Connected-diagram expansions of effective Hamiltonians in incomplete model spaces. I. Quasicomplete and isolated incomplete model spaces

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Abstract
In this and the following paper, we formulate a Fock space theory for incomplete model spaces (IMS) that applies both to coupled‐cluster expansions and to perturbation theory. We stress in this paper that the concept of the ‘‘connected’’ nature of extensive quantities like an effective Hamiltonian Heff is more fundamental than the ‘‘linkedness’’ that is conventionally used in many‐body perturbation theory. The ‘‘connectedness’’ of Heff follows when the wave operator W is multiplicatively separable into noninteracting subsystems. This is ensured by writing W as an exponential Fock space operator with the exponent connected. It is demonstrated in particular that the connectedness of the exponent in W requires that the normalization condition of W be separable as well. Unlike the situation in a complete model space, the definition of ‘‘diagonal’’ or ‘‘nondiagonal’’ operators depends generally on the particular m‐valence IMS. There are, however, special categories of IMS, the ‘‘quasicomplete’’ and the ‘‘isolated’’ model spaces, for which these definitions are possible without reference to the particular IMS. The formal properties of these IMS are discussed. It is shown that for the quasicomplete model space, the intermediate normalization is not separable, while it is so for the isolated model space.