Novel tensor numerical methods are based on representation of d-variate functions and opera-tors on large n?d grids in the rank-structured tensor formats which provide O(dn) complexityof numerical calculations instead of O(nd) by conventional methods. A starting point wasour Hartree-Fock solver based on the tensor-structured calculation of the two-electron inte-grals, the Laplace and nuclear potential operators, using a Gausssian-type basis, discretizedon n × n × n 3D Cartesian grids [1-4]. The low-rank tensor representations of the Newtonkernel and electron density enable a 3D grid-based tensor convolution in O(n log n) complex-ity . The two-electron integrals tensor is computed in a form of Cholesky factorization by”1D density fitting“ . Efficient algorithms are developed for reducing the ranks of canonicaltensors at necessary steps. In calculation of all 3D operators in the Hartree-Fock equation the3D analytical integration is completely avoided, since it is substituted by the grid-based ten-sor algorithms of 1D complexity. This enables calculations in Matlab using 3D grids of sizeof the order of 1015. Tensor factorization of the two-electron integrals was a prerequisite forthe recent efficient method for calculation of the excitation energies for molecules by iterativesolution of the Bethe-Salpeter eigenvalue problem .
Further developments concern crystalline systems, where one of t