The zero-flux surface and the topological and quantum definitions of an atom in a molecule

The partitioning of a charge distribution by surfaces exhibiting a local zero flux in the gradient vector field of the electron density leads to an exhaustive and disjoint division of the system into a set of mono-nuclear regions or atoms, provided the only local attractors present in the system are isolated nuclear attractors and the electronic energy is less than that required to produce the plasma state. The existence of non-isolated attractors, whose limited occurrence is confined primarily to excited state charge distributions of one-electron systems, is shown to be readily encompassed within the topological theory of molecular structure, a theory whose purpose is to relate a system's properties to the observed topology of its density distribution. The zero-flux surface serves as the necessary boundary condition for the application of Schwinger's principle of stationary action to define the physics of an atom in a molecule as an open system. Schwinger's principle requires the use of a special class of trial functions: those whose variation is to be equated to the action of smooth, continuous changes in the coordinates of the physical system caused by the action of generators of infinitesimal transformations, the very requirement needed to ensure the applicability of the zero-flux surface condition as the defining constraint of an open system.