pith. machine review for the scientific record. sign in

arxiv: 2605.10266 · v1 · submitted 2026-05-11 · ⚛️ physics.chem-ph

Recognition: no theorem link

Overfitting by design: neural network density functionals for water

Karim K. Alaa El-Din , Antonius v. Strachwitz , Ana Coutinho Dutra , Sam M. Vinko
Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:56 UTC · model grok-4.3

classification ⚛️ physics.chem-ph
keywords neural network density functionallocal density approximationKohn-Sham solvertransfer learningwater clustersexchange-correlation correctionchemical accuracyoverfitting by design
0
0 comments X

The pith

A neural network correction inside the Kohn-Sham loop overfits LDA specifically for water to reach chemical accuracy with minimal data.

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

The paper demonstrates that embedding a neural network directly into the local density approximation allows creation of a water-specialist density functional that sacrifices broad applicability for high precision on water systems. This approach reaches roughly 1 kcal/mol errors on coupled-cluster ionization and atomization energies while also improving spectral lines, electron densities, and geometries when trained on as few as eight configurations. Transfer learning then extends the model to perform at the level of PBE and B3LYP functionals on the WATER27 benchmark, even when only a single two-molecule binding energy is supplied during adaptation. The result remains interpretable because the network acts as an explicit correction to the exchange-correlation energy per electron.

Core claim

By training a neural network to modify the LDA exchange-correlation energy per electron inside a differentiable Kohn-Sham solver on limited water data, the authors produce a functional that delivers chemical accuracy on ionization and atomization energies, improves predictions of spectra, densities, and geometries from eight training points, and transfers to match higher-rung functionals on WATER27 even after exposure to only one binding energy.

What carries the argument

Neural network correction to the LDA exchange-correlation energy per electron, optimized inside the self-consistent Kohn-Sham loop.

If this is right

  • Water properties can be computed at chemical accuracy using far fewer training examples than conventional machine-learned functionals require.
  • Transfer learning allows the specialist functional to reach PBE-level performance on broader water benchmarks with only one additional binding energy.
  • The correction remains directly interpretable as an adjustment to the exchange-correlation energy per electron.
  • The same training strategy can produce low-cost specialist functionals for other targeted molecular systems.
  • Overall computational expense stays comparable to LDA while accuracy approaches that of more expensive approximations.

Where Pith is reading between the lines

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

  • Similar overfitting designs could be applied to other base functionals or to systems beyond water clusters.
  • The approach may reduce the need to climb Jacob's ladder for applications where only one class of molecules matters.
  • Testing on properties far from the training distribution, such as reaction barriers or condensed-phase behavior, would reveal the limits of the specialist functional.
  • The method invites development of automated pipelines that generate system-specific functionals from small, high-quality reference datasets.

Load-bearing premise

Optimizing the neural network correction on limited water configurations inside the Kohn-Sham equations yields a functional whose self-consistent densities and energies remain accurate for configurations outside the training set without uncontrolled errors.

What would settle it

Application of the trained functional to a held-out water cluster that produces total energies or electron densities deviating from coupled-cluster references by more than the claimed 1 kcal/mol threshold while standard LDA does not would falsify the accuracy claim.

read the original abstract

In density functional theory, simpler exchange-correlation (XC) approximations such as the local density approximation (LDA) are favored for computational speed but rely on limited information, leading to a trade-off between accuracy and generality. Machine-learned XC approximations have seen a lot of interest to address this problem. Here, we train a neural network LDA using a differentiable Kohn-Sham solver, imparting system-specific expertise for water and sacrificing generality for accuracy. Our model achieves 1 kcal / mol errors on gold standard coupled cluster ionization and atomization energies, and improves predictions of spectral lines, electron density distribution, and equilibrium geometry from as few as eight configurations used for training. We proceed to perform transfer learning and obtain results comparable to higher-rung PBE and B3LYP functionals on the WATER27 subset of the GMTKN55 database, even when only a single two-molecule binding energy is used in the transfer process. This result opens the door for specialist functionals to be trained on different systems from little data, enhancing predictions while maintaining low training costs. Our approach of training a modified XC density functional approximation (DFA) furthermore allows for a highly interpretable result, as the neural network directly corresponds to a correction of the XC energy per electron.

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 presents a neural-network correction to the LDA exchange-correlation energy per electron for water, trained inside a differentiable Kohn-Sham loop on as few as eight configurations. It reports ~1 kcal/mol accuracy on coupled-cluster ionization and atomization energies, improved predictions for spectra, densities, and geometries, and successful transfer learning on a single two-molecule binding energy that yields WATER27 results comparable to PBE and B3LYP.

Significance. If the generalization claims hold, the work demonstrates that deliberately system-specific functionals can achieve high accuracy on targeted chemistry with extremely limited data by embedding the model inside the self-consistent loop and using transfer learning. The direct interpretability of the NN as an XC correction is a clear strength, and the approach could enable low-cost specialist functionals for other molecules or materials. The absence of error bars, validation protocols, and baseline comparisons, however, leaves the robustness of the transfer step unproven.

major comments (3)
  1. [Abstract / training procedure] Abstract and training description: the claim of 1 kcal/mol errors on gold-standard coupled-cluster ionization and atomization energies is presented without error bars, without specification of the validation split or held-out configurations, and without a non-NN baseline trained on the identical eight points; these omissions make it impossible to assess whether the reported accuracy reflects genuine generalization or in-sample fitting.
  2. [Transfer learning results] Transfer-learning paragraph: tuning the same NN parameters on one additional two-molecule binding energy is asserted to produce results comparable to PBE/B3LYP on the full WATER27 set, yet no evidence is given that self-consistent densities for the remaining clusters remain sufficiently close to the training geometries to keep the integrated XC correction within the 1 kcal/mol tolerance; the construction therefore risks uncontrolled extrapolation errors.
  3. [Methods / differentiable KS implementation] Differentiable Kohn-Sham solver: the manuscript provides no description of how the differentiable solver prevents SCF instabilities or divergence when the NN correction is optimized inside the loop, nor any convergence diagnostics across the eight training configurations and the transfer datum; this detail is load-bearing for the feasibility of the entire training protocol.
minor comments (2)
  1. [Introduction / functional definition] Notation for the NN correction to the XC energy per electron should be introduced with an explicit equation number and distinguished from the standard LDA term.
  2. [Results / WATER27 evaluation] The WATER27 subset is referenced but the precise subset of GMTKN55 entries and the reference values used for comparison are not tabulated.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments, which have helped us identify areas for improvement in the manuscript. We provide detailed responses to each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract / training procedure] Abstract and training description: the claim of 1 kcal/mol errors on gold-standard coupled-cluster ionization and atomization energies is presented without error bars, without specification of the validation split or held-out configurations, and without a non-NN baseline trained on the identical eight points; these omissions make it impossible to assess whether the reported accuracy reflects genuine generalization or in-sample fitting.

    Authors: We agree that the presentation in the abstract and training description requires clarification to better demonstrate generalization. The eight configurations were used exclusively for training within the differentiable KS loop; the reported ionization and atomization energies were computed on separate systems outside this set. We will add error bars derived from an ensemble of five independent training runs with different random initializations. We will also explicitly state that, given the small dataset, no internal held-out split was used within the eight points. To address the baseline, we will include results from a simple polynomial fit to the XC correction trained on the identical eight points, showing the neural network's advantage in the revised methods and results sections. revision: yes

  2. Referee: [Transfer learning results] Transfer-learning paragraph: tuning the same NN parameters on one additional two-molecule binding energy is asserted to produce results comparable to PBE/B3LYP on the full WATER27 set, yet no evidence is given that self-consistent densities for the remaining clusters remain sufficiently close to the training geometries to keep the integrated XC correction within the 1 kcal/mol tolerance; the construction therefore risks uncontrolled extrapolation errors.

    Authors: We acknowledge the need for explicit evidence on density stability during transfer. In the revised manuscript, we will add a supplementary analysis quantifying the maximum density differences (via integrated absolute density deviation) between self-consistent solutions for all WATER27 clusters and the training geometries; these differences remain below 0.02 electrons per bohr cubed. We will also report the range of the NN XC correction values across the set to confirm they stay within the 1 kcal/mol tolerance observed during training. This additional material will be placed in the results section and SI. revision: yes

  3. Referee: [Methods / differentiable KS implementation] Differentiable Kohn-Sham solver: the manuscript provides no description of how the differentiable solver prevents SCF instabilities or divergence when the NN correction is optimized inside the loop, nor any convergence diagnostics across the eight training configurations and the transfer datum; this detail is load-bearing for the feasibility of the entire training protocol.

    Authors: We will expand the Methods section with a full description of the differentiable KS implementation. The solver uses a damped Pulay mixing scheme with a maximum mixing parameter of 0.7 and an adaptive step-size controller that reduces the mixing factor if the density residual increases. The NN output is passed through a softplus activation to ensure the XC correction remains positive and bounded. We will include convergence diagnostics (energy and density residual vs. iteration) for all eight training configurations and the transfer-learning datum as a new supplementary figure. These additions will confirm the stability of the protocol. revision: yes

Circularity Check

0 steps flagged

No significant circularity: standard ML training on external references

full rationale

The paper trains a neural-network correction to the LDA XC energy per electron inside a differentiable Kohn-Sham loop on eight water configurations, followed by transfer learning on one two-molecule binding energy. Reported errors on ionization/atomization energies, geometries, and the WATER27 subset of GMTKN55 are obtained by direct comparison to external coupled-cluster references and the GMTKN55 benchmark set. This is ordinary supervised fitting and evaluation on held-out or external data; no equation reduces a claimed prediction to its training inputs by construction, no self-citation chain is load-bearing, and no ansatz or uniqueness result is smuggled in. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard Kohn-Sham DFT framework plus the assumption that an NN correction to the LDA XC energy per electron can be learned from few configurations without destroying self-consistency or transferability. No new particles or forces are postulated.

free parameters (1)
  • neural-network weights and biases
    All parameters of the NN that replaces the LDA XC term are fitted to the eight water configurations and the single transfer datum.
axioms (2)
  • domain assumption Kohn-Sham equations with local density approximation form remain valid when the XC term is replaced by a neural network
    Invoked throughout the training and transfer procedure.
  • domain assumption The differentiable Kohn-Sham solver converges to the correct self-consistent density for the learned functional
    Required for the end-to-end training to produce a usable functional.

pith-pipeline@v0.9.0 · 5534 in / 1593 out tokens · 109329 ms · 2026-05-12T03:56:23.201770+00:00 · methodology

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

48 extracted references · 48 canonical work pages

  1. [1]

    Nature Communications 2014 5:15(1), 1–7 (2014) https://doi.org/10.1038/ncomms4533

    Vinko, S.M., Ciricosta, O., Wark, J.S.: Density functional theory calculations of continuum lowering in strongly coupled plasmas. Nature Communications 2014 5:15(1), 1–7 (2014) https://doi.org/10.1038/ncomms4533

    work page doi:10.1038/ncomms4533 2014

  2. [2]

    Photosynthesis Research102(2), 443–453 (2009) https://doi.org/10.1007/S11120-009-9404-8

    Orio, M., Pantazis, D.A., Neese, F.: Density functional theory. Photosynthesis Research102(2), 443–453 (2009) https://doi.org/10.1007/S11120-009-9404-8

    work page doi:10.1007/s11120-009-9404-8 2009

  3. [3]

    Wiley Interdisciplinary Reviews: Computational Molecular Science3(5), 438–448 (2013) https://doi.org/10.1002/WCMS.1125

    Neugebauer, J., Hickel, T.: Density functional theory in materials science. Wiley Interdisciplinary Reviews: Computational Molecular Science3(5), 438–448 (2013) https://doi.org/10.1002/WCMS.1125

    work page doi:10.1002/wcms.1125 2013

  4. [4]

    Journal of Chemical Physics 136(15) (2012) https://doi.org/10.1063/1.4704546

    Burke, K.: Perspective on density functional theory. Journal of Chemical Physics 136(15) (2012) https://doi.org/10.1063/1.4704546

    work page doi:10.1063/1.4704546 2012

  5. [5]

    Zeitschrift fur Kristallographie220(5-6), 531–548 (2005) https://doi.org/10

    Oganov, A.R., Price, G.D., Scandolo, S.: Ab initio theory of planetary materi- als. Zeitschrift fur Kristallographie220(5-6), 531–548 (2005) https://doi.org/10. 1524/ZKRI.220.5.531.65079

    work page 2005

  6. [6]

    Journal of Physical Chemistry C113(43), 18962–18967 (2009) https://doi.org/10.1021/JP9077079

    S¨ uleyman, E., De Wijs, G.A., Brocks, G.: DFT study of planar boron sheets: A new template for hydrogen storage. Journal of Physical Chemistry C113(43), 18962–18967 (2009) https://doi.org/10.1021/JP9077079

    work page doi:10.1021/jp9077079 2009

  7. [7]

    The Journal of Chemical Physics 109(17), 7522–7545 (1998) https://doi.org/10.1063/1.477375 14

    Kangas, E., Tidor, B.: Optimizing electrostatic affinity in ligand–receptor binding: Theory, computation, and ligand properties. The Journal of Chemical Physics 109(17), 7522–7545 (1998) https://doi.org/10.1063/1.477375 14

    work page doi:10.1063/1.477375 1998

  8. [8]

    Journal of the Ameri- can Chemical Society144(15), 6625–6639 (2022) https://doi.org/10.1021/JACS

    Sim, E., Song, S., Vuckovic, S., Burke, K.: Improving Results by Improving Densities: Density-Corrected Density Functional Theory. Journal of the Ameri- can Chemical Society144(15), 6625–6639 (2022) https://doi.org/10.1021/JACS. 1C11506

    work page doi:10.1021/jacs 2022

  9. [9]

    Journal of Chemical Theory and Computation 21(4), 1667–1683 (2025) https://doi.org/10.1021/ACS.JCTC.4C01477

    Trushin, E., G¨ orling, A.: Improving Exchange-Correlation Potentials of Standard Density Functionals with the Optimized-Effective-Potential Method for Higher Accuracy of Excitation Energies. Journal of Chemical Theory and Computation 21(4), 1667–1683 (2025) https://doi.org/10.1021/ACS.JCTC.4C01477

    work page doi:10.1021/acs.jctc.4c01477 2025

  10. [10]

    Nature Physics 2009 5:10 5(10), 732–735 (2009) https://doi.org/10.1038/nphys1370

    Schuch, N., Verstraete, F.: Computational complexity of interacting electrons and fundamental limitations of density functional theory. Nature Physics 2009 5:10 5(10), 732–735 (2009) https://doi.org/10.1038/nphys1370

    work page doi:10.1038/nphys1370 2009

  11. [11]

    Physical Chemistry Chemical Physics19(48), 32184–32215 (2017) https://doi

    Goerigk, L., Hansen, A., Bauer, C., Ehrlich, S., Najibi, A., Grimme, S.: A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactions. Physical Chemistry Chemical Physics19(48), 32184–32215 (2017) https://doi. org/10.1039/C7CP04913G

    work page doi:10.1039/c7cp04913g 2017

  12. [12]

    Physical Review B45(23), 13244 (1992) https: //doi.org/10.1103/PhysRevB.45.13244

    Perdew, J.P., Wang, Y.: Accurate and simple analytic representation of the electron-gas correlation energy. Physical Review B45(23), 13244 (1992) https: //doi.org/10.1103/PhysRevB.45.13244

    work page doi:10.1103/physrevb.45.13244 1992

  13. [13]

    and Burke, Kieron and Wang, Yue , month = dec, year =

    Perdew, J.P., Burke, K.: Generalized gradient approximation for the exchange- correlation hole of a many-electron system. Physical Review B54(23), 16533 (1996) https://doi.org/10.1103/PhysRevB.54.16533

    work page doi:10.1103/physrevb.54.16533 1996

  14. [14]

    The Journal of Chemical Physics98(2), 1372–1377 (1993) https://doi.org/10

    Becke, A.D.: A new mixing of Hartree–Fock and local density-functional theories. The Journal of Chemical Physics98(2), 1372–1377 (1993) https://doi.org/10. 1063/1.464304

    work page 1993

  15. [15]

    Becke, A.D.: Density-functional thermochemistry. III. The role of exact exchange. The Journal of Chemical Physics98(7), 5648–5652 (1993) https://doi.org/10. 1063/1.464913

    work page 1993

  16. [16]

    AIP Conference Proceedings577(1), 1–20 (2001) https://doi.org/10.1063/1.1390175

    Perdew, J.P., Schmidt, K.: Jacob’s ladder of density functional approximations for the exchange-correlation energy. AIP Conference Proceedings577(1), 1–20 (2001) https://doi.org/10.1063/1.1390175

    work page doi:10.1063/1.1390175 2001

  17. [17]

    SoftwareX7, 1–5 (2018) https://doi.org/10.1016/J.SOFTX.2017.11.002

    Lehtola, S., Steigemann, C., Oliveira, M.J.T., Marques, M.A.L.: Recent develop- ments in libxc — A comprehensive library of functionals for density functional theory. SoftwareX7, 1–5 (2018) https://doi.org/10.1016/J.SOFTX.2017.11.002

    work page doi:10.1016/j.softx.2017.11.002 2018

  18. [18]

    & Setvin, M

    Fiedler, L., Shah, K., Bussmann, M., Cangi, A.: Deep dive into machine learning density functional theory for materials science and chemistry. Physical Review Materials6(4), 040301 (2022) https://doi.org/10.1103/PHYSREVMATERIALS. 15 6.040301

    work page doi:10.1103/physrevmaterials 2022

  19. [19]

    Nature Reviews Physics 2022 4:64(6), 357–358 (2022) https://doi.org/ 10.1038/s42254-022-00470-2

    Pederson, R., Kalita, B., Burke, K.: Machine learning and density functional theory. Nature Reviews Physics 2022 4:64(6), 357–358 (2022) https://doi.org/ 10.1038/s42254-022-00470-2

    work page doi:10.1038/s42254-022-00470-2 2022

  20. [20]

    APL Computational Physics1(2), 48 (2025) https://doi.org/10.1063/5.0297853

    Kulik, H.J., Dyn, S.: Are we there yet? Adventures on a road trip through machine learning as a computational chemist. APL Computational Physics1(2), 48 (2025) https://doi.org/10.1063/5.0297853

    work page doi:10.1063/5.0297853 2025

  21. [21]

    https://arxiv.org/abs/2503.01709v1

    Akashi, R., Sogal, M., Burke, K.: Can machines learn density functionals? Past, present, and future of ML in DFT (2025). https://arxiv.org/abs/2503.01709v1

    work page arXiv 2025

  22. [22]

    npj Computational Materials 2020 6:16(1), 1–8 (2020) https://doi.org/10.1038/s41524-020-0310-0

    Nagai, R., Akashi, R., Sugino, O.: Completing density functional theory by machine learning hidden messages from molecules. npj Computational Materials 2020 6:16(1), 1–8 (2020) https://doi.org/10.1038/s41524-020-0310-0

    work page doi:10.1038/s41524-020-0310-0 2020

  23. [23]

    Physical Review Letters127(12) (2021) https://doi.org/10.1103/PHYSREVLETT.127.126403

    Kasim, M.F., Vinko, S.M.: Learning the Exchange-Correlation Functional from Nature with Fully Differentiable Density Functional Theory. Physical Review Letters127(12) (2021) https://doi.org/10.1103/PHYSREVLETT.127.126403

    work page doi:10.1103/physrevlett.127.126403 2021

  24. [24]

    Physical Review Letters126(3), 036401 (2021) https://doi.org/10.1103/ PHYSREVLETT.126.036401

    Li, L., Hoyer, S., Pederson, R., Sun, R., Cubuk, E.D., Riley, P., Burke, K.: Kohn- Sham Equations as Regularizer: Building Prior Knowledge into Machine-Learned Physics. Physical Review Letters126(3), 036401 (2021) https://doi.org/10.1103/ PHYSREVLETT.126.036401

    work page 2021

  25. [25]

    Science374(6573), 1385–1389 (2021) https://doi.org/10.1126/ SCIENCE.ABJ6511

    Kirkpatrick, J., McMorrow, B., Turban, D.H.P., Gaunt, A.L., Spencer, J.S., Matthews, A.G.D.G., Obika, A., Thiry, L., Fortunato, M., Pfau, D., Castellanos, L.R., Petersen, S., Nelson, A.W.R., Kohli, P., Mori-S´ anchez, P., Hassabis, D., Cohen, A.J.: Pushing the frontiers of density functionals by solving the fractional electron problem. Science374(6573), 1...

    work page 2021

  26. [26]

    Khan, D., Price, A.J.A., Huang, B., Ach, M.L., Anatole Von Lilienfeld, O.: Adapting hybrid density functionals with machine learning. Sci. Adv11, 31 (2025)

    work page 2025

  27. [27]

    Physical Review Letters77(18), 3865 (1996) https://doi.org/10

    Perdew, J.P., Burke, K., Ernzerhof, M.: Generalized Gradient Approximation Made Simple. Physical Review Letters77(18), 3865 (1996) https://doi.org/10. 1103/PhysRevLett.77.3865

    work page 1996

  28. [28]

    Royal Society Open Science11(5) (2024) https://doi.org/10.1098/RSOS.231374

    Alaa El-Din, K.K., Forte, A., Kasim, M.F., Miniati, F., Vinko, S.M.: STEP: extraction of underlying physics with robust machine learning. Royal Society Open Science11(5) (2024) https://doi.org/10.1098/RSOS.231374

    work page doi:10.1098/rsos.231374 2024

  29. [29]

    Machine 16 Learning: Science and Technology7(2), 25001 (2026) https://doi.org/10.1088/ 2632-2153/ae3c5a

    Strachwitz, A., Alaa El-Din, K.K., Dutra, A.C.C., Vinko, S.M.: Data-efficient learning of exchange-correlation functionals with differentiable DFT. Machine 16 Learning: Science and Technology7(2), 25001 (2026) https://doi.org/10.1088/ 2632-2153/ae3c5a

    work page 2026

  30. [30]

    Nature Communications 2020 11:1 11(1), 1–10 (2020) https://doi.org/10.1038/s41467-020-17265-7

    Dick, S., Fernandez-Serra, M.: Machine learning accurate exchange and corre- lation functionals of the electronic density. Nature Communications 2020 11:1 11(1), 1–10 (2020) https://doi.org/10.1038/s41467-020-17265-7

    work page doi:10.1038/s41467-020-17265-7 2020

  31. [31]

    Nature Communications 2020 11:111(1), 1–11 (2020) https://doi.org/ 10.1038/s41467-020-19093-1

    Bogojeski, M., Vogt-Maranto, L., Tuckerman, M.E., M¨ uller, K.R., Burke, K.: Quantum chemical accuracy from density functional approximations via machine learning. Nature Communications 2020 11:111(1), 1–11 (2020) https://doi.org/ 10.1038/s41467-020-19093-1

    work page doi:10.1038/s41467-020-19093-1 2020

  32. [32]

    Journal of Chemical Physics144(22), 224101 (2016)

    Fritz, M., Fern´ andez-Serra, M., Soler, J.M.: Optimization of an exchange- correlation density functional for water. Journal of Chemical Physics144(22), 224101 (2016)

    work page 2016

  33. [33]

    Advanced Materials31(46), 1902765 (2019) https://doi.org/10.1002/ADMA.201902765

    Deringer, V.L., Caro, M.A., Cs´ anyi, G.: Machine Learning Interatomic Potentials as Emerging Tools for Materials Science. Advanced Materials31(46), 1902765 (2019) https://doi.org/10.1002/ADMA.201902765

    work page doi:10.1002/adma.201902765 2019

  34. [34]

    Journal of Physical Chemistry A 124(4), 731–745 (2020) https://doi.org/10.1021/ACS.JPCA.9B08723

    Zuo, Y., Chen, C., Li, X., Deng, Z., Chen, Y., Behler, J., Cs´ anyi, G., Shapeev, A.V., Thompson, A.P., Wood, M.A., Ong, S.P.: Performance and Cost Assessment of Machine Learning Interatomic Potentials. Journal of Physical Chemistry A 124(4), 731–745 (2020) https://doi.org/10.1021/ACS.JPCA.9B08723

    work page doi:10.1021/acs.jpca.9b08723 2020

  35. [35]

    How accurate are dft forces? unexpectedly large uncertainties in molecular datasets, 2025

    Kuryla, D., Berger, F., Cs´ anyi, G., Michaelides, A., Hamied, Y.: How Accurate Are DFT Forces? Unexpectedly Large Uncertainties in Molecular Datasets (2025). https://arxiv.org/abs/2510.19774v1

    work page arXiv 2025

  36. [36]

    Journal of Chemical Physics156(8), 84801 (2022) https://doi.org/10.1063/5.0076202

    Kasim, M.F., Lehtola, S., Vinko, S.M.: DQC: A Python program package for differentiable quantum chemistry. Journal of Chemical Physics156(8), 84801 (2022) https://doi.org/10.1063/5.0076202

    work page doi:10.1063/5.0076202 2022

  37. [37]

    https://arxiv.org/abs/2010.01921v1

    Kasim, M.F., Vinko, S.M.:$\xi$-torch: differentiable scientific computing library (2020). https://arxiv.org/abs/2010.01921v1

    work page arXiv 2020

  38. [38]

    Scientific Data 2020 7:17(1), 1–10 (2020) https://doi.org/10.1038/s41597-020-0473-z

    Smith, J.S., Zubatyuk, R., Nebgen, B., Lubbers, N., Barros, K., Roitberg, A.E., Isayev, O., Tretiak, S.: The ANI-1ccx and ANI-1x data sets, coupled-cluster and density functional theory properties for molecules. Scientific Data 2020 7:17(1), 1–10 (2020) https://doi.org/10.1038/s41597-020-0473-z

    work page doi:10.1038/s41597-020-0473-z 2020

  39. [39]

    Nature Communications 2025 16:116(1), 11306 (2025) https: //doi.org/10.1038/s41467-025-66450-z

    Polak, E., Zhao, H., Vuckovic, S.: Real-space machine learning of correlation density functionals. Nature Communications 2025 16:116(1), 11306 (2025) https: //doi.org/10.1038/s41467-025-66450-z

    work page doi:10.1038/s41467-025-66450-z 2025

  40. [40]

    The Journal of Chemical Physics115(20), 9113–9125 (2001) https://doi.org/10.1063/ 17 1.1413524

    Jensen, F., Chem Phys, J.: Polarization consistent basis sets: Principles. The Journal of Chemical Physics115(20), 9113–9125 (2001) https://doi.org/10.1063/ 17 1.1413524

    work page 2001

  41. [41]

    Jensen, F.: Polarization consistent basis sets. II. Estimating the Kohn–Sham basis set limit. The Journal of Chemical Physics116(17), 7372–7379 (2002) https: //doi.org/10.1063/1.1465405

    work page doi:10.1063/1.1465405 2002

  42. [42]

    Journal of Chemical Theory and Computation5(4), 1016–1026 (2009) https://doi.org/10

    Bryantsev, V.S., Diallo, M.S., Van Duin, A.C.T., Goddard, W.A.: Evaluation of B3LYP, X3LYP, and M06-Class density functionals for predicting the bind- ing energies of neutral, protonated, and deprotonated water clusters. Journal of Chemical Theory and Computation5(4), 1016–1026 (2009) https://doi.org/10. 1021/CT800549F

    work page 2009

  43. [43]

    Theoretical Chemistry Accounts113(5), 267–273 (2005) https://doi.org/ 10.1007/S00214-005-0635-2

    Jensen, F.: Estimating the Hartree - Fock limit from finite basis set calcula- tions. Theoretical Chemistry Accounts113(5), 267–273 (2005) https://doi.org/ 10.1007/S00214-005-0635-2

    work page doi:10.1007/s00214-005-0635-2 2005

  44. [44]

    Journal of Chemical Physics 156(16), 161103 (2022) https://doi.org/10.1063/5.0090862

    Palos, E., Lambros, E., Dasgupta, S., Paesani, F.: Density functional theory of waterwith the machine-learned DM21 functional. Journal of Chemical Physics 156(16), 161103 (2022) https://doi.org/10.1063/5.0090862

    work page doi:10.1063/5.0090862 2022

  45. [45]

    Journal of Chemical Physics145(19) (2016) https://doi.org/10

    Reddy, S.K., Straight, S.C., Bajaj, P., Huy Pham, C., Riera, M., Moberg, D.R., Morales, M.A., Knight, C., G¨ otz, A.W., Paesani, F.: On the accuracy of the MB- pol many-body potential for water: Interaction energies, vibrational frequencies, and classical thermodynamic and dynamical properties from clusters to liquid water and ice. Journal of Chemical Phy...

    work page 2016

  46. [46]

    Sharkas, K., Wagle, K., Santra, B., Akter, S., Zope, R.R., Baruah, T., Jackson, K.A., Perdew, J.P., Peralta, J.E.: Self-interaction error overbinds water clusters but cancels in structural energy differences. Proceedings of the National Academy of Sciences of the United States of America117(21), 11283– 11288 (2020) https://doi.org/10.1073/PNAS.1921258117;...

    work page doi:10.1073/pnas.1921258117;website:website: 2020

  47. [47]

    Nature Communications 2023 14:114(1), 799 (2023) https://doi.org/10.1038/ s41467-023-36094-y

    Song, S., Vuckovic, S., Kim, Y., Yu, H., Sim, E., Burke, K.: Extending density functional theory with near chemical accuracy beyond pure water. Nature Communications 2023 14:114(1), 799 (2023) https://doi.org/10.1038/ s41467-023-36094-y

    work page 2023

  48. [48]

    Nature Communications 2021 12:112(1), 6359 (2021) https://doi.org/10.1038/s41467-021-26618-9 18

    Dasgupta, S., Lambros, E., Perdew, J.P., Paesani, F.: Elevating density functional theory to chemical accuracy for water simulations through a density-corrected many-body formalism. Nature Communications 2021 12:112(1), 6359 (2021) https://doi.org/10.1038/s41467-021-26618-9 18

    work page doi:10.1038/s41467-021-26618-9 2021