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Improvement of the extended P+QQ interaction by modifying the monopole field

The extended pairing plus $QQ$ interaction with the $J$-independent isoscalar proton-neutron force as an average monopole field, which has succeeded in describing collective yrast states of $N \approx Z$ even-$A$ nuclei, is improved. The improvement is accomplished by adding small monopole terms (relevant to spectroscopy) to the average monopole field (indispensable to the binding energy). This modification extends the applicability of the interaction to nuclei with $N \approx 28$ such as $^{48}$Ca, and moreover improves energy levels of noncollective states. The modified interaction successfully describes not only even-$A$ but also odd-$A$ nuclei in the $f_{7/2}$ shell region. Results of exact shell model calculations in the model space ($f_{7/2}$, $p_{3/2}$, $p_{1/2}$) (its usefulness has been demonstrated previously) are shown for $A$=47, 48, 49, 50 and 51 nuclei.