Pith. sign in

REVIEW 3 major objections 6 minor 1 cited by

Learning the Electronic Hamiltonian of Large Atomic Structures

T0 review · 3 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper claims that a strictly local, equivariant graph neural network, trained on sliced graphs with virtual boundary nodes, can predict the density functional theory electronic Hamiltonian of amorphous materials containing thousands…

desk verdict A well-executed local equivariant GNN for large disordered materials with a genuine partitioning trick, but the headline eigenvalue error likely compares the wrong spectral object and needs fixing. read the letter →

arxiv 2501.19110 v2 pith:DPLRXUJY submitted 2025-01-31 cond-mat.mtrl-sci cond-mat.dis-nn

Chen Hao Xia , Manasa Kaniselvan , Alexandros Nikolaos Ziogas , Marko Mladenović , Rayen Mahjoub , Alexander Maeder , Mathieu Luisier This is my paper · ORCID
classification cond-mat.mtrl-scicond-mat.dis-nn
keywords Hamiltonianpredictionequivariantgraphneuralnetworksaugmentedpartitioningvirtualboundarynodesamorphousmaterialsdensityfunctionaltheoryquantumtransportnear-sightedness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper attempts to show that graph neural networks can learn the ground-state electronic Hamiltonian H of realistically large disordered materials (amorphous HfO2, GeSbTe, and PtGe with 1,008 to 3,000 atoms) at accuracies previously reached only for molecules and small unit cells. The authors introduce augmented partitioning: the large periodic structure is split into slices, and atoms outside each slice but within the interaction cutoff are added as virtual nodes and edges, so each slice keeps its full connectivity while fitting in GPU memory. A strictly local equivariant message-passing network, which sees only one hop of virtual neighbors, predicts H blocks, and the assembled matrix reproduces the DFT eigenvalue spectrum to within 0.53% relative L1 error. The predicted Hamiltonian also drives quantum transport calculations whose currents track DFT currents across five orders of magnitude, indicating that the approach may extend ML electronic-structure prediction to disorder, interfaces, and defects.

What carries the argument

The central mechanism is augmented partitioning combined with an SO(2)-equivariant architecture. Hamiltonian blocks between orbitals of angular momenta lα and lβ are transformed from the uncoupled basis to a coupled angular-momentum basis via Clebsch-Gordan coefficients; the network then updates node and edge embeddings through eSCN-style convolutions, whose cost scales as O($l_max^{3}$), plus multi-headed equivariant attention, and maps the outputs back to the uncoupled basis to reconstruct H. Training graphs are sliced along the longest cell axis, and atoms outside a slice but within rcut are included as virtual nodes whose outputs are discarded, so each slice sees its full connectivity while staying within GPU memory. Near-sightedness evidence, in which single atomic perturbations affect onsite blocks by at most 0.24% at 8 Å, is what licenses the strict locality of the model.

What would settle it

A direct test is to compute the reference DFT Hamiltonian of one 3,000-atom structure with the full interaction range and with the same Hamiltonian truncated at 8 angstroms, then compare the eigenvalue spectra; the paper's single-perturbation decays do not measure the accumulated effect of all surrounding atoms, so this comparison would settle whether 8 angstroms is sufficient.

Watch

Extended reading notes

Core claim

The central claim is that Hamiltonian blocks Hi,i and Hi,j are determined by local atomic environments, and that a network which never sees beyond the 1-hop neighborhood (containing all atoms within an rcut of 8 to 12 Å) can predict H for large amorphous structures as accurately as full-graph training. Evidence comes from experiments on amorphous HfO2, GeSbTe, and PtGe with test structures of 1,008 to 3,000 atoms and up to 1.79 million edges: node and edge block errors fall in the range 0.77 to 2.16 mEh and 0.08 to 0.16 mEh respectively, with total errors between 2.17 and 2.58 meV. The paper further reports that partitioning a graph into as many as 27 slices changes accuracy negligibly compared to full-graph training, and that diagonalizing the predicted Hamiltonian reproduces the reference eigenvalue spectrum within 0.530% relative L1 error (0.446% when only occupied states are included).

Load-bearing premise

The whole method rests on assuming that every piece of the Hamiltonian depends only on atoms within roughly 8 to 12 angstroms, so that one shell of virtual neighbor atoms is enough to reproduce the behavior of the full material.

Editorial extensions

If this is right

  • Training can be done on small slices of a huge graph without hurting test accuracy, removing the memory bottleneck that prevents full-graph GNN training on 3,000-atom systems.
  • Predicted Hamiltonians can replace DFT Hamiltonians as inputs to quantum transport simulations, reproducing current trends across five orders of magnitude and matching physically relevant features such as band edges.
  • Raising the cutoff from 8 to 12 Å captures all nonzero matrix elements and lowers total error to 2.58 meV, showing that accuracy improves when memory allows larger neighborhoods.
  • Models trained on one sub-stoichiometric composition transfer to other compositions: training on a-HfO1.8 predicts a-HfO1.7, a-HfO1.8, and a-HfO1.9 with errors between 2.34 and 2.45 mEh on node blocks.
  • Inference scales linearly with the number of atoms, unlike DFT's cubic scaling, so the approach targets the size regime where reference calculations cost hours per structure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: because the partitioning machinery is material-agnostic, it should transfer to other near-sighted electronic-structure targets, such as overlap matrices, electron-phonon couplings, or phonon dispersions, with little modification.
  • Editorial inference: the strictly local premise implies a direct scaling path to even larger systems; since partitions exchange only loss values during training, scaling to hundreds of slices on many GPUs is a natural next benchmark.
  • Editorial inference: a clean validation experiment the paper does not report is computing cumulative truncation error by comparing full and rcut-limited reference Hamiltonians on the same 3,000-atom structure, which would directly test whether 8 Å locality holds for all surrounding atoms, not just single perturbations.
  • Editorial inference: the same predicted Hamiltonian could serve as an initial guess to warm-start DFT self-consistent field iterations, reducing the number of SCF cycles needed to reach convergence in large disordered systems.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper presents a strictly local, SO(2)-equivariant graph neural network for predicting the DFT Hamiltonian matrix H of large, disordered materials, together with an 'augmented partitioning' training scheme in which slice sub-graphs are supplemented by virtual nodes and edges that preserve the one-hop connectivity across partition boundaries. The model is trained on slices of one large amorphous structure per material (a-HfO2, a-GST, a-PtGe), validated on slices of a second structure, and tested on full unseen periodic cells with up to 3,000 atoms and 1.79 million edges. Reported element-wise errors on H are 2.17-2.58 meV; for a 3,000-atom a-HfO2 test structure the eigenvalues of the predicted H are claimed to match the reference within 0.530% relative L1 error (0.446% for occupied states). The predicted Hamiltonians are also fed into Landauer-Büttiker transport simulations of TiN/a-HfOx/Ti/TiN stacks, reproducing current differences spanning five orders of magnitude to within roughly 30%. Ablations cover virtual nodes (Table 2), partition thickness (Tables 3 and 11), cutoff radius (Tables 8-9), and vacancy stoichiometry transfer (Tables 5, 10, and 12); the backbone is benchmarked on MD17 molecules (Table 13). Code and data are publicly released.

Significance. If the claims hold, this is a substantial practical advance: it is, to my knowledge, the first demonstration of Hamiltonian prediction with an equivariant GNN at the 3,000-atom scale with on-the-fly graph partitioning, and the device-transport application shows a credible path to ML-driven electronic device modeling. The paper deserves credit for shipping open code and datasets, for a thorough ablation record (virtual-node gain, partition-granularity invariance, rcut sensitivity, vacancy-concentration transfer), and for validating against held-out CP2K DFT labels and an external current reference rather than only against training-style metrics. The main caveat is the eigenvalue-spectrum validation, which currently diagonalizes an eigenproblem that does not match the paper's own DFT formalism; this is a load-bearing gap for the abstract's quantitative headline, although it appears fixable within the scope of the manuscript.

major comments (3)
  1. [Section 4.4 / Figure 5; Section 2; Appendix D.4] The eigenvalue-spectrum validation diagonalizes only the symmetrized Hamiltonian (H = 1/2(H + H†)) and never incorporates the overlap matrix S, even though Section 2 defines the Kohn-Sham eigenproblem in the contracted Gaussian basis as the generalized problem Hψ = εSψ and Appendix D.4 states that S is generated together with H. For the SZV/DZVP bases used in this work S is not the identity, so the eigenvalues of the bare H matrices are not, in general, the Kohn-Sham eigenvalues, and the reported 0.530% relative L1 error (and the 0.446% occupied-state variant) is a spectral error of a different eigenproblem. Because the eigenvalue spectrum is the headline large-scale accuracy claim in the abstract and Section 1, this needs to be fixed: either solve Hψ = εSψ with the ground-truth S (which, as Section 2 notes, is determined by the geometry alone) for both the predicted and reference Hamiltonians and report those errors, or give quantitative evidence that S is close to the identity for these bases.
  2. [Section 3.3; Appendix F; Tables 3, 8-9] The claim that training on one-hop-augmented slices is equivalent to full-graph inference rests on the locality assumption that each Hamiltonian block is determined by the geometry within rcut. The evidence provided — single-perturbation decays of 0.12-0.24% at 8 Å (Appendix F), eigenvalue insensitivity to rcut ≥ 8 Å (Figure 9), and partition-size insensitivity (Table 3) — is appropriate but indirect: the perturbation analysis probes the reference Hamiltonian, not the model's accumulated error from the entire out-of-receptive-field environment, and Tables 8-9 show that increasing rcut from 8 to 12 Å still improves edge errors (0.16 to 0.09 mEh for HfO2), indicating the model is not saturated at the smaller cutoff. A direct test of receptive-field sufficiency — for example, a two-message-passing-layer ablation at fixed rcut, or a comparison of slice-augmented and full-graph inference on a structure small enough to fit in memory — would substantiate the equivalence claim quantitatively.
  3. [Abstract; Section 4.4; Table 4] The eigenvalue-spectrum validation is performed on a single test structure (a-HfO2 structure 3). The paper's justification — that a-HfO2 has the highest element-wise error in Table 4 and thus bounds the other materials — is reasonable, but the abstract's unqualified '≤0.53% error in the eigenvalue spectra' claim is stronger than what is demonstrated. The claim should either be scoped to the a-HfO2 test structure or accompanied by eigenvalue spectra for a-GST and a-PtGe.
minor comments (6)
  1. [Section 4.4 / Figure 5 caption] The occupation threshold separating occupied from unoccupied states is 0.306 Eh in the main text but 0.305 Eh in the Figure 5 caption; the two numbers should be made consistent.
  2. [Appendix I.1] The sentence '0.13-10.18 mEh respectively' contains a typo; the intended range is presumably 0.13-0.18 mEh, matching Table 10.
  3. [Appendix D.1] The Hubbard correction is specified as U = 7 eV on the 3d orbital of Ti for the a-HfO2 DFT reference, but the a-HfO2 structures contain no Ti atoms; this appears to be an error carried over from the electrode workflow and should be corrected or clarified.
  4. [Section 4.5] The unit 'node hours' should be written as 'node-hours' (or 'core-hours') for clarity.
  5. [Abstract / Table 1] The abstract cites 'up to 3,000 nodes, 500,000+ edges', but Table 1 reports a structure with 1,792,760 edges (rcut = 12 Å); the abstract's edge count should be updated or the statement reworded.
  6. [Appendix I.3 / Table 13] The MD17 rows for malondialdehyde and uracil use a different validation split (50 vs 500) and different hyperparameters (f64, 2 MP layers) than the main experiments; a sentence clarifying that the backbone was re-tuned for MD17 would prevent misinterpretation of the comparison.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; the Hamiltonian predictions are validated against independent DFT labels and downstream eigenvalue/current errors are computed from the predicted H, not fitted to the reported targets.

full rationale

The derivation chain is self-contained with respect to the paper's own claims. The network is trained with an MSE loss on DFT-computed Hamiltonian blocks (CP2K PBE/PBE+U) and validated on held-out full structures from the same pipeline, plus an independent MD17/QHNet benchmark and quantum-transport currents compared with DFT references. The reported eigenvalue and current errors are post-hoc diagnostics computed from the predicted H, not quantities that were fit or defined in terms of the reported targets. The augmented-partitioning scheme changes the training graph but not the inference graph; one-hop virtual nodes are an implementation device, and the locality assumption is tested by single-perturbation decay analyses in Appendix F. The paper's self-citations (e.g., Kaniselvan et al., Ducry et al.) are used for structure generation and context, not as a uniqueness theorem or as the evidence for the central prediction. The only notable caveat is a correctness risk rather than a circularity: Section 2 defines DFT eigenvalues via Hψ=εSψ, while Section 4.4 diagonalizes H alone after symmetrization without applying S; if S is materially different from identity in the SZV/DZVP basis, the reported 0.530% eigenvalue error may not correspond to the physical Kohn-Sham spectrum. This does not make any prediction equivalent to its inputs, so the circularity score remains 0.

Assumptions & free parameters 5 free parameters · 5 assumptions · 2 invented entities

The method rests on four premises: DFT ground truth (standard practice), near-sightedness of H in a localized basis (demonstrated in Appendix F), representativeness of a single amorphous realization (asserted in Section 3.4, tested on few structures), and the treatment of the overlap matrix in the spectral comparison (unstated in Section 4.4). Hand-set quantities that affect the reported accuracy are rcut, slice thickness, the number of message passing layers, architecture hyperparameters, and the DFT Hubbard U values; none is fitted to the test targets.

free parameters (5)
  • interaction cutoff radius rcut = 8 Å (12 Å for the best HfO2 and GST results; 10 Å for PtGe)
    Hand-set graph truncation radius. Ablations (Tables 8-9) and the rcut convergence of the eigenvalue spectrum (Figure 9) justify the choice, but the reported accuracy is for the best cutoff per material.
  • partition slice thickness tslice and slice count Nt = 3 Å / 18 slices (a-HfO2), 5 Å / 10 slices (a-PtGe), 5 Å / 6 slices (a-GST)
    Chosen for GPU memory. Tables 3 and 11 show accuracy is insensitive to Nt within the tested range.
  • number of message passing layers = 1 (stated for the HfO2 ablation; not reported in Table 7 for the main models)
    Sets the receptive field exactly to one hop, which is the content of the strict locality premise.
  • architecture hyperparameters (lmax, embedding size, attention heads) = lmax=4, Mmax=4, embedding size 16, heads 2, FFN 64
    Fixed across all main experiments (Table 7); not fitted to the test targets.
  • Hubbard U values in the DFT reference = 10 eV on O 2p; 7 eV on Ti 3d (stated in Appendix D.1 for bulk a-HfO2, which contains no Ti)
    Hand-set DFT corrections that define the training labels; the Ti value appears to be a leftover from the device context.
assumptions (5)
  • standard math The Hamiltonian matrix is a function of atomic positions and identities through the Hohenberg-Kohn mapping F({ri}, {Zi}) -> H (Section 2).
    Foundational DFT statement that makes supervised learning of H well-posed.
  • domain assumption Near-sightedness: H blocks are determined by the local environment within rcut 8-12 Å, including decay of perturbation effects (Appendix F).
    Empirically supported by Figures 8-10 (perturbation response at 8 Å is at most 0.24%), and it underpins both the rcut truncation and the one-hop virtual-node training. The cumulative truncation error from all remote atoms is not directly measured.
  • domain assumption One large amorphous realization per material contains a representative distribution of local motifs (Section 3.4).
    Stated in Section 3.4; tested on one or two unseen structures per material (Table 4 and Table 10).
  • domain assumption CP2K DFT with the PBE functional, Hubbard corrections, and SZV/DZVP Gaussian bases provides the reference H and S (Appendix D).
    Standard practice for label generation; the overlap matrix S is introduced in Section 2 but its use in the Section 4.4 eigenvalue comparison is unstated.
  • ad hoc to paper The eigenvalue spectrum can be obtained by diagonalizing a symmetrized H (Section 4.4, Figure 5).
    Valid only if S is the identity or is folded in; the manuscript does not state which, while Section 2 defines the reference problem as H psi = epsilon S psi.
invented entities (2)
  • Virtual nodes and virtual edges
    purpose: Augment each graph slice with one-hop neighbors from outside the slice so message passing sees the exact boundary connectivity while outputs from virtual nodes are discarded (Algorithm 1, Figure 3c).
    An algorithmic construct, not a physical postulate. The supporting ablation (Table 2) is internal to the paper, though the mechanism is fully specified and shipped in code.
  • Ghost atoms (atomic number 0) representing oxygen vacancies independent evidence
    purpose: Used both in the DFT label generation to mitigate basis set superposition error and in the graph so the network learns to output zero orbital blocks at vacancy sites (Appendix D.2).
    A computational device with an established physical motivation (Senent and Wilson 2001); the vacated site has a real effect on the reference Hamiltonian, so the entity is grounded outside the model.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Learning the Electronic Hamiltonian of Large Atomic Structures." pith.science (2026). https://pith.science/paper/DPLRXUJY

@misc{pith2026250119110,
  author       = {Pith},
  title        = {Pith review of: Learning the Electronic Hamiltonian of Large Atomic Structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DPLRXUJY}},
  note         = {Machine review of arXiv:2501.19110}
}
abstract

Graph neural networks (GNNs) have shown promise in learning the ground-state electronic properties of materials, subverting ab initio density functional theory (DFT) calculations when the underlying lattices can be represented as small and/or repeatable unit cells (i.e., molecules and periodic crystals). Realistic systems are, however, non-ideal and generally characterized by higher structural complexity. As such, they require large (10+ Angstroms) unit cells and thousands of atoms to be accurately described. At these scales, DFT becomes computationally prohibitive, making GNNs especially attractive. In this work, we present a strictly local equivariant GNN capable of learning the electronic Hamiltonian (H) of realistically extended materials. It incorporates an augmented partitioning approach that enables training on arbitrarily large structures while preserving local atomic environments beyond boundaries. We demonstrate its capabilities by predicting the electronic Hamiltonian of various systems with up to 3,000 nodes (atoms), 500,000+ edges, ~28 million orbital interactions (nonzero entries of H), and $\leq$0.53% error in the eigenvalue spectra. Our work expands the applicability of current electronic property prediction methods to some of the most challenging cases encountered in computational materials science, namely systems with disorder, interfaces, and defects.

Figures

Figures reproduced from arXiv: 2501.19110 by the authors.

Figure 1
Figure 1. Schematic of the mapping between the atomic graph and the blocks of the Hamiltonian matrix H in the localized orbital ba￾sis of choice. Each orbital block represents the couplings between atomic orbitals on the same atom (Hi,i, diagonal) or between two different atoms within rcut (Hi,j , off-diagonal). equation at the core of DFT takes the form of a generalized eigenvalue problem: Hψ = εSψ. Here, the Hamiltonian mat… view at source ↗
Figure 2
Figure 2. Illustration of the differences between ideal periodic (left) and compositionally (middle) or structurally (right) disordered materials, whose features can only be described by large unit cells. In all examples, a black box delimits the smallest repeatable unit cell, while the circles/lines correspond to atoms/bonds. this cell to avoid dangling bonds. However, the assumed periodicity can alter the disordered nature … view at source ↗
Figure 3
Figure 3. (a) Data transformation of the Hamiltonian matrix. Blocks of orbital interactions are first extracted from H and reshaped into input tensors, which are transformed into the coupled basis. The tensor corresponding to each node and edge is expanded into a dimension En/Ee, and this set of initial embeddings for every node and edge is sent into (b) the node update block. The node features (nX) are updated using a messag… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Atomic structure example for the three materials we con￾sider. In each case, the black dashed box illustrates the boundaries of the repeating unit cell. has a different local atomic environment, and (2) each struc￾ture contains a large sampling of different motifs. A w…
Figure 5
Figure 5. Figure 5: (a) Eigenvalue spectrum of the predicted (Hpred) and reference (HGT ) Hamiltonian matrices. The alpha value indi￾cates the scatter point transparency. (Hi,j ) pred is symmetrized before diagonalization with H = 1 2 (H + H† ). The rel￾ative L1/L2 errors in the eigenvalu…
Figure 6
Figure 6. Figure 6: TiN/a-HfO2/Ti/TiN structures with different oxygen va￾cancy distributions (top), which lead to distinct electrical transmis￾sion curves (bottom). The positions of vacant oxygen atoms in the former are indicated with red points. The grey boxes in the transmission plots …
Figure 7
Figure 7. Figure 7: a) O-coordination of Hf atoms (number of O atoms bonding a Hf atom) for each of the a-HfO2 structures, showing a distribution around a coordination number of 6, and variation between the structures. b) The radial distribution function (g(r) = dn(r) dr Vdomain 4πr2Natom…
Figure 8
Figure 8. Figure 8: Summary of the Hamiltonian matrix properties for each of the materials in our custom dataset. Shown are (top row) atomic structures, (middle row) frequency of matrix elements as a function of interatomic distances, and (bottom row) distribution of the maximum element o…
Figure 9
Figure 9. Figure 9: Eigenvalues of the ground-truth Hamiltonian matrix, showing (top) the full eigenvalue spectrum and (bottom) the spectrum around the bandgap, for the three structures, in the order of their appearance in [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Scatter plots showing the decay of MAE with increasing distance from different perturbations, including (a) 0.1 A translation ˚ of a Hf atom, (b) replacement of O atom with a vacancy, and (c) replacement of O atom with Hf atom. H. Compute environment and runtime compa…
Figure 11
Figure 11. Figure 11: (a) Time per epoch and (b) speedup resulting from the use of increasing numbers of slices Nt. Median values are shown, while the error bands are one standard deviation. Experiments were run on NVIDIA A100 GPUs with # ranks set to Nt. Measurements are only shown up to …

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Reconstructing the Hamiltonian from the local density of states using neural networks

    cond-mat.dis-nn 2025-09 conditional novelty 5.0 of 10

    A CNN maps single-energy LDOS maps back to the disorder potential in tight-binding models with test MSE 0.016 (1D) and 0.005 (2D).

Reference graph

Works this paper leans on

55 extracted references · 26 canonical work pages · cited by 1 Pith paper

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

  2. [2]

    I., Zaanen, J., and Andersen, O

    Anisimov, V. I., Zaanen, J., and Andersen, O. K. Band theory and mott insulators: Hubbard u instead of stoner i. Physical Review B, 44: 0 943--954, Jul 1991

    work page 1991

  3. [3]

    and De Paula, J

    Atkins, P. and De Paula, J. Atkins' physical chemistry. Oxford University Press, London, England, 9 edition, November 2009

    work page 2009

  4. [4]

    Graph neural network for hamiltonian-based material property prediction

    Bai, H., Chu, P., Tsai, J.-Y., Wilson, N., Qian, X., Yan, Q., and Ling, H. Graph neural network for hamiltonian-based material property prediction. Neural Computing and Applications, 34 0 (6): 0 4625–4632, November 2021. ISSN 1433-3058. doi:10.1007/s00521-021-06616-0. URL http://dx.doi.org/10.1007/s00521-021-06616-0

    work page doi:10.1007/s00521-021-06616-0 2021

  5. [5]

    L., Chen, H., Csányi, G., Ortner, C., and Faber, F

    Batatia, I., Schaaf, L. L., Chen, H., Csányi, G., Ortner, C., and Faber, F. A. Equivariant matrix function neural networks, 2023. URL https://arxiv.org/abs/2310.10434

    arXiv 2023

  6. [6]

    P., Musaelian, A., Simm, G

    Batatia, I., Batzner, S., Kovács, D. P., Musaelian, A., Simm, G. N. C., Drautz, R., Ortner, C., Kozinsky, B., and Csányi, G. The design space of e(3)-equivariant atom-centred interatomic potentials. Nature Machine Intelligence, 7 0 (1): 0 56–67, January 2025. ISSN 2522-5839. doi:10.1038/s42256-024-00956-x. URL http://dx.doi.org/10.1038/s42256-024-00956-x

    doi:10.1038/s42256-024-00956-x 2025

  7. [7]

    Density-functional method for nonequilibrium electron transport

    Brandbyge, M., Mozos, J.-L., Ordejón, P., Taylor, J., and Stokbro, K. Density-functional method for nonequilibrium electron transport. Physical Review B, 65 0 (16), March 2002. ISSN 1095-3795. doi:10.1103/physrevb.65.165401. URL http://dx.doi.org/10.1103/PhysRevB.65.165401

    doi:10.1103/physrevb.65.165401 2002

  8. [8]

    Resistive switching effects of hfo2 high-k dielectric

    Chan, M., Zhang, T., Ho, V., and Lee, P. Resistive switching effects of hfo2 high-k dielectric. Microelectronic Engineering, 85 0 (12): 0 2420–2424, December 2008. ISSN 0167-9317. doi:10.1016/j.mee.2008.09.021. URL http://dx.doi.org/10.1016/j.mee.2008.09.021

    work page doi:10.1016/j.mee.2008.09.021 2008

Show all 55 references
  1. [9]

    Christensen, A. S. and von Lilienfeld, A. Revised MD17 dataset (rMD17) . 7 2020. doi:10.6084/m9.figshare.12672038.v3. URL https://figshare.com/articles/dataset/Revised_MD17_dataset_rMD17_/12672038

    2020 doi

  2. [10]

    R., De Vita, A., Comisso, A., Bernstein, N., and Payne, M

    Cs \'a nyi, G., Winfield, S., Kermode, J. R., De Vita, A., Comisso, A., Bernstein, N., and Payne, M. C. Expressive programming for computational physics in fortran 95+. IoP Comput. Phys. Newsletter, pp.\ Spring 2007, 2007

    2007

  3. [11]

    Electro-thermal transport in disordered nanostructures: a modeling perspective

    Ducry, F., Aeschlimann, J., and Luisier, M. Electro-thermal transport in disordered nanostructures: a modeling perspective. Nanoscale Advances, 2 0 (7): 0 2648–2667, 2020. ISSN 2516-0230. doi:10.1039/d0na00168f. URL http://dx.doi.org/10.1039/d0na00168f

    2020 doi

  4. [12]

    G., and Smidt, T

    Fang, S., Geiger, M., Checkelsky, J. G., and Smidt, T. Phonon predictions with e(3)-equivariant graph neural networks, 2024. URL https://arxiv.org/abs/2403.11347

    2024 arXiv

  5. [13]

    General framework for e(3)-equivariant neural network representation of density functional theory hamiltonian

    Gong, X., Li, H., Zou, N., Xu, R., Duan, W., and Xu, Y. General framework for e(3)-equivariant neural network representation of density functional theory hamiltonian. Nature Communications, 14 0 (1), May 2023. ISSN 2041-1723. doi:10.1038/s41467-023-38468-8. URL http://dx.doi.o...

    2023 doi

  6. [14]

    E., Christensen, R., Dułak, M., Friis, J., Groves, M

    Hjorth Larsen, A., Jørgen Mortensen, J., Blomqvist, J., Castelli, I. E., Christensen, R., Dułak, M., Friis, J., Groves, M. N., Hammer, B., Hargus, C., Hermes, E. D., Jennings, P. C., Bjerre Jensen, P., Kermode, J., Kitchin, J. R., Leonhard Kolsbjerg, E., Kubal, J., Kaasbjerg, ...

    2017

  7. [15]

    and Kohn, W

    Hohenberg, P. and Kohn, W. Inhomogeneous electron gas. Physical Review, 136 0 (3B): 0 B864–B871, November 1964. ISSN 0031-899X. doi:10.1103/physrev.136.b864. URL http://dx.doi.org/10.1103/PhysRev.136.B864

    1964 doi

  8. [16]

    P., Hautier, G., Chen, W., Richards, W

    Jain, A., Ong, S. P., Hautier, G., Chen, W., Richards, W. D., Dacek, S., Cholia, S., Gunter, D., Skinner, D., Ceder, G., and Persson, K. A. Commentary: The materials project: A materials genome approach to accelerating materials innovation. APL Materials, 1 0 (1): 0 011002, 07...

    2013 doi

  9. [17]

    An atomistic model of field-induced resistive switching in valence change memory

    Kaniselvan, M., Luisier, M., and Mladenović, M. An atomistic model of field-induced resistive switching in valence change memory. ACS Nano, 17 0 (9): 0 8281–8292, March 2023. ISSN 1936-086X. doi:10.1021/acsnano.2c12575. URL http://dx.doi.org/10.1021/acsnano.2c12575

    2023 doi

  10. [18]

    and Sham, L

    Kohn, W. and Sham, L. J. Self-consistent equations including exchange and correlation effects. Physical Review, 140 0 (4A): 0 A1133–A1138, November 1965. ISSN 0031-899X. doi:10.1103/physrev.140.a1133. URL http://dx.doi.org/10.1103/PhysRev.140.A1133

    1965 doi

  11. [19]

    V., Fons, P., Frenkel, A

    Kolobov, A. V., Fons, P., Frenkel, A. I., Ankudinov, A. L., Tominaga, J., and Uruga, T. Understanding the phase-change mechanism of rewritable optical media. Nature Materials, 3 0 (10): 0 703–708, September 2004. ISSN 1476-4660. doi:10.1038/nmat1215. URL http://dx.doi.org/10.1...

    2004 doi

  12. [20]

    u hne, T. D., Iannuzzi, M., Ben, M. D., Rybkin, V. V., Seewald, P., Stein, F., Laino, T., Khaliullin, R. Z., Sch\

    K\" u hne, T. D., Iannuzzi, M., Ben, M. D., Rybkin, V. V., Seewald, P., Stein, F., Laino, T., Khaliullin, R. Z., Sch\" u tt, O., Schiffmann, F., Golze, D., Wilhelm, J., Chulkov, S., Bani-Hashemian, M. H., Weber, V., Bor s tnik, U., Taillefumier, M., Jakobovits, A. S., Lazzaro,...

    2020

  13. [21]

    Deep-learning density functional theory hamiltonian for efficient ab initio electronic-structure calculation

    Li, H., Wang, Z., Zou, N., Ye, M., Xu, R., Gong, X., Duan, W., and Xu, Y. Deep-learning density functional theory hamiltonian for efficient ab initio electronic-structure calculation. Nature Computational Science, 2 0 (6): 0 367–377, June 2022. ISSN 2662-8457. doi:10.1038/s435...

    2022 doi

  14. [22]

    Pytorch distributed: Experiences on accelerating data parallel training, 2020

    Li, S., Zhao, Y., Varma, R., Salpekar, O., Noordhuis, P., Li, T., Paszke, A., Smith, J., Vaughan, B., Damania, P., and Chintala, S. Pytorch distributed: Experiences on accelerating data parallel training, 2020. URL https://arxiv.org/abs/2006.15704

    2020 arXiv

  15. [23]

    Equiformerv2: Improved equivariant transformer for scaling to higher-degree representations, 2023

    Liao, Y.-L., Wood, B., Das, A., and Smidt, T. Equiformerv2: Improved equivariant transformer for scaling to higher-degree representations, 2023. URL https://arxiv.org/abs/2306.12059

    2023 arXiv

  16. [24]

    Miret, S., Lee, K. L. K., Gonzales, C., Mannan, S., and Krishnan, N. M. A. Energy & force regression on dft trajectories is not enough for universal machine learning interatomic potentials, 2025. URL https://arxiv.org/abs/2502.03660

    2025 arXiv

  17. [25]

    J., Kornbluth, M., and Kozinsky, B

    Musaelian, A., Batzner, S., Johansson, A., Sun, L., Owen, C. J., Kornbluth, M., and Kozinsky, B. Learning local equivariant representations for large-scale atomistic dynamics. Nature Communications, 14 0 (1): 0 579, 2023

    2023

  18. [26]

    The orca program system

    Neese, F. The orca program system. WIREs Computational Molecular Science, 2 0 (1): 0 73–78, June 2011. ISSN 1759-0884. doi:10.1002/wcms.81. URL http://dx.doi.org/10.1002/wcms.81

    2011 doi

  19. [27]

    S., Caturla, M., Diaz de la Rubia, T., and Tarus, J

    Nordlund, K., Ghaly, M., Averback, R. S., Caturla, M., Diaz de la Rubia, T., and Tarus, J. Defect production in collision cascades in elemental semiconductors and fcc metals. Phys. Rev. B, 57: 0 7556--7570, Apr 1998. doi:10.1103/PhysRevB.57.7556. URL https://link.aps.org/doi/1...

    1998 doi

  20. [28]

    and Zitnick, C

    Passaro, S. and Zitnick, C. L. Reducing so (3) convolutions to so (2) for efficient equivariant gnns. In International conference on machine learning, pp.\ 27420--27438. PMLR, 2023

    2023

  21. [29]

    P., Burke, K., and Ernzerhof, M

    Perdew, J. P., Burke, K., and Ernzerhof, M. Generalized gradient approximation made simple [phys. rev. lett. 77, 3865 (1996)]. Phys. Rev. Lett., 78: 0 1396--1396, Feb 1997

    1996

  22. [30]

    Electronic switching in phase-change memories

    Pirovano, A., Lacaita, A., Benvenuti, A., Pellizzer, F., and Bez, R. Electronic switching in phase-change memories. IEEE Transactions on Electron Devices, 51 0 (3): 0 452–459, March 2004. ISSN 0018-9383. doi:10.1109/ted.2003.823243. URL http://dx.doi.org/10.1109/TED.2003.823243

    2004

  23. [31]

    Repa, G. M. and Fredin, L. A. Predicting electronic structure of realistic amorphous surfaces. Advanced Theory and Simulations, 6 0 (11), June 2023. ISSN 2513-0390. doi:10.1002/adts.202300292. URL http://dx.doi.org/10.1002/adts.202300292

    2023 doi

  24. [32]

    u tt, K. T., Gastegger, M., Tkatchenko, A., M\

    Sch\" u tt, K. T., Gastegger, M., Tkatchenko, A., M\" u ller, K.-R., and Maurer, R. J. Unifying machine learning and quantum chemistry with a deep neural network for molecular wavefunctions. Nature Communications, 10 0 (1), November 2019. ISSN 2041-1723. doi:10.1038/s41467-019...

    2019 doi

  25. [33]

    Senent, M. L. and Wilson, S. Intramolecular basis set superposition errors. International Journal of Quantum Chemistry, 82 0 (6): 0 282--292, 2001. doi:https://doi.org/10.1002/qua.1030. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/qua.1030

    2001 doi

  26. [34]

    Stillinger, F. H. and Weber, T. A. Computer simulation of local order in condensed phases of silicon. Phys. Rev. B, 31: 0 5262--5271, Apr 1985. doi:10.1103/PhysRevB.31.5262. URL https://link.aps.org/doi/10.1103/PhysRevB.31.5262

    1985 doi

  27. [35]

    V., and Shluger, A

    Strand, J., Kaviani, M., Gao, D., El-Sayed, A.-M., Afanas’ev, V. V., and Shluger, A. L. Intrinsic charge trapping in amorphous oxide films: status and challenges. Journal of Physics: Condensed Matter, 30 0 (23): 0 233001, May 2018. ISSN 1361-648X. doi:10.1088/1361-648x/aac005....

    2018 doi

  28. [36]

    S., Blom, A., Markussen, T., Wellendorff, J., Schneider, J., Gunst, T., Verstichel, B., Khomyakov, P

    Søren Smidstrup, K. S., Blom, A., Markussen, T., Wellendorff, J., Schneider, J., Gunst, T., Verstichel, B., Khomyakov, P. A., Vej-Hansen, U. G., Brandbyge, M., et al. Quantumatk: An integrated platform of electronic and atomic-scale modelling tools. J. Phys: Condens. Matter (A...

    2020 doi

  29. [37]

    Tafen, D. N. and Drabold, D. A. Realistic models of binary glasses from models of tetrahedral amorphous semiconductors. Physical Review B, 68 0 (16), October 2003. ISSN 1095-3795. doi:10.1103/physrevb.68.165208. URL http://dx.doi.org/10.1103/PhysRevB.68.165208

    2003 doi

  30. [38]

    Tensor field networks: Rotation- and translation-equivariant neural networks for 3d point clouds, 2018

    Thomas, N., Smidt, T., Kearnes, S., Yang, L., Li, L., Kohlhoff, K., and Riley, P. Tensor field networks: Rotation- and translation-equivariant neural networks for 3d point clouds, 2018. URL https://arxiv.org/abs/1802.08219

    2018 arXiv

  31. [39]

    P., Aktulga, H

    Thompson, A. P., Aktulga, H. M., Berger, R., Bolintineanu, D. S., Brown, W. M., Crozier, P. S., in 't Veld, P. J., Kohlmeyer, A., Moore, S. G., Nguyen, T. D., Shan, R., Stevens, M. J., Tranchida, J., Trott, C., and Plimpton, S. J. LAMMPS - a flexible simulation tool for partic...

    2022

  32. [40]

    Se (3)-equivariant prediction of molecular wavefunctions and electronic densities

    Unke, O., Bogojeski, M., Gastegger, M., Geiger, M., Smidt, T., and M \"u ller, K.-R. Se (3)-equivariant prediction of molecular wavefunctions and electronic densities. Advances in Neural Information Processing Systems, 34: 0 14434--14447, 2021

    2021

  33. [41]

    L., Islam, M

    Urquiza, M. L., Islam, M. M., van Duin, A. C. T., Cartoixà, X., and Strachan, A. Atomistic insights on the full operation cycle of a hfo2-based resistive random access memory cell from molecular dynamics. ACS Nano, 15 0 (8): 0 12945–12954, July 2021. ISSN 1936-086X. doi:10.102...

    2021 doi

  34. [42]

    and Hutter, J

    VandeVondele, J. and Hutter, J. Gaussian basis sets for accurate calculations on molecular systems in gas and condensed phases . J. Chem. Phys., 127 0 (11): 0 114105, 09 2007. ISSN 0021-9606

    2007

  35. [43]

    Graph Attention Networks

    Veli c kovi \'c , P., Cucurull, G., Casanova, A., Romero, A., Li \`o , P., and Bengio, Y. Graph Attention Networks . In ICLR, 2018

    2018

  36. [44]

    R., Kyrillidis, A., Kim, N

    Wan, C., Li, Y., Wolfe, C. R., Kyrillidis, A., Kim, N. S., and Lin, Y. PipeGCN : Efficient full-graph training of graph convolutional networks with pipelined feature communication. March 2022

    2022

  37. [45]

    Deeph-2: Enhancing deep-learning electronic structure via an equivariant local-coordinate transformer, 2024 a

    Wang, Y., Li, H., Tang, Z., Tao, H., Wang, Y., Yuan, Z., Chen, Z., Duan, W., and Xu, Y. Deeph-2: Enhancing deep-learning electronic structure via an equivariant local-coordinate transformer, 2024 a . URL https://arxiv.org/abs/2401.17015

    2024 arXiv

  38. [46]

    Universal materials model of deep-learning density functional theory hamiltonian

    Wang, Y., Li, Y., Tang, Z., Li, H., Yuan, Z., Tao, H., Zou, N., Bao, T., Liang, X., Chen, Z., Xu, S., Bian, C., Xu, Z., Wang, C., Si, C., Duan, W., and Xu, Y. Universal materials model of deep-learning density functional theory hamiltonian. Science Bulletin, 69 0 (16): 0 2514-...

    2024 doi

  39. [47]

    Wang, Z. Q. and Stroud, D. Monte carlo studies of liquid semiconductor surfaces: Si and ge. Phys. Rev. B, 38: 0 1384--1391, Jul 1988. doi:10.1103/PhysRevB.38.1384. URL https://link.aps.org/doi/10.1103/PhysRevB.38.1384

    1988 doi

  40. [48]

    An efficient method to generate amorphous structures based on local geometry

    Youn, Y., Kang, Y., and Han, S. An efficient method to generate amorphous structures based on local geometry. Computational Materials Science, 95: 0 256–262, December 2014. ISSN 0927-0256. doi:10.1016/j.commatsci.2014.07.053. URL http://dx.doi.org/10.1016/j.commatsci.2014.07.053

    2014 doi

  41. [49]

    Qh9: A quantum hamiltonian prediction benchmark for qm9 molecules

    Yu, H., Liu, M., Luo, Y., Strasser, A., Qian, X., Qian, X., and Ji, S. Qh9: A quantum hamiltonian prediction benchmark for qm9 molecules. Advances in Neural Information Processing Systems, 36: 0 40487--40503, 2023 a

    2023

  42. [50]

    Efficient and equivariant graph networks for predicting quantum hamiltonian

    Yu, H., Xu, Z., Qian, X., Qian, X., and Ji, S. Efficient and equivariant graph networks for predicting quantum hamiltonian. In International Conference on Machine Learning, pp.\ 40412--40424. PMLR, 2023 b

    2023

  43. [51]

    Yuxing Zhou, Wei Zhang, E. M. . V. L. D. Device-scale atomistic modelling ofphase-change memory materials. Nature Electronics, 6 0 (10): 0 746--754, 2023

    2023

  44. [52]

    L., K., P., C., W., M., L., and Emboras, A

    Zellweger, T., F., A. L., K., P., C., W., M., L., and Emboras, A. Amorphous germanium as multi-functional switching layer for electro- optical memristors. In Book of Abstracts: MEMRISYS 2024, pp.\ 73, 2024. URL https://www.memrisys2024.org/_files/ugd/1724d9_7d4ffc094d9045ca89a...

    2024

  45. [53]

    Self-consistency training for density-functional-theory hamiltonian prediction, 2024

    Zhang, H., Liu, C., Wang, Z., Wei, X., Liu, S., Zheng, N., Shao, B., and Liu, T.-Y. Self-consistency training for density-functional-theory hamiltonian prediction, 2024. URL https://arxiv.org/abs/2403.09560

    2024 arXiv

  46. [54]

    Transferable equivariant graph neural networks for the hamiltonians of molecules and solids

    Zhong, Y., Yu, H., Su, M., Gong, X., and Xiang, H. Transferable equivariant graph neural networks for the hamiltonians of molecules and solids. npj Computational Materials, 9 0 (1), October 2023. ISSN 2057-3960. doi:10.1038/s41524-023-01130-4. URL http://dx.doi.org/10.1038/s41...

    2023 doi

  47. [55]

    K., Zhang, L., and Gu, Q

    Zhouyin, Z., Gan, Z., Pandey, S. K., Zhang, L., and Gu, Q. Learning local equivariant representations for quantum operators, 2024. URL https://arxiv.org/abs/2407.06053

    2024 arXiv

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.