Combinatorics of Supergiant Fullerenes: Enumeration of Polysubstituted Isomers, Chirality, Nuclear Magnetic Resonance, Electron Spin Resonance Patterns, and Vibrational Modes from C70 to C150000.

Abstract

We have developed combinatorial techniques for the enumeration of isomers of polysubstituted giant fullerenes through icosahedral C150000 and applied the techniques to chirality of the isomers, NMR spectroscopy, and group theoretical analysis of the vibrational modes of supergiant fullerenes. We have employed a combination of distance-degree vectorial sequences, self-returning walk sequences followed by our generalization of Sheehan's version of Pólya's theorem, and Möbius inversion technique extended to all irreducible representations of the point groups of giant fullerenes. The concept of shell equivalence classes was utilized to analyze supergiant fullerenes. We have applied these techniques to golden fullerenes in the series C60m2 for m of up to 50 or C150000 as well as giant fullerenes in the series C180m2 and C70(D5h). We have employed computational and combinatorial tools to enumerate both chiral and achiral isomers of substituted and hetero giant fullerenes as well as NMR-generating functions for the giant fullerenes. The techniques also provide efficient tools to enumerate all of the vibrational modes of giant fullerenes in terms of the shell partitions. General combinatorial formulae are obtained for larger polysubstituted golden fullerenes of the series C60m2 for any m, and thus the techniques are applied to larger fullerenes such as C150000. New insights into chirality measures, NMR, ESR hyperfine structures, and vibrational modes of supergiant fullerenes are provided using the novel combinatorial techniques.