The CCD equation
The CCD equations can be written as
$$
\begin{align}
\left(\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b\right)t_{ij}^{ab}
= \langle ab \vert \hat{v} \vert ij \rangle & \nonumber
\\ +\frac{1}{2}\sum_{cd} \langle ab \vert \hat{v} \vert cd \rangle
t_{ij}^{cd}+\frac{1}{2}\sum_{kl} \langle kl \vert \hat{v} \vert ij
\rangle t_{kl}^{ab}+\hat{P}(ij\vert ab)\sum_{kc} \langle kb \vert
\hat{v} \vert cj \rangle t_{ik}^{ac} & \nonumber
\\ +\frac{1}{4}\sum_{klcd} \langle kl \vert \hat{v} \vert cd \rangle
t_{ij}^{cd}t_{kl}^{ab}+\hat{P}(ij)\sum_{klcd} \langle kl \vert
\hat{v} \vert cd \rangle t_{ik}^{ac}t_{jl}^{bd}& \nonumber
\\ -\frac{1}{2}\hat{P}(ij)\sum_{klcd} \langle kl \vert \hat{v} \vert
cd \rangle t_{ik}^{dc}t_{lj}^{ab}-\frac{1}{2}\hat{P}(ab)\sum_{klcd}
\langle kl \vert \hat{v} \vert cd \rangle t_{lk}^{ac}t_{ij}^{db},&
\tag{9}
\end{align}
$$
for all \( i < j \) and all \( a < b \), using the standard notation that
\( a,b,... \) are particle states and \( i,j,... \) are hole states. With
the CCD correlation energy given by
$$
\begin{equation}
\Delta E_{CCD} = \frac{1}{4} \sum_{ijab}
\langle ij \vert\hat{v}\vert ab\rangle t^{ab}_{ij}.
\tag{10}
\end{equation}
$$