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

## TL;DR

This paper presents a tensor-based grid approach for efficiently solving the linearized Hartree-Fock equation on 3D lattice systems, leveraging block-circulant and Toeplitz structures for computational speedup.

## Contribution

It introduces a novel tensor technique exploiting block-circulant and Toeplitz structures to efficiently solve 3D lattice Hartree-Fock problems with reduced computational complexity.

## Key findings

- Efficient FFT-based diagonalization for periodic systems.
- Fast matrix-vector multiplication for finite systems.
- Successful numerical validation on large lattice chains.

## 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 $L_1\times L_2\times L_3$ 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 $L$.