Alternative approaches to onion-like icosahedral fullerenes

The fullerenes of the C60 series (C60, C240, C540, C960, C1500, C2160 etc.) form onion-like shells with icosahedral Ih symmetry. Up to C2160, their geometry has been optimized by Dunlap & Zope from computations according to the analytic density-functional theory and shown by Wardman to obey structural constraints derived from an affine-extended Ih group. In this paper, these approaches are compared with models based on crystallographic scaling transformations. To start with, it is shown that the 56 symmetry-inequivalent computed carbon positions, approximated by the corresponding ones in the models, are mutually related by crystallographic scalings. This result is consistent with Wardman's remark that the affine-extension approach simultaneously models different shells of a carbon onion. From the regularities observed in the fullerene models derived from scaling, an icosahedral infinite C60 onion molecule is defined, with shells consisting of all successive fullerenes of the C60 series. The structural relations between the C60 onion and graphite lead to a one-parameter model with the same Euclidean symmetry P63mc as graphite and having a c/a = τ(2) ratio, where τ = 1.618… is the golden number. This ratio approximates (up to a 4% discrepancy) the value observed in graphite. A number of tables and figures illustrate successive steps of the present investigation.