On the iterative diagonalization of matrices in quantum chemistry: Reconciling preconditioner design with Brillouin-Wigner perturbation theory

Iterative diagonalization of large matrices to search for a subset of eigenvalues that may be of interest has become routine throughout the field of quantum chemistry. Lanczos and Davidson algorithms hold a monopoly, in particular, owing to their excellent performance on diagonally dominant matrices. However, if the eigenvalues happen to be clustered inside overlapping Gershgorin disks, the convergence rate of both strategies can be noticeably degraded. In this work, we show how Davidson, Jacobi-Davidson, Lanczos, and preconditioned Lanczos correction vectors can be formulated using the reduced partitioning procedure, which takes advantage of the inherent flexibility promoted by Brillouin-Wigner perturbation (BW-PT) theory's resolvent operator. In doing so, we establish a connection between various preconditioning definitions and the BW-PT resolvent operator. Using Natural Localized Molecular Orbitals (NLMOs) to construct Configuration Interaction Singles (CIS) matrices, we study the impact the preconditioner choice has on the convergence rate for these comparatively dense matrices. We find that an attractive by-product of preconditioning the Lanczos algorithm is that the preconditioned variant only needs 21%-35% and 54%-61% of matrix-vector operations to extract the lowest energy solution of several Hartree-Fock- and NLMO-based CIS matrices, respectively. On the other hand, the standard Davidson preconditioning definition seems to be generally optimal in terms of requisite matrix-vector operations.