[Paper Review] Electrostatic interactions in atomistic and machine-learned potentials for polar materials

Lorenzo Monacelli, Nicola Marzari|arXiv (Cornell University)|Dec 2, 2024

Machine Learning in Materials ScienceMaterials Science3 citations

TL;DR

This paper proposes a first-principles model to accurately include long-range electrostatic interactions in atomistic and machine-learned potentials for polar materials, using only physical observables like Born effective charges and dielectric tensor. It successfully reproduces LO-TO splitting and long-wavelength phonon dispersions in cubic BaTiO₃ without retraining or additional first-principles calculations.

ABSTRACT

Long-range electrostatic interactions critically affect polar materials. However, state-of-the-art atomistic potentials, such as neural networks or Gaussian approximation potentials employed in large-scale simulations, often neglect the role of these long-range electrostatic interactions. This study introduces a novel model derived from first principles to evaluate the contribution of long-range electrostatic interactions to total energies, forces, and stresses. The model is designed to integrate seamlessly with existing short-range force fields without further first-principles calculations or retraining. The approach relies solely on physical observables, like the dielectric tensor and Born effective charges, that can be consistently calculated from first principles. We demonstrate that the model reproduces critical features, such as the LO-TO splitting and the long-wavelength phonon dispersions of polar materials, with benchmark results on the cubic phase of barium titanate (BaTiO$_3$).

Motivation & Objective

• To address the lack of long-range electrostatic interactions in state-of-the-art machine-learned and atomistic potentials for polar materials.
• To develop a method that captures non-analytic behavior at Γ-point phonons (LO-TO splitting) without requiring retraining or additional first-principles data.
• To ensure compatibility with existing short-range force fields by relying only on equilibrium-state physical observables such as dielectric tensor and Born effective charges.
• To maintain translational invariance and satisfy acoustic sum rules through a redefined centroid position in the energy expression.
• To enable accurate simulation of thermodynamic and spectroscopic properties in polar insulators using efficient, scalable force fields.

Proposed method

• Derives the long-range electrostatic energy from first principles using dipole-dipole interactions, with the electric field expressed via Ewald summation.
• Expresses the electrostatic energy in terms of atomic displacements, effective charges, and the dielectric tensor, using a Fourier-space formulation.
• Applies a first-order expansion of the sine function in the dipole interaction to eliminate dependence on arbitrary charge definitions.
• Redefines the centroid position of each atom to enforce translational invariance and satisfy the acoustic sum rule, ensuring zero net force on the center of mass.
• Integrates the electrostatic contribution as a post-processing correction to existing force fields, using only standard DFT-calculated observables.
• Implements the model via algorithmic differentiation, ensuring correct derivatives for forces and stress tensors, including corrections from the redefined centroid.

Fig. 1: Phonon band structure of cubic \ch BaTiO3, comparison between DFT, the GAP without long-range (LR) interaction gigliThermodynamicsDielectricResponse2022 , and the same potential with long-range (LR) interactions adding to the forces Eq. ( 11 ) Negative frequencies are imaginary numbers, mean

Experimental results

Research questions

• RQ1 Can long-range electrostatic interactions be accurately captured in machine-learned potentials without retraining or additional training data?
• RQ2 How can the non-analytic behavior of the dynamical matrix at q=0 (LO-TO splitting) be modeled using only physical observables from equilibrium DFT calculations?
• RQ3 Can a physically consistent, translationally invariant electrostatic energy model be constructed that preserves acoustic sum rules?
• RQ4 To what extent can this model reproduce key phonon dispersion features like LO-TO splitting in polar perovskites such as BaTiO₃?
• RQ5 Can the method be seamlessly integrated into existing short-range force fields without modifying the underlying model architecture or requiring new ab initio data?

Key findings

• The model accurately reproduces the LO-TO splitting in cubic BaTiO₃, a hallmark of long-range electrostatics, using only DFT-calculated effective charges and dielectric tensor.
• The electrostatic energy expression is invariant under global translations due to the redefined centroid, ensuring satisfaction of the acoustic sum rule.
• The method successfully captures the non-analytic q-dependence of the dynamical matrix at the Γ point, which is critical for phonon dispersion and thermodynamic properties.
• The model achieves this without requiring partial atomic charges, charge equilibration, or additional training data, preserving compatibility with existing machine-learned potentials.
• The implementation via algorithmic differentiation correctly computes forces and stresses, including corrections from the centroid redefinition, ensuring consistency with conservation laws.
• Benchmark results on BaTiO₃ show excellent agreement with first-principles phonon dispersions, confirming the model's accuracy and physical consistency.