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Constant-Potential Machine Learning Molecular Dynamics Simulations Reveal Potential-Regulated Cu Cluster Formation on MoS$_{2}$

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read By feeding the applied electric potential into a machine-learned force field as an explicit input, the paper makes constant-potential molecular dynamics practical and predicts that adsorbed Cu atoms on 1T'-MoS2 aggregate into small steric…

desk verdict A promising but under-validated method: adding U as an NN input is new, but the -0.1 V clustering result relies on extrapolated potential labels and single trajectories. read the letter →

arxiv 2411.14732 v1 pith:QYA6HA4G submitted 2024-11-22 physics.chem-ph cond-mat.mtrl-sci

Jingwen Zhou , Yunsong Fu , Ling Liu , Chungen Liu This is my paper · ORCID
classification physics.chem-phcond-mat.mtrl-sci
keywords electricpotentialinputmachinelearningforcefieldconstant-potentialmoleculardynamicsCuclusterformationMoS2electrodesingle-clustercatalystsatomicneuralnetworkelectrochemicalinterfaces
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 tries to establish that the applied electric potential can be treated as an ordinary input feature of a machine-learned force field, so that one trained network supplies energies and forces at any potential and molecular dynamics can be run at constant potential for nanoseconds. It applies this explicit-electric-potential model to Cu atoms adsorbed on 1T'-MoS2, and the constant-potential trajectories show that below roughly -0.1 V vs the standard hydrogen electrode the Cu atoms migrate and assemble into small steric clusters, while at +0.1 V and above they remain mostly dispersed as single atoms. If the model is right, the expensive constant-potential calibration step of ab initio electrochemistry can be replaced by a single interpolation, and electrode potential becomes a practical dial for preparing single-cluster catalysts.

What carries the argument

The central object is the explicit-electric-potential atomic neural network, which appends the global scalar $U$ to the smooth SO(3) power-spectrum descriptors of every atom. The argument rests on the established near-quadratic dependence of the system energy on $U$ over a broad range, which lets one network interpolate between training structures generated at different fixed charges in an implicit electrolyte. The network's output is a sum of atomic energies, so the force on each atom is obtained by differentiating through the descriptors; this is what converts potential-dependent energetics into constant-potential trajectories.

What would settle it

Run constant-potential DFT molecular dynamics for the same Cu/MoS2 system at +0.1 V and -0.1 V for a few tens of picoseconds and compare the Cu-Cu radial distribution and the cluster-size populations with the EEP-MLFF trajectories: a 2.4 Å Cu-Cu peak at -0.1 V in the ML run but not in the DFT run, or vice versa at +0.1 V, would show the potential label or the interpolation is wrong.

Watch

Extended reading notes

Core claim

The central claim is that the total energy of an electrochemical interface can be written as $E_{\rm tot} = \sum_i E_i(D_i(R_1,\dots,R_N), U)$, where $D_i$ are atom-centered structural descriptors and $U$ is the applied electric potential shared by all atoms. Because the descriptors depend smoothly on nuclear coordinates, the potential-dependent forces are exact derivatives of this energy, so the trained network can drive molecular dynamics at fixed $U$ with no additional self-consistency loop. Running this constant-potential MLMD on a 96-Cu/1T'-MoS2 slab, the paper finds a potential-driven crossover: at negative potentials the Cu-S bonds weaken and Cu-Cu bonding in close-contact configurations strengthens, so Cu atoms leave their adsorption sites and form tilted dimers, trimers, and small steric clusters below -0.1 V, whereas at positive potentials they stay as single atoms or flat short-range arrays.

Load-bearing premise

The voltage value assigned to each training snapshot really represents the electrode potential, and it remains meaningful when the trained model is used for the much larger copper-on-MoS2 slab.

Editorial extensions

If this is right

  • Constant-potential molecular dynamics becomes affordable for nanosecond-scale, multi-hundred-atom electrochemical interfaces, because adding $U$ as an input costs almost nothing at inference time.
  • The Cu/1T'-MoS2 system is predicted to have a voltage threshold near -0.1 V vs SHE, below which single Cu atoms convert into small steric clusters rather than larger aggregates.
  • Because cluster size and orientation respond to potential, electrode bias can be used as a synthesis handle for single-cluster catalysts, with more negative potentials favoring Cu2 and Cu3 units.
  • The electronic-structure analysis implies the aggregation is driven by potential-tuned competition between Cu-S and Cu-Cu bonding, not simply by faster diffusion.

Reading between the lines

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

  • The same architecture should transfer to other global electrochemical control variables, such as pH or applied strain, provided the energy depends smoothly on them; the paper does not test this.
  • The -0.1 V threshold and the dominance of Cu2/Cu3 over larger clusters are likely sensitive to Cu coverage, surface defects, and simulation time; the paper uses one slab at roughly 22% coverage.
  • A denser potential grid in the training set could reveal whether the quadratic energy assumption degrades outside -0.5 V to +0.6 V, and whether a single network trained on multiple coverages can reproduce coverage-dependent cluster statistics.
  • The predicted crossover between thermodynamic control at positive potentials and kinetic control at negative potentials could be tested by seeding pre-formed larger clusters at negative potentials and checking whether they shrink to Cu2/Cu3 over time.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes EEP-MLFF, an atomic neural-network force field that takes the electrode potential U as an explicit input in addition to atom-centered structural descriptors. The model is trained on DFT data for Cu/1T'-MoS2 systems under implicit solvation (VASPsol), with charge introduced to mimic varying potentials. The authors report MLFF energy/force RMSEs of 8 meV/atom and 0.08 eV/A, a VDOS comparison against DFT, and a CP-MLMD trajectory at +0.685 V that preserves dispersed Cu atoms. The central application is constant-potential molecular dynamics on a 96-Cu slab at several potentials, which shows Cu aggregation into small steric clusters at potentials below -0.1 V vs SHE. The mechanism is interpreted through pCOHP analyses showing potential-dependent Cu-Cu and Cu-S bonding.

Significance. If the central claim is correct, the paper offers a computationally efficient route to nanosecond-scale constant-potential molecular dynamics for electrochemical interfaces, and it makes a falsifiable prediction of a voltage threshold for Cu cluster formation on 1T'-MoS2. The reported accuracy metrics, the VDOS validation, the use of an independent DFT-based pCOHP analysis to rationalize the aggregation, and the explicit potential-dependent architecture are all strengths that make the work potentially useful to the electrocatalysis and ML force-field communities. However, the significance is conditional: the constant-potential interpretation hinges on whether the U labels derived from fixed-charge calculations are transferable to the large, negatively biased, high-coverage systems used in the production simulations.

major comments (4)
  1. [Validation of the Machine-Learning Force Field; Eq. (1)] The central claim that Eq. (1) represents a constant-potential potential-energy surface is not established by the training set. The text states that 'different amounts of charge are introduced into the systems to simulate varying electric potential conditions,' and that the data comprise only neutral, +0.5e, and -0.5e fixed-charge structures. The potential U is evidently assigned after the fact from the VASPsol potential of each fixed-charge calculation, rather than imposed as an independent control. No same-geometry charge series and no constant-potential training trajectories are provided. Consequently, the U-dependence of the neural network is learned by regressing across configurations where U is correlated with coverage, charge, and local geometry, and the production runs at -0.1 V and -0.3 V on the 96-Cu slab are extrapolations outside the sampled (charge, coverage, U) manifold. I ask the authors to provide explicit evidence that the learned U-dependence is transferable: for example, compare EEP-MLFF energies and forces against constant-potential DFT/AIMD for representative configurations at negative potentials, or train with the total charge as an additional input and demonstrate equivalence to a grand-canonical description.
  2. [Evolution From Single Atoms to Single Clusters] The -0.1 V threshold is the main quantitative prediction, but it rests on single CP-MLMD trajectories per potential without uncertainty estimates. Figure 3 reports deterministic time evolutions of N_SA and N_SC from one trajectory at each potential, and the discussion of the 'turning point potential between +0.1 V and -0.1 V' is based on those singles runs. I request multiple independent trajectories (different initial velocities or seeds) per potential, with error bars on the cluster populations, to establish that the aggregation transition is statistically robust rather than a fluctuation of one trajectory.
  3. [Evolution From Single Atoms to Single Clusters; Figure S6] The cluster definition depends on the hand-chosen Cu-Cu distance threshold of 3.2 A. The text and Figure S6 justify this threshold by showing that 2.4 A fails to identify flat-oriented Cu arrays, but they do not test the sensitivity of the central -0.1 V threshold to the cluster criterion. For example, a threshold of 3.0 A or 3.5 A could shift the potential at which SC-Cu species are counted, or change the relative populations of Cu2 and Cu3. Please report the cluster populations as a function of the distance threshold, or provide a physical justification that the reported threshold is the uniquely appropriate one for the 'single cluster' definition.
  4. [Methodology; Eq. (1) and constant-potential implementation] The sentence stating that 'the constant-potential treatment is established on the presumption that system charge fluctuations can be fully suppressed within each time step of MD simulation' is unclear and potentially misleading. In constant-potential AIMD the electron number is an additional dynamical variable that fluctuates to keep the potential fixed, not a quantity that is suppressed. If the EEP-MLFF instead fixes U and computes forces from a U-dependent PES, the authors should clarify how this relates to the grand-canonical Legendre transform and how the lack of explicit charge fluctuation is justified for the Cu/MoS2 system. This clarification is needed to support the use of the word 'constant-potential' throughout the manuscript.
minor comments (4)
  1. [Methodology] There is a typo in the second paragraph of the Methodology section: 'large-scale simulations of electrochemical systhave recently seen' should read 'systems have recently seen.'
  2. [Acknowledgement] The Acknowledgement section contains 'NSF (Grand No. 22073041)'; 'Grand' should be 'Grant.'
  3. [Figure 2 caption] The phrase 'the signatures of the vibrational spectrum of Cu4/MoS2 evaluated with the constant-potential molecular dynamics simulation' is grammatically awkward; consider 'vibrational density of states' or 'vibrational signatures' for clarity.
  4. [Supporting Information] Several critical technical details are deferred to the SI, including the mapping from fixed charge to U, the neural-network hyperparameters, and the definition of the potential window. Since the main text refers to these details to justify the constant-potential interpretation, the SI should be made available with the manuscript and its contents explicitly cross-referenced in the main text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the -0.1 V clustering threshold is an emergent MLMD outcome, and the self-citations are background only.

full rationale

The derivation chain is not circular. The EEP-MLFF model (Eq. 1) expresses the total energy as a sum of atomic neural-network contributions depending on structural descriptors and a global potential U; forces are obtained by differentiating that expression (Eq. 2). This is a model definition, not a claim that the model's predictions are equivalent to its training labels. The training set is built from fixed-charge VASPsol AIMD structures at neutral, +0.5 e, and -0.5 e conditions, with U assigned from each calculation; this is an approximation about how to label potentials, but it is not a fitted parameter that is then renamed as a prediction. The central result—the -0.1 V threshold for Cu clustering—is an emergent statistical outcome of constant-potential MLMD trajectories on a 96-Cu slab (Figure 3), not a quantity fit to the training data. The validations (energy/force RMSE, VDOS against DFT, and the +0.685 V dispersed-trajectory comparison against EXAFS) are independent checks. The pCOHP analysis is a separate DFT calculation supporting the proposed mechanism. The paper does cite prior work by the same authors (Refs. 12, 13, 30), but only as background for the quadratic energy–potential relationship and for potential-dependent cluster expansions; these citations are not load-bearing for the main claim and do not reduce the derivation to a self-citation chain. Whether fixed-charge U labels transfer to a true grand-canonical constant-potential ensemble at negative bias and 96-Cu coverage is a validity and extrapolation question, not circularity.

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

No new physical particles, forces, or conserved quantities are introduced. EEP-MLFF is a computational model, and SC-Cu is a structural classification, not an invented physical entity.

free parameters (3)
  • EEP-MLFF neural network weights and biases = not reported (optimized; test RMSE 8 meV/atom energy, 0.08 eV/A force)
    The entire potential-dependent energy surface is a fit to DFT data; the network parameters are the fitted quantities that determine all MD results.
  • Cu-Cu cluster distance threshold = 3.2 Å
    Chosen by hand to define SC-Cu clusters; the paper reports that a stricter 2.4 Å threshold fails to identify flat-oriented arrays (Figure S6), so the reported cluster populations depend on this choice.
  • Electric potential labels U of training structures = range -0.5 V to +0.6 V vs SHE
    Assigned to fixed-charge VASPsol structures using a procedure described only in the SI; these labels are the input variable on which the entire potential dependence rests.
assumptions (4)
  • domain assumption The energy of an electrochemical system varies quadratically with electric potential over the training range.
    Invoked in the Introduction (refs 11-18) to justify adding U as an input; if the true dependence is not quadratic in the relevant regimes, interpolation by the neural network may be inaccurate.
  • domain assumption System charge fluctuations are fully suppressed within each MD time step, so a fixed-U MLMD trajectory represents a constant-potential ensemble.
    Stated explicitly in the Methodology section, citing ref 33; the model does not evolve electron number, so this approximation is load-bearing.
  • domain assumption DFT with the VASPsol implicit solvation model at fixed charge provides reliable reference energies and forces for Cu/MoS2.
    All training labels come from these calculations; the accuracy of the MLFF cannot exceed the accuracy of this reference.
  • domain assumption Smooth SO(3) power spectrum descriptors are complete enough to distinguish all relevant atomic environments.
    Chosen in the Methodology section; if the descriptors are not complete, the learned forces are not well-defined.

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Cite this review

Pith. "Pith review of Constant-Potential Machine Learning Molecular Dynamics Simulations Reveal Potential-Regulated Cu Cluster Formation on MoS$_{2}$." pith.science (2026). https://pith.science/paper/QYA6HA4G

@misc{pith2026241114732,
  author       = {Pith},
  title        = {Pith review of: Constant-Potential Machine Learning Molecular Dynamics Simulations Reveal Potential-Regulated Cu Cluster Formation on MoS$_2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QYA6HA4G}},
  note         = {Machine review of arXiv:2411.14732}
}
abstract

Electrochemical processes play a crucial role in energy storage and conversion systems, yet their computational modeling remains a significant challenge. Accurately incorporating the effects of electric potential has been a central focus in theoretical electrochemistry. Although constant-potential ab initio molecular dynamics (CP-AIMD) has provided valuable insights, it is limited by its substantial computational demands. Here, we introduce the Explicit Electric Potential Machine Learning Force Field (EEP-MLFF) model. Our model integrates the electric potential as an explicit input parameter along with the atom-centered descriptors in the atomic neural network. This approach enables the evaluation of nuclear forces under arbitrary electric potentials, thus facilitating molecular dynamics simulations at a specific potential. By applying the proposed machine learning method to the Cu/1T$^{\prime}$-MoS$_{2}$ system, molecular dynamics simulations reveal that the potential-modulated Cu atom migration and aggregation lead to the formation of small steric Cu clusters (Single Clusters, SCs) at potentials below -0.1 V. The morphological transformations of adsorbed Cu atoms are elucidated through electronic structure analyses, which demonstrates that both Cu-S and Cu-Cu bonding can be effectively tuned by the applied electric potential. Our findings present an opportunity for the convenient manufacture of single metal cluster catalysts through potential modulation. Moreover, this theoretical framework facilitates the exploration of potential-regulated processes and helps investigate the mechanisms of electrochemical reactions.

Figures

Figures reproduced from arXiv: 2411.14732 by the authors.

Figure 1
Figure 1. The neural network architecture of the EEP-MLP model. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Validation of the EEP-MLFF model in reproducing the result [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Illustration depicting the morphological evolution of SA-Cu [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The influence of electric potential on the morphology of ad [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Characterization of the potential effect on the electron [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.