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

Abstract

The partitioning of a charge distribution by surfaces exhibiting a local zero¯ux in the gradient vector ®eld 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 con®ned 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 zeroux surface serves as the necessary boundary condition for the application of Schwinger's principle of stationary action to de®ne 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 in®nitesimal transformations, the very requirement needed to ensure the applicability of the zero-¯ux surface condition as the de®ning constraint of an open system. The zero-¯ux surfaceThe quantum theory of atoms in molecules [1] is widely employed in the study of the experimentally measured and theoretically determined properties of molecules and crystals, as exempli®ed in a recent review by Spackman describing its application to the analysis of charge densities obtained from X-ray studies [2]. Within this theory, an atom is de®ned as a open system, one that is bounded by a surface S(r s ) of local zero¯ux in the gradient vector ®eld of the electron density q(r), as given in Eq. (1):where n(r) is a unit vector normal to the surface at r. At the meeting on`Chemical Bonding' in La Colle-surLoup France, my talk emphasized that the adoption of the quantum theory of atoms in molecules requires the replacement of the model of structure that imparts an existence to a bond separate from the atoms it links ± the ball and stick model or its orbital equivalents of atomic and overlap contributions ± with the concept of bonding between atoms; two atoms are bonded if they share an interatomic surface and are thus linked by a bond path. It was emphasized that the quantum mechanics of a proper open system not only enables one to de®ne the properties of atoms that are bonded to one another thereby assessing their degree of interaction, but provides, in addition, a characterization of the interaction through the theorems that govern the local behavior of the electron density [3]. This paper takes the opportunity to review and to consider in more detail the topological and mathematical implications of the zero-¯ux surface and its role in establishing the quantum mechanics of an open system, a move prompted in part in response to questions raised at the meeting...

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