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 t
o 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.
