Any MPNN, including MACE, is invariant under permutation of the same chemical species, using the same function for force predictions. Employing orbits expands the number of different chemical species in the models, increasing the number of distinct functions used. To benefit from such an increase, the training set needs to be large enough for every orbit to be well sampled; otherwise, the parameters of those functions cannot be accurately trained. As a demonstration, Figure (b) shows a relationship between the accuracy of different types of the MACE models and the total number of trainable parameters they have, where the variability in accuracy is depicted as error bars. The figure corresponds precisely to the same models as Figure (a). Notably, orbitbased MACE models achieve comparable accuracy with significantly fewer parameters than the default MACE models. For example, to reach a target accuracy of 0.4 kcal/(mol•Å), an orbit-based MACE model requires approximately 9 × 10 4 parameters. In contrast, the default MACE model, which relies on atomic species, requires around 7 × 10 5 parameters-an order of magnitude more.