Single switch surface hopping for molecular quantum dynamics

The aim of this text is to present a surface hopping approximation for molecular quantum dynamics obeying a Schrödinger equation with crossing eigenvalue surfaces. After motivating Schrödinger equations with matrix valued potentials, we describe the single switch algorithm and present some numerical results. Then we discuss the algorithm's mathematical justification and describe extensions to more general situations, where three eigenvalue surfaces intersect or the eigenvalues are of multiplicity two. We emphasize the generality of this surface hopping approximation for non-adiabatic transitions.