---
OA_place: publisher
OA_type: hybrid
_id: '20702'
abstract:
- lang: eng
text: Qualitative and quantitative orbital properties such as bonding/antibonding
character, localization, and orbital energies are critical to how chemists understand
reactivity, catalysis, and excited-state behavior. Despite this, representations
of orbitals in deep learning models have been very underdeveloped relative to
representations of molecular geometries and Hamiltonians. Here, we apply state-of-the-art
equivariant deep learning architectures to the task of assigning global labels
to orbitals, namely energies characterizations, given the molecular coefficients
from Hartree–Fock or density functional theory. The architecture we have developed,
the Cartesian Equivariant Orbital Network (CEONET), shows how molecular orbital
coefficients are readily featurized as equivariant node features common to all
graph-based machine-learned potentials. We find that CEONET performs well at predicting
difficult quantitative labels such as the orbital energy and orbital entropy.
Furthermore, we find that the CEONET representation provides an intuitive latent
space for differentiating orbital character for the qualitative assignment of
e.g. bonding or antibonding character. In addition to providing a useful representation
for further integrating deep learning with electronic structure theory, we expect
CEONET to be useful for automatizing and interpreting the results of advanced
electronic structure methods such as complete active space self-consistent field
theory. In particular, the ability of CEONET to infer multireference character
via the orbital entropy paves the way toward the machine-learned selection of
active spaces.
acknowledgement: This work is supported as part of the Catalyst Design for Decarbonization
Center, an Energy Frontier Research Center funded by the U.S. Department of Energy,
Office of Science, Basic Energy Sciences under award no. DE-SC0023383. We thank
the Research Computing Center at the University of Chicago and for access to computational
resources. Additionally, this research used the Savio computational cluster resource
provided by the Berkeley Research Computing program at the University of California
(UC), Berkeley (supported by the UC Berkeley Chancellor, Vice Chancellor for Research,
and Chief Information Officer). Furthermore, we thank Matthew Hennefarth and Matt
Hermes for useful discussions.
article_number: e2510235122
article_processing_charge: Yes (in subscription journal)
article_type: original
author:
- first_name: Daniel S.
full_name: King, Daniel S.
last_name: King
- first_name: Daniel
full_name: Grzenda, Daniel
last_name: Grzenda
- first_name: Ray
full_name: Zhu, Ray
last_name: Zhu
- first_name: Nathaniel
full_name: Hudson, Nathaniel
last_name: Hudson
- first_name: Ian
full_name: Foster, Ian
last_name: Foster
- first_name: Bingqing
full_name: Cheng, Bingqing
id: cbe3cda4-d82c-11eb-8dc7-8ff94289fcc9
last_name: Cheng
orcid: 0000-0002-3584-9632
- first_name: Laura
full_name: Gagliardi, Laura
last_name: Gagliardi
citation:
ama: King DS, Grzenda D, Zhu R, et al. Cartesian equivariant representations for
learning and understanding molecular orbitals. Proceedings of the National
Academy of Sciences. 2025;122(48). doi:10.1073/pnas.2510235122
apa: King, D. S., Grzenda, D., Zhu, R., Hudson, N., Foster, I., Cheng, B., &
Gagliardi, L. (2025). Cartesian equivariant representations for learning and understanding
molecular orbitals. Proceedings of the National Academy of Sciences. National
Academy of Sciences. https://doi.org/10.1073/pnas.2510235122
chicago: King, Daniel S., Daniel Grzenda, Ray Zhu, Nathaniel Hudson, Ian Foster,
Bingqing Cheng, and Laura Gagliardi. “Cartesian Equivariant Representations for
Learning and Understanding Molecular Orbitals.” Proceedings of the National
Academy of Sciences. National Academy of Sciences, 2025. https://doi.org/10.1073/pnas.2510235122.
ieee: D. S. King et al., “Cartesian equivariant representations for learning
and understanding molecular orbitals,” Proceedings of the National Academy
of Sciences, vol. 122, no. 48. National Academy of Sciences, 2025.
ista: King DS, Grzenda D, Zhu R, Hudson N, Foster I, Cheng B, Gagliardi L. 2025.
Cartesian equivariant representations for learning and understanding molecular
orbitals. Proceedings of the National Academy of Sciences. 122(48), e2510235122.
mla: King, Daniel S., et al. “Cartesian Equivariant Representations for Learning
and Understanding Molecular Orbitals.” Proceedings of the National Academy
of Sciences, vol. 122, no. 48, e2510235122, National Academy of Sciences,
2025, doi:10.1073/pnas.2510235122.
short: D.S. King, D. Grzenda, R. Zhu, N. Hudson, I. Foster, B. Cheng, L. Gagliardi,
Proceedings of the National Academy of Sciences 122 (2025).
corr_author: '1'
das_tickbox: '1'
dataavailabilitystatement: Code has been deposited to https://github.com/GagliardiGroup/CEONet
(83). Data has been deposited to https://doi.org/10.5281/zenodo.16934624 (84).
date_created: 2025-11-30T23:02:06Z
date_published: 2025-12-02T00:00:00Z
date_updated: 2026-08-07T10:27:53Z
day: '02'
ddc:
- '540'
department:
- _id: BiCh
doi: 10.1073/pnas.2510235122
external_id:
pmid:
- '41269783'
file:
- access_level: open_access
checksum: 58051539a884c7a97306fd3afdb539ac
content_type: application/pdf
creator: dernst
date_created: 2025-12-01T08:41:32Z
date_updated: 2025-12-01T08:41:32Z
file_id: '20719'
file_name: 2025_PNAS_King.pdf
file_size: 27607870
relation: main_file
success: 1
file_date_updated: 2025-12-01T08:41:32Z
has_accepted_license: '1'
intvolume: ' 122'
issue: '48'
language:
- iso: eng
license: https://creativecommons.org/licenses/by-nc-nd/4.0/
month: '12'
oa: 1
oa_version: Published Version
pmid: 1
publication: Proceedings of the National Academy of Sciences
publication_identifier:
eissn:
- 1091-6490
publication_status: published
publisher: National Academy of Sciences
quality_controlled: '1'
related_material:
link:
- relation: software
url: 'https://github.com/GagliardiGroup/CEONet '
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Cartesian equivariant representations for learning and understanding molecular
orbitals
tmp:
image: /images/cc_by_nc_nd.png
legal_code_url: https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
name: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
(CC BY-NC-ND 4.0)
short: CC BY-NC-ND (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 122
year: '2025'
...