A canonical approach for computing the eigenvalues of the ...

969

A canonical approach for computing the eigenvalues of the Schrödinger equation for double-well potentials M. Korek and K. Fakhreddine

Abstract: The problem of obtaining the eigenvalues of the Schrödinger equation for a doublewell potential function is considered. By replacing the differential Schrödinger equation by P a Volterra integral equation the wave function will be given by 9v = 1i=0 ai fi where the coefficients ai are obtained from the boundary conditions and the fi are two well-defined canonical functions. Using these canonical functions, we define an eigenvalue function F (E) = 0; its roots E1 , E2 , ... are the eigenvalues of the corresponding double-well potential. The numerical application to analytical potentials (either symmetric or asymmetric) and to a P numerical potential of the (2)1 + u state of the molecule Na2 shows the validity and the high accuracy of the present formulation. PACS No.: 03.65Ge Résumé : Nous cherchons à obtenir les valeurs propres de l’équation de Schrödinger pour un potentiel à double puits. En remplaçant l’équation différentielle de Schrödinger par P1 une équation intégrale de Volterra, la fonction d’onde prend la forme 9v = i=0 ai fi où les coefficients ai sont fixés par les conditions initiales et les fi sont des fonctions canoniques bien définies. Utilisant ces fonctions canoniques, nous obtenons une équation caractéristique F (E) = 0 dont les racines E1 , E2 , ... sont les valeurs propres du problème à double puits. La validité et la haute précision de la méthode ont été vérifiées en l’appliquant numériquement à certains potentiels analytiques (symétriques et asymétriques) et au potentiel P connu numériquement pour l’état (2)1 + u de la molécule de Na2 . [Traduit par la rédaction]

1. Introduction For many decades the double-well potential (DWP) has been used to model various phenomena [1–15] encountered in physics and chemistry. These include, for example, the quantum theories of measurement [6,9], molecular structure [13,14], coherent tunneling [11], the role of exchange forces [3,4], and the inversion spectrum of NH3 [2]. Various methods have appeared in the literature to solve the DWP problem [16–28] with varying degrees of satisfaction and success. A DWP can be represented by a numerical potential [29,30] or by symmetric or asymmetric analytical

Received September 27, 1999. Accepted September 15, 2000. Published on the NRC Research Press Web site on January 9, 2001. M. Korek.1 Faculty of Science, Beirut Arab University, P.O. Box 11-5020, Beirut, Lebanon. K. Fakhreddine. Faculty of Science, Lebanese University, Nabatieh, Lebanon. 1

Author to whom all correspondence should be addressed. FAX: (961) 1 818 402; e-mail:

Can. J. Phys. 78: 969–976 (2000)

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potential functions [31,32]. The DWP problem may be written as a more general one, that of the radial Schrödinger equation d2 y(r) + (E − U )y(r) = 0 dr 2

(1)

where U (r) is a DWP potential function, and y(r) is the eigenfunction that obeys the boundary conditions y(r) −→ 0 r→a

y(r) −→ 0

and

(2)

r→b

where a = 0 and b = ∞ for an asymmetric DWP and b = −a = ∞ for a symmetric potential. The canonical function method [33,34] is a powerful alternative to the standard treatments of the DWP problem where the computation of the wave function y(r) (implying an initial value problem) is replaced by two canonical functions f0 (r0 ; r) and f1 (r0 ; r) that are particular solutions of (1) with the initial values f0 (r0 , r0 ) = 1 f1 (r0 , r0 ) = 0

f00 (r0 , r0 ) = 0 f10 (r0 , r0 ) = 1

(3a) (3b)

r0 being an arbitrary origin and 0 < r0 < ∞. The method is “canonical” in the sense that it dissociates the determination of the eigenvalue from that of the eigenfunction (by calculating these canonical functions f0 and f1 instead of y(r) explicitly), and allows the derivation of an eigenvalue equation F (E) = 0 related uniquely to the given potential function U (r). The solution of this equation using the boundary conditions (2) of y(r) gives the eigenvalues E0 , E1 , E2 , ..., En , ... of the potential considered. A DWP function can be represented by a numerical potential [29,30] or an analytical potential which can be symmetric [31] as i h 2 2 (4) U (r) = D 1 − e−ω(r+r a ) + e−ω(r−r a ) Or asymmetric [32] as i h 2 2 U (r) = D 1 − e−B(r−r a ) + A e−C(r−r b )

(5)

The aim of this work is to show that the canonical functions method [33], already used with success for diatomic potentials in many problems of molecular physics [35–38], can be extended to different types of DWP functions either numerical or analytical (symmetric or asymmetric). Applications to these potentials are presented along with comparisons to previous works.

2. Theory For a given DWP function U (r) and for an arbitrary value of the “parameter” E, a second type of Volterra equation [39] can be associated with the differential equation (1) as 0

Zr

y(r) = y(E, r0 ) + ry (E, r0 ) −

(r − t)[E − U (r)] dt

(6)

r0

in the sense that any solution of (6) is a solution of (1) and r0 is an arbitrary origin a < r0 < b. Using the properties of the Volterra integral equation the solution of (6) is given by [37] y(E, r) =

1 X

an fn

(7)

n=0

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where a0 = y(E, r0 )

a1 = y 0 (E, r0 )

(8)

and the functions f0 and f1 are two particular solutions of (1) with the initial conditions f0 (E, r0 , r0 ) = f10 (E, r0 , r0 ) = 1 f1 (E, r0 , r0 ) = f00 (E, r0 , r0 ) = 0

(9a) (9b)

By imposing the boundary conditions (2) on y(E, r) at r = a and r = b, we find a0 f0 (E, r0 , a) + a1 f1 (E, r0 , a) = 0 a0 f0 (E, r0 , b) + a1 f1 (E, r0 , b) = 0

(10a) (10b)

For an arbitrary value of E, we associate with the potential U (r) the eigenvalue function F (E) defined by F (E) = g1 (E) − g0 (E) where g1 (E) =

f0 (E, r0 , b) , f1 (E, r0 , b)

g0 (E) =

f0 (E, r0 , a) f1 (E, r0 , a)

(11)

If the eigenfunction of (1) (as well as its derivatives) is continuous on the r-axis and obeys the boundary conditions (2), the zeros of the eigenvalue function F (E) = 0 correspond to the eigenvalues E0 , E1 , E2 , ..., En , ... of the given DWP. Therefore, one may write f0 (En , r0 , a) f0 (En , r0 , b) = f1 (En , r0 , b) f1 (En , r0 , a)

(12)

For an arbitrary value of the “parameter” E, the canonical functions f0 and f1 are computed for r ≥ r0 ; using a convenient integrator; (1) is integrated starting at r0 for the functions f0 and f1 simultaneously with the initial values defined in (9). We move toward a large value of r; the integration is stopped when the function g1 reaches a constant limit g1 (E) (11). This integration, repeated for r ≤ r0 , allows the determination of g0 (E) within the computer precision. The eigenvalue problem is then reduced to the calculation of f0 and f1 for each value of E and to finding the zeros of the eigenvalue function F (E) = g1 (E) − g0 (E) when the parameter E varies. This function F (E) behaves like tan(αE + β) and the determination of its zeros is done using conventional procedures. For the symmetric DWP b = −a; if we consider r0 = rmax (rmax is the abscissa of the maximum of the DWP function), g0 (E) = g1 (E) (11), i.e., F (E) = 0 for any value of E (eigenvalue or not). Thus, for a given symmetric DWP, one must take r0 6= rmax (r0 should be carefully optimized), since with this dissymmetry of the potential function with respect to r0 , the calculation of the eigenvalues is similar to that for the asymmetric potential.

3. Numerical application To test the validity, generality, and accuracy of the present formulation for the eigenvalue problem of a DWP, we considered three different types of DWP, numerical [30] and analytical, symmetric [31], or asymmetric [32]. The determination of the eigenvalues E0 , E1 , E2 , ..., En , ... is done by looking for successive solutions of F (E) = 0 when E varies from zero to infinity. We start the calculation at an arbitrary r0 to ©2000 NRC Canada

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compute f0 (E, r0 , r) and f1 (E, r0 , r) and by replacing (1) with a convenient difference equation (that of the integral superposition difference equation [40]) reformulated for the DWP. Once r0 (according to the potential considered), we divide the r-axis into intervals  we choose  Ip = rp , rp+1 , where rp and rp+1 are the abscissas of two consecutive points for the numerical potential U num and any two “close” points for the analytical potential
U ana . In the interval Ip , U num is interpolated using a cubic spline in x = r − rp 0 ≤ x < rp+1 − rp . U ana is proved to be a power series in x truncated to polynomials. For the three potentials considered, one may write U (x) =

∞ X

γm (p)x m

0 ≤ x ≤ hp

(13)

m=0

where the step hp = rp+1 − rp . In this case the integral superposition difference equation for the DWP becomes a simple Taylor series [40]: yp+1 = yp + hp yp0 + 0 = yp0 + yp+1

∞ X n=2

∞ X n=2

Cn (p)hnp

(14a)

nCn (p)hn−1 p

(14b)

where yp = y(rp ) and Cn (p) are given by the recursion (n + 2)(n + 1)Cn+2 (p) = −ECn (p) +

n X m=0

(p)

Cm (p)γn−p

(15)

with C0 (p) = yp

and

C1 (p) = yp0

To improve the accuracy of the numerical integration of the Schrödinger equation for a DWP, and to reduce the “local truncation error,” we used a variable step algorithm with local error control [40]. One may consider (14) and write yp+1 = yp + hp yp0 +

N X n=2

Cn (p)hnp + RN

(16)

where RN is the error neglected when the series in (14) is truncated, to RN we impose |RN | ≤ ε

(17)

ε being a chosen tolerance (usually the computer precision). For U num where the intervals hp are imposed, N is deduced from (17) [40]; we take |Cn (p)hnp | = ε2

(18)

and N varies from one interval to another. For U ana one may fix N and deduce hp as  hp =

εε |CN (p)|

1/N (19) ©2000 NRC Canada

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973 Table 1. Values of the eigenvalues of the asymmetric analytical double-well potential (5) calculated using the present method compared with those calculated by Johnson [32]. n

Johnson [32]

Present work

| 1E | × 106 E

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 302.500 3 205.307 4 227.339 5 144.251 6 064.241 7 092.679 7 614.622 8 911.545 9 095.696 10 208.350 10 869.289 11 482.475 12 353.799 12 972.473 13 690.455 14 435.350

1 302.498 972 63 3 205.303 782 31 4 227.336 543 97 5 144.243 754 36 6 064.225 881 61 7 092.664 815 25 7 614.603 506 89 8 911.513 342 95 9 095.679 497 32 10 208.318 142 1 10 869.255 077 0 11 482.457 956 3 12 353.766 422 6 12 972.453 117 0 13 690.436 602 2 14 435.321 044 1

0.768 0.998 0.591 1.419 2.490 1.864 2.429 3.557 1.814 3.124 3.118 1.489 2.638 1.534 1.344 2.008

Using this local error control in Ip for the DWP, (14) are replaced by yp+1 = yp + hp yp0 + 0 = yp0 + yp+1

N X n=2

N X n=2

Cn (p)hnp

nCn (p)hn−1 p

(20a)

(20b)

These equations allow the propagation from one point to another using a local series (with variable step and (or) variable order). The local error control is not restricted for the integrator, it is used for the approximation of the analytic potential by a Taylor series. To apply the present formulation to analytical asymmetric potential, we used the DWP discussed and studied by Wicke and Harris [41] and by Johnson [32] (5) (the parameters of this potential and reduced mass are given in ref. 32). We considered r0 equal to the abscissa of the maximum of this potential. The limits of integration for all calculations were rmin = 1.02 Å and rmax = 1.95 Å while these values for Johnson [32] are rmin = 1.0 Å and rmax = 2.6 Å. The comparison of the eigenvalues of this DWP calculated by the present method (column 2, Table 1) to those calculated by Johnson [32] (column 3, Table 1) shows excellent agreement for all the levels considered. For the analytical double-well symmetric potential we used the Gaussian potential given by Hamilton and Light [31] (4) where ω = 0.1 , ra = 5.0, and D = 12.0. We considered r0 = rmax + 10−6 where rmax is the abscissa of the maximum of the potential. The comparison of the eigenvalues calculated by the present method (Table 2, column 3) to those calculated by Hamilton and Light [31] (Table 2, column 2) shows an excellent agreement for all the levels considered. We also applied the present method to the asymmetric numerical potential given by Cooper et al. [30]. By taking r0 equal to the abscissa of the outer well we calculated the eigenvalues for different values of r (Table 3, column 2). The comparison of these values to those obtained by a theoretical calculation [42] (Table 3, column 2) shows a very good agreement for all the calculated values, while ©2000 NRC Canada

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Can. J. Phys. Vol. 78, 2000 Table 2. Values of the eigenvalues of the symmetric analytical double-well potential (4) calculated by the present method compared with those calculated by Hamilton and Light [31]. n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Hamilton and Light [31] −11.245 199 313 −11.245 199 313 − 9.773 496 902 − 9.773 496 902 − 8.381 307 510 − 8.381 307 491 − 7.072 039 562 − 7.072 038 846 − 5.849 958 02 − 5.849 940 0 − 4.720 829 36 − 4.720 509 6 − 3.694 518 38 − 3.690 475 6 − 2.798 251 92 − 2.763 219 7 − 2.089 661 3 − 1.924 577 − 1.462 202 9 − 0.149 254 − 0.777 081 − 0.457 88 − 0.177 181 − 0.003 41

Present work −11.245 199 312 999 −11.245 199 312 986 − 9.773 496 902 0000 − 9.773 496 902 4256 − 8.381 307 504 6697 − 8.381 307 491 0744 − 7.072 039 562 0000 − 7.072 038 842 5456 − 5.849 958 029 7256 − 5.849 940 000 0000 − 4.720 829 438 9858 − 4.720 509 598 1823 − 3.694 518 379 6659 − 3.690 475 429 1720 − 2.798 251 905 4419 − 2.763 219 769 7340 − 2.089 661 073 5071 − 1.924 577 036 3140 − 1.462 202 686 4940 − 1.149 254 149 1320 − 0.777 081 556 9842 − 0.457 879 988 9440 − 0.177 181 028 8888 − 0.003 409 978 9142

Table 3. Values of the eigenvalues of the asymmetric numerical double-well potential calculated by the present method compared with experimental [30] and theoretical results [42] for 2(1) 6u+ state of Na2 . r

Jeung

Cooper

(Å)

Present work

[42]

et al. [30]

2.646 3.175 3.704 4.233 4.763 5.292 5.821 6.879 7.938

31 993.225 29 061.920 28 464.119 28 843.500 29 175.576 28 715.283 28 315.518 28 009.829 28 609.504

31 986 29 067 28 474 28 869 29 155 28 782 28 321 27 991 28 606

32 756 29 105 28 455 28 838 29 130 28 796 28 281 27 889 28 452

the comparison of our values with those obtained by Cooper et al. [30] (Table 3, column 4) shows an overall good agreement. We then calculated the eigenvalues EvJ by taking r0 equal to the abscissa of the inner well r0 = 3.688 Å, and in this case we obtained E00 = 28585.920 cm −1 , E10 = 28795.169 cm−1 , ©2000 NRC Canada

Korek and Fakhreddine

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E20 = 28769.642 cm−1 , E30 = 28830.935 cm−1 , and E40 = 29014.803 cm−1 . These values are also in good agreement with those of Cooper et al. [30]. Thus, the eigenvalues of a DWP function calculated using the present canonical approach are in very good agreement with other results; the advantages of this formulation are (i) the variable step method can be used to reach the computer precision, (ii) the boundary limits can be obtained at shorter values of r than for the other methods, (iii) the generality of this method which can be applied to different types of DWP, and (iv) in our technique we know when the boundary limits are reached, whereas in the other techniques this is often done by trial and error.
2 The present method has also been applied to another symmetric potential [32] U (x) = x 2 − 1 and numerical potentials [29,42]; similar results (not shown) are obtained.

4. Conclusion Using the canonical functions approach, we present in this work an alternative powerful method for solving the double-well potential problem. The application of the present formulation to different types of DWP functions either numerical or analytical (symmetric or asymmetric) with the variable-step technique shows its generality and its high accuracy where computer precision is reached. The extension of this work to the calculation of the rotational constants and the centrifugal distortion constants of a DWP function is being undertaken.

References 1. G. Herzberg. Spectra of diatomic molecules. Van Nostrand, Toronto. 1950. 2. R.P. Feynman, R.B. Leighton, and M. Sands. The Feynman lectures in physics. Vol 3. Addison-Wesley, Reading, Mass. 1965. 3. P.R. Certain. J. Chem. Phys. 49, 35 (1968). 4. R.J. Danburg and R.K. Propin. J. Chem. Phys. 55, 612 (1971). 5. M. Simonius. Phys. Rev. Lett. 40, 980 (1978). 6. W.H. Zurek. Phys. Rev. D: Part. Fields, 24, 1516 (1981). 7. R.A. Harris and L. Stodolsky. J. Chem. Phys. 74, 2145 (1981). 8. S. Chakravarty and A. Legett. Phys. Rev. Lett. 52, 5 (1984). 9. E. Joos and H.D. Zeh. Z. Phys. B: Condens. Matter, 59, 223 (1985). 10. A.G. Zawadzki and J.T. Hynes. Chem. Phys. Lett. 113, 476 (1985). 11. H. Dekker. Phys. Lett. 114A, 295 (1986). 12. H. Knowent. Phys. Lett. 118A, 467 (1986). 13. P. Claverie and G.J. Lasinio. Phys. Rev. A: Gen. Phys. 33, 2245 (1986). 14. B.R. Fisher and P. Mittelstaedt. Phys. Lett. 147A, 411 (1990). 15. P. Dutta and S.P. Bhattacharyya. Phys. Lett. 163A, 193 (1992). 16. G.J. Lasinio, F. Martinelly, and E. Scoppola. Phys. Rep. 77, 313 (1981). 17. G.J. Lasinio, F. Martinelly, and E. Scoppola. Commun. Math. Phys. 80, 223 (1981). 18. S. Graffi, V. Grecchi, and G.J. Lasinio. J. Phys. A: Math. Gen. 17, 2935 (1981). 19. B. Simon. J. Funct. Anal. 63, 123 (1985). 20. R.L. Somorjai and D.F. Hornig. J. Chem. Phys. 36, 1980 (1962). 21. C.E. Reid. J. Mol. Spectrosc. 36, 183 (1970). 22. K. Banerjee and S.P. Bhatnagar. Phys. Rev. D: Part. Fields, 18, 4767 (1978). 23. M. Rasavy. Am. J. Phys. 48, 285 (1980). 24. B.R. Hosltein. Am. J. Phys. 56, 338 (1988). 25. M.R.M. Witwit. J. Phys. A: Math. Gen. 25, 503 (1992). ©2000 NRC Canada

976 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.

Can. J. Phys. Vol. 78, 2000 M. Bansal, S. Srivastava, Mamta, and Vishwamitter. Chem. Phys. Lett. 195, 505 (1992). D. Goorvitch and D.C. Galant. J. Quantum. Spectrosc. Radiat. Transfer, 48, 467 (1992). W. Kolos and J. Rychlewsky. J. Mol. Spectrosc. 177, 146 (1996). W. Kolos and J. Rychlewsky. J. Mol. Spectrosc. 50, 3228 (1968). D.L. Cooper, R.F. Barrow, J. Verges, C. Effantin, and J. D’Incan. Can. J. Phys. 62, 1543 (1984). I.P. Hamilton and J.C. Light. J. Chem. Phys. 84, 306 (1986). B.R. Johnson. J. Chem. Phys. 67, 4086 (1977). H. Kobeissi, M. Dagher, M. Korek, and A. Chaalan. J. Comput. Chem. 4, 216 (1983). H. Kobeissi and M. Korek. J. Phys. B: At. Mol. Opt. Phys. 18, 1155 (1985). H. Kobeissi and M. Korek. J. Phys. (Paris), 44, 1257 (1983). M. Korek. Can. J. Phys. 75, 795 (1997). H. Kobeissi and M. Korek. Int. J. Quant. Chem. 22, 23 (1982). H. Kobeissi, M. Korek, and M. Dagher. J. Mol. Spectrosc. 138, 1 (1989). L. Piskonov. Calcul differentiel et integral. Mir, Moscow. 1969. M. Korek and H. Kobeissi. J. Comput. Chem. 13, 1103 (1992). B.G. Wicke and D.O. Harris. J. Chem. Phys. 64, 5236 (1976). G. Jeung. J. Phys. B: At. Mol. Phys. 16, 4289 (1983).

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