Pith. sign in

REVIEW 3 major objections 6 minor 1 cited by

Automated Fitting of Neural Network Potentials at Coupled Cluster Accuracy: Protonated Water Clusters as Testing Ground

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read A neural network potential fitted automatically to ~50,000 coupled cluster energies reproduces protonated water clusters from H3O+ to H9O4+ within 0.06 kJ/mol per atom.

desk verdict Solid, carefully validated NNP at CCSD(T) accuracy, though the active-learning convergence criterion is a heuristic and the MD validation is a short spot check. read the letter →

arxiv 1908.08734 v2 pith:VGSAVMQT submitted 2019-08-23 physics.chem-ph cs.LGstat.ML

Christoph Schran , Jörg Behler , Dominik Marx This is my paper · ORCID
classification physics.chem-phcs.LGstat.ML
keywords neuralnetworkpotentialsactivelearningcoupledclusterpotentialenergysurfaceprotonatedwaterclusterspathintegralmoleculardynamicsCCSD(T*)-F12a
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

The paper attempts to show that a single neural network potential can stand in for expensive coupled cluster calculations across an entire family of molecules, not just one fixed structure. Using an automated active-learning loop, the authors fit one potential to CCSD(T*)-F12a/aug-cc-pVTZ reference energies for H2O, H3O+, H5O2+, H7O3+, and H9O4+, reaching a fitting error of 0.06 kJ/mol per atom. They then demonstrate that the same potential reproduces stationary-point energies, harmonic frequencies, reaction paths, and energies sampled by classical and path integral molecular dynamics. The practical payoff, if the claim holds, is that simulations with essentially converged electronic structure accuracy become routine for these clusters, with each energy evaluation about eight orders of magnitude cheaper than the reference calculation.

What carries the argument

The mechanism that carries the argument is committee-based active learning combined with extrapolation detection. At each iteration, two independent neural network potentials are trained on the current reference set and used to predict energies of a large pool of unlabeled configurations from DFT-based classical and path integral trajectories; the 20 configurations with largest disagreement between the two networks are selected for new coupled cluster calculations. A second selection strategy flags configurations whose atom-centered symmetry function values fall outside the range spanned by the training set, which are encountered when preliminary NNP-based simulations explore new regions. The underlying architecture is a high-dimensional neural network potential: atom-centered symmetry functions convert the structure into invariant input vectors, atomic neural networks output atomic energy contributions that sum to the total energy, and the whole function is analytically differentiable so forces are available. This machinery is what lets the training set grow automatically and keeps the number of expensive reference calculations near a practical minimum.

What would settle it

Take the published final training set, run long NNP-PIMD simulations of H9O4+ at 300 K and 600 K, and compute CCSD(T*)-F12a/aug-cc-pVTZ energies for, say, one thousand uncorrelated frames that were never part of training; if the per-atom RMSE on these frames exceeds about 0.1 kJ/mol, the claim that the potential is converged for production simulations would be disproven. A cheaper check would be to compute explicit coupled cluster forces for a sample of tetramer configurations and compare them with NNP forces, since the paper only tests forces explicitly on H3O+.

Watch

Extended reading notes

Core claim

The central claim is that the automated fitting procedure, not a hand-curated grid of configurations, produces a neural network potential whose errors sit at the intrinsic uncertainty of the coupled cluster reference. Trained on roughly 50,000 energy-only reference points and tested on 5,470 held-out points, the single NNP reaches 0.06 kJ/mol per atom RMSE on the training set and 0.08 kJ/mol per atom on the test set, with binding energies of all optimized clusters matching the reference to within 0.1 kcal/mol and harmonic frequencies within about 10 cm−1. The paper further claims the potential remains accurate far from equilibrium, along proton-transfer coordinates and minimum energy paths for isomerization, and that the same functional form supports classical molecular dynamics from 300 to 600 K and path integral simulations down to 1.67 K. In short, the authors claim to have made coupled-cluster-level force evaluation cheap and automatic for finite protonated water clusters.

Load-bearing premise

The load-bearing premise is that the disagreement between two independently trained neural networks, plus the symmetry-function range check, reliably points to the configurations where the potential most needs new coupled cluster data; if both networks share a systematic bias, the active-learning loop could stop early and the reported test error would be optimistic.

Editorial extensions

If this is right

  • A single set of neural network parameters can describe several cluster sizes and isomers at coupled cluster accuracy, so separate fits per molecule are unnecessary.
  • Classical and path integral simulations of these clusters can run at essentially converged electronic structure accuracy, including quantum nuclear effects at temperatures from 1.67 K to 600 K.
  • Energy-only training is sufficient once the reference set is dense; adding force labels to the hydronium data did not improve the fit, so the expensive computation of coupled cluster forces can be avoided.
  • The automated selection protocol reduces the number of reference calculations needed compared with grid-based fitting, making larger finite systems a feasible next step.

Reading between the lines

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

  • The two-network disagreement test is not tied to the neural network architecture in any essential way, so the same automated loop could be applied to other machine-learned potentials such as Gaussian approximation potentials or moment tensor potentials, with the symmetry-function extrapolation test replaced by an equivalent representation-space distance.
  • The paper's validation re-evaluates only the last 100 frames of 25 ps trajectories; a stricter test would be to compute coupled cluster energies for thousands of uncorrelated frames from long production runs, and the reported test-set error suggests such a check would likely pass but is not explicitly demonstrated.
  • If the procedure transfers to larger clusters, one could imagine building a CCSD(T)-quality description of bulk proton transport from cluster-derived training data, because the symmetry functions used here already cover the full cluster with cutoff radii beyond the molecular size.
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. The paper presents an automated active-learning procedure for constructing a high-dimensional neural network potential (NNP) for protonated water clusters (H3O+ through H9O4+ plus H2O) trained to CCSD(T*)-F12a/aug-cc-pVTZ reference energies. The procedure iteratively selects training configurations using two strategies: (i) disagreement between two independently fitted NNPs and (ii) detection of symmetry-function values outside the range of the current training set. The final NNP achieves a training-set RMSE of 0.06 kJ/mol per atom and a test-set RMSE of 0.08 kJ/mol per atom over the full data set. The fit is validated by comparison to coupled-cluster reference calculations for binding energies of stationary points, harmonic frequencies, potential-energy scans, minimum-energy paths, and short segments of classical and path-integral molecular dynamics trajectories. The authors conclude that the automated procedure enables fast and accurate construction of coupled-cluster-quality potential energy surfaces for finite clusters.

Significance. If the reported accuracy and validation hold, this work is a significant methodological advance: it demonstrates that an automated, committee-based active-learning scheme can produce a single NNP that covers multiple cluster sizes at essentially coupled-cluster accuracy with a small number of reference calculations relative to grid-based approaches. The paper's strengths include a clear train/test split with the test set withheld from the fit, validation against reference data for diverse properties (binding energies, frequencies, scans, MEPs, MD/PIMD), and the public provision of the coupled-cluster reference data in the Supporting Information. The automated nature of the fitting procedure, if reliable, would make such potentials practical for systems beyond the present test case. However, the load-bearing assumption that committee disagreement is a calibrated proxy for true error is not directly tested, and the molecular dynamics validation covers only a very short trajectory window.

major comments (3)
  1. [Sec. II and Sec. IVA] The claim that the automated procedure selects configurations 'in an unbiased and efficient way' rests on the assumption that committee disagreement (Strategy I) and symmetry-function range (Strategy II) are reliable proxies for the NNP error relative to the CCSD(T) reference. The manuscript does not calibrate these proxies: no comparison is shown between committee disagreement and the actual NNP error on the held-out test set, and the stopping criterion ('until the differences between the networks are converged') is only qualitative. Because the test set is a random split of the reference data that were themselves selected by these heuristics, the test RMSE does not by itself certify accuracy in configurations that the procedure might systematically ignore. The external validations (scans, MEPs, MD) are reassuring, but they sample a limited set of low-dimensional paths. Please provide a quantitative calibration of committee disagreement versus true error (e.g., a plot on the test set), define a concrete convergence threshold, and show that further iterations do not change predictions on a fixed validation set; alternatively, temper the 'unbiased' claim to 'heuristic-driven' and discuss the associated risk.
  2. [Sec. IVD and Fig. 7] The MD/PIMD validation is limited to the last 100 steps of each 25 ps trajectory, which at the stated time steps corresponds to 50 fs of classical MD (0.5 fs step) and 25 fs of PIMD (0.25 fs step). This is a very short window and does not sample the full range of configurations explored in the trajectories, including the isomerization event described for the 600 K run (ending near the Ring isomer). Thus the statement that the NNP 'reliably describes protonated water clusters during classical MD and quantum PIMD simulations at various conditions' is only directly demonstrated for a small fraction of the simulation. Please extend the coupled-cluster recomputation to more points along the trajectories (or provide statistical measures over the full trajectory) and clarify precisely which frames were compared.
  3. [Abstract and Table I] The abstract and Sec. IVA quote a 'fitting error of 0.06 kJ/mol per atom' without specifying that this is the training-set RMSE; the independent test-set RMSE for the full data set is 0.08 kJ/mol per atom (Table I). While both values are excellent, the abstract should be precise about which error is being reported, and the difference between training and test error (about 25%) is worth a sentence because it bears on the generalization claim.
minor comments (6)
  1. [Sec. II, first paragraph] The phrase 'and recall only that' appears to be a grammatical error; it should likely read 'and recall that' or 'and recall only that' with proper punctuation.
  2. [Sec. III, MD setup] The thermostat is referred to as 'Nos–Hover chain thermostat'; this should be 'Nosé–Hoover chain thermostat'.
  3. [Sec. IVC, final paragraph] The phrase 'protonated water tertramer' contains a typo; it should be 'tetramer'.
  4. [Sec. IVA, first paragraph] The text states that 'at least 100 000 uncorrelated structures are extracted from these simulations for each cluster', but the trajectory lengths given in Sec. III (at least 100 ps AIMD and 25 ps AI-PIMD with configurations spaced 10 fs) would yield about 10,000 and 2,500 structures per cluster, respectively. Please check this number and reconcile the discrepancy.
  5. [Eq. (1) and Table II] Eq. (1) defines E_bind using n water monomers, but for H3O+ (n=1) the formula gives zero, whereas Table II reports -171.4 kcal/mol for M=1 and the table caption notes that E_bind is the energy difference between H3O+ and water. Please clarify the definition for the monomer case.
  6. [Sec. IVA, speed comparison] The text says the NNP is 'about 10^8 times faster' than the reference calculation; the stated values (7 h vs. 0.1 ms) correspond to about 2.5×10^8, so the order of magnitude is correct, but the wording could be made more precise.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the NNP accuracy claim is supported by direct comparison to independent CCSD(T) reference data, not by construction or self-citation.

full rationale

The paper's central claim is that a single neural network potential reproduces CCSD(T*)-F12a/aug-cc-pVTZ energies for protonated water clusters. The coupled cluster reference energies are obtained from explicit Molpro calculations and are not derived from, or defined in terms of, the NNP. The reported fitting error of 0.06 kJ/mol per atom is exactly what it claims to be, a fit error on the training set, and the test-set RMSE of 0.08 kJ/mol per atom is computed on a randomly held-out 10% of the reference data that were not used in the fit (Table I, Fig. 3), providing an independent, in-distribution accuracy check. Validations on stationary-point energies, harmonic frequencies, potential energy scans, MEPs, and MD/PIMD trajectories all compare the surrogate against freshly computed CCSD(T) energies along structures that were not used in the final training set; even though some of those structures were generated by the NNP, the comparison is made to an external ab initio reference and therefore does not reduce to the fit by construction. The active-learning selection criteria (committee disagreement in Strategy I and symmetry-function range extrapolation in Strategy II) are heuristics for choosing training configurations; they are methodological assumptions about data selection, not equations that force the reported accuracy. The convergence criterion based on two-network disagreement could in principle underestimate shared bias, but that is a generalization or calibration risk, not a circularity of the kind where the prediction is equivalent to its input. Self-citations to Refs. 47 and 48 document prior uses of NNPs for coupled-cluster-level cluster simulations, and Refs. 23 and 43 are the original sources of the committee-based training-set selection; none of these citations is used as a uniqueness theorem or as the sole justification for the present fit's accuracy, which is established by direct comparison to CCSD(T) reference data. No load-bearing step reduces to a self-citation or to the fitted inputs. Accordingly, the paper shows no significant circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central result rests on the fitted NNP weights and on several domain assumptions about the reference electronic structure method and the sampling strategy. No new physical entities are introduced.

free parameters (3)
  • Neural network weights = optimized on training set (about 50,000 configurations)
    The atomic neural network weights for all element-specific NNs are fitted to the CCSD(T) reference energies via extended Kalman filtering. They are the model parameters that define the PES and the reported errors depend entirely on them.
  • Symmetry function parameters = taken from Ref. 73 (optimized for water)
    The parameters of the atom-centered symmetry functions (radial and angular functions, cutoff 6.5 Angstrom) are not fitted in this work but are adopted from a prior water NNP study. They are hand-picked hyperparameters of the model.
  • Network architecture hyperparameters = two hidden layers with 30 nodes each, tanh activation
    The size and activation of the neural networks are chosen ad hoc and are not systematically optimized; they define the flexibility of the surrogate model.
assumptions (4)
  • standard math Born-Oppenheimer approximation
    The paper assumes the potential energy surface fully determines nuclear structure and dynamics, invoked in the Introduction.
  • domain assumption CCSD(T*)-F12a/aug-cc-pVTZ energies are an essentially converged reference for the PES
    The entire training and validation rest on treating these coupled cluster calculations as the ground truth, as stated in Sec. III. If this level of theory is biased for, e.g., proton transfer barriers, the NNP inherits the bias.
  • domain assumption RPBE-D3 DFT sampling at 300 K covers the relevant configuration space of all cluster sizes and conditions
    The initial reference structures are generated from DFT-based AIMD and AI-PIMD at one temperature. Strategy II extends the boundaries, but the initial ensemble constrains what can be discovered.
  • ad hoc to paper Committee disagreement and symmetry-function range predict NNP error against the CCSD(T) reference
    The automated selection (Strategy I and II) assumes that high disagreement between two NNPs, or leaving the symmetry-function range, marks configurations where the NNP is least reliable. This is a heuristic active-learning assumption that is not proven for this system.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Automated Fitting of Neural Network Potentials at Coupled Cluster Accuracy: Protonated Water Clusters as Testing Ground." pith.science (2026). https://pith.science/paper/VGSAVMQT

@misc{pith2026190808734,
  author       = {Pith},
  title        = {Pith review of: Automated Fitting of Neural Network Potentials at Coupled Cluster Accuracy: Protonated Water Clusters as Testing Ground},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VGSAVMQT}},
  note         = {Machine review of arXiv:1908.08734}
}
abstract

Highly accurate potential energy surfaces are of key interest for the detailed understanding and predictive modeling of chemical systems. In recent years, several new types of force fields, which are based on machine learning algorithms and fitted to ab initio reference calculations, have been introduced to meet this requirement. Here we show how high-dimensional neural network potentials can be employed to automatically generate the potential energy surface of finite sized clusters at coupled cluster accuracy, namely CCSD(T*)-F12a/aug-cc-pVTZ. The developed automated procedure utilizes the established intrinsic properties of the model such that the configurations for the training set are selected in an unbiased and efficient way to minimize the computational effort of expensive reference calculations. These ideas are applied to protonated water clusters from the hydronium cation, H$_3$O$^+$, up to the tetramer, H$_9$O$_{4}^{+}$, and lead to a single potential energy surface that describes all these systems at essentially converged coupled cluster accuracy with a fitting error of 0.06 kJ/mol per atom. The fit is validated in detail for all clusters up to the tetramer and yields reliable results not only for stationary points, but also for reaction pathways, intermediate configurations, as well as different sampling techniques. Per design the NNPs constructed in this fashion can handle very different conditions including the quantum nature of the nuclei and enhanced sampling techniques covering very low as well as high temperatures. This enables fast and exhaustive exploration of the targeted protonated water clusters with essentially converged interactions. In addition, the automated process will allow one to tackle finite systems much beyond the present case.

Figures

Figures reproduced from arXiv: 1908.08734 by the authors.

Figure 1
Figure 1. Representation of the structure–energy relation in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Schematic illustration of the four general steps [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Correlation of the energy per atom from ex [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Comparison of the harmonic normal mode frequen [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Scans of the potential energy profile obtained [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Potential energy profiles of the H7O + 3 (top) and H9O + 4 (bottom) clusters along selected minimum energy paths (MEP) for important isomerization reactions (repre￾sented by the respective reactant, transition and product state configurations) obtained using the zero t…
Figure 7
Figure 7. Figure 7: Potential energy along classical MD (left) and one replica of quantum PIMD (right) trajectories of the protonated [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

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. Quantum Nature of the Hydrogen Bond from Ambient Conditions down to Ultra-low Temperatures

    physics.chem-ph 2019-08 conditional novelty 6.0 of 10

    Path integral simulations from 300 K to 1.67 K show that oxygen quantum delocalization can match or exceed proton delocalization in finite water clusters at ultra-low temperatures, while the effect is absent in ice.

Reference graph

Works this paper leans on

84 extracted references · 30 canonical work pages · cited by 1 Pith paper

  1. [1]

    author author R. J. \ Bartlett \ and\ author M. Musia ,\ 10.1103/RevModPhys.79.291 journal journal Rev. Mod. Phys. \ volume 79 ,\ pages 291 ( year 2007 ) NoStop

    doi:10.1103/revmodphys.79.291 2007

  2. [2]

    Hobza ,\ 10.1021/ar200255p journal journal Acc

    author author P. Hobza ,\ 10.1021/ar200255p journal journal Acc. Chem. Res. \ volume 45 ,\ pages 663 ( year 2012 ) NoStop

    work page doi:10.1021/ar200255p 2012

  3. [3]

    Marx \ and\ author J

    author author D. Marx \ and\ author J. Hutter ,\ http://ebooks.cambridge.org/ref/id/CBO9780511609633 title Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods \ ( publisher Cambridge University Press ,\ address Cambridge ,\ year 2009 ) NoStop

    2009

  4. [4]

    Spura , author H

    author author T. Spura , author H. Elgabarty , \ and\ author T. D. \ K \" u hne ,\ 10.1039/C4CP05192K journal journal Phys. Chem. Chem. Phys. \ volume 17 ,\ pages 14355 ( year 2015 ) NoStop

    doi:10.1039/c4cp05192k 2015

  5. [5]

    Mouhat , author S

    author author F. Mouhat , author S. Sorella , author R. Vuilleumier , author A. M. \ Saitta , \ and\ author M. Casula ,\ 10.1021/acs.jctc.7b00017 journal journal J. Chem. Theory Comput. \ volume 13 ,\ pages 2400 ( year 2017 ) NoStop

    work page doi:10.1021/acs.jctc.7b00017 2017

  6. [6]

    Haycraft , author J

    author author C. Haycraft , author J. Li , \ and\ author S. S. \ Iyengar ,\ 10.1021/acs.jctc.6b01107 journal journal J. Chem. Theory Comput. \ volume 13 ,\ pages 1887 ( year 2017 ) NoStop

    work page doi:10.1021/acs.jctc.6b01107 2017

  7. [7]

    Babin , author C

    author author V. Babin , author C. Leforestier , \ and\ author F. Paesani ,\ 10.1021/ct400863t journal journal J. Chem. Theory Comput. \ volume 9 ,\ pages 5395 ( year 2013 ) NoStop

    doi:10.1021/ct400863t 2013

  8. [8]

    Bukowski , author K

    author author R. Bukowski , author K. Szalewicz , author G. C. \ Groenenboom , \ and\ author A. Van Der Avoird ,\ 10.1126/science.1136371 journal journal Science \ volume 315 ,\ pages 1249 ( year 2007 ) NoStop

    work page doi:10.1126/science.1136371 2007

Show all 84 references
  1. [9]

    Wang , author B

    author author Y. Wang , author B. C. \ Shepler , author B. J. \ Braams , \ and\ author J. M. \ Bowman ,\ 10.1063/1.3196178 journal journal J. Chem. Phys. \ volume 131 ,\ pages 054511 ( year 2009 ) NoStop

    2009 doi

  2. [10]

    Babin , author G

    author author V. Babin , author G. R. \ Medders , \ and\ author F. Paesani ,\ 10.1021/jz3017733 journal journal J. Phys. Chem. Lett. \ volume 3 ,\ pages 3765 ( year 2012 ) NoStop

    2012 doi

  3. [11]

    author author G. A. \ Cisneros , author K. T. \ Wikfeldt , author L. Ojam \" a e , author J. Lu , author Y. Xu , author H. Torabifard , author A. P. \ Bart \' o k , author G. Cs \' a nyi , author V. Molinero , \ and\ author F. Paesani ,\ 10.1021/acs.chemrev.5b00644 journal jou...

    doi

  4. [12]

    author author J. P. \ Heindel , author Q. Yu , author J. M. \ Bowman , \ and\ author S. S. \ Xantheas ,\ 10.1021/acs.jctc.8b00598 journal journal J. Chem. Theory Comput. \ volume 14 ,\ pages 4553 ( year 2018 ) NoStop

    2018 doi

  5. [13]

    author author R. E. \ Kozack \ and\ author P. C. \ Jordan ,\ 10.1063/1.461957 journal journal J. Chem. Phys. \ volume 96 ,\ pages 3131 ( year 1992 ) NoStop

    1992 doi

  6. [14]

    author author M. P. \ Hodges \ and\ author A. J. \ Stone ,\ 10.1063/1.478580 journal journal J. Chem. Phys. \ volume 110 ,\ pages 6766 ( year 1999 ) NoStop

    1999 doi

  7. [15]

    Lobaugh \ and\ author G

    author author J. Lobaugh \ and\ author G. A. \ Voth ,\ 10.1063/1.470962 journal journal J. Chem. Phys. \ volume 104 ,\ pages 2056 ( year 1996 ) NoStop

    1996 doi

  8. [16]

    author author U. W. \ Schmitt \ and\ author G. A. \ Voth ,\ 10.1021/jp9818131 journal journal J. Phys. Chem. B \ volume 102 ,\ pages 5547 ( year 1998 ) NoStop

    1998 doi

  9. [17]

    Vuilleumier \ and\ author D

    author author R. Vuilleumier \ and\ author D. Borgis ,\ 10.1016/S0009-2614(97)01365-1 journal journal Chem. Phys. Lett. \ volume 284 ,\ pages 71 ( year 1998 ) NoStop

    1998 doi

  10. [18]

    James \ and\ author D

    author author T. James \ and\ author D. J. \ Wales ,\ 10.1063/1.1869987 journal journal J. Chem. Phys. \ volume 122 ,\ pages 134306 ( year 2005 ) NoStop

    2005 doi

  11. [19]

    Brancato \ and\ author M

    author author G. Brancato \ and\ author M. E. \ Tuckerman ,\ 10.1063/1.1902924 journal journal J. Chem. Phys. \ volume 122 ,\ pages 224507 ( year 2005 ) NoStop

    2005 doi

  12. [20]

    Wu , author H

    author author Y. Wu , author H. Chen , author F. Wang , author F. Paesani , \ and\ author G. A. \ Voth ,\ 10.1021/jp076658h journal journal J. Phys. Chem. B \ volume 112 ,\ pages 467 ( year 2008 ) NoStop

    2008 doi

  13. [21]

    Kumar , author R

    author author R. Kumar , author R. A. \ Christie , \ and\ author K. D. \ Jordan ,\ 10.1021/jp8066475 journal journal J. Phys. Chem. B \ volume 113 ,\ pages 4111 ( year 2009 ) NoStop

    2009 doi

  14. [22]

    author author C. M. \ Handley \ and\ author P. L. \ Popelier ,\ 10.1021/jp9105585 journal journal J. Phys. Chem. A \ volume 114 ,\ pages 3371 ( year 2010 ) NoStop

    2010 doi

  15. [23]

    Behler ,\ 10.1039/C1CP21668F journal journal Phys

    author author J. Behler ,\ 10.1039/C1CP21668F journal journal Phys. Chem. Chem. Phys. \ volume 13 ,\ pages 17930 ( year 2011 a ) NoStop

    2011 doi

  16. [24]

    Behler ,\ 10.1063/1.4966192 journal journal J

    author author J. Behler ,\ 10.1063/1.4966192 journal journal J. Chem. Phys. \ volume 145 ,\ pages 170901 ( year 2016 ) NoStop

    2016 doi

  17. [25]

    author author A. P. \ Bart \' o k , author S. De , author C. Poelking , author N. Bernstein , author J. R. \ Kermode , author G. Cs \' a nyi , \ and\ author M. Ceriotti ,\ 10.1126/sciadv.1701816 journal journal Sci. Adv. \ volume 3 ,\ pages e1701816 ( year 2017 ) NoStop

    doi

  18. [26]

    author author K. T. \ Butler , author D. W. \ Davies , author H. Cartwright , author O. Isayev , \ and\ author A. Walsh ,\ 10.1038/s41586-018-0337-2 journal journal Nature \ volume 559 ,\ pages 547 ( year 2018 ) NoStop

    2018 doi

  19. [27]

    author author T. B. \ Blank , author S. D. \ Brown , author A. W. \ Calhoun , \ and\ author D. J. \ Doren ,\ 10.1063/1.469597 journal journal J. Chem. Phys. \ volume 103 ,\ pages 4129 ( year 1995 ) NoStop

    1995 doi

  20. [28]

    author author F. V. \ Prudente , author P. H. \ Acioli , \ and\ author J. J. \ Neto ,\ 10.1063/1.477550 journal journal J. Chem. Phys. \ volume 109 ,\ pages 8801 ( year 1998 ) NoStop

    1998 doi

  21. [29]

    Lorenz , author A

    author author S. Lorenz , author A. Gro , \ and\ author M. Scheffler ,\ 10.1016/j.cplett.2004.07.076 journal journal Chem. Phys. Lett. \ volume 395 ,\ pages 210 ( year 2004 ) NoStop

    2004 doi

  22. [30]

    Lorenz , author M

    author author S. Lorenz , author M. Scheffler , \ and\ author A. Gross ,\ 10.1103/PhysRevB.73.115431 journal journal Phys. Rev. B \ volume 73 ,\ pages 115431 ( year 2006 ) NoStop

    2006 doi

  23. [31]

    Manzhos , author X

    author author S. Manzhos , author X. Wang , author R. Dawes , \ and\ author T. Carrington ,\ 10.1021/jp055253z journal journal J. Phys. Chem. A \ volume 110 ,\ pages 5295 ( year 2006 ) NoStop

    2006 doi

  24. [32]

    Behler \ and\ author M

    author author J. Behler \ and\ author M. Parrinello ,\ 10.1103/PhysRevLett.98.146401 journal journal Phys. Rev. Lett. \ volume 98 ,\ pages 146401 ( year 2007 ) NoStop

    2007 doi

  25. [33]

    author author A. P. \ Bart \' o k , author M. C. \ Payne , author R. Kondor , \ and\ author G. Cs \' a nyi ,\ 10.1103/PhysRevLett.104.136403 journal journal Phys. Rev. Lett. \ volume 104 ,\ pages 136403 ( year 2010 ) NoStop

    2010 doi

  26. [34]

    Rupp , author A

    author author M. Rupp , author A. Tkatchenko , author K. R. \ M \" u ller , \ and\ author O. A. \ Von Lilienfeld ,\ 10.1103/PhysRevLett.108.058301 journal journal Phys. Rev. Lett. \ volume 108 ,\ pages 058301 ( year 2012 ) NoStop

    2012 doi

  27. [35]

    author author A. V. \ Shapeev ,\ 10.1137/15M1054183 journal journal Multiscale Model. Simul. \ volume 14 ,\ pages 1153 ( year 2015 ) NoStop

    2015 doi

  28. [36]

    Li , author J

    author author Z. Li , author J. R. \ Kermode , \ and\ author A. De Vita ,\ 10.1103/PhysRevLett.114.096405 journal journal Phys. Rev. Lett. \ volume 114 ,\ pages 096405 ( year 2015 ) NoStop

    2015 doi

  29. [37]

    author author A. P. \ Thompson , author L. P. \ Swiler , author C. R. \ Trott , author S. M. \ Foiles , \ and\ author G. J. \ Tucker ,\ 10.1016/j.jcp.2014.12.018 journal journal J. Comput. Phys. \ volume 285 ,\ pages 316 ( year 2015 ) NoStop

    2014 doi

  30. [38]

    u tt , author F. Arbabzadah , author S. Chmiela , author K. R. \ M \

    author author K. T. \ Sch \" u tt , author F. Arbabzadah , author S. Chmiela , author K. R. \ M \" u ller , \ and\ author A. Tkatchenko ,\ 10.1038/ncomms13890 journal journal Nat. Commun. \ volume 8 ,\ pages 13890 ( year 2017 ) NoStop

    doi

  31. [39]

    u tt , \ and\ author K. R. \ M \

    author author S. Chmiela , author A. Tkatchenko , author H. E. \ Sauceda , author I. Poltavsky , author K. T. \ Sch \" u tt , \ and\ author K. R. \ M \" u ller ,\ 10.1126/sciadv.1603015 journal journal Sci. Adv. \ volume 3 ,\ pages e1603015 ( year 2017 ) NoStop

    doi

  32. [40]

    Faraji , author S

    author author S. Faraji , author S. A. \ Ghasemi , author S. Rostami , author R. Rasoulkhani , author B. Schaefer , author S. Goedecker , \ and\ author M. Amsler ,\ 10.1103/PhysRevB.95.104105 journal journal Phys. Rev. B \ volume 95 ,\ pages 104105 ( year 2017 ) NoStop

    doi

  33. [41]

    author author O. T. \ Unke \ and\ author M. Meuwly ,\ 10.1021/acs.jctc.9b00181 journal journal J. Chem. Theory Comput. \ volume 15 ,\ pages 3678 ( year 2019 ) NoStop

    2019 doi

  34. [42]

    Behler ,\ 10.1002/anie.201703114 journal journal Angew

    author author J. Behler ,\ 10.1002/anie.201703114 journal journal Angew. Chemie - Int. Ed. \ volume 56 ,\ pages 12828 ( year 2017 ) NoStop

    2017 doi

  35. [43]

    Artrith \ and\ author J

    author author N. Artrith \ and\ author J. Behler ,\ 10.1103/PhysRevB.85.045439 journal journal Phys. Rev. B \ volume 85 ,\ pages 045439 ( year 2012 ) NoStop

    2012 doi

  36. [44]

    Gastegger , author J

    author author M. Gastegger , author J. Behler , \ and\ author P. Marquetand ,\ 10.1039/c7sc02267k journal journal Chem. Sci. \ volume 8 ,\ pages 6924 ( year 2017 ) NoStop

    2017 doi

  37. [45]

    author author V. L. \ Deringer , author C. J. \ Pickard , \ and\ author G. Cs \' a nyi ,\ 10.1103/PhysRevLett.120.156001 journal journal Phys. Rev. Lett. \ volume 120 ,\ pages 156001 ( year 2018 ) NoStop

    2018 doi

  38. [46]

    author author E. V. \ Podryabinkin \ and\ author A. V. \ Shapeev ,\ 10.1016/j.commatsci.2017.08.031 journal journal Comput. Mater. Sci. \ ( year 2017 ),\ 10.1016/j.commatsci.2017.08.031 NoStop

    2017 doi

  39. [47]

    Schran , author F

    author author C. Schran , author F. Uhl , author J. Behler , \ and\ author D. Marx ,\ 10.1063/1.4996819 journal journal J. Chem. Phys. \ volume 148 ,\ pages 102310 ( year 2018 ) NoStop

    2018 doi

  40. [48]

    Schran , author F

    author author C. Schran , author F. Brieuc , \ and\ author D. Marx ,\ 10.1021/acs.jctc.8b00705 journal journal J. Chem. Theory Comput. \ volume 14 ,\ pages 5068 ( year 2018 ) NoStop

    2018 doi

  41. [49]

    Chmiela , author H

    author author S. Chmiela , author H. E. \ Sauceda , author K. R. \ M \" u ller , \ and\ author A. Tkatchenko ,\ 10.1038/s41467-018-06169-2 journal journal Nat. Commun. \ volume 9 ,\ pages 3887 ( year 2018 ) NoStop

    2018 doi

  42. [50]

    author author J. S. \ Smith , author B. T. \ Nebgen , author R. Zubatyuk , author N. Lubbers , author C. Devereux , author K. Barros , author S. Tretiak , author O. Isayev , \ and\ author A. E. \ Roitberg ,\ 10.1038/s41467-019-10827-4 journal journal Nat. Commun. \ volume 10 ,...

    doi

  43. [51]

    author author S. K. \ Natarajan , author T. Morawietz , \ and\ author J. Behler ,\ 10.1039/C4CP04751F journal journal Phys. Chem. Chem. Phys. \ volume 17 ,\ pages 8356 ( year 2015 ) NoStop

    2015 doi

  44. [52]

    author author B. J. \ Braams \ and\ author J. M. \ Bowman ,\ 10.1080/01442350903234923 journal journal Int. Rev. Phys. Chem. \ volume 28 ,\ pages 577 ( year 2009 ) NoStop

    2009 doi

  45. [53]

    author author A. P. \ Bart \' o k , author M. J. \ Gillan , author F. R. \ Manby , \ and\ author G. Cs \' a nyi ,\ 10.1103/PhysRevB.88.054104 journal journal Phys. Rev. B \ volume 88 ,\ pages 054104 ( year 2013 ) NoStop

    2013 doi

  46. [54]

    Hellstr \" o m \ and\ author J

    author author M. Hellstr \" o m \ and\ author J. Behler ,\ 10.1021/acs.jpclett.6b01448 journal journal J. Phys. Chem. Lett. \ volume 7 ,\ pages 3302 ( year 2016 ) NoStop

    2016 doi

  47. [55]

    Quaranta , author M

    author author V. Quaranta , author M. Hellstr \" o m , \ and\ author J. Behler ,\ 10.1021/acs.jpclett.7b00358 journal journal J. Phys. Chem. Lett. \ volume 8 ,\ pages 1476 ( year 2017 ) NoStop

    2017 doi

  48. [56]

    Behler ,\ 10.1063/1.3553717 journal journal J

    author author J. Behler ,\ 10.1063/1.3553717 journal journal J. Chem. Phys. \ volume 134 ,\ pages 074106 ( year 2011 b ) NoStop

    2011 doi

  49. [57]

    author author H. S. \ Seung , author M. Opper , \ and\ author H. Sompolinsky ,\ in\ 10.1145/130385.130417 booktitle Proc. Fifth Annu. ACM Work. Comput. Learn. Theory \ ( publisher ACM Press ,\ address New York, New York, USA ,\ year 1992 )\ pp.\ pages 287--294 NoStop

  50. [58]

    @noop title CP2K , freely available at the URL https://www.cp2k.org, released under GPL license , \ ( year 2019 ) NoStop

    2019

  51. [59]

    Hutter , author M

    author author J. Hutter , author M. Iannuzzi , author F. Schiffmann , \ and\ author J. Vandevondele ,\ 10.1002/wcms.1159 journal journal Wiley Interdiscip. Rev. Comput. Mol. Sci. \ volume 4 ,\ pages 15 ( year 2014 ) NoStop

    2014 doi

  52. [60]

    Hammer , author L

    author author B. Hammer , author L. B. \ Hansen , \ and\ author J. K. \ N rskov ,\ 10.1103/PhysRevB.59.7413 journal journal Phys. Rev. B \ volume 59 ,\ pages 7413 ( year 1999 ) NoStop

    1999 doi

  53. [61]

    Grimme , author J

    author author S. Grimme , author J. Antony , author S. Ehrlich , \ and\ author H. Krieg ,\ 10.1063/1.3382344 journal journal J. Chem. Phys. \ volume 132 ,\ pages 154104 ( year 2010 ) NoStop

    2010 doi

  54. [62]

    VandeVondele , author M

    author author J. VandeVondele , author M. Krack , author F. Mohamed , author M. Parrinello , author T. Chassaing , \ and\ author J. Hutter ,\ 10.1016/j.cpc.2004.12.014 journal journal Comput. Phys. Commun. \ volume 167 ,\ pages 103 ( year 2005 ) NoStop

    2004 doi

  55. [63]

    Goedecker , author M

    author author S. Goedecker , author M. Teter , \ and\ author J. Hutter ,\ 10.1103/PhysRevB.54.1703 journal journal Phys. Rev. B \ volume 54 ,\ pages 1703 ( year 1996 ) NoStop

    1996 doi

  56. [64]

    Morawietz \ and\ author J

    author author T. Morawietz \ and\ author J. Behler ,\ 10.1021/jp401225b journal journal J. Phys. Chem. A \ volume 117 ,\ pages 7356 ( year 2013 ) NoStop

    2013 doi

  57. [65]

    Forster-Tonigold \ and\ author A

    author author K. Forster-Tonigold \ and\ author A. Gro ,\ 10.1063/1.4892400 journal journal J. Chem. Phys. \ volume 141 ,\ pages 064501 ( year 2014 ) NoStop

    2014 doi

  58. [66]

    Morawietz , author A

    author author T. Morawietz , author A. Singraber , author C. Dellago , \ and\ author J. Behler ,\ 10.1073/pnas.1602375113 journal journal Proc. Natl. Acad. Sci. \ volume 113 ,\ pages 8368 ( year 2016 ) NoStop

    2016 doi

  59. [67]

    Ceriotti \ and\ author D

    author author M. Ceriotti \ and\ author D. E. \ Manolopoulos ,\ 10.1103/PhysRevLett.109.100604 journal journal Phys. Rev. Lett. \ volume 109 ,\ pages 100604 ( year 2012 ) NoStop

    2012 doi

  60. [68]

    author author T. B. \ Adler , author G. Knizia , \ and\ author H. J. \ Werner ,\ 10.1063/1.2817618 journal journal J. Chem. Phys. \ volume 127 ,\ pages 221106 ( year 2007 ) NoStop

    2007 doi

  61. [69]

    Knizia , author T

    author author G. Knizia , author T. B. \ Adler , \ and\ author H. J. \ Werner ,\ 10.1063/1.3054300 journal journal J. Chem. Phys. \ volume 130 ,\ pages 054104 ( year 2009 ) NoStop

    2009 doi

  62. [70]

    author author R. A. \ Kendall , author T. H. D. \ Jr , \ and\ author R. J. \ Harrison ,\ 10.1063/1.462569 journal journal J. Chem. Phys. \ volume 96 ,\ pages 6796 ( year 1992 ) NoStop

    1992 doi

  63. [71]

    author author D. E. \ Woon \ and\ author T. H. \ Dunning Jr. ,\ 10.1063/1.466439 journal journal J. Chem. Phys. \ volume 100 ,\ pages 2975 ( year 1994 ) NoStop

    1994 doi

  64. [72]

    u tz , author P. Celani , author W. Gy\

    author author H.-J. \ Werner , author P. J. \ Knowles , author G. Knizia , author F. R. \ Manby , author M. Sch\" u tz , author P. Celani , author W. Gy\"orffy , author D. Kats , author T. Korona , author R. Lindh , author A. Mitrushenkov , author G. Rauhut , author K. R. \ Sh...

    2015

  65. [73]

    Morawietz , author O

    author author T. Morawietz , author O. Marsalek , author S. R. \ Pattenaude , author L. M. \ Streacker , author D. Ben-Amotz , \ and\ author T. E. \ Markland ,\ 10.1021/acs.jpclett.8b00133 journal journal J. Phys. Chem. Lett. \ volume 9 ,\ pages 851 ( year 2018 ) NoStop

    doi

  66. [74]

    Shah , author F

    author author S. Shah , author F. Palmieri , \ and\ author M. Datum ,\ 10.1016/S0893-6080(05)80139-X journal journal Neural Networks \ volume 5 ,\ pages 779 ( year 1992 ) NoStop

    1992 doi

  67. [75]

    author author T. B. \ Blank \ and\ author S. D. \ Brown ,\ 10.1002/cem.1180080605 journal journal J. Chemom. \ volume 8 ,\ pages 391 ( year 1994 ) NoStop

    1994 doi

  68. [76]

    author author J. B. \ Witkoskie \ and\ author D. J. \ Doren ,\ 10.1021/ct049976i journal journal J. Chem. Theory Comput. \ volume 1 ,\ pages 14 ( year 2005 ) NoStop

    2005 doi

  69. [77]

    a t G \

    author author J. Behler et al. ,\ @noop title RuNNer - A Neural Network Code for High-Dimensional Potential-Energy Surfaces, Universit \" a t G \" o ttingen 2019 (GPL3 license). \ NoStop

    2019

  70. [78]

    E , author W

    author author W. E , author W. Ren , \ and\ author E. Vanden-Eijnden ,\ 10.1103/PhysRevB.66.052301 journal journal Phys. Rev. B \ volume 66 ,\ pages 523011 ( year 2002 ) NoStop

    2002 doi

  71. [79]

    author author H. C. \ Andersen ,\ 10.1016/0021-9991(83)90014-1 journal journal J. Comput. Phys. \ volume 52 ,\ pages 24 ( year 1983 ) NoStop

    1983 doi

  72. [80]

    author author M. J. \ Gillan ,\ 10.1088/0022-3719/20/24/005 journal journal J. Phys. C \ volume 20 ,\ pages 3621 ( year 1987 ) NoStop

    1987 doi

  73. [81]

    author author G. A. \ Voth , author D. Chandler , \ and\ author W. H. \ Miller ,\ 10.1063/1.457242 journal journal J. Chem. Phys. \ volume 91 ,\ pages 7749 ( year 1989 ) NoStop

    1989 doi

  74. [82]

    Walker \ and\ author A

    author author B. Walker \ and\ author A. Michaelides ,\ 10.1063/1.3505038 journal journal J. Chem. Phys. \ volume 133 ,\ pages 174306 ( year 2010 ) NoStop

    2010 doi

  75. [83]

    Ceriotti , author M

    author author M. Ceriotti , author M. Parrinello , author T. E. \ Markland , \ and\ author D. E. \ Manolopoulos ,\ 10.1063/1.3489925 journal journal J. Chem. Phys. \ volume 133 ,\ pages 124104 ( year 2010 ) NoStop

    2010 doi

  76. [84]

    Morawietz , author V

    author author T. Morawietz , author V. Sharma , \ and\ author J. Behler ,\ 10.1063/1.3682557 journal journal J. Chem. Phys. \ volume 136 ,\ pages 064103 ( year 2012 ) NoStop

    2012 doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.