Propagation through Conical Crossings: an Asymptotic Semigroup

We consider the standard model problem for a conical intersection of electronic surfaces in molecular dynamics. Our main result is the construction of a semigroup in order to approximate the Wigner function associated with the solution of the Schrödinger equation at leading order in the semiclassical parameter. The semigroup stems from an underlying Markov process which combines deterministic transport along classical trajectories within the electronic surfaces and random jumps between the surfaces near the crossing. Our semigroup can be viewed as a rigorous mathematical counterpart of so-called trajectory surface hopping algorithms, which are of major importance in chemical physics’ molecular simulations. The key point of our analysis, the incorporation of the non-adiabatic transitions, is based on the Landau-Zener type formula of Fermanian-Kammerer and Gérard [FeGe1] for the propagation of two-scale Wigner measures through conical crossings.