Optimizing the cell parameters is important in view of the study of high-pressure phases. LDA was used only for supercell calculations, because of its computational efficiency, otherwise GGA was used. Of course, in this case, comparisons were made only between cells with the same size.

Comparisons between impurity-containing $\alpha$-T boron (impurity $X$=C or N) were also performed when necessary. In this case, compositions ${\rm B_{50}X_{2}}$ were assumed, where $X$-atoms were place on the $2b$ sites and the interstitial B atoms were placed either at the $8h$ or the $8i$ sites.

Results and Discussion
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The structure of $\alpha$-tetragonal boron {#sec:bondnature}
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### Formation energy of interstitial sites {#sec:formaton-energy}

The formation energies of various IS were extensively studied by Hayami and Otani.[@Hayami10] In this work we want to study the roles of IS in more detail. We constructed a B$_{50}$ structure, where the $2b$ sites are fully occupied, and then placed B atoms at various IS, one by one. The results for $\Delta E_{f}$ are shown in Table \[tbl:HfofB50\]. However, care is needed for the interpretation of these values because usually an energy barrier is to be expected between two local energy minimum sites. Between the $4c$ and $8i$ sites, Hayami 