Post-processing of the plane-wave approximation of Schrödinger equations. Part II: Kohn–Sham models

Abstract

Abstract In this article, we provide a priori estimates for a perturbation-based post-processing method of the plane-wave approximation of nonlinear Kohn–Sham local density approximation (LDA) models with pseudopotentials, relying on Cancès et al. (2020, Post-processing of the plane-wave approximation of Schrödinger equations. Part I: linear operators. IMA Journal of Numerical Analysis, draa044) for the proofs of such estimates in the case of linear Schrödinger equations. As in Cancès et al. (2016, A perturbation-method-based post-processing for the plane-wave discretization of Kohn–Sham models. J. Comput. Phys., 307, 446–459), where these a priori results were announced and tested numerically, we use a periodic setting and the problem is discretized with plane waves (Fourier series). This post-processing method consists of performing a full computation in a coarse plane-wave basis and then to compute corrections based on the first-order perturbation theory in a fine basis, which numerically only requires the computation of the residuals of the ground-state orbitals in the fine basis. We show that this procedure asymptotically improves the accuracy of two quantities of interest: the ground-state density matrix, i.e. the orthogonal projector on the lowest $N$ eigenvectors, and the ground-state energy.