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arxiv: 2604.27910 · v1 · submitted 2026-04-30 · ❄️ cond-mat.mtrl-sci · physics.chem-ph· physics.comp-ph

Fragment-Constrained Charge Equilibration for Charge-Aware Machine Learning Potentials at Electrochemical Interfaces

Akhil Reddy Peeketi , Blas P Uberuaga , Travis E Jones This is my paper

Pith reviewed 2026-05-07 06:23 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci physics.chem-phphysics.comp-ph
keywords machine learning interatomic potentialscharge equilibrationelectrochemical interfacesfragment constraineddouble layer gradientreactive simulationselectrode electrolyte separation
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The pith

A differentiable fragment-constrained charge solver lets machine-learned potentials maintain separate electrode and electrolyte electrochemical potentials at interfaces.

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

The paper develops Soft-FQEq, a charge solver layer for machine learning interatomic potentials that enforces charge conservation separately within electrode and electrolyte fragments. Global charge equilibration forces a single electrochemical potential across all atoms, which erases the gradient needed to describe the electric double layer. By deriving soft fragment boundaries directly from atomic positions in a differentiable way, the method works for systems where bonds break and form. When the model is trained on density functional theory energies, forces, and charges for an iridium oxide water interface containing ions, it produces the expected potential gradient from electrode to electrolyte. Switching the same trained weights to a global solver at inference time flattens that gradient to nearly uniform values, showing that the fragment constraint is required to sustain the separation.

Core claim

The paper establishes that a machine learning potential equipped with fragment-constrained charge equilibration recovers a clear electrode-to-electrolyte gradient in the per-atom electrochemical potential. The identical model weights, when evaluated with global charge equilibration instead, produce an essentially uniform potential profile. This contrast demonstrates that global equilibration cannot sustain the interfacial charge separation required for realistic double-layer behavior, whereas the fragment formulation restores it even in the presence of reactive chemistry.

What carries the argument

Soft-FQEq is a differentiable solver that ingests four scalar outputs per atom from a shared neural network feature extractor—electronegativity, source charge, short-range energy, and a soft bond-connectivity measure—and returns equilibrated charges plus distinct chemical potentials for each geometry-identified fragment.

If this is right

  • Machine learning potentials can now simulate reactive bond rearrangements at electrochemical interfaces without allowing spurious charge transfer between electronically disconnected regions.
  • Per-fragment chemical potentials become directly available from the model, providing a built-in diagnostic for local electrochemical driving forces.
  • Training on combined density functional theory energies, forces, and atomic charges is now feasible for systems containing distinct electronic domains such as metal oxide electrodes in contact with aqueous electrolytes.
  • Charge-aware potentials can be applied to dynamic, non-predefined topologies, extending their range to processes where molecular fragments change identity over time.

Where Pith is reading between the lines

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

  • The geometry-only fragmentation rule may allow the same solver to be attached to other neural network architectures without requiring explicit molecular topology input.
  • If the recovered gradient remains stable across varied training sets, the method could reduce reliance on separate continuum electrostatic models for interface simulations.
  • Testing whether the soft connectivity continues to separate fragments correctly in organic electrolytes or during faradaic reactions would probe the assumption's generality beyond the presented oxide-water case.
  • The per-fragment potentials could serve as local order parameters for enhanced sampling of charge-transfer events in larger-scale reactive simulations.

Load-bearing premise

A soft bond-connectivity function computed solely from atomic geometry can group atoms into fragments that correctly reflect the electronic separation between electrode and electrolyte regions, even while bonds rearrange during reactions.

What would settle it

Evaluating the trained model on held-out interface configurations and finding that the average per-atom electrochemical potential shows no statistically significant difference between electrode atoms and adjacent electrolyte atoms would falsify the claim that the fragment-constrained solver recovers the required gradient.

Figures

Figures reproduced from arXiv: 2604.27910 by Akhil Reddy Peeketi, Blas P Uberuaga, Travis E Jones.

Figure 1
Figure 1. Figure 1: Snapshot of an IrO2(110)/aqueous NaClO4 interface, our prototype training system for Soft-FQEq. The interface partitions into three structurally distinct regions (electrode, inner and outer Helmholtz layers, and diffuse/bulk electrolyte) and, at molecular resolution, into fragments for per-fragment charge equilibration: the connected electrode slab (one fragment), each intact water molecule, and each indiv… view at source ↗
Figure 2
Figure 2. Figure 2: Architecture of fragment-constrained charge equilibration on a HIP-NN feature net view at source ↗
Figure 3
Figure 3. Figure 3: Per-atom µphys,i = χi + [Aq]i along the interface normal z for a charge-neutral IrO2/H2O/NaClO4 slab, referenced to the bulk-water mean (z ∈ [30, 60] Å). The two shaded regions separate the electrode (gray, z ≲ 11.5 Å) from the electrolyte (light blue, z ≳ 11.5 Å). Soft-FQEq (blue) shows a flat electrode plateau, a sharp interfacial drop, and a bulk-water plateau near zero; shaded band is the 5-seed ensemb… view at source ↗
Figure 4
Figure 4. Figure 4: Response of µphys(z) to electrode charging. Nine ion compositions from 4 Na+ to 4 ClO4 – per electrode face (placed in the Stern layer) span a range of model-predicted surface charge densities σ, with each composition averaged over 5 random seeds. Shaded regions and species inclusion match view at source ↗
read the original abstract

Predictive simulation of electrochemical interfaces requires atomistic models that capture reactive bond rearrangements, long-range electrostatics, and charge distributions reflecting the electronic distinctness of electrode and electrolyte. Existing charge-aware machine-learned interatomic potentials (MLIPs) built on global charge equilibration (QEq) settle electrode and electrolyte at a common electrochemical potential, leaving no room for the interfacial gradient that the double layer requires and admitting spurious charge transfer between electronically disconnected regions. Per-fragment charge equilibration is the established remedy in classical molecular dynamics, but reliance on predefined molecular topology has confined it to non-reactive systems. We lift this restriction by making fragment identification itself a differentiable function of atomic geometry, yielding soft fragment-constrained charge equilibration (Soft-FQEq) -- a solver layer that restores fragment-resolved charge conservation in reactive MLIPs. The layer consumes four scalar MLP readouts from a shared atomic-feature network -- per-atom electronegativity, source charge, short-range energy, and a soft bond connectivity -- and returns equilibrated charges together with per-fragment chemical potentials. We implement Soft-FQEq as an extension of the hippynn framework on a HIP-NN feature network and train it on DFT energies, forces, and DDEC6 charges for IrO2/H2O/Na+/ClO4- interfaces. The trained model recovers a clear electrode-to-electrolyte gradient in the per-atom electrochemical potential. With the same trained weights but the fragment-constrained solver replaced by global QEq at inference, this gradient collapses to an essentially uniform profile, directly showing that the gradient cannot be sustained within global QEq while the fragment formulation recovers it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces Soft-FQEq, a differentiable solver layer for fragment-constrained charge equilibration in charge-aware MLIPs. A shared HIP-NN feature network produces four scalar outputs per atom (electronegativity, source charge, short-range energy, and soft bond connectivity). These feed the Soft-FQEq solver to yield equilibrated charges and per-fragment chemical potentials while allowing reactive bond rearrangements. Trained on DFT energies, forces, and DDEC6 charges for IrO2/H2O/Na+/ClO4- interfaces, the model produces a clear electrode-to-electrolyte gradient in the per-atom electrochemical potential. Replacing the solver at inference with global QEq (identical MLP weights) causes the gradient to collapse to a uniform profile, which the authors interpret as direct evidence that global QEq cannot sustain the required interfacial potential difference.

Significance. If the learned soft connectivity reliably partitions the system into electronically distinct fragments, the work would be significant for electrochemical interface modeling. It removes the topology restriction that has limited per-fragment QEq to non-reactive classical MD, while retaining the ability to train end-to-end on DFT targets. The solver-swap experiment with fixed weights provides a clean isolation of the solver effect and directly addresses a known limitation of global-QEq MLIPs. The approach could enable ML-driven simulations that capture double-layer electrostatics without ad-hoc charge constraints.

major comments (3)
  1. [results section describing the per-atom electrochemical potential gradient] The central demonstration (abstract and results on the IrO2/H2O interface) shows recovery of the electrode-to-electrolyte gradient under Soft-FQEq and its collapse under global QEq. However, this contrast is only informative if the soft bond connectivity matrix defines fragments whose boundaries coincide with the physical electrode/electrolyte division. Because the connectivity is a learned, geometry-only function with no electronic-structure input and the training objective contains no explicit term enforcing electronic distinctness, nothing guarantees that the connected components respect the intended partition rather than forming mixed electrode–adsorbate clusters or splitting the electrode. Without visualizations of the connectivity matrix, statistics on cross-boundary bonds, or overlap metrics with known physical regions, the observed gradient could be an artifact of whatever ad-hoc
  2. [training and implementation details (hippynn extension)] The manuscript reports training on DFT energies, forces, and DDEC6 charges but provides no quantitative error metrics, training or validation curves, loss-component weights, or test-set performance for the IrO2/H2O/Na+/ClO4- system. In the absence of these data it is impossible to judge whether the MLP has converged to a solution that genuinely supports the solver comparison or whether the gradient recovery is sensitive to under-training or overfitting.
  3. [Soft-FQEq solver layer description] The soft bond connectivity is described as a differentiable function of atomic geometry with a single free softness parameter. No analysis is given of how this parameter is chosen or regularized, nor of its sensitivity: small changes could alter fragment boundaries and thereby change the recovered chemical-potential gradient. Because the gradient is the headline observable, this parameter choice is load-bearing and requires explicit validation.
minor comments (2)
  1. Clarify the precise mathematical definition of the soft bond connectivity readout (including the functional form, cutoff, and how it is normalized into a connectivity matrix) so that the solver layer can be reproduced independently.
  2. The abstract states that the model 'recovers a clear electrode-to-electrolyte gradient'; a quantitative measure (e.g., slope or potential difference across the interface) would strengthen the claim and allow direct comparison with global QEq.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which identify key areas where additional evidence and documentation will strengthen the manuscript. We address each major point below and commit to the indicated revisions.

read point-by-point responses

  1. Referee: The central demonstration (abstract and results on the IrO2/H2O interface) shows recovery of the electrode-to-electrolyte gradient under Soft-FQEq and its collapse under global QEq. However, this contrast is only informative if the soft bond connectivity matrix defines fragments whose boundaries coincide with the physical electrode/electrolyte division. Because the connectivity is a learned, geometry-only function with no electronic-structure input and the training objective contains no explicit term enforcing electronic distinctness, nothing guarantees that the connected components respect the intended partition rather than forming mixed electrode–adsorbate clusters or splitting the electrode. Without visualizations of the connectivity matrix, statistics on cross-boundary bonds, or overlap metrics with known physical regions, the observed gradient could be an artifact of whatever ad-hoc

    Authors: We agree that explicit validation of the learned fragment boundaries is essential to rule out artifacts. Although training on DDEC6 charges (which reflect physical charge distributions) provides implicit guidance toward appropriate partitions, we will add in the revision: (i) visualizations of the soft connectivity matrix for representative interface snapshots, (ii) statistics on the fraction of cross-boundary soft bonds, and (iii) overlap metrics between the connected components and the known physical electrode/electrolyte regions. These additions will confirm that the fragments align with the physical division and that the gradient recovery is not an artifact of the solver-swap experiment. revision: yes

  2. Referee: The manuscript reports training on DFT energies, forces, and DDEC6 charges but provides no quantitative error metrics, training or validation curves, loss-component weights, or test-set performance for the IrO2/H2O/Na+/ClO4- system. In the absence of these data it is impossible to judge whether the MLP has converged to a solution that genuinely supports the solver comparison or whether the gradient recovery is sensitive to under-training or overfitting.

    Authors: We acknowledge that these quantitative details were omitted from the original submission. In the revised manuscript we will add a dedicated subsection (or supplementary material) containing: mean absolute errors for energies, forces, and charges on training/validation/test splits; training and validation loss curves; the relative weights assigned to each loss term; and confirmation that the model converged without overfitting. This information will allow readers to assess the quality of the underlying MLP and the reliability of the solver comparison. revision: yes

  3. Referee: The soft bond connectivity is described as a differentiable function of atomic geometry with a single free softness parameter. No analysis is given of how this parameter is chosen or regularized, nor of its sensitivity: small changes could alter fragment boundaries and thereby change the recovered chemical-potential gradient. Because the gradient is the headline observable, this parameter choice is load-bearing and requires explicit validation.

    Authors: The softness parameter was selected to maintain differentiability while producing sufficiently distinct fragments; a small value was used to sharpen boundaries. To address the referee’s concern we will include in the revision a sensitivity analysis that varies the parameter over a physically reasonable range and reports the resulting changes (or stability) in the electrochemical potential gradient. We will also document the selection criterion and any regularization applied during training. revision: yes

Circularity Check

0 steps flagged

No circularity: central result follows from external DFT training targets and inference-time solver swap

full rationale

The paper trains the MLP end-to-end with the differentiable Soft-FQEq solver on independent DFT energies, forces, and DDEC6 charges. The headline demonstration applies the identical trained weights once with the fragment-constrained solver (recovering the electrode-electrolyte gradient in per-fragment chemical potentials) and once with global QEq (yielding a uniform profile). This contrast is not forced by construction: the training objective does not presuppose the gradient, the soft connectivity is optimized to reproduce external charge distributions rather than being defined to enforce electrode-electrolyte separation, and no self-citation, uniqueness theorem, or ansatz is invoked to justify the observed difference. The derivation chain remains self-contained against the external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The approach rests on the ad hoc assumption that geometry-derived soft connectivity can substitute for predefined molecular topology while preserving physical charge separation. Standard charge conservation and DFT reference data are used but the soft partitioning is new to the paper.

free parameters (1)
  • softness parameter for bond connectivity
    Controls how sharply or diffusely fragments are defined from atomic geometry; required to make the connectivity differentiable and is likely set or fitted as part of the model.
axioms (2)
  • domain assumption Charge must be conserved independently within each geometry-defined fragment.
    Core principle of per-fragment QEq extended to the soft case.
  • ad hoc to paper A differentiable function of atomic positions can identify fragments that correspond to electronically distinct regions.
    This is the key modeling choice that enables reactivity.
invented entities (1)
  • Soft bond connectivity readout no independent evidence
    purpose: Provides a continuous, differentiable measure of atomic connectivity to define fragments without fixed topology.
    New output head introduced in the MLP to support the Soft-FQEq layer.

pith-pipeline@v0.9.0 · 5613 in / 1629 out tokens · 125956 ms · 2026-05-07T06:23:52.131728+00:00 · methodology

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