Spatial Signatures of Electron Correlation in Least-Squares Tensor Hyper-Contraction

TL;DR

This paper identifies a consistent error pattern in LS-THC tensor factorization related to electron correlation, revealing a missing pair-point kernel crucial for accurate wavefunction representation in quantum chemistry.

Contribution

The study introduces a novel theoretical framework to analyze LS-THC errors and uncovers a correlation feature indicating a missing pair-point kernel in the method.

Findings

Discovered a correlation feature in LS-THC errors across multiple molecules and basis sets.

Identified the importance of pairs of grid points near atoms within 1-2 Bohr radii.

Proposed directions for future LS-THC improvements based on these insights.

Abstract

Least Squares Tensor Hypercontraction (LS-THC) has received some attention in recent years as an approach to reduce the significant computational costs of wavefunction based methods in quantum chemistry. However, previous work has demonstrated that the LS-THC factorization performs disproportionately worse in the description of wavefunction components (e.g. cluster amplitudes $\hat{T}_2$) than Hamiltonian components (e.g. electron repulsion integrals $(pq|rs)$). This work develops novel theoretical methods to study the source of these errors in the context of the real-space $\hat{T}_2$ kernel, and reports, for the first time, the existence of a "correlation feature" in the errors of the LS-THC representation of the "exchange-like" correlation energy $E_X$ and $\hat{T}_2$ that is remarkably consistent across ten molecular species, three correlated wavefunctions, and four basis sets. This correlation feature portends the existence of a "pair-point kernel" missing in the usual LS-THC representation of the wavefunction, which critically depends upon pairs of grid points situated close to atoms and with inter-pair distances between one and two Bohr radii. These findings point the way for future LS-THC developments to address these shortcomings.