Communications in Mathematical Sciences

Volume 13 (2015)

Number 7

A posteriori error estimator for adaptive local basis functions to solve Kohn–Sham density functional theory

Pages: 1741 – 1773



Jason Kaye (Courant Institute of Mathematical Sciences, New York University, New York, N.Y., U.S.A.)

Lin Lin (Department of Mathematics, University of California at Berkeley; and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, Calif., U.S.A.)

Chao Yang (Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, Calif., U.S.A.)


Kohn–Sham density functional theory is one of the most widely used electronic structure theories. The recently developed adaptive local basis functions form an accurate and systematically improvable basis set for solving Kohn–Sham density functional theory using discontinuous Galerkin methods, requiring a small number of basis functions per atom. In this paper, we develop residual-based, a posteriori error estimates for the adaptive local basis approach, which can be used to guide non-uniform basis refinement for highly inhomogeneous systems such as surfaces and large molecules. The adaptive local basis functions are non-polynomial basis functions, and standard a posteriori error estimates for hp-refinement using polynomial basis functions do not directly apply. We generalize the error estimates for hp-refinement to non-polynomial basis functions. We demonstrate the practical use of the a posteriori error estimator in performing three-dimensional Kohn–Sham density functional theory calculations for quasi-2D aluminum surfaces and a single-layer graphene oxide system in water.


Kohn–Sham density functional theory, $\textit{a posteriori}$ error estimator, adaptive local basis function, discontinuous Galerkin method

2010 Mathematics Subject Classification

65N15, 65N25, 65N30, 65Z05

Published 19 August 2015