Electronic structure, bonding nature, and charge transfer in ${\mathrm{Ba}\mathrm{@}\mathrm{Si}}_{20}$ and ${\mathrm{Si}}_{20}$ clusters: An ab initio study

First-principles electronic structure calculations are carried out for the ${\mathrm{Si}}_{20}$ and endohedral ${\mathrm{Ba}\mathrm{@}\mathrm{Si}}_{20}$ clusters, which appear in the silicon clathrate I and II. We analyze electronic structures, binding energies, charge transfers, and exohedral binding energies of single Si atoms on the ${\mathrm{Ba}\mathrm{@}\mathrm{Si}}_{20}$ clusters. We compare the electronic structures of the ${\mathrm{Ba}\mathrm{@}\mathrm{Si}}_{20}$ cluster with that of the corresponding hollow ${\mathrm{Si}}_{20}$ cluster. We find that, unlike the hollow $I_h$ ${\mathrm{Si}}_{20}$ cluster which undergoes a structural change, the endohedral Ba atom in the ${\mathrm{Ba}\mathrm{@}\mathrm{Si}}_{20}$ cluster stabilizes the cage structure with $C_i$ symmetry slightly distorted from $I_h$ symmetry. The computed geometrical charge transfer is $2.006e$ and the physical one is $0.063e$ for the endohedral Ba atom. The eigenstates of both the clusters can be understood on the basis of a simple model previously used to explain the spherical carbon fullerenes. The charge distribution, corresponded to the highest valence electron state, relates to the ${\mathrm{sp}}^3$ bonding nature of the cage Si atoms of the ${\mathrm{Ba}\mathrm{@}\mathrm{Si}}_{20}$ cluster. The top positions of the cage Si atoms on this cluster is the most unstable among the positions investigated despite the ${\mathrm{sp}}^3$ bonding of the cage atoms; the position on the center of edge is the most stable for the exohedral Si atom; the hollow site at the center of the pentagonal ring is the second. The two exohedral positions, top and $6c$ sites, around the ${\mathrm{Ba}\mathrm{@}\mathrm{Si}}_{20}$ cluster in the clathrate I are the most unstable among the positions. We discuss the mechanism of the growth of the clathrates I.