when the main peak in the spectral function is located well away from the Fermi level, i.e. | ˜ ϵ$_{0}$ − µ $^{eq}$|  ≫ linewidth of the peaks. For strong electron-vibron coupling, the differences between different levels of approximations are more pronounced. All G$_{lin}$ values are smaller than the exact MCIST result. This is essentially due to the fact that perturbation expansion gives only an approximate value of the polaron shift. The differences from the exact result are more important for SCBA-based calculations (difference of ∼ 10% with exact result) and of course calculations performed up to second order give a better polaron shift and hence better values for the linear con- ductance ( ∼ 5% difference only). 15 G lin /G 0 off-resonant off-resonant resonant weak e-vib strong e-vib (e-h symmetric) Fig. 10 Fig. 11 Fig. 12 SCBA 0.002654 0.004470 0.8379 ∗ SCBA+DX 0.002666 0.004606 - SCBA+DX+DPH 0.003186 - - SC(BA+DX) 0.002668 0.004606 1.1296 SC(BA+DPH) - - 0.7194 SC(BA+DX+DPH) - 0.004638 0.8545 MCIST 0.002621 0.004936 n.a. no e-vib 0.002021 0.002021 1. ∗ in principle here G$_{lin}$ should be G$_{0}$ but is not because we are using a tiny but finite η = 0 . 030 value. See appendix B for detailed explanations. In the quasi-resonant case when | ˜ ϵ$_{0}$ − µ $^{eq}$| ∼ (1 to 2) linewidth, the effects described above will be more pronounced, because the linear conductance is no longer given by the extreme tail of the main peak crossing the Fermi level. TABLE I: Linear conductance G lin = dI/dV | V → 0 obtained from the value of the equilibrium spectral density at ω = µ $^{eq}$. G lin is given in units of the quantum of conductance G 0 = 2 e $^{2}$/h . The equilibrium spectral functions are shown in Figs. 10, 11 and 12) and were done for different different lev- els of approximation as explained in the corresponding figure captions. For the resonant case in the absence of the Hartree po- tential, the SCBA-based calculations should give a per- fect linear conductance. The reason why G$_{lin}$ = G$_{0}$ is because the calculations were performed with a tiny but finite value of η as explained in detail in Appendix B. Hence one may say that, for the chosen set of parame- ters, the value G$_{lin}$ = 0 . 838 represents an upper bound for the linear conductance (the corresponding numerical perfect conductance). And as explained in Appendix B, linear conductance values can only be compared between calculations performed with the same value of η , which is what we have done. The linear conductance is strongly renormalized (de- creased and increased) when including only one of the two second-order diagrams (DPH and DX respectively). In contrast, it becomes close again to the expected quantum of conductance when the calculations are performed with both second-order diagrams. This dependence of the con- ductance is well understood from the behaviour of the spectral functions shown in Figure 12. SC(BA+DPH) calculations introduce an extra broadening and a cor- responding decrease of amplitude of the peaks in the spectral functions, hence a decrease of G$_{lin}$ . However, the strong narrowing, with increased amplitude, of the peaks due to SC(BA+DX) calculations leads to an in- crease of G$_{lin}$ . The SC(BA+DX) calculations give a lin- ear conductance larger than that obtained from SCBA, i.e. G$_{lin}$ > G$_{0}$ , which is an unphysical result for our non-degenerate electronic level in the central part of the system. This implies a strong constraint on the validity of the values of G$_{lin}$ obtained from a many-body perturbation expansion: in order to conserve the Fermi-liquid prop- erties (see Appendix B) in the electron-hole symmetric resonant regime, one has to perform the calculations by including all diagrams order by order. Results obtained from partial resummation of a subset of diagrams will probably give an incorrect linear conductance. Hence calculations performed by fully renormalizing the vibron propagators 24,32 or all crossing diagrams $^{84}$, For the off-resonant regime at the weak electron-vibron coupling, there are not many differences in the linear conductance G$_{lin}$ values calculated for all the different diagrams, as one might expect from the spectral func- tion behaviour shown in figure 10. This is especially true Table III E shows the different values for the linear conductance G$_{lin}$ obtained from the equilibrium spectral functions shown in figures 10, 11 and 12. where G$_{0}$ = 2 e $^{2}$/h is the quantum of conductance. is obtained from where Γ = Γ$_{L}$Γ$_{R}$ / (Γ$_{L}$ + Γ$_{R}$) and Γ$_{α}$ = − 2 ℑ m [Σ r $_{α}$]. The linear conductance In our model, there is a direct proportionality between the left and right leads’ self-energies Σ r L,R because, the central region is coupled to the leads via the single hop- ping matrix element and we have chosen identical leads. Then the current (Eq. (20)) can be recast as follows (for the details of the derivation, see for example Refs.[64,70]): We now briefly discuss the effects of electron-vibron interaction taken at different levels of the diagrammatic expansion on the linear-response transport properties of the single-molecule nanojunction. Before doing so, we explain how to derive the linear conductance from the value of the spectral functions calculated at equilibrium. E. Linear response transport properties G$_{lin}$/G$_{0}$ = − 2Γ( µ $^{eq}$) ℑ m [ G $^{r}$( µ $^{eq}$)] = 2Γ( µ $^{eq}$) A ( µ $^{eq}$) (31) G$_{lin}$ = dI dV |$_{V}$ → 0 I = − 4 e/h ∫ d ω ( f$_{L}$ ( ω ) − f$_{R}$ ( ω )) Γ( ω ) ℑ m [ G $^{r}$( ω )] , (30)