The CCD approximation
The operator \( \hat{P}(ij\vert ab) \) is defined as
$$
\hat{P}(ij\vert ab) = (1-\hat{P}_{ij})(1-\hat{P}_{ab}).
$$
Recall also that the unknown amplitudes \( t_{ij}^{ab} \)
represent anti-symmetrized matrix elements, meaning that they obey the same symmetry relations as the two-body interaction, that is
$$
t_{ij}^{ab}=-t_{ji}^{ab}=-t_{ij}^{ba}=t_{ji}^{ba}.
$$
The two-body matrix elements are also anti-symmetrized, meaning that
$$
\langle ab \vert \hat{v} \vert ij \rangle = -\langle ab \vert \hat{v} \vert ji \rangle= -\langle ba \vert \hat{v} \vert ij \rangle=\langle ba \vert \hat{v} \vert ji \rangle.
$$
The non-linear equations for the unknown amplitudes \( t_{ij}^{ab} \) are solved iteratively. We discuss the implementation of these equations below.