Here, \( T \) is an operator that induces correlations. We can now demand that the correlate state (11) becomes and eigenstate of the Hamiltonian \( H_N \), i.e. \( H_N\vert \Psi\rangle = E\vert \Psi\rangle \). This view, while correct, is not the most productive one. Instead, we left-multiply the Schroedinger equation with \( e^{-T} \) and find
$$ \begin{equation} \tag{12} \overline{H_N}\vert \Phi_0\rangle = E_c \vert \Phi_0\rangle . \end{equation} $$Here, \( E_c \) is the correlation energy, and the total energy is \( E=E_c+E_{HF} \).