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

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 $\epsilon$-rank bounds for the blocks of BSE matrix. The numerics indicates that the reduced BSE eigenvalue problem with small matrices enables calculation of the lowest part of the excitation spectrum with sufficient accuracy.

References in corpus (4)

• Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States
• Structure Preserving Parallel Algorithms for Solving the Bethe-Salpeter Eigenvalue Problem
• Calculating excitation energies by extrapolation along adiabatic connections
• Electronic excitation energies of molecular systems from the Bethe-Salpeter equation: Example of the H2 molecule

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• Fast optical absorption spectra calculations for periodic solid state systems
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• Range-separated tensor formats for numerical modeling of many-particle interaction potentials
• Accelerating Optical Absorption Spectra and Exciton Energy Computation for Nanosystems via Interpolative Separable Density Fitting
• Computing the density of states for optical spectra by low-rank and QTT tensor approximation
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• Doubling algorithm for the discretized Bethe-Salpeter eigenvalue problem