A theory of molecular structure is presented. The theory demonstrates that the concepts of atoms and bonds may be rigorously defined and given physical expression in terms of the topological properties of the observable distribution of charge for a molecular system. As a consequence of these definitions, one in turn obtains a definition of structure and a predictive theory of structural stability. The theory is linked to quantum mechanics by demonstrating that the atoms so defined represent a class of open quantum subsystems with a unique set of variationally defined properties.

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This article describes an algorithm for the calculation of the average properties of an atom in a molecule. The atom is defined within the topological theory of molecular structure, a theory which defines atoms, bonds, structure, and structural stability in terms of the topological properties of a system's charge distribution. The average properties of the atom so defined are uniquely determined by quantum mechanics. Results for a number of hydrocarbon molecules, obtained by the program PROAIM (properties of atoms in molecules) which implements this algorithm, are given. In general, this program enables one to calculate the average energy of an atom in a molecule to an accuracy of fl kcal/mol.

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, per), 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 Rene Thorn'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 per) via the associated gradient vector field yo per), reduces to the identification of the critical points in per). Two types of critical points assume special roles in the analysis. A (3, -3) critical point, a maximum in p(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, -I) 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 andlor arrangement of bond paths. The making andlor 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 per), 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 Hj and H 2 0) is provided by the unfolding of a particular type of catastrophe, the elliptic umbilic.

In this paper we review and exemplify a new and rigorous approach to the problem of molecular structure and its morphogenesis: the theory of quantum topology. The basis for this approach is provided by the topology of the total charge density in a given molecular system. The essential observation is that the only local maxima of a ground state distribution occur at the positions of the nuclei. The nuclei are therefore identified as point attractors of the gradient vector field of the charge density. The associated basins partition the molecular system into atomic fragments. Each atom is a stable structural unit defined as the union of an attractor and its basin. The common boundary of two neighbouring atomic fragments, the interatomic surface, contains a particular critical point, which generates a pair of gradient paths linking the two neighbouring attractors. The union of this pair of gradient paths and their endpoints is called a bond path. The network of bond paths defines a molecular graph of the system. Having defined a unique molecular graph for any molecular geometry, the total configuration space is partitioned into a finite number of regions. Each region is associated with a particular structure defined as an equivalence class of molecular graphs. A chemical reaction in which chemical bonds are broken and/or formed is therefore a trajectory in configuration space which must cross one of the boundaries between two neighbouring structural regions. These boundaries form the catastrophe set of the system which, like a phase diagram in thermodynamics, denotes the points of “balance” between neighbouring structures. A general analysis of the structural changes in an ABC type system is given in detail together with specific examples of all possible structural elements in a molecular system. The properties of the topologically defined atoms and their temporal changes are identified within a general formulation of subspace quantum mechanics. It is shown that the quantum mechanical partitioning of a system into subsystems coincides with the topological partitioning: both are defined by the same set of “zero flux” surfaces. Consequently the total energy, or any other property, is partitioned into additive atomic contributions. We show that, in general, a definite structure can be assigned to a given molecular system. Quantum mechanically this structure is associated with an open neighbourhood of the most probable nuclear geometry. Finally we generalize the notion of molecular structure to non‐isolated molecules and, in contrast to recent work by Woolley, we conclude that molecular structure exists in spite of intermolecular interactions and not as a result of them.