Single-reference coupled-cluster theory based on the multi-purpose cluster operator

In this paper, we develop a theoretical framework that extends single-reference (SR) coupled-cluster (CC) theory beyond its conventional role of describing a single electronic state-typically the lowest-energy state within the symmetry sector defined by the reference determinant. Rather than viewing the SR-CC cluster operator solely as a device for reproducing one target state, we consider more general constructions in which different components of the cluster operator play distinct roles, ranging from encoding states of different symmetry than the reference to enabling the SR-CC ansatz to describe multiple states simultaneously. These developments lead to a new class of SR-CC downfolding formalisms in which the resulting active-space effective Hamiltonians are capable of concurrently representing multiple correlated states non-orthogonal to the reference function. We establish three theorems that formalize this extension and demonstrate that standard CC downfolding emerges as a special case of the proposed framework. Finally, we introduce a Hermitian variant based on a unitary CC representation, which enables realistic simulations of ground and excited states while reducing the quantum resources required.