The CCD equations, depicted in Fig xx, are nonlinear in the cluster amplitudes. How do we solve \( \overline{H}_{ij}^{ab}=0 \)? We subtract \( (f_a^a +f_b^b -f_i^i -f_j^j)t_{ij}^{ab} \) from both sides of \( \overline{H}_{ij}^{ab}=0 \) (because this term is contained in \( \overline{H}_{ij}^{ab} \)) and find
$$ \begin{align*} (f_i^i +f_j^j -f_a^a -f_b^b)t_{ij}^{ab} &= (f_i^i +f_j^j -f_a^a -f_b^b)t_{ij}^{ab} +\overline{H}_{ij}^{ab} \end{align*} $$