Full major-shell calculation for states that are degenerate in a single-$j$-shell calculation

In a previous work~\footnote{A.~Escuderos, B.F.~Bayman, L.~Zamick, and S.J.Q.~Robinson, Phys. Rev. C {\bf 72}, 054301 (2005)}, we explained why certain states were degenerate in the single $j$ shell for an interaction in which the isospin $T=0$ two-body matrix elements were set to zero. The degeneracy splitting was recovered by reintroducing the full interaction. In this work, we perform a full $fp$-shell calculation with the FPD6 interaction to obtain these energy splittings; the interaction obtained by setting the $T=0$ matrix elements to zero but keeping the $T=1$ ones unchanged will be called T0FPD6. Comparing the results with FPD6 and T0FPD6, we can see that most of the splitting in a complete shell calculation (but not all) comes from the $T=0$ part of the interaction. For example, the $(9^+_1-10^+_1)$ splitting in $^{44}$Ti is 1.214~MeV for FPD6, but it is only 0.094~MeV for T0FPD6. In $^{47}$V, the $(29/2^-_1-31/2^-_1)$ splitting is 0.780~MeV with FPD6, in agreement with the experimental value of 0.765~MeV, but T0FPD6 yields only 0.072~MeV. In general, we observe a continuity in the splittings between the single-$j$ and the full-$fp$ calculations; only in two cases we see an inversion of the states. These two cases involve low angular momentum states ($1/2^-$ in $^{43}$Sc and $3^+$ in $^{44}$Ti), for which there tends to be much more configuration mixing.