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This paper introduces and analyses the new grid-based tensor approach forapproximate solution of the eigenvalue problem for linearized Hartree-Fockequation applied to the 3D lattice-structured and periodic systems. The set oflocalized basis functions over spatial $(L_1,L_2,L_3)$ lattice in a boundingbox (or supercell) is assembled by multiple replicas of those from the unitcell. All basis functions and operators are discretized on a global 3D tensorgrid in the bounding box which enables rather general basis sets. In theperiodic case, the Galerkin Fock matrix is shown to have the three-level blockcirculant structure, that allows the FFT-based diagonalization. The proposedtensor techniques manifest the twofold benefits: (a) the entries of the Fockmatrix are computed by 1D operations using low-rank tensors represented on a 3Dgrid, (b) the low-rank tensor structure in the diagonal blocks of the Fockmatrix in the Fourier space reduces the conventional 3D FFT to the product of1D FFTs. We describe fast numerical algorithms for the block circulantrepresentation of the core Hamiltonian in the periodic setting based onlow-rank tensor representation of arising multidimensional functions. Latticetype systems in a box with open boundary conditions are treated by our previoustensor 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. Thenumerical simulations for box/periodic $(L,1,1)$ lattice systems in a 3Drectangular "tube" with $L$ up to several hundred confirm the theoreticalcomplexity bounds for the tensor-structured eigenvalue solvers in the limit oflarge $L$.

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