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

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 C_60m² for m of up to 50 or C₁₅₀₀₀₀ as well as giant fullerenes in the series C_180m² and C₇₀(D_5h). 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 C_60m² 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.

Combinatorics of Edge Symmetry: Chiral and Achiral Edge Colorings of Icosahedral Giant Fullerenes: C80, C180, and C240

We develop the combinatorics of edge symmetry and edge colorings under the action of the edge group for icosahedral giant fullerenes from C80 to C240. We use computational symmetry techniques that employ Sheehan’s modification of Pόlya’s theorem and the Möbius inversion method together with generalized character cycle indices. These techniques are applied to generate edge group symmetry comprised of induced edge permutations and thus colorings of giant fullerenes under the edge symmetry action for all irreducible representations. We primarily consider high-symmetry icosahedral fullerenes such as C80 with a chamfered dodecahedron structure, icosahedral C180, and C240 with a chamfered truncated icosahedron geometry. These symmetry-based combinatorial techniques enumerate both achiral and chiral edge colorings of such giant fullerenes with or without constraints. Our computed results show that there are several equivalence classes of edge colorings for giant fullerenes, most of which are chiral. The techniques can be applied to superaromaticity, sextet polynomials, the rapid computation of conjugated circuits and resonance energies, chirality measures, etc., through the enumeration of equivalence classes of edge colorings.

The role of aromaticity in determining the molecular structure and reactivity of (endohedral metallo)fullerenes

The molecular structure and chemical reactivity of endohedral metallofullerenes can be greatly predicted and rationalized by their local and global aromaticity.

Aromaticity and kinetic stability of fullerene C₃₆ isomers and their molecular ions

The aromatic character of fullerene C(36) isomers was examined by the Hess-Schaad resonance energy (HSRE), topological resonance energy (TRE) and the percentage topological resonance energy (%TRE) models. According to the nucleus-independent chemical shift (NICS) at the cage center, C(36) fullerene isomers must be highly aromatic with negative values. However, they are predicted to be antiaromatic with negative HSREs and negative TREs. The TRE method revealed that they all are aromatized by acquiring two or more electrons. NICSs at the cage center and the 2(N + 1)(2) rule cannot be used as an indicator of the aromatic stabilization for C(36) isomers and their molecular ions. We utilized the bond resonance energy (BRE) model to estimate the kinetic stability of C(36) isomers and their molecular ions. C(36) isomers are only stabilized kinetically in penta- and hexavalent molecular anions. All the results indicate that aromaticity and kinetic stability are closely related to the cyclic motion of π electrons.

Increasing Stability of the Fullerenes with the Number of Carbon Atoms: The Experimental Evidence

The values of the molar standard enthalpies of formation, Delta(f)H(o)(m)(C(76), cr) = (2705.6 +/- 37.7) kJ x mol(-1), Delta(f)H(o)(m)(C(78), cr) = (2766.5 +/- 36.7) kJ x mol(-1), and Delta(f)H(o)(m)(C(84), cr) = (2826.6 +/- 42.6) kJ x mol(-1), were determined from the energies of combustion, measured by microcombustion calorimetry on a high-purity sample of the D(2) isomer of fullerene C(76), as well as on a mixture of the two most abundant constitutional isomers of C(78) (C(2nu)-C(78) and D(3)-C(78)) and C(84) (D(2)-C(84), and D(2d)-C(84). These values, combined with the published data on the enthalpies of sublimation of each cluster, lead to the gas-phase enthalpies of formation, Delta(f)H(o)(m)(C(76), g) = (2911.6 +/- 37.9) kJ x mol(-1); Delta(f)H(o)(m)(C(78), g) = (2979.3 +/- 37.2) kJ x mol(-1), and Delta(f)H(o)(m)(C(84), (g)) = (3051.6 +/- 43.0) kJ x mol(-1), results that were found to compare well with those reported from density functional theory calculations. Values of enthalpies of atomization, strain energies, and the average C-C bond energy were also derived for each fullerene. A decreasing trend in the gas-phase enthalpy of formation and strain energy per carbon atom as the size of the cluster increases is found. This is the first experimental evidence that these fullerenes become more stable as they become larger. The derived experimental average C-C bond energy E(C-C) = 461.04 kJ x mol(-1) for fullerenes is close to the average bond energy E(C-C) = 462.8 kJ x mol(-1) for polycyclic aromatic hydrocarbons (PAHs).

Photostability via a Sloped Conical Intersection: A CASSCF and RASSCF Study of Pyracylene

An extensive ab initio study of the ground- and excited-state potential energy surfaces of pyracylene is presented in this work. CASSCF calculations show that there is an accessible sloped S0/S1 conical intersection, which leads to ultrafast internal conversion and explains the observed photostability. RASSCF calculations (using a well-defined subset of the CASSCF configurations) are shown to be able to reproduce CASSCF results satisfactorily and will therefore be useful for larger systems where CASSCF is currently too expensive. MRCI and MRPT2 energy corrections are computed to assess the ionic character of the excited states. Finally, MMVB calculations are also benchmarked against CASSCF, to assess the reliability of this parametrized method for excited states of large conjugated polycyclic aromatic hydrocarbons.

The evolution of the football structure for the C 60 molecule: a retrospective

By chance in 1970, we conjectured the possibility of the football-shaped C 60 molecule, now known as buckminsterfullerene, while considering superaromatic molecules having three-dimensional π-electron delocalization. A translation of the original description, initially written in Japanese, is given. The processes leading to scientific discoveries are analysed in the light of our missed opportunity.

Chemistry of the fullerenes: the manifestation of strain in a class of continuous aromatic molecules

Within the wr-orbital axis vector theory, the total rehybridization required for closure of the fullerenes is approximately conserved. This result allows the development of a structure-based index of strain in the fullerenes, and it is estimated that about 80 percent of the heat of formation of the carbon atoms in C60 may be attributed to a combination of v strain and steric inhibition of resonance. Application of this analysis to the geometries of structurally characterized organometallic derivatives of C60 and C70 shows that the reactivity exhibited by the fullerenes may be attributed to the relief of a combination of local and global strain energy. C60 is of ambiguous aromatic character with anomalous magnetic properties but with the reactivity of a continuous aromatic molecule, moderated only by the tremendous strain inherent in the spheroidal structure.