TRAVELING DISTRIBUTED APPROXIMATING FUNCTION-APPROACH TO WAVE PACKET PROPAGATION - EXPLICIT INCLUSION OF A LOCAL WAVE VELOCITY

The distributed approximating function (DAF) approach to wave packet propagation, designed to treat a restricted class of wave packets which can be approximated to a specified level of accuracy by a polynomial of degree M under the envelope of the DAF, is modified to take explicit account of the group velocity of the specific wave packet describing the collision system of interest. Because the DAFs are exactly and analytically freely propagatible, they yield an accurate analytical fit of the freely propagated wave packet also, which expression can be used to obtain the coordinate representation ''traveling'' DAF (TDAF) class free propagator. TDAFs are similar to the original DAFs studied earlier, which we here refer to as ''stationary'' DAFs (SDAFs). The TDAF yields a more efficient propagation of the wave packet under consideration than does the corresponding SDAF. This reflects the explicit inclusion of the wave packet oscillations associated with a characteristic velocity in the DAF. The structure of the TDAF class free propaptor is analyzed in both the coordinate and momentum representations. The TDAF is generalized to include a position-dependent (local) wave velocity. Two ways in which this can be done are discussed and compared. One of these we label ''progression'' (corresponding to an evolution from the initial state to a final state) dynamics and the other ''regression'' (corresponding to moving backward from the final to an initial state) dynamics. Also, a ''grid transfer'' method is introduced to separate incident and scattered pieces of the packet in regions where they overlap, so that the TDAF concept can be applied effectively. Finally, two simple, but computationally demanding, 1-D problems are discussed which illustrate the utility of the method. The TDAF procedure is shown to be highly accurate on a discretized grid, even for an electron transmission problem that required 120 000 time steps (for a total of 60 ps) to complete the scattering. It is found that the TDAF method can greatly reduce the required computational effort by narrowing the spatial bandwidth of the DAF class free propagator.