Relating the molecular topology and local geometry: Haddon's pyramidalization angle and the Gaussian curvature

The pyramidalization angle and spherical curvature are well-known quantities used to characterize the local geometry of a molecule and to provide a measure of regio-chemical activity of molecules. In this paper, we give a self-contained presentation of these two concepts and discuss their limitations. These limitations can bypass, thanks to the introduction of the notions of angular defect and discrete Gauss curvature coming from discrete differential geometry. In particular, these quantities can be easily computed for arbitrary molecules, trivalent or not, with bond of equal lengths or not. All these quantities have been implemented. We then compute all these quantities over the Tománek database covering an almost exhaustive list of fullerene molecules. In particular, we discuss the interdependence of the pyramidalization angle with the spherical curvature, angular defect, and hybridization numbers. We also explore the dependence of the pyramidalization angle with respect to some characteristics of the molecule, such as the number of atoms, the group of symmetry, and the geometrical optimization process.