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We study the crossover from quasi free electron to small polaron in theHolstein model for a single electron by means of both exact and self-consistentcalculations in one dimension and on an infinite coordination lattice. We showthat the crossover occurs when both strong coupling and multiphonon conditionsare fulfilled leading to different relevant coupling constants in adiabatic andanti-adiabatic region of the parameters space. We also show that theself-consistent calculations obtained by including the first electron-phononvertex correction give accurate results in a sizeable region of the phasediagram well separated from the polaronic crossover.Comment: 6 pages, revtex (europhys.sty,euromacr.tex); 3 postscript figure

The small-polaron crossover: Comparison between exact results and vertex correction approximation