 Solution

Adopt $\mathcal{H}=-2J({\mathbf{S}}_1+{\mathbf{S}}_2)\cdot \mathbf{s}-2J^{\prime }{\mathbf{S}}_1\cdot {\mathbf{S}}_2$, 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 giveThe five allowed multiplets areTheir degeneracies $2,2,4,4,6$ sum to 18, matching $3\times 3\times 2$. For dominant ferromagnetic $J$, the ground multiplet has $S_{12}=2$, $S=5/2$. A weak antiferromagnetic $J^{\prime }$ does not overcome the two strong metal–radical bonds.