  The quantum-classical Liouville equation describes the dynamics of a quantum
subsystem coupled to a classical environment. It has been simulated using
various methods, notably, surface-hopping schemes. A representation of this
equation in the mapping Hamiltonian basis for the quantum subsystem is derived.
The resulting equation of motion, in conjunction with expressions for quantum
expectation values in the mapping basis, provide another route to the
computation of the nonadiabatic dynamics of observables that does not involve
surface-hopping dynamics. The quantum-classical Liouville equation is exact for
the spin-boson system. This well-known model is simulated using an
approximation to the evolution equation in the mapping basis and close
agreement with exact quantum results is found.
