Model calculations for ⁵⁰Ca and the convergence properties of the perturbation expansion for the effective interaction

Using the results of the "exact" shell-model calculation of McGrory et al. for the Ca isotopes as "experimental data", we performed model calculations for ⁵⁰Ca in order to test the convergence properties of different perturbation schemes for computing the effective interaction V. As such, our aim was not to describe the physical spectrum of ⁵⁰Ca. Three different model spaces were investigated: (i) p _{3 2} (ii) p _{3 2} + p _{1 2} and (iii) p _{3 2} + p _{1 2} + f _{5 2}. In the p _{3 2} model space three different first-order interactions were used in computing the perturbation expansion for V through third order: (a) V ≡ {G + G}_3p-1hKB, (b) a standard G-matrix computed from V in (a), and (c) a doubly partitioned G-matrix computed from V by excluding excitations into the p _{1 2}level. The results for the doubly partitioned G yielded the best convergence through third order and the best agreement with the J = 0, T = 1 ground state of ⁵⁰Ca. The core polarization was found to be repulsive for a pure T = 1 system, so that certain third-order terms did not cancel against the large vertex renormalization terms in our special case. The [2, 1] padé approximant for each interaction appeared to indicate reasonable convergence in all three cases through third order. The results for the expanded (p _{3 2} + p _{1 2}) and (p _{3 2} + p _{1 2} + f _{5 2}) model spaces were difficult to interpret in a meaningful way, because of small energy denominators. Our results would indicate that one should compute in a doubly partitioned space when there are single-particle levels lying close in energy to the model space levels.