Tensor-structured factorized calculation of two-electron integrals in a general basis
Venera Khoromskaia, Boris N. Khoromskij, and Reinhold Schneider
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Submission date: 22. May. 2012 (revised version: November 2012)
MSC-Numbers: 65F30, 65F50, 65N35, 65F10
Keywords and phrases: Hartree-Fock equation, Coloumb and exchange matrices, two-electron integrals, tensor approximation, truncated Cholesky decomposition, reduced higher order SVD
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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(Nb2) of 3D convolutions, defined over cross-products of Nb basis functions, to O(Nb) scaling. The tensor-structured numerical integration with the 3D Newton convolving kernel is performed in 1D complexity, thus enabling high resolution over fine 3D 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 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 1D samplings on a tensor grid. Numerical tests for some moderate size compact molecules demonstrate the expected asymptotic performance.