On the effective rank of canonical polyadic decomposition of electron repulsion integrals

TL;DR

This paper investigates the effective rank of the canonical polyadic decomposition of electron repulsion integrals, revealing it cannot grow linearly with system size and providing a lower bound related to the number of atomic orbitals.

Contribution

It establishes a mathematical lower bound for the effective rank of CPD in quantum chemistry, challenging assumptions of linear growth with system size.

Findings

Effective rank does not grow linearly with system size.

Derived a lower bound proportional to N_AO^2 / log^7 N_AO.

Linear relationship between rank and N_AO cannot be universally assumed.

Abstract

In this paper, we study the effective rank of the canonical polyadic decomposition applied to the electron repulsion integrals, ubiquitous in quantum chemistry. We demonstrate, both mathematically and numerically, that in general the effective rank of this decomposition cannot grow linearly as a function of the system size. Moreover, we derive a lower bound for the effective rank in the form $\propto N_{\mathrm{AO}}^2/\log_2^7 N_{\mathrm{AO}}$, where $N_{\mathrm{AO}}$ is the number of atomic orbitals in the molecule, under mild conditions imposed on the decomposition threshold $\epsilon$. As a result, while a subquadratic growth of the CPD rank is not excluded, a linear relationship between the rank and $N_{\mathrm{AO}}$ cannot hold universally. The implications of these findings for the use of the canonical polyadic format to represent electron repulsion integrals in quantum chemistry are analyzed.