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PreprintCC BY 4.0arXiv

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

  • Aleksandra Oszmian
  • Michał Lesiuk
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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 ∝ NAO2/log27 NAO, where NAO is the number of atomic orbitals in the molecule, under mild conditions imposed on the decomposition threshold ε. As a result, while a subquadratic growth of the CPD rank is not excluded, a linear relationship between the rank and NAO 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.

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Cite

BibTeXRIS
BibTeX
@misc{oszmian2026effective,
  title = {{On the effective rank of canonical polyadic decomposition of electron repulsion integrals}},
  author = {Aleksandra Oszmian and Michał Lesiuk},
  howpublished = {arXiv},
  year = {2026},
  month = jul,
  doi = {10.48550/arxiv.2605.14608},
  eprint = {2605.14608},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.14608},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the effective rank of canonical polyadic decomposition of electron repulsion integrals
AU  - Aleksandra Oszmian
AU  - Michał Lesiuk
PY  - 2026
DA  - 2026/07/21
PB  - arXiv
DO  - 10.48550/arxiv.2605.14608
UR  - https://arxiv.org/abs/2605.14608
AB  - 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 ∝ N_AO^2/log_2^7 N_AO, where N_AO is the number of atomic orbitals in the molecule, under mild conditions imposed on the decomposition threshold ε. As a result, while a subquadratic growth of the CPD rank is not excluded, a linear relationship between the rank and N_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.
ER  -