Truncation of Configuration Space and the Nature of Effective Two-Body Interaction

Using the renormalized $G$ matrix (the $G_r$) results of Kuo (for the $d-s$ shell) and of Kuo and Brown (for the $f-p$ shell), a re-renormalized effective $G$ matrix (the $G_{\mathrm{eff}}$) for only two active orbits is calculated numerically, which would be valid for use in ${\left(d_{\frac{5}{2}}{s}_{\frac{1}{2}}\right)}^n$ and ${\left(f_{\frac{7}{2}}{p}_{\frac{3}{2}}\right)}^n$ configurations. The operators $G_r$ and $G_{\mathrm{eff}}$ as well as the bare $G$ matrix (the $G_b$) of Kuo and Brown, which thus define the transformation chain $G_b\to G_r\to G_{\mathrm{eff}}$, are then analyzed in the light of truncation of the shell-model space. The analysis shows that the effect of this truncation can largely be taken into account by the addition of a pairing-force term to the effective two-body interaction. The relative change in $T=1\mathrm{}$ and $T=0\mathrm{}$ parts of the interaction due to the truncation of configuration space in various ways is also discussed. Finally, a comparison between the available empirically fitted effective interactions and our theoretical $G_{\mathrm{eff}}$ interaction is carried out.