We now come to a key element of coupled-cluster theory: the cluster operator (15) consists of sums of terms that consist of particle creation and hole annihilation operators (but no particle annihilation or hole creation operators). Thus, all terms that enter \( T \) commute with each other. This means that the commutators in the Baker-Campbell-Hausdorff expansion (23) can only be non-zero because each \( T \) must connect to \( H_N \) (but no \( T \) with another \( T \)). Thus, the expansion is finite.