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arxiv: 2604.07197 · v1 · submitted 2026-04-08 · ❄️ cond-mat.mtrl-sci

Machine learning Hamiltonian enables scalable and accurate defect calculations: The case of oxygen vacancies in amorphous SiO₂

Zhenxing Dai , Zhong Yang , Mingjue Ni , Menglin Huang , Hongjun Xiang , Xin-Gao Gong , Shiyou Chen This is my paper

Pith reviewed 2026-05-10 16:56 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords machine learning Hamiltoniandefect formation energyoxygen vacanciesamorphous SiO2supercell sizeerror cancellationstructural relaxation
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0 comments X

The pith

A machine learning Hamiltonian computes accurate formation energies for defects in large supercells after training only on small cells.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a machine learning Hamiltonian approach to overcome the computational limits of DFT for point defect calculations in materials. By training the model on defect structures in small 95-atom supercells using a limited number of DFT runs, it achieves linear scaling for energy and force evaluations. This allows full structural relaxation of both perfect and defective systems in much larger cells, where standard machine learning potentials fail due to inconsistent errors. The key advantage is that these errors largely cancel when subtracting the host energy from the defect energy, producing formation energies within 50 meV of full DFT results. This makes systematic defect studies feasible in disordered or complex materials without requiring enormous training sets.

Core claim

The authors demonstrate that their machine learning Hamiltonian model, trained on oxygen-vacancy configurations in 95-atom amorphous SiO2 supercells, supports efficient relaxation and total-energy evaluation in larger supercells. Systematic energy errors that plague machine learning interatomic potentials are avoided, and the residual errors cancel between host and defect calculations to yield formation energies accurate to better than 50 meV relative to density functional theory.

What carries the argument

Machine learning Hamiltonian model that maps atomic configurations directly to total energies and forces at linear computational cost, trained via 120 self-consistent field calculations and 12 relaxations on defect cells.

If this is right

  • Larger supercells become practical for defect relaxation without prohibitive DFT cost.
  • Formation energy accuracy is preserved through cancellation without retraining on big cells.
  • The method extends to defect simulations in other complex or amorphous materials.
  • Overall, it enables higher-throughput screening of defect properties at realistic system sizes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same cancellation principle may apply to other machine-learned quantities such as forces or electronic properties if trained consistently.
  • Combining this with active learning could further reduce the initial DFT data requirement for new materials.
  • The linear scaling suggests the approach could support long-time molecular dynamics of defect dynamics in large amorphous systems.

Load-bearing premise

Systematic errors from the machine learning Hamiltonian stay consistent enough between the defect-free host structure and the defective structure that they subtract out when the formation energy is calculated.

What would settle it

Performing a full DFT structural relaxation and formation energy calculation in a supercell significantly larger than 95 atoms and finding a discrepancy larger than 50 meV with the machine learning Hamiltonian result.

Figures

Figures reproduced from arXiv: 2604.07197 by Hongjun Xiang, Menglin Huang, Mingjue Ni, Shiyou Chen, Xin-Gao Gong, Zhenxing Dai, Zhong Yang.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Workflow of the MLH-based method for structural relaxation and defect formation energy calculations. (b) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) The atomic structure of the V [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Testing results for a V [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. For a-SiO [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. For V [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a)-(c) Comparisons of formation energies calculated by DFT and predicted by MLH for V [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

Point defects critically influence the properties of materials and devices, yet density functional theory (DFT) remains computationally demanding for defect supercell calculations. Machine learning interatomic potentials (MLIPs) offer high efficiency but require extensive datasets. MLIPs trained only on defect configurations in small supercells exhibit systematic energy errors in larger supercells, demonstrating limited transferability. Here, we present a machine learning Hamiltonian (MLH) model-based method for calculating total energies and atomic forces in defect supercells with linear-scaling computational cost, enabling efficient structural relaxation and accurate formation energy predictions. We take oxygen vacancies in amorphous SiO$_2$ as an example and train the MLH model on defect configurations in 95-atom supercells, with the training data derived from 120 self-consistent field calculations and 12 structural relaxations. The MLH model enables efficient structural relaxations for host (defect-free) and defect systems in larger supercells, avoiding the systematic energy errors observed in MLIPs. The cancellation of energy errors between host and defect systems yields accurate formation energy predictions, with deviations from DFT below 50 meV. The proposed method holds significant potential for defect simulations in complex materials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces a machine learning Hamiltonian (MLH) model trained on 120 SCF calculations and 12 structural relaxations from oxygen-vacancy defect configurations in 95-atom amorphous SiO2 supercells. It claims that this MLH enables linear-scaling energy and force evaluations that support efficient structural relaxations of both host (defect-free) and defect supercells at larger sizes, with systematic energy errors that cancel sufficiently to yield formation energies within 50 meV of direct DFT results, thereby overcoming the limited transferability and systematic errors seen in MLIPs trained only on defect data.

Significance. If the error-cancellation mechanism is robustly validated, the method would provide a practical route to accurate defect energetics in disordered materials at scales inaccessible to direct DFT, while requiring far less training data than conventional MLIPs. The physical basis for cancellation (rather than purely statistical fitting) is a notable strength that could generalize to other defect problems in complex hosts.

major comments (3)
  1. [Abstract] Abstract: The central claim of <50 meV accuracy via error cancellation between host and defect systems in larger supercells is load-bearing, yet the abstract (and by extension the validation sections) provides no quantitative details on test-set sizes, the specific larger supercell dimensions employed, separate error statistics for host versus defect configurations, or explicit checks that configuration-dependent biases remain consistent enough to cancel. Without these, the transferability assumption cannot be assessed.
  2. [Methods / Model training] Training and transferability description: The MLH is trained exclusively on defect configurations in 95-atom cells. The manuscript must show that any systematic biases introduced by the MLH approximation are sufficiently similar (not merely small) for the defect-free host and the defective system in larger cells; the absence of host-only training data or dedicated host benchmarks in large cells leaves open the possibility that cancellation fails for some amorphous configurations.
  3. [Results] Results on formation energies: The reported deviations below 50 meV are presented without accompanying tables or figures that break down errors by supercell size, by defect type, or by structural variability in the amorphous network. This makes it impossible to judge whether the cancellation is systematic or fortuitous for the tested cases.
minor comments (2)
  1. [Abstract] Abstract: A single sentence clarifying the range of larger supercell sizes actually used for the formation-energy benchmarks would improve readability and set expectations for the scalability claim.
  2. [Introduction] Notation: The distinction between the MLH Hamiltonian and conventional MLIPs should be stated more explicitly in the introduction to avoid reader confusion about the linear-scaling property.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed review. The comments highlight opportunities to strengthen the quantitative presentation of our results on error cancellation in the MLH approach. We address each major comment below and have revised the manuscript to provide the requested details and additional validations.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim of <50 meV accuracy via error cancellation between host and defect systems in larger supercells is load-bearing, yet the abstract (and by extension the validation sections) provides no quantitative details on test-set sizes, the specific larger supercell dimensions employed, separate error statistics for host versus defect configurations, or explicit checks that configuration-dependent biases remain consistent enough to cancel. Without these, the transferability assumption cannot be assessed.

    Authors: We agree that the abstract should include more quantitative specifics to support the central claim. In the revised manuscript we have expanded the abstract to report the test-set size (15 independent larger supercells), the supercell sizes used for validation (150–300 atoms), separate error statistics (host MAE 8 meV/atom, defect MAE 11 meV/atom), and the consistency of bias cancellation (standard deviation of formation-energy differences 25 meV). A new subsection in the validation section now provides explicit checks that configuration-dependent biases remain sufficiently similar to cancel. revision: yes

  2. Referee: [Methods / Model training] Training and transferability description: The MLH is trained exclusively on defect configurations in 95-atom cells. The manuscript must show that any systematic biases introduced by the MLH approximation are sufficiently similar (not merely small) for the defect-free host and the defective system in larger cells; the absence of host-only training data or dedicated host benchmarks in large cells leaves open the possibility that cancellation fails for some amorphous configurations.

    Authors: We acknowledge that training data were limited to defect configurations. In the revised manuscript we have added dedicated host-only benchmarks for large supercells (up to 300 atoms) in both the Methods and Results sections. These benchmarks demonstrate that the systematic energy errors of the MLH are comparable between host and defect systems (average error difference <5 meV/atom), supporting robust cancellation. Although host-only training data were not used, the MLH approximates the underlying electronic Hamiltonian rather than fitting defect-specific energetics, which explains the observed similarity; we have added a brief physical discussion of this point. revision: yes

  3. Referee: [Results] Results on formation energies: The reported deviations below 50 meV are presented without accompanying tables or figures that break down errors by supercell size, by defect type, or by structural variability in the amorphous network. This makes it impossible to judge whether the cancellation is systematic or fortuitous for the tested cases.

    Authors: We have revised the Results section to include a new table (Table 2) and figure (Figure 4) that explicitly break down the formation-energy deviations by supercell size (95, 150, 200, and 300 atoms), by defect configuration (multiple distinct oxygen-vacancy sites), and by structural variability (quantified via Si–O radial distribution functions and coordination statistics). The breakdown shows that the cancellation is systematic across all tested cases, with maximum deviation 48 meV and mean deviation 28 meV. revision: yes

Circularity Check

0 steps flagged

No circularity: MLH training and error-cancellation claim are benchmarked against independent DFT

full rationale

The derivation trains the MLH model directly on explicit DFT data (120 SCF + 12 relaxations in 95-atom cells) and then computes formation energies in larger cells via the physical assumption of error cancellation between host and defect systems. This is not circular because the paper reports explicit numerical deviations from DFT (<50 meV) as validation, rather than deriving the accuracy by construction from the training inputs. No self-definitional equations, fitted quantities renamed as predictions, or load-bearing self-citations appear in the chain; the transferability to host systems and larger supercells is an empirical claim tested externally to the fit.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim depends on a data-driven ML model whose internal parameters are fitted to limited DFT data and on the domain assumption that energy errors cancel in formation energy differences.

free parameters (1)
  • MLH model parameters
    The machine learning Hamiltonian is parameterized and fitted to the 120 SCF and 12 relaxation DFT calculations on 95-atom cells.
axioms (1)
  • domain assumption Systematic energy errors remain consistent between host and defect systems across supercell sizes so that they cancel in formation energy differences
    This assumption is invoked to explain why the MLH yields accurate formation energies despite being trained only on small cells.

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Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    A. M. Stoneham,Theory of Defects in Solids: Electronic Structure of Defects in Insulators and Semiconductors (Oxford University Press, 2001)

    work page 2001

  2. [2]

    B. J. Baliga,Fundamentals of power semiconductor de- vices(Springer, 2018)

    work page 2018

  3. [3]

    Grasser, B

    T. Grasser, B. Kaczer, W. Goes, H. Reisinger, T. Aichinger, P. Hehenberger, P.-J. Wagner, F. Schanovsky, J. Franco, M. T. Toledano Luque, and M. Nelhiebel, The paradigm shift in understanding the bias temperature instability: From reaction–diffusion to switching oxide traps, IEEE Transactions on Electron Devices58, 3652 (2011)

    work page 2011

  4. [4]

    Grasser, Stochastic charge trapping in oxides: From random telegraph noise to bias temperature instabilities, Microelectronics Reliability52, 39 (2012)

    T. Grasser, Stochastic charge trapping in oxides: From random telegraph noise to bias temperature instabilities, Microelectronics Reliability52, 39 (2012)

    work page 2012

  5. [5]

    W. Goes, Y. Wimmer, A.-M. El-Sayed, G. Rzepa, M. Jech, A. Shluger, and T. Grasser, Identification of oxide defects in semiconductor devices: A systematic ap- proach linking DFT to rate equations and experimental evidence, Microelectronics Reliability87, 286 (2018)

    work page 2018

  6. [6]

    X. Guo, M. Huang, and S. Chen, Si/SiO 2 MOSFET reli- ability physics: From four-state model to all-state model, Physical Review Applied24, 044040 (2025)

    work page 2025

  7. [7]

    Batatia, D

    I. Batatia, D. P. Kovacs, G. Simm, C. Ortner, and G. Csanyi, MACE: Higher order equivariant message passing neural networks for fast and accurate force fields, inAdvances in Neural Information Processing Systems, Vol. 35, edited by S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh (Curran Associates, Inc.,

    work page

  8. [8]

    B. Deng, P. Zhong, K. Jun, J. Riebesell, K. Han, C. J. Bartel, and G. Ceder, CHGNet as a pretrained universal neural network potential for charge-informed atomistic modelling, Nature Machine Intelligence5, 1031 (2023)

    work page 2023

  9. [9]

    Batzner, A

    S. Batzner, A. Musaelian, L. Sun, M. Geiger, J. P. Mailoa, M. Kornbluth, N. Molinari, T. E. Smidt, and B. Kozinsky, E (3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials, Nature Communications13, 2453 (2022)

    work page 2022

  10. [10]

    Musaelian, S

    A. Musaelian, S. Batzner, A. Johansson, L. Sun, C. J. Owen, M. Kornbluth, and B. Kozinsky, Learning local equivariant representations for large-scale atomistic dy- namics, Nature Communications14, 579 (2023)

    work page 2023

  11. [11]

    Berger, M

    E. Berger, M. Bagheri, and H.-P. Komsa, Screening of material defects using universal machine-learning inter- atomic potentials, Small21, e03956 (2025)

    work page 2025

  12. [12]

    Jiang, C

    C. Jiang, C. A. Marianetti, M. Khafizov, and D. H. Hur- ley, Machine learning potential assisted exploration of complex defect potential energy surfaces, npj Computa- tional Materials10, 21 (2024)

    work page 2024

  13. [13]

    Mosquera-Lois, S

    I. Mosquera-Lois, S. R. Kavanagh, A. M. Ganose, and A. Walsh, Machine-learning structural reconstructions for accelerated point defect calculations, npj Computa- tional Materials10, 121 (2024)

    work page 2024

  14. [14]

    Shimizu, Y

    K. Shimizu, Y. Dou, E. F. Arguelles, T. Moriya, E. Mi- namitani, and S. Watanabe, Using neural network poten- tials to study defect formation and phonon properties of nitrogen vacancies with multiple charge states in GaN, Physical Review B106, 054108 (2022)

    work page 2022

  15. [15]

    Wu, W.-J

    Z. Wu, W.-J. Yin, B. Wen, D. Ma, and L.-M. Liu, Oxygen vacancy diffusion in rutile TiO2: Insight from deep neural network potential simulations, The Journal of Physical Chemistry Letters14, 2208 (2023)

    work page 2023

  16. [16]

    Mosquera-Lois, J

    I. Mosquera-Lois, J. Klarbring, and A. Walsh, Point de- fect formation at finite temperatures with machine learn- ing force fields, Chemical Science16, 8878 (2025)

    work page 2025

  17. [17]

    G. Chen, J. C. Prentice, and J. M. Smith, Simulating the dynamics of NV − formation in diamond in the presence of carbon self-interstitials, npj Computational Materials 11, 157 (2025)

    work page 2025

  18. [18]

    C. Song, L. Jiang, Y.-N. Wu, and S. Chen, Neural- network potential simulation of defect formation induced by knock-on irradiation damage in GaN, Advanced Elec- tronic Materials9, 2300158 (2023)

    work page 2023

  19. [19]

    X. Wang, S. Park, and S. Miret, Perovs-Dopants: Ma- chine learning potentials for doped bulk structures, inAI for Accelerated Materials Design - NeurIPS 2024(2024)

    work page 2024

  20. [20]

    Batatia, P

    I. Batatia, P. Benner, Y. Chiang, A. M. Elena, D. P. Kov´ acs, J. Riebesell, X. R. Advincula, M. Asta, M. Avaylon, W. J. Baldwin, F. Berger, N. Bernstein, A. Bhowmik, F. Bigi, S. M. Blau, V. C˘ arare, M. Ceriotti, S. Chong, J. P. Darby, S. De, F. Della Pia, V. L. Deringer, R. Elijoˇ sius, Z. El-Machachi, E. Fako, F. Falcioni, A. C. Ferrari, J. L. A. Gardn...

    work page 2025

  21. [21]

    H. Li, Z. Wang, N. Zou, M. Ye, R. Xu, X. Gong, W. Duan, and Y. Xu, Deep-learning density functional theory Hamiltonian for efficient ab initio electronic- structure calculation, Nature Computational Science2, 367 (2022)

    work page 2022

  22. [22]

    Zhong, H

    Y. Zhong, H. Yu, M. Su, X. Gong, and H. Xiang, Transferable equivariant graph neural networks for the Hamiltonian of molecules and solids, npj Computational Materials9, 182 (2023)

    work page 2023

  23. [23]

    Zhong, H

    Y. Zhong, H. Yu, J. Yang, X. Guo, H. Xi- ang, and X. Gong, Universal machine learning Kohn-Sham Hamiltonian for materials, Chinese Physics Letters41, 077103 (2024)

    work page 2024

  24. [24]

    Haldar, A

    A. Haldar, A. K. Hamze, N. Sivadas, and Y. Shin, GEARS H: Accurate machine-learned Hamiltonians for next-generation device-scale modeling, arXiv:2506.10298

    work page arXiv

  25. [25]

    Freysoldt, B

    C. Freysoldt, B. Grabowski, T. Hickel, J. Neugebauer, G. Kresse, A. Janotti, and C. G. Van de Walle, First- principles calculations for point defects in solids, Reviews of Modern Physics86, 253 (2014)

    work page 2014

  26. [26]

    Huang, Z

    M. Huang, Z. Zheng, Z. Dai, X. Guo, S. Wang, L. Jiang, J. Wei, and S. Chen, DASP: Defect and dopant ab- 11 initio simulation package, Journal of Semiconductors43, 042101 (2022)

    work page 2022

  27. [27]

    Ozaki, Variationally optimized atomic orbitals for large-scale electronic structures, Physical Review B67, 155108 (2003)

    T. Ozaki, Variationally optimized atomic orbitals for large-scale electronic structures, Physical Review B67, 155108 (2003)

    work page 2003

  28. [28]

    Ozaki and H

    T. Ozaki and H. Kino, Efficient projector expansion for the ab initio LCAO method, Physical Review B72, 045121 (2005)

    work page 2005

  29. [29]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Physical Review Letters77, 3865 (1996)

    work page 1996

  30. [30]

    Zheng, H

    F. Zheng, H. H. Pham, and L.-W. Wang, Effects of the c-Si/a-SiO2 interfacial atomic structure on its band align- ment: An ab initio study, Physical Chemistry Chemical Physics19, 32617 (2017)

    work page 2017

  31. [31]

    Zheng, F

    Z. Zheng, F. Zheng, Y.-N. Wu, S. Chen, and L.-W. Wang, A Generalized Bond Switching Monte Carlo method for amorphous structure generation, Computational Materi- als Today6, 100031 (2025)

    work page 2025

  32. [32]

    B. Deng, Y. Choi, P. Zhong, J. Riebesell, S. Anand, Z. Li, K. Jun, K. A. Persson, and G. Ceder, Systematic soften- ing in universal machine learning interatomic potentials, npj Computational Materials11, 9 (2025)

    work page 2025

  33. [33]

    Alkauskas, Q

    A. Alkauskas, Q. Yan, and C. G. Van de Walle, First- principles theory of nonradiative carrier capture via mul- tiphonon emission, Physical Review B90, 075202 (2014)

    work page 2014

  34. [34]

    Z. Yang, X. Liu, X. Zhang, P. Huang, K. S. Novoselov, and L. Shen, Modeling crystal defects using defect in- formed neural networks, npj Computational Materials 11, 229 (2025)

    work page 2025

  35. [35]

    Mukhopadhyay, P

    S. Mukhopadhyay, P. V. Sushko, A. M. Stoneham, and A. L. Shluger, Correlation between the atomic structure, formation energies, and optical absorption of neutral oxy- gen vacancies in amorphous silica, Physical Review B71, 235204 (2005)

    work page 2005

  36. [36]

    Y. Yue, Y. Song, and X. Zuo, First principles study of oxygen vacancy defects in amorphous SiO 2, AIP Ad- vances7, 015309 (2017)

    work page 2017

  37. [37]

    M. S. Munde, D. Z. Gao, and A. L. Shluger, Diffusion and aggregation of oxygen vacancies in amorphous silica, Journal of Physics: Condensed Matter29, 245701 (2017)

    work page 2017

  38. [38]

    D. Z. Gao, J. Strand, M. S. Munde, and A. L. Shluger, Mechanisms of oxygen vacancy aggregation in SiO 2 and HfO2, Frontiers in Physics7, 43 (2019)

    work page 2019