SCDM-k: Localized orbitals for solids via selected columns of the density matrix

Abstract

The recently developed selected columns of the density matrix (SCDM) method [J. Chem. Theory Comput. 11, 1463, 2015] is a simple, robust, efficient and highly parallelizable method for constructing localized orbitals from a set of delocalized Kohn-Sham orbitals for insulators and semiconductors with $\Gamma$ point sampling of the Brillouin zone. In this work we generalize the SCDM method to Kohn-Sham density functional theory calculations with k-point sampling of the Brillouin zone, which is needed for more general electronic structure calculations for solids. We demonstrate that our new method, called SCDM-k, is by construction gauge independent and is a natural way to describe localized orbitals. SCDM-k computes localized orbitals without the use of an optimization procedure, and thus does not suffer from the possibility of being trapped in a local minimum. Furthermore, the computational complexity of using SCDM-k to construct orthogonal and localized orbitals scales as O(N log N ) where N is the total number of k-points in the Brillouin zone. SCDM-k is therefore efficient even when a large number of k-points are used for Brillouin zone sampling. We demonstrate the numerical performance of SCDM-k using systems with model potentials in two and three dimensions.Comment: 25 pages, 7 figures; added more background sections, clarified presentation of the algorithm, revised the presentation of previous work, added a more high level overview of the new algorithm, and mildly clarified the presentation of the results (there were no changes to the numerical results themselves