Correlation effects in momentum space for the electronic shells in Be

Abstract

Our previous partitioning technique for analyzing correlation effects within the individual electronic shells of Be in position space has been applied here to a corresponding examination in momentum space. Comparability with the results in position space was ensured by considering the same configuration-interaction (CI) wave functions and Hartree–Fock (HF) description as before; the functions were transformed into momentum space by applying the standard Dirac procedure. By analogy with position space, Coloumb shifts Δf(p12) vs p12 were derived for the K(1S), L(1S), KL(1S), and KL(3S) shells in Be. Selected one- and two-particle momentum expectation values are also reported along with various radial and angular correlation coefficients. The opposing effects of radial and angular correlation in momentum space for Be gave a K-shell Coulomb shift which, being small, suggested a rough balance between these correlation components. For the L shell, however, the large excess of angular correlation produced a ‘‘shift’’ which, relative to the K shell, was of considerable magnitude. Thus, when forming the sum total Coulomb shift for the whole atom, the L-shell component virtually masked all contributions arising from the other shells. This is in direct contrast with position space where the intershell effects in Be produced some detailed structure in the total Coulomb hole Δf(r12) vs r12. The characteristics of the momentum curves for the various shells are rationalized and their relationship with the corresponding results in position space is discussed.