  Mixed quantum-classical methods provide powerful algorithms for the
simulation of quantum processes in large and complex systems. The
forward-backward trajectory solution of the mixed quantum-classical Liouville
equation in the mapping basis [J. Chem. Phys. 137, 22A507 (2012)] is one such
scheme. It simulates the dynamics via the propagation of forward and backward
trajectories of quantum coherent state variables, and the propagation of bath
trajectories on a mean-field potential determined jointly by the forward and
backward trajectories. An analysis of the properties of this solution,
numerical tests of its validity and an investigation of its utility for the
study of nonadiabtic quantum processes are given. In addition, we present an
extension of this approximate solution that allows one to systematically
improve the results. This extension, termed the jump forward-backward
trajectory solution, is analyzed and tested in detail and its various
implementations are discussed.
