Spin-orbit coupling effect on the 2$^{3}\Pi $ state of $^{39}$K$^{85}$Rb

Recently we investigated the spin-orbit components ($\Omega =$ 0$^{+}$, 0$^{-}$, 1, and 2) of the 2$^{3}\Pi $ state of $^{39}$K$^{85}$Rb by using experimental spectroscopy of ultracold molecules formed by photoassociation [1]. The separations ($\Delta (E_{\Omega = 1}-E_{\Omega = 0})$ and $\Delta (E_{\Omega = 2}-E_{\Omega = 1}))$ between $\Omega $ components were unequal due to second-order perturbations by other electronic states. In the present work we investigate the spin-orbit coupling effect on the 2 $^{3}\Pi$ state of $^{\mathrm{39}}$K$^{\mathrm{85}}$Rb in the framework of 1$^{\mathrm{st}}$ and 2$^{\mathrm{nd}}$ order non-degenerate perturbation theory based on an \textit{ab initio }method. Required potential energy curves and electronic spin-orbit coupling matrix elements are evaluated over a wide range of internuclear distance in the basis of the spin-averaged wavefunctions corresponding to the pure Hund's case (a) coupling scheme. We compare the experimental spin-orbit splittings of the 2 $^{3}\Pi$ state with its \textit{ab initio} counterparts, which agree well and elucidate the pronounced 2$^{\mathrm{nd}}$ order perturbation effects caused by nearby electronic states. \\[4pt] [1] J. T. Kim \textit{et al}., New J. of Phys. \textbf{11}, 055020 (2009).