Microscopic theory of dielectric screening and lattice dynamics. I. Local-field corrections and dielectric constants

We show how local-field corrections in solids may be treated by a very general factorization scheme for $\epsilon (\overrightarrow{\mathrm{Q}},{\overrightarrow{\mathrm{Q}}}^{\prime })$ from which practically all existing models of dielectric screening and lattice dynamics may be derived as special cases, including the shell model, the breathing-shell model, and the bond-charge model, as well as generalizations of these models which result from the introduction of a "screening medium." The latter arise naturally in our formalism from a portion of $\epsilon (\overrightarrow{\mathrm{Q}},{\overrightarrow{\mathrm{Q}}}^{\prime })$ which is purely diagonal. It is shown that the formalism also allows for charge-transfer and multipole effects. In this first paper we derive explicit expressions for the elements of $\epsilon (\overrightarrow{\mathrm{q}}+\overrightarrow{\mathrm{G}},\overrightarrow{\mathrm{q}}+{\overrightarrow{\mathrm{G}}}^{\prime })$ and its inverse and show that they have the correct analytic behavior as $\overrightarrow{\mathrm{q}}\to 0$. For insulators of cubic and tetrahedral symmetry, explicit microscopic expressions are derived for the high-frequency dielectric constant ${\epsilon }_{\infty }$ thus realizing a generalization of the Lorentz-Lorenz formula, and for the local field produced by an applied field.