  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.
