In what follows, we will consider the coupled cluster doubles (CCD) approximation. This approximation is valid in cases where the system cannot exhibit any particle-hole excitations (such as nuclear matter when formulated on a momentum-space grid) or for the pairing model (as the pairing interactions only excites pairs of particles). In this case \( t_i^a=0 \) for all \( i, a \), and \( \overline{H}_i^a=0 \). The CCD approximation is also of some sort of leading order approximation in the Hartree-Fock basis (as the Hartree-Fock Hamiltonian exhibits no particle-hole excitations).
!splitkLet us consider the matrix element \( \overline{H}_{ij}^{ab} \). Clearly, it consists of all diagrams (i.e. all combinations of \( T_2 \), and a single \( F \) or \( V \) that have two incoming hole lines and two outgoing particle lines. Write down all these diagrams.
We start systematically and consider all combinations of \( F \) and \( V \) diagrams with 0, 1, and 2 cluster amplitudes \( T_2 \).