  We investigate the properties of the molecular quantum dot
(Holstein-Anderson) model using numerical and analytical techniques. Path
integral Monte Carlo simulations for the cumulants of the distribution function
of the phonon coordinate reveal that at intermediate temperatures the effective
potential for the oscillator exhibits two minima rather than a single one,
which can be understood as a signature of a bistability effect. A
straightforward adiabatic approximation turns out to adequately describe the
properties of the system in this regime. Upon lowering the temperature the two
potential energy minima of the oscillator merge to a single one at the
equilibrium position of the uncoupled system. Using the parallels to the X-ray
edge problem in metals we derive the oscillator partition function. It turns
out to be identical to that of the Kondo model, which is known to possess a
universal low energy fixed point characterized by a single parameter -- the
Kondo temperature $T_K$. We derive an analog of $T_K$ for the molecular quantum
dot model, present numerical evidence pointing towards the appearance of the
Kondo physics and discuss experimental implications of the discovered
phenomena.
