  This article presents a systematic theoretical enquiry concerning the
conceptual foundations and the nature of phonon-mediated electron-electron
interactions. Starting from the fundamental many-body Hamiltonian, we propose a
simple scheme to decouple the electrons and nuclei of a crystalline solid via
effective interactions. These effective interactions, which we express in terms
of linear response functions, are completely symmetric under the exchange of
electrons and nuclei. Correspondingly, we derive concrete formulae for both the
effective electron interaction mediated by phonons and the effective core
interaction mediated by electrons. In particular, we rederive from our
fundamental ansatz the well-known general expressions of the effective
electron-electron interaction in terms of the elastic Green function and the
phonon dispersion relation. We further show that the effective core interaction
coincides in the instantaneous limit with the dynamical matrix as calculated in
electronic structure theory. If combined with the Kubo formalism, our general
formulae lend themselves to the calculation of effective interactions from
first principles. By showing the compatibility of our approach with the
functional integral formalism, this work also paves the way for the derivation
of ab initio initial interactions for functional renormalization group
applications.

Materials Science;Strongly Correlated Electrons;Superconductivity