The concept of an effective Hamiltonian in Fock space is introduced. It is based on the division of the entire one-particle space into subspaces of ‘‘active’’ and ‘‘inactive’’ orbitals. The effective Fock space Hamiltonian has—for active model states—the same eigenvalues as the full Hamiltonian. The theory outlined in this context differs from that of paper I mainly in a different definition of the ‘‘diagonal part’’ of an operator, and in the fact that the ‘‘quasidegenerate case’’ applies throughout. The separation theorem, and as a consequence the connected diagram theorem, is shown to hold, in a more limited sense though, even for those normalizations where it did not in the context of universal wave and energy operators. Unlike in the theory of the ‘‘universal’’ operators of paper I the Fock space and n-particle Hilbert space approaches with analogous normalizations are no longer equivalent. In particular, the Primas normalization with a fully Lie-algebraic structure does not lead to a connected diagram expansion if it is formulated in n-particle Hilbert space, only so in a Fock space formulation. In n-particle Hilbert space with the present definition of the diagonal part of an operator the normalizations b (‘‘canonical’’) and c (‘‘Primas’’) happen to agree. As an alternative to the construction of the wave and energy operator W and L by perturbation theory the nonperturbative approach is presented as a generalization of the coupled-cluster method, in detail both in the intermediate and in the unitary normalization. In the unitary variant only a linear system for σ (the logarithm of the wave operator) has to be solved in order to get L correct through fifth order in perturbation theory with important contributions of higher orders included. A generalization of the Hartree–Fock method to Fock space theory is outlined, which guarantees stationarity of all (active) eigenstates with respect to one-particle transformations. A generalized electron pair theory is also defined. An analysis of the necessary computational steps shows that the nonperturbative approaches do not require significantly more computational effort than perturbation theory to the corresponding order. As a numerical example the H2 molecule in a small basis is discussed.

The known CEPA variants CEPA (v) with v = 0, 1, 2, 3 and two new ones with v = 4, 5 are compared both formally and for various numerical examples with CP-MET. The main conclusions are: 1. In those situations where both CP-MET and the CEPA variants are justified (i.e. for "good" closed shell states) the correlation energies obtained with the 7 different schemes differ very little (by something like +2%), with CEPA (1) closest to CP-MET (difference usually a fraction of 1%) and CEPA (4) nearly as close; this is rather insensitive to whether one uses canonical or localized orbitals. Even CEPA (3) is not too far from CP-MET, which confirms an earlier suggestion of Kelly. 2. In those cases where one of the 7 schemes fails (e.g. due to near degeneracy as in covalent molecules at large internuclear distances) the other 6 usually fail as well, though CEPA (0) is then somewhat poorer than the other schemes. Then no longer CEPA (1) but rather CEPA (3) is closest to CP-MET and then all schemes converge much better in a localized representation. 3. CEPA (2) usually leads to best agreement with experiment since it simulates to some extent triple substitutions. In none of the studied examples does CP-MET show a significant superiority as compared to the other schemes. Possible improvements to extend the domain of applicability of these methods are discussed.Key words: Electron correlation -Coupled electron pair approximation (CEPA) -Coupled cluster (CC) -Coupled pair many electron theory (CP-MET) -Many-body perturbation theory (MBPT) -M~ller-Plesset perturbation theory (MP-PT) -Near degeneracy.

No 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.

No abstract