Perturbation energy expansions based on two-component relativistic Hamiltonians

The approximate elimination of the small-component approach provides ansätze for the relativistic wave function. The assumed form of the small component of the wave function in combination with the Dirac equation define transformed but exact Dirac equations. The present derivation yields a family of two-component relativistic Hamiltonians which can be used as zeroth-order approximation to the Dirac equation. The operator difference between the Dirac and the two-component relativistic Hamiltonians can be used as a perturbation operator. The first-order perturbation energy corrections have been obtained from a direct perturbation theory scheme based on these two-component relativistic Hamiltonians. At the two-component relativistic level, the errors of the relativistic correction to the energies are proportional to α4Z4, whereas for the relativistic energy corrections including the first-order perturbation theory contributions, the errors are of the order of α6Z6–α8Z8 depending on the zeroth-order Hamiltonian.