Exchange Potentials in a Nonuniform Electron Gas

Off-diagonal matrix elements of the exchange operator are computed for a degenerate electron gas having a small sinusoidal density modulation. The extreme nonlocal character of exchange is shown explicitly by its wave-vector dependence. The Slater exchange approximation severely underestimates the off-diagonal action of the exact exchange operator (by a numerical factor approaching $\infty$ for long-wavelength modulations). Such errors are largely compensated by neglect of the correlation potential.