Identities, perturbative expansions, and recursion relations for mapping operators, effective Hamiltonians, and effective operators

Abstract

Perturbative and complete model space linked diagrammatic expansions for the canonical effective operatorWe derive perturbation expansions for the mapping operators (k,l) that transform a full Hilbert space time-independent Hamiltonian H and operator A into, respectively, a finite (multidimensional) space effective Hamiltonian h and effective operator a. The eigenvalues of h are identical to some of those of H, and a produces exact matrix elements of A for the corresponding states. Our derivations are substantially both more general and simpler than most literature ones and yield simple linear recursive expressions for k and I. Both these mapping solutions and new identities involving h, a, k, and I straightforwardly produce new recursive relations for h and the first known recursive a expressions. We apply our results to the Bloch, Kato, and all norm-preserving formalisms, including the canonical one. The new h and a identities are also shown to be suitable for iterative and muItireference coupled cluster approaches.which is equivalent to the canonical case of Eq. (4.5), Le., KeHe = HKe . Mukherjee and co-workers also apply special versions of Eq. (4.5) to open-shell CC formalisms.44-46

General calculation of 4f-5d transition rates for rare-earth ions using many-body perturbation theory

“…A systematical method to take all those contributions into account in a reduced space is effective operator method. 13,14,15 In this method, the full Hilbert space timeindependent Hamiltonian H is transformed into an effective Hamiltonian H eff , which acts on the reduced space (referred to as model space) and gives upon diagonalization a set of exact eigenvalues and model space eigenvectors. For a time-independent operator O, an effective operator O eff may be introduced that gives the same matrix elements between the model space eigenvectors of H eff as those of the original operator O between the corresponding true eigenvectors of H. Effective Hamiltonians H eff and transition operators O eff are often constructed by many-body perturbation theory (MBPT) with order by order approximation 15 and then represented by connected diagrams similar to Feynman diagrams.…”

Irreducible Green functions method and many-particle interacting systems on a lattice

“…For many practically interesting cases (e.g. in quantum chemistry problems ), the standard schemes of perturbation expansion must be reformulated greatly [11] - [15]. Moreover, many-body systems on a lattice have their own specific features and in some important aspects differ greatly from continuous systems.…”

Simplified diagrammatic expansion for effective operators

“…Nonetheless, There are common-used effective Hamiltonians, one hermitian and the other nonhermitian are known to contain only connected diagrams. [16] The rules to generate diagrams and to evaluate them are wellknown. Factorization theorem is shown to be able to combine diagrams having the same set of vertexes and lines but different relative orderings of vertexes, and hence reduces the number of high-order diagrams.…”

“…The hermitian (canonical) effective transition operator, which works together with the hermitian (canonical) effective Hamiltonian, has been presented in detail by Hurtubise and co-workers in a series of papers. [1,9,16] Duan and Reid [19] constructed a simple connected nonhermitian O eff that works together with the connected nonhermitian effective Hamiltonian and showed how to construct a connected expansion. Since it is well-known that hermitian effective Hamiltonian up to third order can be obtained from trivial symmetrization, [6] and this can also be shown for effective transition operators (up to third order in V , the perturbation in Hamiltonian), and also for higher-order calculations usually coupled-cluster methods come into play, most researches required only limited order diagram calculations of nonhermitian effective Hamiltonian and nonhermitian effective transition operators.…”

“…However, there are two problems with the effective operator methods: too many diagrams, and the rules to calculate energy denominators for effective transition operators being different from those for effective Hamiltonian. [16] Our aim in this paper is to modify the diagram representation of many-body perturbation expansion of effective Hamiltonian and effective transition operators, so that the energy denominator rules for effective transition operators and effective Hamiltonian are the same, and the number of diagrams is greatly reduced and becomes handleable for third order terms of nonhermitian effective transition operators. In section II we present our diagrammatic representation of many-body operators and the corresponding evaluation rules for perturbation diagrams; in section III we illustrate the generalized factorization theorem for the effective transition operators; and in Section IV we show how to denote several diagrams of the same set of vertexes and lines with only one diagram to group and reduce the number of diagrams.…”