  In ab initio calculations of superconducting properties, the Coulomb
repulsion is accounted for at the GW level and is usually computed in RPA,
which amounts to neglecting vertex corrections both at the polarizability level
and in the self-energy. Although this approach is unjustified, the brute force
inclusion of higher order corrections to the self-energy is computationally
prohibitive. We propose to use a generalized GW self-energy, where vertex
corrections are incorporated into W by employing the Kukkonen and Overhauser
(KO) ansatz for the effective interaction between two electrons in the electron
gas. By computing the KO interaction in the adiabatic local density
approximation for a diverse set of conventional superconductors, and using it
in the Eliashberg equations, we find that vertex corrections lead to a
systematic decrease of the critical temperature (Tc), ranging from a few
percent in bulk lead to more than 40% in compressed lithium. We propose a set
of simple rules to identify those systems where large Tc corrections are to be
expected and hence the use of the KO interaction is recommended. Our approach
offers a rigorous extension of the RPA and GW methods for the prediction of
superconducting properties at a negligible extra computational cost.
