The Coulomb part of the Hamiltonian can be written as
\begin{eqnarray} H = (U_S-\alpha J_S)\frac{N(N-1)}{2} -\alpha' J_S \vec{S}^2 +\cdots = (U_S-\alpha J_S)\frac{N(N-1)}{2} -\alpha' J_S (S_x^2+S_y^2+S_z^2)+\cdots \end{eqnarray}for the $t2g$ subshell $\alpha=1.172$,$\alpha'=1.543$ and for $e_g$ it is $\alpha=1.517$, $\alpha'=1.773$.
Within cLDA we usually evaluate
\begin{eqnarray} U_S-\alpha J_S = \frac{\partial E[n+1/2]}{\partial n}-\frac{\partial E[n-1/2]}{\partial n} =\varepsilon_\uparrow(n_\uparrow=\frac{n+1}{2},n_\downarrow=\frac{n}{2})-\varepsilon_\uparrow(n_\uparrow=\frac{n+1}{2},n_\downarrow=\frac{n}{2}-1) \end{eqnarray}Withing cDMFT, we do not have Janak's theorem, therefore we evaluate
\begin{eqnarray} U_S-\alpha J_S = E[n+1]-2E[n]-E[n-1] \end{eqnarray}From the interacting Hamiltonian it follows that
\begin{eqnarray} -J = \frac{1}{2 S_z}\frac{\partial E[n,S_z]}{\partial S_z}= \frac{1}{2 S_z}\left[ \frac{\partial E[n,\frac{1}{2}(n_\uparrow-n_\downarrow)]}{\partial n_\uparrow}- \frac{\partial E[n,\frac{1}{2}(n_\uparrow-n_\downarrow)]}{\partial n_\downarrow} \right]=\varepsilon_\uparrow(n,S_z=1/2)-\varepsilon_\downarrow(n,S_z=1/2) \end{eqnarray}This is usually evaluated as
\begin{eqnarray} -J = \varepsilon_\uparrow(n_\uparrow=n/2+1/2,n_\downarrow=n/2-1/2)-\varepsilon_\downarrow(n_\uparrow=n/2+1/2,n_\downarrow=n/2-1/2) \end{eqnarray}within cLDA. In cDMFT, the Janak's theorem has not been proved, therefore we will use the energy difference, rather than their derivatives, to compute interaction parameters. We have
\begin{eqnarray} -J = \frac{1}{S^2}\left( E[n,Sz=S]-E[n,Sz=0] \right)= E[n,Sz=1]-E[n,Sz=0] \end{eqnarray}One way to evaluate this is to compute:
\begin{eqnarray} -J =E[n_\uparrow=n/2+1,n_\downarrow=n/2-1]-E[n_\uparrow=n/2,n_\downarrow=n/2] \end{eqnarray}