Spatial Signatures of Electron Correlation in Least-Squares Tensor Hypercontraction

Least-squares tensor hypercontraction (LS-THC) has received some attention in recent years as an approach to reduce the significant computational costs of wave function-based methods in quantum chemistry. However, previous work has demonstrated that LS-THC factorization performs disproportionately worse in the description of wave function components (e.g., cluster amplitudes <i>T̂</i>₂) than Hamiltonian components (e.g., electron repulsion integrals (<i>pq</i>|<i>rs</i>)). This work develops novel theoretical methods to study the source of these errors in the context of the real-space <i>T̂</i>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 <i>E</i>_<i>X</i> and <i>T̂</i>₂ that is remarkably consistent across ten molecular species, three correlated wave functions, and four basis sets. This correlation feature portends the existence of a "pair point kernel" missing in the usual LS-THC representation of the wave function, which critically depends upon pairs of grid points situated close to atoms and with interpair distances between one and two Bohr radii. These findings point the way for future LS-THC developments to address these shortcomings.