the possibility of hydrogen incorporation is not excluded. Their chemical analysis, however, did not indicate H content as high as it influences the lattice parameter.

On the basis of the above analysis, we can summarize the current state of $\alpha$-T boron research as follows: If the recently synthesized $\alpha$-T borons are pure ones, there is little doubt that they are non-stoichiometric; in contrast theory predicts stoichiometric B$_{52}$ to be the lowest-energy structure. The question is thus, if the deviation from stoichiometry in real crystals is an intrinsic property (lowest-energy state) or not. For $\beta$-R boron[@Setten07; @Widom08; @Ogitsu09] and ${\rm B_{13}C_{2}}$ [@Shirai14], the deviation is an intrinsic property. If this is not the case for $\alpha$-T boron, the deviation must be an extrinsic property, caused by entropic effects at high temperatures. Then the degree of deviation would depend on the preparation conditions. Accordingly, one should characterize the real crystals with respect to the preparation conditions.

Computational Methods {#sec:method}
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The electronic structures of $\alpha$-T boron were studied by density functional theory using a pseudopotential method and the Osaka2k code.[@Osaka2k] It uses the LDA parameterization by Perdew and Zunger [@PZ81], the Perdew-Burke-Ernzerhof form of the generalized gradient approximation (GGA) [@PBE96] and Troullier-Martins 