Extension of the mapped Fourier method to time-dependent problems.
A numerical method is described for integration of the time-dependent Schrödinger equation within the presence of a Coulomb field. Because of the singularity at r=0, the wave packet has to be represented on a grid with a high density of points near the origin; at the same time, because of the long-range character of the Coulomb potential, the grid must extend to large values 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 criterion. Following those workers, the unequally spaced grid points are mapped to an equally spaced grid, allowing use of fast Fourier transform propagation 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 short-time segments of the electronic wave-packet autocorrelation function; high accuracy is obtained by using the filter-diagonalization method. As a second test, the ionization rate of the hydrogen atom resulting from a half-cycle pulse is calculated. These results are in excellent agreement with earlier calculations.
Kleinekathöfer
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Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel.
Tannor
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