Digital orbitals: connecting the tight-binding and plane-wave approaches through sampled-real-space models

Abstract

Discrete approximations to continuous systems have proven valuable in many areas of physics and engineering. Discrete real-space calculations are often used in electronic structure calculations for molecules, and have been used in electronic structure calculations for crystals. Besides their computational utility, they can be used to connect the plane-wave and tight-binding methods of crystalline electronic structure calculations. A fully discrete-space approach results in exactly equivalent, finite, plane-wave and tight-binding bases for the same crystal because the two bases therefore span exactly the same space. Beyond connecting the two methods, the fully discrete approach addresses important issues in tight-binding. First, it results in a fully gauge invariant position operator capable of sub-cell spatial resolution. Second, it explains the observation in tight-binding that a small basis with long-range couplings and a large basis with short-range couplings can often reproduce technologically relevant bands with comparable accuracy.