Frozen Gaussian approximation with surface hopping for mixed quantum-classical dynamics: A mathematical justification of fewest switches surface hopping algorithms

We develop a surface hopping algorithm based on frozen Gaussian approximation for semiclassical matrix Schr\"odinger equations, in the spirit of Tully's fewest switches surface hopping method. The algorithm is asymptotically derived from the Schr\"odinger equation with rigorous approximation error analysis. The resulting algorithm can be viewed as a path integral stochastic representation of the semiclassical matrix Schr\"odinger equations. Our results provide mathematical understanding to and shed new light on the important class of surface hopping methods in theoretical and computational chemistry.