Systematic study of structures and stabilities of fullerenes

We describe a scheme to generate the ground-state network for carbon clusters. Combining this scheme with an accurate tight-binding potential for carbon, we have systematically studied the ground-state structures of fullerenes from ${\mathrm{C}}_{20}$ to ${\mathrm{C}}_{94}$. Clusters of sizes 60, 70, and 84 are found to be energetically more stable than their neighbors. The occurrence of the more abundant fullerene sizes is related to the fragmentation stability and chemical reactivity of the clusters obtained from our calculations.