Numerical study of the iterative solution of the one-electron Dirac equation based on ‘direct perturbation theory’

The one-electron Dirac equation is solved in an iterative manner starting with the solution of the Schrödinger equation. The method is applied in a basis of atom-centred Gaussian-type functions to the ground state of selected hydrogen-like ions up to Eka Pt109+ and the heavy quasi-molecules Th2179+, NiPb109+ and Th3269+ (in D∞h and D3h symmetry). An overall 8-figure accuracy in the absolute relativistic energies is achieved. The iterative procedure converges better than linearly for light systems and linearly for systems containing nuclear charges greater than Z 2∼ 40.