core of the interaction, are introduced by a radial distance-dependent shift pushing nucleons apart from each other if they are within the range of the core. Tensor correlations between two nucleons are generated by a spatial shift perpendicular to the radial direction. For a given bare potential, the corresponding correlation functions are determined by an energy minimization in the two-body system for each $(S,T)$ channel.

Matrix elements of an operator $O$ with correlated many-body states $|\tilde{\Psi}\rangle$ can be equivalently written as matrix elements of a “correlated" (transformed) operator $\tilde{O}=C^{\dagger}OC$ and uncorrelated many-body states $|\Psi\rangle$, thanks to the unitarity of the correlation operator. Thus, one can work in simple Hilbert spaces (simple states) using correlated operators, as well as with bare operators and explicitly correlated states. By applying the unitary transformation to a bare NN interaction, a phase-shift equivalent correlated interaction is obtained, which is suitable for use in tractable model spaces [@RNH2004; @RHP2005; @RPP2006]. The same transformation can then be applied to any other operator under study, as is needed for a consistent UCOM treatment.

There are a couple of important issues arising in actual applications. In an $A$-body system the correlated operator contains irreducible contributions to all particle numbers. 