Design Space of Self--Consistent Electrostatic Machine Learning Interatomic Potentials

Abstract

Machine learning interatomic potentials (MLIPs) have become widely used tools in atomistic simulations. For much of the history of this field, the most commonly employed architectures were based on short-ranged atomic energy contributions, and the assumption of locality still persists in many modern foundation models. While this approach has enabled efficient and accurate modelling for many use cases, it poses intrinsic limitations for systems where long-range electrostatics, charge transfer, or induced polarization play a central role. A growing body of work has proposed extensions that incorporate electrostatic effects, ranging from locally predicted atomic charges to self-consistent models. While these models have demonstrated success for specific examples, their underlying assumptions, and fundamental limitations are not yet well understood. In this work, we present a framework for treating electrostatics in MLIPs by viewing existing models as coarse-grained approximations to density functional theory (DFT). This perspective makes explicit the approximations involved, clarifies the physical meaning of the learned quantities, and reveals connections and equivalences between several previously proposed models. Using this formalism, we identify key design choices that define a broader design space of self-consistent electrostatic MLIPs. We implement salient points in this space using the MACE architecture and a shared representation of the charge density, enabling controlled comparisons between different approaches. Finally, we evaluate these models on two instructive test cases: metal-water interfaces, which probe the contrasting electrostatic response of conducting and insulating systems, and charged vacancies in silicon dioxide. Our results highlight the limitations of existing approaches and demonstrate how more expressive self-consistent models are needed to resolve failures.

Figures (18)

• Figure 1: Required properties of the density expansion and AIM Fukui functions. (a): The potential at $r_1$ can be calculated by replacing all the nuclei and electrons outside of some radius $R$ by the coarse-grained density. The shaded regions indicate some excess charge due to, for example, defects. The coarse grained density smooths out all the detail of the atoms, leaving only the net charge or dipole around the defects. (b): The AIM Fukui function $\tilde{f}_k$ must give rise to the same far-field potential as the basis function $\phi_k$. On the left is shown a basis function for charge coefficient $p_k$ which is spherically symmetric, and on the right is shown two possible AIM Fukui functions which might be associated with $p_k$. The one at the top has more structure, but will still give rise to the same electric potential outside the dashed line as the $\phi_k$. The one at the bottom has a significant dipole moment, meaning the potential outside the dashed line will not match that of $\phi_k$.
• Figure 2: (a): Derivative discontinuity of the energy of an isolated system, (b) discontinuity of the total charge as a function of chemical potential for an isolated system. (c): Infinite susceptibility of one variable, $p_k$, as one would find in a perfect conductor, appears as zero curvature in the energy. (d): Similarly, zero susceptibility appears as zero response to a change in the potential corresponding to that variable.
• Figure 3: This paper investigates the theory and practical aspects of two alternative methods for creating MLIPs with a rich description of electrostatics. The figure should be understood along with sections \ref{['sec:energy_min']}--\ref{['sec:scf:theory:equivalence']}. The two methods are referred to as the energy functional and fixed point methods. In this cartoon, $\mathbf{p}_i$ represents a set of descriptors of the charge density associated with atom $i$, for instance a list of partial multipole moments. $\mathbf{v}_i$ represents a vector of descriptors of the effective potential around atom $i$. The lowest row, labelled 'empirical models' compares the classical charge equilibration approach and a linear polarizable force field (in which $\mathbf{m}_i$ denotes an atomic dipole on atom $i$, and $\mathbf{E}_i$ denotes the electric field at atom $i$). In DFT, the left and right hand side are equivalent. However, when we approximate the functions involved with machine learned surrogates, and switch to a coarse-grained charge density, the two approaches are no longer equivalent in principle.
• Figure 4: Overview of the implementation of each approach.
• Figure 5: Training Methods for our self-consistent models.