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

Non-equilibrium transport with self-consistent renormalised contacts for a single-molecule nanodevice with electron-vibron interaction