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Physical Review

Methods for Calculating Ground-State Correlations of Vibrational Nuclei

R. E. Johnson, R. M. Dreizler, Abraham Klein

Published Oct 20, 1969• 36 citations
Publisher

Abstract

A general method for studying ground-state correlations is described. It is shown that the usual random-phase approximation overestimates depletion of the uncorrelated state by a factor of 2, in agreement with a result found by Rowe, using the number-operator method. The correctness of the latter method is then independently verified. A precise division of the ground-state correlations into collective and noncollective parts is shown for a simple model, but it is argued that the division is not valid in general.

DOI
10.1103/physrev.186.1289

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Physical Review C•Aug 1, 1972•2 citations
A general method for studying ground-state correlations is described. It is shown that the usual random-phase approximation overestimates depletion of the uncorrelated state by a factor of 2, in agreement with a result found by Rowe, using the number-operator method. The correctness of the latter method is then independently verified. A precise division of the ground-state correlations into collective and noncollective parts is shown for a simple model, but it is argued that the division is not valid in general.
The short range repulsion between nucleons is treated by a unitary correlation operator which shifts the nucleons away from each other whenever their uncorrelated positions are within the replusive core. By formulating the correlation as a transformation of the relative distance between particle pairs, general analytic expressions for the correlated wave functions and correlated operators are given. The decomposition of correlated operators into irreducible n-body operators is discussed. The one- and two-body-irreducible parts are worked out explicitly and the contribution of three-body correlations is estimated to check convergence. Ground state energies of nuclei up to mass number A=48 are calculated with a spin-isospin-dependent potential and single Slater determinants as uncorrelated states. They show that the deduced energy- and mass-number-independent correlated two-body Hamiltonian reproduces all "exact" many-body calculations surprisingly well.
A general theory of long range correlations in the ground state of a closed shell nucleus is outlined, using a number of ideas introduced or adumbrated, but not always successfully developed, in previous work of the authors and others. These include the following: (i) Any operator can be formally expressed as a series in particle-hole (p-h) excitation and destruction operators which should converge rapidly for a suitable choice of single-particle basis at the closed shells. By means of these relations and with the help of sum rules all observables can be computed without the explicit use of wave functions. (ii) For a fixed choice of single-particle basis, the ground state energy can be computed from a variational principle in which in lowest nontrivial approximation, in consequence of (i), the parameters are amplitudes of the same type as occur in the random phase approximation (RPA). The precise relation to RPA is fully detailed. (iii) The single particle basis chosen is the natural orbital basis of Lowdin, which presents itself as the obvious choice both for formulation of the dynamics and for the algorithm of solution. (iv) Though the theory deals with pairs of fermion operators, particular attention is paid that all Pauli principle restrictions are satisfied. (v) It is verified that in the perturbation limit, known results are reproduced. The method is generally applicable to any system for which the correlations can be viewed as particle-hole excitations with respect to a reference Slater determinant.
It is well known that the second-order ground-state correlation energy calculated in the usual random-phase approximation (RPA) is off by a factor of 2. It has also been noted that there is a factor-of-2 discrepancy in the correlation corrections to the Hartree-Fock occupation probabilities. The reason for these phenomena is traced to the treatment of the exchange terms in the interaction matrix element. An illustration based on a simple single multipole interaction shows that the factor of 2 arises from a coherent sum of small higher-order terms. Results based on the consideration of one or a few eigenmodes at a time are seen to be free of this difficulty. The correlation energy, occupation-probability corrections, and equations of motion are reexamined in the Beliaev-Zelevinsky boson expansion, as an illustration of clearly defined higher-order treatments.
Abstract The article discusses nuclear ground state correlations (GSCs), which are particularly clearly manifested when using the formalism of quantum Green’s functions (GFs) and Feynman diagrams. In addition to the one-particle-one-hole ground state correlations in the well-known random phase approximation (RPA) method, there also exist and make a noticeable quantitative contribution GSCs that arise from using more complex configurations than in RPA, specifically, configurations containing phonons and those containing only three quasiparticles, phonon–GSCs and GSC3s, respectively. The latter case of GSC3 is discussed in detail in this paper. It has been shown that GSC3s make a significant quantitative contribution to problems related to explaining certain characteristics of low-lying excited states (phonons) and transitions between them in even–even nuclei.
We present a generalization of the Hartree-Fock (HF) method incorporating two-body correlations. The correlations are introduced by means of a unitary operator eiF{e}^{\mathrm{iF}}eiF, where FFF is a Hermitian one- and two-body operator to be determined by the variational principle. Assuming that the matrix elements of FFF are small so that powers of FFF higher than the second may be neglected, a set of linear equations determining FFF is obtained. The relation between our method and methods used to treat singular interactions is discussed. The method is applied to two recently suggested models and is compared with the HF method, second-order perturbation theory, and with the random-phase approximation. It is found that our method yields very good results within the range of validity of our approximations, which seems to call forth applications to more realistic problems. Possible applications to the study of collective phenomena in nuclei are indicated.