Self-consistent band structures, charge distributions, and optical-absorption spectra in MgO, α-${\mathrm{Al}}_2$${\mathrm{O}}_3$, and ${\mathrm{MgAl}}_2$${\mathrm{O}}_4$

The electronic structures, the charge-density distributions, and the optical-absorption spectra in MgO, α-${\mathrm{Al}}_2$ ${\mathrm{O}}_3$, and ${\mathrm{MgAl}}_2$ ${\mathrm{O}}_4$ crystals are studied by means of the first-principles self-consistent orthogonalized linear combination of atomic orbitals method. The calculated band structures of MgO and α-${\mathrm{Al}}_2$ ${\mathrm{O}}_3$ are compared with experimental data and previous calculations. It is concluded that the electronic structure and optical properties of the spinel ${\mathrm{MgAl}}_2$ ${\mathrm{O}}_4$ cannot be taken as a simple average of that of MgO and α-${\mathrm{Al}}_2$ ${\mathrm{O}}_3$. By direct space integration of the calculated charge, it is shown that the ionic nature of these crystals should be best described by the formulas ${\mathrm{Mg}}^{1.83+}$ ${\mathrm{O}}^{1.83\mathrm{-}}$, ${\mathrm{Al}}_2^{2.63+}$ ${\mathrm{O}}_3^{1.75\mathrm{-}}$, and ${\mathrm{Mg}}^{1.79+}$ ${\mathrm{Al}}_2^{2.63+}$ ${\mathrm{O}}_4^{1.76\mathrm{-}}$. The charge-density maps also reveal that there are open channels of very low electron density in ${\mathrm{MgAl}}_2$ ${\mathrm{O}}_4$, which may be important in ionic conductivity. The calculated optical dielectric functions for the three crystals are compared with recent vacuum ultraviolet data for an energy range up to 40 eV and show good overall agreement. The existence of an excitonic peak near the absorption edge tends to sharpen the edge, thereby increasing the gap as compared with the one-electron local-density calculations. For absorption spectra beyond the excitonic peak, most of the experimentally resolved structures are reproduced by the calculations without the need of an adjustment to widen the gap. It appears that the high conduction-band states in these three crystals, calculated by using the local-density approximation, are reasonably accurate.