ar X iv : 1 40 5 . 07 41 v 1 [ m at h . N A ] 4 M ay 2 01 4 A multi - band semiclassical model for surface hopping quantum dynamics ∗

In the paper we derive a semiclassical model for surface hopping allowing quantum dynamical non-adiabatic transition between different potential energy surfaces in which cases the classical Born-Oppenheimer approximation breaks down. The model is derived using the Wigner transform and Weyl quantization, and the central idea is to evolve the entire Wigner matrix rather than just the diagonal entries as was done previously in the adiabatic case. The off-diagonal entries of the Wigner matrix suitably describe the non-adiabatic transition, such as the Berry connection, for avoided crossings. We study the numerical approximation issues of the model, and then conduct numerical experiments to validate the model. This work was partially supported by NSF grants DMS-1114546 and DMS-1107291: NSF Research Network in Mathematical Sciences KI-Net: Kinetic description of emerging challenges in multiscale problems of natural sciences. Department of Mathematics, University of California, Santa Barbara, CA 93106, USA ([email protected]) Department of Mathematics, Institute of Natural Sciences and MOE-LSE, Shanghai Jiao Tong University, Shanghai 200240, China; and Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA ([email protected]) Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125, USA ([email protected]) Institut de Physique et Chimie des Matériaux de Strasbourg, CNRS and University of Strasbourg, 23, rue du Loess, F-67034 Strasbourg, France ([email protected])