Parameterizing empirical interatomic potentials for predicting thermophysical properties via an irreducible derivative approach: the case of ThO$_2$ and UO$_2$

Technical Summary: Parameterizing Empirical Interatomic Potentials via an Irreducible Derivative Approach for ThO₂ and UO₂

Problem Statement
The accuracy of classical molecular dynamics (MD) simulations for predicting thermophysical properties is fundamentally limited by the quality of the empirical interatomic potentials (EIPs) employed. Traditional EIPs are often parameterized by fitting to experimental data, such as lattice parameters, elastic constants, and thermal expansion. However, experimental data is frequently limited in quantity and quality, particularly for complex defect structures, and often fails to directly probe the details of the Born-Oppenheimer potential. Consequently, traditional parameterization strategies often yield potentials that are inadequate for accurately predicting phonon dispersions and higher-order phonon interactions, which are critical for understanding thermal transport and defect formation in nuclear fuel materials like Thorium dioxide (ThO₂) and Uranium dioxide (UO₂). While machine-learning interatomic potentials (MLIPs) offer higher accuracy, their computational cost and lack of physical interpretability remain barriers. There is a need for a robust training methodology that leverages first-principles data to capture phonon-related properties without the computational overhead of MLIPs.

Methodology
The authors introduce a training approach for EIPs based on the direct comparison of irreducible derivatives (IDs) between the target EIP and the Born-Oppenheimer potential derived from Density Functional Theory (DFT).

• Training Data: Instead of relying solely on energies and forces, the dataset includes second- and third-order displacement IDs and second-order strain IDs. These derivatives fully exploit space group symmetry, allowing for training without redundant information. The training set comprises:
  • Second-order displacement IDs within a $2 \times 2 \times 2$ supercell.
  • Third-order displacement IDs within a conventional cubic supercell.
  • Second-order strain IDs.
  • Lattice parameters at 0 K, normalized Born effective charges, and Frenkel pair (FP) formation energies.
• Potential Form: The study parameterizes an EIP for ThO₂ and UO₂ based on the existing CRG potential (which uses Buckingham, Morse, and EAM terms) but enhances it by incorporating a core-shell model. In this model, each ion splits into a massive core and a massless shell connected by a harmonic spring, a feature essential for capturing specific optical phonon modes.
• Optimization: A two-stage optimization process is employed. Stage 1 uses local optimization (Nelder-Mead) to refine parameters and weights, adding training data progressively from basic properties to complex derivatives. Stage 2 utilizes global optimization (particle swarm algorithm) to finalize the parameter set.
• Validation: The resulting potentials (denoted as PW) are validated against DFT calculations and experimental data for phonon dispersion, elastic constants, thermal conductivity (calculated via the Boltzmann Transport Equation in the relaxation time approximation), thermal expansion, and defect formation energies.

Key Contributions and Results

• Improved Phonon Dispersion: The PW potentials significantly outperform the widely used CRG potential. While the CRG potential overpredicts optical phonon branches and fails to capture the frequency gap between the highest optical branches, the PW successfully reproduces the DFT-calculated dispersion and the optical branch gap, attributed to the inclusion of the core-shell model.
• Thermal Conductivity: The PW yields substantial improvements in thermal conductivity predictions compared to the CRG. At 1000 K, the error in predicted thermal conductivity for ThO₂ and UO₂ is reduced from 54% and 63% (CRG) to 17% and 9% (PW), respectively, when compared to experimental data.
• Defect Formation Energies: Despite being trained on only two data points for defect formation (Th/U FPs in a conventional cell), the PW accurately predicts the formation energies for O, Th, and U FPs in larger supercells, with errors within 9% of DFT values.
• Elastic Constants: The PW provides better predictions for certain elastic constants ($C_{11}$ and $C_{44}$ in ThO₂; $C_{44}$ in UO₂) compared to the CRG, even though the CRG was directly fitted to experimental elastic constants. Discrepancies in other constants are noted to stem from differences between DFT and experimental values rather than potential failure.
• Thermal Expansion: The PW shows good agreement with experimental thermal expansion data up to 2500 K for both materials.

Significance and Claims
The paper claims that training EIPs based on irreducible derivatives is a promising direction for developing accurate potentials focused on phonon-related thermophysical properties. By utilizing space group symmetry to minimize redundant training data and incorporating higher-order derivatives, the approach achieves high fidelity to first-principles data. The authors assert that their method yields an EIP that reliably characterizes DFT data and outperforms traditional potentials in predicting phonon dispersions and thermal conductivity. Furthermore, the study demonstrates that enhancing the analytical functional form with a core-shell model is essential for capturing specific optical phonon modes in actinide oxides. The authors suggest this ID-based approach could be useful in the broader context of machine learning-based potentials, offering a pathway to combine the efficiency of analytical forms with the accuracy of first-principles training.