Relativistic ab initio treatment of the second-order spin-orbit splitting of the $a{}^3{\Sigma }_u^{+}$ potential of rubidium and cesium dimers

We have calculated the splitting between the $0_u^{-}$ and $1_u$ components of the $a{}^3{\Sigma }_u^{+}$ state of ${\mathrm{Rb}}_2$ and ${\mathrm{Cs}}_2$ using a relativistic ab initio configuration-interaction valence bond method. This so-called second-order spin-orbit splitting is entirely due to relativistic correlations within the molecule. Our ab initio nonperturbative splitting is 5 and 2 times larger than perturbative splittings at the inner turning point of the $a{}^3{\Sigma }_u^{+}$ potential for ${\mathrm{Rb}}_2$ and ${\mathrm{Cs}}_2,$ respectively. In addition, close-coupled nuclear dynamics calculations that estimate the effect of this splitting on experimentally accessible quantities are presented. The splitting affects the collisional loss rate of magnetically trapped ultracold Rb and Cs atoms and the spectroscopic determination of the vibrational structure of ${\mathrm{Rb}}_2$ and ${\mathrm{Cs}}_2$ dimers. Agreement with the experimental collisional loss rates of Cs is found.