  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.
