### Abstract

The time-dependent Schroedinger equation provides the fundamental description of quantum mechanical molecular dynamics. Its multiscale character suggests a splitting in two coupled subproblems, the so called Born-Oppenheimer approximation: One solves a family of stationary Schroedinger equations in the electronic degrees of freedom (one equation for each nucelonic configuration) and subsequently a time-dependent Schroedinger equation in the nucleonic degrees of freedom, whose potential has been determined by the electronic problem. This splitting fails to provide an approximation, if different electronic eigenvalues are not uniformly separated for all nucleonic configurations. The talk explains a microlocal point of view on this non-adiabatic coupling between electronic and nucleonic degrees of freedom and derives an associated deterministic surface hopping algorithm. Its numerical realization crucially relies on the sampling of highly oscillatory initial data on high-dimensional configuration spaces, which can be tackled by a Monte Carlo approach. The presented results are joint work with C. Fermanian, S. Kube, and M. Weber.