pith. sign in

arxiv: 2412.03281 · v3 · pith:GOPTSHN4new · submitted 2024-12-04 · ⚛️ physics.chem-ph

Fast and flexible long-range models for atomistic machine learning

Philip Loche , Kevin K. Huguenin-Dumittan , Melika Honarmand , Qianjun Xu , Egor Rumiantsev , Wei Bin How , Marcel F. Langer , Michele Ceriotti This is my paper

Pith reviewed 2026-05-23 08:23 UTC · model grok-4.3

classification ⚛️ physics.chem-ph
keywords atomistic machine learninglong-range interactionsEwald summationparticle-mesh Ewaldmolecular dynamicsequivariant descriptorsPyTorchJAX
0
0 comments X

The pith

A framework ports Ewald summation, PME and P3M algorithms into PyTorch and JAX so atomistic ML models can treat long-range electrostatics directly.

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

Most atomistic machine learning models decompose energy into short-ranged, atom-centered terms under a locality ansatz, which limits their ability to capture long-range effects such as electrostatics. The paper develops a modular library that embeds classical non-bonded solvers—Ewald summation, particle-mesh Ewald, and particle-particle/particle-mesh Ewald—into differentiable programming frameworks. It also supplies purified descriptors that exclude an atom’s immediate neighborhood to suit general long-range tasks. The resulting implementations support accurate force evaluation, automatic differentiation for mixing with local models, and flexible construction of more complex architectures. Benchmarks demonstrate their use in molecular dynamics, potential training, and computation of long-range equivariant descriptors.

Core claim

The central claim is that established algorithms for evaluating non-bonded interactions can be brought into atomistic machine learning through reference implementations in PyTorch and an experimental JAX version, delivering accurate long-range forces and purified descriptors that operate seamlessly with existing local ML schemes.

What carries the argument

Ewald summation and its particle-mesh variants adapted for automatic differentiation, together with purified descriptors that disregard immediate atomic neighborhoods.

If this is right

  • Accurate physical long-range forces become available as building blocks for semi-empirical baseline potentials.
  • Long-range and local ML components combine directly through automatic differentiation without custom glue code.
  • Complex model architectures can treat physical long-range interactions as modular components.
  • Molecular dynamics simulations can incorporate the long-range terms while remaining fully differentiable.
  • Long-range equivariant descriptors of atomic structures can be evaluated efficiently inside the same framework.

Where Pith is reading between the lines

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

  • The same infrastructure could be reused for other slowly decaying potentials such as dispersion or polarization terms without new solver development.
  • Purified descriptors may prove useful for any property that depends on distant rather than nearest-neighbor environments.
  • Because the code is open and modular, it lowers the barrier for embedding classical physics priors into larger ML pipelines.
  • The approach suggests that other mature classical algorithms could be ported in the same differentiable style to expand the range of physics-informed ML components.

Load-bearing premise

That classical long-range solvers can be incorporated into ML frameworks without introducing accuracy or efficiency trade-offs that would need separate validation.

What would settle it

A side-by-side test on a charged periodic system in which the new implementation either deviates from reference Ewald energies and forces or runs slower than a conventional non-ML Ewald code.

Figures

Figures reproduced from arXiv: 2412.03281 by Egor Rumiantsev, Kevin K. Huguenin-Dumittan, Marcel F. Langer, Melika Honarmand, Michele Ceriotti, Philip Loche, Qianjun Xu, Wei Bin How.

Figure 1
Figure 1. Figure 1: A schematic representation of the main building [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic representation of the atoms contribut [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Empirical versus target accuracy after tuning cal [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Benchmark for computational cost of long-range calculators to run a single point evaluation of the energy, force and [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A: Time series of the first 200 ps showing the fluc [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Training curves for the charge of the sodium [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Schematic representation of an architecture that [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A schematic representation of the architecture of a [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
read the original abstract

Most atomistic machine learning (ML) models rely on a locality ansatz, and decompose the energy into a sum of short-ranged, atom-centered contributions. This leads to clear limitations when trying to describe problems that are dominated by long-range physical effects - most notably electrostatics. Many approaches have been proposed to overcome these limitations, but efforts to make them efficient and widely available are hampered by the need to incorporate an ad hoc implementation of methods to treat long-range interactions. We develop a framework aiming to bring some of the established algorithms to evaluate non-bonded interactions - including Ewald summation, classical particle-mesh Ewald (PME), and particle-particle/particle-mesh (P3M) Ewald - into atomistic ML. We provide a reference implementation for pyTorch as well as an experimental one for JAX. Beyond Coulomb and more general long-range potentials, we introduce purified descriptors which disregard the immediate neighborhood of each atom, and are more suitable for general long-ranged ML applications. Our implementations are fast, feature-rich, and modular: They provide an accurate evaluation of physical long-range forces that can be used in the construction of (semi)empirical baseline potentials; they exploit the availability of automatic differentiation to seamlessly combine long-range models with conventional, local ML schemes; and they are sufficiently flexible to implement more complex architectures that use physical interactions as building blocks. We benchmark and demonstrate our torch-pme and jax-pme libraries to perform molecular dynamics simulations, to train ML potentials, and to evaluate long-range equivariant descriptors of atomic structures.

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

0 major / 3 minor

Summary. The manuscript develops a modular framework and reference libraries (torch-pme, jax-pme) that port established long-range electrostatic algorithms (Ewald summation, classical PME, P3M) into atomistic ML models. It supplies PyTorch and JAX implementations, introduces purified descriptors that exclude immediate atomic neighborhoods, and demonstrates use cases including MD simulations, training of ML potentials, and evaluation of long-range equivariant descriptors, with emphasis on automatic differentiation for seamless combination with local models and modularity for more complex architectures.

Significance. If the reported accuracy, efficiency, and integration hold, the work provides a practical solution to a recognized limitation of locality-based atomistic ML models. Explicit credit is due for the open reference implementations, the multiple use-case benchmarks, and the focus on autodiff compatibility, all of which lower barriers to adoption and support reproducibility.

minor comments (3)
  1. [Abstract, §1] The abstract and introduction would benefit from a single consolidated table or paragraph that lists the key performance metrics (timings, energy/force errors) reported in the benchmarks, rather than scattering them across later sections.
  2. [§2.3] Notation for the purified descriptors (e.g., the precise cutoff or weighting function used to disregard the local neighborhood) should be defined once in a dedicated subsection and then used consistently in all subsequent equations and figures.
  3. [§4] Figure captions for the MD and training benchmarks should explicitly state the system sizes, number of independent runs, and hardware used, to allow direct comparison with other long-range ML implementations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their supportive summary, recognition of the significance of the work, and recommendation for minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; applies established external algorithms

full rationale

The paper's derivation chain consists of porting well-established, externally validated algorithms (Ewald summation, classical PME, P3M) into ML frameworks, along with the introduction of purified descriptors. These components rely on independent mathematical and computational foundations outside the present work, with no equations or claims reducing to self-referential definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The manuscript supplies reference implementations and benchmarks against external standards, rendering the central claims self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that locality is the primary limitation and that established physical long-range solvers can be dropped in without new modeling errors. No free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The locality ansatz in atomistic ML produces clear limitations for long-range physical effects such as electrostatics.
    Opening sentence of the abstract; treated as given.

pith-pipeline@v0.9.0 · 5847 in / 1243 out tokens · 27312 ms · 2026-05-23T08:23:52.630354+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

89 extracted references · 89 canonical work pages · 1 internal anchor

  1. [1]

    Behler \ and\ author M

    author author J. Behler \ and\ author M. Parrinello ,\ title title Generalized neural-network representation of high-dimensional potential-energy surfaces , \ @noop journal journal Physical Review Letters \ volume 98 ,\ pages 146401 ( year 2007 ) NoStop

    work page 2007

  2. [2]

    author author A. P. \ Bartók , author M. C. \ Payne , author R. Kondor ,\ and\ author G. Csányi ,\ title title Gaussian approximation potentials: The accuracy of quantum mechanics, without the electrons , \ @noop journal journal Physical Review Letters \ volume 104 ,\ pages 136403 ( year 2010 ) NoStop

    work page 2010

  3. [3]

    Rupp , author A

    author author M. Rupp , author A. Tkatchenko , author K.-R. \ Müller ,\ and\ author O. A. \ von Lilienfeld ,\ title title Fast and accurate modeling of molecular atomization energies with machine learning , \ @noop journal journal Physical Review Letters \ volume 108 ,\ pages 058301 ( year 2012 ) NoStop

    work page 2012

  4. [4]

    author author K. T. \ Schütt , author H. E. \ Sauceda , author P.-J. \ Kindermans , author A. Tkatchenko ,\ and\ author K.-R. \ Müller ,\ title title Schnet – a deep learning architecture for molecules and materials , \ @noop journal journal The Journal of Chemical Physics \ volume 148 ,\ pages 241722 ( year 2018 ) NoStop

    work page 2018

  5. [5]

    author author K. T. \ Schütt , author P.-J. \ Kindermans , author H. E. \ Sauceda , author S. Chmiela , author A. Tkatchenko ,\ and\ author K.-R. \ Müller ,\ title title Schnet: A continuous-filter convolutional neural network for modeling quantum interactions , \ @noop journal journal Advances in Neural Information Processing Systems \ volume 30 ( year 2...

    work page 2017

  6. [6]

    Klicpera , author S

    author author J. Klicpera , author S. Giri , author J. T. \ Margraf ,\ and\ author S. Günnemann ,\ title title Directional message passing for molecular graphs , \ @noop journal journal International Conference on Learning Representations \ ( year 2020 ) NoStop

    work page 2020

  7. [7]

    Zhang , author J

    author author L. Zhang , author J. Han , author H. Wang , author R. Car ,\ and\ author W. E ,\ title title Deep potential molecular dynamics: A scalable model with the accuracy of quantum mechanics at the cost of classical molecular dynamics , \ @noop journal journal Physical Review Letters \ volume 120 ,\ pages 143001 ( year 2018 ) NoStop

    work page 2018

  8. [8]

    Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds

    author author N. Thomas , author T. E. \ Smidt , author S. Kearnes , author L. Yang , author L. Li , author K. Kohlhoff ,\ and\ author P. Riley ,\ title title Tensor field networks: Rotation- and translation-equivariant neural networks for 3d point clouds , \ @noop journal journal arXiv preprint arXiv:1802.08219 \ ( year 2018 ) NoStop

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  9. [9]

    Musaelian , author S

    author author A. Musaelian , author S. Batzner , author A. Johansson , author L. Sun , author C. J. \ Owen , author M. Kornbluth ,\ and\ author B. Kozinsky ,\ title title Learning local equivariant representations for large-scale atomistic dynamics , \ https://doi.org/10.1038/s41467-023-36329-y journal journal Nat Commun \ volume 14 ,\ pages 579 ( year 20...

    work page doi:10.1038/s41467-023-36329-y 2023

  10. [10]

    Glielmo , author P

    author author A. Glielmo , author P. Sollich ,\ and\ author A. De Vita ,\ title title Accurate interatomic force fields via machine learning with covariant kernels , \ @noop journal journal Physical Review B \ volume 95 ,\ pages 214302 ( year 2017 ) NoStop

    work page 2017

  11. [11]

    Grisafi , author D

    author author A. Grisafi , author D. M. \ Wilkins , author G. Csányi ,\ and\ author M. Ceriotti ,\ title title Symmetry-adapted machine learning for tensorial properties of atomistic systems , \ @noop journal journal Physical Review Letters \ volume 120 ,\ pages 036002 ( year 2018 ) NoStop

    work page 2018

  12. [12]

    author author D. M. \ Wilkins , author A. Grisafi , author Y. Yang , author K. U. \ Lao , author R. A. \ DiStasio Jr. ,\ and\ author M. Ceriotti ,\ title title Accurate molecular polarizabilities with coupled cluster theory and machine learning , \ @noop journal journal Proceedings of the National Academy of Sciences \ volume 116 ,\ pages 3401--3406 ( yea...

    work page 2019

  13. [13]

    Brockherde , author L

    author author F. Brockherde , author L. Vogt , author L. Li , author M. E. \ Tuckerman , author K. Burke ,\ and\ author K.-R. \ Müller ,\ title title Bypassing the kohn-sham equations with machine learning , \ @noop journal journal Nature Communications \ volume 8 ,\ pages 872 ( year 2017 ) NoStop

    work page 2017

  14. [14]

    Fabrizio , author A

    author author A. Fabrizio , author A. Grisafi , author B. Meyer ,\ and\ author C. Corminboeuf ,\ title title Electron density learning of non-covalent systems , \ @noop journal journal Chemical Science \ volume 10 ,\ pages 9424--9432 ( year 2019 ) NoStop

    work page 2019

  15. [15]

    author author A. J. \ Lewis , author A. Grisafi , author M. Ceriotti ,\ and\ author M. Rossi ,\ title title Learning electron density in the condensed phase , \ @noop journal journal Journal of Chemical Theory and Computation \ volume 17 ,\ pages 6725--6737 ( year 2021 ) NoStop

    work page 2021

  16. [16]

    author author K. T. \ Schütt , author M. Gastegger , author A. Tkatchenko , author K.-R. \ Müller ,\ and\ author R. J. \ Maurer ,\ title title Unifying machine learning and quantum chemistry with a deep neural network for molecular wavefunctions , \ @noop journal journal Nature Communications \ volume 10 ,\ pages 5024 ( year 2019 ) NoStop

    work page 2019

  17. [17]

    Cignoni , author D

    author author E. Cignoni , author D. Suman , author J. Nigam , author L. Cupellini , author B. Mennucci ,\ and\ author M. Ceriotti ,\ title title Electronic excited states from physically constrained machine learning , \ https://doi.org/10.1021/acscentsci.3c01480 journal journal ACS Central Science \ volume 10 ,\ pages 637--648 ( year 2024 ) ,\ https://ar...

    work page doi:10.1021/acscentsci.3c01480 2024

  18. [18]

    Carleo \ and\ author M

    author author G. Carleo \ and\ author M. Troyer ,\ title title Solving the quantum many-body problem with artificial neural networks , \ @noop journal journal Science \ volume 355 ,\ pages 602--606 ( year 2017 ) NoStop

    work page 2017

  19. [19]

    Hermann , author Z

    author author J. Hermann , author Z. Schätzle ,\ and\ author F. Noé ,\ title title Deep-neural-network solution of the electronic schrödinger equation , \ @noop journal journal Nature Chemistry \ volume 12 ,\ pages 891--897 ( year 2020 ) NoStop

    work page 2020

  20. [20]

    author author O. T. \ Unke , author M. Bogojeski , author M. Gastegger , author M. Geiger , author T. Smidt ,\ and\ author K.-R. \ M \"u ller ,\ https://doi.org/10.48550/arXiv.2106.02347 title SE (3)-equivariant prediction of molecular wavefunctions and electronic densities , \ ( year 2021 a ),\ https://arxiv.org/abs/2106.02347 arXiv:2106.02347 NoStop

    work page doi:10.48550/arxiv.2106.02347 2021

  21. [21]

    Fulton \ and\ author J

    author author W. Fulton \ and\ author J. Harris ,\ @noop title Representation Theory: A First Course ,\ series Graduate Texts in Mathematics , Vol.\ volume 129 \ ( publisher Springer-Verlag ,\ address New York ,\ year 1991 ) NoStop

    work page 1991

  22. [22]

    \ Serre ,\ @noop title Linear Representations of Finite Groups ,\ series Graduate Texts in Mathematics , Vol

    author author J.-P. \ Serre ,\ @noop title Linear Representations of Finite Groups ,\ series Graduate Texts in Mathematics , Vol. volume 42 \ ( publisher Springer-Verlag ,\ address New York ,\ year 1977 ) NoStop

    work page 1977

  23. [23]

    author author M. Tinkham ,\ @noop title Group Theory and Quantum Mechanics \ ( publisher Dover Publications ,\ address Mineola, NY ,\ year 2003 )\ note originally published in 1964 by McGraw-Hill NoStop

    work page 2003

  24. [24]

    Kohn ,\ title title Density functional and density matrix method scaling linearly with the number of atoms , \ https://doi.org/10.1103/PhysRevLett.76.3168 journal journal Phys

    author author W. Kohn ,\ title title Density functional and density matrix method scaling linearly with the number of atoms , \ https://doi.org/10.1103/PhysRevLett.76.3168 journal journal Phys. Rev. Lett. \ volume 76 ,\ pages 3168--3171 ( year 1996 ) NoStop

    work page doi:10.1103/physrevlett.76.3168 1996

  25. [25]

    Prodan \ and\ author W

    author author E. Prodan \ and\ author W. Kohn ,\ title title Nearsightedness of electronic matter , \ https://doi.org/10.1073/pnas.0505436102 journal journal Proceedings of the National Academy of Sciences \ volume 102 ,\ pages 11635--11638 ( year 2005 ) ,\ https://arxiv.org/abs/https://www.pnas.org/content/102/33/11635.full.pdf https://www.pnas.org/conte...

    work page doi:10.1073/pnas.0505436102 2005

  26. [26]

    author author C. J. \ Fennell \ and\ author J. D. \ Gezelter ,\ title title Is the ewald summation still necessary? pairwise alternatives to the accepted standard for long-range electrostatics , \ @noop journal journal Journal of Chemical Physics \ volume 124 ,\ pages 234104 ( year 2006 ) NoStop

    work page 2006

  27. [27]

    Born \ and\ author K

    author author M. Born \ and\ author K. Huang ,\ title title Dynamical theory of crystal lattices , \ @noop journal journal Oxford University Press \ ( year 1954 ) NoStop

    work page 1954

  28. [28]

    author author J. D. \ Jackson ,\ @noop title Classical Electrodynamics Third Edition ,\ edition 3rd \ ed.\ ( publisher Wiley ,\ address New York ,\ year 1998 ) NoStop

    work page 1998

  29. [29]

    author author S. Grimme ,\ title title Semiempirical gga-type density functional constructed with a long-range dispersion correction , \ @noop journal journal Journal of Computational Chemistry \ volume 27 ,\ pages 1787--1799 ( year 2006 ) NoStop

    work page 2006

  30. [30]

    Tkatchenko \ and\ author M

    author author A. Tkatchenko \ and\ author M. Scheffler ,\ title title Accurate molecular van der waals interactions from ground-state electron density and free-atom reference data , \ @noop journal journal Physical Review Letters \ volume 102 ,\ pages 073005 ( year 2009 ) NoStop

    work page 2009

  31. [31]

    author author R. A. \ DiStasio , author V. V. \ Gobre ,\ and\ author A. Tkatchenko ,\ title title Many-body van der waals interactions in molecules and condensed matter , \ @noop journal journal Journal of Physics: Condensed Matter \ volume 26 ,\ pages 213202 ( year 2014 ) NoStop

    work page 2014

  32. [32]

    author author J. M. \ Borwein , author M. L. \ Glasser , author R. C. \ McPhedran , author J. G. \ Wan ,\ and\ author I. J. \ Zucker ,\ @noop title Lattice Sums Then and Now ,\ Encyclopedia of Mathematics and its Applications\ ( publisher Cambridge University Press ,\ year 2013 ) NoStop

    work page 2013

  33. [33]

    author author P. P. \ Ewald ,\ title title Die Berechnung optischer und elektrostatischer Gitterpotentiale , \ @noop journal journal Annalen der Physik \ volume 369 ,\ pages 253–287 ( year 1921 ) NoStop

    work page 1921

  34. [34]

    author author M. P. \ Allen \ and\ author D. J. \ Tildesley ,\ https://doi.org/10.1093/oso/9780198803195.001.0001 title Computer Simulation of Liquids \ ( publisher Oxford University Press ,\ year 2017 ) NoStop

    work page doi:10.1093/oso/9780198803195.001.0001 2017

  35. [35]

    author author B. R. A. \ Nijboer \ and\ author F. W. \ De Wette ,\ title English title On the calculation of lattice sums , \ https://doi.org/10.1016/S0031-8914(57)92124-9 journal journal Physica \ volume 23 ,\ pages 309--321 ( year 1957 ) NoStop

    work page doi:10.1016/s0031-8914(57)92124-9 1957

  36. [36]

    author author D. E. \ Williams ,\ title title Accelerated convergence of crystal-lattice potential sums , \ https://doi.org/10.1107/S0567739471000998 journal journal Acta Crystallographica Section A \ volume 27 ,\ pages 452--455 ( year 1971 ) NoStop

    work page doi:10.1107/s0567739471000998 1971

  37. [37]

    author author D. E. \ Williams ,\ title title Accelerated Convergence Treatment of R -n Lattice Sums , \ https://doi.org/10.1080/08893118908032944 journal journal Crystallography Reviews \ volume 2 ,\ pages 3--23 ( year 1989 ) ,\ note publisher: Taylor & Francis \_eprint: https://doi.org/10.1080/08893118908032944 NoStop

    work page doi:10.1080/08893118908032944 1989

  38. [38]

    Karasawa \ and\ author W

    author author N. Karasawa \ and\ author W. A. \ Goddard III ,\ title title Acceleration of convergence for lattice sums , \ @noop journal journal The Journal of Physical Chemistry \ volume 93 ,\ pages 7320--7327 ( year 1989 ) NoStop

    work page 1989

  39. [39]

    author author R. W. \ Hockney \ and\ author J. W. \ Eastwood ,\ title title Computer simulation using particles , \ @noop journal journal CRC Press \ ( year 1981 ) NoStop

    work page 1981

  40. [40]

    Deserno \ and\ author C

    author author M. Deserno \ and\ author C. Holm ,\ title title How to mesh up Ewald sums. I . A theoretical and numerical comparison of various particle mesh routines , \ https://doi.org/10.1063/1.477414 journal journal J. Chem. Phys. \ volume 109 ,\ pages 7678--7693 ( year 1998 ) NoStop

    work page doi:10.1063/1.477414 1998

  41. [41]

    Darden , author D

    author author T. Darden , author D. York ,\ and\ author L. Pedersen ,\ title title Particle mesh ewald: An n·log(n) method for ewald sums in large systems , \ @noop journal journal Journal of Chemical Physics \ volume 98 ,\ pages 10089--10092 ( year 1993 ) NoStop

    work page 1993

  42. [42]

    Essmann , author L

    author author U. Essmann , author L. Perera , author M. L. \ Berkowitz , author T. Darden , author H. Lee ,\ and\ author L. G. \ Pedersen ,\ title title A smooth particle mesh ewald method , \ @noop journal journal The Journal of chemical physics \ volume 103 ,\ pages 8577--8593 ( year 1995 ) NoStop

    work page 1995

  43. [43]

    author author J. R. \ Winkler ,\ title title Numerical recipes in C : The art of scientific computing, second edition , \ https://doi.org/10.1016/0160-9327(93)90069-F journal journal Endeavour \ volume 17 ,\ pages 201 ( year 1993 ) NoStop

    work page doi:10.1016/0160-9327(93)90069-f 1993

  44. [44]

    Kawata \ and\ author U

    author author M. Kawata \ and\ author U. Nagashima ,\ title title Particle mesh ewald method for three-dimensional systems with two-dimensional periodicity , \ @noop journal journal Chemical Physics Letters \ volume 340 ,\ pages 165--172 ( year 2001 ) NoStop

    work page 2001

  45. [45]

    Gautschi ,\ @noop title Numerical analysis \ ( publisher Springer Science & Business Media ,\ year 2011 ) NoStop

    author author W. Gautschi ,\ @noop title Numerical analysis \ ( publisher Springer Science & Business Media ,\ year 2011 ) NoStop

    work page 2011

  46. [46]

    Greengard \ and\ author V

    author author L. Greengard \ and\ author V. Rokhlin ,\ title title A fast algorithm for particle simulations , \ @noop journal journal Journal of Computational Physics \ volume 73 ,\ pages 325--348 ( year 1987 ) NoStop

    work page 1987

  47. [47]

    Gibbon \ and\ author G

    author author P. Gibbon \ and\ author G. Sutmann ,\ title title Long-range interactions in many-particle simulation , \ @noop journal journal Quantum Simulations of Complex Many-Body Systems: From Theory to Algorithms, NIC Series \ ,\ pages 467--506 ( year 2004 ) NoStop

    work page 2004

  48. [48]

    Deng , author C

    author author Z. Deng , author C. Chen , author X.-G. \ Li ,\ and\ author S. P. \ Ong ,\ title title An electrostatic spectral neighbor analysis potential for lithium nitride , \ https://doi.org/10.1038/s41524-019-0212-1 journal journal npj Computational Materials \ volume 5 ,\ pages 1--8 ( year 2019 ) NoStop

    work page doi:10.1038/s41524-019-0212-1 2019

  49. [49]

    author author V. L. \ Deringer , author M. A. \ Caro ,\ and\ author G. Cs \'a nyi ,\ title title A general-purpose machine-learning force field for bulk and nanostructured phosphorus , \ https://doi.org/10.1038/s41467-020-19168-z journal journal Nat Commun \ volume 11 ,\ pages 5461 ( year 2020 ) NoStop

    work page doi:10.1038/s41467-020-19168-z 2020

  50. [50]

    u tt , author H. E. \ Sauceda ,\ and\ author K.-R. \ M \

    author author O. T. \ Unke , author S. Chmiela , author M. Gastegger , author K. T. \ Sch \"u tt , author H. E. \ Sauceda ,\ and\ author K.-R. \ M \"u ller ,\ title title Spookynet: Learning force fields with electronic degrees of freedom and nonlocal effects , \ @noop journal journal Nature communications \ volume 12 ,\ pages 7273 ( year 2021 b ) NoStop

    work page 2021

  51. [51]

    author author S. P. \ Niblett , author M. Galib ,\ and\ author D. T. \ Limmer ,\ title title Learning intermolecular forces at liquid vapor interfaces , \ https://doi.org/10.1063/5.0067565 journal journal J. Chem. Phys. \ volume 155 ,\ pages 164101 ( year 2021 ) NoStop

    work page doi:10.1063/5.0067565 2021

  52. [52]

    Kabylda , author J

    author author A. Kabylda , author J. T. \ Frank , author S. S. \ Dou , author A. Khabibrakhmanov , author L. M. \ Sandonas , author O. T. \ Unke , author S. Chmiela , author K.-R. \ Muller ,\ and\ author A. Tkatchenko ,\ title title Molecular simulations with a pretrained neural network and universal pairwise force fields , \ https://doi.org/10.26434/chem...

    work page doi:10.26434/chemrxiv-2024-bdfr0 2024

  53. [53]

    author author O. T. \ Unke \ and\ author M. Meuwly ,\ title title Physnet: A neural network for predicting energies, forces, dipole moments, and partial charges , \ https://doi.org/10.1021/acs.jctc.9b00181 journal journal Journal of Chemical Theory and Computation \ volume 15 ,\ pages 3678--3693 ( year 2019 ) ,\ note pMID: 31042390 ,\ https://arxiv.org/ab...

    work page doi:10.1021/acs.jctc.9b00181 2019

  54. [54]

    author author A. E. \ Sifain , author N. Lubbers , author B. T. \ Nebgen , author J. S. \ Smith , author A. Y. \ Lokhov , author O. Isayev , author A. E. \ Roitberg , author K. Barros ,\ and\ author S. Tretiak ,\ title title Discovering a transferable charge assignment model using machine learning , \ https://doi.org/10.1021/acs.jpclett.8b01939 journal jo...

    work page doi:10.1021/acs.jpclett.8b01939 2018

  55. [55]

    Gao \ and\ author R

    author author A. Gao \ and\ author R. C. \ Remsing ,\ title title Self-consistent determination of long-range electrostatics in neural network potentials , \ https://doi.org/10.1038/s41467-022-29243-2 journal journal Nature Communications \ volume 13 ,\ pages 1572 ( year 2022 ) NoStop

    work page doi:10.1038/s41467-022-29243-2 2022

  56. [56]

    Peng , author L

    author author Y. Peng , author L. Lin , author L. Ying ,\ and\ author L. Zepeda-N \'u \ n ez ,\ title title Efficient long-range convolutions for point clouds , \ https://doi.org/10.1016/j.jcp.2022.111692 journal journal Journal of Computational Physics \ volume 473 ,\ pages 111692 ( year 2023 ) NoStop

    work page doi:10.1016/j.jcp.2022.111692 2022

  57. [57]

    Zhang , author H

    author author L. Zhang , author H. Wang , author M. C. \ Muniz , author A. Z. \ Panagiotopoulos , author R. Car ,\ and\ author W. E ,\ title title A deep potential model with long-range electrostatic interactions , \ https://doi.org/10.1063/5.0083669 journal journal J. Chem. Phys. \ volume 156 ,\ pages 124107 ( year 2022 ) NoStop

    work page doi:10.1063/5.0083669 2022

  58. [58]

    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 ,\ title title High accuracy and transferability of a neural network potential through charge equilibration for calcium fluoride , \ https://doi.org/10.1103/PhysRevB.95.104105 journal journal Phys...

    work page doi:10.1103/physrevb.95.104105 2017

  59. [59]

    author author T. W. \ Ko , author J. A. \ Finkler , author S. Goedecker ,\ and\ author J. Behler ,\ title title A fourth-generation high-dimensional neural network potential with accurate electrostatics including non-local charge transfer , \ https://doi.org/10.1038/s41467-020-20427-2 journal journal Nat Commun \ volume 12 ,\ pages 398 ( year 2021 a ) NoStop

    work page doi:10.1038/s41467-020-20427-2 2021

  60. [60]

    author author T. W. \ Ko , author J. A. \ Finkler , author S. Goedecker ,\ and\ author J. Behler ,\ title title General- Purpose Machine Learning Potentials Capturing Nonlocal Charge Transfer , \ https://doi.org/10.1021/acs.accounts.0c00689 journal journal Acc. Chem. Res. \ volume 54 ,\ pages 808--817 ( year 2021 b ) NoStop

    work page doi:10.1021/acs.accounts.0c00689 2021

  61. [61]

    Gubler , author J

    author author M. Gubler , author J. A. \ Finkler , author M. R. \ Sch \"a fer , author J. Behler ,\ and\ author S. Goedecker ,\ title title Accelerating Fourth-Generation Machine Learning Potentials Using Quasi-Linear Scaling Particle Mesh Charge Equilibration , \ https://doi.org/10.1021/acs.jctc.4c00334 journal journal J. Chem. Theory Comput. \ ( year 20...

    work page doi:10.1021/acs.jctc.4c00334 2024

  62. [62]

    Grisafi \ and\ author M

    author author A. Grisafi \ and\ author M. Ceriotti ,\ title title Incorporating long-range physics in atomic-scale machine learning , \ https://doi.org/10.1063/1.5128375 journal journal J. Chem. Phys. \ volume 151 ,\ pages 204105 ( year 2019 ) NoStop

    work page doi:10.1063/1.5128375 2019

  63. [63]

    Grisafi , author J

    author author A. Grisafi , author J. Nigam ,\ and\ author M. Ceriotti ,\ title title Multi-scale approach for the prediction of atomic scale properties , \ https://doi.org/10.1039/D0SC04934D journal journal Chem. Sci. \ volume 12 ,\ pages 2078--2090 ( year 2021 ) NoStop

    work page doi:10.1039/d0sc04934d 2078

  64. [64]

    author author K. K. \ Huguenin-Dumittan , author P. Loche , author N. Haoran ,\ and\ author M. Ceriotti ,\ title title Physics- Inspired Equivariant Descriptors of Nonbonded Interactions , \ https://doi.org/10.1021/acs.jpclett.3c02375 journal journal J. Phys. Chem. Lett. \ ,\ pages 9612--9618 ( year 2023 ) NoStop

    work page doi:10.1021/acs.jpclett.3c02375 2023

  65. [65]

    Chong , author F

    author author S. Chong , author F. Bigi , author F. Grasselli , author P. Loche , author M. Kellner ,\ and\ author M. Ceriotti ,\ title title Prediction rigidities for data-driven chemistry , \ https://doi.org/10.1039/D4FD00101J journal journal Faraday Discussions \ ( year 2024 ),\ 10.1039/D4FD00101J NoStop

    work page doi:10.1039/d4fd00101j 2024

  66. [66]

    author author B. Cheng ,\ https://doi.org/10.48550/arXiv.2408.15165 title Latent Ewald summation for machine learning of long-range interactions , \ ( year 2024 ),\ https://arxiv.org/abs/2408.15165 arXiv:2408.15165 NoStop

    work page doi:10.48550/arxiv.2408.15165 2024

  67. [67]

    Faller , author M

    author author C. Faller , author M. Kaltak ,\ and\ author G. Kresse ,\ https://doi.org/10.48550/arXiv.2406.17595 title Density- Based Long-Range Electrostatic Descriptors for Machine Learning Force Fields , \ ( year 2024 ),\ https://arxiv.org/abs/2406.17595 arXiv:2406.17595 NoStop

    work page doi:10.48550/arxiv.2406.17595 2024

  68. [68]

    author author G. Hummer ,\ title title The numerical accuracy of truncated ewald sums for periodic systems with long-range coulomb interactions , \ https://doi.org/https://doi.org/10.1016/0009-2614(95)00117-M journal journal Chemical Physics Letters \ volume 235 ,\ pages 297--302 ( year 1995 ) NoStop

    work page doi:10.1016/0009-2614(95)00117-m 1995

  69. [69]

    author author H. G. \ Petersen ,\ title title Accuracy and efficiency of the particle mesh Ewald method , \ https://doi.org/10.1063/1.470043 journal journal The Journal of Chemical Physics \ volume 103 ,\ pages 3668--3679 ( year 1995 ) NoStop

    work page doi:10.1063/1.470043 1995

  70. [70]

    Deserno \ and\ author C

    author author M. Deserno \ and\ author C. Holm ,\ title title How to mesh up Ewald sums. II . An accurate error estimate for the particle–particle–particle-mesh algorithm , \ https://doi.org/10.1063/1.477415 \ volume 109 ,\ pages 7694--7701 NoStop

    work page doi:10.1063/1.477415

  71. [71]

    author author M. E. \ Tuckerman , author B. J. \ Berne ,\ and\ author A. Rossi ,\ title title Molecular dynamics algorithm for multiple time scales: Systems with disparate masses , \ https://doi.org/10.1063/1.460004 journal journal The Journal of Chemical Physics \ volume 94 ,\ pages 1465--1469 ( year 1991 ) NoStop

    work page doi:10.1063/1.460004 1991

  72. [72]

    Kapil , author J

    author author V. Kapil , author J. VandeVondele ,\ and\ author M. Ceriotti ,\ title title Accurate molecular dynamics and nuclear quantum effects at low cost by multiple steps in real and imaginary time: Using density functional theory to accelerate wavefunction methods , \ https://doi.org/10.1063/1.4941091 journal journal The Journal of Chemical Physics ...

    work page doi:10.1063/1.4941091 2016

  73. [73]

    author author J. E. \ House ,\ title title Chapter 7 - Ionic bonding and structures of solids , \ in\ https://doi.org/10.1016/B978-0-12-814369-8.00007-8 booktitle Inorganic Chemistry ( Third Edition ) ,\ editor edited by\ editor J. E. \ House \ ( publisher Academic Press ,\ year 2020 )\ pp.\ pages 229--274 NoStop

    work page doi:10.1016/b978-0-12-814369-8.00007-8 2020

  74. [74]

    Glasser ,\ title title Solid- State Energetics and Electrostatics : Madelung Constants and Madelung Energies , \ https://doi.org/10.1021/ic2023852 journal journal Inorg

    author author L. Glasser ,\ title title Solid- State Energetics and Electrostatics : Madelung Constants and Madelung Energies , \ https://doi.org/10.1021/ic2023852 journal journal Inorg. Chem. \ volume 51 ,\ pages 2420--2424 ( year 2012 ) NoStop

    work page doi:10.1021/ic2023852 2012

  75. [75]

    author author H. J. C. \ Berendsen , author J. R. \ Grigera ,\ and\ author T. P. \ Straatsma ,\ title title The missing term in effective pair potentials , \ https://doi.org/10.1021/j100308a038 journal journal J. Phys. Chem. \ volume 91 ,\ pages 6269--6271 ( year 1987 ) NoStop

    work page doi:10.1021/j100308a038 1987

  76. [76]

    Fraux , author P

    author author G. Fraux , author P. Loche , author F. Bigi , author J. W. \ Abbott , author D. Tisi , author A. Goscinski ,\ and\ author M. Ceriotti ,\ https://github.com/metatensor/metatensor title Github repository: Metatensor , \ ( year 2024 a ) NoStop

    work page 2024

  77. [77]

    author author A. P. \ Thompson , author H. M. \ Aktulga , author R. Berger , author D. S. \ Bolintineanu , author W. M. \ Brown , author P. S. \ Crozier , author P. J. \ in 't Veld , author A. Kohlmeyer , author S. G. \ Moore , author T. D. \ Nguyen , author R. Shan , author M. J. \ Stevens , author J. Tranchida , author C. Trott ,\ and\ author S. J. \ Pl...

    work page doi:10.1016/j.cpc.2021.108171 2021

  78. [78]

    Bussi , author D

    author author G. Bussi , author D. Donadio ,\ and\ author M. Parrinello ,\ title title Canonical sampling through velocity rescaling , \ https://doi.org/10.1063/1.2408420 journal journal The Journal of Chemical Physics \ volume 126 ,\ pages 014101 ( year 2007 ) NoStop

    work page doi:10.1063/1.2408420 2007

  79. [79]

    Vega \ and\ author J

    author author C. Vega \ and\ author J. L. F. \ Abascal ,\ title title Simulating water with rigid non-polarizable models: A general perspective , \ https://doi.org/10.1039/C1CP22168J journal journal Phys. Chem. Chem. Phys. \ volume 13 ,\ pages 19663--19688 ( year 2011 ) NoStop

    work page doi:10.1039/c1cp22168j 2011

  80. [80]

    author author J. Behler ,\ title title Four Generations of High-Dimensional Neural Network Potentials , \ https://doi.org/10.1021/acs.chemrev.0c00868 journal journal Chemical Reviews \ volume 121 ,\ pages 10037--10072 ( year 2021 ) NoStop

    work page doi:10.1021/acs.chemrev.0c00868 2021

Showing first 80 references.