A reduced basis approach for calculation of the Bethe-Salpeter excitation energies by using low-rank tensor factorisations

Abstract

The Bethe-Salpeter equation (BSE) is a reliable model for estimating the absorption spectra in molecules and solids on the basis of accurate calculation of the excited states from first principles. Direct diagonalisation of the BSE matrix is practically intractable due to O(N6) complexity scaling in the size of the atomic orbital basis set, N. In this paper, we introduce and analyse a reduced basis approach to computation of the Bethe-Salpeter excitation energies which can lead to a relaxation of the numerical costs down to O(N3). The BSE operator is specified in terms of the two-electron integrals in the Hartree-Fock molecular orbital basis and the respective energies, calculated by the tensor-based solver described in previous works. The reduced basis method includes two steps. First, the diagonal plus low-rank approximation to fully populated blocks in the BSE matrix is calculated, enabling an easier partial eigenvalue solver for a large auxiliary system relying only on matrix-vector multiplications with rank-structured matrices. Second, a small subset of eigenvectors from the auxiliary eigenvalue problem is selected to build a projection of the exact BSE system onto this reduced basis set. Numerical tests demonstrate the ϵ-rank bounds for the blocks of the BSE matrix on examples of some compact molecules. The accuracy of the reduced basis approach vs. the effective matrix rank is illustrated.