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General Learning of the Electric Response of Inorganic Materials

Bradley A. A. Martin, Alex M. Ganose, Venkat Kapil, Tingwei Li, Keith T. Butler

TL;DR

The paper tackles predicting dielectric and ferroelectric responses across a broad chemical space with a single, differentiable energy functional. It introduces MACE-Field, a field-aware extension of an equivariant interatomic potential that injects a uniform electric field into latent multipole features while keeping the readout scalar, enabling exact differentiation to obtain polarisation, Born effective charges, and polarisability. By cross-chemistry fine-tuning on MP-Dielectric and MP-Ferroelectric datasets and incorporating a replay set, the authors demonstrate accurate $Z^*$, $\boldsymbol\alpha$, and derived dielectric properties, plus the ability to perform finite-field MD and time-domain spectroscopy for materials like BaTiO$_3$ and $\alpha$-quartz. The approach preserves the underlying MACE performance and supports high-throughput dielectric screening across thousands of inorganic solids, offering a practical route toward integrated, physics-consistent material discovery for ferroelectrics and dielectrics.

Abstract

We present MACE-Field, a field-aware $O(3)$-equivariant interatomic potential that provides a compact, derivative-consistent route to dielectric properties (such as polarisation $\mathbf P$, Born effective charges $Z^*$ and polarisability $\boldsymbolα$) and finite-field simulations across chemistry for inorganic solids. MACE-Field preserves the standard MACE readout and can inherit existing MACE foundation weights, turning pretrained models into field-aware ones with minimal change. To demonstrate, we fine-tune MACE-MP-0 on multiple heads covering BECs and polarisabilities ($\sim$6k MP dielectrics spanning 81 elements), polarisations (2.5k MP nonpolar-to-polar polarisation branches), and energies, forces, and stresses (10,000 structure-replay set from MPtraj), resulting in a field-aware foundation model, MACE-Field-MP-0. We show that MACE-Field can evaluate polarisation branches and spontaneous polarisations, predict $Z^*$ and dielectric constants across diverse chemistries, and reproduce finite-field MD simulations, such as BaTiO$_3$ polarisation hysteresis and the IR/Raman and dielectric spectra of $α$-quartz, benchmarking against Allegro-pol and DFPT.

General Learning of the Electric Response of Inorganic Materials

TL;DR

The paper tackles predicting dielectric and ferroelectric responses across a broad chemical space with a single, differentiable energy functional. It introduces MACE-Field, a field-aware extension of an equivariant interatomic potential that injects a uniform electric field into latent multipole features while keeping the readout scalar, enabling exact differentiation to obtain polarisation, Born effective charges, and polarisability. By cross-chemistry fine-tuning on MP-Dielectric and MP-Ferroelectric datasets and incorporating a replay set, the authors demonstrate accurate , , and derived dielectric properties, plus the ability to perform finite-field MD and time-domain spectroscopy for materials like BaTiO and -quartz. The approach preserves the underlying MACE performance and supports high-throughput dielectric screening across thousands of inorganic solids, offering a practical route toward integrated, physics-consistent material discovery for ferroelectrics and dielectrics.

Abstract

We present MACE-Field, a field-aware -equivariant interatomic potential that provides a compact, derivative-consistent route to dielectric properties (such as polarisation , Born effective charges and polarisability ) and finite-field simulations across chemistry for inorganic solids. MACE-Field preserves the standard MACE readout and can inherit existing MACE foundation weights, turning pretrained models into field-aware ones with minimal change. To demonstrate, we fine-tune MACE-MP-0 on multiple heads covering BECs and polarisabilities (6k MP dielectrics spanning 81 elements), polarisations (2.5k MP nonpolar-to-polar polarisation branches), and energies, forces, and stresses (10,000 structure-replay set from MPtraj), resulting in a field-aware foundation model, MACE-Field-MP-0. We show that MACE-Field can evaluate polarisation branches and spontaneous polarisations, predict and dielectric constants across diverse chemistries, and reproduce finite-field MD simulations, such as BaTiO polarisation hysteresis and the IR/Raman and dielectric spectra of -quartz, benchmarking against Allegro-pol and DFPT.

Paper Structure

This paper contains 33 sections, 41 equations, 15 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Theory
  3. Modern theory of polarisation (Berry phase)
  4. Finite electric fields and DFPT/finite-field formalisms
  5. Macroscopic response tensors as energy derivatives
  6. Ferroelectrics and spontaneous polarisation
  7. From linear dielectric response to spectra
  8. Multipole expansion and inspiration for equivariant field coupling
  9. Methodology
  10. Overview
  11. Architecture and field coupling
  12. Differentiable observables and units
  13. Target properties
  14. Dielectric constants
  15. Branch-invariant polarisation supervision
  16. ...and 18 more sections

Figures (15)

  • Figure 1: MACE-Field architecture. At message-passing layer $t$, MACE produces equivariant latent features $h^{(t)}_{\alpha,kLM}$ (generalised “multipoles” of order $L$). A uniform external field $\mathbf E$ (irrep $l{=}1$) couples to these features via a fully-connected tensor product (Clebsch--Gordan contraction) to form $\Delta h^{(t)}_{\alpha,kLM}$; an irrep-wise linear map $W^{(t)}$ and a residual update yield field-aware features $\tilde{h}^{(t)}_{\alpha,kLM}$. Scalar components ($L{=}0$) are read out at each layer and summed to give a rotationally invariant electric enthalpy $\mathcal{F}$. All dielectric observables are obtained as exact derivatives of this single scalar: polarisation $\mathbf P=-\Omega^{-1}\partial \mathcal{F}/\partial \mathbf E$, Born effective charges $Z^{*}_{\alpha,ij}=\Omega\,\partial P_i/\partial R_{\alpha j}/e=\partial F_{\alpha j}/\partial E_i$, and electronic polarisability $\alpha_{ij}=\partial P_i/\partial E_j$. Hidden layers remain $O(3)$-equivariant; only the final readout in layer $t=T$ is strictly invariant.
  • Figure 2: Elemental coverage of the datasets used in this work. (a) Smidt et al. ferroelectric distortion-path set smidt-2020-ferrodb covering 61 elements. (b) MP-Dielectric (DFPT BECs and electronic polarisabilities) covering 81 elements. Colour encodes the per-dataset normalised frequency (fraction of structures containing each element, see colour bars); grey indicates no examples.
  • Figure 3: MPtrj replay-set parity for MACE-Field-MP-0. Parities on the Materials Project replay set of 10,000 sub-selected structures. Colour bars show the log number of data points. Top-left: Energy parity (eV). Top-right: Forces parity with all components combined (eV/Å). Bottom: Stress parity for diagonal components (left) and off-diagonal components (right) (eV/Å$^3$).
  • Figure 4: MP-Dielectric parity for MACE-Field-MP-0. Parities for Born effective charges and electronic polarisabilities from the MP-Dielectric dataset (train, validation and test combined). Colour bars show the log number of data points. Top: Born effective charge tensor ($e$) parity for diagonal (left) and off-diagonal (right) components. Bottom: Polarisability tensor ($e/(\text{V}\,\text{\AA})$) parity for diagonal (left) and off-diagonal (right) components.
  • Figure 5: Density distribution of BECs and electronic dielectric constants. KDE density curves for the Born effective charges $Z^*$ (left) and the high-frequency dielectric constants $\varepsilon_{\infty}$ (right). Solid blue: distribution predicted by the fine-tuned MACE-Field model on the sub-selected MPtrj dataset. Dashed orange: distribution predicted by the fine-tuned MACE-Field model on the MP-Dielectric dataset. Dotted green: distribution from the reference DFPT MP-Dielectric dataset.
  • ...and 10 more figures