The fine-structure splitting of the thallium atomic ground state: LS- versus jj-coupling

We examine the spin-orbit splitting of the 2P electronic ground state of the Tl atom by two different approaches, both of which incorporate relativistic effects: namely, a four-component method involving the Dirac-Coulomb-Breit Hamiltonian, and a two-component method which makes use of the Douglas-Kroll transformed no-pair Hamiltonian. The former method operates by definition in the jj-coupling scheme, while the latter uses LS-coupled states. The four-component method is found to work well provided that separate orbital optimizations are performed for the 12(I) and 32(I) 2P sublevels. In the LS-coupling scheme consideration of the spin-orbit splitting by perturbation theory fails, while inclusion of spin-orbit interaction in a variational procedure yields a good result. The reason for the failure of perturbation theory lies in the large differences of the spatial extensions of the p32 and p12 valence orbitals of the Tl atom.