for reconciling the electron deficiency and the covalency. Using this mechanism, difficult problems with the electronic properties of B$_{13}$C$_{2}$ were resolved.[@Shirai14; @Ektrarawong15]

The above-mentioned mechanism should naturally work for $\alpha$-T boron, too. The initially assumed structure of $\alpha$-T boron is B$_{50}$, which is composed of four icosahedra and two $2b$-site B atoms (Fig. \[fig:struct\]). The $2b$ site is almost perfectly occupied (see Table \[tbl:alpha-tetra\]). Therefore B$_{50}$ was believed to be stoichiometric. Later we will show, however, that this model should be revised. Although the hypothetical structure B$_{50}$ has an even number of electrons and condition (i) does not hold, the valence-electron counting indicates that the covalent condition (ii) is not satisfied, because the system is short of ten electrons to completely fill up the valence band. This is the reason why Longuet-Higgins and Roberts cast doubt on the existence of B$_{50}$.[@LH55] Their conclusion is essentially correct even in modern DFT calculations. See, for example, Fig. 3 of Ref. 23 where the top five bands of the valence band are unoccupied.

Aside from the $2b$ site, many IS, which are partially occupied, have been reported (see Fig. \[fig:struct\]). The occupancies are listed in Table \[tbl:alpha-tetra\]. As described in the Introduction, the 