Explicit Electric Potential-Embedded Machine Learning Framework: A Unified Description from Atomic to Electronic Scales

To further develop accurate and large-scale simulations of electrochemical interfaces, we propose a unified explicit electric potential framework to simultaneously predict atomic forces and electron density distributions. The framework consists of three components: data generation, model training, and application. The data generation component, implemented in Hy-DFT, efficiently regulates the potential during constant-potential ab initio molecular dynamics (CP-AIMD), reducing the number of single-point calculations required for convergence. The model training component includes two modules: Potential-Embedded MACE (PE-MACE) and Potential-Embedded Electron Density Prediction (PE-EDP). PE-MACE implements an explicit electric potential machine learning force field (EEP-MLFF) based on the MACE architecture. We develop PE-EDP to overcome the limitation of EEP-MLFF in describing atom forces. PE-EDP, also based on equivariant graph neural networks, predicts electron density distributions under arbitrary potentials. Using the Pt(111)/water interface as a model system, both PE-MACE and PE-EDP show high accuracy on training and test sets. Radial distribution functions from CP-MLMD agree well with CP-AIMD, and long-timescale simulations reveal potential-induced reorganization of interfacial water. Planar-integrated charge profiles and Bader analysis from PE-EDP are consistent with DFT results. These results demonstrate that the framework can simultaneously describe atomic dynamics and electron density distributions under arbitrary potentials, providing a useful tool for studying electrochemical interfaces.