  The quantum-classical Liouville equation provides a description of the
dynamics of a quantum subsystem coupled to a classical environment.
Representing this equation in the mapping basis leads to a continuous
description of discrete quantum states of the subsystem and may provide an
alternate route to the construction of simulation schemes. In the mapping basis
the quantum-classical Liouville equation consists of a Poisson bracket
contribution and a more complex term. By transforming the evolution equation,
term-by-term, back to the subsystem basis, the complex term (excess coupling
term) is identified as being due to a fraction of the back reaction of the
quantum subsystem on its environment. A simple approximation to
quantum-classical Liouville dynamics in the mapping basis is obtained by
retaining only the Poisson bracket contribution. This approximate mapping form
of the quantum-classical Liouville equation can be simulated easily by
Newtonian trajectories. We provide an analysis of the effects of neglecting the
presence of the excess coupling term on the expectation values of various types
of observables. Calculations are carried out on nonadiabatic population and
quantum coherence dynamics for curve crossing models. For these observables,
the effects of the excess coupling term enter indirectly in the computation and
good estimates are obtained with the simplified propagation.
