  We present an application of a new formalism to treat the quantum transport
properties of fully interacting nanoscale junctions [Phys. Rev. B {\bf 84},
235428 (2011)]. We consider a model single-molecule nanojunction in the
presence of two kinds of electron-vibron interactions. In terms of electron
density matrix, one interaction is diagonal in the central region and the
second is off-diagonal in between the central region and the left electrode. We
use a non-equilibrium Green's function technique to calculate the system's
properties in a self-consistent manner. The interaction self-energies are
calculated at the Hartree-Fock level in the central region and at the Hartree
level for the crossing interaction. Our calculations are performed for
different transport regimes ranging from the far off-resonance to the
quasi-resonant regime, and for a wide range of parameters. They show that a
non-equilibrium (i.e. bias dependent) static (i.e. energy independent)
renormalisation is obtained for the nominal hopping matrix element between the
left electrode and the central region. Such a renormalisation is highly
non-linear and non-monotonic with the applied bias, however it always lead to a
reduction of the current, and also affects the resonances in the conductance.
Furthermore, we show that the relationship between the non-equilibrium charge
susceptibility and dynamical conductance still holds even in the presence of
crossing interaction.
