Correlation Potentials in a Nonuniform Electron Gas

A microscopic theory of the correlation operator, directed toward conduction electrons in a metal, is developed. The off-diagonal matrix elements, between electron states $\overrightarrow{\mathrm{k}}$ and $\overrightarrow{\mathrm{k}}+\overrightarrow{\mathrm{q}}$, that arise when the electron density has a modulation of wave vector $\overrightarrow{\mathrm{q}}$, determine the correlation contribution to band structure. These are computed explicitly. They depend dramatically on $\overrightarrow{\mathrm{k}}$ (i.e., nonlocal behavior) and on $\overrightarrow{\mathrm{q}}$. The sum of exchange and correlation operators is also nonlocal, but is less singular than either individually. An alternative division of exchange plus correlation into screened-exchange plus Coulomb-hole operators is made. It is shown that the (often ignored) Coulomb-hole operator is usually much larger than the screened-exchange operator.