Convergence Rate of Intermediate-State Summations in the Effective Shell-Model Interaction

Abstract

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 ($10\ensuremath{\hbar}\ensuremath{\Omega}$ to $12\ensuremath{\hbar}\ensuremath{\Omega}$ 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 ($2\ensuremath{\hbar}\ensuremath{\Omega}$) 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.