Quantum Topology: Theory of Molecular Structure and its Change

Abstract 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.