The band structures of α-rhombohedral ${\mathrm{B}}_{12}$ (α-r-${\mathrm{B}}_{12}$) and α-tetragonal ${\mathrm{B}}_{50}$ (α-t-${\mathrm{B}}_{50}$) crystals have been calculated by means of the first-principles orthogonalized linear combination of atomic orbitals method. It is shown that α-r-${\mathrm{B}}_{12}$ is a semiconductor with an indirect band gap of 1.70 eV and α-t-${\mathrm{B}}_{50}$ is a metal with a semiconductorlike band structure near the Fermi level. The intra-icosahedral and inter-icosahedral bondings in these two crystals are elucidated by resolving the density of states into various partial components, and by studying the valence-charge-density distributions. There is strong evidence for weak three-center bonding in the B crystals, but the traditional view regarding the bonding pattern in icosahedral ${\mathrm{B}}_{12}$ may be oversimplified. The total energies of α-r-${\mathrm{B}}_{12}$ and α-t-${\mathrm{B}}_{50}$ are calculated as a function of crystal volumes without change in the symmetry of the crystals. The equilibrium volume, the bulk moduli, and the cohesive energies obtained are in good agreement with available experimental data and some other recent calculations. Using the wave functions obtained from the band-structure calculation, the frequency-dependent interband optical conductivities in α-r-${\mathrm{B}}_{12}$ and α-t-${\mathrm{B}}_{50}$ crystals are also calculated. The real and imaginary parts of the dielectric-function curve are shown as a function of photon energy up to 40 eV for α-r-${\mathrm{B}}_{12}$ and up to 10 eV for α-t-${\mathrm{B}}_{50}$. They are significant differences in the optical properties of these two crystals because of the different nature of the band structures. For α-r-${\mathrm{B}}_{12}$, a large plasmon excitation at 31 eV is predicted.

• Received 29 July 1991

DOI:https://doi.org/10.1103/PhysRevB.45.5895