We investigated double perovskite compounds of the form Sr$_2$XOsO$_6$ (X = Li, Na, Ca) using the fullpotential linearized augmented plane wave (FP-LAPW) method. For the exchange-correlation energy, Wu andCohen generalized gradient approximation (WC-GGA), Perdew, Burke and Ernzerhof GGA (PBE-GGA), Engel and Vosko GGA (EV-GGA), and GGA plus Hubbard U-parameter (GGA $+$ U) were used. The calculated structuralparameters are in good agreement with the existing experimental results. Calculation of different elastic constants and elastic moduli reveals that these compounds are elastically stable and possess ductile nature. The GGA $+$ Uapproach yields quite accurate results of the bandgap as compared with the simple GGA schemes. The density of states plot shows that Sr-4d, Os-5d and O-2p states predominantly contribute to the conduction and valence bands.Further, our results regarding to the magnetic properties of these compounds reveal their ferromagnetic nature. In addition, these compounds seem to possess half-metallic properties, making them useful candidates for applicationsin spintronics devices.

The structural parameters, elastic constants, electronic and optical properties of the bi-alkali antimonides (Na$_2$KSb, Na$_2$RbSb, Na$_2$CsSb, K$_2$RbSb, K$_2$CsSb and Rb$_2$CsSb) were calculated using state-of-the-art density functional theory. Different exchange-correlation potentials were adopted to predict the physical properties of these compounds. The calculated structural parameters are found in good agreement with the available experimental and theoretical results. All the compounds are mechanically stable. The compounds Na$_2$KSb, K$_2$RbSb, K$_2$CsSb and Rb$_2$CsSb have direct bandgaps, in which chemical bonding among the cations and anions is mainly ionic. Furthermore, the optical properties of these compounds are described in detail in terms of the dielectric function, refractive index, reflectivity, optical conductivity and absorption coefficient.