THE SMALL-POLARON CROSSOVER - COMPARISON BETWEEN EXACT RESULTS AND VERTEX CORRECTION APPROXIMATION

We study the crossover from quasi-free electron to small polaron in th e Holstein model for a single electron by means of both exact and self -consistent calculations in one dimension and on an infinite coordinat ion lattice. We show that the crossover occurs when both strong coupli ng (lambda > 1) and multiphonon (alpha(2) > 1) conditions are fulfille d leading to different relevant coupling constants (lambda) in the adi abatic and (alpha(2)) antiadiabatic 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 polaron ic crossover.