Symmetry adapted Fourier solution of the time-dependent Schrödinger equation

Abstract

The application of the time-dependent Schrödinger equation/Fourier method to triatomic systems in hyperspherical coordinates is described. In particular, we consider the high symmetry situation of three identical particles, and focus on the reduced dimensional dynamics in the hyperspherical coordinate χ, which contains all the symmetry of the general problem. The periodic structure of the wave functions in the χ coordinate leads to a discrete spectrum in k space. The additional finite symmetry of the wave functions in this coordinate gives rise to selection rules, analogous to rotational selection rules for symmetric top molecules, which ensure that only one out of six of these integer k values has nonzero intensity. It is possible to reduce the range of the wave function (and the calculation) by a factor of 6 by selecting only one replica of the coordinate space wave function. The FFT automatically increases the spacing in k space by a factor of 6, eliminating the lines in k space with nonzero amplitude. For each different symmetry of wave function in coordinate space it is necessary to shift the compacted grid in k space by a different amount, in accord with k space selection rules.