Algebraic propagator approaches and intermediate-state representations .1. The biorthogonal and unitary coupled-cluster methods

As a general common concept, underlying diverse methods used to compute generalized electronic excitations in atoms and molecules, intermediate-state representations (ISR's), are considered and analyzed. Essentially, an ISR results by representing the excitation energy operator in terms of so-called correlated excited states (CES's) or states derived thereof. Three different ISR schemes are compared, namely the biorthogonal coupled-cluster (BCC) representation used in both the coupled-cluster linear response and equation-of-motion coupled-cluster methods, a unitary coupled-cluster (UCC) representation, and the excitation class orthogonalized (EGO) representation resulting from a Gram-Schmidt orthogonalization procedure for the CES. Moreover, the relationship between the BCC scheme and the symmetry-adapted-cluster-configuration-interaction method is discussed. The relevance of the ISR schemes, as opposed to the much simpler configuration-interaction (CI) expansions, arises from two basic properties referred to as separability and compactness. The former property is a sufficient condition for size-consistent results, while the latter allows one to use smaller explicit configuration spaces than in comparable CI treatments. We show that the ECO and UCC representations are both separable and compact, whereas a somewhat restricted compactness property applies in the BCC case.