Suggestion for Choosing the Single-Particle Energies in Doubly Closed-Shell Nuclei

The single-particle energies in a nucleus of $A\pm 1$ particles as taken from experiment are seen to be shifted when used for the calculation of the $A$-particle-system excited states. This shift can be described as a change in the particle-hole gap arising from an isospin-isospin interaction of the excited nucleon with the remaining core of $A-1\mathrm{}$ particles. It lowers the $T=0\mathrm{}$ states of ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ by about 1.5 MeV and of ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ by 3.75 MeV, and it raises the $T=1\mathrm{}$ states one third this amount. This shift gives an account of discrepancies which are observed in all existing Tamm-Dancoff or random-phase approximation calculations of doubly closed shell nuclei.