Frozen Gaussian approximation with surface hopping: a rigorous algorithm for non-adiabatic dynamics

The numerical simulation of many-body quantum dynamics has been a long standing challenge. Semiclassical methods, like ab initio molecular dynamics based on Born-Oppenheimer approximation, provide practical strategies to overcome the curse of dimensionality. The Born-Oppenheimer approximation breaks down however in the non-adiabatic regime. In this talk, we discuss some recent mathematical understanding and development of semiclassical methods for non-adiabatic quantum dynamics. In particular, we will present a rigorous surface hopping algorithm in the spirit of fewest switches surface hopping by John Tully (Yale Chemistry).