A comparison of pseudospectral methods for the solution of the Schrödinger equation; the Lennard-Jones (n, 6) potential

Abstract

The eigenvalues of the Schrödinger equation for the Lennard-Jones (n,6) potential with n=8, 10 and 12, have been determined by numerous workers with different numerical methods. In this paper, we consider a pseudospectral method based on a grid defined by the quadrature points associated with nonclassical polynomials. The polynomials are orthogonal with respect to the weight function defined by the ground state wave function for a Morse potential that approximates the Lennard-Jones potential. The Morse potential, unlike the Lennard-Jones potential, belongs to the class of potentials in SUperSYmmetric (SUSY) quantum mechanics. The ground state wavefunction is thus known and used as the weight function to define the nonclassical grid. An alternate supersymmeric weight function that approximates the Lennard-Jones potential is also considered. The results obtained in this way are compared with the results obtained with a direct numerical integration of the radial Schrödinger equation, a pseudospectral Fourier method and Jeffreys-Wentzel-Kramers-Brouillon (JWKB) approximations as well as other approximate analytical methods. The convergence of the eigenvalues with the nonclassical quadrature grid is demonstrated to be rapid and very competitive with the other methods.