🔬 materials science

A Linear-Scaling, Charge-Aware Foundation Potential for Atomistic Simulations

This paper introduces charge-equilibrated TensorNet (QET), a linear-scaling, charge-aware foundation potential that overcomes the limitations of existing models by accurately capturing electrostatic interactions to predict complex ionic structures and reactive processes in energy storage and catalysis systems.

Original authors: Tsz Wai Ko, Runze Liu, Adesh Rohan Mishra, Zihan Yu, Ji Qi, Shyue Ping Ong

Published 2026-08-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Tsz Wai Ko, Runze Liu, Adesh Rohan Mishra, Zihan Yu, Ji Qi, Shyue Ping Ong

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Matter is built from atoms, but what holds those atoms together and makes them react is often a matter of invisible forces. In the world of materials science, one of the most critical forces is electrostatics—the push and pull between electrically charged particles. When atoms swap electrons or shift their charge, they change how they bond, how they move, and how they behave. This is the engine behind everything from a battery storing energy to a catalyst speeding up a chemical reaction. For decades, scientists have tried to simulate these behaviors on computers to predict new materials without having to build them in a lab first. However, there has been a persistent trade-off. The most accurate methods are so slow they can only handle tiny groups of atoms for fleeting moments. The faster methods, which can handle millions of atoms, often ignore the complex dance of electric charges, leading to predictions that look right but behave wrong when electricity is involved.

Researchers at the University of California, San Diego, have introduced a new approach that bridges this gap. They developed a computer model called QET, which stands for charge-equilibrated TensorNet. This model is designed to be both fast and aware of electric charges. It allows scientists to simulate large systems, like the inside of a battery or a molten salt, while accurately tracking how electrons move between atoms. The team showed that their model could predict the structure of complex liquids and the way certain materials crystallize with a level of detail that previous fast models missed. By making these simulations both efficient and electrically aware, they have opened the door to designing better energy storage devices and understanding chemical reactions in ways that were previously too computationally expensive to attempt.

The core of the problem the team tackled is that electric forces act over long distances. In a traditional simulation, if you want to know how one atom feels the pull of another far away, you have to calculate the interaction between every single pair of atoms. As the number of atoms grows, the time required to do these calculations explodes, making it impossible to simulate large systems. To get around this, many models simply ignore these long-range forces or use shortcuts that treat atoms as having a fixed charge. But in reality, atoms are dynamic; they gain or lose electrons depending on their neighbors. The new QET model solves this by using a clever mathematical trick to estimate how charges distribute across a system instantly, without needing to solve a massive, time-consuming equation for every step of the simulation. This allows the model to scale linearly, meaning that doubling the number of atoms only doubles the time it takes to run the simulation, rather than making it exponentially slower.

To test if this new approach actually worked, the researchers trained the model on a massive dataset of over 100,000 atomic configurations, covering 86 different chemical elements. They then put the model through a series of rigorous tests. In one experiment, they simulated a liquid mixture of sodium chloride and calcium chloride, a system known for its complex ionic behavior. Previous fast models predicted a density and structure that did not match experimental reality, either overestimating or underestimating how tightly the atoms packed together. The QET model, however, reproduced the correct density and structure, matching what is seen in real-world experiments. This success was not just a matter of getting the numbers right; it meant the model correctly understood how the different ions were arranging themselves in space, a feat that required it to accurately track the shifting electric charges between them.

The team also looked at a material used in phase-change memory, a type of computer storage that works by switching between amorphous and crystalline states. They simulated how this material, made of germanium, antimony, and tellurium, would crystallize when heated. Models that ignored the nuances of charge transfer failed to capture the correct speed and pattern of this crystallization, sometimes predicting that the material would stay disordered or crystallize in the wrong way. The QET model, by contrast, predicted a crystallization process that closely matched the behavior seen in more expensive, high-precision simulations. It correctly identified that the atoms were forming specific crystalline domains, driven by the subtle ways their electric charges influenced their movement. This suggests that the model can capture the fundamental physics driving phase transitions, which is crucial for designing faster and more reliable memory devices.

Perhaps the most promising application demonstrated in the study involves the interface between a lithium metal electrode and a solid electrolyte, a key component in next-generation solid-state batteries. In these batteries, a thin layer called the solid-electrolyte interphase forms between the electrode and the electrolyte, and its properties determine how well the battery works. The researchers used their model to simulate this interface and observed how it evolved over time. They found that the model correctly predicted the formation of specific chemical compounds, such as lithium phosphide and lithium sulfide, which are known to be the products of the reaction. Crucially, when they applied an electric voltage to the simulation to mimic a charging or discharging battery, the model showed how the chemical reactions sped up or slowed down depending on the voltage. It even predicted how the thickness of the protective layer would change under different electrical conditions, a level of detail that is vital for engineers trying to prevent batteries from failing.

The significance of this work lies in its ability to remove a fundamental bottleneck in materials science. For years, scientists had to choose between simulating large systems with simple physics or small systems with complex physics. This new model demonstrates that it is possible to have both. By accurately capturing how charges move and interact without sacrificing speed, it provides a powerful tool for exploring the vast landscape of chemical possibilities. The researchers have made their model and the data it was trained on available to the scientific community, inviting others to use it to design new catalysts, better batteries, and more efficient materials. While the model is not perfect and still relies on approximations for certain extreme conditions, it represents a major step forward in making large-scale, charge-aware simulations a practical reality for solving real-world energy and materials challenges.

Problem Statement

Electrostatics are fundamental to describing charge transfer, reactivity, and non-bonded interactions in materials. While ab initio methods like Density Functional Theory (DFT) accurately capture these interactions, their computational cost ( $O(N_e^3)$ ) and poor scaling render them impractical for large-scale, long-time simulations. Machine Learning Interatomic Potentials (MLIPs) and Foundation Potentials (FPs) offer a solution by providing linear scaling ( $O(N_{atoms})$ ) and DFT-level accuracy at a fraction of the cost.

However, a critical limitation persists: most existing MLIPs and FPs either neglect explicit electrostatic interactions or rely on charge equilibration (Qeq) schemes that introduce prohibitive computational bottlenecks.

Graph-based limitations: Extending interaction ranges via message-passing steps is inefficient, leads to over-smoothing, and increases model parameters.
Qeq scaling issues: Traditional Qeq schemes require solving a dense $(N_{atoms}+1) \times (N_{atoms}+1)$ linear system. Direct solutions scale as $O(N_{atoms}^3)$ , and even iterative solvers with Ewald summation often result in quadratic or quasi-linear scaling ( $O(N_{atoms} \log N_{atoms})$ ) with significant overhead.
Differentiation overhead: In Qeq-based MLIPs, calculating forces and stresses requires backpropagation through the charge-equilibration procedure, introducing additional computational costs and memory consumption that hinder large-scale reactive molecular dynamics (MD).

Consequently, there is a lack of a scalable, charge-aware FP that can accurately model non-local charge transfer across the periodic table without sacrificing computational efficiency.

Methodology

1. The QET Architecture
The authors introduce Charge-Equilibrated TensorNet (QET), an equivariant, charge-aware architecture built upon the TensorNet framework. QET achieves linear scaling by replacing conventional electrostatic solvers with an Analytically Solvable Charge-Equilibration (LQeq) scheme.

Effective Electronegativity ( $\chi'_i$ ): The model predicts environment-dependent electronegativities ( $\chi_i$ ) and hardnesses ( $\eta_i$ ) using Multi-Layer Perceptrons (MLPs) fed by graph-convoluted atomic features. To handle long-range interactions within the cutoff, an effective electronegativity is defined:
$\chi'_i = \chi_i + \frac{q_i}{2\sigma_i\sqrt{\pi}} + \sum_{j>i}^{N_{atoms}} \frac{\text{erf}\left(\frac{R_{ij}}{\sqrt{2(\sigma_i^2+\sigma_j^2)}}\right)}{R_{ij}} q_j$
where $\sigma$ represents the width of Gaussian-distributed charges.
Analytic Charge Solution: Unlike standard Qeq which requires iterative matrix inversion, QET derives an elegant analytic solution for partial charges ( $q_i$ ) based on $\chi'_i$ and $\eta_i$ , constrained by the total system charge ( $Q_{tot}$ ):
$q_i = -\eta_i^{-1} \left[ \chi'_i - \frac{Q_{tot} + \sum \chi'_i \eta_i^{-1}}{\sum \eta_i^{-1}} \right]$
This formulation ensures strictly linear scaling ( $O(N_{atoms})$ ) and avoids the $O(N^3)$ bottleneck.
Energy Prediction: The predicted charges are used to calculate a truncated Coulomb potential ( $V_i$ ) acting on each atom. Atomic features, charges, and the Coulomb potential are concatenated and fed into a gated MLP to predict atomic energies.

2. The MatQ Dataset
To train a general-purpose charge-aware FP, the authors constructed the Materials Charge (MatQ) dataset.

Composition: ~120,000 structures covering 86 elements.
Sources: Structural deformations of Materials Project crystals and high-temperature (1000K/3000K) AIMD trajectories from the OMat24 validation set.
Labels: Energies, forces, stresses, and atomic charges computed via DDEC6 partitioning of DFT electron densities (PBE functional).

3. Training and Fine-tuning
Models were trained using a loss function combining energy, force, stress, and charge errors. The authors also demonstrated fine-tuning capabilities on specific reactive interfaces (Li/Li $_6$ PS $_5$ Cl) and the ability to simulate systems under applied electrochemical potentials by shifting the electronegativities of electrode atoms.

Key Contributions

Linear-Scaling Charge Equilibration: The development of the LQeq scheme, which provides an analytic solution for charge equilibration, removing the computational bottleneck associated with traditional Qeq in MLIPs.
QET Foundation Potential: The release of a pre-trained, charge-aware FP (QET-MatQ) that maintains the high accuracy of TensorNet while explicitly modeling charge transfer.
Qualitative Improvements in Electrostatic Systems: Demonstration that charge-awareness leads to qualitatively different and more accurate predictions in systems dominated by electrostatics, where charge-agnostic models fail.
Simulation of Applied Potentials: A framework to simulate interfacial reactions under constant electrochemical potentials by modifying atomic electronegativities, enabling the study of voltage-dependent chemistry.

Results

1. Benchmark Performance

Accuracy: On standard benchmarks (C10H $_2$ , Ag $_3^{\pm}$ , Na $_9$ Cl $_8^+$ , Au/MgO), QET achieves RMSEs < 1 meV/atom for energies and < 20 meV/Å for forces, comparable to state-of-the-art charge-informed models (4G-HDNNP, CELLI) but with superior scaling.
Charge Prediction: The LQeq scheme predicts charges comparable to conventional Qeq and latent-charge methods, with a mean absolute error (MAE) of ~26.8 me on the MatQ test set.

2. Qualitative Differences in Simulations

Ag $_3^{\pm}$ Clusters: QET correctly predicts distinct equilibrium geometries and energy differences for Ag $_3^+$ and Ag $_3^-$ , matching DFT. TensorNet (charge-agnostic) predicts identical geometries for both charge states.
NaCl–CaCl $_2$ Ionic Liquid: QET-MatQ reproduces the correct liquid density and structure factor (specifically low-Q scattering related to long-range packing) observed in experiments. Charge-agnostic models (TensorNet, GRACE) either overestimate or underestimate density and fail to capture long-range structural correlations.
Ge $_1$ Sb $_2$ Te $_4$ Phase Change Memory: QET-MatQ accurately captures the crystallization kinetics and final crystalline ratio of amorphous Ge $_1$ Sb $_2$ Te $_4$ , outperforming several other FPs. The model correctly predicts the oxidation states (Ge/Sb positive, Te negative) and produces extended crystalline domains consistent with AIMD studies.

3. Reactive Interface Simulations

Li/Li $_6$ PS $_5$ Cl Interface: Fine-tuned QET successfully simulates the solid-electrolyte interphase (SEI). Unlike TensorNet, which erroneously forms elemental phosphorus (P-P bonds) due to a lack of charge interactions, QET correctly predicts the formation of phosphide-like structures (Li $_3$ P, Li $_2$ S, LiCl).
Applied Potentials: Under applied biases (+4V, -4V), QET demonstrates voltage-dependent interfacial growth. A cathodic bias (-4V) accelerates SEI thickening, while an anodic bias (+4V) delays it. The model correctly captures the redistribution of charges across the electrode and interphase.

4. Scaling and Efficiency

QET exhibits linear scaling with system size, comparable to standard TensorNet.
In contrast, combining TensorNet with conventional Qeq results in prohibitive non-linear scaling, making it impractical for realistic system sizes.
For solid electrolytes, the inclusion of explicit long-range Ewald summation provided only marginal improvements over the truncated Coulomb potential used in QET, suggesting QET captures the dominant electrostatic interactions governing ion migration.

Significance and Claims

The paper claims that QET removes a fundamental constraint in large-scale atomistic simulations: the trade-off between accurate electrostatic modeling and computational scalability. By establishing a general, data-driven framework for charge-aware FPs, the authors position QET as a transformative tool for:

Energy Storage: Accurately modeling electrode-electrolyte interfaces and voltage-dependent chemistry in batteries.
Catalysis: Describing charge transfer in heterogeneous catalysis and adsorbate-catalyst interactions.
Materials Discovery: Enabling the screening of vast chemical spaces where charge states and electrostatics are critical.

The authors emphasize that while QET inherits some limitations of Qeq (e.g., potential overpolarization errors), it represents a significant step forward by enabling linear-scaling, charge-aware simulations that were previously computationally intractable. They call for the community to include atomic charges and magnetic moments in future datasets to further advance physically grounded foundation potentials.

Problem Statement

Methodology

Key Contributions

Results

Significance and Claims

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