Iterative solution of the one-electron Dirac equation based on the Bloch equation of the `direct perturbation theoryʹ

The one-electron Dirac equation is solved in an iterative manner starting with the solution of the Schrödinger equation. The method is based on `direct perturbation theoryʹ for relativistic effects generalized to the case of a set of near-degenerate strongly interacting states. Relativistic energies are obtained by solving a Schrödinger-like equation within a non-relativistic model space. The effective hermitian Hamiltonian and symmetric metric operator are expressed with the help of a Møller waveoperator Ω, which generates the complete four-component Dirac wavefunction from the non-relativistic two-component one. The corresponding Bloch equation is solved to infinite order by iteration in a basis of atom-centered Gaussian-type functions for the ground state of hydrogen-like ion Eka Pt109+ and for a few states of the heavy quasi-molecule Th179+2.