Pith. sign in

REVIEW 38 references

Mandala learns the sparse quantum operators of electronic structure—not just energies and forces—so band structures and related observables come from the same model.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · grok-4.5

2026-07-31 03:59 UTC pith:K7J56DUS

load-bearing objection Solid CPC-style software paper: real integration of sparse multi-operator learning with observable guidance, but the demos lean on reference S/D for the hard cases and are not competitive benchmarks.

arxiv 2607.24997 v1 pith:K7J56DUS submitted 2026-07-27 cond-mat.mtrl-sci physics.chem-ph

MANDALA: An E(3)-Equivariant Graph Neural Network Framework for Learning Electronic-Structure Operators with Observable Guidance

Bartosz Brzoza , Wiktoria Szopa , Zakaria Elabid , Vincent Martinetto , Varadarajan Rengaraj , Mani Lokamani , Thomas D. K\"uhne , Attila Cangi This is my paper
classification cond-mat.mtrl-sci physics.chem-ph
keywords Electronic structureDensity functional theoryEquivariant graph neural networksSparse matrix learningOperator-derived observablesHamiltonian learningScientific machine learning
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Standard machine-learned potentials speed up atomistic simulation by predicting energies and forces, but they discard the operator-level information needed for band structures, densities of states, and charge analysis. Mandala is a modular software framework that instead learns the block-sparse Hamiltonian, overlap, and density matrices that arise in localized-orbital density-functional theory, using rotation-and-reflection equivariant graph neural networks. From those predicted matrices it evaluates physical observables directly—band energy, electron count, density of states, and band structure—and can fold those observables back into training. The same pipeline supports multiple chemistry backends, heterogeneous materials, and many architecture variants without rewriting the workflow. The point is to keep electronic structure inside large-scale modeling rather than treating it as something only full quantum calculations can provide.

Core claim

A single modular framework can represent Hamiltonian, overlap, and density matrices as atom-pair sparse blocks, map those blocks into E(3)-irreducible features, train equivariant graph networks on them, and obtain operator-derived observables from the same predictions—so electronic-structure emulation and observable-guided learning share one scalable implementation rather than separate surrogate models.

What carries the argument

Block-sparse operator learning with BlockIrrepMapper: atom-pair matrix blocks are converted to and from E(3) irreducible representations so an equivariant message-passing network predicts symmetry-adapted coefficients; differentiable sparse traces then yield band energy and electron count, and optional spectral losses compare generalized eigenvalues on a k-mesh.

Load-bearing premise

That cutting interactions at a finite distance in fixed, real, non-spin-polarized orbital bases, then lightly cleaning the matrices afterward, is enough for spectra and electron counts to stay scientifically trustworthy even when the matrices are not forced to obey every quantum consistency rule.

What would settle it

Train and evaluate on a system outside the demonstrated regimes—for example a spin-polarized magnetic defect or a material whose Hamiltonian range exceeds the chosen cutoff—and check whether predicted band structures and densities of states still match reference DFT within the paper’s reported error levels after the same postprocessing.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Large-scale atomistic workflows can report DOS and band structure from learned operators instead of only energy and force.
  • Training can trade a little matrix-element error for much better eigenvalues or band energy by turning on observable guidance.
  • New DFT codes and new equivariant architectures can be swapped in without rebuilding the sparse-operator pipeline.
  • Joint Hamiltonian–density–overlap prediction becomes a practical multitask setup on one shared latent representation.
  • Inference on thousands of atoms is limited mainly by graph preparation and memory, not by rewriting the electronic-structure method.

Where Pith is reading between the lines

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

  • If total-energy contributions beyond band energy are added as planned, the same operator stack could supply forces and stresses that compete with conventional MLIPs while still exposing electronic spectra.
  • Enforcing density-matrix idempotency or H–S–D consistency inside the loss may be the next bottleneck once element-wise matrix error is already small.
  • Materials problems driven by charge transfer, defects, or field response are the natural first applications where operator learning would change the scientific question, not only the speed.

Editorial analysis

A structured set of objections, weighed in public.

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

Circularity Check

0 steps flagged

No circularity: operator learning is supervised by external DFT references; observables are standard traces and eigenvalues, not fitted quantities relabeled as predictions.

full rationale

Mandala is a methods/software paper whose load-bearing chain is (1) parse localized-orbital DFT outputs into sparse H/S/D blocks, (2) train E(3)-equivariant GNNs against those external reference matrices, and (3) evaluate band energy, electron count, DOS, and band structure from the usual quantum-mechanical definitions Tr(DH), Tr(DS), and generalized eigenvalues. None of these steps defines a target in terms of itself or fits a parameter and then reports a closely related quantity as an independent law. Gauge alignment μ_H is explicitly reference-dependent and restricted to error reporting (Eqs. 21–22; §2.1.1), not used at inference to force agreement. Observable guidance adds MSE terms on operator-derived scalars against DFT references (Eqs. 61–64); that is ordinary multi-task supervision, not circular construction. Mixed-mode demos that contract H_pred with D_ref or use reference S for spectra (§4.1, §7) are scope/validation limitations, not circular reductions. No self-citation uniqueness theorem or ansatz-smuggling chain carries the central claim. The derivation is self-contained against external DFT benchmarks.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 1 invented entities

Load-bearing content is mostly standard KS-DFT linear algebra plus equivariant ML engineering choices. The central claim rests on domain assumptions about localized-orbital sparsity and on many configurable hyperparameters, not on new physical entities. Free parameters are training/architecture choices that affect demonstrated accuracy but are not hidden constants presented as theory.

free parameters (5)
  • Real-space graph/operator cutoff r_c = 8–11 Å (dataset-dependent)
    Controls which atom-pair blocks and edges are learned; accuracy and cost depend on capturing operator range (§2.2, §3.8). Material-specific values 8–11 Å in demos.
  • Loss weights λ_X, λ_E, λ_N, λ_spec and schedules = e.g. energy guidance 0 / 1e-4 / 1e-3; Si joint model 3e-3
    Trade off matrix vs observable fidelity; ablations show 1e-4 energy weight helps SiO2 band energy while 1e-3 hurts (§3.4, §5.1).
  • Hidden irrep layout, ℓ_max, depth, learning rate, head design = e.g. ℓ_max 4–6; 2 message layers; LR 2e-4 to 5e-3
    Architecture and optimizer choices selected per showcase model (Appendix A); broad search surface admitted (§3.6).
  • Spectral window W, taper τ, k-mesh, Huber δ = W=10 eV, τ=2 eV, δ=0.1 eV (ablation)
    Defines which eigenvalues enter L_spec; ablation uses 2×2×2, W=10 eV, τ=2 eV, δ=0.1 eV (§3.5, §5.2).
  • Overlap conditioning floor ε_S = 1e-6
    Numerical floor max(s_i, 1e-6) for PSD overlap in eigensolves (§3.3).
axioms (6)
  • domain assumption Kohn–Sham DFT in a localized nonorthogonal AO basis yields generalized eigenproblem HC=SCε and operators H,S,D with Ne=Tr(DS), Eb=Tr(DH).
    Foundation of the entire operator representation (§2.1).
  • domain assumption Relevant operator content is block-sparse within a finite real-space cutoff including periodic images.
    Justifies graph edges and sparse learning domain (§2.1 end, §2.2, §3.1).
  • domain assumption E(3) equivariance via irreps/tensor products (e3nn) is the correct geometric inductive bias for AO matrix blocks.
    Architectural premise (§2.3–2.5); standard in modern atomistic ML but still an assumption about sufficiency.
  • domain assumption Hamiltonian gauge freedom is only H→H−αS for error reporting; learned H keeps training-backend energy reference.
    §2.1.1; affects how matrix errors are reported vs deployed spectra.
  • ad hoc to paper Demonstrations restricted to real, non-spin-polarized matrices without SOC are representative enough for the claimed workflow.
    Explicit scope (§2.2, §3.8, §6); limits generality of capability claims.
  • ad hoc to paper Post hoc symmetrization, optional density rescaling, and reference-overlap spectral comparisons suffice without full N-representability constraints.
    Authors state remaining constraints are not enforced (§3.3).
invented entities (1)
  • Mandala framework (Snapshot, BlockIrrepMapper, E3GNN multi-head stack, observable-guided training loop) independent evidence
    purpose: Unify parsing, sparse irrep operator learning, and operator-derived supervision/analysis in one codebase.
    Software invention, not a new physical particle/force; existence is the artifact itself (GitHub/RODARE).

pith-pipeline@v1.2.0-grok45-kimik3 · 28608 in / 4109 out tokens · 77771 ms · 2026-07-31T03:59:41.117212+00:00 · methodology

0 comments
read the original abstract

Electronic-structure calculations based on Kohn-Sham density functional theory remain indispensable in computational materials science and chemistry. Their computational cost, however, limits accessible system sizes and simulation times. At the same time, conventional machine-learning interatomic potentials (MLIPs), which are becoming the workhorse of large-scale materials modeling, usually target only energies and forces. They therefore leave out the quantum-operator-level information required to reconstruct band structures, densities of states, spatial charge distributions, and other electronic observables. \texttt{Mandala} fills this methodological gap. It is a modular software framework for learning block-sparse electronic-structure matrices with E(3)-equivariant graph neural networks. The framework is built around a unified representation of atom-resolved Hamiltonian, overlap, and density matrices, together with reusable abstractions for basis conversion, sparse block handling, irreducible representation mapping, graph construction, model definition, and training. This design allows \texttt{Mandala} to support heterogeneous chemical compositions, a wide range of neural architecture variants within one workflow, and multiple electronic-structure backends. \texttt{Mandala} evaluates selected observables directly from the predicted operators, including band energy, electron count, density of states, and band structure. This connects electronic-structure learning and observable-guided modeling while retaining a representation tied to quantum-mechanical operators rather than only scalar or vector targets as in MLIPs. In this form, \texttt{Mandala} is intended to complement atomistic interatomic potential workflows by resolving electronic structure and operator-derived observables within one scalable implementation.

Figures

Figures reproduced from arXiv: 2607.24997 by Attila Cangi, Bartosz Brzoza, Mani Lokamani, Thomas D. K\"uhne, Varadarajan Rengaraj, Vincent Martinetto, Wiktoria Szopa, Zakaria Elabid.

Figure 1
Figure 1. Figure 1: End-to-end Mandala workflow. Outputs from localized-orbital electronic￾structure backends are standardized as Snapshot objects and assembled into graph samples with sparse operator targets. An E(3)-equivariant graph neural network maps these sam￾ples to Hamiltonian, overlap, and density blocks. The predicted operators support direct matrix and observable supervision, numerical safeguards and normalization,… view at source ↗
Figure 2
Figure 2. Figure 2: Data loading and graph-sample construction. Backend-specific parsers convert [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: E(3)-equivariant model architecture and operator-specific output heads. Periodic [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Observable-guided training from predicted sparse operators. Hamiltonian, over [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Physics-aware postprocessing and electronic-structure analysis. Predicted [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Inference scaling for periodic silicon supercells on one NVIDIA B200 GPU. [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Hamiltonian fragments for the first six atoms of a representative ZnCu [PITH_FULL_IMAGE:figures/full_fig_p032_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Hamiltonian-element correlation for the representative ZnCu [PITH_FULL_IMAGE:figures/full_fig_p033_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Density of states and its prediction error for a representative glassy SiO [PITH_FULL_IMAGE:figures/full_fig_p034_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Band structure and density of states for a representative silicon validation [PITH_FULL_IMAGE:figures/full_fig_p036_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The central density matrix of the eight-atom representative silicon validation [PITH_FULL_IMAGE:figures/full_fig_p036_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Relative block error by atom-pair distance for the representative silicon struc [PITH_FULL_IMAGE:figures/full_fig_p037_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Held-out test Hamiltonian and per-atom band-energy MAE for five matched [PITH_FULL_IMAGE:figures/full_fig_p038_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Held-out test Hamiltonian and eigenvalue MAE for ten matched [PITH_FULL_IMAGE:figures/full_fig_p039_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Hamiltonian MAE for distance-envelope factorization on ZnCu [PITH_FULL_IMAGE:figures/full_fig_p039_15.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

38 extracted references · 11 canonical work pages

  1. [1]

    Hohenberg, W

    P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Physical Review 136 (3B) (1964) B864–B871.doi:10.1103/PhysRev.136.B864. URLhttps://doi.org/10.1103/PhysRev.136.B864

    doi:10.1103/physrev.136.b864 1964

  2. [2]

    W. Kohn, L. J. Sham, Self-consistent equations including exchange and correlation effects, Physical Review 140 (4A) (1965) A1133–A1138.doi: 10.1103/PhysRev.140.A1133. URLhttps://doi.org/10.1103/PhysRev.140.A1133

    doi:10.1103/physrev.140.a1133 1965

  3. [3]

    Fiedler, K

    L. Fiedler, K. Shah, M. Bussmann, A. Cangi, Deep dive into ma- chine learning density functional theory for materials science and chem- istry, Physical Review Materials 6 (4) (2022) 040301.doi:10.1103/ PhysRevMaterials.6.040301. URLhttps://doi.org/10.1103/PhysRevMaterials.6.040301

    work page doi:10.1103/physrevmaterials.6.040301 2022

  4. [4]

    Behler, M

    J. Behler, M. Parrinello, Generalized neural-network representation of high-dimensional potential-energy surfaces, Physical Review Letters 98 (14) (2007) 146401.doi:10.1103/PhysRevLett.98.146401. URLhttps://doi.org/10.1103/PhysRevLett.98.146401

    doi:10.1103/physrevlett.98.146401 2007

  5. [5]

    A. P. Bart´ ok, M. C. Payne, R. Kondor, G. Cs´ anyi, Gaussian ap- proximation potentials: The accuracy of quantum mechanics, with- out the electrons, Physical Review Letters 104 (13) (2010) 136403. doi:10.1103/PhysRevLett.104.136403. URLhttps://doi.org/10.1103/PhysRevLett.104.136403

    doi:10.1103/physrevlett.104.136403 2010

  6. [6]

    Zhang, J

    L. Zhang, J. Han, H. Wang, R. Car, W. E, Deep potential molecular dynamics: A scalable model with the accuracy of quantum mechan- ics, Physical Review Letters 120 (14) (2018) 143001.doi:10.1103/ PhysRevLett.120.143001. URLhttps://doi.org/10.1103/PhysRevLett.120.143001

    doi:10.1103/physrevlett.120.143001 2018

  7. [7]

    Batzner, A

    S. Batzner, A. Musaelian, L. Sun, M. Geiger, J. P. Mailoa, M. Ko- rnbluth, N. Molinari, T. E. Smidt, B. Kozinsky, E(3)-equivariant graph neural networks for data-efficient and accurate interatomic po- tentials, Nature Communications 13 (1) (2022) 2453.doi:10.1038/ s41467-022-29939-5. URLhttps://doi.org/10.1038/s41467-022-29939-5 44

    doi:10.1038/s41467-022-29939-5 2022

  8. [8]

    Batatia, D

    I. Batatia, D. P. Kov´ acs, G. N. C. Simm, C. Ortner, G. Cs´ anyi, MACE: Higher order equivariant message passing neural networks for fast and accurate force fields, in: Advances in Neural Information Pro- cessing Systems, Vol. 35, 2022, pp. 11423–11436.arXiv:2206.07697, doi:10.52202/068431-0830. URLhttps://proceedings.neurips.cc/paper_files/paper/2022/ h...

    Pith/arXiv arXiv 2022

  9. [9]

    D. Pfau, J. S. Spencer, A. G. D. G. Matthews, W. M. C. Foulkes, Ab initio solution of the many-electron schr¨ odinger equation with deep neural networks, Physical Review Research 2 (3) (2020) 033429.doi: 10.1103/PhysRevResearch.2.033429. URLhttps://doi.org/10.1103/PhysRevResearch.2.033429

    doi:10.1103/physrevresearch.2.033429 2020

  10. [10]

    Hermann, Z

    J. Hermann, Z. Sch¨ atzle, F. No´ e, Deep-neural-network solution of the electronic schr¨ odinger equation, Nature Chemistry 12 (10) (2020) 891– 897.doi:10.1038/s41557-020-0544-y. URLhttps://doi.org/10.1038/s41557-020-0544-y

    doi:10.1038/s41557-020-0544-y 2020

  11. [11]

    Brockherde, L

    F. Brockherde, L. Vogt, L. Li, M. E. Tuckerman, K. Burke, K.-R. M¨ uller, Bypassing the Kohn–Sham equations with machine learning, Nature Communications 8 (1) (2017) 872.doi:10.1038/s41467-017-00839-3. URLhttps://doi.org/10.1038/s41467-017-00839-3

    doi:10.1038/s41467-017-00839-3 2017

  12. [12]

    Grisafi, A

    A. Grisafi, A. Fabrizio, B. Meyer, D. M. Wilkins, C. Corminboeuf, M. Ceriotti, Transferable machine-learning model of the electron den- sity, ACS Central Science 5 (1) (2019) 57–64, pMID: 30693325.doi: 10.1021/acscentsci.8b00551. URLhttps://doi.org/10.1021/acscentsci.8b00551

    doi:10.1021/acscentsci.8b00551 2019

  13. [13]

    J. A. Ellis, L. Fiedler, G. A. Popoola, N. A. Modine, J. A. Stephens, A. P. Thompson, A. Cangi, S. Rajamanickam, Accelerating finite-temperature Kohn–Sham density functional theory with deep neural networks, Phys- ical Review B 104 (3) (2021) 035120.doi:10.1103/PhysRevB.104. 035120. URLhttps://doi.org/10.1103/PhysRevB.104.035120

    doi:10.1103/physrevb.104 2021

  14. [14]

    J. A. Rackers, L. Tecot, M. Geiger, T. E. Smidt, A recipe for cracking the quantum scaling limit with machine learned electron densities, Machine 45 Learning: Science and Technology 4 (1) (2023) 015027.doi:10.1088/ 2632-2153/acb314. URLhttps://doi.org/10.1088/2632-2153/acb314

    doi:10.1088/2632-2153/acb314 2023

  15. [15]

    Alghadeer, A

    M. Alghadeer, A. Al-Aswad, F. H. Alharbi, Highly accurate machine learning model for kinetic energy density functional, Physics Letters A 414 (2021) 127621.doi:10.1016/j.physleta.2021.127621. URLhttps://doi.org/10.1016/j.physleta.2021.127621

    arXiv 2021

  16. [16]

    S. A. Ghasemi, T. D. K¨ uhne, Artificial neural networks for the kinetic energy functional of non-interacting fermions, The Journal of Chemical Physics 154 (7) (2021) 074107.doi:10.1063/5.0037319. URLhttps://doi.org/10.1063/5.0037319

    work page doi:10.1063/5.0037319 2021

  17. [17]

    X. Shao, L. Paetow, M. E. Tuckerman, M. Pavanello, Machine learning electronic structure methods based on the one-electron reduced density matrix, Nature Communications 14 (1) (2023) 6281.doi:10.1038/ s41467-023-41953-9. URLhttps://doi.org/10.1038/s41467-023-41953-9

    doi:10.1038/s41467-023-41953-9 2023

  18. [18]

    Chandrasekaran, D

    A. Chandrasekaran, D. Kamal, R. Batra, C. Kim, L. Chen, R. Ram- prasad, Solving the electronic structure problem with machine learn- ing, npj Computational Materials 5 (1) (2019) 22.doi:10.1038/ s41524-019-0162-7. URLhttps://doi.org/10.1038/s41524-019-0162-7

    work page doi:10.1038/s41524-019-0162-7 2019

  19. [19]

    Fiedler, N

    L. Fiedler, N. A. Modine, K. D. Miller, A. Cangi, Machine learning the electronic structure of matter across temperatures, Physical Review B 108 (12) (2023) 125146.doi:10.1103/PhysRevB.108.125146. URLhttps://doi.org/10.1103/PhysRevB.108.125146

    work page doi:10.1103/physrevb.108.125146 2023

  20. [20]

    Fiedler, N

    L. Fiedler, N. A. Modine, S. Schmerler, D. J. Vogel, G. A. Popoola, A. P. Thompson, S. Rajamanickam, A. Cangi, Predicting electronic structures at any length scale with machine learning, npj Computational Materials 9 (1) (2023) 115.doi:10.1038/s41524-023-01070-z. URLhttps://doi.org/10.1038/s41524-023-01070-z

    work page doi:10.1038/s41524-023-01070-z 2023

  21. [21]

    Kocer, T

    E. Kocer, T. W. Ko, J. Behler, Neural network potentials: A concise overview of methods, Annual Review of Physical Chemistry 73 (2022) 46 163–186.doi:10.1146/annurev-physchem-082720-034254. URLhttps://doi.org/10.1146/annurev-physchem-082720-034254

    doi:10.1146/annurev-physchem-082720-034254 2022

  22. [22]

    Batzner, A

    S. Batzner, A. Musaelian, B. Kozinsky, Advancing molecular simula- tion with equivariant interatomic potentials, Nature Reviews Physics 5 (2023) 437–438.doi:10.1038/s42254-023-00615-x. URLhttps://doi.org/10.1038/s42254-023-00615-x

    work page doi:10.1038/s42254-023-00615-x 2023

  23. [23]

    T. W. Ko, S. P. Ong, Recent advances and outstanding challenges for machine learning interatomic potentials, Nature Computational Science 3 (12) (2023) 998–1000.doi:10.1038/s43588-023-00561-9. URLhttps://doi.org/10.1038/s43588-023-00561-9

    doi:10.1038/s43588-023-00561-9 2023

  24. [24]

    H. Li, Z. Wang, N. Zou, M. Ye, R. Xu, X. Gong, W. Duan, Y. Xu, Deep-learning density functional theory hamiltonian for efficient ab ini- tio electronic-structure calculation, Nature Computational Science 2 (6) (2022) 367–377.doi:10.1038/s43588-022-00265-6. URLhttps://doi.org/10.1038/s43588-022-00265-6

    doi:10.1038/s43588-022-00265-6 2022

  25. [25]

    H. Li, Z. Tang, X. Gong, N. Zou, W. Duan, Y. Xu, Deep- learning electronic-structure calculation of magnetic superstructures, Nature Computational Science 3 (4) (2023) 321–327.doi:10.1038/ s43588-023-00424-3. URLhttps://doi.org/10.1038/s43588-023-00424-3

    work page doi:10.1038/s43588-023-00424-3 2023

  26. [26]

    X. Gong, H. Li, N. Zou, R. Xu, W. Duan, Y. Xu, General framework for E(3)-equivariant neural network representation of density functional theory hamiltonian, Nature Communications 14 (1) (2023) 2848.doi: 10.1038/s41467-023-38468-8. URLhttps://doi.org/10.1038/s41467-023-38468-8

    doi:10.1038/s41467-023-38468-8 2023

  27. [27]

    Z. Tang, H. Li, P. Lin, X. Gong, G. Jin, L. He, H. Jiang, X. Ren, W. Duan, Y. Xu, A deep equivariant neural network approach for ef- ficient hybrid density functional calculations, Nature Communications 15 (1) (2024) 8815.doi:10.1038/s41467-024-53028-4. URLhttps://doi.org/10.1038/s41467-024-53028-4

    doi:10.1038/s41467-024-53028-4 2024

  28. [28]

    X. Gong, S. G. Louie, W. Duan, Y. Xu, Generalizing deep learn- ing electronic structure calculation to the plane-wave basis, Na- ture Computational Science 4 (10) (2024) 752–760.doi:10.1038/ 47 s43588-024-00701-9. URLhttps://doi.org/10.1038/s43588-024-00701-9

    work page doi:10.1038/s43588-024-00701-9 2024

  29. [29]

    Z. Tang, H. Chen, Y. Li, Y. Qian, Y. Wang, W. Fu, J. Li, C. Si, W. Duan, J. Chen, Y. Xu, Deep-learning electronic structure calcu- lations, Nature Computational Science 5 (12) (2025) 1133–1146.doi: 10.1038/s43588-025-00932-4. URLhttps://doi.org/10.1038/s43588-025-00932-4

    work page doi:10.1038/s43588-025-00932-4 2025

  30. [30]

    C. Qian, V. Vitartas, J. R. Kermode, R. J. Maurer, Equivari- ant electronic hamiltonian prediction with many-body message pass- ing, npj Computational Materials 12 (1) (2026) 169.doi:10.1038/ s41524-026-02020-1. URLhttps://doi.org/10.1038/s41524-026-02020-1

    work page doi:10.1038/s41524-026-02020-1 2026

  31. [31]

    M. Born, R. Oppenheimer, Zur quantentheorie der molekeln, Annalen der Physik 389 (20) (1927) 457–484.doi:10.1002/andp.19273892002. URLhttps://doi.org/10.1002/andp.19273892002

    doi:10.1002/andp.19273892002 1927

  32. [32]

    Geiger, T

    M. Geiger, T. Smidt, e3nn: Euclidean neural networks (2022).arXiv: 2207.09453,doi:10.48550/arXiv.2207.09453. URLhttps://doi.org/10.48550/arXiv.2207.09453

    Pith/arXiv arXiv doi:10.48550/arxiv.2207.09453 2022

  33. [33]

    Ozaki, Variationally optimized atomic orbitals for large-scale elec- tronic structures, Physical Review B 67 (15) (2003) 155108.doi: 10.1103/PhysRevB.67.155108

    T. Ozaki, Variationally optimized atomic orbitals for large-scale elec- tronic structures, Physical Review B 67 (15) (2003) 155108.doi: 10.1103/PhysRevB.67.155108. URLhttps://doi.org/10.1103/PhysRevB.67.155108

    doi:10.1103/physrevb.67.155108 2003

  34. [34]

    Ozaki, H

    T. Ozaki, H. Kino, Numerical atomic basis orbitals from H to Kr, Physical Review B 69 (19) (2004) 195113.doi:10.1103/PhysRevB. 69.195113. URLhttps://doi.org/10.1103/PhysRevB.69.195113

    doi:10.1103/physrevb 2004

  35. [35]

    V. Blum, R. Gehrke, F. Hanke, P. Havu, V. Havu, X. Ren, K. Reuter, M. Scheffler, Ab initio molecular simulations with numeric atom- centered orbitals, Computer Physics Communications 180 (11) (2009) 2175–2196.doi:10.1016/j.cpc.2009.06.022. URLhttps://doi.org/10.1016/j.cpc.2009.06.022

    doi:10.1016/j.cpc.2009.06.022 2009

  36. [36]

    Q. Sun, X. Zhang, S. Banerjee, P. Bao, M. Barbry, N. S. Blunt, N. A. Bogdanov, G. H. Booth, J. Chen, Z.-H. Cui, J. J. Eriksen, Y. Gao, 48 S. Guo, J. Hermann, M. R. Hermes, K. Koh, P. Koval, S. Lehtola, Z. Li, J. Liu, N. Mardirossian, J. D. McClain, M. Motta, B. Mussard, H. Q. Pham, A. Pulkin, W. Purwanto, P. J. Robinson, E. Ronca, E. R. Say- futyarova, M....

    2020

  37. [37]

    Biewald, Experiment tracking with weights and biases, software avail- able from wandb.com (2020)

    L. Biewald, Experiment tracking with weights and biases, software avail- able from wandb.com (2020). URLhttps://www.wandb.com/

    2020

  38. [38]

    Fiedler, N

    L. Fiedler, N. Hoffmann, P. Mohammed, G. A. Popoola, T. Yovell, V. Oles, J. A. Ellis, S. Rajamanickam, A. Cangi, Training-free hyper- parameter optimization of neural networks for electronic structures in matter, Machine Learning: Science and Technology 3 (4) (2022) 045008. doi:10.1088/2632-2153/ac9956. URLhttps://doi.org/10.1088/2632-2153/ac9956 49 Appen...

    work page doi:10.1088/2632-2153/ac9956 2022