EL must converge to a limit for sufficiently large values of L owing to the well-known fact that a discrete spectrum for each v exists in the case of the corresponding unbounded problem Therefore, the eigenvalues EL of a confined oscillator are upper bounds for the asymptotic eigenvalues EŽ From a computational point of view it is essential to find a specific boundary value, which may be referred to as the critical boundary parameter L cr, such that for a given as small as pleased, EŽ L EŽ L Ž cr for all large values of The strictly decreasing property of eigenvalues then implies that the absolute error in determining asymptotic eigen- cr 94 VOL 6, NO 5

7 BESSEL BASIS WITH APPLICATIONS values E is less than, that is, E L E cr It is clear that a critical value of L so defined represents numerically the infinity depending on, the quantum numbers and the potential function in question Hence the present approach based mainly on estimating such Lcr values by numerical experiments 4 Results and Discussion In this work, we explored a new way of computing the energy eigenvalues of N-dimensional polynomial oscillators For a complete analysis of the system, let us first consider the exceptional case of one dimension For N, it is shown from Ž 5 that l is either or, and the equation reduces to the usual one-dimensional Schrodinger equation: d VŽ x Ž x EŽ x, dx x L, L, L, 4 where r has been replaced by x, x r Corresponding to l and l, we see from Ž 8 that, respectively, and hence that the unperturbed equation Ž 7 is altered to the differential equation satisfied by the circular functions Therefore, the one-dimensional problem may be treated applying trigonometric basis sets, as was done in three earlier studies of the first author 5 Notice that such simple trigonometric functions can be derived within our general for- malism since, for, ( xj Ž x cos x, ( xj Ž x sin x Ž 4 representing the symmetric and antisymmetric states, respectively It is important to observe that the symmetric energy levels E Ž n of the one-dimensional problem are single in N-dimensional system However, the antisymmetric levels E Ž n are doubly degenerate since stands also for the three-dimensional oscillators with l As outlined in the introduc- tion, we deduce from 8 and 9 that E Ž E Ž5 n, n, Ž Ž5 Ž7 En,En,En, Ž 4 Ž Ž5 Ž7 En, len, len, l Žl Žl Žl En, En, En, when N is odd In particular, we have En, Ž En Ž If N is even, it can be shown that En, Ž is single in the system, and the remaining eigenvalues are degenerate as follows: E Ž E Ž4 n, n, Ž Ž4 Ž6 En,En,En, Ž 44 Ž Ž4 Ž6 En, len, len, l Žl Žl Žl En, En, En, As a significant consequence of Ž 44 and Ž 4, the full spectra of N-dimensional oscillators can be furnished by determining the spectra of two- and three-dimensional problems The former rests on an expansion of the wave function in terms of the Bessel functions of integer order, whereas the latter is treated by an expansion of the spherical Bessel functions We now understand that the spectrum of a two-dimensional system considered numerically in 6 is representative for N- dimensional spaces as well We, therefore, present computer results here for N to cope with also the Ž N -dimensional spaces according to Ž 4, for completion In Tables IV, the eigenvalues accurate to - digits have been tabulated in a systematic manner to illustrate the efficiency of the present basis set, for a wide class of polynomial potentials of degree up to In the tables, n and l stand for the quantum numbers of the state, and Lcr denotes the radius of the finite spherical surface which represents numerically the unbounded domain to the accuracy quoted The number of basis elements or the truncation order M of the variational matrix required to get the prescribed accuracy is also included in the tables Within the framework of the above general remarks, Table I tests the rate of convergence of the method in calculating the ground-state eigenvalue E Ž of the harmonic oscillator, which is known, INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 94

8 TAŞELI AND ZAFER TABLE I Convergence rate of the method as functions of L () and M for the ground-state energy E, of the ( ) three-dimensional harmonic oscillator, V r =r () L M E, exactly from Ž 9 It is shown that significant digits have been achieved when M 4 and L 9 If we continue to increase L, we obtain the same accuracy at the cost of using a higher truncation size M Thus, in this case, we identify 9 as the critical or optimum boundary value L cr In a gen- eral situation, the behavior of Lcr is virtually the same, further discussion on which can be found in 6 In Table II, we consider a polynomial potential of degree K containing only the dominant term, K V r v r 45 K for K,,, 4, and Since a linear scaling transformation transforms any case to v K via the relation, ŽK E v v EŽ, Ž 46 K K TABLE II () ( ) K Critical values L and the energy eigenvalues E of the potential, V r =r, as a function of K cr n, l () K n l En, l Lcr M VOL 6, NO 5

9 BESSEL BASIS WITH APPLICATIONS TABLE III () ( ) 4 Critical values I and the energy eigenvalues E of the potential, Vr =r +vr, as a function of v cr n, l 4 4 () v4 n l En, l Lcr M we present eigenvalues of r K for some pairs of Ž n, l It is seen that there is no accuracy loss for higher values of K although the required truncation sizes increase Tables III and IV are concerned with the poten- K tials of the form, V r vr vkr, with two limiting values of K, and The eigenvalues Ev,v has the scaling property K ' Ž K K K E v,v v E, v v 47 so that we set v in our calculations and tabulate the results for a wide range of v K from to The last table is devoted to a general polyno mial potential, V r vr vr 4 vr 6 v8r, with more than one effective coupling constants Of course, there is a huge number of combinations of the set of parameters v, v, v, v so that only the ground-state energies of some selected potentials are recorded in order not to overfill the content of the study with tabular material anymore Further results are available from the authors The potentials 4 and 5 in Table V are sextic and octic oscillators, respectively, in which the coupling constants differ from to It is noteworthy that the method is constantly efficient for any polynomial having small or large parameters Furthermore, results with such a high accuracy are being reported for the first time, which may be used as a guide for future numerical calculations The tabulated eigenvalues are in good agreement with those already available in the literature, 5, 6 For a specific example, if we recall that the ground-state eigenvalue for the potential 4 was given to seven digits in Ref, employ- INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 94

10 TAŞELI AND ZAFER TABLE IV Critical values L () ( ) and the energy eigenvalues E of the potential, V r =r +v r, as a function of v cr n, l () v n l En, l Lcr M ing a Hill determinant of size 5 5, the success of our method becomes quite impressive as it uses only a truncation order of 4 in calculating the same eigenvalue to digits The potentials 6 in Table V are different in nature since they describe multiwell quartic, sextic, and octic oscillators in Cartesian coordinates It should be mentioned that the variational method with Bessel basis set attains the same high accuracy for these potentials as well As another remark, the trivial eigenvalue ordering properties E E l l and E n, l n, l n, l E n n for all N have been confirmed by n, l our calculations Therefore, Eqs 4 and 44 now imply that Ž N ŽN E E N N Ž 48 n, l n, l for all pairs of Ž n, l Furthermore, we deduce from computer experiments that, if we characterize the energy levels En, l as groups denoted by the num- ber m, where m n l, then the eigenvalues in such a group may be ordered according to the rule E E E E 49, m, m m, m, for all N The connection between hydrogen atom in three dimensions and the isotropic harmonic oscillator in two and four dimensions has been a subject 944 VOL 6, NO 5

11 BESSEL BASIS WITH APPLICATIONS TABLE V Specimen calculations for the ground-state eigenvalues E, () of polynomial potentials ( ) V r =v r +vr +v r +v r as a function of the set { v, v, v, v } of coupling constants { } () Potential v, v 4, v 6, v8 E, Lcr M,,, {,,, } {,,, } {,,, } {,,, } {,,, } {,,, } {,,, } {,,, } {,,, } { } {,,, } {,,, } {,,, } { } 5 { } 6 { },,, ,,, ,,, of interest in the past 7 In order to obtain explicit passage formulas between the spectra of hydrogen-like and anharmonic oscillator systems, we now recall the perturbed Coulomb problem with a general confinement potential defined by the equation d N d lž ln dr r dr r Z K i Ý cr i RŽ r ERŽ r, r, r 4 where Z and the ci are the nuclear charge parame- ter and the potential constants, respectively In Ž 4, E denotes the eigenvalue of the problem in N-dimensional space for an angular-momentum state l Transforming the independent variable from r to s, where s' r,, s, Ž 4 we obtain d N d 4lŽ ln ds s ds s K Ý i i v s RŽ s ERŽ s Ž 4 in which a new eigenvalue parameter E has been defined by the parameter such that 8ZE, 4 and that the potential has been rearranged with the coefficients v8e, vi8c, i,,, K Ž 44 Note also that the sign, positive or negative, of depends on E Thus, for negative values of E we may replace by in Ž 4 without any further modification Now, introducing a new wave function Ž s such that Ž s srž s,eq Ž 4 takes INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 945

12 TAŞELI AND ZAFER the form d N d lž ln ds s ds s K Ý i i v s Ž s EŽ s, Ž 45 which is equivalent completely to 5 with NN4, l l 46 This connection in a higher-dimensional space between the anharmonic oscillators and the confinement potentials suggests evidently that, after solving Eq Ž 45, we may generate accordingly some solutions for the perturbed Coulomb potential Ž 4 as well Actually, from Ž 4, Ž 44, and ŽN4 46, we see that any eigenvalue En,l of a polynomial oscillator of degree K is related explicitly to an eigenvalue En, Ž N l of a confinement potential of degree K by the expression Ž N En, l Z; c, c,,ck 8vZ with ŽN4 E Ž v ; v,v,,v Ž 47 n,l 4 6 K i i c E 8 v ZE, i,,, K 48 The relation in Ž 47 needs a correct interpretation Ž N4 It should be noticed that every eigenvalue En,l of a polynomial oscillator with a fixed set of parameters v, v,, v 4 4 K corresponds only to one eigenvalue En, Ž N l of various perturbed Coulomb problems whose potential constants are calculated by Ž 48, except the charge parameter Z Thus, there is no one-to-one correspondence between the eigenvalues of a prescribed general anharmonic oscillator and those of a perturbed Coulomb potential with a definite set of parameters Z, c, c,,c 4 K The sole exceptional case is that of the harmonic oscillator, ie, v4 v6 v K for which the hydrogen atom limit, c c ck, is determined uniquely ŽN4 from 48 In this case E Ž v ;,,, n,l ŽN4 E Ž v v Ž nln from Ž 9 n,l ', and hence the discrete spectrum of N-dimensional hydrogen atom, Ž N ŽN E Z;,,, E Ž Z n,l n, l Z Ž 49 nl Ž N is obtainable exactly from 47 In particular, we reproduce the energy eigenvalues, E, n,l,, Ž 4 n, l Ž nl of the standard three-dimensional case when Z Finally, it should be stated that our main goal is to attack the more challenging problem of unisotropic potentials mentioned in the introduction The first nontrivial problems of this kind are the two-dimensional Ž Ž 4 4 V x, y v x y v x y ax y 4, a Ž 4 and three-dimensional Ž V x, y, z v x y z v Ž x y z ax y ax z ay z, a Ž 4 quartic oscillators It is apparent that the case a corresponds to the isotropic oscillators dealt with in the present article Writing, for instance, Ž 4 in the cylindrical polar coordinates 4 V r, v r vr 4 v4 a r 4 cos sin Ž 4 we may notice the importance of developing such an exact method for the isotropic case, which is necessary and very promising in the investigation of unisotropic potentials along this line as well Work is in progress toward this aim and will be reported in due course References A Rampal and K Datta, J Math Phys 4, 86 Ž 98 R N Chaudhuri and M Mondal, Phys Rev A 5, 85 Ž 995 E R Vrscay and J Cizek, J Math Phys 7, 85 Ž E R Vrscay, Theor Chim Acta 7, 65 Ž VOL 6, NO 5

13 BESSEL BASIS WITH APPLICATIONS 5 H Taşeli, Int J Quant Chem 57, 6 Ž H Taşeli and A Zafer, Int J Quant Chem 6, 759 Ž F M Fernandez and E A Castro, Phys Rev A 4, 88 Ž 98 8 E R Vrscay, Int J Quant Chem, 6 Ž B G Adams, J Cizek, and J Paldus, Int J Quant Chem, 5 Ž 986 G N Watson, A Treatise on the Theory of Bessel Functions Ž Cambridge University Press, Cambridge, 96 I N Sneddon, Special Functions of Mathematical Physics and Chemistry Ž Oliver and Boyd, Edinburgh, 966 M Abramowitz and I A Stegun, Handbook of Mathematical Functions Ž Dover, New York, 97 H Taşeli, J Comput Phys, 5 Ž 99 4 H Taşeli, Int J Quant Chem 46, 9 Ž 99 5 H Taşeli, Int J Quant Chem 6, 64 Ž M Lakshmanan, P Kaliappan, K Larsson, F Carlsson, and P O Froman, Phys Rev A 49, 96 Ž 994 Žand references cited therein 7 M Kibler, A Ronveaux, and T Negadi, J Math Phys 7, 54 Ž 986 Ž and references cited therein INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 947