  Electron-phonon interactions are of great importance to a variety of physical
phenomena, and their accurate description is an important goal for
first-principles calculations. Isolated examples of materials and molecular
systems have emerged where electron-phonon coupling is enhanced over density
functional theory (DFT) when using the Green's-function-based ab initio GW
method, which provides a more accurate description of electronic correlations.
It is however unclear how general this enhancement is, and how employing
high-end quantum chemistry methods, which further improve the description of
electronic correlations, might further alter electron-phonon interactions over
GW or DFT. Here, we address these questions by computing the renormalization of
the highest occupied molecular orbital energies of Thiel's set of organic
molecules by harmonic vibrations using DFT, GW and equation-of-motion
coupled-cluster calculations. We find that GW can increase the magnitude of the
electron-phonon coupling across this set of molecules by an average factor of
1.1-1.8 compared to DFT, while equation-of-motion coupled-cluster leads to an
increase of 1.4-2. The electron-phonon coupling predicted with the ab initio GW
method is generally in much closer agreement to coupled cluster values compared
to DFT, establishing GW as an accurate way of computing electron-phonon
phenomena in molecules and beyond at a much lower computational cost than
higher-end quantum chemistry techniques.
