A many-body perturbation theory approach to the electron-phonon interaction with density-functional theory as a starting point

The electron-phonon interaction plays a crucial role in many fields of physics and chemistry. Nevertheless, its actual calculation by means of modern many-body perturbation theory is weakened by the use of model Hamiltonians that are based on parameters difficult to extract from the experiments. Such shortcoming can be bypassed by using density-functional theory to evaluate the electron-phonon scattering amplitudes, phonon frequencies and electronic bare energies. In this work, we discuss how a consistent many-body diagrammatic expansion can be constructed on top of density-functional theory. In that context, the role played by screening and self-consistency when all the components of the electron-nucleus and nucleus-nucleus interactions are taken into account is paramount. A way to avoid over-screening is notably presented. Finally, we derive cancellations rules as well as internal consistency constraints in order to draw a clear, sound and practical scheme to merge many-body perturbation and density-functional theory.