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MACE-OFF: Transferable Short Range Machine Learning Force Fields for Organic Molecules

Dávid Péter Kovács, J. Harry Moore, Nicholas J. Browning, Ilyes Batatia, Joshua T. Horton, Yixuan Pu, Venkat Kapil, William C. Witt, Ioan-Bogdan Magdău, Daniel J. Cole, Gábor Csányi

arXiv:2312.15211v5physics.chem-ph

TL;DR

Accurate water description is a key requirement for bio-organic force fields, while folding larger peptides is a difficult test for purely local models. MACE-OFF introduces transferable short-range force fields with controllable computational cost and reports accurate torsion and intermolecular-force predictions, while remaining limited to neutral, non-radical, non-reactive systems.

  • Problem

    Accurate water description is a key requirement for bio-organic force fields, and folding larger peptides tests whether purely local models capture complex hydrogen bonding interactions.

  • Method

    MACE-OFF is a family of purely local, transferable force fields parameterized for ten organic-chemistry elements, with small, medium, and large variants controlling expressivity and computational cost.

  • Results

    MACE-OFF achieves mean barrier height errors of 0.3, 0.4, and 0.5 kcal/mol for its large, medium, and small models, respectively, while medium and large models yield intermolecular force errors of around 5–15 meV/Å.

  • Takeaways & Limitations

    The models support simulations spanning molecular liquids, crystals, drug-like molecules, and biopolymers, with smaller variants suited to large-scale simulations and larger variants to higher-accuracy small-system calculations.

  • Takeaways & Limitations

    Explicit long-range interactions are absent, limiting applicability to neutral, non-radical, non-reactive systems.

Abstract

from arXiv · show

Classical empirical force fields have dominated biomolecular simulation for over 50 years. Although widely used in drug discovery, crystal structure prediction, and biomolecular dynamics, they generally lack the accuracy and transferability required for first-principles predictive modeling. In this paper, we introduce MACE-OFF, a series of short range transferable force fields for organic molecules created using state-of-the-art machine learning technology and first-principles reference data computed with a high level of quantum mechanical theory. MACE-OFF demonstrates the remarkable capabilities of short range models by accurately predicting a wide variety of gas and condensed phase properties of molecular systems. It produces accurate, easy-to-converge dihedral torsion scans of unseen molecules, as well as reliable descriptions of molecular crystals and liquids, including quantum nuclear effects. We further demonstrate the capabilities of MACE-OFF by determining free energy surfaces in explicit solvent, as well as the folding dynamics of peptides.Finally, we simulate a fully solvated small protein, observing accurate secondary structure and vibrational spectrum. These developments enable first-principles simulations of molecular systems for the broader chemistry community at high accuracy and relatively low computational cost.

I. Introduction

MACE-OFF is introduced as a family of short-range, transferable force fields for neutral closed-shell organic systems, addressing accuracy and transferability limitations of empirical approaches. The models use local equivariant many-body representations and are validated across molecular, condensed-phase, and biomolecular applications.

  • Motivation: Empirical force fields remain widely used because bio-organic simulations prioritize large systems and long timescales, despite sacrificing accuracy for speed.
  • Contribution: MACE-OFF comprises purely local, transferable force fields parameterized for H, C, N, O, F, P, S, Cl, Br, and I.
  • Contribution: The models target neutral closed-shell systems and describe both intra- and intermolecular interactions across liquids, crystals, drug-like molecules, and biopolymers.
  • Validation: The models are validated on torsion barriers, optimized geometries, crystal properties, Raman spectra, liquid properties, water structure, and further molecular simulations.
  • MACE architecture: MACE maps atomic positions and chemical elements to potential energy while decomposing total energy into atomic site energies for linear scaling with system size.
  • MACE architecture: Two message-passing layers build environment-dependent node features from neighbor displacements, learnable radial functions, and equivariant many-body combinations.
  • MACE architecture: Higher-order symmetric features are formed through tensor products and Clebsch-Gordan contractions, with maximum body-order ν fixed at 3 for these models.

B. Training data

MACE-OFF is trained on high-level quantum-mechanical energies and forces from SPICE, augmented with larger molecules, water clusters, and additional SPICE configurations. The resulting model variants trade computational cost against accuracy and are evaluated on molecular properties including torsion scans and intermolecular forces.

  • Training data: The core training set uses SPICE, with 95% allocated to training and validation and 5% held out for molecule-level testing.Conformers of the same molecule were kept out of both training/validation and test splits.
  • Training data: SPICE was augmented with 50–90 atom QMugs molecules to improve learning of intramolecular non-bonded interactions.The added geometries were generated with GFN2-xTB molecular dynamics and re-evaluated using the SPICE quantum-mechanical level.
  • Training data: The dataset was further expanded with water clusters, while MACE-OFF24(M) adds 208,000 configurations from SPICE version 2, including solvated molecules and amino acid-ligand pairs.These additions target improved descriptions of water and broader chemical configurations.
  • Model details: MACE-OFF23 models are offered in small, medium, and large variants, with increasing model size improving accuracy while increasing computational cost.The small model targets large-scale simulations, the medium model balances speed and accuracy, and the large model targets small systems or highest accuracy.
  • Validation: On held-out tests, the large model generally reaches 0.5–1.0 meV/atom energy errors and 15–20 meV/˚A force errors, below the 1 kcal/mol chemical-accuracy limit.MACE-OFF23(M) and MACE-OFF23(L) produce intermolecular force errors of around 5–15 meV/˚A.
  • Validation: For TorsionNet-500, medium and large models achieve barrier-height errors of around 0.25 kcal/mol and geometry deviations of about 0.025 ˚A.The comparison includes Sage, GFN2-xTB, AIMNet2, and the three MACE-OFF23 variants; the other models used different DFT training levels.

2. Biaryl fragments

MACE-OFF is evaluated on torsional profiles and condensed-phase molecular properties, including vibrational spectra, crystal sublimation enthalpies, liquid densities, and heats of vaporization.

  • Biaryl fragments: 78 biaryl molecules test torsional profiles for rotatable bonds between aromatic rings, a chemistry frequently occurring in drug-like molecules.The benchmark uses coupled-cluster reference data and a subset of 78 molecules for comparison with ANI-1ccx.
  • Biaryl fragments: 0.3 kcal/mol mean absolute error is achieved by the large MACE model for biaryl barrier heights, versus 0.4 and 0.5 kcal/mol for medium and small models.The DFT torsion drives have a 0.2 kcal/mol mean barrier height error relative to coupled-cluster data, setting the theoretical reference for models trained at that DFT level.
  • Condensed phase properties: Classical and quantum infrared spectra of paracetamol agree with experiment up to around 2000 cm^-1, while quantum nuclear treatment improves high-frequency O–H and N–H modes.The classical spectrum is blue-shifted for high-frequency stretches because it lacks anharmonic zero-point fluctuations; the quantum spectrum retains a net blue shift and slight intensity discrepancies.
  • Condensed phase properties: 1.8 kcal/mol mean absolute error is obtained for sublimation enthalpies of 23 molecular crystals, with MACE-OFF23(M) improving significantly over ANI-2x.The comparison is against experimentally measured sublimation enthalpies; the small model is intermediate in accuracy, while the large model gives no significant improvement.
  • Condensed phase properties: 0.09 g/cm3 mean absolute error is obtained for molecular liquid densities, while heats of vaporization show a systematic offset of approximately 2 kcal/mol.ANI-2x has density MAE and RMSE errors twice as large as MACE-OFF23(M); the vaporization offset may reflect interactions beyond the cutoff or insufficient training coverage.

D. Biomolecular simulations

MACE-OFF models reproduce key water, peptide, and protein behaviors in biomolecular simulations, including structural, thermodynamic, and dynamical properties. The models also extrapolate to larger biomolecular systems, while water-density accuracy improves with a larger receptive field.

  • Water structure and dynamics: MACE-OFF23(M) predicts water radial distribution functions comparably to TIP3P and MB-pol, whereas ANI-2X significantly over-structures the RDF.The MACE result arises from purely local interactions trained on non-periodic water clusters containing up to 50 molecules.
  • Water structure and dynamics: A 6 Å layerwise cutoff brings predicted room-temperature water density within 2% of experiment and keeps the full-temperature-range error within 5%.The original 5 Å MACE-OFF23(M) model overestimates liquid-water density by approximately 20%.
  • Ala3 free-energy surface: 2.2 × 10^6 steps/day for solvated Ala3 enables sufficient free-energy-surface sampling, despite being slower than MM at around 150 × 10^6 steps/day.The corresponding vacuum throughput is 9.6 × 10^6 steps/day on an NVIDIA 80GB A100 GPU via OpenMM.
  • Ala3 free-energy surface: MACE-OFF24(M) identifies the same four Ala3 free-energy minima as AMBER14SB/TIP3P and predicts the anti-parallel β-sheet to be 0.1 kcal/mol above PPII, versus 1 kcal/mol for AMBER.Relative depths of the α-helical conformations agree between the two simulations.
  • Ala3 free-energy surface: MACE-OFF24(M) accurately predicts Ala3 3J coupling constants comparably to AMBER14SB, with both outperforming ANI-2x against experiment.The couplings were computed from the Ramachandran distribution of an unbiased 20 ns NPT trajectory.
  • Folding dynamics of Ala15: Ala15 folds within 200 ps, proceeds through a “wavy” intermediate, and oscillates between α- and 310-helices with the α-helix predominant.The oscillatory behavior agrees with other computational studies and experiments on alanine-rich polypeptides.
  • Folding dynamics of Ala15: The folded Ala15 structure remains stable to around 500 K before unfolding into a primarily random coil as temperature rises to 900 K.The temperature ramp runs from 300 K to 900 K over 3 ns.
  • Protein simulation in explicit solvent: In a 1 ns fully solvated Crambin simulation, deviations from the minimized structure remain below 1 Å, no bonds break, and secondary-structure motifs remain intact.The simulation reaches 3×10^5 steps per day on a single NVIDIA A100 80GB GPU and tests a 42-residue protein with four charged residues.

E. Computational performance

The paper evaluates MACE-OFF computational performance across OpenMM and LAMMPS and describes optimized implementations. It presents high-throughput GPU and CPU pathways while emphasizing accuracy, speed, and model-size trade-offs.

  • Benchmarking setup: NVT water simulations benchmark MACE-OFF with OpenMM and LAMMPS across boxes ranging from 10^2 to 10^5 atoms.OpenMM measurements use a single NVIDIA A100 80GB GPU, while LAMMPS results include CPU and GPU evaluation.
  • Optimized implementations: Optimized MACE-OFF variants use custom C++, CUDA, and/or Kokkos implementations, including the inference-only cuda mace library and planned symmetrix implementations.These variants supplement the native PyTorch implementations of MACE-OFF23.
  • Optimized implementations: The optimized kernels accelerate radial functions, spherical harmonics, equivariant operations, and product-basis calculations using cubic splines, unified kernels, tensor cores, and coefficient sparsity.The implementation uses sphericart for optimized real spherical harmonics.
  • Benchmarking setup: The OpenMM benchmark uses liquid water at 1 g/cm3 and 300 K with a 1 fs timestep on one NVIDIA 80GB A100 GPU.Forces are evaluated in double precision and integrated in single precision.
  • Overall performance: The authors report broad applicability, transferability, and high computational speed for purely short-range MACE-OFF models, with architecture, training data, and loss function contributing to improvements in accuracy and extrapolation.The paper compares these capabilities with pioneering ANI models but does not provide a full ablation study across model designs.
  • Overall performance: The small, medium, and large models show systematically improving gas-phase accuracy, while medium models provide a compromise between condensed-phase accuracy and computational expense.All force fields were released so users can choose a model suited to their application.
  • Scope boundary: The present models are limited to neutral, non-radical, non-reactive systems because they lack explicit long-range interactions.A next-generation model is being developed to include charges and extend coverage to additional biologically relevant systems.

Data Availability

The paper provides public access to its training data, torsion-drive dataset, and MACE-OFF models. Supplementary materials document training, benchmarking, simulation, and implementation details.

  • Public resources: The training data are publicly available through the listed DOI, and the torsion-drive dataset is available on Zenodo.The paper also provides a GitHub repository for the MACE-OFF model series.
  • Public resources: The MACE-OFF model series is available through the project’s GitHub repository.The repository is listed alongside the training data and torsion-drive dataset.
  • Supplementary documentation: Supplementary information covers LAMMPS performance, model training and SPICE filtering, test-set errors, torsion scans, lattice enthalpies, and crystal geometries.These sections document both model construction and condensed-phase validation.
  • Supplementary documentation: Additional supplementary sections describe condensed-phase simulations, biological simulations, and dipole-model simulation details and results.These materials include associated error tables and biological-simulation protocols.

Supporting Information for:

The supporting information documents MACE-OFF training, dataset cleaning, and test-set error analysis, including separate treatment of intermolecular force errors.

  • Training: MACE-OFF training uses two phases that shift emphasis from force accuracy to energy accuracy.The first phase uses force weight 1000 and energy weight 40; the second uses force weight 10 and energy weight 1000.
  • Dataset cleaning: Recomputed high-error configurations showed that about a third agreed with MACE rather than the original DFT labels.The authors attributed this discrepancy to limitations in the underlying electronic-structure calculations.
  • Dataset cleaning: The dataset was purified using force filters to remove configurations with extreme or nonphysical forces.The procedure removed configurations exceeding maximum-force or total-force thresholds, including 10,372 configurations from SPICE version 2 subsets.
  • Error analysis: Test-set force errors include intermolecular components computed from translational and rotational molecular force contributions.The analysis identifies molecules, sums atomic forces, redistributes translation by atomic masses, and combines translational and rotational terms.
  • Error analysis: The supporting tables report test-set errors across dimers, dipeptides, solvated amino acids, water, QMugs, and tripeptides for multiple MACE models.The reported values include, for example, 23M intermolecular errors of 5.70, 12.25, and 11.00 across listed categories.

i |f DFT

This section explains how intermolecular force errors are separated and evaluates MACE-OFF on torsional potential-energy surfaces and molecular-crystal sublimation enthalpies.

  • Intermolecular force errors: DFT force data can violate translational and rotational symmetry, producing apparent intermolecular errors even for isolated molecules.MACE forces obey these symmetries, so the discrepancy is assigned to the intermolecular component.
  • Torsion scans: TorsionNet-500 and biaryl torsion benchmarks were recomputed at ωB97M-D3(BJ)/def2-TZVPPD-level theory to test rotatable-bond potential-energy surfaces.Dihedral angles were scanned in 15° increments, with constrained geometry optimizations at each grid point.
  • Torsion scans: The torsion evaluation compares force-field-optimized geometries and energies against DFT reference geometries to construct potential-energy surfaces.Separate constrained optimizations were performed for each assessed force field while holding the target torsion fixed.
  • Lattice enthalpies: ANI-2x crystal geometries could not be fully relaxed because the model cannot predict stresses, so DFT-equilibrium cells were kept fixed.This constrains the comparison between ANI-2x and the MACE models for crystal properties.

S5 Crystal structure geometries

MACE-OFF reproduces molecular-crystal lattice constants and provides condensed-phase simulations of organic liquids, while density errors remain linked to training-set coverage.

  • Crystal structures: MACE accurately reproduces molecular-crystal lattice constants despite being trained purely on molecular dimers.The comparison uses MACE-OFF23(L) relaxed crystal structures against experimental lattice parameters.
  • Crystal structures: Additional trimer data or higher-level coupled-cluster reference theory could further improve agreement with experiment.This is presented as a possible improvement rather than an evaluated intervention.
  • Liquid properties: Condensed-phase simulations used NPT molecular dynamics with a 1 fs timestep and temperatures generally set to 298 K.Compounds with lower boiling points were simulated at 10 K below their experimental boiling points.
  • Liquid properties: Large-model condensed-phase performance is comparable to the medium model, including similar errors for compounds predicted poorly by the medium model.This indicates that the dominant error source is the dataset rather than model size.
  • Liquid properties: MACE-OFF24(M) reduces heat-of-vaporization MAE from 2.18 to 1.75 kcal/mol but slightly decreases liquid-density accuracy.The extended-cutoff model also includes additional nonbonded training data.
  • Liquid properties: Density outliers concentrate in low-boiling-point ethers, polychlorinated hydrocarbons, and dibromo compounds, consistent with sparse functional-group sampling.The training set contains only 84 examples of dibromo-containing dimers.

S7 Biological Simulations

The biological simulations apply MACE-OFF to peptide enhanced sampling, peptide dynamics, and a solvated protein, with analyses of structure and vibrational behavior.

  • Peptides: Enhanced sampling used metadynamics with the two backbone torsions of the central alanine residue as collective variables.The simulation lasted 1 ns with temperature bias factor 10, barrier height 1 kJ/mol, and hill deposition every 100 steps.
  • Peptides: Ala15 dynamics were propagated for 500 ps in vacuo using Langevin dynamics with a 1 fs timestep.The initial structure was an extended peptide generated with PyMOL.
  • Peptides: The previous 5 Å model failed to reproduce the experimentally observed oscillation between the α-helix and 310 helix.This comparison is made against MACE-OFF24(M).
  • Analysis: The supporting figures report Ala15 secondary structure and temperature dependence, and Crambin RMSD relative to the first trajectory frame.Crambin RMSD is calculated by averaging two halves of a 1 ns trajectory on Cα atoms.

S8 Dipole models

MACE-OFF23-µ extends the MACE-OFF series to predict molecular total dipoles, enabling vibrational-spectrum calculations for organic molecules. On held-out SPICE data, it shows strong agreement with reference dipole moments and supports zero-shot prediction on unseen molecules.

  • Spectral calculation: Vibrational spectra are estimated by combining a regular MACE-OFF energy-and-force model with dipole autocorrelation functions from the dipole model.
  • Training data: The model was trained on the SPICE subset with available total dipole references, excluding the QMugs and water subsets.
  • Evaluation: Held-out SPICE test sets show small RMSEs and strong alignment between reference and predicted total-dipole distributions.The evaluation targets dipole moments and infrared spectra of organic molecules.
  • Evaluation: The results suggest promising zero-shot prediction of dipole moments and infrared spectra for unseen organic molecules.

S8.3 Vibrational spectroscopy of paracetamol

MACE-OFF23(S) reproduces high- and low-frequency features of paracetamol form II Raman spectra, with quantum nuclear effects improving agreement with experiment. The remaining overall vibrational-frequency shift is approximately 2–3%.

  • Quantum and non-Condon effects: Quantum nuclear effects are incorporated through a PIGS-derived effective potential energy surface added to MACE-OFF23(S).A separate equivariant MACE model represents paracetamol polarizability and captures non-Condon effects.
  • Raman-spectrum prediction: MACE-OFF23(S) predicts the high- and low-frequency regions of paracetamol form II Raman spectra with overall good agreement with experimental band positions.
  • Quantum and non-Condon effects: Classical predictions are consistently shifted relative to experiment, whereas PIGS-based quantum nuclear predictions improve the agreement.
  • Spectral features: A broad band near 3300 cm^-1 appears only when quantum nuclear effects are included.
  • Spectral features: Low-frequency modes agree semi-quantitatively with experiment without requiring a quantum nuclear description.
  • Limitation: The predicted vibrational frequencies retain an overall 2–3% shift relative to experiment.

S8.4 Vibrational spectroscopy of liquid water

MACE-OFF23(S) captures characteristic vibrational features of ambient liquid water using MACE models of the potential energy surface, dipole, and polarizability. Incorporating quantum nuclear effects brings predictions into agreement with experiment across the frequency range, while a 2–3% blue shift remains.

  • Spectral calculation: MACE-OFF23(S) models water vibrational spectra using dipole and polarizability predictions together with molecular-dynamics trajectories.The spectra include IR, isotropic Raman, and anisotropic Raman components.
  • Quantum nuclear effects: Quantum nuclear effects are incorporated with the PIGS approach because quantum nuclear motion substantially affects water dynamics.
  • Comparison with experiment: After incorporating quantum nuclear effects, MACE-OFF23(S) predictions agree with experiment across the entire frequency range.
  • Limitation: The predicted water spectra retain an overall 2–3% blue shift relative to experiment.
  • Spectral features: The model qualitatively captures 0–1000 cm^-1 hydrogen-bonding fingerprints in anisotropic Raman spectra and bimodal isotropic Raman stretching bands.The bimodal stretching feature is associated with hydrogen-bonded defects in room-temperature water.
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