Tensor hypercontraction. II. Least-squares renormalization - Parrish, Robert M.; Hohenstein, Edward G.; Martínez, Todd J.; Sherrill, C. David (The Journal of chemical physics, 2012)
The least-squares tensor hypercontraction (LS-THC) representation for the electron repulsion integral (ERI) tensor is presented. Recently, we developed the generic tensor hypercontraction (THC) ansatz, which represents the fourth-order ERI tensor as a product of five second-order tensors [E. G. Hohenstein, R. M. Parrish, and T. J. Martínez, J. Chem. Phys. 137, 044103 (2012)]10.1063/1.4732310. Our initial algorithm for the generation of the THC factors involved a two-sided invocation of overlap-metric density fitting, followed by a PARAFAC decomposition, and is denoted PARAFAC tensor hypercontraction (PF-THC). LS-THC supersedes PF-THC by producing the THC factors through a least-squares renormalization of a spatial quadrature over the otherwise singular 1/r12 operator. Remarkably, an analytical and simple formula for the LS-THC factors exists. Using this formula, the factors may be generated with \documentclass[12pt]{minimal}\begin{document}${\cal O}(N^5)$\end{document}O(N5) effort if exact integrals are decomposed, or \documentclass[12pt]{minimal}\begin{document}${\cal O}(N^4)$\end{document}O(N4) effort if the decomposition is applied to density-fitted integrals, using any choice of density fitting metric. The accuracy of LS-THC is explored for a range of systems using both conventional and density-fitted integrals in the context of MP2. The grid fitting error is found to be negligible even for extremely sparse spatial quadrature grids. For the case of density-fitted integrals, the additional error incurred by the grid fitting step is generally markedly smaller than the underlying Coulomb-metric density fitting error. The present results, coupled with our previously published factorizations of MP2 and MP3, provide an efficient, robust \documentclass[12pt]{minimal}\begin{document}${\cal O}(N^4)$\end{document}O(N4) approach to both methods. Moreover, LS-THC is generally applicable to many other methods in quantum chemistry.