Figure 1

Structure of $\alpha$-rhombohedral boron: (a) the eight ${\mathrm{B}}_{12}$ icosahedra locate at the vertices of a rhombohedral unit cell with the lattice parameters of $a=4.974\AA$ and $\alpha =58.16°$. The peculiar delocalized three-center bonds are denoted with the red triangles; (b) the polar atoms are labeled with numbers of from 1 to 6, and the equatorial atoms with numbers of from 7 to 12 in ${\mathrm{B}}_{12}$ icosahedron.Reuse & Permissions

Figure 2

Typical color-contour maps of charge densities of chemical bonds in $\alpha$-rhombohedral boron: (a) on the color-contour map in (100) plane of $\alpha$-rhombohedral boron, the bright red ${\mathrm{B}}_p\mathrm{\text{−}}{\mathrm{B}}_p$ bonds (indicated by arrow $\mathrm{u}$) in type-I are strong two-center covalent bonds, their charge density centers located at the middle of their bonds; the dark red ${\mathrm{B}}_p\mathrm{\text{−}}{\mathrm{B}}_p$ bonds (indicated by arrow $\mathrm{w}$) are type-III bonds, their charge density weighted centers in (100) planes moving clearly from the bonding axis towards center $\mathrm{C}$ of icosahedron; (b) the charge densities of the peculiar delocalized three-center ${\mathrm{B}}_e\mathrm{\text{−}}{\mathrm{B}}_e$ bonds in type-II show threefold rotation symmetry because three ${\mathrm{B}}_e\mathrm{\text{−}}{\mathrm{B}}_e$ bond lengths are the same; (c) the charge densities of the bonds in type-III of ${\mathrm{B}}_p\mathrm{\text{−}}{\mathrm{B}}_p\mathrm{\text{−}}{\mathrm{B}}_e$ triangle exhibit the feature of asymmetry; (d) the charge densities of the bonds in type-III of ${\mathrm{B}}_p\mathrm{\text{−}}{\mathrm{B}}_e\mathrm{\text{−}}{\mathrm{B}}_e$ triangle also exhibit the feature of asymmetry.Reuse & Permissions

Figure 3

Plot of $f_g$, $f_z$, $f_c$, and $f_h$ versus $f_i$ values for zinc blende structure. Note that the $f_h/f_h^{1.36}$ ratio for the chosen 20 materials lies on (or close to) the line of slope 1.Reuse & Permissions