Energetics of fullerenes with heptagonal rings

The energetic costs of widening the fullerene definition to include carbon cages with heptagonal as well as pentagonal and hexagonal faces are investigated theoretically. Relative energies of all 426 hypothetical C-40 cages that can be assembled from pentagonal, hexagonal and heptagonal faces are calculated within two independent semi-empirical models. All isomers are found to lie in local minima on the potential surface. The QCFF/PI (quantum consistent force field/pi) and DFTB (density functional tight binding) approaches agree in predicting that no cage with one or more heptagons is of lower energy than the best classical C-40 fullerene, but that many such cases are more stable than many C-40 fullerenes. All one-heptagon C-40 cages are predicted to lie within the range of energies spanned by the classical fullerene isomers. Energy penalties of 90-150 kJ mol(-1) per heptagon are suggested by the DFTB calculations, and penalties about half as large again by the QCFF/PI model. The energy variation across the range of fullerenes and pseudo-fullerenes is rationalised by an extension of the isolated-pentagon rule: when heptagons are present, structures of low energy are those that maximise the number of pentagon-heptagon contacts subject to prior minimisation of the number of pentagon-pentagon adjacencies.