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 Poincare–Hopf relation for molecules and clusters and the Euler–Hopf relation in solids. Thom's theory of elementary catastrophes combined with the Poincare–Hopf relation provides the inspiration for the new quantum topology. An alternative use of the Poincare–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.