A foundation machine learning potential with polarizable long-range interactions for materials modelling

TL;DR

By just learning from energies and forces, MLIPs can accurately capture electrostatics and predict atomic charges, as well as being efficiently finetuned to achieve high-level accuracy for specific challenging systems.

Abstract

Long-range interactions are essential determinants of chemical system behaviour across diverse environments. We present a foundation framework that integrates explicit polarizable long-range physics with an equivariant graph neural network potential. It employs a physically motivated polarizable charge equilibration scheme that directly optimizes electrostatic interaction energies rather than partial charges. The foundation model, trained across the periodic table up to Pu, demonstrates strong performance across key materials modelling challenges. It effectively captures long-range interactions that are challenging for traditional message-passing mechanisms and accurately reproduces polarization effects under external electric fields. We have applied the model to mechanical properties, ionic diffusivity in solid-state electrolytes, ferroelectric phase transitions, and reactive dynamics at electrode-electrolyte interfaces, highlighting the model's capacity to balance accuracy and computational efficiency. Furthermore, we show that as a foundation model, it can be efficiently finetuned to achieve high-level accuracy for specific challenging systems.

Figures (6)

• Figure 1: a, Overview of the framework. The framework takes atomic numbers ($Z_i$) and coordinates ($\bm{r}_i$) as inputs and integrates neural network block and explicit polarizable long-range interactions block. For the neural network block, atomic numbers are converted into feature vectors via one-hot encoding, while the coordinates are transformed into a representation of the local environment using radial basis functions and spherical harmonics to capture the distance ($|\bm{R}_{ij}|$) and directional ($\bm{R}_{ij}/|\bm{R}_{ij}|$) information. These embeddings are processed through an equivariant graph neural network to output the scalar MLIP energy $E_0$. Explicit, physically-grounded calculations are performed for long-range effects. An electrostatic energy component ($E_{\rm PQEq}$) is calculated using the polarizable charge equilibration (PQEq) method, and a dispersion energy component ($E_{\rm D3}$) is calculated using the DFT-D3 method. The total energy is the sum of these components. Gradients of the energy yield forces and stress on the system. The long-range block also provides partial charges as an auxiliary output. b, A flowchart detailing the iterative self-consistent procedure for the PQEq method. Starting with atomic core positions ($\bm{r}_{ic}$), atomic numbers ($Z_i$), and predetermined atomic parameters (electronegativities $\chi_i^0$, chemical hardness $\eta_i^0$, and spring constant $K_s^i$), the method uses a core-shell Gaussian model to build and solve a system of linear equations. This loop is repeated until the partial charges ($q_i$) and shell positions ($\bm{r}_{is}$) converge, yielding the final electrostatic energy ($E_{\rm PQEq}$) and the converged partial charges. c, Partition of core-shell Gaussian charge model used in long-range electrostatics. The partial charge ($q_i$) of each atom is represented as sum of the core ($q_{ic}$) and shell ($q_{is}$) charge. The harmonic spring constant, $K_s^i$, couples the core and shell.
• Figure 2: a, The interaction energies for Na-Na (green) and Na-Cl (purple) dimers predicted by our model, w/o-lr model, and reference calculations using DFT. b, Evaluation of polarizable interactions in water molecules. The water molecule model oriented in the yz-plane, with oxygen and hydrogen atoms shown in red and white, respectively. The energy response curves as a function of external electric field ($\bm \epsilon$) strength applied along the x-axis. Results are compared among our model with polarizable long-range interactions (blue), the QEq-based model (yellow), and the reference DFT calculations (green) serving as the reference values. c-e, Comparison of bulk modulus from first principles calculations and our model: c, Voigt approach, d, Reuss method, and c, Hill average.
• Figure 3: a, Crystal structure of $\rm{Ia\bar{3}d}$-Li$_7$La$_3$Zr$_2$O$_{12}$ and Arrhenius plots depicting the lithium-ion diffusion coefficients across varying temperatures. The dark blue polyhedron signifies La located at the 24(c) site and the light brown polyhedron indicates Zr at the 16(a) site. Li fraction occupies the 24(d) and 96(h) sites. Predicted diffusion coefficients of AIMD ref39, our model, and w/o-lr model complete with error bars, are presented to calculate activation energies. b, 2-ns mean square displacements using our model of lithium-ion in c-LLZO with different temperatures ranging from 800 K to 1800 K in an increment of 200K. The linear dashed grey lines, with a slope of 1, are also plotted.
• Figure 4: The temperature dependence of a, lattice constants, and b, local polarizations of unit cells exhibit notable changes during the phase transitions observed from MD simulations on 10×10×10 supercell of BaTiO$_3$.
• Figure 5: a, The initial structure of $\beta$-Li$_3$PS$_4$ [010]/Li [001] interface. The elements are colour-coded: Li in green, S in purple, and P in pink. A segment of the primary reaction zone, ranging from 8 to 26 nanometres is displayed. b, Snapshots at different times during the MD simulations. The partial charges on lithium ions are represented with colour coding to enhance the visibility of structural transformations during the formation of SEI. The SEI layer, approximately 8.5 nm in thickness, forms after 4 ns MD simulations, comprising an amorphous $\beta$-Li$_3$PS$_4$/Li$_2$S interface ( 2 nm), a crystalline Li$_2$S layer ( 4.5 nm), and an amorphous Li$_2$S (Li$_3$P)/Li interface layer ( 2 nm) in sequence. c, The distributions of the Li partial atomic charges along the x-direction at initial, 20, 100, 2000, and 4000 ps state. d, The radial distribution function plots of Li-S and Li-P within the electrolyte and SEI layer.