A quick tour of Coupled Cluster theory
In our FCI discussion the correlation energy is defined as, with a two-body Hamiltonian,
$$
\Delta E=\sum_{ai}\langle i| \hat{f}|a \rangle C_{i}^{a}+
\sum_{abij}\langle ij | \hat{v}| ab \rangle C_{ij}^{ab}.
$$
In Coupled cluster theory it becomes (irrespective of level of truncation of \( T \))
$$
\Delta E=\sum_{ai}\langle i| \hat{f}|a \rangle t_{i}^{a}+
\sum_{abij}\langle ij | \hat{v}| ab \rangle t_{ij}^{ab}.
$$