Effective Hamiltonian for near-degenerate states in direct relativistic perturbation theory. I. Formalism

Abstract

Direct perturbation theory (DPT) for relativistic effects is generalized to the case of a set of near-degenerate strongly interacting states. This situation, where the standard approach breaks down, is quite common in atoms and especially in molecules. We introduce a new partitioning of the Dirac equation and apply the Mo/ller–Bloch approach. An effective Schrödinger-like equation within a nonrelativistic model space of near-degenerate states is derived. The effective Hamiltonian and metric operators are expressed with the help of a Mo/ller wave operator Ω, which generates the complete four-component Dirac wave function from the nonrelativistic Schrödinger wave function in the finite model space. The corresponding Bloch equation can be solved numerically in a basis set to infinite order by iteration. Also explicit formulas are derived for different orders of Heff and Seff. They can be used to determine the relativistic energies to different orders either directly by diagonalization, or by a perturbation approach.