A complete description of two outgoing electrons following an ionizing collision between a single electron and an atom

or molecule has long stood as one of the unsolved fundamental problems in quantum collision theory.

In this technical paper we describe our use of distributed memory parallel computers to calculate a fully converged wave function describing the electron-impact ionization of hydrogen. Our approach hinges on a transformation of the Schr\"{o}dinger equation that simplifies the boundary conditions but requires solving very ill-conditioned systems of a few million complex, sparse linear equations.

We developed a two-level iterative algorithm that requires repeated solution of sets of a few hundred thousand linear equations. These are solved directly by $LU$-factorization using a specially tuned, distributed memory parallel version of the sparse $LU$-factorization library SuperLU. In smaller cases, where direct solution is technically possible, our iterative algorithm still gives significant savings in time despite lower megaflop rates.