  We analyze the effects of a distortion of the nuclear potential of a
molecular quantum dot (QD), as well as a shift of its equilibrium position, on
nonequilibrium-vibration-assisted tunneling through the QD with a single level
($\epsilon_d$) coupled to the vibrational mode. For this purpose, we derive an
explicit analytical expression for the Franck-Condon (FC) factor for a
displaced-distorted oscillator surface of the molecule and establish rate
equations in the joint electron-phonon representation to examine the
current-voltage characteristics and zero-frequency shot noise, and skewness as
well. Our numerical analyses shows that the distortion has two important
effects. The first one is that it breaks the symmetry between the excitation
spectra of the charge states, leading to asymmetric tunneling properties with
respect to $\epsilon_d>0$ and $\epsilon_d<0$. Secondly, distortion (frequency
change of the oscillator) significantly changes the voltage-activated cascaded
transition mechanism, and consequently gives rise to a different nonequilibrium
vibrational distribution from that of the case without distortion. Taken in
conjunction with strongly modified FC factors due to distortion, this results
in some new transport features: the appearance of strong NDC even for a
single-level QD with symmetric tunnel couplings; a giant Fano factor even for a
molecule with an extremely weak electron-phonon interaction; and enhanced
skewness that can have a large negative value under certain conditions.
