where the quantum subsystem belonging to the Hilbert space H Q , with Hamiltonian ĤQ and identity ÎQ , and the classical subsystem (typically nuclear DOF) belonging to the Hilbert space H C , with Hamiltonian ĤC and identity ÎC , interact through ĤQC . In reactive molecular systems, the Hamiltonian ĤC is usually anharmonic and cannot easily be directly considered as ĤE . When possible, the system dynamics of Eq. 5 can be evaluated fully quantum mechanically; however, when the classical subsystem is too large, the semiclassical approximation can be made. Under the semiclassical approximation, the classical subsystem evolves using classical mechanics while the quantum subsystem evolves using quantum mechanics.