The CCD approximation
This quantity becomes
$$
E_{CCD}=E_{\mathrm{ref}}+\frac{1}{4}\sum_{abij}\langle ij \vert \hat{v} \vert ab \rangle t_{ij}^{ab},
$$
where the latter is the correlation energy from this level of approximation of CC theory.
Similarly, the expression for the amplitudes reads
$$
\langle \Phi_{ij}^{ab} \vert \exp{-\left(\hat{T}_2\right)}\hat{H}_N\exp{\left(\hat{T}_2\right)}\vert \Phi_0\rangle = 0.
$$