  A quantum-classical limit of the canonical equilibrium time correlation
function for a quantum system is derived. The quantum-classical limit for the
dynamics is obtained for quantum systems comprising a subsystem of light
particles in a bath of heavy quantum particles. In this limit the time
evolution of operators is determined by a quantum-classical Liouville operator
but the full equilibrium canonical statistical description of the initial
condition is retained. The quantum-classical correlation function expressions
derived here provide a way to simulate the transport properties of quantum
systems using quantum-classical surface-hopping dynamics combined with sampling
schemes for the quantum equilibrium structure of both the subsystem of interest
and its environment.
