Deriving the nuclear shell model from first principles*

The results of an 18-nucleon No Core Shell Model calculation, performed in a large basis space using a bare, $soft$ NN interaction, can be projected into the $0\hbar\omega$ space, i.e., the $sd$-shell. Because the 16 nucleons in the $^{16}$O $core$ are $frozen$ in the $0\hbar\omega$ space, all the correlations of the 18-nucleon system are captured by the two valence, $sd$-shell nucleons. By the projection, we obtain microscopically the $sd$-shell 2-body effective interactions, the core energy and the $sd$-shell s.p. energies. Thus, the input for standard shell-model calculations can be determined microscopically by this approach. If the same procedure is then applied to 19-nucleon systems, the $sd$-shell 3-body effective interactions can also be obtained, indicating the importance of these 3-body effective interactions relative to the 2-body effective interactions. Applications to A=19 and heavier nuclei with different intrinsic NN interactions will be presented and discussed.