Approximations to the full CCD equations
The next approximation is to include the two-hole term in Eq.
(1), a term which allow us to make a link with Green's function theory with two-particle and two-hole correlations. This means that we solve
$$
\begin{equation}
0 = \langle ab \vert \hat{v} \vert ij \rangle + \left(\epsilon_a+\epsilon_b-\epsilon_i-\epsilon_j\right)t_{ij}^{ab}+\frac{1}{2}\sum_{cd} \langle ab \vert \hat{v} \vert cd \rangle t_{ij}^{cd}+\frac{1}{2}\sum_{kl} \langle kl \vert \hat{v} \vert ij \rangle t_{kl}^{ab}.
\tag{3}
\end{equation}
$$
This equation is solved the same way as we would do for Eq.
(2). The final step is then to include all terms in Eq.
(1).