  We present a formalism that explicitly unifies the commonly used
Nakajima-Zwanzig approach for reduced density matrix dynamics with the more
versatile Mori theory in the context of nonequilibrium dynamics. Employing a
Dyson-type expansion to circumvent the difficulty of projected dynamics, we
obtain a self-consistent equation for the memory kernel which requires only
knowledge of normally evolved auxiliary kernels. To illustrate the properties
of the current approach, we focus on the spin-boson model and limit our
attention to the use of a simple and inexpensive quasi-classical dynamics,
given by the Ehrenfest method, for the calculation of the auxiliary kernels.
For the first time, we provide a detailed analysis of the dependence of the
properties of the memory kernels obtained via different projection operators,
namely the thermal (Redfield-type) and population based (NIBA-type) projection
operators. We further elucidate the conditions that lead to short-lived memory
kernels and the regions of parameter space to which this program is best
suited. Via a thorough analysis of the different closures available for the
auxiliary kernels and the convergence properties of the self-consistently
extracted memory kernel, we identify the mechanisms whereby the current
approach leads to a significant improvement over the direct usage of standard
semi- and quasi-classical dynamics.
