Caroline Lasser, Stefan Teufel
Propagation through Concial Crossings: an Asymptotic Semigroup
(500K, pdf)
ABSTRACT. 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
Schr\"odinger 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
C.\ Fermanian Kammerer and P.\ G\'erard for the propagation of
two-scale Wigner measures through conical crossings.