Structural homeomorphism between the electronic charge density and the nuclear potential of a molecular system

This paper is a study of the topological relationship between two scalar fields of a molecular system, the electronic charge density and the Coulombic field generated by the atomic nuclei—the nuclear potential. Because of the essential observation that the only local maxima of ground-state charge distributions occur at the positions of the nuclei, the nuclei are the point attractors of the gradient vector fields derived from the charge density and from the nuclear potential. The basins associated with the set of point attractors in either field partition a molecular system into nucleus-dominated regions. The common boundary of any two such neighboring regions contains a particular critical point which generates a pair of gradient paths linking the neighboring attractors. The union of this pair of gradient paths and their end points is called an interaction line. The network of interaction lines defines an elementary graph of the molecular system which identifies the dominant physical interactions in both the charge density and the nuclear potential. Having defined a unique elementary graph for either scalar field for any molecular geometry, the authors partition the total nuclear-configuration space into a finite number of regions. Each region is associated with a particular structure defined as an equivalence class of elementary graphs. The representation of this structural partitioning of nuclear-configuration space is called a structure diagram, which is analogous to a thermodynamic phase diagram. Bader, Nguyen-Dang, and Tal have previously shown that chemical concepts like bonds and molecular structure can be rigorously defined through such a topological analysis of the electronic charge distribution in a molecule. In this paper the authors trace the fundamental role of the nuclear potential in determining the topological properties of this charge distribution. Through a detailed study it is demonstrated that the structure diagrams of the charge density and of the nuclear potential are homeomorphic for the ${\mathrm{H}}_2$O system. It is conjectured that this homeomorphism exists in general for any molecular system.