Atomic Cluster Expansion Framework

• Atomic Cluster Expansion is a hierarchical, systematically improvable framework that constructs invariant basis sets to represent atomic environments and many-body interactions.
• The method underpins modern machine-learned interatomic potentials, driving advances in force field development and active learning for materials modeling.
• ACE decomposes total energy into site energies, expanding them via body-ordered N-atom clusters to achieve accurate and efficient simulations.

The Atomic Cluster Expansion (ACE) is a hierarchical, systematically improvable framework for constructing invariant, complete, and efficient basis sets for the representation of atomic environments and many-body interactions. ACE has become a central methodology in modern machine-learned interatomic potentials, with widespread adoption in force field development for materials modeling, active learning, electronic structure, and beyond (Ortner, 2023).

1. Mathematical Structure and Basis Construction

Atomic Cluster Expansion is founded on a decomposition of the total energy of a system into site energies: $E = \sum_i E_i$ Each site energy $E_i$ is formally expanded as a sum over all body-ordered $N$-atom clusters in the neighborhood of atom $i$: [ E_{i} = U_{0}(Z