Simple Model of Electronic Correlation in Insulators

To facilitate calculation of the effects of electron-electron interaction in insulators, a dielectric model is developed analogous to one used by Overhauser to discuss correlation in metals. In this model the dynamical many-electron problem is replaced by a field-theoretic problem in which an electron interacts with a "plasmon" field representing the valence-charge distribution. For small wave vector $\overrightarrow{\mathrm{q}}$ the plasmon dispersion $\omega \left(\overrightarrow{\mathrm{q}}\right)$ approaches the bulk plasma frequency; at large $\overrightarrow{\mathrm{q}}$, $\omega \left(\overrightarrow{\mathrm{q}}\right)\to \frac{\hslash {\overrightarrow{\mathrm{q}}}^2}{2m}$, corresponding to single-particle excitations. To determine the electron-plasmon coupling Poisson's equation and the sum rule on oscillator strength are employed. Calculated self-energies of an electron (or hole) due to correlation are large, typically $\frac{1}{3}$ Ry, though substantially reduced by recoil effects if the electron mass is small. Exchange effects lead to further reduction. The crucial importance of short-range (large-$\overrightarrow{\mathrm{q}}$) dielectric behavior is emphasized.