Using the first-principles density functional approach, we comparatively studied the electronic and magnetic properties of two series of ordered double perovskite, $\mathrm{B}{\mathrm{a}}_{2}B\mathrm{Os}{\mathrm{O}}_{6}\phantom{\rule{4pt}{0ex}}(B=\mathrm{Sc},\text{Y},\phantom{\rule{4pt}{0ex}}\text{and In})$ and ${A}_{2}\mathrm{ScOs}{\mathrm{O}}_{6}\phantom{\rule{4pt}{0ex}}(A=\text{Ba},\text{Sr},\phantom{\rule{4pt}{0ex}}\text{and Ca})$. The electronic structure calculations indicate that the Os ion bears the nominal ${t}_{2g}^{3}$ state with oxidation state $5+$ and is the only magnetically active one in all compounds. In nice agreement with the experimental observations, all of them stabilized in the type I antiferromagnetic alignment, which is mediated by the moderate nearest-neighbor antiferromagnetic interactions through the $\mathrm{Os}\text{\ensuremath{-}}\mathrm{O}\text{\ensuremath{-}}B\text{\ensuremath{-}}\mathrm{O}\text{\ensuremath{-}}\mathrm{Os}90$\ifmmode^\circ\else\textdegree\fi{} route and accompanied by mild next-nearest-neighbor antiferromagnetic coupling via the $\mathrm{Os}\text{\ensuremath{-}}\mathrm{O}\text{\ensuremath{-}}B\text{\ensuremath{-}}\mathrm{O}\text{\ensuremath{-}}\mathrm{Os}180$\ifmmode^\circ\else\textdegree\fi{} route, in spite of the geometric frustration. The mechanism of the antiferromagnetic interaction has been proposed based on the density of states analysis. In addition, the computed magnetic moment on Os ions, $\ensuremath{\sim}2\phantom{\rule{0.16em}{0ex}}{\ensuremath{\mu}}_{B}$, is remarkably smaller than the ideal spin-only value. In conjunction with the modest spin-orbit coupling effect, strong covalency between the $\mathrm{Os}\text{\ensuremath{-}}5d$ and $\mathrm{O}\text{\ensuremath{-}}2p$ states reduces the magnetic moments. More importantly, our results demonstrated that the magnitude of the magnetic couplings as well as magnetic transition temperature ${T}_{N}$ are directly proportional to the reciprocal of the absolute value of energy mismatch between the $B\text{\ensuremath{-}}nd$ and occupied $\mathrm{Os}\text{\ensuremath{-}}{t}_{2g}$ band; in addition, they are also a function of the Sc-O-Os bond angles, i.e., the buckling of the octahedral connectivity. Moreover, comparison of these two series helps to identify the energy mismatch as the efficient factor to adjust the magnetic interactions.
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