High order finite difference algorithms for solving the Schrödinger equation in molecular dynamics

Abstract

The view of considering global Pseudospectral methods (Sinc and Fourier) as the infinite order limit of local finite difference methods, and vice versa, finite difference as a certain sum acceleration of the pseudospectral methods is exploited to investigate high order finite difference algorithms for solving the Schrödinger equation in molecular dynamics. A Morse type potential for iodine molecule is used to compare the eigenenergies obtained by a Sinc Pseudospectral method and a high order finite difference approximation of the action of the kinetic energy operator on the wave function. Two-dimensional and three-dimensional model potentials are employed to compare spectra obtained by fast Fourier transform techniques and variable order finite difference. It is shown that it is not needed to employ very high order approximations of finite differences to reach the numerical accuracy of pseudospectral techniques. This, in addition to the fact that for complex configuration geometries and high dimensionality, local methods require less memory and are faster than pseudospectral methods, put finite difference among the effective algorithms for solving the Schrödinger equation in realistic molecular systems.