CSP-MACE-Å for Faster Crystal Structure Prediction

• CSP-MACE-Å is a machine learning interatomic potential that decomposes lattice energy into intramolecular and intermolecular components for efficient crystal structure prediction.
• It integrates a high-capacity MACE-POLAR model, an analytical XDM dispersion correction, and a learned residual to replicate DFT-level intermolecular energetics at much faster speeds.
• The method enables large-scale candidate screening and harmonic free-energy reranking, significantly accelerating CSP workflows while maintaining near-DFT accuracy.

Searching arXiv for the cited paper and closely related models to ground the article in current literature. CSP-MACE-Å is a machine learning interatomic potential for crystal structure prediction (CSP) that is intended to replace density functional theory (DFT) in the evaluation and reranking of molecular crystal candidates. Its defining feature is an additive decomposition of the total lattice energy into intramolecular and intermolecular components, with the intermolecular term itself partitioned into a MACE-POLAR contribution, an analytical XDM-style dispersion correction, and a learned residual $\Delta$ model trained to reproduce B86bPBE-XDM intermolecular energies. In the reported evaluation, the method achieves performance comparable to PBE DFT with the Neumann-Perrin dispersion correction on an AstraZeneca set and performance close to B86bPBE-XDM DFT on a blind-test set, while running multiple orders of magnitude faster than DFT (Midgley et al., 27 May 2026).

1. Definition and scope

CSP-MACE-Å is presented as a machine learning interatomic potential designed for crystal structure prediction workflows in which large numbers of candidate molecular crystal structures must be relaxed, ranked, and, in some cases, reranked by approximate free energies rather than lattice energies alone (Midgley et al., 27 May 2026). The central objective is to provide DFT-level ranking fidelity at substantially lower computational cost.

The model is formulated around the decomposition

$E_{\rm total} = E_{\rm intra} + E_{\rm inter}\,.$

Here, $E_{\rm intra}$ is the intramolecular, or “monomer,” contribution, defined as the sum of gas-phase energies of each molecule in the cell, while $E_{\rm inter}$ is the intermolecular, or “binding,” contribution (Midgley et al., 27 May 2026). This decomposition is not merely notational; it structures both model construction and training. The intramolecular part is handled by a MACE-POLAR model trained on single-molecule data, whereas the intermolecular part is assembled from physically and statistically distinct terms.

This suggests that CSP-MACE-Å is tailored to the particular error structure of molecular-crystal energetics: intramolecular conformational energetics, short-to-mid-range intermolecular interactions, long-range dispersion, and residual DFT-level corrections are treated as separable modeling targets rather than collapsed into a single universal lattice-energy predictor.

2. Energy decomposition and model architecture

The intramolecular component $E_{\rm intra}$ is predicted by the MACE-POLAR architecture in its “medium” size configuration (Midgley et al., 27 May 2026). MACE-POLAR is described as consisting of a stack of higher-order equivariant message-passing layers, a long-range electrostatics module using Ewald or multipole summation of learned partial charges, and a final feed-forward network mapping per-atom embeddings to atomic energy contributions $e_i$, such that

$$E_{\rm intra}=\sum_{i\in\,\text{all atoms in isolated molecules}} e_i(\mathbf{h}_i)\,.$$

In this definition, the molecular energies are evaluated in the gas phase and summed across the molecules in the crystal cell (Midgley et al., 27 May 2026).

The intermolecular contribution is split into three parts:

$$E_{\rm inter} = E_{\rm inter}^{\rm MACE\text{-}POLAR} + E^{\rm XDM}_{\rm disp} + \Delta E_{\rm inter}\,.$$

The first term, $E_{\rm inter}^{\rm MACE\text{-}POLAR}$, is defined as the total MACE-POLAR energy on the crystal minus the gas-phase $E_{\rm intra}$ contribution (Midgley et al., 27 May 2026). The second term is a fixed, pairwise-additive XDM-style dispersion correction with the functional form

$E_{\rm total} = E_{\rm intra} + E_{\rm inter}\,.$0

where $E_{\rm total} = E_{\rm intra} + E_{\rm inter}\,.$1 is the interatomic distance, $E_{\rm total} = E_{\rm intra} + E_{\rm inter}\,.$2, $E_{\rm total} = E_{\rm intra} + E_{\rm inter}\,.$3, and $E_{\rm total} = E_{\rm intra} + E_{\rm inter}\,.$4 are fixed averaged XDM coefficients, and $E_{\rm total} = E_{\rm intra} + E_{\rm inter}\,.$5 is the sum of van der Waals radii for atoms $E_{\rm total} = E_{\rm intra} + E_{\rm inter}\,.$6 and $E_{\rm total} = E_{\rm intra} + E_{\rm inter}\,.$7 (Midgley et al., 27 May 2026).

The third term is a learned residual:

$E_{\rm total} = E_{\rm intra} + E_{\rm inter}\,.$8

Residual targets of the same form are also used for forces and virials (Midgley et al., 27 May 2026). In effect, the model places physically motivated structure into the intermolecular energy and reserves machine learning capacity for the remaining discrepancy with B86bPBE-XDM.

A plausible implication is that the architecture is designed to reduce the burden on the residual learner by isolating the most systematic long-range component analytically and the monomer physics through a separate high-capacity intramolecular model.

3. Training data and optimization procedure

The intramolecular model is trained on the OMol25 dataset, which contains 100 million single-molecule DFT calculations at the $E_{\rm total} = E_{\rm intra} + E_{\rm inter}\,.$9B97M-V/def2-TZVPD level of theory (Midgley et al., 27 May 2026). The chemistry is described as having broad coverage of small and medium-sized organic molecules, including heterocycles and aliphatics, and the targets are total energies, atomic forces, and optionally dipoles (Midgley et al., 27 May 2026). These characteristics establish the monomer term as a gas-phase, electronically detailed component trained on very large-scale DFT data.

The residual intermolecular model is trained on 50 000 periodic B86bPBE-XDM crystal DFT calculations with energies, forces, and stresses (Midgley et al., 27 May 2026). The residuals are computed as

$E_{\rm intra}$0

The $E_{\rm intra}$1 model itself is a MACE-POLAR “medium” architecture trained on these residuals with mean-squared-error losses on energy, forces, and stress (Midgley et al., 27 May 2026). The reported optimization details are Adam, a typical learning rate of $E_{\rm intra}$2, early stopping after convergence on a held-out 5% set, and a batch size of approximately one crystal per batch (Midgley et al., 27 May 2026).

The separation between OMol25-based monomer training and crystal-based residual training is a central procedural feature. It assigns broad intramolecular chemical coverage to the monomer model and crystal-specific DFT correction to the intermolecular residual model. This suggests a hybrid transfer strategy in which single-molecule electronic structure data and periodic crystal data are combined without forcing a single model to learn both regimes from scratch.

4. Evaluation protocol and quantitative performance

The reported evaluation uses two test suites (Midgley et al., 27 May 2026). The first, the “AZ Set,” comprises 19 AstraZeneca compounds, described as 17 small molecules plus 1 salt, each with approximately 1 000 crystal candidates from a force-field prescreen. The second, the “Blind-Test Set,” comprises 28 compounds from the first through seventh CSP blind tests and includes neutral organics, salts, cocrystals, flexible molecules, and polymorphs (Midgley et al., 27 May 2026). The key metrics are the rank of the experimental structure, where 1 is best, and $E_{\rm intra}$3, the energy gap in kJ/mol to the lowest-energy candidate (Midgley et al., 27 May 2026).

The average results reported for the two sets are as follows.

Evaluation set	Model	$E_{\rm intra}$4, $E_{\rm intra}$5
AZ Set (19 compounds)	PBE + Neumann–Perrin (E)	3.68, 0.68 kJ/mol
AZ Set (19 compounds)	CSP-MACE-Å (E)	3.58, 0.66 kJ/mol
AZ Set (19 compounds)	CSP-MACE-Å (A)	2.11, 0.36 kJ/mol
AZ Set (19 compounds)	UMA-OMC (E)	25.26, 1.75 kJ/mol
AZ Set (19 compounds)	UMA-OMC (A)	23.26, 1.59 kJ/mol
AZ Set (19 compounds)	MACE-POLAR-1 (E)	5.58, 1.32 kJ/mol
AZ Set (19 compounds)	MACE-POLAR-1 (A)	3.00, 0.58 kJ/mol
Blind-Test Set (28 compounds)	B86bPBE-XDM (E)	3.25, 0.91 kJ/mol
Blind-Test Set (28 compounds)	CSP-MACE-Å (E)	3.86, 0.94 kJ/mol
Blind-Test Set (28 compounds)	CSP-MACE-Å (A)	2.96, 0.80 kJ/mol
Blind-Test Set (28 compounds)	UMA-OMC (E)	3.89, 1.11 kJ/mol
Blind-Test Set (28 compounds)	UMA-OMC (A)	3.50, 0.89 kJ/mol
Blind-Test Set (28 compounds)	MACE-POLAR-1 (E)	7.46, 1.85 kJ/mol
Blind-Test Set (28 compounds)	MACE-POLAR-1 (A)	5.43, 1.18 kJ/mol

On the AstraZeneca set, CSP-MACE-Å with lattice energies alone is reported to be comparable to PBE DFT with the Neumann-Perrin dispersion correction, with average rank 3.58 versus 3.68 and average $E_{\rm intra}$6 0.66 versus 0.68 kJ/mol (Midgley et al., 27 May 2026). On the blind-test set, CSP-MACE-Å with lattice energies is reported to be close to B86bPBE-XDM DFT, with average rank 3.86 versus 3.25 and average $E_{\rm intra}$7 0.94 versus 0.91 kJ/mol (Midgley et al., 27 May 2026).

Across the full evaluation suite, CSP-MACE-Å is reported to outperform both MACE-POLAR-1 and UMA-OMC (Midgley et al., 27 May 2026). Because the comparison spans neutral organics, salts, cocrystals, flexible molecules, and polymorph systems, the performance claim is not restricted to a narrow crystal subclass.

5. Harmonic free-energy reranking and temperature dependence

A major aspect of the reported workflow is reranking by the Helmholtz free energy under the harmonic approximation rather than by static lattice energy alone (Midgley et al., 27 May 2026). The free energy is evaluated as

$E_{\rm intra}$8

where $E_{\rm intra}$9 are phonon frequencies obtained by finite-difference supercell diagonalization (Midgley et al., 27 May 2026).

In both evaluation sets, reranking with harmonic free energies improves performance relative to ranking by energy alone (Midgley et al., 27 May 2026). Across both sets, reranking by $E_{\rm inter}$0 reduces the average rank by approximately 30% and lowers $E_{\rm inter}$1 by approximately 40% (Midgley et al., 27 May 2026). Numerically, on the AstraZeneca set CSP-MACE-Å improves from average rank 3.58 and $E_{\rm inter}$2 0.66 kJ/mol in energy ranking to 2.11 and 0.36 kJ/mol in harmonic free-energy ranking; on the blind-test set it improves from 3.86 and 0.94 kJ/mol to 2.96 and 0.80 kJ/mol (Midgley et al., 27 May 2026).

The method was also applied to five polymorphic systems, including sulfathiazole, mexiletine·HCl, and AZD1305, over the temperature range 0–600 K under the harmonic model (Midgley et al., 27 May 2026). CSP-MACE-Å is reported to reproduce the qualitative ordering and phase-transition trends in all five cases, with errors in exact transition temperatures of approximately 20–50 K (Midgley et al., 27 May 2026).

These results place harmonic phonon contributions at the center of the model’s practical utility rather than as a peripheral post-processing step. A plausible implication is that the main gain from CSP-MACE-Å is not only cheaper lattice-energy evaluation but also the expansion of free-energy-based reranking to a materially larger subset of candidate structures.

6. Comparative position among related machine-learning potentials

The comparison against MACE-POLAR-1 and UMA-OMC clarifies the design choices of CSP-MACE-Å (Midgley et al., 27 May 2026). MACE-POLAR-1 is described as lacking an explicit long-range dispersion term and as being trained only on gas-phase OMol25; it underbinds crystals, with $E_{\rm inter}$3 kJ/mol error and rank $E_{\rm inter}$4 in the AZ set (Midgley et al., 27 May 2026). UMA-OMC is described as being trained on $E_{\rm inter}$5 million PBE-D3 crystal configurations and as capturing dispersion implicitly, but showing larger errors, with $E_{\rm inter}$6 kJ/mol and rank $E_{\rm inter}$7 on the AZ set (Midgley et al., 27 May 2026).

Within this comparative framing, CSP-MACE-Å is characterized by a hybrid design: high-quality intramolecular modeling, explicit XDM dispersion, and a crystal-trained residual correction (Midgley et al., 27 May 2026). The reported results indicate that this combination yields near-DFT crystal-ranking accuracy while avoiding two distinct failure modes suggested by the baselines: insufficient intermolecular attraction in the absence of explicit long-range dispersion, and lower ranking fidelity when dispersion is learned only implicitly from crystal data.

A common misconception would be to interpret CSP-MACE-Å as a pure end-to-end lattice-energy network. The reported construction does not support that reading. Instead, it is an additive composite potential whose components reflect distinct physical and data-driven roles: monomer energetics, analytical dispersion, and residual correction to B86bPBE-XDM-level intermolecular energetics (Midgley et al., 27 May 2026).

7. Computational implications for crystal structure prediction workflows

The reported acceleration relative to DFT is substantial (Midgley et al., 27 May 2026). Single-point evaluation plus geometry optimization under CSP-MACE-Å is stated to be approximately $E_{\rm inter}$8–$E_{\rm inter}$9 faster than DFT, while free-energy evaluation including phonons is approximately $E_{\rm intra}$0–$E_{\rm intra}$1 faster (Midgley et al., 27 May 2026). This speedup enables relaxation and energy ranking of $E_{\rm intra}$2 candidates, in contrast to $E_{\rm intra}$3 for DFT, and renders free-energy reranking of the top 50–100 structures feasible (Midgley et al., 27 May 2026).

In the context of CSP, where force-field prescreens often produce large candidate pools and the experimentally realized polymorph may not rank first under approximate lattice energies, this computational regime changes the practical allocation of accuracy. Rather than restricting high-level calculations to a very small set of structures, CSP-MACE-Å permits a larger portion of the search space to be evaluated at DFT-comparable accuracy and a nontrivial subset to be reranked by harmonic free energies.

The practical significance described for this workflow is “greater confidence when derisking solid forms” (Midgley et al., 27 May 2026). This suggests a role not only in academic benchmark studies but also in industrial solid-form assessment, where salts, cocrystals, and polymorph stability trends are operationally important. The reported results indicate that CSP-MACE-Å is positioned as a surrogate for DFT within CSP pipelines rather than merely as a prescreening heuristic, with its principal contribution lying in the combination of DFT-level ranking behavior, harmonic free-energy capability, and multiple-orders-of-magnitude speedup (Midgley et al., 27 May 2026).