Relativistic pseudopotential theories and corrections to the Hartree-Fock method

Abstract

The fully relativistic Dirac-Fock generalizations of non-relativistic Phillips-Kleinman and generalized Phillips-Kleinman pseudopotentials are used to derive pseudopotential theories based on the Schrödinger hamiltonian which incorporate relativistic effects.

In the theory in which all the orbitals are treated to the lowest non-vanishing order (1/c ²), the relativistic corrections to the energies are obtained by taking the expectation value of a perturbing operator over the non-relativistic wavefunctions. This perturbation consists of coulomb and exchange operators built from the relativistically induced changes in the charge distributions of other orbitals and two new operators arising from the pseudopotential, in addition to the standard mass-velocity, spin-orbit coupling and Darwin terms. The leading corrections to Hartree-Fock theory are obtained by omitting the pseudopotential operators and correct two expressions reported previously. The relative contributions of the standard operators and the coulomb plus exchange terms provide a quantitative measure of the relative importance of the direct and indirect relativistic effects. It is shown that, for a nodeless pseudo-orbital having a low amplitude in the inner spatial regions, the direct contribution is dominated by the two perturbations originating in the pseudopotential. The further operators arising for open shell systems are derived.

The possibility of extending the theory to higher order than 1/c ² is discussed and it is concluded that whilst this is not possible in Hartree-Fock theory, non-divergent higher-order corrections to the pseudopotential can be derived. A pseudopotential theory is developed in which the core orbitals are treated exactly, the pseudopotential contributions to valence orbitals to order 1/c ⁴ and the remaining valence orbital contributions to order 1/c ².