The effective interaction appropriate to a ${(2\mathrm{}s-1d)}^2$ model space is studied for the $J^{\pi }=0^{+}$ states of ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}$. Perturbation theory and various Padé approximants are compared with exact reslts obtained by solving large shell-model problems that realistically include many 3p-1h and 4p-2h states. We analyze two cases that differ only in the choice of the $2s-1d$ single-particle energies. In one, there is a collective 4p-2h intruder state, as well as several intruders at negative values of the coupling parameter. The perturbation theory expansion for the effective interaction is found to diverge in this case. The other case has no intruders and the perturbation expansion seems to converge. In both cases, third-order perturbation theory is found to be more accurate than second order, and gives matrix elements correct to 200 keV. The intruder states do not seem to be responsible for the fact that third-order terms are often larger than second-order terms. The $N+1, N$ Padé approximants of low orders are less accurate than third-order perturbation theory. However, the operator-valued [1,2] Padé approximant is accurate to 130 keV, for reasons that are not yet understood.

NUCLEAR STRUCTURE Effective interactions; tested perturbation theory and Padé approximants; ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}$ $J^{\pi }=0^{+}$ large-matrix calculations; included intruder states.

• Received 29 September 1975

DOI:https://doi.org/10.1103/PhysRevC.13.412