Microscopic Theory of Force Constants in the Adiabatic Approximation

TLDR

The microscopic quantum‑mechanical expressions for the Born‑von Karman force constants in arbitrary solids are derived in terms of the complete inverse dielectric function of the electrons, treating many‑body effects exactly while applying only the Born‑Oppenheimer approximation. The derived force constants satisfy Born’s invariance conditions, reproduce the phenomenological dynamical matrix for perfect crystals, yield an effective charge expression, enforce an acoustic sum rule that guarantees vanishing acoustic‑mode frequencies at long wavelength, and preserve phenomenological relations such as the generalized Lyddane‑Sachs‑Teller relation.

Abstract

The microscopic quantum-mechanical expressions for the Born-von Karman force constants in an arbitrary solid, crystalline or amorphous, are derived in terms of the complete inverse dielectric function ${\ensuremath{\epsilon}}^{\ensuremath{-}1}(\mathrm{r}, {\mathrm{r}}^{\ensuremath{'}})$ of the electrons The many-body nature of the electrons is treated exactly; only the Born-Oppenheimer approximation is made. Born's translation and rotation invariance conditions are shown to be satisfied by the microscopic force constants. In the case of a perfect crystal, it is shown for the first time that the microscopic formulas recapture completely the phenomenological form of the dynamical matrix; in particular, the microscopic expression for the effective charge in an insulator is found. We prove that the charge neutrality of the system implies the "effective charge neutrality" condition and that, consequently, all acoustic-mode frequencies vanish at long wavelength. This condition may be stated as a useful property of ${\ensuremath{\epsilon}}^{\ensuremath{-}1}$ which we term the acoustic sum rule. Many results of the phenomenological theory, e.g., the generalized Lyddane-Sachs-Teller relation, carry over exactly to the microscopic theory.