Messages in MACE can be of arbitrary body order, constructed in a computationally efficient way based on Atomic Cluster Expansion framework. As a result, MACE models can reach a given accuracy with fewer message-passing iterations than if conventional 2-body messages were used. Spherical tensors of arbitrary order are used as messages further improving the expressivity of the structure description and thus the model's performance. Detailed equations defining MACE architecture may be found in reference 55. Key MACE parameters for the model used in this work are reported in Table .