  We apply weak-coupling perturbation theory to the Holstein molecular crystal
model in order to compute an electron-phonon correlation function
characterizing the shape and size of the polaron lattice distortion in one,
two, and three dimensions. This correlation function is computed exactly to
leading order in the electron-phonon coupling constant, permitting a complete
description of correlations in any dimension for both isotropic and arbitrarily
anisotropic cases. Using this exact result, the width of the polaron is
characterized along arbitrary directions. The width of the polaron thus
determined disagrees in every dimension with some well-known characterizations
of polarons, signalling in particular the breakdown of the adiabatic
approximation and the characterizations of self-trapping associated with it.
