Success and Limitations of Two- and Four-Quasiparticle Tamm-Dancoff Theories of Vibrational States: Applications to Even Tin Isotopes with a Realistic Nucleon-Nucleon Potential

Extensive calculations of the spectra of the even Sn isotopes with the two- and four-quasiparticle (qp) Tamm-Dancoff theories are performed. The effective nuclear forces are the Tabakin and the Yale-Shakin potentials renormalized for core polarization involving up to all the subshells between the magic numbers 8 and 126. Several different sets of single-particle energies are considered, one of which corresponds to a Woods-Saxon potential. Over-all semiquantitative agreement with the data is obtained. We show how relative large values of the quadrupole moment of the $2_1^{}{}_{}{}^{+}$ state [such as $Q\left({2_1}^{+}\right)=+0.125\mathrm{}$ b] may arise from large two-qp-four-qp cross terms even in cases where four-qp components are small. Many aspects of the theory are critically examined; in particular, several different approximations in projecting out spurious kets of the nucleon-number nonconservation. Projected improvements and developments of the theory are outlined.