Solution of the perturbed eigenvalue equation by the low-rank perturbation method

Abstract

In its simplest form, the low-rank perturbation (LRP) method solves the perturbed matrix eigenvalue equationsAΨ ≡ (B + V)Ψ =εΨ, whereA, B andV are nth-order Hermitian matrices, and where the eigenstates and the eigenvalues of the unperturbed matrixB are known. The method can be applied to arbitrary perturbations V, but it is numerically most efficient if the rank ρ ofV is “small”. A special case of low-rank perturbations are localized perturbations (e.g. replacement of one atom with another, creation and destruction of a chemical bond, local interaction of large molecules, etc.). In the case of local perturbations with a fixed localizability I, the operation count for the calculation of a single eigenvalue and/or a single eigenstate isO(l ² n). In the more general case of a delocalized perturbation with a fixed rank p, the operation count for the derivation of all eigenvalues and/or all eigenstates is O(π ² n ²). For largen, the performance of the LRP method is hence at least one order of magnitude better than the performance of other methods. The obtained numerical results demonstrate that the LRP method is numerically reliable, and that the performance of the method is in accord with predicted operation counts.