NURBS-based non-periodic finite element framework for Kohn-Sham density functional theory calculations

Abstract

A real-space non-periodic computational framework is developed for Kohn-Sham density functional theory (DFT). The electronic structure calculation framework is based on the finite element method (FEM) where the underlying basis is chosen as non-uniform rational B-splines (NURBS) which display continuous higher-order derivatives. The framework is formulated within a unified presentation that can simultaneously address both all-electron and pseudopotential settings in radial and three-dimensional cases. The canonical Kohn-Sham equation and the Poisson equation are discretized on different meshes in order to ensure that the underlying variational structural of Kohn-Sham DFT is preserved within the weak formulation of FEM. The discrete generalized eigenvalue problem emanating from the Kohn-Sham equation is solved efficiently based on the Chebyshev-filtered subspace iteration method. Numerical investigations in the radial case demonstrate all-electron and local pseudopotential capabilities on single atoms. In the three-dimensional case, all-electron and nonlocal pseudopotential computations on single atoms and small molecules are followed by local and nonlocal pseudopotential studies on larger systems. At all stages, special care is taken to demonstrate optimal convergence rates towards the ground state energy with chemical accuracy. Comparisons with classical Lagrange basis sets indicate the significantly higher per-degree-of-freedom accuracy displayed by NURBS. Specifically, cubic NURBS discretizations can offer a faster route to a prescribed accuracy than even sixth-order Lagrange discretizations on comparable meshes, thereby indicating considerable efficiency gains which are possible with these higher-order basis sets within effective numerical implementations.