Atomic Cluster Expansion (ACE)

• Atomic Cluster Expansion (ACE) is a symmetry-adapted framework that decomposes the total energy into local atomic contributions using invariant many-body basis functions.
• ACE enables systematic and efficient fitting of quantum-mechanical data through regularization and active learning, ensuring high predictive accuracy.
• ACE is applied to a wide range of systems, from elemental metals and alloys to complex molecular and condensed matter structures, supporting scalable simulations.

The Atomic Cluster Expansion (ACE) is a systematically improvable, universally complete, and symmetry-adapted framework for representing the potential energy surface of many-atom systems. As a foundational tool in machine-learned interatomic potentials, ACE underpins recent advances in modeling accuracy and transferability for condensed matter, molecular systems, and even quantum wavefunctions. The core philosophy of ACE is to decompose the total energy into local atomic contributions, each parameterized as a linear (or sparsely nonlinear) combination of many-body rotationally, translationally, and permutationally invariant basis functions. This approach enables the systematic capture of arbitrary multi-atom correlations, efficient linear fitting to quantum-mechanical reference data, and robust extrapolation beyond the training set.

1. Mathematical Structure and Symmetry Adaptation

ACE expresses the total potential energy as

$E_{\text{tot}}\;=\;\sum_{i=1}^N E_i,$

where each atomic energy $E_i$ is expanded in a linear basis of many-body “cluster” functions that capture up to $\nu_{\max}$-body correlations among an atom’s neighbors. These local expansions take the generic form

$$E_i = \sum_{\nu=1}^{\nu_{\max}}\sum_{n_1\ldots n_\nu,\,l_1\ldots l_\nu} c_{n_1\ldots n_\nu; l_1\ldots l_\nu} \, B_{n_1\ldots n_\nu; l_1\ldots l_\nu}(i),$$

where $B_{n_1\ldots n_\nu; l_1\ldots l_\nu}(i)$ are scalar, rotationally invariant basis functions constructed as products of radial and angular basis projections over all $\nu$-tuples of neighbors of atom $i$. Radial basis functions (e.g., orthonormal polynomials, Chebyshev, or B-splines) resolve $r_{ij}$ up to a cutoff $r_{\rm cut}$, and angular dependence is encoded via (real) spherical harmonics $Y_{lm}$. These are coupled via Clebsch–Gordan algebra to obtain overall $l=0$ scalars, ensuring invariance under global rotations, translations, and permutation of like atoms (Ortner, 2023, Li et al., 2023).

The completeness and systematic improvability of ACE follow from its polynomial structure over neighbor densities; the expansion converges to any smooth, symmetry-adapted function given sufficiently large basis cutoffs in body order, radial, and angular indices.

2. Parameterization, Regularization, and Training

Fitting an ACE potential consists of generating a diverse set of atomic configurations via ab initio electronic structure methods (e.g. DFT), evaluating energies, forces, and (optionally) virials, and then solving a regularized least squares problem to determine the expansion coefficients. The loss function typically combines errors in total energy and atomic forces,

$$\mathcal{L} = \sum_k \Big[ w_E \big(E_k^{\text{ACE}} - E_k^{\text{DFT}}\big)^2 + w_F \sum_i \|\mathbf{F}_{ik}^{\text{ACE}} - \mathbf{F}_{ik}^{\text{DFT}}\|^2 \Big] + \lambda \|\mathbf{c}\|^2,$$

where $w_E$, $w_F$ are relative weights, and $\lambda$ is a Tikhonov regularizer to control overfitting given the typically large number of basis functions (Li et al., 2023, Ibrahim et al., 20 Jun 2024, Araki et al., 2023).

Active learning using extrapolation grades (D-optimality based "MaxVol" selection) enables iterative refinement of the training set by automatically identifying environments where the model is extrapolating and thus requires higher-fidelity quantum data (Lysogorskiy et al., 2022, Ibrahim et al., 20 Jun 2024, Wang et al., 18 Jun 2025). This efficiently extends coverage to rare or complex configurations, such as high-energy defects or out-of-equilibrium structures.

3. Practical Implementation in Materials Modeling

ACE has been deployed across an exceptionally broad variety of materials systems, including elemental metals (Mg, W, Cu, AlN) (Ibrahim et al., 2023, Pan et al., 1 Jul 2024, Yang et al., 2023), transition metal alloys (Fe–Co, Pt–Rh, Cu–W) (Li et al., 2023, Liang et al., 2023, Pan et al., 1 Jul 2024), oxides (BaTiO$_3$) (Grünebohm et al., 23 May 2025), carbon allotropes and layered materials (diamond, graphene, amorphous carbon, multilayer graphene) (Qamar et al., 2022, Wang et al., 18 Jun 2025), water and aqueous systems (Ibrahim et al., 20 Jun 2024), and lattices with multi-degree-of-freedom order parameters (e.g., magnetic ACE for Fe) (Rinaldi et al., 2023).

The following table summarizes the key basis and hyperparameter choices in recent ACE applications:

System	Body order ($\nu_{\max}$)	Radial cutoff [Å]	Total basis size	Reference
Mg (hcp/all phases)	4	8.2	724	(Ibrahim et al., 2023)
Cu–W (alloys)	4	7.0	$\sim$1000--10000	(Pan et al., 1 Jul 2024)
Water (ice, liquid)	5	6.0	1,774	(Ibrahim et al., 20 Jun 2024)
Fe–Co (alloy)	3--4	$\sim$6	$\sim$1,000	(Li et al., 2023)
Carbon (all phases)	5	5.0	488	(Qamar et al., 2022)
BaTiO$_3$ (ferroelectric)	4	6.0	%%%%18$w_F$19%%%%	(Grünebohm et al., 23 May 2025)

Energy and force RMSEs of ACE models, after sufficiently comprehensive parameterization, routinely fall below 1--5 meV/atom and 50 meV/Å, enabling predictive-quality molecular dynamics, free-energy calculations, and the computation of sensitive structural, vibrational, and thermomechanical properties (Ortner, 2023, Pan et al., 1 Jul 2024, Ibrahim et al., 20 Jun 2024).

4. Variants, Extensions, and Generalizations

ACE admits a hierarchy of extensions that broaden its applicability:

• Orthonormal/non-self-interacting ACE: By explicitly removing self-interaction terms (where the same neighbor is counted multiple times within a cluster basis function), improved numerical conditioning, coefficient decay, and regularization are achieved (Ho et al., 3 Jan 2024).
• Non-collinear Magnetic ACE: Simultaneous expansion in both atomic positions and vectorial/tensorial degrees of freedom (e.g., magnetic moments) enables accurate modeling of coupled spin-lattice dynamics and complex magnetic phase behavior (Rinaldi et al., 2023).
• Charge-constrained ACE: Augmenting the ACE basis with per-site scalar charges (and optionally higher multipoles), along with global Coulomb or QEq-type terms, captures long-range polarization and charge-transfer phenomena, allowing rigorous variational coupling between short-range ACE and global electrostatics (Rinaldi et al., 6 Nov 2024, Goff et al., 2023).
• Generalized properties: ACE expansions can fit not only scalar energies but also vector and tensor properties (dipoles, Born charges, stresses), operator matrix elements (tight-binding Hamiltonians), and can parameterize fermionic wavefunctions for quantum Monte Carlo with anti-symmetry enforced via determinantal factors (Drautz et al., 2022, Zhou et al., 2023, Ortner, 2023).
• Finnis–Sinclair-type nonlinear embeddings: Embedding densities and mildly nonlinear transformations (e.g., $E_i = A_i^{(1)} + \sqrt{A_i^{(2)}}$) improve performance in metallic systems but preserve linear fitting (Qamar et al., 2022, Ibrahim et al., 2023).

5. Computational Efficiency and Scalability

The computational cost of ACE scales linearly with the number of atoms $N$, and for fixed maximum body order, the dominant prefactor is set by the number of retained basis functions (often $10^3$--$10^4$). Production-level implementations (PACE, LAMMPS, ACE.jl) attain per-step costs only a few times above classical EAM/MEAM and several orders of magnitude below DFT or neural network potentials. This allows nanosecond-scale molecular dynamics simulations of $10^5$--$10^6$ atoms with DFT-quality accuracy (Li et al., 2023, Araki et al., 2023, Wang et al., 18 Jun 2025).

Active-learning-enabled workflow, robust regularization, and extrapolation grade metrics further facilitate deployment in regimes with incomplete training data, maintaining accuracy and ensuring robustness to extrapolation (Lysogorskiy et al., 2022, Ibrahim et al., 20 Jun 2024, Wang et al., 18 Jun 2025).

6. Limitations and Outlook

The main limitations of current ACE approaches include the potential combinatorial growth of feature sets at high body-order or angular truncation, leading to increased parameter count and possible numerical ill-conditioning. Recent work on orthogonal, non-self-interacting bases, basis sparsification using D-optimality and active learning, and smoothness-aware regularization partially mitigates these issues (Ho et al., 3 Jan 2024, Lysogorskiy et al., 2022, Ibrahim et al., 20 Jun 2024).

In ionic, highly polarizable, or long-range correlated materials, nonlocal physics such as Coulomb or van der Waals interactions must either be included via model extension (e.g., explicit charge models, additive D3 dispersion) or supplemented by hybrid approaches (Rinaldi et al., 6 Nov 2024, Grünebohm et al., 23 May 2025, Wang et al., 18 Jun 2025). For high-accuracy modeling of strongly correlated or open-shell systems, direct integration with quantum Hamiltonian parameterization or wavefunction techniques is ongoing.

ACE’s linear, universally complete, and symmetry-adapted methodology now provides a foundation for both general-purpose and highly specialized machine-learned potentials in chemistry, physics, and materials science. Ongoing research targets deeper integration with equivariant graph neural architectures, automated and on-the-fly database construction, multi-scale coupling, and robust extension to open quantum systems and reactive chemistries (Ortner, 2023, Lysogorskiy et al., 2022, Rinaldi et al., 6 Nov 2024).