Long-wavelength behavior of the exchange-correlation kernel in the Kohn-Sham theory of periodic systems

The polarization dependence of the exchange-correlation (xc) energy functional of periodic insulators within Kohn-Sham (KS) density-functional theory requires a ${O(1/q}^2)$ divergence in the xc kernel for small vectors q. This behavior, exemplified for a one-dimensional model semiconductor, is also observed when an insulator happens to be described as a KS metal, or vice versa. Although it can occur in the exchange-only kernel, it is not found in the usual local, semilocal, or even nonlocal approximations to KS theory. We also show that the test-charge and electronic definitions of the macroscopic dielectric constant differ from one another in exact KS theory, but are equivalent in the above-mentioned approximations.