Approximations to the full CCD equations
With this starting point, we are now ready to solve Eq.
(1) iteratively. Before we attack the full equations, it is however instructive to study a truncated version of the equations. We will first study the following approximation where we take away all terms except the linear terms that involve the single-particle energies and the the two-particle intermediate excitations, that is
$$
\begin{equation}
0 = \langle ab \vert \hat{v} \vert ij \rangle + \left(\epsilon_a+\epsilon_b-\epsilon_i-\epsilon_j\right)t_{ij}^{ab}+\frac{1}{2}\sum_{cd} \langle ab \vert \hat{v} \vert cd \rangle t_{ij}^{cd}.
\tag{2}
\end{equation}
$$