Relativistic one-electron Hamiltonians `for electrons onlyʹ and the variational treatment of the Dirac equation Original Research Article

A relation χ=Xϕ between the upper and lower components of the Dirac spinor ψ=(ϕ,χ), with X satisfying a nonlinear operator equation, guarantees that the expectation value 〈D〉ψ of the Dirac operator is bounded from below by the exact electronic ground state energy. Unfortunately X can, except for special cases, not be constructed in closed form. It is relatively easy to satisfy the condition χ=Xϕ if ϕ has the correct behaviour near a point nucleus, i.e. if it goes in the spherical average as rν with ν slightly smaller than 0. To satisfy χ=Xϕ for trial functions regular at the position of the nuclei is possible in principle, but very difficult in practice. If one satisfies the condition χ=Xϕ only approximatively, one may still get a lower bound, i.e. a variationally stable approach, but the lower bound is below its exact counterpart, i.e. the exact electronic ground state. Examples of variationally stable approaches are analyzed, in particular the regularized stationary direct perturbation theory (SDPT), the so-called regular approximation (RA), and the Douglas-Kroll-Hess transformation (DKH), of which a few new features are revealed. Finally the possibility of a minimax principle for the Dirac equation is discussed.