8.1 Introduction

In this chapter, the energetic stability of isomers of $\mathrm {C}_{24}$, $\mathrm {C}_{26}$, $\mathrm {C}_{28}$ and $\mathrm {C}_{32}$ carbon clusters is studied using density functional and quantum Monte Carlo methods, with the aim of identifying the smallest energetically stable fullerene. Calculating the energy of small carbon clusters is a strong test of electronic structure methods due to their widely differing geometries and the energetic importance of electron correlation. Recent experimental results comparing the relative abundance of small carbon clusters, including fullerenes, demonstrate a sensitive dependence on experimental conditions and many experimental results are in disagreement with previous electronic structure calculations. By using diffusion Monte Carlo, electron correlation is treated consistently, and accurate relative energies are obtained for systems of up to 128 valence electrons. The results also bear on the existence of fullerene cluster based solids, which at the time of writing have been the subject of considerable theoretical speculation but have not been synthesised in quantity.