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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

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Volume 17, Issue 3

Issues

Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach

Venera Khoromskaia
  • Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig; and Max Planck Institute for Dynamics of Complex Systems, Magdeburg, Germany
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/ Boris N. Khoromskij
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  • Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig; and Max Planck Institute for Dynamics of Complex Systems, Magdeburg, Germany
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Published Online: 2017-05-31 | DOI: https://doi.org/10.1515/cmam-2017-0004

Abstract

This paper introduces and analyzes the new grid-based tensor approach to approximate solutions of the elliptic eigenvalue problem for the 3D lattice-structured systems. We consider the linearized Hartree–Fock equation over a spatial L1×L2×L3 lattice for both periodic and non-periodic problem setting, discretized in the localized Gaussian-type orbitals basis. In the periodic case, the Galerkin system matrix obeys a three-level block-circulant structure that allows the FFT-based diagonalization, while for the finite extended systems in a box (Dirichlet boundary conditions) we arrive at the perturbed block-Toeplitz representation providing fast matrix-vector multiplication and low storage size. The proposed grid-based tensor techniques manifest the twofold benefits: (a) the entries of the Fock matrix are computed by 1D operations using low-rank tensors represented on a 3D grid, (b) in the periodic case the low-rank tensor structure in the diagonal blocks of the Fock matrix in the Fourier space reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems in a box with Dirichlet boundary conditions are treated numerically by our previous tensor solver for single molecules, which makes possible calculations on rather large L1×L2×L3 lattices due to reduced numerical cost for 3D problems. The numerical simulations for both box-type and periodic L×1×1 lattice chain in a 3D rectangular “tube” with L up to several hundred confirm the theoretical complexity bounds for the block-structured eigenvalue solvers in the limit of large L.

Keywords: Tensor Structured Numerical Methods for PDEs; 3D Grid-Based Tensor Approximation, Hartree–Fock Equation; Linearized Fock Operator; Periodic Systems; Lattice Sum of Potentials; Block Circulant/Toeplitz Matrix; Fast Fourier Transform

MSC 2010: 65F30; 65F50; 65N35; 65F10

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About the article

Received: 2017-01-30

Revised: 2017-03-25

Accepted: 2017-03-27

Published Online: 2017-05-31

Published in Print: 2017-07-01


Citation Information: Computational Methods in Applied Mathematics, Volume 17, Issue 3, Pages 431–455, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2017-0004.

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