Cartesian equivariant representations for learning and understanding molecular orbitals

article Cartesian equivariant representations for learning and understanding molecular orbitals published yes Daniel S. King author Daniel Grzenda author Ray Zhu author Nathaniel Hudson author Ian Foster author Bingqing Cheng author cbe3cda4-d82c-11eb-8dc7-8ff94289fcc90000-0002-3584-9632 Laura Gagliardi author BiCh department 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. https://research-explorer.ista.ac.at/download/20702/20719/2025_PNAS_King.pdf application/pdfno National Academy of Sciences2025 eng Proceedings of the National Academy of Sciences 1091-6490 4126978310.1073/pnas.2510235122 12248 https://github.com/GagliardiGroup/CEONet 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. <a href="https://doi.org/10.1073/pnas.2510235122">https://doi.org/10.1073/pnas.2510235122</a>. 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. 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. 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:<a href="https://doi.org/10.1073/pnas.2510235122">10.1073/pnas.2510235122</a>. 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:<a href="https://doi.org/10.1073/pnas.2510235122">10.1073/pnas.2510235122</a> D.S. King, D. Grzenda, R. Zhu, N. Hudson, I. Foster, B. Cheng, L. Gagliardi, Proceedings of the National Academy of Sciences 122 (2025). 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. <a href="https://doi.org/10.1073/pnas.2510235122">https://doi.org/10.1073/pnas.2510235122</a> 207022025-11-30T23:02:06Z2026-02-16T12:31:49Z