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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

Empirical force fields often lack the accuracy and transferability sought for first-principles molecular modeling, while purely local models face a difficult test in larger peptides. This paper introduces MACE-OFF, short-range transferable force fields trained on quantum-mechanical energies and forces, and reports accurate results across torsion-barrier and intermolecular-force evaluations.

  • Problem

    Purely local models face a difficult transferability test in larger peptides, where complex hydrogen bonding must stabilize secondary structure.

    Source: §3 · p. 11

  • Method

    MACE-OFF develops short-range transferable force fields for neutral closed-shell organic systems, trained to reproduce quantum-mechanical energies and forces.

    Source: §I · p. 2, B. Training data · p. 3

  • Results

    Across torsion-barrier and intermolecular-force evaluations, MACE-OFF reports accurate predictions; MACE-OFF23(L) has a mean absolute barrier-height error of 0.3 kcal/mol.

    Source: A. Extended SPICE Test Set · p. 4, §2 · p. 5

  • Takeaways & Limitations

    The force fields enable simulation of diverse chemical systems, from molecular liquids and crystals to drug-like molecules and biopolymers.

    Source: §I · p. 2

    Without explicit long-range interactions, the model's applicability is limited to neutral, non-radical, non-reactive systems.

    Source: §IV · p. 13

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 · II. Methods · A. The MACE architecture

MACE-OFF is a family of short-range transferable force fields for neutral, closed-shell organic systems, designed to extend short-range modeling across molecular liquids, crystals, drug-like molecules, and biopolymers. The paper describes its validation tasks and a two-layer equivariant MACE architecture that builds local atomic energies and derives forces and stresses from the total energy.

I. Introduction

  • Empirical force fields remain common for bio-organic systems because simulations prioritize large systems and long timescales, despite sacrificing accuracy for computational speed.Source: §I · p. 1
  • MACE-OFF comprises purely local transferable force fields parameterized for H, C, N, O, F, P, S, Cl, Br, and I to describe intra- and intermolecular interactions in neutral closed-shell systems.The stated application scope ranges from molecular liquids and crystals to drug-like molecules and biopolymers.Source: §I · p. 2
  • The models are validated on torsion barriers, geometry optimizations, molecular-crystal properties, liquid properties, peptide free-energy surfaces and folding, solvated crambin, and implementation scaling.The listed crystal tasks include lattice parameters, enthalpies of formation, and Raman spectra with nuclear quantum effects; the crambin simulation contains 18,000 atoms.Source: §I · p. 2
  • The paper focuses on extending short-range potentials, while noting that general-purpose biomolecular force fields require explicit long-range Coulomb interactions.Explicit polarizable electrostatics are identified as a possible future addition.Source: §I · p. 2

A. The MACE architecture

  • MACE maps atomic positions and elements to potential energy by summing atomic site energies, with atoms connected in a local graph defined by cutoff r_cut and linear scaling with system size.Its two message-passing layers yield an effective receptive field of 2 × r_cut.Source: A. The MACE architecture · pp. 2–3
  • MACE constructs equivariant neighbor features using learnable radial functions, spherical harmonics, and Clebsch–Gordan couplings, then forms atomic bases and higher-order product features.The maximum body order is fixed at three, corresponding to four-body terms including the central atom.Source: A. The MACE architecture · p. 3
  • Element-dependent message updates recursively modify node features, and the first layer omits the feature-update term so isolated-atom energies can be set exactly.Site energies combine rotationally invariant first-layer features with a second-layer multilayer-perceptron readout.Source: A. The MACE architecture · p. 3
  • Forces and periodic stresses are obtained as analytical derivatives of total potential energy with respect to atomic positions and the deformation tensor.Source: A. The MACE architecture · p. 3

B. Training data … 1. TorsionNet-500

MACE-OFF training combines SPICE data with larger molecules and water clusters, while model variants trade computational cost for accuracy. The models show low errors on held-out tests and reproduce drug-like torsion scans closely.

B. Training data

  • The training set uses about 85% of SPICE v1, with 95% of that data assigned to training and validation and 5% to testing at the molecule level.The subset contains neutral molecules with H, C, N, O, F, P, S, Cl, Br, and I, excluding ion pairs, and reference energies and forces use ωB97M-D3(BJ)/def2-TZVPPD.Source: B. Training data · p. 3
  • SPICE is augmented with 50–90-atom QMugs molecules, water clusters up to 50 molecules, and COMP6 tripeptides for testing, while SPICE itself contains molecules of up to 50 atoms.QMugs geometries are sampled using GFN2-xTB molecular dynamics and re-evaluated at the SPICE quantum-chemistry level; COMP6 data are not used for training.Source: B. Training data · p. 4

C. Model details

  • MACE-OFF23 is provided in small, medium, and large variants, with increasing size improving accuracy at higher computational cost; MACE-OFF24(M) adds a 6 Å cutoff and SPICE v2 configurations.The small model suits large-scale simulations, the medium balances speed and accuracy, and the large targets small systems or highest accuracy; SPICE v2 contributes 208,000 additional configurations and water clusters up to 90 atoms.Source: C. Model details · p. 4, B. Training data · p. 4

A. Extended SPICE Test Set

  • The large MACE-OFF23 model achieves held-out errors around 0.5–1.0 meV/atom and 15–20 meV/Å, below the 43 meV chemical-accuracy limit.Energy and force errors decrease as model size increases.Source: A. Extended SPICE Test Set · p. 4
  • MACE-OFF23 models trained on dipeptides extrapolate to tripeptides with no loss of accuracy on the held-out test set.The tripeptide test evaluates larger fragments with more complex interactions.Source: A. Extended SPICE Test Set · p. 4
  • MACE-OFF23(M) and MACE-OFF23(L) have intermolecular force RMSEs around 5–15 meV/Å, about 1.5–3 times below total-force errors.These errors are about 5–10% of typical DFT intermolecular force magnitudes; the passage argues this metric is more appropriate for condensed-phase applications than total-force errors.Source: A. Extended SPICE Test Set · p. 4

B. Dihedral scans

  • Drug-like dihedral scans are challenging because constrained geometry optimizations can fail to converge when the potential energy surface is not smooth.TorsionNet-500 contains torsion drives for 500 molecules spanning pharmaceutically relevant chemical space; the profiles were recreated at the SPICE DFT level rather than the original B3LYP/6-31G(d) level.Source: B. Dihedral scans · pp. 4–5, §1 · p. 5

1. TorsionNet-500

  • MACE-OFF23 medium and large models achieve mean torsion barrier errors around 0.25 kcal/mol and optimized-geometry deviations about 0.025 Å across TorsionNet-500 scans.MACE geometries are reported as almost indistinguishable from DFT-optimized structures; other models used different DFT training levels, which may contribute to differences.Source: §1 · p. 5

2. Biaryl fragments … 2. Condensed phase properties of organic liquids

MACE-OFF23 models show accurate predictions across biaryl torsions, paracetamol spectra, molecular-crystal sublimation enthalpies, and organic-liquid properties. The results also identify functional-group coverage and intermolecular interactions as areas for improvement.

C. Molecular crystals and organic liquids

  • The study uses MACE-OFF medium models for molecular-crystal vibrational and thermal properties based on relative accuracy and computational expense, with small- and large-model experiments reported in the Supporting Information.For 109 liquid benchmark molecules, throughput on one 80GB NVIDIA A100 GPU was 2.1×106, 1.1×106, and 2.8×105 steps/day for the small, medium, and large models, respectively.Source: C. Molecular crystals and organic… · p. 6, §2 · p. 7

2. Biaryl fragments

  • 0.3 kcal/mol: MACE-OFF23(L) has a mean absolute biaryl barrier-height error, compared with 0.4 and 0.5 kcal/mol for the medium and small models.The small model is more accurate than the next-best non-DFT methods, ANI-1ccx and GFN2-xTB, despite those models being parameterized using coupled-cluster references; MACE-OFF23 errors are robust with essentially no outliers.Source: §2 · pp. 5–6, Figure 2 · p. 7

3. Infrared spectroscopy of a drug molecule

  • Quantum nuclear dynamics produces a paracetamol infrared spectrum in good agreement with experiment, except for a net blue shift and slightly discrepant high-frequency stretching intensities.Classical and quantum spectra agree with experiment up to around 2000 cm−1; classical high-frequency O–H and N–H modes are blue-shifted, while the quantum treatment improves agreement.Source: §3 · p. 6, Figure 3 · p. 8

1. Lattice enthalphies

  • A 1.8 kcal/mol MAE: MACE-OFF23(M) captures trends in sublimation enthalpies for 23 molecular crystals and improves significantly over ANI-2x.Its MAE is comparable to several dispersion-corrected density functionals; the small model is intermediate in accuracy, while the large model gives no significant improvement over the medium model.Source: §1 · pp. 6–7, Figure 4 · p. 8

2. Condensed phase properties of organic liquids

  • 0.09 g/cm3 MAE: MACE-OFF23(M) predicts molecular-liquid densities with good experimental correlation, while ANI-2x has MAE and RMSE errors twice as large.The result indicates that the model generalizes from intermolecular interactions between training-set dimers to the condensed phase.Source: Figure 5 · p. 8, §2 · p. 8
  • MACE-OFF23(M) predictions of heats of vaporization correlate strongly with experiment but show a systematic offset of approximately 2 kcal/mol.The authors hypothesize that the offset reflects under-predicted intermolecular interactions, including interactions beyond the model’s cutoff and interactions insufficiently represented in training data.Source: Figure 5 · p. 8, §2 · p. 9
  • Density outliers chiefly involve low-boiling ether-containing, polychlorinated, or dibromo-containing compounds, implicating boiling-point under-prediction or insufficient training coverage of nonbonded interactions.For example, the training set contains only 84 dibromo-containing dimers; the authors note the need for greater coverage of specific functional groups.Source: §2 · p. 8
  • The small model over-predicts experimental density with a 0.23 g/cm3 MAE, while neither the large model nor a 6 Å cutoff model significantly improves on the medium model.The authors attribute poorly predicted densities to under-represented functional groups and suggest larger training sets and explicit long-range Coulomb interactions for future accuracy gains.Source: §2 · p. 9

D. Biomolecular simulations … 4. Protein simulation in explicit solvent

The biomolecular simulations test MACE-OFF’s transferability across water, peptides, and a fully solvated protein, showing that a local model trained on quantum-mechanical data can reproduce several structural and dynamical properties.

1. Water structure and dynamics

  • MACE-OFF23(M) predicted water’s radial distribution function comparably to TIP3P and MB-pol, whereas ANI-2X significantly over-structured it.MB-pol’s radial distribution function is indistinguishable from experiment. MACE learned these interactions locally from non-periodic clusters of up to 50 water molecules, and the model also showed good agreement with experiment for liquid-water vibrational density of states.Source: §1 · p. 9, Figure 6 · p. 10
  • Extending the cutoff from 5 Å to 6 Å brought predicted room-temperature water density within 2% of experiment and within 5% across the temperature range.MACE-OFF23(M) overestimated liquid-water density by approximately 20%; MACE-OFF24(M) also incorporated additional SPICE training data, with test-set errors reported as in agreement with MACE-OFF23(M).Source: §1 · p. 9, Figure 7 · p. 10

2. Ala3 Free Energy Surface

  • MACE-OFF24(M) reproduced Ala3’s four local free-energy minima; the anti-parallel β-sheet was 0.1 kcal/mol above PPII, versus 1 kcal/mol with AMBER14SB.The relative depths of the α-helical conformations agreed between the simulations.Source: §2 · p. 10, Figure 8 · p. 11
  • MACE-OFF24(M) predicted Ala3 3J coupling constants accurately and comparably to AMBER14SB, while both significantly outperformed ANI-2x.The values were computed from a 20 ns unbiased NPT ensemble; MACE-OFF24(M) ran at 2.2 × 10^6 steps/day for the solvated system, compared with around 150 × 10^6 steps/day for MM.Source: §2 · pp. 9–10, Table 3 · p. 10

3. Folding dynamics of Ala15

  • MACE-OFF24(M) folded Ala15 within 200 ps via a “wavy” intermediate, then oscillated between 310 and α-helices, with α-helix predominant.The oscillatory behavior has also been observed experimentally in alanine-rich polypeptides.Source: §3 · p. 11, Figure 9 · p. 11
  • Ala15 remained folded to around 500 K before unfolding into a primarily random coil as temperature rose toward 900 K.The temperature was increased from 300 K to 900 K in steps over 3 ns.Source: §3 · p. 11, Figure 10 · p. 11

4. Protein simulation in explicit solvent

  • MACE-OFF24(M) maintained crambin’s secondary structure and backbone fluctuations below 1 Å during a 1 ns simulation, with no bond breaking.The fully solvated 42-residue protein contained four charged residues; throughput was 3×10^5 steps per day on one NVIDIA A100 80GB GPU. The results illustrate the potential of bottom-up models to learn intermolecular interactions, while the authors note that broader evidence of improvement over classical force fields is still needed.Source: §4 · pp. 11–12
  • MACE-OFF24(M) reproduced crambin’s water peaks at 1640 cm-1 and 3200-3600 cm-1 reasonably accurately, but 125 ps was insufficient to converge the THz spectrum.The spectrum came from the final 125 ps of a 1.6 ns, approximately 18,000-atom simulation; a substantially longer trajectory is needed for quantitative experimental comparison. No nuclear quantum effects were included.Source: §4 · p. 12, Figure 11 · p. 12

E. Computational performance · IV. Outlook

MACE-OFF combines improved accuracy and extrapolation over ANI with high computational speed. The outlook identifies the models’ current scope and plans to add explicit charge descriptions for charged biological systems.

E. Computational performance

  • NVT water simulations evaluated performance using both OpenMM and LAMMPS, with LAMMPS results covering CPU and GPU evaluation.Source: E. Computational performance · p. 12
  • Alongside native PyTorch models, optimized implementations include inference-only cuda mace and C++ and Kokkos implementations planned for the symmetrix library.cuda mace uses custom CUDA kernels; the C++ and Kokkos implementations are also optimized variants.Source: E. Computational performance · p. 12
  • Optimized variants use cubic splines for radial functions, unified kernels for basis calculations, accelerated spherical harmonics, custom linear-operation kernels, tensor cores, and sparse product-basis calculations.The sparse generalized Clebsch-Gordon coefficients accelerate product calculations across body order and coupling terms.Source: E. Computational performance · pp. 12–13
  • Figure 12 reports OpenMM performance for liquid water at 1 g/cm3 and 300 K with a 1 fs timestep on one NVIDIA 80GB A100 GPU.Forces were evaluated in double precision and integrated in single precision; cuda mace is labeled OPT.Source: E. Computational performance · p. 12, Figure 12 · p. 13

IV. Outlook

  • MACE-OFF’s architecture, training data, and loss improve accuracy and extrapolation over ANI, while retaining high computational speed.Source: §IV · p. 13
  • Small, medium, and large MACE-OFF models show systematically improving accuracy against gas-phase quantum-mechanical data; medium models balance accuracy and computational expense across condensed-phase tests.The medium models were investigated against crystalline, liquid, and biomolecular condensed-phase data, and the force fields are available for users to select by application.Source: §IV · p. 13
  • Without explicit long-range interactions, current models are limited to neutral, non-radical, non-reactive systems; planned charge descriptions target protonation states, charged nucleic acids, and counter-ions.The paper notes AIMNet2’s extension of ANI to charged species and long-range interactions.Source: §IV · p. 13

Data Availability

The paper makes its training data, torsion-drive dataset, and MACE-OFF models publicly available, and lists the supplementary information’s training, evaluation, simulation, and performance details.

Data Availability

  • Training data, the torsion-drive dataset, and the MACE-OFF model series are publicly available through linked repositories.The passage provides links for the training data (DOI: 10.17863/CAM.107498), torsion-drive dataset (Zenodo record 11385284), and models (GitHub).Source: Data Availability · p. 14

Supplementary Information

  • Supplementary information covers model training and SPICE filtering, test-set and intramolecular force errors, torsion scans, lattice enthalpy calculations, and crystal geometries.Source: Supplementary Information · p. 14
  • It also reports condensed-phase and biological simulation details, dipole-model results, and computational performance in LAMMPS.Source: Supplementary Information · p. 14

Supporting Information for:

The supporting information details MACE-OFF’s staged training and dataset filtering, and describes how test-set errors—including intermolecular-force errors—were evaluated and summarized.

S1 MACE-OFF Training Details

  • Training used an initial force-weighted phase, then shifted to energy weighting after force-error convergence and reduced the learning rate.The initial force and energy weights were 1000 and 40; the second phase used weights of 10 and 1000 and a learning rate of 0.00025.Source: S1 MACE-OFF Training Details · p. 19
  • Re-evaluation of the highest-error configurations found that about one third matched MACE predictions rather than their original DFT labels, motivating removal of 808 configurations.The removed configurations exceeded a maximum force-error threshold of 2 eV/Å; many contained phosphorus or iodine.Source: S1 MACE-OFF Training Details · p. 19
  • SPICE version 2 filtering removed nonphysical or force-inconsistent configurations using maximum-force and total-force thresholds, eliminating 10,372 configurations.The filters were 15 eV/Å for maximum force and 0.1 eV/Å for total force; outliers included distorted geometries and phosphorus-containing compounds.Source: S1 MACE-OFF Training Details · pp. 19–20

S2 Test set errors

  • Intermolecular forces were calculated by combining molecular translational and rotational force contributions, with vibrational contributions obtained as their difference from the total.The procedure sums atomic forces within each molecule and derives rotational contributions from the molecular torque and moments of inertia.Source: S2 Test set errors · pp. 21, 23
  • Table S1 summarizes test-set errors, including intermolecular-force errors, which were evaluated as RMSEs between DFT forces and MACE predictions.The supplied text does not provide an unambiguous mapping of the table’s numerical values to its datasets and models.Source: S2 Test set errors · p. 21, Table S1 · p. 22

i |f DFT … S4 Lattice enthalpies

The supplementary analyses explain an apparent force-error artifact, describe torsion-scan procedures, and outline how lattice enthalpies were calculated and assessed. They also report the scope of MACE-OFF24(M) test errors and a limitation of ANI-2x for crystal relaxation.

i(f DFT

  • An apparent intermolecular error in isolated-molecule tests arises because DFT forces violate translational and rotational symmetry, whereas MACE forces obey both, making the difference identical across the three MACE models.The passage attributes the small error to nonzero total molecular force and torque in the DFT data, not to actual intermolecular interactions.Source: i(f DFT · p. 23

S2.1 MACE-OFF24(M) Test Errors

S3 Torsion angle scanning calculations

  • TorsionNet-500 and biaryl torsion benchmark datasets were recomputed at the SPICE level of QM theory to assess MACE models on drug-like rotatable-bond potential energy surfaces.Source: S3 Torsion angle scanning calculations · p. 25
  • The torsion scans used 15° increments, with constrained geometry optimizations performed from published optimized geometries using TorsionDrive and QCEngine.DFT energies and forces for constrained optimizations were calculated using PSI4.Source: S3 Torsion angle scanning calculations · p. 25
  • For each assessed force-field model, constrained optimizations began from DFT reference geometries with the target torsion fixed, and final geometry RMSD and energy were recorded.Source: S3 Torsion angle scanning calculations · p. 25

S4 Lattice enthalpies

  • The lattice-enthalpy protocol combines crystal electronic, zero-point, and finite-temperature vibrational contributions with ideal-gas enthalpies; vibrational contributions used 3 × 3 × 3 supercells.The ideal-gas final term was 4RT for systems except linear carbon dioxide, for which it was 3.5RT.Source: S4 Lattice enthalpies · p. 26
  • The X23 molecular-crystal assessment relaxed cells and then used phonon and normal-mode calculations to obtain finite-temperature vibrational and zero-point enthalpy contributions.Source: S4 Lattice enthalpies · p. 26
  • ANI-2x crystal geometries could not be fully relaxed because the model cannot predict stresses, so its cell was fixed at the DFT equilibrium values.Source: S4 Lattice enthalpies · pp. 26–27

S5 Crystal structure geometries · S6 Condensed phase simulations

MACE-OFF reproduces molecular-crystal lattice constants when trained on molecular dimers, while the condensed-phase evaluation compares density and heat-of-vaporization errors across models and tests model size and training-data coverage.

S5 Crystal structure geometries

  • MACE accurately reproduces molecular-crystal lattice constants using models trained purely on molecular dimers, compared with experimental values reported in Ref.15.Source: S5 Crystal structure geometries · p. 28

S6 Condensed phase simulations

S7 Biological Simulations … S7.3 Crambin in explicit solvent

The supplied biological-simulation passages describe Ala3 metadynamics and Ala15 folding protocols, report that MACE-OFF23(M) missed the observed Ala15 helix oscillation, and detail Crambin simulation and analysis methods.

S7.1 Ala3 Free Energy Surface

  • Ala3 enhanced sampling used metadynamics with the central alanine’s two backbone torsions as collective variables.The simulation lasted 1 ns, with a temperature bias factor of 10, a 1 kJ/mol barrier height, and hills deposited every 100 steps.Source: S7.1 Ala3 Free Energy Surface · p. 31

S7.2 Ala15 peptide folding

  • Ala15 folding dynamics began from an extended structure and used Langevin dynamics for 500 ps in vacuo, with a 1 fs timestep through OpenMM.The initial structure was generated using PyMOL.Source: S7.2 Ala15 peptide folding · p. 31

S7.2.1 Secondary structure prediction with MACE-OFF23(M)

  • MACE-OFF23(M), unlike MACE-OFF24(M), failed to reproduce the experimentally observed oscillation between α-helix and 3₁₀-helix states in Ala15.Figure S1 identifies the STRIDE labels used for the secondary-structure predictions and notes their temperature dependence.Source: S7.2.1 Secondary structure prediction… · p. 31, Figure S1 · p. 32

S7.3 Crambin in explicit solvent

  • Crambin was prepared from PDB structure 1EJG using pdbfixer18 and solvated in an orthorombic box.Source: S7.3 Crambin in explicit solvent · p. 32
  • Crambin dynamics used an NPT ensemble at 1 atm with an OpenMM Monte Carlo barostat and a 1 fs timestep for integration and trajectory writing.The velocity autocorrelation function’s Fourier transform was used to calculate the power spectrum with Travis defaults.Source: S7.3 Crambin in explicit solvent · pp. 32–33
  • Figure S2 reports Crambin RMSD averaged over two halves of a 1 ns trajectory, relative to the first frame and calculated on Cα atoms.The caption specifies the RMSD calculation but provides no numerical RMSD values in the supplied passage.Source: Figure S2 · p. 33

S8 Dipole models · S8.1 MACE-OFF23-µ · S8.2 Computational details

The supplement introduces MACE-OFF23-µ for molecular dipoles and evaluates its held-out SPICE performance, then details PIGS-based quantum-nuclear simulations and polarizability-based Raman calculations.

S8.1 MACE-OFF23-µ

  • MACE-OFF23-µ predicts molecular total dipole moments and can be paired with a regular MACE-OFF energy-and-force model to estimate vibrational spectra from dipole autocorrelation.Source: S8.1 MACE-OFF23-µ · p. 34
  • The dipole model retains MACE architecture and hyperparameters except for an L = 1 per-atom tensor readout, making the medium architecture the smallest possible model.Source: S8.1 MACE-OFF23-µ · p. 34
  • Training used the SPICE4 subset with available total dipoles, excluding QMugs and water, and evaluation used held-out SPICE test sets.Source: S8.1 MACE-OFF23-µ · p. 34
  • Small RMSEs and close alignment of reference and predicted dipole distributions indicate high-fidelity predictions across available SPICE test sets, supporting zero-shot dipole and infrared-spectrum prediction for unseen organic molecules.Figure S3 displays the reference and predicted distributions and reports RMSEs for individual test sets.Source: S8.1 MACE-OFF23-µ · p. 34, Figure S3 · p. 35

S8.2 Computational details

  • For the paracetamol form II vibrational spectrum, the workflow performed 20 ps of path-integral MD using MACE-OFF23(S) at 500 K with a 0.50 fs timestep.It used the BAOAB integration scheme extended to path-integral MD26 and path-integral Langevin equation thermostats27 with a 100 fs time constant.Source: S8.2 Computational details · p. 36
  • For paracetamol form II, a PIGS model fitted to path-integral centroid positions and forces captures the difference between the potential surface and the 500 K centroid potential of mean force.Adding this model to MACE-OFF23(S) yields a transferable effective potential surface encoding quantum nuclear effects.Source: S8.2 Computational details · p. 36
  • Time-correlation functions of tensors on the trajectories yielded isotropic and anisotropic Raman spectra24; the powder Raman spectrum was estimated as a linear combination of them29.Source: S8.2 Computational details · p. 36
  • For dipole and polarizability predictions, the MACE architecture differs from energy models by using vector and tensor readouts instead of atomic site energies; a relatively small model achieves high accuracy.Source: S8.2 Computational details · pp. 36–37

S8.3 Vibrational spectroscopy of paracetamol

MACE-OFF23(S) predicts paracetamol form II Raman band positions with overall good agreement to experiment; quantum nuclear effects improve agreement, while low-frequency modes agree semi-quantitatively without them.

S8.3 Vibrational spectroscopy of paracetamol

  • Quantum nuclear effects are incorporated with an effective MACE potential energy surface and PIGS, while a separate equivariant MACE polarizability model accounts for non-Condon effects.Source: S8.3 Vibrational spectroscopy of… · p. 37
  • MACE-OFF23(S) predicts both Raman frequency regions with overall good agreement in band positions; experimental directional spectra and the estimated powder spectrum preclude comparing intensities.The spectra shown are for paracetamol form II.Source: S8.3 Vibrational spectroscopy of… · p. 38
  • Classical predictions are consistently shifted from experiment, whereas PIGS quantum nuclear effects improve agreement and capture the broad band around 3300 cm−1.Source: S8.3 Vibrational spectroscopy of… · p. 38
  • Low-frequency modes agree semi-quantitatively with experiment and do not require a quantum nuclear description.Source: S8.3 Vibrational spectroscopy of… · p. 38
  • Despite a 2–3 % overall vibrational-frequency shift, MACE-OFF23(S) with quantum nuclear and non-Condon effects shows promise for characterizing molecular crystals.Source: S8.3 Vibrational spectroscopy of… · p. 38

S8.4 Vibrational spectroscopy of liquid water · S9 Computational performance in LAMMPS

MACE-OFF23(S) reproduces experimental liquid-water vibrational spectra with quantum nuclear effects included, while LAMMPS performance is evaluated across CPU and GPU implementations and scaling setups.

S8.4 Vibrational spectroscopy of liquid water

  • PIGS incorporates quantum nuclear effects into water dynamics, and MACE models of dipoles and polarizabilities provide the IR and Raman selection rules.The spectra are computed using MACE-OFF23(S) for the potential energy surface and a single MACE model for water’s dipole and polarizability.Source: S8.4 Vibrational spectroscopy of… · p. 38, Figure S5 · p. 40
  • MACE-OFF23(S) qualitatively captures water’s 0–1000 cm−1 hydrogen-bonding fingerprints and the bimodal isotropic Raman stretching band linked to hydrogen-bonded defects.The defects are present in room-temperature water.Source: S8.4 Vibrational spectroscopy of… · p. 39
  • Across the entire frequency range, MACE-OFF23(S) spectra agree with experiment with an overall 2–3% blue shift.The spectra include IR, isotropic Raman, and anisotropic Raman at ambient temperature and density.Source: S8.4 Vibrational spectroscopy of… · pp. 38–39, Figure S5 · p. 40

S9 Computational performance in LAMMPS

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