Long-Range Electrostatics and Born Effective Charges from Invariant Moment Tensor Potentials with Latent Ewald Summation (MTP-LES)

Technical Summary: Long-Range Electrostatics and Born Effective Charges from Invariant Moment Tensor Potentials with Latent Ewald Summation (MTP–LES)

Problem Statement

Machine-learning interatomic potentials (MLIPs) have achieved high accuracy at low computational cost but are fundamentally limited by their reliance on local, short-range descriptors (typically 4–6 Å cutoffs). This locality assumption fails to capture long-range electrostatic interactions and polarization responses essential for modeling polar liquids, ferroelectric oxides, and finite molecular systems. While recent frameworks like Latent Ewald Summation (LES) have demonstrated that per-atom latent charges can be inferred from energy and force training alone—recovering properties like Born effective charges (BECs) and dipole moments without explicit charge supervision—existing implementations have been restricted to graph-based or equivariant architectures (e.g., CACE-LR, MACE-LES, NequIP-LES). These architectures explicitly encode directional information, raising the question of whether such features are strictly necessary for recovering physically meaningful latent electrostatic responses, or if the key lies in how charge degrees of freedom are constrained and stabilized.

Methodology

The authors introduce MTP–LES, a framework that couples the Latent Ewald Summation (LES) formalism with Moment Tensor Potentials (MTP), which utilize rotationally invariant descriptors. Unlike previous LES implementations that rely on equivariant message passing, MTP–LES employs scalar contractions of moment tensors. To enable stable learning of latent charges within this invariant descriptor space, the authors implement three specific architectural modifications:

1. Independent Branches: The model utilizes separate neural network branches for short-range energy ($E_{SR}$) and latent charge prediction ($q_i$). This prevents the charge degrees of freedom from being "slaved" to the local energy representation.
2. Species-Dependent Bias and Neutrality Projection: A learnable, species-dependent charge bias ($b_{s_i}$) initializes the charge channel with chemically sensible ordering. Crucially, the model enforces explicit charge neutrality for every configuration by projecting the mean charge to zero ($\sum_i q_i = 0$). This removes unphysical residual cell charges that arise in unconstrained models.
3. Optional Dipole Loss: For finite molecular systems (non-periodic), where the long-range Ewald contribution is absent and energy/force training alone may not sufficiently constrain the absolute charge scale, an optional auxiliary loss term is added to match the total molecular dipole moment against reference values (computed via MBIS charges).

The total energy is defined as $E_{total} = E_{SR} + E_{LR}$, where $E_{LR}$ is computed via Ewald summation for periodic systems or damped pairwise Coulomb summation for non-periodic systems. Forces and BECs are obtained via automatic differentiation of the total energy and polarization, respectively.

Key Results

The MTP–LES framework was evaluated across four diverse material systems:

• Ferroelectric PbTiO$_3$: On a stringent benchmark using SCAN-DFT data, MTP–LES achieved a diagonal BEC correlation of $R = 0.994$, matching the established CACE-LR baseline ($R = 0.993$) on identical data. The model successfully recovered anomalous Ti and O charge responses (e.g., $Z^*_{Ti} \approx +6$) without any charge or BEC supervision. While the absolute RMSE for BEC magnitudes was higher than CACE-LR, the tensor correlations and anomalous species ordering were preserved.
• Liquid Water: The model reproduced structural correlations (oxygen-oxygen radial distribution function) and the infrared absorption spectrum (librational, bending, and stretching modes) consistent with experiment. Adding the LES module improved force accuracy by approximately 21% compared to an MTP-only baseline. The recovered BECs fell within the range of other published LES-augmented benchmarks (MACE-LES, NequIP-LES, etc.).
• Polar Dipeptides: For finite systems in vacuum, the inclusion of a dipole-matching loss stabilized the latent charge scale. The model achieved high accuracy in predicting molecular dipoles ($R^2 = 0.996$) and BECs ($R^2 = 0.93$), outperforming the ablation without dipole supervision.
• Hafnia (HfO$_2$): Applied to a newly curated dataset of monoclinic HfO$_2$, the model preserved structural fidelity (reproducing the sevenfold Hf–O coordination) and recovered anomalous Hf Born charges ($Z^* \approx +5$) without prior LES baselines for this system.
• Computational Efficiency: Timing benchmarks on liquid water systems (up to 15,360 atoms) showed that the LES correction introduces only a 3.4% wall-time overhead relative to the MTP-only baseline, while preserving near-linear scaling with system size.

Significance and Claims

The paper claims that accurate latent-charge recovery in MLIPs is not solely determined by the use of graph-based or equivariant architectures. Instead, the critical factors are how charge degrees of freedom are exposed, constrained, and stabilized. By decoupling the charge and energy branches, enforcing charge neutrality, and initializing with a chemically sensible prior, MTP–LES demonstrates that rotationally invariant, descriptor-based potentials can effectively model long-range electrostatics and polarization responses.

This work extends the applicability of the LES framework to a computationally efficient, descriptor-based architecture, offering a route to incorporate long-range electrostatics into MLIPs without the computational cost of equivariant feature propagation or the need for explicit charge labels during training. The authors position MTP–LES as a viable alternative for simulating polar, ionic, and ferroelectric materials where long-range interactions are physically significant.