Computing Localized Representations of the Kohn--Sham Subspace Via Randomization and Refinement

Abstract

Localized representation of the Kohn--Sham subspace plays an important role in quantum chemistry and materials science. The recently developed selected columns of the density matrix (SCDM) method [A. Damle, L. Lin, and L. Ying, J. Chem. Theory Comput., 11 (2015), pp. 1463--1469}] is a simple and robust procedure for finding a localized representation of a set of Kohn--Sham orbitals from an insulating system. The SCDM method allows the direct construction of a well conditioned (or even orthonormal) and localized basis for the Kohn--Sham subspace. The SCDM algorithm avoids the use of an optimization procedure and does not depend on any adjustable parameters. The most computationally expensive step of the SCDM method is a column pivoted QR factorization that identifies the important columns for constructing the localized basis set. In this paper, we develop a two-stage approximate column selection strategy to find the important columns at much lower computational cost. We demonstrate the effectiveness of this process using the dissociation process of a BH$_{3}$NH$_{3}$ molecule, an alkane chain, and a supercell with $256$ water molecules. Numerical results for the large collection of water molecules show that the two-stage localization procedure can be more than 30 times faster than the original SCDM algorithm and compares favorably with the popular Wannier90 package.