8.8 Results and analysis

8.8.1 $\mathrm {C}_{24}$

The results shown in figure 8.7 confirm that the treatment of correlation has a profound effect on the relative energies. All of the density functionals give different orderings of the energies, and none gives the same ordering as DMC. The graphitic sheet is placed lowest in energy by DMC, in agreement with each of the density functionals except BLYP, which places the ring lowest in energy. The low energy of the $\mathrm {C}_{24}$ graphitic sheet is expected because the structure accommodates a large number (7) of hexagonal rings without significant strain. This structure is expected to be the smallest stable graphitic fragment, as for smaller clusters ring structures should be lower in energy. Neither DMC or the density functionals find the $\mathrm {C}_{24}$ fullerene to be energetically stable.

Figure 8.7: Energies of the $\mathrm {C}_{24}$ isomers relative to the $\mathrm{D}_6$ fullerene.

8.8.2 $\mathrm {C}_{26}$

The DMC and DFT results shown in figure 8.8 for $\mathrm {C}_{26}$ display many of the features established by the $\mathrm {C}_{24}$ results. A similar degree of scatter between the different methods is evident. Despite the large strains apparent in the fullerene structure, the fullerene is placed lowest in energy by the DMC and the LDA calculations. The BLYP functional favours the low coordination ring placing it low in energy, and the gradient corrected functionals behave as in $\mathrm {C}_{24}$.

The multi-determinant wavefunction gives a slightly lower DMC energy than the single determinant wavefunction, confirming that CI wavefunctions have better nodal surfaces than HF wavefunctions. However, the ring and sheet-like isomers remain very close in energy, approximately $2.5$ eV above the fullerene.

Figure 8.8: Energies of the $\mathrm {C}_{26}$ isomers relative to the fullerene.

8.8.3 $\mathrm {C}_{28}$

The DMC results for $\mathrm {C}_{28}$, figure 8.9, again highlight a wide variation in the performance of different DFT functionals. The LDA and B3LYP functionals predict the same ordering as DMC, but the PBE and BLYP functionals predict different orderings.

Spin-polarized DFT calculations show the ground state of the $\mathrm{T}_{d}$ symmetry fullerene to be a spin-polarized $^5A_2$ state. DMC predicts that this spin-polarized fullerene is the lowest energy isomer of $\mathrm {C}_{28}$, and this is supported by each of the density functionals except BLYP.

Figure 8.9: Energies of the $\mathrm {C}_{28}$ isomers relative to the spin-polarized fullerene.

The spin-polarized fullerene has four unpaired electrons and is therefore reactive. This property has been exploited in atom trapping experiments in which $\mathrm {C}_{28}$ fullerenes containing a single atom have been prepared by laser vaporization of a graphite-MO$_2$ (M = Ti, Zr, Hf or U) composite rod. [162] The prediction that the fullerene is the most stable isomer of $\mathrm {C}_{28}$ indicates that isolated fullerenes might be produced, thereby facilitating the investigation of possible $\mathrm {C}_{28}$ fullerene solids, which have been discussed but not yet produced. [162,163] (A $\mathrm{C}_{36}$ fullerene solid has recently been reported. [164])

The DMC data strongly indicates that, of those considered, the $\mathrm{T}_{d}$ fullerene is the most stable $\mathrm {C}_{28}$ isomer. The $\mathrm{T}_{d}$ fullerene has the lowest DMC energy in both spin-polarized and non spin-polarized states, and is substantially more stable than the sheet and ring. DMC calculations finds the spin-polarized state to be 2.08(20) eV more stable than the unpolarized state, which is 1.10(26)eV more stable than the sheet. Small changes in the geometries are therefore unlikely to change the energetic ordering.

8.8.4 $\mathrm {C}_{32}$

DMC calculations for a monocyclic ring and fullerene of $\mathrm {C}_{32}$ showed that the fullerene is 8.4(4) eV per molecule lower in energy. This is consistent with the observation of a large abundance of $\mathrm {C}_{32}$ fullerenes in recent experiments. [143]

8.8.5 Ring and fullerene binding energies

In figure 8.10 the DMC binding energies of the ring and fullerene structures for $\mathrm {C}_{24}$, $\mathrm {C}_{26}$, $\mathrm {C}_{28}$ and $\mathrm {C}_{32}$ are shown. The binding energy of the fullerenes rises much more rapidly with cluster size than for the rings because of the large amount of strain energy in the smaller fullerenes. The DMC binding energy per atom of the $\mathrm {C}_{32}$ fullerene is $\approx 1$ eV per atom smaller than the experimental binding energy of $\mathrm {C}_{60}$.

Figure 8.10: DMC binding energies of the ring ($\circ$) and fullerene ($\Box$) isomers in eV per atom. The lines are drawn for guidance only.