The molecular structure hypothesis - that a molecule is a collection of atoms linked by a network of bonds - was forged in the crucible of nineteenth century experimental chemistry. It has continued to serve as the principal means of ordering and classifying the observations of chemistry. The difficulty with this hypothesis was that it was not related directly to quantum mechanics, the physics which governs the motions of the nuclei and electrons that make up the atoms and the bonds. Indeed there was, and with some there still is, a prevailing opinion that these fundamental concepts, while unquestionably useful, were beyond theoretical definition. We have in chemistry an understanding based on a classification scheme that is both powerful and at the same time, because of its empirical nature, limited.

Richard Feynman and Julian Schwinger have given us a reformulation of physics that enables one to pose and answer the questions "what is an atom in a molecule and how does one predict its properties?" These questions were posed in my laboratory where it was demonstrated that this new formulation of physics, when applied to the observed topology of the distribution of electronic charge in real space, yields a unique partitioning of some total system into a set of bounded spatial regions. The form and properties of the groups so defined faithfully recover the characteristics ascribed to the atoms and functional groups of chemistry. By establishing this association, the molecular structure hypothesis is freed from its empirical constraints and the full predictive power of quantum mechanics can be incorporated into the resulting theory - a theory of atoms in molecules and crystals.

The theory recovers the central operational concepts of the molecular structure hypothesis, that of a functional grouping of atoms with an additive and characteristic set of properties, together with a definition of the bonds that link the atoms and impart the structure. Not only does the theory thereby quantify and provide the physical understanding of the existing concepts of chemistry, it makes possible new applications of theory. These new applications will eventually enable one to perform on a computer, in a manner directly paralleling experiment, everything that can now be done in the laboratory, but more quickly and more efficiently, by linking together the functional groups of theory. These applications include the design and synthesis of new molecules and new materials with specific desirable properties.

The theory of atoms in molecules enables one to take advantage of the single most important observation of chemistry, that of a functional group with a characteristic set of properties. This document outlines and illustrates the topological basis of the theory and its relation to the quantum mechanics of an open system.