Computational chemists are constantly seeking faster, more intuitive methods to analyze the complex electronic structures that govern how molecules behave. To meet this challenge, one of the primary research goals of the Laura Gagliardi Group at the University of Chicago has been to automate the analysis of molecular orbitals (MOs)—the quantum-level electron structures critical for understanding bonding and reactivity. The ability to automate this analysis accurately directly dictates how effectively chemists can design new materials and discover new drugs.
This critical area of research is now receiving a powerful boost from research on the Cartesian Equivariant Orbital Network (CEONet), a new AI tool developed by Daniel King. King began this work as a graduate student in the Gagliardi Group, collaborating with the Ian Foster Group at the University of Chicago and, more recently, with Bingqing Cheng at UC Berkeley, where King is now a postdoctoral fellow. Ultimately, CEONet offers a major step forward, providing a more rapid and insightful approach to the foundational analysis of molecular electronic structure.
Solving the Parity Problem with Physics-Aware AI
Without careful construction, standard AI tools struggle to adhere to core quantum rules such as symmetry. In this project, a primary hurdle was orbital parity: a fundamental quantum rule that says flipping the mathematical sign of an orbital doesn't change its physical reality. To an AI, however, an orbital and its sign-flipped counterpart look like two entirely different sets of numerical inputs (one positive, one negative). This ambiguity causes instability, forcing the model to try and predict two different outcomes for what is physically the same state.
King, the first author of the publication, explains the dilemma: "In quantum mechanics, the orbitals famously always have at least two answers. So you need to design your model in a way so that those two inputs reach the same answer."
The researchers solved the problem by designing their CEONet model with equivariant deep learning, a method that builds "physics-awareness" directly into the AI architecture. This means the fundamental rules of nature—like knowing a molecule's properties don't change just because you rotate it—are hardwired into its code, allowing it to master the sign-flip problem and achieve physically consistent results.
Intuitive and Accurate
CEONet’s mastery of the sign problem achieved a major functional goal: predicting complex molecular properties, such as orbital energy, with chemical accuracy. The model's utility lies in developing superhuman intuition about molecular orbitals.
King highlights this unique feature, noting the AI’s speed and insight compared to an expert scientist: "What's nice about this model is it evaluates these functions at a glance. A chemist would spend time to first visualize these orbitals and then predict properties. CEONet infers this complicated operation just from quickly looking at the orbital itself.”
This insight is essential for predicting the behavior of catalysts relevant to CD4DC and the reactivity of industrially important compounds. Moreover, it directly enables the model's most powerful application: accelerating complex quantum methods. CEONet accurately predicts orbital entropy, which flags the most complex, unpredictable electronic behavior. This is vital for automating the initial time-consuming step of quantum calculations.
"I think this is something that you could easily integrate into predictions to automate the interpretation of an electronic structure at large scale,” says King.
King hopes CEONet will act as a significant leap toward broader automation, establishing a clear path toward routinely applying advanced, formerly inaccessible electronic structure methods at scale in quantum chemistry.
This research is funded by the EFRC, Catalysts Design for Decarbonization Energy (CD4DC).
Citation: D.S. King, D. Grzenda, R. Zhu, N. Hudson, I. Foster, B. Cheng, & L. Gagliardi, Cartesian equivariant representations for learning and understanding molecular orbitals, Proc. Natl. Acad. Sci. U.S.A. 122 (48) e2510235122, https://doi.org/10.1073/pnas.2510235122 (2025).