Quantum chemistry in Fock space. II. Effective Hamiltonians in Fock space

Abstract

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.

In-medium similarity renormalization group for closed and open-shell nuclei

“…[129][130][131] ). Using a broad definition of diagonality is ill-advised because we must avoid to induce strong in-medium 3N, .…”

Quasi-variational coupled cluster theory

“…"Alternative" CC methods have been proposed, including extended coupled cluster 34 (ECC), quadratic coupled cluster 35 (QCC), improved coupled cluster 36 (ICC), unitary coupled cluster [37][38][39][40] (UCC), and variational coupled cluster 41 (VCC), and a number of studies 20,[28][29][30][31][32][33] have confirmed the superiority of VCC and related approaches over TCC for the treatment of problems involving strong static correlation. This can be attributed to the upper bound property of VCC, which ensures that calculated VCC energies are always higher than FCI energies, which is not the case for TCC.…”

Convergence studies in the theory of effective Hamiltonians

“…(1) for n energy levels. In (8), H'ff is an effective Hamiltonian defined in the n-dimensional model space and CR is the corresponding wave operator acting on the right in the model space. H ' f f is chosen in such a way that the n solutions of H'ff in the model space are the projection in this space of n exact solutions of H. Within such a choice, R obeys…”