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 $$ \begin{equation} \Delta E_{CCD} = \frac{1}{4} \sum_{ijab} \langle ij \vert\hat{v}\vert ab\rangle t^{ab}_{ij}. \tag{10} \end{equation} $$