Eigenvalues of the Two-Dimensional Schrodinger Equation with Nonseparable Potentials

Transcription

1 Eigenvalues of the Two-Dimensional Schrodinger Equation with Nonseparable Potentials H. TASELI" AND R. EID Department of Mathematics, Middle East Technical University, Ankara, Turkey Received June 6, 95; accepted August 9, 95 ABSTRACT The energy eigenvalues of coupled oscillators in two dimensions with quartic and sextic couplings have been calculated to a high accuracy. For this purpose, unbounded domain of the wave function has been truncated and various combination of trigonometric functions are employed as the basis sets in a Rayleigh-Ritz variational method. The method is applicable to the multiwell oscillators as well John Wiley & Sons, Inc. m 1. Introduction W e examine the two-dimensional Schrodinger equation H q = E q in Cartesian coordinates with a Hamiltonian of the perturbed oscillator form where the wave function Y'( x, y) usually satisfies the condition lim q(r) = 0, r = (x, y), x, y E (-a,m). llrll+ m *To whom correspondence should be addressed. (1 2) The particular example which we consider in this study is that of the Schrodinger equation with the potential M i /.\ M = 1,2,... (1.3) in which the coupling constants qi and will be freely chosen so that, in general, we deal with a nonseparable problem. A separable equation, namely, a circularly symmetric oscillator problem is under discussion when ai-,,j = ai; i = 1,2,..., M, j = 0,1,..., i. (1.4) International Journal of Quantum Chemistry, Vol. 59, 3-1 (96) 0 96 John Wiley & Sons, Inc. CCC I 96 I

2 TASELI AND EID Numerical evaluations will be carried out for M = 1,2, and 3. For M = 1, the potential (1.3) reduces to the trivial case of the harmonic oscillator, where The sextic oscillator, where (1.8) which admits exact solution in the form of expcnentially weighted Hermite s polynomials. We take into account this case only for testing the accuracy of our method. The case of M = 2, in which has received less attention in two dimensions. A nonnegative sextic anharmonicity is at hand if we assume that corresponds to the quartic oscillator that has been studied from different points of view [l-81. The main interest is in the case of investigating bound states where the system parameters in pure quartic coupling satisfy the inequalities If, in addition, the harmonicity constants are all positive, we have a single-well potential with only a minimum located at the origin. However, negative values of v2, v2 < 0, enable us to consider two-well quartic oscillators (Fig. 1). Numerical results for a typical example of such oscillators are also included in this work. As far as we know, eigenvalues of two-well potentials in two dimensions have not been reported previously. making the full potential bounded below. Moreover, the potential function V3( x, y) with nonnegative harmonic and quartic terms has an obvious single minimum at the origin, and otherwise is positive with no extrema. On the other hand, it may be shown, analogues to the sextic oscillators in one dimension [9], that the same potential with a strictly negative quartic coupling, v, < 0, possesses three minima provided that vi > 3v2v6. The investigation of eigenvalue problems of this kind is, however, left to a future study. Energy eigenvalues of the aforementioned nontrivial systems are determined by using a twodimensional Rayleigh-Ritz variational method in which the function to be determined depends upon two independent variables. The crucial point of the method lies in the consideration of a truncated domain of the independent variables x and y such that and the modification of the usual boundary condition given in (1.2). Therefore, we encounter the mathematical problem which consists of finding the solution of H 9 = ElIr subject to Dirichlet boundary conditions X FIGURE 1. A two-well oscillator in two dimensions. for all values of x and y on the surfaces bounding the finite rectangular region being considered. The motivation stems from the success of the similar approximation for solving one-dimensional 4 VOL. 59, NO. 3

3 NONSEPARABLE POTENTIALS IN TWO DIMENSIONS Schrodinger equation [9,101. Indeed, the accuracy of the simple technique presented by TaSeli [9] was very impressive in one dimension and suggests evidently the introduction of trigonometric basis in two dimensions as well. Hence we utilize the exact solutions of the unperturbed Schrodinger equation of the form xcos[(i + ijfy], -V2W = AW (1.12) satisfying the boundary conditions in (l.ll), where V2 is the well-known Laplace s operator. In other words, the eigenvalue problem in which the potential is in the form of a rectangular box with impenetrable walls has been considered. In this way, we derive four different enumerable infinite sets of eigensolutions. Such solutions in terms of circular functions and the variational formulations are given in Section 2. In Section 3, we introduce the numerical applications. The rest of the work contains some analytical results, to make the numerical procedure plausible, and the discussion of the energy levels crossings with further concluding remarks. and 2 r2 2 A,, = ( k + 1) - + ( I + 1) 7, a2 (2.4) where the A,, are the corresponding eigenvalues. Thus we have P 2. Trigonometric Basis Separating the variables x and y, it is a simple matter to construct the eigenfunctions and the eigenvalues of the boundary value problem defined in (1.12) and (1.11). Actually, we obtain, for k, I = O,l,..., the following four normalized sequences of orthogonal eigenfunctions: xcos I + - -y, [i :I; I xsin (I + 1)-y, [ p 1 2 r2 + ( I + 1) 2, (2.2) P E [ En,] = El, Ell El2 El3 E E E E, (2.7) E30 E31 E32 E33 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 5

4 TASELI AND EID it is then possible to split [En,] into four matrices which, on evaluating the double integrals, yields the system of algebraic equations m m where Elklrnn is defined by m, n = 0,1,..., (2.12) (2.8) F) +<[(k owing to the reflection symmetry of the wave function wx, y) = W - X, y) = w x, -y) =?P(-x, -y) (2.9) being considered in this study. According to the above decomposition of the spectrum, we conjecture that sets can be used separately to determine the four blocks of eigenvalues in (2.8), respectively. Now the wave function is postulated to be of the form r a m where fkl are the linear combination coefficients to be determined. We assume that the double-infinite series on the right-hand side converges uniformly to the sum Wx, y). If we substitute Wx, y) into the Schrodinger equation H? = E9, we may multiply both sides by +,Jx, y) and integrate term by term with respect to x and y over their intervals to obtain the relation In this definition RP) stands for the definite integral T RV) = ; x2'cos kxdx, (2.14) which, after integration by parts 2r times, is expressible explicitly as ' r-1 1 (- & ) j (2.15) (2r - 2i - i=o for k > 0 and r = 1,2,..., M. In particular, we find that Alternatively, Rr) may be evaluated recursively by the relation k2r'," = 2r(-l)k7r2r-2-2r(2r - l)rc,'-'), r = 1,2,..., M (2.) for any fixed k > 0, with the initial condition that R(kO) = 0. On the other hand, the integer parameters sl, s2, pl, and p2 in Hklrnn are either zero or T1, 6 VOL.59, NO. 3

5 NONSEPARABLE POTENTIALS IN TWO DIMENSIONS depending on the basis set under consideration. We must take altered to the standard matrix eigenvalue problem of the form and s1 = s2 = 1, p1 = p2 = 0, (2.) N2 C ( A,, - E6ij)gj = s1 0, = 1, s2 = -1, p1 = 0, p2 = 1, j= 1 (2.) s1 = -1, s2 = 1, p1 = 1, p2 = 0, (2*) s1 = s2 = -1, p1 = p2 = 1, (2.) for the sets 0, 0, 0, respectively. Notice also the block symmetry of Hklmn Hklmn = Hmnk/ (2.) in the first two and the last two indices. At the computational side of this work, we consider a truncated wave function so that the resulting infinite system of algebraic equations in (2.12) is replaced by the finite system N-1 N-1 C C (Hklmn - E'km'/n)fk/ = 0, k=o l=o m,n=0,1,..., N -1 (2.) of order N2, where N is the size of truncation. It is interesting to note that if we now introduce an integer transformation T, T: Ni + N2 defined by T = {(i, j) E N2: i = mn + n + 1 and j = kn V(k, I, m, n) E Ni) (2.24) Hklmn and 6,,S,, are reduced to Aij and S,,, respectively, where N = {1,2,...} is the set of natural numbers and No = N U {O). Here the matrix [ Aii] of order N2 may be called the reduced variational matrix, and [ aijl is the identity matrix of the same order since the transformed indices i and j differ from 1 to N2 for k, I, m, n = 0,1,..., N - 1. The block symmetry of Hklmn implies immediately the symmetry [Aij] = [Aji] of the reduced variational matrix. Similarly, the transformation S, S: No X No + N S={jeN: j=kn+i+lv(k,l)~n,xn,} (2.25) i = 1,2,..., N2 (2.26) to determine numerically the energy spectrum of the Schrodinger equation using available routines such as TRED2 and TQL2 [ll]. 3. Application The present method is applied to calculate bound-state energies of anharmonic oscillators for a wide range of coupling constants. We first consider the harmonic oscillator () whose exact eigenvalues are known as E = En, = &[(2n + l)& + ( + 1)&] (3.1) in the unbounded domain of x and y. The problem is interesting from the viewpoint of testing our approximation for each basis set. Results accurate to digits are given in Table I for a,, = aol = v2 = 1. In the numerical tables, n and 1 show the quantum numbers of the state and N stands for the truncation size which is sufficient to reach the desired accuracy. In each case, we report eigenvalues to digits and set p = a. Therefore, the sole boundary parameter a, for which the presented accuracy is achieved, is defined as the critical value a,, similar to that of the one-dimensional Schrodinger equation [9]. It should be noted that a systematic investigation of the potentials considered here requires too many numerical tables since the number of coupling parameters is rather large with infinitely many combinations. So we present extensive data only for potentials with the interchange symmetry V(X, y) = V(y, XI, () which explains why we solve the problem on a square domain putting p = a. For such an oscillator the coefficients in (1.8) satisfy the relations transforms fkl with k, 2 = 0,1,..., N - 1 into g, with j = 1,2,..., N2. Hence the system in (2.) is allj = aol, 4.0 = a = a12r a30 = a 03~ () INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1 87

6 TASELI AND EID TABLE I Eigenvalues of the harmonic oscillator, V(x, y) = x2 + y2. Basis set acr k I N and the corresponding potential, on making a simple scaling transformation, is equivalent to V( x, y) = x2 + y2 + c4( x4 + 2ax2y2 + y4> + c6(x6 + 3bx4y2 + 3bx2y4 + y6), (3.4) where a, b, c4, and c6 are the new system parameters. In the case of the quartic coupling, when c6 = 0, we see from (1.7) that c4 > 0, -1 I a 5 1. (3.5) Numerical results are included in Tables II-VI for the values of c4 = 1, lo3, lo6, and 00 corresponding to each a = -l,o, and 1, covering the range of a. Notice that c4 -, 00 limit represents the pure quartic oscillator problem ( -V2 + x4 + 2ax2y2 + y4)q = 89, (3.6) where E = c:138. For this reason eigenvalues for cq > 1 have been scaled by cq1i3 to infer how rapidly the c4 -, m limit values of energies are achieved as c4 increases. Tables VII-X are devoted to the more general problem of the sextic oscillators in two dimensions by taking into account the potential in (3.4) with c4 = 0. It may be shown that the conditions in (1.9) now imply the relations c6 > 0, (3b + l)(b - li3 I 0 (3.7) so that the parameter b varies between - I b I 1. Thus the entire range of b is covered by setting b = - i,o, i, 5, and 1 while c6 changes from to m. Here, c6 -, 00 limit Hamiltonian corresponds to the pure sextic oscillator (-V2 + x6 + 3bx4y2 + 3bx2y4 + y6)'p = ' (3.8) with E = c;i42f". As in the case of the quartic coupling eigenvalues in the numerical tables for c6 > 1 are those scaled by ct Discussion In this work, the spectra of unbounded oscillators in two dimensions are obtained by way of increasing systematically the boundary value a, regarding it as a nonlinear optimization parameter. It was shown numerically that there exists a critical value acr at which the low-lying state energies are equal to those of a limit to a prescribed accuracy. In order to justify the procedure analytically, we first prove the domain monotonicity of the eigenvalues. Let us reconsider the Schrodinger equation [ -V2 + V (X, y) - E ]~(x, y) = 0, X, y E l. -a, ai (4.1) in which the potential, as does the wave function, satisfies both reflection and interchange symmetries with the accompanying Dirichlet boundary conditions 8 VOL.59, NO. 3

7 NONSEPARABLE POTENTIALS IN TWO DIMENSIONS TABLE II Eigenvalues of the quartic oscillator for c, = 10-3 as a function of a. a %r N n I E", Basis set INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1 89

8 TASELI AND EID TABLE 111 Eigenvalues of the quartic oscillator for c, = 1 as a function of a. a %r N n I E", Basis set VOL. 59, NO. 3

9 NONSEPARABLE POTENTIALS IN TWO DIMENSIONS TABLE IV Eigenvalues of the quartic oscillator for ca = lo3 as a function of a. a acr N n I c, "3E ", Basis set a37 91 o i o I a INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1

10 TA$ELI AND EID TABLE V Eigenvalues of the quartic oscillator for c, = lo6 as a function of a. a %I N n I c, "3E ", Basis set VOL. 59, NO. 3

11 NONSEPARABLE POTENTIALS IN TWO DIMENSIONS TABLE VI Eigenvalues of the quartic oscillator for ca + ~0 as a function of a. a %r N I c, "3E ", Basis set INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 3

12 TA!$ELI AND EID TABLE VII Eigenvalues of the sextic oscillator for c, = 10-4 as a function of b. b %r N n I E, Basis set _ VOL. 59, -NO. 3

13 NONSEPARABLE POTENTIALS IN TWO DIMENSIONS TABLE Vlll Eigenvalues of the sextic oscillator for c6 = 1 as a function of b. b %r N n i E", Basis set ~ INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1 95

14 TASELl AND EID TABLE IX Eigenvalues of the sextic oscillator for c6 = lo4 as a function of b. b %-r N n I c, "4E,, Basis set VOL. 59, NO. 3

15 NONSEPARABLE POTENTIALS IN TWO DIMENSIONS TABLE X Eigenvalues of the sextic oscillator for c6 as a function of b. b %r N n I c6- "4E, Basis set INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 7

16 TASELI AND EID for all y and W(x, -a) = 0, Wx, a) = 0 (4.3) for all x. It is obvious that the eigenvalues and the eigenfunctions in (4.1) depend on the boundary parameter a. Therefore, we may denote W = W( x, y, a), E = (a). (4.4) Now, differentiating (4.1) with respect to a, multiplying by W and integrating the result over the range of the independent variables, we obtain de d dw da da T(x, y, a) dxdy. () We may assume, without any loss of generality, that the wave function is normalized, (W, W) = 1, so that de - = ([ -V2 + V (x, y) - da where the derivative operators with respect to a and the Laplace operator in () have been interchanged. On integrating by parts, the right-hand side of (4.6) can be put into a form 2, (g, (L W) = boundary terms + L*Y) da (4.7) wherein the last inner product vanishes from (4.1) since the operator L = -0' + V(x, y) - E is formally self-adjoint, L* = L. Hence, writing explicitly the boundary terms we arrive at the relation de dw d Y -."jla dy?")la dw d Y - dx. (4.8) da dx aa dxda X=-a dyda Y=-a To derive a more neat expression for de/da, we consider the total differential of W. From (4.41, we have dw dw dw dy = -dx + -dy + -da. (4.9) dx JY da If x is a function of a, x = x(a) say, then dx = (dx/da) da so that where the meaning of dw/da + [(dx/da) (dy/dx)] is surely that of the partial derivative of W with respect to a when y is kept constant, dy = 0. Therefore, for x = da), dw dw dw dx (4.11) da aa dx da Now, differentiation of the conditions given in (4.2) with respect to a by taking x = -a and x = a, respectively, it follows that Ya(-a, y) - Wx(-a, y) = 0, Wa(a, y) + Wx(a, y) = 0, (4.12) where the subscripts denote partial derivatives. Likewise for y = y(a), from (4.9) and (4.3), we have Wa(x, -01) - Ty(x, -a) = 0, Wa( x, a) + WY( x, a) = 0. (4.13) Substituting (4.12) and (4.13) into (4.8) and using the symmetries of the wave function, we see that (4.8) can be written in the form (4.14) It is clear that de/da is strictly negative which implies that E( a) decreases monotonically to its limit E(m) as a In other words, the larger region has smaller eigenvalues, E(a) 2 E(m). (4.15) This means that E(a)'s are upper bounds to the asymptotic eigenvalues; a property which is well known for the corresponding one-dimensional problems,1. As a consequence of (4.14) and (4.151, the difference IEJa,) - En[(a,)l = E,, for any quantum state (n, I ) measures the inaccuracy of the results. In our calculations we estimate acr values in such a way that E,! is less than 10-*", and therefore present eigenvalues to significant figures. 8 VOL. 59, NO. 3

17 NONSEPARABLE POTENTIALS IN TWO DIMENSIONS On the other hand, for any a, fixed, the relations EII/(a,, N) > Enl(a,, N + 11, EJa,) = lim Enl(a,, N) (4.16) N-m are known from the variational principle. Therefore, stable digits of eigenvalues conforming between two consecutive values of N, imply the accuracy of En,(az). In the tables, the recorded truncation orders N are those for which I En[(&,, N + 1) - Enl(~,, N)I < lop2 (4.) showing that the Enl(a,) s are correct up to digits. and are used in evaluating the energy levels with the same parity, namely, both even or both odd. The eigenvalues, however, having different parity, one even and one odd, may be calculated by either or since they are doubly degenerate for all oscillators considered numerically in this work. In Table I, we see that the exact eigenstates En, = 2(n ) of the harmonic oscillator, V(x, y) = x2 + y2, are reproduced. This problem might have been treated in cylindrical polar coordinates by the separation of variables. In general, the separation process may cause a loss of some solutions. Indeed, solving the radial Schrodinger equation we could find the exact eigenvalues to be 2(2n + I + 1) [13]. Here, E2,,, is determined rather than En,, and hence the solutions corresponding to M odd are lost in the separation process. A similar situation occurs for the quartic and the sextic oscillators defined in (3.4) when a = 1 and b = 1, respectively, for which the systems have a special circular symmetry. For this reason, the eigenvalues of the circularly symmetric quartic oscillators characterized by En, in [3] and [13] should be compared with those of E2n,l in this study. The structure of the present trigonometric basis sets provides a very natural and a simple way to order the eigenvalues according to the quantum numbers. If the energy levels En, are characterized as groups denoted by the number rn = n + I, it is then easier to understand their certain ordering properties. In the case of the circularly symmetric oscillators we deduce, from Tables 11-VI for a = 1 and from Tables VII-X for b = 1, that if rn is even - E0,m = E 1,m-1 < E 2,m-z - Es,m-3 < **. < E m-2,2 - L 1, l < Em,0 (4.) while if m is odd the numbers of degeneracies being 2,2,..., 2,l and 2,2,..., 2, respectively. As another special case we have two uncoupled quartic anharmonic oscillators when a = 0. Similarly, for b = 0, the system becomes one of two independent sextic oscillators. For these cases, it is clear that En, = En + E,, M, I = 0,1,... (4.) where the Ek s are the kth eigenvalues of the corresponding one-dimensional problems. Thus the accuracy of the present results may be confirmed by means of (4.) on recalling, especially, the numerical results of Banejee [14] and TaSeli and Demiralp [15] given in one dimension. The equation (4.) implies also that En, = El, which can be seen immediately from our tables. Furthermore, the group characterized by rn has the energy levels in ascending order of magnitude for rn even, 2k say, and for rn odd, 2k + 1 say, - Ek,k+l = E k+l,k < E k-l,k+2 - Ek+2,k-l <. < E1,2k = E2k,/ < E0,2k+l - E2k+1,0 (4.) with the degeneracies equal, respectively, to 1,2,2,...,2 and 2,2,...,2. The eigenvalues of the sextic oscillator in the group 2k + 1 remain doubly degenerate and unsplit as b varies from - to 1. For the group 2k, however, the doubly degenerate levels at b = 0 and b = 1 split into two levels such that Ek,k < E k-l,k+l < E k+l,k-l < Ek-2,k+2 < ** < 1,2k--l < 2k-1,l < 0,2k < 2k,0 (4.) when - is b < 0 and 0 < b < 1. Although the numerical results are not quoted here, it seems that the eigenvalues of the quartic oscillators with -1 < a < 0 and 0 < a < 1 show the same trend. As another important feature of energy levels crossings, we examine the ordering of eigenvalues INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1 99

18 TA$ELI AND EID belonging to two different groups. Let rn and rn, denote two groups such that rn = n + 1 and rn, = n, + I,. If rn > rn, then we infer that En, > Enl,ll* rn = 0,1,..., M, (4.24) where M is a finite number representing the group M. It is noteworthy that the number M, up to which eigenvalues satisfy (4.241, depends mainly on the potential function being considered. Obviously, M tends to infinity only for the exceptional case of harmonic oscillators. Hence in the near harmonic regime of eigenvalues, we observe that M is indeed a very large number. It is, however, considerably small in the boundary layer and in the pure anharmonic regime. More specifically, as can be seen from Table I11 for the quartic coupling with a = - 1 and c4 = 1,, in group rn = 4 is less than Eo,3 in group rn, = 3 in spite of rn > rn,. Therefore, M = 3 for this potential. We notice, from Table VI for a = -1, that M becomes 2 as c4 + m since E,,, < E,,,. In general, the number M is more or less the same whenever the coupling constant c In the case of uncoupled quartic oscillator, a = 0, the eigenvalues may be ordered according to (4.24) for rn = 0,1,...,7 since E4,, < E,, which implies that M = 7. For the circularly symmetric oscillators with c4 2 1, by noting that inequality (4.24) is invalid for the first time at rn = 11 where E,,,, < E,,,, we find that M = 10. Eigenvalues of the sextic oscillators behave very similarly. For c6 2 1, it is concluded that M is at least 3 and at most 7 as b varies from - 5 to 1. We believe that this interesting structure of the energy spectrum of a coupled system is discussed for the first time. Numerical computations are performed using quadruple precision arithmetic on an IBM AIX computer system. Over a wide range of the coupling constants, we see that a truncation size of N about is sufficient in determining eigenvalues to significant digits, which consumes approximately 4 CPU minutes. Only the cases of a = -1 and b = - 3 require to take higher truncation orders. This is related to the fact that the rate of convergence of our algorithm is relatively slower for the corresponding potentials. The comments, given in [9], on the critical values of OL are completely representative in the twodimensional case as well. Therefore, we do not TABLE XI Eigenvalues of V(x, y) = -x2 - y (x2 + y2i2. %T N n I En, -k 2.5 Basis set VOL. 59, NO. 3

19 NONSEPARABLE POTENTIALS IN TWO DIMENSIONS duplicate here the fairly detailed discussion of [9] about acr. As an attempt of applying the method to an eigenvalue problem of a different nature we finally consider two-well quartic oscillators where V(X, y) = -x2 - y2 + c4(x4 + 2ux2y2 + y4). (4.25) Numerical results of a particular example, c4 = 0.1 and u = 1, are given in Table XI, which encourage us to employ trigonometric basis sets in solving more complex systems having both symmetrical and unsymmetrical potentials with one or more than one minima. References 1. T. Banks, C. M. Bender, and T. T. Wu, Phys. Rev. D 8,3346 (73). 2. I. C. Percival and N. Pomphrey, Mol. Phys. 31, 97 (76). 3. F. T. Hioe, D. MacMillen, and E. W. Montroll, Phys. Rep. 43, 305 (78). 4. R. A. Pullen and A. R. Edmonds, J. Phys. A: Math. Gen. 14, L477 (81). 5. N. Ari and M. Demiralp, J, Math. Phys. 26, (85). 6. F. M. Fernandez, A. M. Meson, and E. A. Castro, Phys. Lett, 112A, 107 (85). 7. J. Killingbeck and M. N. Jones, J. Phys. A: Math. Gen., 705 (86). 8. E. R. Vrscay and C. R. Handy, J. Phys. A: Math. Gen.,8 (89). 9. H. Taqeli, Int. J. Quantum Chem. 46, 3 (93). 10. H. Taqeli, J. Comput. Phys. 101, 252 (92). 11. H. W. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 861, p F. M. Fernandez and E. A. Castro, Int. J. Quantum Chem., 533 (81). 13. H. Taqeli, Int. J. Quantum Chem. 57, 63 (96). 14. K. Banejee, Proc. R. Soc. A 364, 265 (78). 15. H. Taqeli and M. Demiralp, J. Phys. A Math. Gen., 3903 (88). INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1