A new paper is out on the different approximation in GW for the electron-phonon couping matrix elements. In particular we tested two approximation the static screening (COHSEX) and the constant screening respect to the phonon displacement. We found that the first approximation miserably fails to reproduce the G0W0 results while the second introduce a negligible error respect to the full calculation. This result open the way to an implementation of the G0W0 corrections within Density Functional Perturbation Theory(DFTP).
Here the reference:
Phys. Rev. B 91, 155109 (2015)