at site $s$). This is seen by inspecting the whole range of $N_{\rm at}$ in Table \[tbl:HfofB50\]. Hayami and Otani studied IS of $\alpha$-T boron over a compositional range $n=0-4$, concluding that B$_{50}+2$B$_{4c}$ is the most stable structure.[@Hayami10] The present calculations confirm their conclusion. Among various combinations of two $4c$-site atoms, the out-of-plane configuration has the lowest energy (For details of the structures, see Supplemental Material). A configuration in which two $4c$-site atoms occupy neighboring sites is highest in energy. There is a general tendency in boron-rich crystals that atoms of the same IS avoid each another. The present result for $4c$ sites is in accordance with this tendency.

The fact that the formation energy is minimized at $n=2$ can be understood by examining the valence filling for different values of $n$. Figure \[fig:POSfilling\] shows the evolution of the density of states (DOS) by successively placing B atoms at $4c$-sites. For B$_{50}$, there are 160 valence states (80 bands), and the top 10 states are unoccupied. Among these 10 unoccupied states, the lower 6 states are the tail part of the valence band and they are mainly contribution derived from the [*inter*]{}icosahedral $t-t$ bonds (in the notations of Ref. ). 