  The crossover from quasi free electron to small polaron in the Holstein model
for a single electron is studied by means of both exact and self-consistent
calculations in one dimension and on an infinite coordination lattice, in order
to understand the role of dimensionality in such a crossover. We show that a
small polaron ground-state occurs when both strong coupling ($\lambda>1$) and
multiphonon ($\alpha^2 >1$) conditions are fulfilled leading to different
relevant coupling constants ($\lambda$) in adiabatic and ($\alpha^2$) anti
adiabatic region of the parameters space. We also show that the self-consistent
calculations obtained by including the first electron-phonon vertex correction
give accurate results in a sizeable region of the phase diagram well separated
from the polaronic crossover.
