Electronic structure and bonding in garnet crystals ${\mathrm{Gd}}_3{\mathrm{Sc}}_2{\mathrm{Ga}}_3{\mathrm{O}}_{12},$ ${\mathrm{Gd}}_3{\mathrm{Sc}}_2{\mathrm{Al}}_3{\mathrm{O}}_{12},$ and ${\mathrm{Gd}}_3{\mathrm{Ga}}_3{\mathrm{O}}_{12}$ compared to ${\mathrm{Y}}_3{\mathrm{Al}}_3{\mathrm{O}}_{12}$

The electronic structure and bonding of ${\mathrm{Gd}}_3{\mathrm{Sc}}_2{\mathrm{Ga}}_3{\mathrm{O}}_{12}$ (GSGG), ${\mathrm{Gd}}_3{\mathrm{Sc}}_2{\mathrm{Al}}_3{\mathrm{O}}_{12}$ (GSAG), and ${\mathrm{Gd}}_3{\mathrm{Ga}}_5{\mathrm{O}}_{12}$ (GGG) crystals with a garnet structure are studied by means of first-principles local-density calculations. The results are compared with a similar calculation on yttrium aluminum garnet [${\mathrm{Y}}_3{\mathrm{Al}}_5{\mathrm{O}}_{12}$ (YAG)]. The calculated equilibrium volumes of the three crystals are close to the measured volumes with a slight overestimation for GGG. GGG also has a smaller bulk modulus than the other three crystals. The calculated density of states and their atomic and orbital decompositions are presented and contrasted. All four crystals show very similar band structures and interatomic bonding. However, it is found that in GSGG and GSAG crystals, the Sc atom at the octahedral site shows a higher covalent character and an increased bond order in comparison to Ga or Al at the same site. This result may provide some insight into the significant difference in the radiation hardness of ${\mathrm{Cr}}^{3+}{:\mathrm{N}\mathrm{d}}^{3+}:\mathrm{G}\mathrm{S}\mathrm{G}\mathrm{G}$ as compared to ${\mathrm{Nd}}^{3+}:\mathrm{Y}\mathrm{A}\mathrm{G}.$