Direct optimization of the atomic-orbital density matrix using the conjugate-gradient method with a multilevel preconditioner

Abstract

Based on a recently proposed exponential parametrization of the one-electron atomic-orbital density matrix of a single-determinant wave function [Chem. Phys. Lett. 327, 397 (2000)], we present an implementation of the direct optimization of the atomic-orbital density matrix as an alternative to the diagonalization of the Fock–Kohn–Sham matrix when solving the Roothaan–Hall self-consistent field equations. The optimization of the density matrix is carried out by the conjugate-gradient method with a multilevel nondiagonal preconditioner and is well suited to linear scaling. Although a diagonal preconditioner may be sufficient for minimal basis sets and large highest occupied molecular orbital–lowest unoccupied molecular orbital (HOMO–LUMO) gaps, a nondiagonal preconditioner is needed in more difficult cases—that is, for basis sets with polarization and diffuse functions and for systems with small HOMO–LUMO gaps. Redundancies of the exponential parametrization of the density matrix are handled by a projection technique, thereby avoiding singular equations in the optimization of the density matrix.