Multi-fidelity Machine Learning Interatomic Potentials for Charged Point Defects

Benchmarks show that four mainstream equivariant foundation potentials (MACE-MH-1, MACE-MPA-0, GRACE-2L-OMAT, MatterSim v1 5M) fail to consistently identify PBE-DFT ground-state structures for VSb(1) in Sb2Se3 across five charge states, with structural RMSD generally 0.2–0.4 Å, well beyond the 0.1 Å threshold for reliable identification. Prior generalization claims for foundation potentials were established mainly on pristine crystals; this work extends the evaluation to charged point defects as an out-of-distribution regime and attributes the failures to a domain gap in training data and the absence of charge-state descriptors in the architectures. PBE-DFT, matching the functional level of the foundation models' training data, serves as reference; the same ShakeNBreak-generated initial configurations are relaxed by each model, and per-charge-state RMSD distributions and potential energy surface shapes are compared.

Adding global charge embeddings to MACE (total charge mapped to a learnable vector added to atomic species embeddings, contributing energy through both message passing and a geometry-independent readout) yields a single joint model that distinguishes all five charge states, with test-set averages of RMSEE = 0.48 meV/atom and RMSEF = 20.15 meV/Å and ground-state RMSD within 0.05 Å for all charge states. Unlike approaches that train separate models per charge state, this work handles multiple charge states within one architecture and shows that different charge states form separable clusters in descriptor space. Based on HSE06 training data with shortest-bond-length-based filtering and an 80/20 split, the work reports training/test error tables, principal component analysis of the latent space, and comparisons of ground-state structures and thermodynamic transition levels.

The charge-embedded model reproduces defect thermodynamics in close agreement with the HSE06 reference: thermodynamic transition levels deviate by at most about 0.015 eV, and the model distinguishes metastable configurations with energy differences as small as 1 meV/atom. Validation of the machine learning potential is extended from energy and force errors to physical quantities such as formation energy diagrams and transition levels, covering five charge states from q = +2 to q = −2. Formation energies and transition levels are computed with the equations given in the text, finite-size electrostatic corrections come from HSE06 single-point calculations on ML-relaxed geometries, and results are tabulated against DFT.

Multi-fidelity training, combining dense PBE sampling with about 10% HSE06 single-point corrections, discovers a lower global minimum missed by conventional HSE06 Γ-point coarse search (about 0.02 eV lower for neutral VSe(1)); this prediction is confirmed by converged k-point HSE06 within 8 meV, and the approach reduces the cost of comprehensive potential energy surface exploration by roughly three orders of magnitude. The conventional two-stage workflow assumes coarse sampling reproduces the high-fidelity surface shape and its global minimum; this work instead learns the functional-dependent difference directly and quantifies the cost reduction. Using neutral VSe(1) as an example, configurations and energies from the multi-fidelity model are compared with the ShakeNBreak workflow, and timing magnitudes for HSE06 and PBE relaxations on a single node are reported.

The results target charged point defects in semiconductors and insulators, with concrete validation on VSb(1) and VSe(1) in Sb2Se3, a 3×1×1 supercell (59 atoms with the vacancy), and charge states from q = +2 to q = −2. The approach suits settings that need hybrid-functional-level potential energy surfaces but where direct searching is too costly, such as defect structure screening, formation energy and transition level prediction, and densely sampled properties such as configuration coordinate diagrams. The text notes that the model is trained at a fixed supercell size, so size consistency across supercell sizes should not be assumed, and transfer across inequivalent defect sites does not hold when the relevant local environments are absent from the training data.

A careful reader would still watch how the charge embedding distributes charge-dependent energy between defect-local and bulk-like atoms, since at a fixed supercell size this distribution is not uniquely determined and size consistency when the number of bulk-like atoms changes remains an open question; the model trained on VSe(1) does not reproduce the DFT potential energy surface for neutral VSe(2), so the conditions for cross-site transfer need further definition; explicit electrostatics schemes (QET, MACE-POLAR, MACE-LES) generally miss ground states in the negative charge states, indicating that modeling the coupling between charge state and local bonding still leaves room for design choices; in addition, the text read here is a fast parse in which figures and some supplementary details are not fully rendered, so conclusions involving specific numbers should be checked against the original figures and tables.