  The crossover between two customary limits of phonon-assisted tunneling, the
adiabatic and antiadiabatic regimes, is studied systematically in the framework
of a minimal model for molecular devices: a resonant level coupled by
displacement to a localized vibrational mode. Conventionally associated with
the limits where the phonon frequency is either sufficiently small or
sufficiently large as compared to the bare electronic hopping rate, we show
that the crossover between the two regimes is governed for strong
electron-phonon interactions primarily by the polaronic shift rather than the
phonon frequency. In particular, the perturbative adiabatic limit is approached
only as the bare hopping rate \Gamma exceeds the polaronic shift, leaving an
extended window of couplings where \Gamma well exceeds the phonon frequency and
yet the physics is basically that of the antiadiabatic regime. We term this
intermediate regime the extended antiadiabatic regime. The effective low-energy
Hamiltonian in the traditional and the extended antiadiabatic regime is shown
to be the (purely fermionic) interacting resonant-level model, with parameters
that we extract from numerical renormalization-group calculations. The extended
antiadiabatic regime is followed in turn by a true crossover region where the
polaron gets progressively undressed. The renormalized tunneling rate, which
serves as the low-energy scale in the problem and thus sets the width of the
tunneling resonance, is found to follow an approximate scaling form on going
from the adiabatic to the antiadiabatic regime. Charging properties are
governed by two distinct mechanisms at the extended antiadiabatic and into the
crossover region, giving rise to distinctive shoulders in the low-temperature
conductance as a function of gate voltage.
