Energies of sp(2) carbon shapes with pentagonal disclinations and elasticity theory

Energies of a certain class of fullerene molecules (elongated, contracted and regular icosahedral fullerenes) are numerically calculated using a microscopic description of carbon-carbon bonding. It is shown how these results can be interpreted and comprehended using the theory of elasticity that describes bending of a graphene plane. Detailed studies of a wide variety of structures constructed by application of the same general principle are performed, and analytical expressions for energies of such structures are derived. Comparison of numerical results with the predictions of a simple implementation of elasticity theory confirms the usefulness of the latter approach.