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Tensor Numerical Approach to Linearized Hartree-Fock Equation for Lattice-type and Periodic Systems

This paper introduces and analyses the new grid-based tensor approach for approximate solution of the eigenvalue problem for linearized Hartree-Fock equation applied to the 3D lattice-structured and periodic systems. The set of localized basis functions over spatial $(L_1,L_2,L_3)$ lattice in a bounding box (or supercell) is assembled by multiple replicas of those from the unit cell. All basis functions and operators are discretized on a global 3D tensor grid in the bounding box which enables rather general basis sets. In the periodic case, the Galerkin Fock matrix is shown to have the three-level block circulant structure, that allows the FFT-based diagonalization. The proposed 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) 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. We describe fast numerical algorithms for the block circulant representation of the core Hamiltonian in the periodic setting based on low-rank tensor representation of arising multidimensional functions. Lattice type systems in a box with open boundary conditions are treated by our previous tensor solver for single molecules, which makes possible calculations on large $(L_1,L_2,L_3)$ lattices due to reduced numerical cost for 3D problems. The numerical simulations for box/periodic $(L,1,1)$ lattice systems in a 3D rectangular "tube" with $L$ up to several hundred confirm the theoretical complexity bounds for the tensor-structured eigenvalue solvers in the limit of large $L$.