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This paper investigates the effective rank of canonical polyadic decomposition (CPD) when applied to electron repulsion integrals in quantum chemistry calculations. The authors prove that the effective rank cannot grow linearly with system size and establish a lower bound proportional to N²ₐₒ/log₂⁷Nₐₒ, where Nₐₒ is the number of atomic orbitals. While subquadratic scaling is not ruled out, the results demonstrate that linear scaling between CPD rank and molecular size cannot hold universally.
Why it matters
These findings have important implications for computational quantum chemistry, as they establish fundamental limitations on how efficiently the CPD format can represent electron repulsion integrals in large molecular systems. Understanding these scaling limits is crucial for developing accurate and computationally tractable methods for studying complex chemical systems.
Understand the Science
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.
Source: On the effective rank of canonical polyadic decomposition of electron repulsion integrals