Black-Box Hartree–Fock Solver by Tensor Numerical Methods

Khoromskaia, Venera

doi:10.1515/cmam-2013-0023

Cited by 17 publications

(39 citation statements)

References 44 publications

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“…In this section we present numerical illustrations to the reduced basis approach for the BSE problem, which use the TEI tensor and molecular orbitals obtained from the solution of the Hartree-Fock equation by the 3D grid-based tensor-structured method [14,15]. All examples below utilize the grid representation of the Galerkin basis functions from Gaussian basis sets of type cc-pDVZ.…”

Section: Numerics For the Reduced Basis Methodsmentioning

confidence: 99%

“…For ab-initio electronic structure calculations we use the tensor-structured Hartree-Fock solver[16,14] implemented in Matlab, employing the rank-structured calculation of the core Hamiltonian and TEI, using the discrete representation of basis functions on n × n × n 3D Cartesian grids. The arising 3D convolution integrals with the Newton kernel are replaced by algebraic operations in 1D complexity.…”

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

2016

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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 ε-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. ). 1DFT or the Hartree-Fock calculations do not allow reliable estimates for excitation energies of molecular structures. One of the approaches providing means for calculation of the excited states in molecules and solids is based on the solution of the Bethe-Salpeter equation (BSE) [27,23,25,28,29]. The alternative methods treat the problem by using the time-dependent DFT or Green's function approach [30,10,6,31,25,29]. The approximate coupled cluster calculations of electronic excitation energies by using rank decompositions have been described in [13].The BSE model, originating from high energy physics and incorporating the many-body perturbation theory and the Green's function formalism, governs calculation of the excited states in a self-consistent way. The BSE approach leads to the challenging computational task on the solution of the eigenvalue problem for a large fully populated matrix, that is in general non-symmetric. It is worth to note that the size of BSE matrix scales quadratically in the size of the basis set, O(N 2 b ), used in ab initio electronic structure calculations. Hence the direct diagonalization is limited by O(N 6 b ) complexity making the problem computationally extensive already for moderate size molecules with the size of the atomic orbitals basis set, N b ≈ 100. Furthermore, the numerical calculation of the matrix elements, based on the precomputed two-electron integrals (TEI) in the Hartree-Fock molecular orbitals basis, has the numerical cost that scales at least as O(N 4...

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Section: Numerics For the Reduced Basis Methodsmentioning

confidence: 99%

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confidence: 99%

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

2016

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“…One of the important results in these developments was the efficient method for summation of the electrostatic potentials in molecular systems. The starting point was the grid based computation of the nuclear potential operator for molecules in the framework of the Hartree-Fock equation [18,15], where the canonical tensor representation of the Newton kernel was applied and the controllable accuracy is provided due to large 3D Cartesian grids. The tensor-structured representation of the Coulomb potential became especially advantageous in grid-based assembled summation of the long-range potentials on large lattices.…”

Section: Tensor Representation Of Multiparticle Interaction Potentialsmentioning

confidence: 99%

Range-separated tensor decomposition of the discretized Dirac delta and elliptic operator inverse

Khoromskij

2020

Journal of Computational Physics

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“…Given the factorized TEI, the update of the Column and exchange parts in the Fock matrix reduces to the cheap algebraic operations. Other steps are tensor calculation of the core Hamiltonian and the efficient MP2 energy correction scheme [44], which all together gave rise to the black-box Hartree-Fock solver [45].…”

Section: Introductionmentioning

confidence: 99%

Tensor numerical methods in quantum chemistry: from Hartree–Fock to excitation energies

Khoromskaia

Khoromskij

2015

Phys. Chem. Chem. Phys.

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We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, first appeared as an accurate tensor calculus for the 3D Hartree potential using 1D complexity operations, and have evolved to entirely grid-based tensor-structured 3D Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core Hamiltonian and two-electron integrals (TEI) in O(n log n) complexity using the rank-structured approximation of basis functions, electron densities and convolution integral operators all represented on 3D n × n × n Cartesian grids. The algorithm for calculating TEI tensor in a form of the Cholesky decomposition is based on multiple factorizations using algebraic 1D "density fitting" scheme, which yield an almost irreducible number of product basis functions involved in the 3D convolution integrals, depending on a threshold ε > 0. The basis functions are not restricted to separable Gaussians, since the analytical integration is substituted by high-precision tensor-structured numerical quadratures. The tensor approaches to post-Hartree-Fock calculations for the MP2 energy correction and for the Bethe-Salpeter excitation energies, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is towards the tensor-based Hartree-Fock numerical scheme for finite lattices, where one of the numerical challenges is the summation of electrostatic potentials of a large number of nuclei. The 3D grid-based tensor method for calculation of a potential sum on a L × L × L lattice manifests the linear in L computational work, O(L), instead of the usual O(L(3) log L) scaling by the Ewald-type approaches.

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Black-Box Hartree–Fock Solver by Tensor Numerical Methods

Cited by 17 publications

References 44 publications

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

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

Range-separated tensor decomposition of the discretized Dirac delta and elliptic operator inverse

Tensor numerical methods in quantum chemistry: from Hartree–Fock to excitation energies

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