Analytic banded approximation for the discretized free propagator

The wave packet that represents a physical molecular scattering system of interest evolves according to the time-dependent Schroedinger equation (TDSE), which is a linear, first order (in time), differential equation. In recent years, there has been considerable interest in the development and application of numerical methods for solving dynamics problems using time-dependent methods. A procedure for obtaining an approximate sparse (banded) matrix for the coordination representation of the free-particle propagator is presented. The technique takes advantage of the fact that the action of the free propagator on Hermite polynomials can be obtained analytically and represents the propagator in terms of its effect on the Hermite basis.