Abstract: Molecular surface area and volume are widely used to describe molecular size and are closely related to intermolecular interactions and thermodynamic properties. Conventional geometric models rely on empirical atomic radii, whereas electronic-structure-based definitions provide a more intrinsic description of molecular boundaries but remain computationally demanding. In this work, we define molecular surface area and volume using an isopotential condition of the Kohn-Sham (KS) one-electron potential and construct a machine learning model to accelerate their evaluation. Reference data were generated from three-dimensional KS potential grids and followed by geometric integration. SchNet was then trained to learn the relationship between molecular structure and the corresponding KS-defined size descriptors. The model achieves high predictive accuracy, with R² values exceeding 0.9900 for surface area and 0.9800 for volume on the test set. The predicted descriptors exhibit strong linear correlations with van der Waals surface area and volume, and display clear relationships with experimental boiling points. These results indicate that the proposed size descriptors retain electronic-structure information while being efficiently predictable from molecular geometry. This work demonstrates that electronic-structure-based molecular size definitions can be combined with machine learning to provide accurate and computationally scalable descriptors for molecular modeling and property prediction.

Key words: Molecular surface area, Molecular volume, Kohn-Sham one-electron potential, Machine learning, SchNet