10. Tensor factorization of grid-based two-electron integrals

Mikaël Frosini, T. Duguet, Pierre Tamagno

Venera Khoromskaia, Boris N. Khoromskij

Venera Khoromskaia, Boris N. Khoromskij

We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, first appeared as an accurate tensor calculus for the 3D Hartree potential using 1D complexity operations, and have evolved to entirely grid-based tensor-structured 3D Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core Hamiltonian and two-electron integrals (TEI) in O(n log n) complexity using the rank-structured approximation of basis functions, electron densities and convolution integral operators all represented on 3D n × n × n Cartesian grids. The algorithm for calculating TEI tensor in a form of the Cholesky decomposition is based on multiple factorizations using algebraic 1D "density fitting" scheme, which yield an almost irreducible number of product basis functions involved in the 3D convolution integrals, depending on a threshold ε > 0. The basis functions are not restricted to separable Gaussians, since the analytical integration is substituted by high-precision tensor-structured numerical quadratures. The tensor approaches to post-Hartree-Fock calculations for the MP2 energy correction and for the Bethe-Salpeter excitation energies, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is towards the tensor-based Hartree-Fock numerical scheme for finite lattices, where one of the numerical challenges is the summation of electrostatic potentials of a large number of nuclei. The 3D grid-based tensor method for calculation of a potential sum on a L × L × L lattice manifests the linear in L computational work, O(L), instead of the usual O(L(3) log L) scaling by the Ewald-type approaches.

T. Daniel Crawford, Roberto Di Remigio

Heinz J. Schulz

Andreas Erbs Hillers‐Bendtsen, Lixin Lu, Todd J. Martı́nez

Tensor hypercontraction provides an attractive four-center two-electron repulsion integral format that can lower the scaling of many electronic structure methods while only requiring O(N^2) memory. However, in its grid-based least-squares incarnation, tensor hypercontraction requires the tedious design of compact spatial quadrature grids to achieve efficiency and accuracy, representing a bottleneck for widespread application. To simplify grid generation, we devise a reweighting scheme in which the grid weights are optimized to ensure accurate reproduction of the atomic orbital overlap matrix by numerical integration. By casting this fitting task as a non-negative least-squares problem, we obtain a black-box methodology that not only yields robust grids for tensor hypercontraction as well as numerical integration of other integrals but also prunes the grids by zeroing quadrature weights for insignificant points.