  Molecular or condensed matter systems are often well approximated by hybrid
quantum-classical models: the electrons retain their quantum character, whereas
the ions are considered to be classical particles. We discuss various
alternative approaches for the computation of equilibrium (canonical) ensemble
averages for observables of these hybrid quantum-classical systems through the
use of molecular dynamics (MD), i.e. by performing dynamics in the presence of
a thermostat and computing time averages over the trajectories. Often, in
classical or ab initio MD, the temperature of the electrons is ignored and they
are assumed to remain at the instantaneous ground state given by each ionic
configuration during the evolution. Here, however, we discuss the general case
that considers both classical and quantum subsystems at finite temperature
canonical equilibrium. Inspired by a recent formal derivation for the canonical
ensemble for quantum classical hybrids, we discuss previous approaches found in
the literature, and provide some new formulas.
