F.2 Electron-Phonon Interaction

where is the density of electrons. The density of the nuclei in the lattice is represented as a sum of local charge densities

where the actual positions of the nuclei , is defined in terms of the equilibrium lattice vector , the basis vector within the unit cell , and the lattice displacement . In case of bare nuclei, would approximately be functions. However, it is more convenient to consider rigid ion cores instead of bare nuclei. In this case denotes the charge density of the ion cores.

For a simple derivation of the electron-phonon interaction, one has to add an additional external source in (F.24) [203], which couples to the charge density of the nuclei and is merely a mathematical trick, see (F.32)-(F.35). With similar steps for deriving (F.7), one can show that under the HAMILTONian in (F.24) the effective potential can be written as

The aim is the calculation of the total linear response of the system, including the contribution from the nuclei, i.e. the variation of the total electrostatic potential with the external potential [205]

Solving with respect to , one obtains

where the

and is the screened interaction. The derivative differs from the purely electronic polarization which is introduced in Appendix F.1.1, owing to the phonon contribution to the total potential. Neglecting this phonon contribution to the polarization function is one of the ingredients of the

With similar steps for deriving (F.5), the density response of the nuclei under the action of can be calculated as

In the last step, the deviation operator is introduced. Furthermore, the relation is used. Now the additional external field comes into play, which allows us to eliminate the mixed electron-nuclei contribution. By steps completely analogous to those used before, one finds

which together with (F.31), yields the result

where the density-density correlation function of the nuclei is defined as

One can again apply the chain rule to (F.33) to eliminate the contribution.

Making use of the relation (F.33) once more, one can solve the resulting equation with respect to and express the solution in terms of the dielectric function. After insertion in (F.28), this yields the total dielectric screening function as

The desired effective electron-electron interaction induced by lattice vibrations is thus finally given by [205]

Therefore, the problem of electron-phonon interaction is reduced to the replacement of the electronically screened interaction introduced in Appendix F.1.1 by the effective interaction (F.37).