An orbital graph neural network reported low errors on three computational chemistry benchmarks: nitrogen-bond dissociation, an 18-species diatomic test and the energy gaps between spin states in five octahedral iron(II) complexes. In the iron test, at equilibrium geometries with the CAS(10,12) active-space setup, its mean absolute error, or average prediction error, was below 1 kcal per mole against CASPT2 references.
OrbGNN is a message-passing graph neural network in which molecular orbitals are nodes and pairwise orbital connectivity is encoded by edges. It implements three connectivity measures: mutual information, mutual correlation and the two-electron cumulant.
The model's targets were correlation energies, which describe linked electron behavior, for the diatomic systems and spin-state energy gaps for the iron systems. Orbital-connectivity features came from DMRG calculations, with the bond dimension set to 100 for diatomic molecules and 200 for Fe(II) complexes. The reference targets used CASSCF for the diatomics and CASPT2 for the iron systems.
Stretching the nitrogen bond
The N2 dissociation benchmark contained 500 geometries with bond lengths from 0.7 to 2.8 angstroms. The data were split 80% for training, 10% for validation and 10% for testing.
On the held-out N2 data, all three descriptors met the study's sub-chemical-accuracy threshold of less than 1 kJ per mole, or 0.38 mEh. The study reports errors in mEh, the energy unit used for these calculations. The cumulant had the lowest MAE at 0.2038 mEh, followed by mutual correlation at 0.2052 mEh and mutual information at 0.2473 mEh. The two-cumulant was selected for later OrbGNN development.
As the nitrogen bond was stretched, orbital connectivity generally increased across orbital pairs. Mutual information showed a decrease in entanglement around 1.9 angstroms, which the study interprets as marking the onset of the dissociative regime.
But the signal was not equally informative everywhere. Errors were elevated for bond lengths below approximately 1.1 angstroms, where the orbital-entanglement features were substantially reduced and less informative.
A wider diatomic test
The broader diatomic benchmark included 18 species and 100 geometries per species across the same 0.7-to-2.8-angstrom range. It used an 80/10/10 training, validation and test split.
Across those molecules, test errors were predominantly between 0.6 and 0.9 mEh. The overall MAE was 0.742 mEh with ANO-RCC-MB and 0.773 mEh with ANO-RCC-DZ, the two basis sets used. No individual molecule showed pronounced deterioration.
The diatomic benchmark used CASSCF reference targets, while the iron benchmark used CASPT2 targets. The two parts of the study therefore compared predictions with different quantum-chemical reference calculations.
A spin-state test for iron complexes
For the iron test, the researchers constructed 25 geometries covering five octahedral Fe(II) complexes with NH3, CO, HCN, CNH and PH3 ligands. The split was 90% for training and 10% for testing.
At equilibrium geometries with CAS(10,12), OrbGNN's spin-gap predictions agreed with CASPT2 references across all five complexes, with a mean absolute error below 1 kcal per mole. Here, CAS(10,12) denotes the active-space choice of 10 electrons in 12 orbitals.
The reported iron finding applies to that active-space and equilibrium-geometry setting, while the N2 caveat shows that performance varied within the dissociation scan.
Results tied to the tested benchmarks
Across the three benchmarks, all three N2 descriptors met the stated sub-chemical-accuracy threshold, diatomic test errors were generally 0.6 to 0.9 mEh, and the Fe(II) spin-gap MAE stayed below 1 kcal per mole.
The study says its software is open source and that the complete data and workflows needed to reproduce its figures are available in the same GitHub repository.
The work acknowledges National Science Foundation CAREER and CAS-Climate financial support, along with University of Tennessee ISAAC computational resources.
Paper data and sources
Original title: OrbGNN: A Wave function-based Machine Learning Interelectronic Representation
Authors: Brody Quebedeaux, Shahzad Akram, Markus Reiher, Konstantinos D. Vogiatzis
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
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