Dynamics of coupled Bohmian and phase-space variables: a moment approach to mixed quantum-classical dynamics

The theoretical framework of the mixed quantum-classical description given by Burghardt and Parlant [J. Chem. Phys. 120, 3055 (2004)] is detailed. A representation in terms of partial hydrodynamic moments is developed, the dynamics of which is determined by a hierarchy of equations derived from the quantum Liouville equation. Exact equations of motion are obtained, whose quantum-classical approximants are associated with a fluid-dynamical trajectory representation which couples classical variables to quantum hydrodynamic variables. The latter evolve under a generalized hydrodynamic force which also depends upon the classical phase-space variables. The hydrodynamic moment description is shown to be closely connected to mixed quantum-classical phase-space methods.