= Solution
Adopt $\mathcal H=-2J(\mathbf S_1+\mathbf S_2)\cdot\mathbf s-2J'\mathbf S_1\cdot\mathbf S_2$, with $S_1=S_2=1$ and ligand <spin> $s=1/2$. Let $\mathbf S_{12}=\mathbf S_1+\mathbf S_2$ and $\mathbf S=\mathbf S_{12}+\mathbf s$. The <angular momentum addition> identities give
$$
E=-J\{S(S+1)-S_{12}(S_{12}+1)-3/4\}-J'\{S_{12}(S_{12}+1)-4\}.
$$
The five allowed multiplets are
$$
\begin{array}{c|c|c}
S_{12}&S&E\\\hline
0&1/2&4J'\\
1&1/2&2J+2J'\\
1&3/2&-J+2J'\\
2&3/2&3J-2J'\\
2&5/2&-2J-2J'
\end{array}
$$
Their degeneracies $2,2,4,4,6$ sum to 18, matching $3\times3\times2$. \b[For dominant ferromagnetic $J$, the ground multiplet has $S_{12}=2$, $S=5/2$.] A weak antiferromagnetic $J'$ does not overcome the two strong metal–radical bonds.
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