To make progress, we have to specify the cluster operator \( T \). In coupled cluster theory, this operator is
$$ \begin{equation} \tag{15} T \equiv \sum_{ia} t_i^a a^\dagger_a a_i + \frac{1}{4}\sum_{ijab}t_{ij}^{ab} a^\dagger_aa^\dagger_ba_ja_i + \cdots + \frac{1}{(N!)^2}\sum_{i_1\ldots i_N a_1 \ldots a_N} t_{i_1\ldots i_N}^{a_1\ldots a_N} a^\dagger_{a_1}\cdots a^\dagger_{a_N} a_{i_N}\cdots a_{i_1} . \end{equation} $$