Conjugate gradient density matrix search: A linear scaling alternative to diagonalization

Abstract

Advances in the computation of the Coulomb, exchange, and correlation contributions to Gaussian-based Hartree-Fock and density functional theory Hamiltonians have demonstrated near-linear scaling with molecular size for these steps. These advances leave the ${\cal O}(N\sp3)$ diagonalization bottleneck as the rate determining step for very large systems. In this work, a conjugate gradient density matrix search (CG-DMS) method has been successfully extended and computationally implemented for use with first principles calculations. A Cholesky decomposition of the overlap matrix and its inverse, which can be formed in near linear time for sparse systems, is used to transform to and back from an orthonormal basis. Linear scaling of CPU time for the density matrix search and crossover of CPU time with diagonalization is demonstrated for polyglycine chains containing up to 493 atoms and water clusters up to 900 atoms.