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Physical Review

Methods for Calculating Ground-State Correlations of Vibrational Nuclei

DAVID J. ROWE

Published Nov 20, 1968• 207 citations
Publisher

Abstract

Various methods are investigated for calculating correlated ground states and their one-particle densities, with a view to including correlation effects in higher random-phase-approximation (RPA) calculations. The problem of the divergent correlations associated with the center-of-mass and other spurious RPA solutions is discussed and it is shown that they can simply be ignored.

DOI
10.1103/physrev.175.1283

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Physical review. B, Condensed matter•Nov 15, 1982•142 citations
Various methods are investigated for calculating correlated ground states and their one-particle densities, with a view to including correlation effects in higher random-phase-approximation (RPA) calculations. The problem of the divergent correlations associated with the center-of-mass and other spurious RPA solutions is discussed and it is shown that they can simply be ignored.
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.
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.
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.
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.
The random phase approximation (RPA) builds in correlations left out by mean-field theory. In full $0\ensuremath{\Elzxh}\ensuremath{\omega}$ shell-model spaces we calculate the Hartree-Fock (HF)+RPA(\mathrm{HF})+\mathrm{RPA}(HF)+RPA binding energy, and compare it to exact diagonalization. We find that often, but not always, the HF+RPA\mathrm{HF}+\mathrm{RPA}HF+RPA gives a good approximation to the ``exact'' ground state energy. In those cases where the RPA is less satisfactory, however, there is no obvious correlation with properties of the HF state, such as deformation or overlap with the exact ground state wave function. This weakens the reliability of the RPA for computing binding energies.
In this second work in a series of papers the coupled-cluster or exp(S)\mathrm{exp}(S)exp(S) formalism is applied to the problem of ground-state correlations in the one-component Fermi plasma. The main approximation employed is the so-called SUB2 approximation for the two-body subsystem correlation operator S2{S}_{2}S2​ which provides a measure of the two-particle---two-hole component in the true ground-state wave function, and in which the coupling to larger subsystems is neglected. The SUB2 equations for Fermi systems are brought into tractable form by a "state-averaging" procedure which we develop by analogy with the comparable Bose equations, and which is shown by comparison with earlier exact results to be accurate in the metallic density regime at about the 1% level. Our final results for metallic densities include the completely integrated and self-consistent effects of the terms which by themselves generate (i) the random-phase approximation (RPA) and its associated exact long-range screening effects, (ii) the extra random-phase approximation exchange terms necessary to keep the RPA explicitly antisymmetric, (iii) the self-consistent particle-particle ladders (LAD) that describe two-particle scattering within the many-body medium and which describe the exact short-range behavior, (iv) a class of particle-hole ladder terms, and (v) the self-consistent hole-potential terms. Particular attention is paid to the important effects caused by the interference at intermediate separations of the long-range RPA and the short-range LAD effects. By comparison with recent and essentially exact stochastic simulations of the many-body Schr\"odinger equation, our results are seen to be accurate to about 1% for metallic densities, and hence to provide what is probably the currently best microscopic description available for this system. We show also that even in the low-density limit the SUB2 approximation provides a good "translationally-invariant-solid" description of both charged Fermi and Bose systems in this exact Wigner crystal regime.