10 side-band peak. This energy difference should be equal to the vibron energy ω$_{0}$   as it is given by exact MCIST calculations. Note that the limits of BA-based calcula- tions were also been studied by Lee et al. in a somewhat different context in Ref. [60]. In conclusion  this means that Hartree-Fock (or BA) based calculations for electron-vibron interaction are only valid for weak coupling  as can be expected from a perturbation-expansion based theory. Hence one needs to include higher-order diagrams in the electron-vibron self- energies to go beyond the commonly-used self-consistent Born approximation (Hartree-Fock) in order to obtain correct results for a wide range of parameters. The ef- fects of the higher-order diagrams (here second-order-DX and DPH diagrams) are explored in detail in the follow- ing sections. Additionally  although MCIST calculations are only valid in the off-resonant transport regime at and near equilibrium  they include all possible higher-order dia- grams (with bare vibron propagator) and hence can be used as a reference for any perturbation-expansion-based NEGF calculations performed at equilibrium or in the quasi-equilibrium regime. In this section  we present results for the spectral func- tions when the second-order diagrams (see Figure 3) are included in the calculations of the Green’s functions. The reader can find more information about the mathemati- cal expressions for the self-energies corresponding to the second-order diagrams in Appendix A. These diagrams fall into two types—the double- exchange DX diagram  corresponding to vertex correc- tions  and the dressed vibron diagram  which includes a single electron-hole bubble  renormalizing the vibron propagator and hence giving rise to polarization effects. We have used three different levels of approximation to calculate these Green’s functions: Firstly  calcula- tions with no self-consistency—the Green’s functions are simply calculated using the diagrams in Figure 2 and Figure 3 using the bare propagator G$_{0}$ as the electron Green’s function. In our model  G$_{0}$ is the Green’s func- tion of the central region connected to the leads with no electron-vibron interactions. This is a first-order pertur- bation expansion for which Σ H F DX  and / or DP e -vib [ G$_{0}$ ]. We use the abbreviations BA (Born approximation for non self-consistent Hartree and Fock diagrams) and BA+DX (DX for double exchange) and BA+DX+DPH (DPH for dressed vibron  for the GW -like diagram) in the follow- ing. of the resonant transport regime (i.e. vibron side-band peaks on both sides of the main peak). This is essentially due to the fact that in MCIST  one does not take prop- erly into account the statistics of the Fermi seas of the left and right leads. See for example Eqs. (28) and (29)  there are no leads’ Fermi distributions in the retarded component of the leads’ self-energies Σ r $_{L R}$. Now  comparing BA-based calculations with the exact MCIST calculations  one can see from figure 9 that the BA-based calculations give the wrong polaron shift  i.e. the normalized position of the main peak ε$_{0}$   especially in the strong electron-vibron coupling regime. Further- more BA-based calculations also give the wrong energy separation between the main peak and the first vibron Secondly  we perform partly self-consistent calcula- tions  where the Green’s functions are calculated with the first loop of self-consistent calculations with the Hartree and Fock diagrams. We use these Green’s functions as a starting point to calculate new  corrected  Green’s func- FIG. 9: (Color online) Equilibrium off-resonant spectral func- tions calculated with the MCIST technique and NEGF-SCBA for the off-resonant regime. The upper panel shows results for weak electron-vibron coupling ( γ 0 /ω 0 = 0 . 5)  while the lower panel shows results for strong electron-vibron coupling ( γ 0 /ω 0 = 0 . 8). MCIST calculations give exact results in this case  and can also be performed at the same level of approx- imation as SCBA (see main text for details). MCIST-SCBA and NEGF-SCBA are virtually identical  but show large dis- crepancies for the peak positions (main peak and more espe- cially for the vibron side-band peaks) in comparison to the exact results. The parameters are ε 0 = +0 . 5 (electron trans- port)  γ 0 = 0 . 2 (upper panel) and γ 0 = 0 . 32 (lower panel)  ω 0 = 0 . 4   t 0 L R = 0 . 15. D. Vertex corrections and polarization effects to the spectral functions 0 0.2 0.4 0.6 0.8 1 1.2 0 10 20 30 40 A( MCIST - exact MCIST - SCBA NEGF - SCBA (a) 0 0.2 0.4 0.6 0.8 1 1.2 0 10 20 30 40 A( MCIST - exact MCIST - SCBA NEGF - SCBA (b)