  The two-site Holstein model represents a first non-trivial paradigm for the
interaction between an itinerant charge with a quantum oscillator, a very
common topic in different ambits. Exact results can be achieved both
analytically and numerically, nevertheless it can be useful to compare them
with approximate, semi-classical techniques in order to highlight the role of
quantum effects. In this paper we consider the adiabatic limit in which the
oscillator is very much slow than the electron. A density matrix approach is
introduced for studying the charge dynamics and the exact results are compared
with two different approximations: a Born-Oppenheimer-based Static
Approximation for the oscillator (SA) and a Quantum-classical (QC) dynamics.
