Dynamical electron-phonon vertex correction

The dynamical screening of the electron--phonon vertex is caused by the retarded oscillations of the electronic charge following the electron--hole scattering with a phonon mode. This retardation induces a frequency dependence of the electron--phonon interaction. Model Hamiltonians and {\em ab--initio} approaches have instilled the idea that this retardation is, in most of the cases, negligible. In this work I demonstrate that the dynamical screening of the electron--phonon vertex cannot be neglected {\em a priori}. By using a perturbative expansion I introduce a controllable and physically sound method to evaluate and include dynamical screening effects. Based on the exact results of the homogeneous electron gas I propose a dynamical vertex correction function $\Gamma_{e-p}(\omega)$ designed to screen the commonly used adiabatic electron--phonon interaction. This function is expressed in terms of adiabatic quantities, that can be easily calculated and used to evaluate the strength of the dynamical corrections, even in realistic materials.