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Calculating branching ratio and spin-orbit coupling from first-principles: A formalism and its application to iridates

We present a simple technique to calculate spin-orbit coupling, $\langle {\bf L}\cdot{\bf S} \rangle$, and branching ratio measured in x-ray absorption spectroscopy. Our method is for first-principles electronic structure calculation and its implementation is straightforward for any of standard formulations and codes. We applied this technique to several different large spin-orbit coupling iridates. The calculated $\langle{ {\bf L}\cdot{\bf S}} \rangle$ and branching ratio of a prototype $j_{\rm eff}$=1/2 Mott insulator, Sr$_2$IrO$_4$, are in good agreement with recent experimental data over the wide range of Rh-doping. Three different double perovskite iridates (namely, Sr$_2$MgIrO$_6$, Sr$_2$ScIrO$_6$, and Sr$_2$TiIrO$_6$) are also well described. This technique can serve as a promising tool for studying large spin-orbit coupling materials from first-principles and for understanding experiments.