  We investigate the dimensionality effects of the Holstein polaron from the
fully quantum regime, where the crossover between large and small polaron
solutions is known to be continuous in all dimensions, into the limit described
by the semiclassical Discrete Nonlinear Schr\"odinger (DNLS) Equation, where
the crossover is continuous in 1D but discontinuous in higher dimensions. We
use exact numerics on one hand and a two variable parametrization of the
Toyozawa ansatz on the other in order to probe the crossover region in all
parameter regimes. We find that a barrier appears also in 1D separating the two
types of solutions, seemingly in contradiction to the common paradigm for the
DNLS according to which the crossover is barrier-free. We quantify the polaron
behavior in the crossover region as a function of the exciton overlap and find
that the barrier remains small in 1D and tunnelling through it is not
rate-limiting.
