transition.

Let us now describe in detail how this sum rule works. The sum rule should work for single-shell as well mixed-shell space.

We first consider the case of $^8\mbox{Be}$. We have the following values in a large space calculation for the $H_{Q}$ interaction corresponding to orbital $M1$ excitations from the $J=0^+$ $T=0$ ground state to all $1^+$ $T=1$ states:

= 1. Energy weighted isovector orbital $M1$ strength: = = $\mu_N^2MeV$\
2. The isoscalar summed strength $B(E2;1,1)$: $237.46$ $e^2fm^4$\
3. The isovector summed strength $B(E2;1,-1)$: $89.611$ $e^2fm^4$\
4. The right hand side ($\frac{9\chi}{16\pi}=0.1032$): $\mu_N^2MeV$.\

We don’t get exact agreement ($14.04$ $\mu_N^2MeV$ vs. $15.25$ $\mu_N^2MeV$) but it is reasonably close. One possible reason for the disagreement is that spurious states have been removed and/or that only $2\hbar\omega$ excitations to the $\Delta N=2$ shell are taken into account [@ibm] [@zr].

There have been other approaches, especially in the context of the Interacting Boson Model [@ibm] which relate the energy weighted orbital magnetic sum to the $B(E2)$ of the lowest $2^+$ state. As a matter of curiosity we shall examine in our calculation what happens if we take only the lowest $2^+$ state in the right hand side of the sum rule ($EWSR$).

