Aurora Costales

The Molecular Electrostatic Potential (MEP) is widely used to understand chemical reactivity and noncovalent interactions. Nevertheless, obtaining a complete topological picture of the MEP, identifying all maxima, minima, and saddle points has remained challenging due to its slow long-range decay and sensitivity to computational setups. In this work, we introduce a robust and efficient algorithm to exhaustively locate all critical points (CPs) of the MEP in three dimensions. Our approach combines Newton's iterative method with a systematic, multi-resolution sampling of physical space that reaches far beyond the molecular boundaries. We test the algorithm on a model function with known topology and apply it to various molecular systems, including neutral molecules, ions, and noncovalent complexes from the S66 and IONIC-HB datasets. Additionally, we explore a tricubic interpolation of the MEP to boost computational speed while preserving topological accuracy. Our findings show that the interpolated MEP reproduces the exact topology in 87% of the cases tested. Discrepancies are minor, appearing only near nuclei or in low-gradient regions (e.g., LiH, N₃⁻). The method satisfies the Poincaré–Hopf index for charged systems and achieves speedups of 2–7× (2.12× on average) compared to exact calculations. As an illustration, Figure 1 displays the full set of CPs for the sulfate anion (SO₄²⁻), featuring twelve (3,–1), eighteen (3,+1), and twelve (3,+3) critical points that reveal the rich topology of its MEP. This work positions MEP topology as a practical and scalable tool for large-scale studies, offering deep insights into electrostatic-driven molecular recognition and noncovalent bonding patterns.