Preconditioned conjugate gradient method for the sparse generalized eigenvalue problem in electronic structure calculations

Abstract

The use of localized basis sets is essential in linear-scaling electronic structure calculations, and since such basis sets are mostly non-orthogonal, it is necessary to solve the generalized eigenvalue problem Hx=εSx. In this work, an iterative method for finding the lowest few eigenvalues and corresponding eigenvectors for the generalized eigenvalue problem based on the conjugate gradient method is presented. The method is applied to first-principles electronic structure calculations within density-functional theory using a localized spherical-wave basis set, first introduced in the context of linear-scaling methods [Comput. Phys. Commun. 102 (1997) 17]. The method exhibits linear convergence of the solution, the rate of which is improved by a preconditioning scheme using the kinetic energy matrix.