Effective operators from exact many-body renormalization

We construct effective two-body Hamiltonians and E2 operators for the p-shell by performing $16\hbar\Omega$ ab initio no-core shell model (NCSM) calculations for A=5 and A=6 nuclei and explicitly projecting the many-body Hamiltonians and E2 operator onto the $0\hbar\Omega$ space. We then separate the effective E2 operator into one-body and two-body contributions employing the two-body valence cluster approximation. We analyze the convergence of proton and neutron valence one-body contributions with increasing model space size and explore the role of valence two-body contributions. We show that the constructed effective E2 operator can be parametrized in terms of one-body effective charges giving a good estimate of the NCSM result for heavier p-shell nuclei.