  We apply weak-coupling perturbation theory and strong-coupling perturbation
theory to the Holstein molecular crystal model in order to elucidate the
effects of anisotropy on polaron properties in D dimensions. The ground state
energy is considered as a primary criterion through which to study the effects
of anisotropy on the self-trapping transition, the self-trapping line
associated with this transition, and the adiabatic critical point. The effects
of dimensionality and anisotropy on electron-phonon correlations and polaronic
mass enhancement are studied, with particular attention given to the polaron
radius and the characteristics of quasi-1D and quasi-2D structures.
Perturbative results are confirmed by selected comparisons with variational
calculations and quantum Monte Carlo data.
