pseudopotentials [@TM91] with the fully separable Kleinman-Bylander form [@KB82]. In all the cases, the kinetic cutoff energy was 40 Ry. Various $k$-point sampling methods were used. For calculations using the primitive unit cell, a $4 \times 4 \times 4$ grid was used and for supercell calculations $\Gamma$-point only sampling. The convergence was well tested in our previous studies.[@Masago06; @Dekura11; @Shirai14] The formation energy $\Delta E_{f}$ is defined as the difference of the total energy with respect to a reference state, which was one form of $\alpha$-T boron B$_{m}$ (the stoichiometric composition, $m$=52, in most cases). Then, $\Delta E_{f}$ of a composition B$_{m+n}$ is obtained by, $$\Delta E_{f}[{\text B}_{m+n}] = E[{\text B}_{m+n}] - \frac{m+n}{m}E[{\text B}_{m}],
  \label{eq:formE}$$ where $E[{\text B}_{m}]$ is the total energy of B$_{m}$. With this definition of $\Delta E_{f}$, a negative value implies a more stable structure. A similar definition is used to indicate the change of the specific volume $\Delta V$, where $E$ is substituted by $V$ in Eq. (\[eq:formE\]) and $\Delta V$ is evaluated in units of eV/GPa. Structural optimizations were performed with respect to atomic positions and cell parameters and no constraints on the crystal symmetry were imposed, except the tetragonal symmetry for the lattice parameters. 