Exhaustive Spatial Sampling for Complete Topology of the Electrostatic Potential

This work presents a robust and efficient algorithm for exhaustively determining the critical points (CPs) of the molecular electrostatic potential (MEP) in 3D space. By combining Newton's method with a systematic physical space sampling strategy, we locate all CPs (maxima, minima, and saddle points) for both exact quantum-chemical MEPs and their tricubic interpolated approximations. The method is validated using a test function with known CPs and applied to a diverse set of molecules, including neutral systems, ions, and noncovalent complexes from the S66 and IONIC-HB datasets. Our results demonstrate that the interpolated MEP faithfully reproduces the topology of the exact potential in most cases, with minor discrepancies arising near nuclear positions or in regions of low gradient. The algorithm's efficiency (2-7 $\times$ faster for interpolated calculations) and robustness make it suitable for large-scale analyses of MEP topologies, offering insights into chemical reactivity and noncovalent interactions.