  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 $L$.
