MiS Preprint Repository

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. This challenging task includes calculation of the BSE operator in terms of two-electron integrals tensor represented in molecular orbital basis, and introduces a complicated algebraic task of solving the arising large matrix eigenvalue problem. The direct diagonalization of the BSE matrix is practically intractable due to $O(N^6)$ complexity scaling in the size of the atomic orbitals basis set, $N$. In this paper, we present a new approach to the computation of Bethe-Salpeter excitation energies which can lead to relaxation of the numerical costs up to $O(N^3)$.

The idea is twofold: first, the diagonal plus low-rank tensor approximations to the fully populated blocks in the BSE matrix is constructed, enabling easier partial eigenvalue solver for a large auxiliary system but with a simplified block structure.

And second, a small subset of eigenfunctions from the auxiliary BSE problem is selected to solve the Galerkin projection of the initial spectral problem onto the reduced basis set. We present numerical tests on BSE calculations for a number of molecules confirming the $\epsilon$-rank bounds for the blocks of BSE matrix. The numerics indicates that the reduced BSE eigenvalue problem with small matrices enables calculation of the low part of the excitation spectrum with sufficient accuracy.