Ab initio calculations of the ${} ^2{\bi P}_{{\b 1}\over {\b 2}} \hbox{-}{} ^2{\bi P}_{{\b 3} \over {\b 2}} $ splitting in the thallium atom

The splitting between the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${} ^2P_{3 \over 2} $\end{document} and the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${} ^2P_{1 \over 2} $\end{document} terms in the thallium atom has been calculated at the perturbation theory level and by spin-orbit CI calculations, using both the Breit-Pauli and the no-pair form of the microscopic spin-orbit Hamiltonian. The importance of the spin-other-orbit contribution to the spin-orbit splitting is investigated, and it is also shown that an averaging procedure of the kinematic factors in the expression for the spin-other-orbit integrals in the no-pair spin-orbit Hamiltonian yields highly accurate results. A slightly modified version of a previously proposed mean-field spin-orbit method is shown to have an accuracy of a few wave numbers. Perturbation theory is found to give a term-splitting which is too low by more than 1000 cm−1, while spin-orbit CI with the no-pair form of the spin-orbit operator with the averaged spin-other-orbit term, and the no-pair mean-field operator, gives results in good agreement with experimental data.