We examine the convergence rate of the sum over intermediate-particle states in the second-order core-polarization diagram of the effective shell-model interaction. For a $G$ matrix resulting from the Reid soft-core interaction, convergence is achieved in an oscillator basis only after the intermediate particle is summed over five or six major-shell excitations ($10\hslash \Omega$ to $12\hslash \Omega$ excitations). The total effect of this diagram on the shell-model spectra of ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}$ is markedly altered from the results of Kuo and Brown and others who only include oscillator excitations of one major shell ($2\hslash \Omega$) in the second-order diagram. We further demonstrate that slow convergence is attributed to strong components of the effective tensor force in the $G$ matrix.

• Received 10 January 1973

DOI:https://doi.org/10.1103/PhysRevC.7.1776