Scheme in $^8Be$
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The $Q \cdot Q$ interaction that we have been using fits in nicely with the $L-S$ supermultiplet scheme of Wigner [@wig]. For the $p$ shell the unitary group $U(3)$ is relevant since there are three states: $L=1$, $M=$1, 0 and -1. A very useful reference for this section is the book by Hammermesh [@ham].

If the Hamiltonian were a Casimir operator of $U(3)$ all states of a given special symmetry $[f]=[f_1,~f_2,~f_3]$ would be degenerate. For the case of $1p$ shell a state with a given particle symmetry $[f_1,~f_2,~f_3]$ is identical to a quantum oscillator symmetry state [@Har; @Elliot] $(\lambda, \mu)=(f_1-f_2,f_2-f_3)$. The states $(\lambda, \mu,L)$ are eigenstates of our $Q \cdot Q$ interaction which is a linear combination of the Casimir operator of $SU(3)$ and an $L\cdot L$ interaction. The latter gives rise to a terminating rotational $L(L+1)$ spectrum for states of different $L$ but with the same $[f]$. Amusingly, as has been pointed out by many, one gets identical bands in all $p$ shell nuclei with this model provided the coefficient of $L\cdot L$ is fixed.

Unlike in the $s,~d$ shell, nothing new is added by using the quantum numbers ($\lambda,~ \mu$) instead of \[$f_1,~f_2,~f_3$\] for 