An efficient procedure for calculating the evolution of the wave function by fast Fourier transform methods for systems with spatially extended wave function and localized potential

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Abstract
Various methods using fast Fourier transform algorithms or other ‘‘grid’’ methods for solving the time‐dependent Schrödinger equation are very efficient if the wave function remains spatially localized throughout its evolution. Here we present and test an extension of these methods which is efficient even if the wave function spreads out, provided that the potential remains localized. The idea is to split the wave function at various times during the propagation into two parts, one localized in the interaction region and the other in the force free region; the first is propagated by a fast Fourier transform method on a grid whose size barely exceeds the interaction region, and the latter by a single application of a free particle propagator. This splitting is performed whenever the interaction region wave function comes close to the end of the grid. The total asymptotic wave function at a given time t is reconstructed by adding coherently all the asymptotic wave function pieces which were split at earlier times, after they have been propagated to the common time t. The method is tested by studying the wave function of a diatomic molecule dissociated by a strong laser field. We compute the rate of energy absorption and dissociation and the momentum distribution of the fragments.