Sum Rules for the Dielectric Screening Properties of Crystals

The question of the screening of contributions from the second-order electron-phonon interaction (i.e., background scattering corrections) in a recent many-body theory of phonon spectra is reexamined. An identity deduced in this latter work requires correction appropriate to the case of unscreened background scattering terms, and a recent sum rule given by Pick et al. is shown to arise as one special case of this corrected identity, which consists of a set of sum rules on the screening function elements in the long-wave-length limit.