Extension of the mapped Fourier method to time-dependent problems

A numerical method is described for integration of the time-dependent Schro dinger equation within the presence of a Coulomb held. Because of the singu larity at r = 0, the wave packet has to be represented on a grid with a hig h density of points near the origin; at the same timet because of the long- range character of the Coulomb potential, the grid must extend to large val ues of r. The sampling points are chosen, following E. Fattal, R. Baer, and R. Kosloff [Phys. Rev. E 53, 1217 (1996)], using a classical phase space c riterion. Following those workers, the unequally spaced grid points are map ped to an equally spaced grid, allowing use of fast Fourier transform propa gation methods that scale as N ln N, where N is the number of grid points. As a first test, eigenenergies for the hydrogen atom are extracted from sho rt-time segments of the electronic wave-packet autocorrelation function; hi gh accuracy is obtained by using the filter-diagonalization method,As a sec ond test, the ionization rate of the hydrogen atom resulting from a half-cy cle pulse is calculated. These results are in excellent agreement with earl ier calculations. [S1063-651X(99)14210-7].