Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach
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| Title: | Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach |
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| Authors: | Khoromskaia, Venera, Khoromskij, Boris N. |
| Source: | Computational Methods in Applied Mathematics ; volume 17, issue 3, page 431-455 ; ISSN 1609-9389 1609-4840 |
| Publisher Information: | Walter de Gruyter GmbH |
| Publication Year: | 2017 |
| Description: | 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}\times L_{2}\times 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}\times L_{2}\times L_{3}} lattices due to reduced numerical cost for 3D problems. The numerical simulations for both box-type and periodic {L\times 1\times 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 . |
| Document Type: | article in journal/newspaper |
| Language: | English |
| DOI: | 10.1515/cmam-2017-0004 |
| DOI: | 10.1515/cmam-2017-0004/xml |
| DOI: | 10.1515/cmam-2017-0004/pdf |
| Availability: | https://doi.org/10.1515/cmam-2017-0004 https://www.degruyter.com/view/journals/cmam/17/3/article-p431.xml https://www.degruyter.com/document/doi/10.1515/cmam-2017-0004/xml https://www.degruyter.com/document/doi/10.1515/cmam-2017-0004/pdf |
| Accession Number: | edsbas.29757566 |
| Database: | BASE |
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