Comparison within the algebraic approximation of configuration interaction and many-body perturbation theory for the Be ground state

Calculations of the electronic energy of the Be-atom ground state are performed using a configuration-interaction technique and a nondegenerate formulation of diagrammatic many-body perturbation theory. Both methods are applied within the algebraic approximation defined by three finite-basis sets. One of the basis sets contains only $s$ functions, and it defines an algebraic problem without degeneracies where agreement between the two methods is excellent. The other two basis sets, although more complete, both contain $p$ functions, and they define algebraic problems involving a near degeneracy where agreement between the two methods is found to be less satisfactory. It is concluded that either a degenerate formulation of perturbation theory is required, or certain diagrams in the model perturbative scheme need to be summed to high order through denominator shifts in order to properly handle the nondynamic-correlation effect found in the Be atom.