Table 1.

Definitions of the operators used in the diagonalization of the CE Hamiltonian [Eqs. 30, 42, 47].

Reference	Definition	Subspace
Equation 30	$\begin{array}{c}{S}_{\Sigma z}=\frac{1}{2}(S_{1z}+S_{2z}),S_{\Sigma x}=\frac{1}{2}(S_1^{+}{S}_2^{+}+S_1^{-}{S}_2^{-}),S_{\Sigma y}=\frac{1}{2i}(S_1^{+}{S}_2^{+}-S_1^{-}{S}_2^{-}),\hfill \end{array}$	|1〉, |4〉, |5〉, |8〉
Equation 42	$\begin{array}{c}{E}_{S\Sigma }=4S_{\Sigma z}^2,S_{\Sigma }^{\alpha }=\frac{1}{2}{E}_{S\Sigma }+S_{\Sigma z},S_{\Sigma }^{\beta }=\frac{1}{2}{E}_{S\Sigma }-S_{\Sigma z},S_{\Sigma }^{+}=S_{\Sigma x}+i{S}_{\Sigma y},S_{\Sigma }^{-}=S_{\Sigma x}-i{S}_{\Sigma y}\hfill \end{array}$
Figure 5	$\begin{array}{c}{S}_{\Delta z}=\frac{1}{2}(S_{1z}-S_{2z}),S_{\Delta x}=\frac{1}{2}(S_1^{+}{S}_2^{-}+S_1^{-}{S}_2^{+}),S_{\Delta y}=\frac{1}{2i}(S_1^{+}{S}_2^{-}-S_1^{-}{S}_2^{+}),\hfill \end{array}$	|2〉, |3〉, |6〉, |7〉
	$\begin{array}{c}{E}_{S\Delta }=4S_{\Delta z}^2,S_{\Delta }^{\alpha }=\frac{1}{2}{E}_{S\Delta }+S_{\Delta z},S_{\Delta }^{\beta }=\frac{1}{2}{E}_{S\Delta }-S_{\Delta z},S_{\Delta }^{+}=S_{\Delta x}+i{S}_{\Delta y},S_{\Delta }^{-}=S_{\Delta x}-i{S}_{\Delta y}\hfill \end{array}$
Equation 47	$\begin{array}{c}{M}_{\Sigma z}=\frac{1}{2}(S_{\Delta z}+E_{S\Delta }{I}_z),M_{\Sigma x}=\frac{1}{2}(S_{\Delta }^{+}{I}^{+}+S_{\Delta }^{-}{I}^{-}),M_{\Sigma y}=\frac{1}{2i}(S_{\Delta }^{+}{I}^{+}-S_{\Delta }^{-}{I}^{-}),\hfill \end{array}$	$|\tilde{2}⟩,|\tilde{7}⟩$
Figure 6	$\begin{array}{c}{E}_{M\Sigma }=4M_{\Sigma z}^2,M_{\Sigma }^{\alpha }=\frac{1}{2}{E}_{M\Sigma }+M_{\Sigma z},M_{\Sigma }^{\beta }=\frac{1}{2}{E}_{M\Sigma }-M_{\Sigma z}\hfill \end{array}$
	$\begin{array}{c}{M}_{\Delta z}=\frac{1}{2}(S_{\Delta z}-E_{S\Delta }{I}_z),M_{\Delta x}=\frac{1}{2}(S_{\Delta }^{+}{I}^{-}+S_{\Delta }^{-}{I}^{+}),M_{\Delta y}=\frac{1}{2i}(S_{\Delta }^{+}{I}^{-}-S_{\Delta }^{-}{I}^{+}),\hfill \\ E_{M\Delta }=4M_{\Delta z}^2,M_{\Delta }^{\alpha }=\frac{1}{2}{E}_{M\Delta }+M_{\Delta z},M_{\Delta }^{\beta }=\frac{1}{2}{E}_{M\Delta }-M_{\Delta z}\hfill \end{array}$	$|\tilde{3}⟩,|\tilde{6}⟩$