Quantum topology phase diagrams for molecules, clusters, and solids

The need to make more quantitative use of the total electronic charge density distribution is demonstrated in this short perspective. This is framed in the perspective of the ground breaking early work of Bader and coworkers, along with mathematicians who captured the essential nature of a molecule in a suitably compact form in real space. We see that this simple form is the Poincaré–Hopf relation for molecules and clusters and the Euler–Hopf relation in solids. Thom's theory of elementary catastrophes combined with the Poincaré–Hopf relation provides the inspiration for the new quantum topology. An alternative use of the Poincaré–Hopf relation, molecular recognition, is discussed. Quantum topology is then used to create a topology phase diagram for both molecules and solids. The author adds their perspectives of the huge potential of the quantum topology approach by demonstrating the ease with which new theoretical ideas can be generated. © 2013 Wiley Periodicals, Inc. The promising early beginnings of the theoretical development of the Poincaré–Hopf relation, abandoned in the early 1980s, have inspired the recent creation of quantum topology phase diagrams to predict new isomer topologies. Alternative uses of the Poincaré–Hopf and Euler–Poincaré relations, for example, for molecular recognition and phase transitions in solids, respectively, are discussed. A perspective is given and it is demonstrated that within the quantum topology framework, new theory can be created to link with traditional chemical ideas.