The CCD approximation
Inserting these equations in the expression for the computation of the energy we have,
with a Hamiltonian defined with respect to a general vacuum (see the exercises in the second quantization part)
$$
\hat{H}=\hat{H}_N+E_{\mathrm{ref}},
$$
with
$$
\hat{H}_N=\sum_{pq}\langle p \vert \hat{f} \vert q \rangle a^{\dagger}_pa_q + \frac{1}{4}\sum_{pqrs}\langle pq \vert \hat{v} \vert rs \rangle a^{\dagger}_pa^{\dagger}_qa_sa_r,
$$
we obtain that the energy can be written as
$$
\langle \Phi_0 \vert \exp{-\left(\hat{T}_2\right)}\hat{H}_N\exp{\left(\hat{T}_2\right)}\vert \Phi_0\rangle =
\langle \Phi_0 \vert \hat{H}_N(1+\hat{T}_2)\vert \Phi_0\rangle = E_{CCD}.
$$