Mixed quantum-classical surface hopping dynamics

An algorithm is presented for the exact solution of the evolution of the de nsity matrix of a mixed quantum-classical system in terms of an ensemble of surface hopping trajectories. The system comprises a quantum subsystem cou pled to a classical bath whose evolution is governed by a mixed quantum-cla ssical Liouville equation. The integral solution of the evolution equation is formulated in terms of a concatenation of classical evolution segments f or the bath phase space coordinates separated by operators that change the quantum state and bath momenta. A hybrid Molecular Dynamics-Monte Carlo sch eme which follows a branching tree of trajectories arising from the action of momentum derivatives is constructed to solve the integral equation. We a lso consider a simpler scheme where changes in the bath momenta are approxi mated by momentum jumps. These schemes are illustrated by considering the c omputation of the evolution of the density matrix for a two-level system co upled to a low dimensional classical bath. (C) 2000 American Institute of P hysics. [S0021-9606(00)50215-3].