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# Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

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Volume 17, Issue 3

# 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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• Other articles by this author:
/ Boris N. Khoromskij
• Corresponding author
• 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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• Other articles by this author:
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 ${L}_{1}×{L}_{2}×{L}_{3}$ 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 ${L}_{1}×{L}_{2}×{L}_{3}$ 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.

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

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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,

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