  We use a 1-dimensional tight binding model with an impurity site
characterized by electron-vibration coupling, to describe electron transfer and
localization at zero temperature, aiming to examine the process of polaron
formation in this system. In particular we focus on comparing a semiclassical
approach that describes nuclear motion in this many vibronic-states system on
the Ehrenfest dynamics level to a numerically exact fully quantum calculation
based on the Bonca-Trugman method [J. Bon\v{c}a and S. A. Trugman, Phys. Rev.
Lett. 75, 2566 (1995)]. In both approaches, thermal relaxation in the nuclear
subspace is implemented in equivalent approximate ways: In the Ehrenfest
calculation the uncoupled (to the electronic subsystem) motion of the classical
(harmonic) oscillator is simply damped as would be implied by coupling to a
markovian zero temperature bath. In the quantum calculation, thermal relaxation
is implemented by augmenting the Liouville equation for the oscillator density
matrix with kinetic terms that account for the same relaxation. In both cases
we calculate the probability to trap the electron in a polaron cage and the
probability that it escapes to infinity. Comparing these calculations, we find
that while both result in similar long time yields for these processes, the
Ehrenfest-dynamics based calculation fails to account for the correct timescale
for the polaron formation. This failure results, as usual, from the fact that
at the early stage of polaron formation the classical nuclear dynamics takes
place on an unphysical average potential surface that reflects the
otherwise-distributed electronic population in the system, while the quantum
calculation accounts fully for correlations between the electronic and
vibrational subsystems.
