Variational formulations of the coupled-cluster method in quantum chemistry

The coupled-cluster method is one of the most successful methods for ab initio electronic structure calculations. For computation of expectation values and properties, two equivalent but very different derivations exist: the Lagrangian formulation of Helgaker and JOrgensen [Adv. Quant. Chem. 19, 183 (1988)] and the bivariational formulation of Arponen [Ann. Phys. 151, 311 (1983)]. The latter is, apparently, less well known among quantum chemists, and we therefore discuss and contrast the two approaches. Some generalisations of coupled-cluster based on the bivariational principle are discussed, including the so-called extended coupled-cluster method.