Propagation through Generic Level Crossings: A Surface Hopping Semigroup

We construct a surface hopping semigroup, which asymptotically describes nuclear propagation through crossings of electron energy levels. The underlying time-dependent Schrödinger equation has a matrix-valued potential, whose eigenvalue surfaces have a generic intersection of codimension two, three or five in Hagedorn’s classification. Using microlocal normal forms reminiscent of the Landau-Zener problem, we prove convergence to the true solution with an error of the order ε1/8, where ε is the semi-classical parameter. We present numerical experiments for an algorithmic realization of the semigroup illustrating the convergence of the algorithm.