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

TL;DR

This paper introduces a reduced basis method using low-rank tensor factorizations to efficiently compute Bethe-Salpeter excitation energies, significantly lowering computational costs while maintaining accuracy.

Contribution

The authors develop a novel reduced basis approach that approximates the BSE matrix with low-rank tensor blocks, enabling faster eigenvalue computations for excited states.

Findings

Achieves up to $O(N^3)$ computational complexity

Accurately computes low-lying excitation energies

Validates approach on molecular BSE calculations

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. 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 relying only on matrix-vector multiplications with rank-structured matrices. And second, a small subset of eigenfunctions from the auxiliary eigenvalue problem is selected to build the Galerkin projection of the exact BSE system onto the reduced basis set. We present numerical tests on BSE calculations for a number of molecules confirming the $\varepsilon$-rank bounds for the blocks of BSE matrix. The numerics indicates that the reduced BSE eigenvalue problem with small matrices enables calculation of the lowest part of the excitation spectrum with sufficient accuracy.