Spatial Signatures\nof Electron Correlation in Least-Squares\nTensor Hypercontraction

Least-squares tensor hypercontraction\n(LS-THC) has received\nsome\nattention in recent years as an approach to reduce the significant\ncomputational costs of wave function-based methods in quantum chemistry.\nHowever, previous work has demonstrated that LS-THC factorization\nperforms disproportionately worse in the description of wave function\ncomponents (e.g., cluster amplitudes <i>T̂</i><sub>2</sub>) than Hamiltonian components (e.g., electron repulsion integrals\n(<i>pq</i>|<i>rs</i>)). This work develops novel\ntheoretical methods to study the source of these errors in the context\nof the real-space <i>T̂</i><sub>2</sub> kernel, and\nreports, for the first time, the existence of a “correlation\nfeature” in the errors of the LS-THC representation of the\n“exchange-like” correlation energy <i>E</i><sub><i>X</i></sub> and <i>T̂</i><sub>2</sub> that is remarkably consistent across ten molecular species, three\ncorrelated wave functions, and four basis sets. This correlation feature\nportends the existence of a “pair point kernel” missing\nin the usual LS-THC representation of the wave function, which critically\ndepends upon pairs of grid points situated close to atoms and with\ninterpair distances between one and two Bohr radii. These findings\npoint the way for future LS-THC developments to address these shortcomings.