Relativistic Hartree–Fock by means of stationary direct perturbation theory. I. General theory

Abstract

Perturbation theory of relativistic effects (if done properly, i.e. avoiding spurious singularities as they arise, e.g. if one starts with the Foldy–Wouthuysen transformation) has many advantages and only few drawbacks compared to a nonperturbative relativistic theory. In this paper relativistic Hartree–Fock theory for closed-shell states is combined with the stationary direct perturbation theory of relativistic effects in powers of the inverse velocity of light c−1 (which in hartree units is equal to the fine structure constant α). After a formulation for an arbitrary electron interaction the explicit expressions for the Dirac–Coulomb Hamiltonian are presented. The first step is always a nonrelativistic Hartree–Fock calculation, followed by the construction of the perturbation adapted zeroth order bispinors in terms of their large and small components φ(0)i and χ(0)i, respectively. The leading relativistic correction to the energy c−2E(2) is obtained as an expectation value involving φ(0)i and χ(0)i. The higher-order corrections c−4E(4) and c−6E(6) require that first the lowest order relativistic corrections φ(2)i and χ(2)i to the Hartree–Fock MOs are constructed by means of a coupled-Hartree–Fock-type approach. A hierarchy of approximations to the full treatment up to O(c−6) is proposed, in three of which the lowest order relativistic correction c−2E(2) to the energy is calculated exactly, while two-electron contributions to c−4E(4) and c−6E(6) are neglected in part or entirely. Finally the use of the Breit interaction and related interactions is discussed.