Atomic Cluster Expansion (ACE)

ACE constructs the total energy of a system as a hierarchical sum of 1-body, 2-body, 3-body, and higher-order contributions. This body-ordered expansion is systematically complete, meaning it can represent any local atomic environment to arbitrary accuracy by including terms up to a chosen correlation order ν.

• Linear in the basis: Once the ACE basis is constructed, the model is linear in its coefficients, enabling rapid fitting via linear regression.
• Controlled truncation: The expansion can be truncated at a specific body order to balance accuracy and computational cost.
• Explicit many-body terms: Unlike message-passing networks that implicitly learn many-body correlations, ACE explicitly parameterizes them.

ACE represents the local atomic density around a central atom using a complete basis of permutationally invariant functions. The atomic density is expanded in a product basis of radial functions and spherical harmonics, and then symmetrized to ensure invariance under permutations of like atoms.

• Atomic base: The one-particle basis is formed by the tensor product of radial basis functions R_nl(r) and spherical harmonics Y_lm(θ, φ).
• Product basis: Higher-order correlations are built by taking tensor products of the atomic base and contracting them to invariants using Clebsch-Gordan coefficients.
• No data-dependent filters: The basis is pre-defined and complete, unlike learned filters in equivariant message-passing networks.

A defining advantage of ACE is that it achieves the accuracy of complex non-linear models while retaining the computational efficiency of a linear model. Once the invariant basis functions B_i are evaluated for an atomic environment, the site energy is simply a linear combination: E = Σ_i c_i B_i.

• Rapid fitting: Coefficients c_i can be solved via linear least-squares regression, avoiding expensive gradient-based optimization.
• Active learning compatibility: The linear form makes uncertainty quantification straightforward using Bayesian linear regression, enabling efficient active learning for training set construction.
• Evaluation speed: Inference requires only the evaluation of pre-defined basis functions and a dot product, making it orders of magnitude faster than deep equivariant models.

While the standard ACE yields invariant scalar features, the framework naturally extends to equivariant ACE for predicting vector and tensor properties such as atomic forces, dipole moments, and polarizabilities. This is achieved by retaining the covariant components of the tensor product rather than fully contracting them to scalars.

• Forces from gradients: Atomic forces are computed analytically as the negative gradient of the invariant energy with respect to atomic positions.
• Equivariant messages: The framework can be recast as an equivariant message-passing scheme, forming the foundation for the MACE architecture.
• Higher-order tensors: By controlling the maximum rotation order L in the basis, the model can represent properties of arbitrary tensorial rank.

ACE provides the mathematical foundation for several state-of-the-art interatomic potentials and descriptors. The MACE (Multi-ACE) architecture generalizes ACE by combining it with equivariant message passing and higher-order tensor products.

• MACE: A highly accurate equivariant potential that uses ACE as its atomic basis within a message-passing framework, achieving state-of-the-art accuracy on materials and molecular benchmarks.
• SOAP and Behler-Parrinello: ACE can be viewed as a systematic generalization of the Smooth Overlap of Atomic Positions (SOAP) descriptor and the Behler-Parrinello symmetry functions.
• Completeness guarantee: Unlike heuristic descriptors, ACE provides a rigorous completeness guarantee for representing local atomic environments.

The computational cost of ACE scales with the number of basis functions, which is controlled by three hyperparameters: the maximum radial basis index n_max, the maximum spherical harmonic degree l_max, and the maximum correlation order ν.

• Basis size control: The total number of invariant basis functions grows combinatorially with ν and l_max, but can be systematically controlled.
• Sparsity exploitation: Many basis functions contribute negligibly and can be pruned using regularization techniques like automatic relevance determination (ARD).
• Linear scaling with atoms: Once the basis is evaluated, the energy calculation scales linearly with the number of atoms in the system.