4 processes. The first of these is the double-exchange self- energy Σ DX e -vib given by Figure 3(a). In the many-body language, it is part of the vertex correction to the Fock diagram. The second is given by Figure 3(b) and cor- responds to the dressed vibron (or GW-like) self-energy Σ DPH e $_{-vib}$, which consists of the vibron propagator renormal- ized by a single electron-hole bubble (the polarization). This is why we refer to the effects of Σ DPH e -vib as polarization effects in the following. The details of how we implement these self-energies are given in appendix A. (a) Σ Fock e -vib = ✁ (b) Σ Hartree e -vib = ✁ FIG. 2: The (a) Fock and (b) Hartree diagrams FIG. 3: The (a) double exchange DX and (b) vibron prop- agator dressed by the e-h bubble diagrams (dressed phonon, DPH) C. Physical properties (a) Σ DX e -vib = ✁ (b) Σ DPH e -vib = ✁ Once we have calculated all the different Green’s func- tions, any of the physical properties of the system, such as the electron density, the electronic current denstiy, the total energy, the current noise, the heat transfer, etc. can be calculated. For example, the electronic current passing through the α contact is given by sites (or energy levels, depending on the representation used to the electrodes) connected to the central region is given by g r 0 $_{α}$. In this paper, we have chosen a simple model, a semi- infinite tight-binding chain with on-site energy ε$_{α}$ and nearest-neighbour hopping integral β$_{α}$ . This model gives a semi-elliptic density of states of the terminal lead site connected to the central region, and so each lead’s Green’s function becomes i.e the first term describes the transfer of an electron from the interacting region to electrode α , while the sec- ond transfers an electron from electrode α to the central region. We can then express this in terms of Green’s functions and derive an expression for the expectation value of the current $^{70}$: with ω = ε$_{α}$ + 2 β$_{α}$ cos k$_{α}$ ( ω ). We have chosen this model because it is one of the most simple, although in prin- ciple and in practice there are no limitations for taking any other more complicated or more realistic models for the lead, such as Bethe lattices with z -coordination, or a nanotip supported by a semi-infinite surface as shown in Ref 64) since all their electronic properties are wrapped up in the lead self-energies Σ x $_{L,R}$( ω ). The self-energy for the interacting central region, Σ$_{int}$ is somewhat more complicated. It consists of the sum of the self-energies due to interactions between the elec- trons and to interactions between the electrons and the quantum vibration modes (vibrons). In this paper we consider only the coupling between each electron and a single vibron of the central region (the molecule), hence the electron-vibron self-energy Σ$_{e}$$_{-vib}$. All physical properties may be expressed in terms of the spectral function A ( ω ) which is at the heart of this paper. The spectral function is related to the imaginary part of the retarded or advanced electron Green’s func- tions, as For the current work, it is necessary to calculate sev- eral types of self energy. Firstly we have the Fock- like self-energy Σ F,r/a/ ≷ e -vib ( ω ), which is a function of en- ergy, and is represented in diagrammatic form by Fig- ure 2(a). We also have the Hartree-like self-energy Σ H,r e -vib which is independent of energy, given by Figure 2(b). Calculations using only the Hartree and Fock di- agrams, and performed in a self-consistent way are usu- ally referred to as the self-consistent Born approximation (SCBA) $^{20,24,26,27,30–34,38,44–46}$. However, as explained in the introduction, we also want to go beyond the SCBA, and thus we will also cal- culate two further self-energies that include two-vibron Furthermore, when the system is at equilibrium ( f$_{L}$ = f$_{R}$ = f $^{eq}$), there are some relationships between the lesser (greater) and the advanced and retarded Green’s functions: For non-interacting systems, it is simply proportional to the density of electronic states n ( ω ) = A ( ω ) /π . For inter- acting systems, it gives information about the excitations (electron or hole) of the system. G <, $^{eq}$( ω ) = − f $^{eq}$( ω ) ( G r, $^{eq}$( ω ) − G a, $^{eq}$( ω )) = 2i f $^{eq}$( ω ) A ( ω ) , (22) A ( ω ) = −ℑ m [ G $^{r}$( ω )] = + ℑ m [ G $^{a}$( ω )] . (21) I$_{α}$ = 2 e ℏ ∫ d ω 2 π Tr { Σ < $_{α}$( ω ) G $^{>}$( ω ) − Σ > $_{α}$( ω ) G $^{<}$( ω ) } = i2 e ℏ ∫ d ω 2 π Tr { f$_{α}$ ( ω )Γ$_{α}$( ω )[ G $^{r}$( ω ) − G $^{a}$( ω )] + Γ$_{α}$( ω ) G $^{<}$( ω ) } . (20) g r 0 $_{α}$( ω ) = exp(i k$_{α}$ ( ω )) /β$_{α}$ (18) I$_{α}$ ( t ) = 2i e ℏ ∑ n V$_{αn}$ 〈 c † $_{α}$( t ) d$_{n}$ ( t ) 〉 − V ∗ $_{αn}$〈 d † $_{n}$( t ) c$_{α}$ ( t ) 〉 , (19)