Carrier transfer in inhomogeneous media

From four non-stochastic rigorous theories leading to a set of equations for site-diagonal as well as site off-diagonal elements of the electron (carrier) density matrix (Time-Convolution Generalized Master Equation, Mori theory, Time-Convolutionless Generalized Master Equation and Tokuyama-Mori), the first two (time non-local) approaches were recently found to be disabled in the second order in the carrier-bath coupling. On the other hand, the latter two (time local) theories are argued here to lead, in the above order and any electronic system with a site-local coupling to bath, to completely identical and physically fully acceptable equations. The theory is applied to a periodic chain and asymmetric dimer. It is shown that, irrespective of problems with introducing <(gamma)over bar>(mn) or gamma(mn) (m not equal n) Haken-Strobl-Reineker parameters, the usual transition from the non-activated to activated (Arrhenius-type) 'up-and-down' asymmetric Marcus long-time reaction kinetics is obtained. The usual long-time markovian description of the population kinetics assumed in the standard theory is, however, found to be unjustified.