where the trace is performed over the classical nuclear DOFs R (or whatever the classical DOFs happen to be). When the mixed quantum-classical approximation is used, 1 the nuclear coordinates evolve classically and the quantum subsystem evolves using the time-dependent Schrödinger equation governed by ĤQ + ĤQC (R). In this case, since only states in the quantum subspace are evolved using quantum mechanics, ρ(t) can be computationally evaluated without performing a trace on the much larger system Hilbert space which greatly improves computational efficiency.