the DOS of ${\rm B_{50}C_{2}}$ and ${\rm B_{50}N_{2}}$ (see the lower panel of Fig. \[fig:POSfilling\]). For ${\rm B_{50}N_{2}}$, 10 valence electrons from ${\rm N_{2}}$ completely fill the unoccupied states of ${\rm B_{50}}$. Therefore, ${\rm B_{50}N_{2}}$ is the most desirable compound for $\alpha$-T type structures. ${\rm B_{50}C_{2}}$ may be the next desirable form. In contrast, in pure $\alpha$-T boron the unoccupied states are successively filled with increasing $n$, but this has the side effect of also creating gap states. For the interstitial configurations examined here, we found no configuration that completely separates the valence and conduction bands by using interstitial B atoms only. Probably, there is really no pure configuration without gap states. The significance of this behavior is discussed in Sec. \[sec:frustration\].

Let us now discuss the energy gap of the B$_{52}$ structure. Hayami and Otani claimed that B$_{52}$ is a narrow energy-gap semiconductor with an energy gap of 0.07 eV. (according to a private communication with the authors Fig. 8 of their paper [@Hayami10], which is expected to indicate this gap, is incorrect.). The corresponding “gap" is seen in Fig. \[fig:POSfilling\]. However, in our opinion the empty bands, indicated by the light-blue area, should rather be regarded as gap 