8.3 Previous work

Carbon clusters have been extensively studied using electronic structure methods such as tight-binding, Hartree-Fock, density functional, quantum chemical and quantum Monte Carlo methods.[144,145,139,146,43,147] The need for calculations with a sophisticated treatment of electron correlation has been clearly illustrated by several previous studies.

A DFT study of $\mathrm{C}_{20}$ isomers showed that the local density approximation (LDA) [88] and the BLYP gradient corrected functional [148,149] gave different energy orderings for the sheet, bowl and fullerene structures, with neither ordering agreeing with the diffusion quantum Monte Carlo (DMC) ordering.[43] Recent calculations for the monocyclic ring and fullerene of $\mathrm {C}_{24}$ have shown substantial differences between the predictions of the LDA, gradient corrected and hybrid density functionals. [150] Second-order Moller-Plesset (MP2) and coupled cluster calculations at the CCSD(T) level were also performed, and the authors concluded that the fullerene lies lower in energy than the ring. Raghavachari et al. [151] and Martin et al. [152] performed DFT calculations on seven isomers of $\mathrm {C}_{24}$, but obtained conflicting energetic orderings.

Grossman et al. [43] have performed pseudopotential DMC calculations for $\mathrm{C}_{20}$ clusters, finding that the fullerene isomer of $\mathrm{C}_{20}$ is not energetically stable. They also demonstrated in tests on a small series of hydrocarbons, $\mathrm{CH}_{4}$ to $\mathrm{C}_{6}\mathrm{H}_{6}$, that pseudopotential DMC give close to ``chemical accuracy'' of 1-2 kcal mol$^{-1}$. In an earlier study, Greef et al.[113] found similar accuracies for a series of silanes, $\mathrm{SiH}$ to $\mathrm{Si}_{3}\mathrm{H}_{8}$. DMC appears an ideal method for both the study of these molecular systems and for benchmarking other electronic structure methods.

Carbon clusters are challenging to model accurately due to the wide range of geometries and the occurrence of single, double, and triple bonds. These differences result in a non-cancellation of errors in relative energies, exaggerating errors or approximations made in calculations. QMC is therefore ideal for systematically investigating the energetic ordering of different isomers due to the consistent treatment of electron correlation. The approach selected here was to use DFT calculations as a basis for selecting low energy structures and determining geometries, followed by DMC calculations for a more accurate treatment of electron correlation and better ground state total energy.