  The formalism of the generalized quantum master equation (GQME) is an
effective tool to simultaneously increase the accuracy and the efficiency of
quasiclassical trajectory methods in the simulation of nonadiabatic quantum
dynamics. The GQME expresses correlation functions in terms of a non-Markovian
equation of motion, involving memory kernels which are typically fast-decaying
and can therefore be computed by short-time quasiclassical trajectories. In
this paper we study the approximate solution of the GQME, obtained by
calculating the kernels with two methods, namely Ehrenfest mean-field theory
and spin mapping. We test the approaches on a range of spin--boson models with
increasing energy bias between the two electronic levels and place a particular
focus on the long-time limits of the populations. We find that the accuracy of
the predictions of the GQME depends strongly on the specific technique used to
calculate the kernels. In particular, spin mapping outperforms Ehrenfest for
all systems studied. The problem of unphysical negative electronic populations
affecting spin mapping is resolved by coupling the method with the master
equation. Conversely, Ehrenfest in conjunction with the GQME can predict
negative populations, despite the fact that the populations calculated from
direct dynamics are positive definite.
