Embedding and characterization of quantum chemical reaction graphs on two-dimensional orientable surfaces

In chemistry it has been a long established practice to represent molecules by graphs, with the constituent atoms and the interconnecting bonds taking the roles of vertices and edges, respectively. Whereas such representations contain a great amount of chemical information, and are of considerable value, nonetheless, they are in general insufficient to account for energy differences between various chemical species, and for the energies of activation required for a chemical reaction. Information on the energy variation of individual molecules during a chemical reaction is fundamentally important for a reliable prediction of favoured reaction mechanisms. In the last decade major advances have been made in the quantum chemical calculation of the energies of molecules, and now it is possible to evaluate the energy content of medium size molecules in any nuclear configuration occuring during a chemical change. Our purpose is to discuss a graph-theoretical representation of the energy function of molecules. With the recent spectacular advances in computer technology the quantum chemical calculation of molecular energy as the function of atomic arrangements is rapidly becoming a routine task, and the proposed graph-theoretical framework is expected to reveal some of the regularities and general trends within families of chemical reactions. Graph theory is a powerful mathematical discipline for a concise description of the most essential interrelations among entities from which a given physical system is composed [for selected applications in the field of the natural sciences, see, e.g.,