Valence universal cluster operator

Mukherjee et. alZ68-69/ formulated an explicitly connected CC theory for complete model space by invoking a valence universal cluster operator Q of the form... 

On the one hand, we can strive for a single cluster operator T, defining the valence universal wave operator U, U = exp(T), which will transform all the model space states ]< > ) into some linear combinations of the exact states jfl i), f = 1,2, , M, which in turn span the target space M, i.e.. 

Clearly, with this choice, not only are the equations for the core cluster amplitudes decoupled from the rest, but also the equations for the T1n cluster— amplitudes have only T m>amplitudes with mvalence values. It should again be noted that O acting on a P for m this approach uses implicitly a valence-universal wave-operator. 

Mukherjee/69/, use of the sufficiency conditions (7.3.9) amounts in effect to assuming that ft is a valence-universal wave-operator. In fact Haque has explicitly demonstrated/123/ that the use of a valence-universal ft in the Fock-space Bloch equation leads automatically to eqn (7.3.9) with the ad-hoc sufficiency requirement. We give the sketch of a general proof here, since it shows that the extra information content of a Fock-space ft, as opposed to a Hilbert space, can be used to advantage for ensuring the connectivity of the cluster amplitudes of S/93/. For a valence-universal ft, the Fock-space Bloch equation (6.1.15) leads to... 

In what follows, it will be useful to classify the parent n-valence IMS belonging to the space P, the complementary"active space, whose union with P forms the CMS, as R and the true virtual space containing inactive labels as Q. The cluster operators inducing the transition P+R are all labelled by valence lines, and will henceforth be denoted as quasi-open. The operators making transitions P- Q will be called open. The operators connecting model spaces only will be called closed. In general, for a valence-universal O, it so happens that products of quasi-open... 

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