MACE-Field is a field-aware extension of the MACE architecture that enables learning electric-field–dependent energy functionals for molecules and periodic materials. From a single scalar electric enthalpy, MACE-Field exposes physically consistent dielectric response properties via automatic differentiation:

• Polarization
  $\mathbf{P} = -\frac{1}{\Omega},\frac{\partial E}{\partial \mathbf{E}}$

• Born effective charges (BECs)
  $Z^\ast_{\kappa,\alpha\beta} = \frac{\partial P_\alpha}{\partial R_{\kappa,\beta}}$

• Polarisability/susceptibility
  $\chi_{\alpha\beta} = \frac{\partial P_\alpha}{\partial E_\beta}$

All quantities are derivative-consistent (Maxwell relations, acoustic sum rule) by construction.

MACE-Field can be:

• trained from scratch, or
• used to fine-tune existing MACE foundation models to become field-aware.

git clone https://github.com/mdi-group/mace-field.git
pip install ./mace

Note
MACE-Field is in the process of being upstreamed into the main MACE repository
ACEsuit#1177

MACE-Field preserves the standard MACE backbone and readout, but injects a uniform electric field (an O(3) irrep 1o) into the latent equivariant features at each interaction layer.

Conceptually:

1. Atomic neighbourhoods are expanded into equivariant “multipole-like” features (as in MACE / ACE).
2. A global electric field couples to these features through symmetry-allowed Clebsch–Gordan tensor products.
3. The final readout remains a scalar energy (electric enthalpy).
4. Dielectric observables are obtained by exact differentiation of this scalar.

At zero field, MACE-Field reduces exactly to standard MACE, enabling foundation-weight reuse.

(See Fig. 1 in Martin et al., 2025 for an architectural schematic.)

MACE-Field uses ASE-readable datasets (typically extended XYZ).

Key names can be overridden via CLI flags.

Training uses the standard MACE CLI with:

• --model MACEField
• --loss universal_field

python -m mace.scripts.run_train \
  --model MACEField \
  --name MACEField_model \
  --train_file data/field_train.xyz \
  --valid_fraction 0.2 \
  --device cuda \
  --r_max 5.0 \
  --num_interactions 2 \
  --num_channels 128 \
  --max_L 1 \
  --loss universal_field \
  --energy_weight 1.0 \
  --forces_weight 100.0 \
  --polarization_weight 1.0 \
  --becs_weight 100.0 \
  --polarizability_weight 100.0

Polarization is multi-valued in periodic systems. During training, MACE-Field compares folded polarization differences, ensuring a branch-invariant loss:

• avoids discontinuities,
• supports ferroelectric distortion paths,
• preserves a conservative derivative definition.

One of MACE-Field’s key strengths is foundation-model inheritance.

You can fine-tune a pretrained MACE model (e.g. mace-mp-0b3) to add polarization, BEC, and polarisability heads.

name: mace-field-mp-0b3-mh
foundation_model: mace-mp-0b3-medium.model
model: MACEField
loss: universal_field
multiheads_finetuning: true

heads:
  mp-dielectric:
    train_file: becs-polarizabilities-train.xyz
    valid_file: becs-polarizabilities-valid.xyz
  mp-polarization:
    train_file: polarizations-train.xyz
    valid_file: polarizations-valid.xyz

pt_train_file: mp_replay.xyz
compute_forces: true
compute_polarization: true
compute_becs: true
compute_polarizability: true

Run:

torchrun --standalone --nproc_per_node=gpu \
  python -m mace.scripts.run_train --config config.yaml

This adds dielectric response without degrading the original energy/force accuracy.

from mace.calculators import MACECalculator

calc = MACECalculator(
    model_path="MACEField.model",
    model_type="MACEField",
    electric_field=[0.0, 0.0, 0.02],  # overrides per-structure field
)

atoms.calc = calc
E = atoms.get_potential_energy()
P = atoms.calc.results["polarization"]
Z = atoms.calc.results["becs"]
alpha = atoms.calc.results["polarizability"]

mace_eval_configs \
  --configs input.xyz \
  --model MACEField.model \
  --output output.xyz \
  --compute_polarization \
  --compute_becs \
  --compute_polarizability

MACE-Field integrates seamlessly with ASE MD drivers.

atoms.info["REF_electric_field"] = [0.0, 0.0, 0.1]

# Update dynamically if required:
calc.electric_field = [0.0, 0.0, Ez_t]

This enables:

• ferroelectric hysteresis loops,
• IR / Raman spectra,
• finite-field response at finite temperature.

• Energy: eV
• Force: eV/Å
• Electric field: V/Å
• Polarization: e/Ų
• BECs: |e|
• Polarisability: e/(V·Å) (often reported in units of ε₀)

If you use this code, please cite:

@misc{martin2025generallearningelectricresponse,
  title={General Learning of the Electric Response of Inorganic Materials},
  author={Martin, Bradley A. A. and Ganose, Alex M. and Kapil, Venkat and Li, Tingwei and Butler, Keith T.},
  year={2025},
  eprint={2508.17870},
  archivePrefix={arXiv},
  primaryClass={cond-mat.mtrl-sci}
}

and the main MACE papers:

@inproceedings{Batatia2022mace,
  title={{MACE}: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields},
  author={Batatia, Ilyes and Kovacs, David Peter and Simm, Gregor N. C. and Ortner, Christoph and Csanyi, Gabor},
  booktitle={Advances in Neural Information Processing Systems},
  year={2022}
}

@misc{Batatia2022Design,
  title = {The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentials},
  author = {Batatia, Ilyes and Batzner, Simon and Kov{'a}cs, D{'a}vid P{'e}ter and Musaelian, Albert and Simm, Gregor N. C. and Drautz, Ralf and Ortner, Christoph and Kozinsky, Boris and Cs{'a}nyi, G{'a}bor},
  year = {2022},
  eprint = {2205.06643},
  archiveprefix = {arXiv}
}

• MACE-Field: [email protected]
• MACE core: [email protected]
• Issues & feature requests: https://github.com/mdi-group/mace-field/issues

MACE-Field is released under the MIT License.
(Some upstream MACE models may use different licenses.)