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

@article{Benner2016ARB,
title={A reduced basis approach for calculation of the Bethe–Salpeter excitation energies by using low-rank tensor factorisations*},
author={Peter Benner and Venera Khoromskaia and Boris N. Khoromskij},
journal={Molecular Physics},
year={2016},
volume={114},
pages={1148 - 1161}
}
• Published 2016
• Physics, Mathematics
• Molecular Physics
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… Expand
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#### References

SHOWING 1-10 OF 97 REFERENCES
Tensor numerical methods in quantum chemistry: from Hartree-Fock to excitation energies.
• Physics, Medicine
• Physical chemistry chemical physics : PCCP
• 2015
The grid-based tensor numerical methods, based on the low-rank representation of the multidimensional functions and integral operators, are resumed and their prospects in real-space electronic structure calculations are discussed. Expand
Møller-Plesset (MP2) energy correction using tensor factorization of the grid-based two-electron integrals
• Mathematics, Computer Science
• Comput. Phys. Commun.
• 2014
A tensor-structured method for calculating the Moller–Plesset (MP2) correction to the Hartree–Fock energy with reduced computational cost and the so-called quantized approximation of the long skeleton vectors comprising tensor factorizations of the main entities allows a reduction in storage costs. Expand
Electronic excitation energies of molecular systems from the Bethe-Salpeter equation: Example of the H2 molecule
• Chemistry, Physics
• 2013
We review the Bethe-Salpeter equation (BSE) approach to the calculation of electronic excitation energies of molecular systems. We recall the general Green's function many-theory formalism and giveExpand
Black-Box Hartree–Fock Solver by Tensor Numerical Methods
• V. Khoromskaia
• Mathematics, Computer Science
• Comput. Methods Appl. Math.
• 2014
This paper presents ab initio Hartree–Fock calculations of the ground state energy for the amino acid molecules, and of the “energy bands” for the model examples of extended (quasi-periodic) systems. Expand
NEW METHOD FOR CALCULATING THE ONE-PARTICLE GREEN'S FUNCTION WITH APPLICATION TO THE ELECTRON-GAS PROBLEM
A set of successively more accurate self-consistent equations for the one-electron Green's function have been derived. They correspond to an expansion in a screened potential rather than the bareExpand
Tensor-Structured Factorized Calculation of Two-Electron Integrals in a General Basis
• Mathematics, Computer Science
• SIAM J. Sci. Comput.
• 2013
The novel multiple tensor factorizations of the TEI unfolding matrix are introduced which decrease the computational demands for the evaluation of TEI in several aspects. Expand
Benchmark Many-Body GW and Bethe-Salpeter Calculations for Small Transition Metal Molecules.
• Chemistry, Medicine
• Journal of chemical theory and computation
• 2014
Calculations validate the accuracy of the parameter-free GW and Bethe-Salpeter formalisms for this class of systems, opening the way to the study of large clusters containing transition metal atoms of interest for photovoltaic applications. Expand
Electronic excitations: density-functional versus many-body Green's-function approaches
• Physics
• 2002
Electronic excitations lie at the origin of most of the commonly measured spectra. However, the first-principles computation of excited states requires a larger effort than ground-state calculations,Expand
An efficient implementation of time-dependent density-functional theory for the calculation of excitation energies of large molecules
• Chemistry
• 1998
Time-dependent density-functional (TDDFT) methods are applied within the adiabatic approximation to a series of molecules including C70. Our implementation provides an efficient approach for treatingExpand
Universal basis sets and Cholesky decomposition of the two-electron integral matrix
Abstract It is shown that, for a one-centre universal even-tempered basis set containing N functions, only N of the N(N + 1)/2 eigenvalues of the two-electron integral matrix are larger than someExpand