Structural and electronic properties of the fullerene isomers of ${\mathrm{Si}}_{38}$: A systematic theoretical study

A systematic study of the structural and electronic properties of the 17 ${\mathrm{Si}}_{38}$ fullerene cage isomers, which are constructed by making all possible permutations among their pentagons and hexagons, is presented. For the full optimization of these structures, a tight-binding molecular-dynamics method is firstly applied, and the resulting structures were further optimized with two more accurate but more time-consuming methods, namely, the generalized tight-binding molecular dynamics and a density-functional theory calculation at the B3LYP level. In addition, the ${\mathrm{Si}}_{20}$ fullerene cage, optimized with the same methods, is also presented for comparison. The results of all these methods are in good agreement with each other, and they all predict the same isomer as the energetically most stable structure among these 17 isomers. In all the fullerene isomers, half of the atoms are nearly coplanar with their three neighbors, and half lie on sharp corners of the polyhedral cage, coinciding with a distinction between half of the atoms adopting an $s{p}^2$-like hybridization and the other half using essentially pure $p$ orbitals in their bonding to nearest neighbors.

Figure 1

(Color online) The optimum structures of the 17 isomers of ${\mathrm{Si}}_{38}$ and the optimum structure of ${\mathrm{Si}}_{20}$ obtained with the three methods.

Figure 2

(Color online) Distribution of bond lengths for the 17 optimized ${\mathrm{Si}}_{38}$ fullerene isomers (solid lines) and for the ${\mathrm{Si}}_{20}$ fullerene (dashed) computed with the three methods (see text for details).

Figure 3

(Color online) Cohesive energies of the 17 ${\mathrm{Si}}_{38}$ isomers found with (a) DFT/B3LYP (solid line) (b) GTBMD (dashed line). The fused-pentagon-quadruple motif is shown in the inset (p, pentagon; h, hexagon).

Figure 4

(Color online) Distribution of the (a) pentagon and (b) hexagon angles for the (I) DFT/B3LYP (solid line), (II) GTBMD (dotted line), and (III) OTBMD (dot-dashed line) calculations. (c) The pentagon (solid line) and the hexagon (dotted line) angle distributions for the DFT calculation, shifting the former by $+12°$ and multiplying the latter by $60∕54$. These distributions have been calculated for all the 17 ${\mathrm{Si}}_{38}$ fullerene isomers.

Figure 5

(Color online) Distribution of the angle sums associated with every atom, for all the atoms of the 17 ${\mathrm{Si}}_{38}$ fullerene isomers, obtained with (a) DFT/B3LYP (solid line) and (b) GTBMD (dotted line).

Figure 6

(Color online) Distribution versus $n$ of (a) the three hybridized bond orbitals for near-planar atoms (solid line), (b) the dangling orbital for the same atoms (dotted line), (c) the bond orbitals for the atoms at sharp corners (dashed line), and (d) the dangling orbitals for the same atoms (dot-dashed line), for the 17 ${\mathrm{Si}}_{38}$ isomers (OTBMD method).