Prediction of electron localization functions from superposed atomic densities for accelerated superhydride discovery

Abstract

The electron localization function (ELF) provides a chemically interpretable representation of bonding and interstitial electron localization, but generating volumetric ELF fields with density functional theory (DFT) is computationally expensive for large candidate libraries. We present ELFNet, a structure-to-field model that predicts DFT-derived ELF grids from computationally inexpensive superposed atomic density (SAD) inputs for high-pressure chemical-template screening. To train the model, we constructed a pressure-targeted Perdew–Burke–Ernzerhof (PBE) ELF dataset containing 326,009 converged unary and binary prototype structures spanning 89 elements and covering pressures from approximately -50 to 500 GPa. ELFNet uses periodic full-grid convolutions and a composite objective designed to recover sparse high-ELF features. We evaluated the model on 50,000 metal-only templates derived from known high-pressure superhydride prototypes. Predicted fields captured chemically meaningful differences in the spatial topology of interstitial localization, while global maximum ELF values and maximum-location errors provided complementary measures of agreement with DFT references. On the reported hardware, the full inference-through-write workflow reduced summed per-structure elapsed time by approximately 350-fold relative to fixed-geometry Vienna Ab initio Simulation Package (VASP) ELF generation. These results establish SAD-to-ELF prediction as a low-cost front-end screening method for prioritizing candidate metal templates before downstream first-principles validation.

Keywords

Electron localization function / materials informatics / deep learning / high-pressure materials / superhydrides / predictive modeling

References

[1] Becke AD.A simple measure of electron localization in atomic and molecular systems.J Chem Phys1990;92:5397-403

[2] Silvi B.Classification of chemical bonds based on topological analysis of electron localization functions.Nature1994;371:683-6

[3] Savin A,Wengert S.ELF: the electron localization function.Angew Chem Int Ed Engl1997;36:1808-32

[4] Hilleke KP,Zurek E.Materials under high pressure: a chemical perspective.Appl Phys A2022;128:441

[5] Ashcroft NW.Hydrogen dominant metallic alloys: high temperature superconductors?.Phys Rev Lett2004;92:187002

[6] Drozdov AP,Troyan IA,Shylin SI.Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system.Nature2015;525:73-6

[7] Duan D,Tian F.Pressure-induced metallization of dense (H₂S)₂H₂ with high-T_c superconductivity.Sci Rep2014;4:6968 PMCID:PMC4225546

[8] Li Y,Liu H,Ma Y.The metallization and superconductivity of dense hydrogen sulfide.J Chem Phys2014;140:174712

[9] Drozdov AP,Minkov VS.Superconductivity at 250 K in lanthanum hydride under high pressures.Nature2019;569:528-31

[10] Somayazulu M,Mishra AK.Evidence for superconductivity above 260 K in lanthanum superhydride at megabar pressures.Phys Rev Lett2019;122:027001

[11] Errea I,Monacelli L.Quantum crystal structure in the 250-kelvin superconducting lanthanum hydride.Nature2020;578:66-9

[12] Wang H,Tanaka K,Ma Y.Superconductive sodalite-like clathrate calcium hydride at high pressures.Proc Natl Acad Sci U S A2012;109:6463-6 PMCID:PMC3340045

[13] Peng F,Pickard CJ,Wu Q.Hydrogen clathrate structures in rare earth hydrides at high pressures: possible route to room-temperature superconductivity.Phys Rev Lett2017;119:107001

[14] Ma L,Xie Y.High-temperature superconducting phase in clathrate calcium hydride CaH₆ up to 215 K at a Pressure of 172 GPa.Phys Rev Lett2022;128:167001

[15] Li Z,Zhang C.Superconductivity above 200 K discovered in superhydrides of calcium.Nat Commun2022;13:2863 PMCID:PMC9126910

[16] Bi J,Zhang P.Giant enhancement of superconducting critical temperature in substitutional alloy (La,Ce)H₉.Nat Commun2022;13:5952 PMCID:PMC9551097

[17] Chen W,Semenok DV.Enhancement of superconducting properties in the La-Ce-H system at moderate pressures.Nat Commun2023;14:2660 PMCID:PMC10170082

[18] Chen LC,Cao ZY.Synthesis and superconductivity in yttrium-cerium hydrides at high pressures.Nat Commun2024;15:1809 PMCID:PMC10901869

[19] Shan P,Yang X.Molecular hydride superconductor BiH₄ with T_c up to 91 K at 170 GPa.J Am Chem Soc2025;147:4375-81

[20] Liu P,Xu Q,Liu Z.Mechanism of high-temperature superconductivity in compressed H₂-molecular-type hydride.Sci Adv2025;11:eadt9411 PMCID:PMC11952093

[21] Zhao W,Duan D.Unlocking the origin of high‐temperature superconductivity in molecular hydrides at moderate pressures.Adv Funct Mater2025;35:2415910

[22] Sun Y.Chemical templates that assemble the metal superhydrides.Chem2023;9:443-59

[23] Belli F,Contreras-García J.Strong correlation between electronic bonding network and critical temperature in hydrogen-based superconductors.Nat Commun2021;12:5381 PMCID:PMC8446067

[24] Sun Y,Chen X.Batch discovery of complex metal superhydrides via an effective machine learning method structured by chemical template concept.J Am Chem Soc2025;147:40407-19

[25] Glass CW,Hansen N.USPEX - evolutionary crystal structure prediction.Comput Phys Commun2006;175:713-20

[26] Wang Y,Zhu L.Crystal structure prediction via particle-swarm optimization.Phys Rev B2010;82:094116

[27] Merchant A,Schoenholz SS,Cheon G.Scaling deep learning for materials discovery.Nature2023;624:80-5 PMCID:PMC10700131

[28] Zeni C,Zügner D.A generative model for inorganic materials design.Nature2025;639:624-32 PMCID:PMC11922738

[29] Hessmann SSP,Gebauer NWA,Oguchi T.Accelerating crystal structure search through active learning with neural networks for rapid relaxations.npj Comput Mater2025;11:44

[30] Kulichenko M,Lubbers N.Uncertainty-driven dynamics for active learning of interatomic potentials.Nat Comput Sci2023;3:230-9 PMCID:PMC10766548

[31] Conway LJ.Accelerating crystal structure prediction using data‐derived potentials: high‐pressure binary hydrides.Ann Phys2026;538:e00608

[32] Jørgensen PB.Equivariant graph neural networks for fast electron density estimation of molecules, liquids, and solids.npj Comput Mater2022;8:183

[33] Lv T,Liang Y,Huang J.Deep charge: deep learning model of electron density from a one-shot density functional theory calculation.Phys Rev B2023;108:235159

[34] Koker T,Taw E,Li L.Higher-order equivariant neural networks for charge density prediction in materials.npj Comput Mater2024;10:161

[35] Qin X,Zhong Z.EAC-Net: predicting real-space charge density via equivariant atomic contributions.J Chem Theory Comput2026;22:4813-21

[36] Li C,Yuan S.Image super-resolution inspired electron density prediction.Nat Commun2025;16:4811 PMCID:PMC12102193

[37] Wang Z,Evans ML.Machine learning-accelerated discovery of A₂BC₂ ternary electrides with diverse anionic electron densities.J Am Chem Soc2023;145:26412-24

[38] Dewaele A,Pickard CJ.Synthesis and stability of xenon oxides Xe₂O₅ and Xe₃O₂ under pressure.Nat Chem2016;8:784-90

[39] Dong X,Goncharov AF.A stable compound of helium and sodium at high pressure.Nat Chem2017;9:440-5

[40] Kaplan AD,Qi J. A foundational potential energy surface dataset for materials. arXiv 2025, arXiv:2503.04070. Available online: https://doi.org/10.48550/arXiv.2503.04070. (accessed 2026-08-20)

[41] Kuner MC,Persson KA,Chrzan DC.MP-ALOE: an r²SCAN dataset for universal machine learning interatomic potentials.npj Comput Mater2025;11:352

[42] Deng B,Jun K.CHGNet as a pretrained universal neural network potential for charge-informed atomistic modelling.Nat Mach Intell2023;5:1031-41

[43] Barros-Luque L,Fu X.The Open Materials 2024 (OMat24) inorganic materials dataset and models.Nat Comput Sci2026;6:642-52

[44] Chen C.A universal graph deep learning interatomic potential for the periodic table.Nat Comput Sci2022;2:718-28

[45] Zhang D,Zhang X.DPA-2: a large atomic model as a multi-task learner.NPJ Comput Mater2024;10:293 PMCID:PMC12369844

[46] Batzner S,Sun L.E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials.Nat Commun2022;13:2453 PMCID:PMC9068614

[47] Musaelian A,Johansson A.Learning local equivariant representations for large-scale atomistic dynamics.Nat Commun2023;14:579 PMCID:PMC9898554

[48] Batatia I,Simm GNC,Csányi G. MACE: higher order equivariant message passing neural networks for fast and accurate force fields. arXiv 2022, arXiv:2206.07697. Available online: https://doi.org/10.48550/arXiv.2206.07697. (accessed 2026-08-20)

[49] Batatia I,Chiang Y.A foundation model for atomistic materials chemistry.J Chem Phys2025;163:184110

[50] Lysogorskiy Y,Drautz R.Graph atomic cluster expansion for foundational machine learning interatomic potentials.npj Comput Mater2026;12:114

[51] Barlow RE,Bremner JM. Statistical inference under order restrictions. The theory and application of isotonic regression. John Wiley & Sons: London and New York, 1972. https://books.google.com/books/about/Statistical_Inference_Under_Order_Restri.html?id=DEamySUDBWcC&utm_source. (accessed 2026-08-20)

[52] Kohn W.Self-consistent equations including exchange and correlation effects.Phys Rev1965;140:A1133

[53] Perdew JP,Ernzerhof M.Generalized gradient approximation made simple.Phys Rev Lett1996;77:3865

[54] Kresse G.Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set.Phys Rev B1996;54:11169

[55] Blöchl PE.Projector augmented-wave method.Phys Rev B1994;50:17953

[56] Kresse G.From ultrasoft pseudopotentials to the projector augmented-wave method.Phys Rev B1999;59:1758

[57] Furness JW,Ning J,Sun J.Accurate and numerically efficient r²SCAN meta-generalized gradient approximation.J Phys Chem Lett2020;11:8208-15

[58] Hirshfeld FL.Bonded-atom fragments for describing molecular charge densities.Theoret Chim Acta1977;44:129-38

[59] He K,Ren S.Deep residual learning for image recognition. In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Las Vegas, USA. Jun 27-30, 2016. IEEE; 2016. pp 770-8.

[60] Wu Y.Group normalization. In Ferrari, V.; Hebert, M.; Sminchisescu, C.; Weiss, Y.; editors. Computer Vision - ECCV 2018. Cham: Springer International Publishing; 2018. pp. 3-19.

[61] Hendrycks D. Gaussian error linear units (GELUs). arXiv 2016, arXiv:1606.08415. Available online: https://doi.org/10.48550/arXiv.1606.08415. (accessed 2026-08-20)

[62] Hu J,Sun G.Squeeze-and-excitation networks. In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, Salt Lake City, USA. Jun 18-23, 2018. IEEE; 2018. pp 7132-41.

[63] Woo S,Lee JY.CBAM: convolutional block attention Module. In: Ferrari, V.; Hebert, M.; Sminchisescu, C.; Weiss, Y.; editors. Computer Vision - ECCV 2018. Cham: Springer International Publishing; 2018. pp. 3-19.

[64] Cipolla R,Kendall A.Multi-task learning using uncertainty to weigh losses for scene geometry and semantics. In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, Salt Lake City, USA. Jun 18-23, 2018. IEEE; 2018. pp. 7482-91.

[65] Reents T,Bercx M,Pizzi G.Score-based diffusion models for accurate crystal-structure inpainting and reconstruction of hydrogen positions.npj Comput Mater2026;12:203