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First, we start by using the following decomposition of the retarded and advanced vibron propagator D r,a 0 : 

<img src='content_image/377969.jpg'>

and the definition of the Hilbert transformation (H.T.): 

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One can then rewrite the Fock retarded (advanced) self-energy in terms of retarded (advanced) and lesser electron Green’s functions only: 

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and 

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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. 

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. 

