Quantum topology of molecular charge distributions. II. Molecular structure and its change

Abstract

This paper illustrates how the concepts of atoms and bonds may be given definite expression in terms of the topological properties of the charge density, ρ (r), and how, as a consequence of these identifications, one is led to a definition of structure and to a phenomenological analysis of structural stability. This approach finds its natural expression in René Thom’s general analysis of structural stability as it applies to a system whose behavior is describable in terms of the gradient of some scalar field. Chemical observations are made in real space, and thus chemical behavior is determined by the morphology of a system’s charge distribution and its evolution with time. The analysis of the topological properties of ρ (r) via the associated gradient vector field ∇ρ (r), reduces to the identification of the critical points in ρ (r). Two types of critical points assume special roles in the analysis. A (3,−3) critical point, a maximum in ρ (r), is an attractor and is identified with the position of a nucleus in the molecular system under study. The basin of the attractor defines the atom associated with the corresponding nucleus. A (3,−1) critical point defines the interatomic surface separating two neighboring atoms, and the bond path linking their nuclei, the line along which the charge density is maximum with respect to lateral displacements. Hence, neighboring atoms are defined to be bonded to one another and the network of bond paths, for a given nuclear configuration, determines its molecular graph. Structure is defined as that set of molecular graphs which contain the same number of bond paths, linking the same nuclei. Thus a change in structure necessitates a change in the number and/or arrangement of bond paths. The making and/or breaking of chemical bonds associated with such a change is topologically a discontinuous process, and the associated change in structure is therefore, abrupt: a continuous change in the nuclear coordinates, the parameters which control the behavior of the system, can lead to a discontinuous change in molecule’s behavior. A point in control space defining the nuclear configuration for which such discontinuous behavior is observed, is called a catastrophe point. The set of catastrophe points thus partitions nuclear configuration space into regions of different structure. The breaking or making of bonds is a catastrophe of the bifurcation type, resulting from the formation of a singularity in ρ (r), whereas the switching of a bond from one nucleus to another is a catastrophe of the conflict type. It is shown that the analytical description of the formation of a three-membered ring structure from all possible neighboring structures (as illustrated for H+3 and H2O) is provided by the unfolding of a particular type of catastrophe, the elliptic umbilic.