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 $T$) $$\Delta E=\sum_{ai}\langle i| \hat{f}|a \rangle t_{i}^{a}+ \sum_{abij}\langle ij | \hat{v}| ab \rangle t_{ij}^{ab}.$$

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