The solution of the CCSD equations, i.e. the second and third line of Eq. (20), and the computation of the correlation energy requires us to compute matrix elements of the similarity-transformed Hamiltonian (13). This can be done with the Baker-Campbell-Hausdorff expansion
$$ \begin{align} \tag{23} \overline{H_N} &= e^{-T} H_N e^T \\ &=H_N + \left[ H_N, T\right]+ \frac{1}{2!}\left[ \left[ H_N, T\right], T\right] + \frac{1}{3!}\left[\left[ \left[ H_N, T\right], T\right], T\right] +\ldots . \tag{24} \end{align} $$