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In this section, we present calculations for the resonant regime at equilibrium. Though we have shown, in sec- tions III A 2 and III B 2, the importance of the Hartree diagram, we will consider below results obtained with- out the Hartree diagram. The calculations were per- formed with the Fock and second-order DX and/or DPH diagrams which conserve (at numerical accuracy) the electron-hole symmetry of the system. 

The reasons why we have choosen to perform this model calculation are twofold: first, even for a single electronic level, we expect to have the maximum possi- ble electron-hole excitations available when the spectral functions are electron-hole symmetric. Hence we expect the polarization effects be to more pronounced in a sys- tem with electron-hole symmetry. Second, the electron- hole symmetric model permits us to emphasize the com- petitive effects between the DX and DPH diagrams as will be shown below. 

Figure 12 shows the equilibrium spectral functions of the resonant regime (at weak/intermediate electron- vibron coupling) obtained from different self-consistent calculations including first and second-order diagrams. 

## 2. Resonant regime at equilibrium 

On one hand, the second-order DPH diagram corre- sponds to a partial dressing of the vibron propagator by one electron-hole bubble. It gives an extra lifetime in the retarded/advanced electron Green’s functions in comparison to Hartree-Fock (SCBA) calculations. Hence the main effect of the DPH diagram is to introduce an extra broadening of the peaks in the spectral functions. Since the DPH diagram is one of the so-called conserving approximations $^{71,81}$, the broadening of the peaks leads to a reduction of their height to keep globally the same total spectral weight. These effects can be seen on the SC(BA+DPH) curve in Figure 12. 

On the other hand, the second-order DX diagram, which is a conserving approximation, has an opposite effect: a strong narrowing of the peaks (especially of the central peak) accompanied with an increase of their height, as can seen on the SC(BA+DX) curve in Figure 12. 

Now it is interesting to see what happens when the calculations are performed with both second-order DX and DPH diagrams. This is shown on the SC(BA+DX+DPH) curve in Figure 12: We obtain a hy- brid behaviour, in the sense that in appearance the cen- tral peak is broadened in comparison to SCBA calcula- tions. However, the height of the peak at the Fermi level is conserved (up to numerical accuracy). This is a very important result which proves that for the model calcu- lation of an electron-hole symmetric system, one has to include both the DX and DPH second-order diagrams in order to conserve the expected Fermi-liquid properties of the system. In this regime both the DX and DPH second- order diagrams play an equally important role which de- termines the linear response properties of the system as shown below in Section III E. 

By comparison, in a diagrammatic treatment of the electron-electron interaction on the electron propagator, the situation is often different: the electron-hole bub- ble diagram that appears here in the DPH contribution to the propagator is large, especially in highly polariz- able metallic and open-shell systems where electron-hole pairs may be created with low energy cost, because the Coulomb interaction operates at all energy scales. In that case, summing the bubble diagrams to infinite order as is done in Hedin’s GW approximation 82,83 is much more important than including the second-order exchange dia- gram. The key difference for the electron-phonon interac- tion is that the vibron frequency ω$_{0}$ imposes a restricted energy scale on the interaction, reducing the importance of the bubble diagrams, and correspondingly increasing the importance of the second-order exchange diagram. 

Figure 13 shows the spectral functions of the off- resonant transport regime at finite bias. The calculations have been performed for weak/intermediate electron- vibron coupling ( γ$_{0}$/ω$_{0}$ = 0 . 50) and for different dia- 

## 3. Off-resonant regime at finite bias 

