Preconditioned Conjugate Gradient Method for the Sparse Generalized Eigenvalue Problem in Electronic Structure Calculations

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 = "Sx. In this work, an iterative method for nd-ing 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 rst-principles electronic structure calculations within density-functional theory using a localized spherical-wave basis set, rst 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.