Gradient Flow Based Kohn--Sham Density Functional Theory Model

Abstract

In this paper, we propose and analyze a gradient flow based model for electronic structure calculations. First, based on an extended gradient flow proposed in this paper, we propose a Kohn--Sham gradient flow based model. We prove that our gradient flow based model is orthogonality preserving, the extended gradient has an exponential decay over time $t$, and the equilibrium point is a local minimizer of the Kohn--Sham energy functional. Then we propose a midpoint scheme to carry out the temporal discretization, which is proven to be orthogonality preserving, too. Based on the midpoint scheme, we design a practical orthogonality preserving iteration scheme which can deal with the propagation of the gradient flow based model and prove that the scheme produces approximations that converge to a local minimizer with some convergence rate under some reasonable assumptions. Finally, we report a number of numerical experiments that validate our theoretical results.