Non-perturbative theory of the electron-phonon coupling and its first-principles implementation

The harmonic approximation of ionic fluctuations and the linear coupling between phonons and electrons provide the standard framework to compute, from first principles, the contribution of nuclear dynamics and its interaction with electrons to materials properties. These approaches become questionable when quantum and anharmonic effects are significant, such as in hydrogenous systems, high-$T_c$ superconductors, and systems close to displacive phase transitions. Here we propose a novel non-perturbative approach to compute the electron-phonon interaction from first principles, including non-linear effects and accounting for the quantum nature of nuclei. The method is based on the $GW^{ph}$ approximation for the electron self-energy, given by the nuclei-mediated electron-electron interaction $W^{ph}$ and the electron Green's function $G$. Electrons are treated at a mean-field level, while nuclear dynamics is described by a Gaussian distribution function that captures anharmonic effects, for example within the self-consistent harmonic approximation. The key quantities of the Gaussian $GW^{ph}$ self-energy are renormalized average vertices, computed in supercells using a stochastic approach based on self-consistent electronic potentials for distorted configurations. To validate the method, $GW^{ph}$ calculations are performed on aluminum, where the results reproduce standard linear electron-phonon theory, and on palladium hydride, where strong non-linear contributions emerge, with corrections comparable in magnitude to the linear-order result.