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 I_h 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 I_h 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 = τ² 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.