Linear relationship between the bond lengths and the Pauling bond orders in fullerene molecules

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96-145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website.

Bond-order discrimination by atomic force microscopy

We show that the different bond orders of individual carbon-carbon bonds in polycyclic aromatic hydrocarbons and fullerenes can be distinguished by noncontact atomic force microscopy (AFM) with a carbon monoxide (CO)-functionalized tip. We found two different contrast mechanisms, which were corroborated by density functional theory calculations: The greater electron density in bonds of higher bond order led to a stronger Pauli repulsion, which enhanced the brightness of these bonds in high-resolution AFM images. The apparent bond length in the AFM images decreased with increasing bond order because of tilting of the CO molecule at the tip apex.

Predicting bond lengths in planar benzenoid polycyclic aromatic hydrocarbons: a chemometric approach

Two hundred and twenty-three aromatic carbon-carbon bond lengths in high precision crystal structures containing 22 planar condensed benzenoid polycyclic aromatic hydrocarbons (PB-PAHs) were related to the Pauling pi-bond order, its analogue corrected to crystal packing effects, the number of hexagonal rings around the bond, and the numbers of carbons atoms around the bond at topological distance one and two. Principal Component Analysis (PCA) showed that the bond lengths in PB-PAHs are at least two-dimensional phenomenon, with well pronounced classification into 12 types of bonds, as confirmed with Hierachical Cluster Analysis (HCA). Consequently, Multiple Linear Regression (MLR) and Partial Least Squares (PLS) models were superior to univariate models, reducing the degeneration of the data set and improving the estimation of Julg's structural aromaticity index. The approximate regression models based on topological descriptors only were built for fast and easy prediction of bond lengths and bond orders in PB-PAHs.