Optical sum rules for inhomogeneous electron systems

Optical properties can be determined by either the transverse dielectric function ${\mathrm{\epsilon }}_{\mathrm{\perp }}$(q,ω) or the longitudinal dielectric function ${\mathrm{\epsilon }}_{\mathcal{n}}$(q,ω) in the long wavelength limit, i.e., q→0. By expressing ${\mathrm{\epsilon }}_{\mathrm{\perp }}$(q→0,ω) in terms of density response functions, we calculate the third, fifth, and seventh frequency moments of Im${\mathrm{\epsilon }}_{\mathrm{\perp }}$(q→0,ω) in addition to the first frequency moment, the well-known conductivity sum rule. While the third and fifth moments reflect the inhomogeneity of the system, correlation effects contribute explicitly to the seventh moment. In addition, the frequency moments of Im[-1/${\mathrm{\epsilon }}_{\mathrm{\parallel }}$(q→0,ω)] can be constructed from these moments. The dielectric functions evaluated in a self-consistent field (SCF) approximation, such as the random phase approximation (RPA) or the adiabatic local density approximation (ALDA), satisfy the first three odd frequency moments provided the single particle states on which the SCF calculation is based are calculated in the corresponding SCF approximation. The seventh frequency moment constitutes a severe requirement that ${\mathrm{\epsilon }}_{\mathrm{\perp }}$(q→0,ω) evaluated in the RPA or the ALDA cannot satisfy. This is demonstrated for nearly free electron systems.