Shell-Model Effective Interactions from the No-Core Shell Model

The extension of the No-Core Shell Model (NCSM) approach [1] to heavier nuclei (i.e., to the sd- and pf-shells) is a challenging problem. Standard shell-model calculations usually employ either empirical or theoretial effective 2-body matrix elements (TBME), which contain excluded many-body correlations. We show how a NCSM investigation in a 4$\hbar\Omega$ model space can yield TB sd-shell ME, which exactly reproduce the many-body correlations present in the original calculation. We demonstrate how the effective Hamiltonian derived in the 4$\hbar\Omega$ NCSM at the 2-body cluster level should be modified to properly account for the many-body correlations produced by truncating to a single major shell. The capability of these standard sd-shell TBME for $^{18}$F, obtained by direct projection, to reproduce the results of large scale NCSM calculations for F, Ne and Na isotopes will be shown.\\ 1. P.Navr\'atil, {\it et al.}, Phys. Rev. C {\bf 62}, 054331 (2000).