Almost variational coupled cluster theory

The energy expectation value of coupled-cluster theory can be formulated as the sum of the energy expression of traditional coupled-cluster (TCC) theory plus a correction term. The latter is simplified if the stationarity conditions of TCC hold. It is then of O(S 4), where S is the coupled-cluster amplitude. The leading error contribution agrees with the leading term of the difference between the TCC energy and the energy expression of extended coupled-cluster (ECC) theory. It is suggested to evaluate this routinely at the end of any TCC calculation as a check of the reliability of the latter. The error of the ECC energy expression with respect to an expectation value is of O(S 6). The error of traditional CCSD with respect to an expectation value is not affected by the inclusion of triple excitations in CCSDT. Approximations to CCSDT are also discussed. A hierarchy of approximations starting from TCC and ending at variational coupled-cluster (VCC) theory, alternative to the previously proposed improved c...