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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.
