19 D r,a 0 ( ω ) = 1 ω − ω$_{0}$ ± i η − 1 ω + ω$_{0}$ ± i η , η → 0 + = P . V . 1 ω − ω$_{0}$ − P . V . 1 ω + ω$_{0}$ ∓ iπ ( δ ( ω − ω$_{0}$ ) − δ ( ω + ω$_{0}$ ) , (B1) H ( y ) = H . T . [ f ( x )] = 1 π P . V . ∫ ∞ −∞ dx f ( x ) x − y . (B2) First, we start by using the following decomposition of the retarded and advanced vibron propagator D r,a 0 : 0 0.2 0.4 0.6 0.8 1 1.2 0 10 20 30 A( MCIST - SCBA NEGF - SCBA [  = 0$^{+}$, FFT/HT] NEGF - SCBA [  = 0.001] NEGF - SCBA [  = 0.025] $_{0}$/ $_{0}$ = 0.5 and the definition of the Hilbert transformation (H.T.): FIG. 14: Equilibrium spectral functions for the off-resonant regime calculated with MCIST and NEGF-SCBA. MCIST calculations are performed at the Hartree-Fock level (BA). NEGF are done using the limit η → 0 + for the imaginary part of the bare vibron GF and also by using a tiny, but finite value for η (0.001 and 0.025). See main text for details. The other parameters are ε 0 = +0 . 5 (electron transport), γ 0 = 0 . 20 (upper panel), ω 0 = 0 . 40 , t 0 L = t 0 R = 0 . 15. For a small enough value of η , one recovers the exact MCIST as well as the η → 0 + NEGF results obtained from Hilbert transform (HT). Larger values of η widen the peaks; however the spectroscopic information remains. For such values of η , one can work with fewer grid points and makes second-order self-energy calculations tractable. One can then rewrite the Fock retarded (advanced) self-energy in terms of retarded (advanced) and lesser electron Green’s functions only: Σ F,r e $_{-vib}$( ω ) = γ 2 0 [ 〈 N$_{ph}$ 〉 G $^{r}$( ω − ω$_{0}$ ) + ( 〈 N$_{ph}$ 〉 + 1) G $^{r}$( ω + ω$_{0}$ ) + 1 2 ( G $^{<}$( ω − ω$_{0}$ ) + G $^{r}$( ω − ω$_{0}$ ) − G $^{<}$( ω + ω$_{0}$ ) − G $^{r}$( ω + ω$_{0}$ ) + i 2 ( H [ G < + G $^{r}$]( ω + ω$_{0}$ ) − H [ G < + G $^{r}$]( ω − ω$_{0}$ ) ] , (B3) Σ F,a e $_{-vib}$( ω ) = γ 2 0 [ 〈 N$_{ph}$ 〉 G $^{a}$( ω − ω$_{0}$ ) + ( 〈 N$_{ph}$ 〉 + 1) G $^{a}$( ω + ω$_{0}$ ) − 1 2 ( G $^{<}$( ω − ω$_{0}$ ) − G $^{a}$( ω − ω$_{0}$ ) − G $^{<}$( ω + ω$_{0}$ ) + G $^{a}$( ω + ω$_{0}$ ) + i 2 ( H [ G < − G $^{a}$]( ω + ω$_{0}$ ) − H [ G < − G $^{a}$]( ω − ω$_{0}$ ) ] . (B4) and However since FFT routines introduce artifical periodic boundary conditions, and one has to make sure that the functions, which decay slowly as 1 /ω (the kernel of the Hilbert transform, and real part of G $^{r,a}$), have sufficiently small values at the boundaries of the energy grid. Oth- erwise, the corresponding discontinuities introduce spu- rious oscillations in the FFT. This means that one has to work with a wide energy grid and with a correspond- ingly large number of grid points to keep a good enough resolution. These expressions are in a form convenient for com- putation, since, as for Σ F,<,> e -vib , they involve the direct evaluation of G r,a,< on a energy grid shifted by ± ω$_{0}$ in addition with the Hilbert transform of G < and G r,a on a shifted energy grid. The Hilbert transformation is actually a conventional convolution product of the trial function with an appro- priate (1 /x )-like kernel. Calculation of convolution prod- ucts can be made faster, instead of scaling as the square of the number of grid points, by the use of FFT routines.