Chapter 13 Perturbation theory based on quasi-relativistic hamiltonians

Abstract

Abstract The approximate elimination of the small component approach provides ansatze for the small component of 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 which are a solid basis for the construction of approximate one- and two-component (quasi-relativistic) Hamiltonians. The present general derivation yields a whole family of quasi-relativistic Hamiltonians. The quasi-relativistic Hamiltonians can be used as zeroth-order approximation to the Dirac equation and the operator difference between the Dirac and the quasi-relativistic Hamiltonians can be used as a perturbation operator. The first-order perturbation energy corrections can be obtained from a direct perturbation theory scheme based on these quasi-relativistic Hamiltonians. These methods can rigorously be tested on one-electron atoms. At the variational quasi-relativistic level of theory, the errors of the relativistic correction to the energies are proportional to α4 Z4, whereas for the relativistic energy corrections including the first-order perturbation theory contributions, the errors are of the order of α6 Z6 to α8Z8 depending on the zeroth-order Hamiltonian. Expressions for the calculation of first-order electric and magnetic properties at the quasi-relativistic level of theory are also discussed.