The topological molecule: Its finite fluxes, exchange stability and minimal surfaces

Abstract

Molecules have at least one nontrivial topological property in common: their minimal surfaces of finite flux. This is why they are stable aggregates of atoms mutually engaged to varying degrees via Coulombic and exchange interactions in fealty to quantum mechanics on otherwise passive nuclear scaffolds. Isolated atoms do not have minimal surfaces but they do undergo exchange interactions. All surfaces implicitly defined by a molecule’s charge density are shown to have zero mean curvature and are consequently minimal surfaces. This finding extends to any potential of a molecule. The minimal surface is of importance in that it is indicative of a vanishing mean curvature whose measurement serves as a way of gauging the charge density or electrostatic potential’s local reliability, a quality assurance protocol absent in conventional crystallography but available to scanning force microscopy. The smaller the mean curvature of an atom, the more bonded is that atom in a molecule. The basis for this discovery is that implicit surfaces admit finite flux to cross them regardless of atomic affiliation, thus engendering exchange, correlation, and chemical bonding between the atoms in the underlying nuclear framework of a molecule. Finite flux in the charge density is a necessary condition for chemical bonding and the stability of molecules and is what makes the electron localization function (ELF) and the exchange-correlation functional (BLYP) useful.