Tensor-Structured Factorized Calculation of Two-Electron Integrals in a General Basis

Abstract

In this paper, the problem of efficient grid-based computation of the two-electron integrals (TEI) in a general basis is considered. We introduce the novel multiple tensor factorizations of the TEI unfolding matrix which decrease the computational demands for the evaluation of TEI in several aspects. Using the reduced higher-order SVD the redundancy-free product-basis set is constructed that diminishes dramatically the initial number $O(N_b^2)$ of three-dimensional ($3$D) convolutions, defined over cross products of $N_b$ basis functions, to $O(N_b)$ scaling. The tensor-structured numerical integration with the $3$D Newton convolving kernel is performed in one-dimensional ($1$D) complexity, thus enabling high resolution over fine $3$D Cartesian grids. Furthermore, using the quantized approximation of long vectors ensures the logarithmic storage complexity in the grid size. Finally, we present and analyze two approaches to compute the Cholesky decomposition of the TEI matrix based on two types of precomputed factorizations. We show that further compression is possible via columnwise quantization of the Cholesky factors. Our “black-box” approach essentially relaxes limitations on the traditional Gaussian-type basis sets, giving an alternative choice of rather general low-rank basis functions represented only by their $1$D samplings on a tensor grid. Numerical tests for some moderate size compact molecules demonstrate the expected asymptotic performance.