Tensor numerical methods in quantum chemistry: from Hartree–Fock to excitation energies

Abstract

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.

Quantized CP approximation and sparse tensor interpolation of function‐generated data

“…We refer to reviews on the multilinear algebra 23,[39][40][41][42] and to recent surveys and monographs on tensor numerical methods and their application in scientific computing. 16,[43][44][45][46] The rest of this paper is organized as follows. Some auxiliary technical results concerning the ALS-canonical algorithm are presented in Section 2.…”

Range-Separated Tensor Format for Many-Particle Modeling

“…Similarly to [33], where the case of lattice-structured systems was analyzed, we show that the interior sum in (4.4) can be obtained from the tensor P 0 traced onto the centers of particles x k , where the term corresponding to x j = x k is removed:…”

“…Consider the calculation of the interaction energy for a charged multiparticle system. In the case of lattice-structured systems, the fast tensor-based computation scheme for the interaction energy was described in [33]. Recall that the interaction energy of the total electrostatic potential generated by the system of N charged particles located at x k ∈ R 3 (k = 1, .…”

“…In particular, such tensor approximations proved to be efficient for real-space calculations in computational quantum chemistry [32,33].…”

Rank Structured Approximation Method for Quasi-Periodic Elliptic Problems

“…Literature survey on the modern tensor numerical methods for multi-dimensional PDEs can be found in [13,14,16]. In the context of problems considered in the paper, we are mainly concerned with another specific feature: very complicated material structure.…”