[L~L]^0 = 0.74536 [0~0]^0~+~0.66666 [2~2]^0$$

where the notation $[L_\pi~L_\nu]^J$ is used.

The $J=2$ wave function is

$$N''\left[\begin{array}{lll}1 & 1 & 0\\0 & 0 &
0\end{array}\right]  \left[\begin{array}{lll}1 & 1 & 2\\0 & 0 &
0\end{array}\right]  [0~2]^2~+~ \left[\begin{array}{lll}1 & 1 &
2\\0 & 0 &
0\end{array}\right]  \left[\begin{array}{lll}1 & 1 & 0\\0 & 0 &
0\end{array}\right] [2~0]^2$$

$$~+~  \left[\begin{array}{lll}1 & 1& 2\\0 & 0 &
0\end{array}\right]  \left[\begin{array}{lll}1 & 1 & 2\\0 & 0 &
0\end{array}\right] \left[\begin{array}{lll}2 & 2 & 2\\0 & 0 &
0\end{array}\right] [2~2]^2$$

i.e.

$$\psi^{J''=2}=0.62361[0~2]^2~+~0.62361[2~0]^2+0.47141[2~2]^2$$

We don’t actually have to specify the $J=1^+$ state in detail to get the ratio. We note that only the component $[2~2]^1$ of the $J=1$ wave function can contribute to the $M1$ transition. The probability of this component factors out in the ratio. With some additional Racah algebra, we can show that $r=\frac{7}{8}=0.875$. This should be compared with the value $r=\frac{1}{2}$ of the simple rotational formula and with the experimental value $r_{exp}=0.8$. We see that we get better agreement by using this projection method.

To complete the story, using results described in the next section, we are able to deduce that for $^8Be$ $B(M1)_{1^+\rightarrow
2_1^+}$=$\frac{7}{4\pi}~\mu_N^2$

Supermultiplet Scheme with a $Q \cdot Q$ interaction
====================================================

A. Supermultiplet 