  We calculate the non-linear tunneling current through a molecule with two
electron-accepting orbitals which interact with an intramolecular vibration. We
investigate the interplay between Coulomb blockade and non-equilibrium
vibration-assisted tunneling under the following assumptions: (i) The Coulomb
charging effect restricts the number of extra electrons to one. (ii) The
orbitals are non-degenerate and couple asymmetrically to the vibration. (iii)
The tunneling induces a non-equilibrium vibrational distribution; we compare
with the opposite limit of strong relaxation of the vibration due to some
dissipative environment. We find that a non-equilibrium feedback mechanism in
the tunneling transitions generates strong negative differential conductance
(NDC) in the model with two competing orbitals, whereas in a one-orbital model
it leads only to weak NDC. In addition, we find another mechanism leading to
weak NDC over a broader range of applied voltages. This pervasive effect is
completely robust against strong relaxation of the vibrational energy.
Importantly, the modulation of the electronic transport is based on an
intramolecular asymmetry. We show that one can infer a non-equilibrium
vibrational distribution when finding NDC under distinct gate- and bias-voltage
conditions. In contrast, we demonstrate that any NDC effect in the one-orbital
case is completely suppressed in the strong relaxation limit.
