Fullerenes are a remarkable class of molecules, as their bond graph structure is captured by simple and beautiful mathematics that is rare in nature, and from which derives many of their properties: from topology, spatial shapes, and indicators of their chemical behaviours. Because of this, their graph theory has been studied in detail over several decades by many authors.
However, their properties in pure graph theory only scratch the surface. Many more beautiful properties of fullerenes - and similar low-dimensional systems - can be derived from relations to discrete manifolds and differential geometry; and even those accessible through the graph theory can be studied in much greater detail using these new tools.
This project introduces new formalisms that form a bridge between chemical graphs methods and full quantum chemistry. This is achieved through geometrical structures, which I have named carbon manifolds, that arise naturally from the mathematics of fullerenes and other graphene-like carbon surfaces. These will be leveraged to devise algorithms and build software enabling the study of electronic wave equations directly on the manifold geometries, radically more efficient than solving full wave equations. The goal is predictive approximation methods millions of times faster than current state of the art ab initio calculations, facilitating systematic screening of entire isomer spaces to identify structures of interest. The same geometric formalism will be leveraged to aid discovery of direct synthesis paths of specific fullerene isomers, a presently unsolved problem. Generalizations are proposed to extend the methods beyond fullerenes to more complex carbon allotropes, as well as low-dimensional structures containing other atoms. The theory is accompanied by software prototypes that I have developed, which I am in the process of adding to the present web site.
James Avery (avery@nbi.ku.dk)