Supersymmetric improvement of the Pekeris approximation for the rotating Morse potential Daniel A. Morales * Facultad de Ciencias, Universidad de Los Andes, Apartado Postal A61, La Hechicera Me ´rida 5101, Venezuela Received 26 March 2004; in final form 22 June 2004 Available online 17 July 2004 Abstract It is demonstrated that the accuracy of the approximate expression of Pekeris for the eigenvalues of the rotating Morse potential can be improved substantially by supersymmetry. A hierarchy of approximate supersymmetric potentials is obtained which allows one to calculate the energies of higher rovibrational states from the energies of lower rovibrational states for which PekerisÕ expres- sion is accurate. This new formulation is tested by calculating the energies of rovibrational states of the hydrogen molecule and com- paring the results with those of accurate numerical calculations. Ó 2004 Elsevier B.V. All rights reserved. 1. Introduction The Morse potential has been for more than seventy years one of the most useful models to describe the spec- tra and other properties of diatomic molecules. It is known that for this potential the Schro ¨ dinger equation can be solved exactly when the angular momentum quantum number l is equal to zero [1]. On the other hand, when inclusion of rotation is needed to describe, for example, the rovibrational energy states of diatomic molecules, some approximations are necessary to obtain analytical or semianalytical solutions to the Schro ¨ dinger equation [2–6]. Several schemes have been presented for obtaining approximate solutions. Among these approx- imations the most widely known and used has been the one developed by Pekeris [2,7,8]. His method is based on the expansion of the centrifugal barrier in a series of ex- ponentials depending on the internuclear distance, up to second order, such that the effective l-dependent poten- tial keeps the same form as the potential with l =0. In this way, a solution of the Schro ¨ dinger equation can be immediately written down, knowing the exact solu- tion for l = 0. However, by construction, this approxima- tion is valid only for low rovibrational energy states. Several other approximations have been developed to find better analytical formulas for the rotating Morse potential. However, all these approximations require the calculation of a state-dependent internuclear dis- tance through the numerical solutions of transcendental equations [3–6]. On the other hand, supersymmetry (SUSY, for short) is a proposed fundamental symmetry of nature which in- terchanges bosons and fermions. The ideas of supersym- metry have been applied to elucidate many problems in non-relativistic quantum mechanics [9–13]. SUSY in quantum mechanics does not refer to the original sym- metry relating bosons and fermions but to transforma- tions between two orthogonal eigenstates of a Hamiltonian with the same degenerate energy eigen- value [9]. The application of SUSY in quantum mechan- ics has yielded the concept of shape-invariance, an integrability condition, which establishes that two SUSY partners potentials are shape invariant if they share the same spatial dependence but with modified pa- rameters (such as well depth or range). It has been 0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.06.109 * Fax: +58-274-2401-286. E-mail address: danoltab@ula.ve (D.A. Morales). www.elsevier.com/locate/cplett Chemical Physics Letters 394 (2004) 68–75