Theory of the Electron–Phonon Interaction

An exact solution to the self-consistent field screening problem is presented in terms of Bloch-state matrix elements of the bare perturbation. This solution is equivalent to one given by Sham and Ziman in terms of Fourier transforms of the bare perturbation, but unlike the latter it avoids the convergence problem in momentum space that arises in the screening of the electron–phonon interaction because of the 'deep' part of the ionic potential. The new form of the solution converts naturally into a pseudopotential formalism. At the same time it provides a framework in which to discuss and extend the two most common calculation schemes for the electron–phonon matrix element, the rigid Schrödinger potential model and the diffraction model. The inverse dielectric tensor is evaluated to first-order in the pseudopotential. Local field effects cause the Q → 0 limit of the electron–phonon interaction to deviate from its conventional value.