5 Actually at each iteration of the calculations, we use a simple mixing scheme of the self-energies obtained at the present iteration and at the previous iteration. This mix- ing scheme permits us to achieve full self-consistency in a maybe slightly longer but more stable iterative process. Note finally that by using Eq. 26, instead of the more general formulation given by Eq. 12, we assume that, after switching on the interactions, there are no bound states in the system (i.e. there are no interaction-induced electron states located outside the spectral supports of the left and right leads) 72,73 which is indeed the case. These relationships are at the centre of the fluctuation- dissipation theorem for equilibrium, which also be recast as a relationship betweeen the greater and lesser Green’s functions: for statistical averages at finite temperature in the grand canonical ensemble. This equation is related to the Kubo-Martin-Schwinger boundary conditions $^{66,71}$. For non-equilibrium conditions, there is no unique Fermi level at finite bias (or no unique temperature if T$_{L}$ = T$_{R}$ ) in the whole system, and the relationships given by Eqs. (22-24) no longer hold. This is an impor- tant feature of the non-equilibrium formalism for which conventional equilibrium statistics need to be reformu- lated. In this section, we present results for the spectral func- tions A ( ω ) for the different transport regimes and for different applied biases in the low vibron-temperature regime. We divide the calculations into four types. Firstly, we calculate the spectral functions at equilibrium for two transport regimes. The first of these is where ε$_{0}$ ≫ µ eq or ε$_{0}$ ≪ µ $^{eq}$, known as the off-resonant regime as in order to create a current between the left and right leads one puts an electron in the empty (for electron transport, ε$_{0}$ ≫ µ $^{eq}$) or full (for hole transport, ε$_{0}$ ≪ µ $^{eq}$) electronic level ε$_{0}$ . The calculations start by constructing the non- interacting Green’s functions of the entire system G x 0 with x being any three of the possible Green’s functions x = { r, a, <, >, t, ˜ t } . The other Green’s functions are obtained by using the relationships between them as shown in Appendix A 1. For example, the retarded Green’s function is given by G r $_{0}$( ω ) = [ g r $_{0}$( ω ) − $^{1}$− Σ r $_{L}$( ω ) − Σ r $_{R}$( ω )] − $^{1}$. One then calculates three “Keldysh components” for the self-energies corresponding to any of the diagrams ⋆ = { F,DX or DPH } : Σ ⋆ ,< e $_{-vib}$, Σ ⋆ ,> e -vib and Σ ⋆ ,t e $_{-vib}$. The Hartree diagram has only one component Σ H,r e -vib as shown above. The second transport regime is when ε$_{0}$ ± linewidth ∼ µ $^{eq}$, known as the resonant transport regime. The linewidth is the width of the peak in the spectral function which arises from the electronic coupling of the central region to the left and right leads. In this case the elec- tronic level ε$_{0}$ is, at equilibrium, half-filled by an electron (and thus also half-filled with a hole). For each of these transport regimes, we calculate the spectral function both at equilibrium (applied bias V = 0) and non-equilibrium at finite bias ( V > 0). The advanced and retarded self-energies Σ ⋆ , { r,a } e -vib are then obtained by simple algebra using the relationships between the different self-energies as explained in Ap- pendix A 1. The new Green’s functions are then calculated by us- ing the Dyson equations for G r,a For each of these four groups (resonant/off-resonant transport regimes, at/out of equilibrium), a large amount of different NEGF calculations have been performed for different values of the relevant parameters and within different levels of approximation (Hartree Fock, Hartree Fock+second-order, partially or fully self-consistent cal- culations). In the rest of this section, we present only a limited and selected number of results which we found the most relevant for each case, and we analyse and com- pare in detail the effects of the different diagrams on the spectral functions of the system at and out of equilib- rium. We also, in section III C, compare perturbation expan- sion based calculations (NEGF) to a reference calcula- tion which is exact in term of electron-vibron coupling but which however is only valid for a specific transport regime. The new self-energies are then recalculated and the pro- cess re-iterates until full self-consistency is achieved. where the total self-energies are given by and the quantum kinetic equations for G <,> G $^{r,a}$( ω ) = [ ω − ε$_{0}$ − Σ r,a $_{total}$( ω )] − $^{1}$, (25) G $^{≷}$( ω ) = G $^{r}$( ω )Σ ≷ $_{total}$( ω ) G $^{a}$( ω ) , (26) Σ x $_{total}$( ω ) = Σ x $_{L}$( ω ) + Σ x $_{R}$( ω ) + ∑ any ⋆ Σ ⋆ ,x e $_{-vib}$( ω ) . (27) G >, $^{eq}$( ω ) = − e ( ω − µ$_{0}$ ) /kT G <, $^{eq}$( ω ) (24) G >, $^{eq}$( ω ) = − ( f $^{eq}$( ω ) − 1) ( G r, $^{eq}$( ω ) − G a, $^{eq}$( ω )) = 2i( f $^{eq}$( ω ) − 1) A ( ω ) . (23) and III. RESULTS: SPECTRAL FUNCTIONS D. Computational aspects