The characterization of atomic interactions
Authors: Bader R. F. W. , Essén H. 
Published: February 2003

The theory of molecular structure determined by the gradient vector field of the charge density ρ identifies the set of atomic interactions present in a molecule. The interactions so defined are characterized in terms of the properties of the Laplacian of the charge density ∇2ρ(r). A scalar field is concentrated in those regions of space where its Laplacian is negative and depleted in those where it is positive. An expression derived from the quantum mechanical stress tensor relates the sign of the Laplacian of ρ to the relative magnitudes of the local contributions of the potential and kinetic energy densities to their virial theorem averages. By obtaining a map of those regions where ∇2ρ(r)&lt;0, the regions where electronic charge is concentrated, one obtains a map of the regions where the potential energy density makes its dominant contributions to the energy of a system. It is demonstrated that atomic interactions fall into two broad general classes, closed-shell and shared interactions, each characterized by a particular set of mechanical properties. Interactions resulting from the sharing of charge density between atoms, covalent and polar bonds, are caused by a contraction of the charge density towards the line of interaction linking the nuclei. The curvatures of ρ perpendicular to the interaction line are dominant, electronic charge is concentrated in the internuclear region, and ∇2ρ&lt;0. These interactions are characterized by the large negative value of the potential energy in the internuclear region. Interactions between closed-shell atoms as found in noble gas repulsive states, in ionic bonds, in hydrogen bonds, and in Van der Waals molecules are governed by the contraction of the charge density towards each of the interacting nuclei. Thus one finds the parallel curvature of ρ to be dominant in these interactions, electronic charge is depleted in the interatomic surface and ∇2ρ&gt;0. The mechanics are characterized by the relatively large value of the kinetic energy, particularly the component parallel to the interaction line. In the closed-shell interactions, the regions of dominant potential energy contributions are separately localized within the boundaries of each of the interacting atoms or molecules. In the shared interactions, a region of low potential energy is contiguous over the basins of both of the interacting atoms. The problem of further classifying a given interaction as belonging to a bound or unbound state of a system is also considered, first from the electrostatic point of view wherein the regions of charge concentration as determined by the Laplacian of ρ are related to the forces acting on the nuclei. This is followed by and linked to a discussion of the energetics of interactions in terms of the regions of dominant potential and kinetic energy contributions to the virial as again determined by the Laplacian of ρ. The properties of the Laplacian of the electronic charge thus yield a unified view of atomic interactions, one which incorporates the understandings afforded by both the Hellmann–Feynman and virial theorems.