Aspects of linked cluster expansion in general model space many-body perturbation and coupled-cluster theory

In this paper, a method of generating separable forms of the wave-operator for incomplete model spaces is discussed. With a time-dependent access to the many-body perturbation and coupled-cluster theories, it is shown how one can extract the regular part of the wave-operator which consists of linked cluster-operators only in the adiabatic limit. The procedure naturally suggests a hierarchy of lower valence model spaces P(k). once a particular m-valence incomplete model space P(m) is specified. The wave-operator Ω and the effective Hamiltonian Heff are linked in this development and are valence-universal in the sense of being valid for all P(k)' s. 0 ⩽ k ⩽ m. We have derived two distinct forms for Ω: (i) Ω = {exp(S)}, with { } as normal order with respect to suitable vacuum, where S are open operators inducing transitions from P(m) to outside it; (ii) ΩN = {exp(S + X)}, where X are additional closed operators which are introduced to maintain isometry of ΩN: P(k)ΩN + ΩN P(k) = P(k). In neither of these choices do we have intermediate normalization. It is also possible to develop an alternative strategy with the complete model spaces, such that an effective valence-universal operator H may be found which generates roots, only a subset of which are equal to the eigenvalues of H. These subsets are the ones that Heff would have furnished. This may thus be viewed as a Fock-space realization of the intermediate Hamiltonian approach.