8.5 Candidate geometries

For clusters containing between 20 and 32 atoms, three types of structural isomer are energetically competitive: fullerenes, planar or near-planar sheets and bowls, and monocyclic rings. Five isomers of $\mathrm {C}_{24}$, three isomers of $\mathrm {C}_{26}$, three isomers of $\mathrm {C}_{28}$, and two isomers of $\mathrm {C}_{32}$ were studied. These are illustrated in figures 8.3, 8.4, 8.5,8.6 respectively.

The following isomers of $\mathrm {C}_{24}$, depicted in figure 8.3, were considered: a monocyclic ring, a flat graphitic sheet, a bowl-shaped structure with one pentagon, and a caged structure with a mixture of square, pentagonal and hexagonal faces, and a fullerene. Other candidate structures, such as bicyclic rings and a 3-pentagon bowl were excluded on the grounds that DFT calculations using several different density functionals have shown them to be significantly higher in energy. [151,152]

Figure 8.3: Structural isomers of $\mathrm {C}_{24}$: (a) ring, (b) $\protect\mathrm{O}_{h}$ symmetry cage, (c) $\protect\mathrm{D}_{6}$ symmetry fullerene, (d,e) (side, front) one-pentagon bowl, (f) graphitic sheet.

Three isomers of $\mathrm {C}_{26}$ were studied. A previous density functional study [155] considered 10 isomers, but the effects of basis set incompleteness are apparent and as only two density functionals were used, the low energy ordering of the structures is difficult to ascribe. Structures in the study included 3 graphitic sheet-like structures, a monocyclic ``cumulenic''^8.1 ring, a cage and a fullerene. The fullerene, ring and one of the sheets were selected for DMC calculations due to their apparent low energy. The fullerene is significantly distorted and has only $\mathrm{C}_{2v}$ symmetry. The sheet resembles the purely graphitic $\mathrm {C}_{24}$ sheet but for the addition of a pentagon.

Figure 8.4: Structural isomers of $\mathrm {C}_{26}$: (a) ring, (b) one-pentagon sheet, (c,d) (side, front) $\mathrm{C}_{2v}$ symmetry cage.

Three structures of $\mathrm {C}_{28}$ were investigated: a monocyclic ring, a graphitic sheet and a fullerene of $\mathrm{T}_{d}$ symmetry (figure 8.5). Other bowl and sheet-like structures [156] were excluded on energetic grounds.

Figure 8.5: Structural isomers of $\mathrm {C}_{28}$: (a) ring, (b) sheet, (c) $\mathrm{T}_{d}$ symmetry fullerene.

Two $\mathrm {C}_{32}$ structures were examined: a monocyclic ring and a fullerene (figure 8.6). HF calculations suggest that the fullerene form of $\mathrm {C}_{32}$ is energetically favourable, [146] demonstrating the reduced role of electron correlation in determining energetic orderings with increased cluster size. These structures were included to provide additional information about trends in binding energies.

Figure 8.6: Structural isomers of $\mathrm {C}_{32}$: (a) ring, (b) fullerene.