Polarizability and dielectric constant in density-functional supercell calculations with discrete k-point samplings

We address the dielectric constant (or the polarizability for an isolated system) as obtained in density-functional supercell calculations with a discrete k-point sampling. We compare a scheme based on conventional perturbational theory to one based on a discrete Berry phase, which can also be used for treating finite electric fields. We show, both analytically and numerically, that the difference between the dielectric constants in the two schemes converges as 1/L², L being the supercell size. Applications to the water molecule and bulk silicon illustrate this behavior. In both cases, the conventional perturbational scheme is found to converge faster with L than the discrete Berry-phase scheme.