Time propagation of extreme two-electron wavefunctions

We develop a method for solving the time-dependent Schrödinger equation for the situation where two electrons interact in extreme circumstances. In particular, the method allows accurate representation of the wavefunction when several 10s or 100s of angular momenta are needed and the spatial region covers several 100s to 1000s of atomic units. The method is based on a discrete variable representation for the cos(θ12) inside the 1/r12 operator. We also discuss a propagator for the radial part of the wavefunction which would allow efficient treatment of two continuum electrons as well as two Rydberg electrons. The method is tested on the case where two continuum electrons are successively launched from small r with zero angular momentum. The first electron has less energy so that the second electron must pass it. We show that the method is stable up to the highest angular momentum we tested: max = 160. Although being a somewhat artificial case of post-collision interaction, this example has interesting features which we explore. 1