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

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.

• The paper introduces Soft-FQEq, a differentiable, fragment-constrained method that overcomes the uniform chemical potential issue in global QEq.
• It integrates atomic feature networks with a smooth MLP-based bond connectivity model to enable reactive simulations and achieve charge MAEs below 0.05 e.
• The study demonstrates that enforcing per-fragment charge conservation is essential for reproducing correct interfacial double-layer gradients and capacitive responses.

Fragment-Constrained Charge Equilibration for Charge-Aware MLIPs at Electrochemical Interfaces

Motivation and Problem Definition

Atomistic modeling of electrochemical interfaces necessitates accurate representation of reactive chemistry, long-range electrostatics, and spatially resolved charge distributions suitable for separating the electrode and electrolyte regions. Conventional charge-aware machine-learned interatomic potentials (MLIPs), typically built on global charge equilibration (QEq), enforce a single system-wide electrochemical potential. This constraint eliminates the inherent spatial gradient of electrochemical potential required to properly represent the electric double layer and enables spurious charge transfer between electronically isolated regions. Classical molecular dynamics with per-fragment charge equilibration addresses this via predefined fragment topology, but such construction is inherently non-reactive and fails when bonds break or form. The core challenge is obtaining a continuous, differentiable formulation of fragment identification that adapts smoothly through bond rearrangements, enabling charge-constrained MLIPs for reactive electrochemical environments.

Methodological Advances: Soft Fragment-Constrained QEq (Soft-FQEq)

The authors present Soft-FQEq, a differentiable, fragment-resolved charge equilibration framework that generalizes fragment identification to a continuous function of atomic geometry. The architecture integrates four scalar readouts from a shared atomic-feature network (HIP-NN): per-atom electronegativity, per-atom source charge, per-atom short-range energy, and per-pair soft bond connectivity. A specially designed multi-layer perceptron (MLP) generates smooth bond-connectivity scores; a graph Laplacian resolvent constructs a soft fragment-membership matrix $C$ with continuous membership values. This matrix allows the definition of fragment-level charge constraints that evolve seamlessly with atom positions, preserving PES smoothness during bond-breaking events and enabling differentiable training.

Charge equilibration is then performed within fragments via an augmented Lagrangian Uzawa solver. Unlike classical QEq, which enforces a single Lagrange multiplier for the total system, Soft-FQEq enforces per-fragment charge conservation, producing fragment-resolved chemical potentials and equilibrated atomic charges. The differentiable nature of all stages ensures that upstream atomic features receive meaningful gradient signals, essential for supervised training and physical interpretability.

Electrostatic interactions are handled using a Gaussian-screened Ewald summation, which yields a well-conditioned, positive-definite Coulomb matrix and stabilizes charge equilibration for periodic boundary conditions and large systems. Auxiliary loss functions supervise the upstream atomic electronegativities and source charges to maintain identifiability and ensure that the fragment-resolved charges and chemical potentials remain physically meaningful throughout training.

Numerical Results and Architectural Ablation

Training was performed on DFT- and DDEC6-derived energies, atomic charges, and forces for IrO$_2$/H$_2$O/Na$^+$/ClO$_4^-$ electrochemical interfaces. The trained Soft-FQEq model accurately predicts atomic charges with mean absolute errors below $0.05\,e$, consistent with the state-of-the-art DDEC6 reference, demonstrating joint accuracy in the learned atomic electronegativities, source charges, fragment-connectivity prediction, and Coulombic decomposition.

A robust spatial gradient in the per-atom electrochemical potential $\muphys[i] = \chi_i + [\mathbf{A}\mathbf{q}]_i$ emerges along the interface normal, clearly separating the electrode from bulk aqueous phases and reproducing the intrinsic potential-of-zero-charge at the metal-water boundary. Notably, replacing Soft-FQEq with global QEq at inference—keeping trained weights fixed—collapses this spatial gradient instantaneously, producing a uniform $\muphys$ across all atoms. This controlled ablation demonstrates the architectural necessity of fragment-constrained equilibration for sustaining the double-layer gradient; training within global QEq or alternative parameterizations does not remedy the fundamental single-$\mu$ pathology.

Soft-FQEq further models the capacitive response of the interface: the electrode chemical potential shifts monotonically with imposed surface charge densities, while the bulk electrolyte maintains consistency, corresponding with expected physical behavior. Fragment assignment correlates accurately with underlying chemical connectivity, trackable even through bond dissociation and forming events due to the soft, continuous construction.

Implications, Structural Claims, and Forward Directions

The work establishes that global QEq structurally cannot support spatial gradients in electrochemical potential across electronically distinct regions, irrespective of fit quality, feature parameterization, or training protocol. This pathology extends universally to all charge-aware MLIPs enforcing a single system-wide chemical potential constraint, including contemporary neural architectures and charge-equilibration variants. The remedy via fragment-constrained QEq, enabled by differentiable fragment identification, is decisively architectural and transferable across MLIP frameworks.

Practically, Soft-FQEq opens the path to simulating electrochemical double layers, constant-potential dynamics, and voltage-controlled interfaces with charge-aware MLIPs capable of reactive chemistry. Its modular solver layer, coupling via scalar network outputs, is agnostic to the backbone feature architecture and can be integrated into advanced message-passing and equivariant networks. The underlying logic generalizes to other systems with electronically distinct fragments (molten salts, ionic liquids, supported clusters), promising broad applicability in complex electrochemical and heterogeneous environments.

Further extensions include production active-learning runs on larger-scale datasets, constant-charge and constant-potential molecular dynamics, generalized GCMC sampling for voltage regimes, and multi-electrode architectures. Scalability improvements via block-sparse resolvent approximation and fast Particle Mesh Ewald (PME) integrations are anticipated.

Conclusion

Soft-FQEq demonstrates that fragment-constrained charge equilibration is essential for charge-aware MLIPs to correctly represent spatially resolved electrochemical potentials and avoid spurious charge redistribution at interfaces. The differentiable fragment-identification and solver architecture allows for both reactive chemistry and the accurate physical modeling of electrochemical double layers. The structural ablation reveals that global QEq fundamentally precludes the correct interfacial potential profile, regardless of parameterization, establishing the necessity of fragment constraints for future MLIP-based electrochemical simulation frameworks.

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