  Our main concern in this work is to show how higher shell admixtures affect
the spectrum of a Q.Q interaction. We first review how, in the valence space,
the familiar SU(3) result for the energy spectrum can be obtained using a
coordinate space Q.Q interaction rather than the Elliott one which is symmetric
in r and p. We then reemphasize that the Elliott spectrum goes as L(L+1) where
L is the orbital angular momentum. While in many cases this is compatible with
the rotational formula which involves I(I+1), where I is the total angular
momentum, there are cases, e.g. odd-odd nuclei, where there is disagreement.
Finally, we consider higher shell admixtures and devise a scheme so as to
obtain results, with the Q.Q interaction, which converge as the model spaces
are increased. We consider not only ground state rotational bands but also
those that involve intruder states.
