Machine-Learned Bond-Order Potential for Exploring the Configuration Space of Carbon

TL;DR

This work develops MLBOP, a general-purpose, transferably accurate interatomic potential for carbon by coupling a machine-learned bond-order framework with a classical two-body energy term and Grimme-D3 dispersion. The authors derive the energy as $E_{tot} = E_{bond} + E_{disp}$, and implement a bond-embedded descriptor to predict environment-dependent bond parameters via MLPs, trained in two stages on a large, diverse DFT dataset. MLBOP demonstrates strong performance across crystals, defects, surfaces, liquids, amorphous phases, phonons, elasticity, clusters, and high-pressure phase behavior, surpassing several state-of-the-art MLIPs in smoothness, transferability, and physical realism, with manageable computational cost. The approach enables comprehensive exploration of carbon's configuration space, including phase diagrams and enthalpy-volume maps of local minima, and holds promise for accelerating discovery of novel carbon materials. Overall, MLBOP provides a robust, interpretable, and efficient pathway to accurate PES coverage with a relatively small parameter set.

Abstract

Construction of transferable machine-learning interatomic potentials with a minimal number of parameters is important for their general applicability. Here, we present a machine-learning interatomic potential with the functional form of the bond-order potential for comprehensive exploration over the configuration space of carbon. The physics-based design of this potential enables robust and accurate description over a wide range of the potential energy surface with a small number of parameters. We demonstrate the versatility of this potential through validations across various tasks, including phonon dispersion calculations, global structure searches for clusters, phase diagram calculations, and enthalpy-volume mappings of local minima structures. We expect that this potential can contribute to the discovery of novel carbon materials.

Figures (14)

• Figure 1: Schematic diagram of the local environment of bond $i$-$j$ surrounded by atoms $k_{1}$, $k_{2}$, $k_{3}$, and $k_{4}$ within the cutoff distance $R_{c}$.
• Figure 2: Schematic structure of the MLBOP model. (a) Construction of the bond-embedded feature vector $\bm{\zeta}_{ij}$ in Eq. \ref{['eq:zeta']}. (b) Computation process of $a_{ij}$ and $b_{ij}$ in Eq. \ref{['eq:aij']} and Eq. \ref{['eq:bij']}.
• Figure 3: Cohesive energy vs nearest-neighbor interatomic distance curves for (a) diamond, (b) graphene, and (c) simple cubic, calculated with MLBOP, DeePMD, ACE, and SchNet trained on the toy datasets, and DFT (PBE). The gray regions indicate the per-atom energy range covered by the training datasets.
• Figure 4: Cohesive energy vs nearest-neighbor interatomic distance curves for the unstable crystals in the Samara Carbon Allotrope Database (SACADA) https://doi.org/10.1002/anie.201600655, calculated with DFT (PBE), C-ACE, and MLBOP trained on the C-ACE dataset. The dispersion energy terms in MLBOP and C-ACE were omitted for comparison with the PBE energies. The boxes on the top right in each panel are magnified views of the low-energy regions.
• Figure 5: Comparison between DFT and MLBOP for the testing dataset. Top: Energy. Bottom: Force. The colors of the circles represent the type of the dataset structure. The MLBOP energies are adjusted by a constant shift to match the DFT energies. Insets show magnified views for the low energy regions. The values in the panels are the mean absolute errors (MAEs) for each structure type.