Weak Collective Effects in the Nuclear Shell Model

Configuration mixing between the odd particle and the core is formulated to account for the weak collective quadrupole effects near closed shells. The configuration mixing sum is evaluated formally for the general case and applied to ${\mathrm{O}}^{17}$. An equivalent formalism in which closure is done over the configuration mixing sum is developed and applied to ${\mathrm{O}}^{17}$, giving good agreement with experiment. The closure result is found to be very insensitive to variation of parameters. The closure formalism is also applied near the lead closed shell, using a nuclear charge density distribution for the core instead of a sum over particle states. The collective effects in this region are well accounted for, and the formalism is compared with that for weak surface coupling.