Convergence and the third-order effective shell-model interaction

Abstract

Several commonly made approximations in microscopic effective interaction calculations have been found to result in serious inaccuracies. Recent improved second-order calculations for $A=18$ nuclei suggested somewhat better order by order convergence in the reaction matrix $G$ than indicated previously. A selected third-order term has been calculated to study further this possibility. The convergence of the intermediate-state summation was also investigated for this diagram. Our improved calculational technique results in a greater reduction of the third-order term than the reduction found earlier for the second-order term. The outlook for the (asymptotic) convergence of the expansion is much improved, but the prospects for a complete third-order calculation of the effective interaction are dim due to the difficulties of the calculation.[NUCLEAR STRUCTURE Effective interaction, theory; convergence of the intermediate-state sums; convergence of the diagram series through third-order in the nuclear reaction matrix; shell-model matrix elements in the $\mathrm{sd}$ shell.]