8.9 Discussion

The QMC and DFT results demonstrate the energetic importance of electron correlation. The very wide variation in energy amongst DFT methods gives a confusing and unclear picture of cluster stability. The QMC results, which we believe to be very accurate, indicate that the $\mathrm {C}_{26}$ and $\mathrm {C}_{28}$ fullerenes should be stable. This is close to the currently quoted range of experimentally determined fullerene stability, around $\mathrm{C}_{30}$ and higher.

Comparing the QMC results with those of the various DFT functionals, several important trends in the relative performance of the different functionals are highlighted. The overall quality of a functional for $\mathrm {C}_{24}$, $\mathrm {C}_{26}$, and $\mathrm {C}_{28}$ clusters is best judged by the agreement with the DMC data for the overall shapes of the relative energy data of figures 8.7 and 8.9. The best agreement is given by the PBE and B3LYP functionals, with the LDA being slightly inferior and the BLYP functional being worst. The tendency of the BLYP functional to favour structures with lower average coordination number and the tendency of the LDA to favour structures with higher average coordination numbers is consistent with the results on $\mathrm{C}_{20}$ reported by Grossman et al. [43]

The LDA, perhaps surprisingly, performs at least as well as the gradient corrected functionals. The LDA appears to favour systems with high average densities, therefore placing the rings anomalously high in energy. The gradient corrected functionals, which normally would be expected to improve the ordering do not consistently reduce the differences between the DFT and DMC results; for $\mathrm {C}_{24}$ the rms deviation for the PBE functional is larger than for the LDA and the overall spread in energies is increased from 3.04 to 3.41 eV. The BLYP functional performs poorly, consistently placing the structures of low average coordination substantially lower in energy relative to DMC and other DFT values. The B3LYP functional, which includes an exact exchange component, performs most consistently, and also gives the same increase in stability of $\mathrm {C}_{28}$ as DMC when spin-polarization is included. The PBE and B3LYP density functionals give the best description of the relative energies of the isomers, while the BLYP functional gives the poorest.

The DMC binding energies show a clear increase in stability with cluster size for both ring and fullerene isomers. The cumulenic $\mathrm {C}_{26}$ ring is found to be more weakly bound per atom than the adjacent polyacetylenic $\mathrm {C}_{24}$ and $\mathrm {C}_{28}$ rings, in agreement with early semi-empirical calculations of $\mathrm{C}_{n}$ rings. [165] However, the difference in binding energies of the fullerene and rings (either cumulenic or polyacetylenic) increases nearly linearly with cluster size.

The final test of the DMC predictions must lie with experiment. It is clear that the actual abundances of different clusters depend sensitively on the precise experimental conditions. The stability of clusters against fragmentation, growth and other chemical reactions is clearly a very complicated subject. One issue is that the clusters are formed at temperatures of order 10$^3$ K and therefore the vibrational contributions to the free energy may be significant. A relatively simple picture emerges from computations of vibrational properties [151,152,156]). Fullerenes are relatively rigid and have smaller vibrational free energies than rings, which have many low-lying vibrational modes. Consequently, the ring isomers are favoured relative to fullerenes at high temperatures. If thermodynamic stability alone were to determine which cluster sizes were observed then only the larger fullerenes would ever be observed, but in a recent experiment the abundance of the $\mathrm {C}_{32}$ fullerene was found to be greater than $\mathrm {C}_{60}$.[143] There is more evidence that thermodynamic stability to rearrangements of clusters of a particular size are important in determining which isomers are observed. For example, in the experimental study of Ref. [143], fullerenes were mostly observed for clusters containing more than about 30 carbon atoms, while for smaller clusters mostly rings were formed. This behaviour closely matches the critical size for fullerene stability predicted by our DMC calculations.

There have been several proposals that cluster solids could be synthesized by surface deposition of fullerenes. [162,163] Our DMC results support these proposals: the $\mathrm {C}_{28}$ fullerene is predicted to be stable, yet it is likely to readily polymerise due to its geometry and $\mathrm{T}_{d}$ spin-polarised ground-state. However, unless sufficiently high abundances of the fullerene are produced, its high reactivity could also hinder production of such solids, as the fullerene is likely to react with other clusters of different size or with any impurities present. The $\mathrm {C}_{26}$ fullerene, although predicted to be stable, appears less likely to polymerise due to its strained structure and less reactive ground state.