Canonical Statistical Averages of Coupled Quantum-Classical Systems

We study the thermal-equilibrium distribution generated by a Nosé-Hoover thermostat coupled to a mixed quantum-classical system, whose time evolution is taken to be either Ehrenfest (E), or Car-Parrinello (CP) type. It is found that neither E or CP dynamics lead to the correct averages for the quantum subsystem. A new classical modified CP dynamics is introduced which mimics the level quantization and yield the correct quantum averages after a suitable reweighting of the trajectories. We support our theoretical formulation with numerical simulations on a simple model system.