It is useful to make approximations to the equations for the amplitudes. The standard method for solving these equations is to set up an iterative scheme where method's like Newton's method or similar root searching methods are used to find the amplitudes. Itreative solvers need a guess for the amplitudes. A good starting point is to use the correlated wave operator from perturbation theory to first order in the interaction. This means that we define the zeroth approximation to the amplitudes as $$ t^{(0)}=\frac{\langle ab \vert \hat{v} \vert ij \rangle}{\left(\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b\right)}, $$ leading to our first approximation for the correlation energy at the CCD level to be equal to second-order perturbation theory without \( 1p-1h \) excitations, namely $$ \Delta E_{\mathrm{CCD}}^{(0)}=\frac{1}{4}\sum_{abij} \frac{\langle ij \vert \hat{v} \vert ab \rangle \langle ab \vert \hat{v} \vert ij \rangle}{\left(\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b\right)}. $$