Block circulant and Toeplitz structures in the linearized Hartree-Fock equation on finite lattices: tensor approach

Venera Khoromskaia and Boris N. Khoromskij

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Submission date: 30. Jan. 2017
Pages: 32
published in: Computational methods in applied mathematics, 17 (2017) 3, p. 431-455 
DOI number (of the published article): 10.1515/cmam-2017-0004
Bibtex
MSC-Numbers: 65F30, 65F50, 65N35, 65F10
Keywords and phrases: Tensor structured numerical methods for PDEs, 3D grid-based tensor approximation, Hartree-Fock equation, linearized Fock operator, periodic systems, block circulant/Toeplitz matrix, fast Fourier transform

Abstract:
This paper introduces and analyses the new grid-based tensor approach to approximate solution 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.

You can download the paper at http://arxiv.org/abs/1702.00339