Approximations to the full CCD equations

Setting the single-particle energies for the hole states equal to an energy variable \( \omega = \epsilon_i+\epsilon_j \), Eq. (2) reduces to the well-known equations for the so-called \( G \)-matrix, widely used in infinite matter and finite nuclei studies. The equation can then be reordered and solved by matrix inversion. To see this let us define the following quantity $$ \tau_{ij}^{ab}= \left(\omega-\epsilon_a-\epsilon_b\right)t_{ij}^{ab}, $$ and inserting $$ 1=\frac{\left(\omega-\epsilon_c-\epsilon_d\right)}{\left(\omega-\epsilon_c-\epsilon_d\right)}, $$ in the intermediate sums over \( cd \) in Eq. (2), we can rewrite the latter equation as $$ \tau_{ij}^{ab}(\omega)= \langle ab \vert \hat{v} \vert ij \rangle + \frac{1}{2}\sum_{cd} \langle ab \vert \hat{v} \vert cd \rangle \frac{1}{\omega-\epsilon_c-\epsilon_d}\tau_{ij}^{cd}(\omega), $$ where we have indicated an explicit energy dependence. This equation, transforming a two-particle configuration into a single index, can be transformed into a matrix inversion problem. Solving the equations for a fixed energy \( \omega \) allows us to compare directly with results from Green's function theory when only two-particle intermediate states are included.