First-principles approach to the electron-phonon interaction

We present a first-principles approach to the calculation of the electron-phonon interaction. This approach solves some theoretical difficulties in the standard derivation of the electron-phonon interaction. We do not make a Born-Oppenheimer approximation from the outset but transform the electronic coordinates to a frame attached to the nuclear framework. Subsequently coupled equations are derived which connect the nuclear density-density correlation function to the electron Green function, the screened interaction, and the vertex. This set of equations is completely equivalent to the full problem and therefore higher-order effects are systematically included. The derived equations are further compared to those obtained from the Frohlich Hamiltonian. It is shown that careless use of this Hamiltonian leads to double counting but also insight is given why use of this Hamiltonian has led to many useful results. Finally a simple method is presented that allows for the inclusion of electron-phonon coupling within a density-functional context.