A quick tour of Coupled Cluster theory
The coupled cluster energy is a function of the unknown cluster amplitudes \( t_{i_1i_2\ldots i_n}^{a_1a_2\ldots a_n} \),
given by the solutions to the amplitude equations
$$
\begin{equation*}
0 = \langle\Phi_{i_1 \ldots i_n}^{a_1 \ldots a_n}\vert \overline{H}\vert \Phi_0\rangle.
\end{equation*}
$$
The similarity transformed Hamiltonian \( \overline{H} \) is expanded using the Baker-Campbell-Hausdorff expression,
$$
\begin{align*}
\overline{H}&= \hat{H}_N + \left[ \hat{H}_N, \hat{T} \right] +
\frac{1}{2} \left[\left[ \hat{H}_N, \hat{T} \right], \hat{T}\right] + \ldots \\
& \quad \frac{1}{n!} \left[ \ldots \left[ \hat{H}_N, \hat{T} \right], \ldots \hat{T} \right] +\dots
\end{align*}
$$
and simplified using the connected cluster theorem
$$
\begin{equation*}
\overline{H}= \hat{H}_N + \left( \hat{H}_N \hat{T}\right)_c + \frac{1}{2} \left( \hat{H}_N \hat{T}^2\right)_c
+ \dots + \frac{1}{n!} \left( \hat{H}_N \hat{T}^n\right)_c +\dots
\end{equation*}
$$