$ab$ plane. Although the $8i$ and $8h$ sites have short bond lengths (1.62 and 1.73 Å, respectively), there are only two of these short bonds, and the overall bonding is weaker than of $4c$-site atoms.

To summarize, the IS of $\alpha$-T boron are not defect states. Their role is not just to fill space between the icosahedra, but it is a more active one, i.e., the enhancement of the host structure by forming tetrahedral bonds. The closer the sites are to the perfect tetrahedral coordination, the stronger is the bonding.

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### Comparison with C- or N-containing $\alpha$-tetragonal boron {#sec:impurity-tetra}

Although the positions of $8h$ sites $(0,1/2,u)$ and $8i$ sites $(u,0,1/2)$ are distinct from $4c$ sites $(0,1/2,0)$ by symmetry, they can be continuously connected to the $4c$ site by varying $u$. However, the bonding environments of $8h$ and $8i$ are very different from that of $4c$. The coordination of the former is two-fold, while the latter is four-fold. For pure $\alpha$-T boron, $\Delta E_{f}$ of the $8i$ and $8h$ sites is bigger than that of the $4c$ sites by 0.03 and 0.11 eV, respectively. On the other hand, for C- or N-containing $\alpha$-T boron, the energy difference between these sites 