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Physical Review C

Cluster Expansions for Correlated Wave Functions of Finite Systems

João da Providênica, Carl M. Shakin

Published Nov 1, 1971• 20 citations
Publisher

Abstract

A theory of correlated wave functions for finite systems is developed which systematically neglects the contribution from three-body cluster terms. The approximation made leads to a derivation of a Bethe-Goldstone equation for finite systems which includes modified occupation probabilities and self-consistent potentials for occupied states. The absence of potentials in unoccupied states follows from the neglect of three-body cluster terms. It is also shown that a careful treatment of the Pauli principle and occupation probabilities allows the application of variational techniques.

DOI
10.1103/physrevc.4.1560

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Physical Review•Dec 16, 1966•184 citations
A theory of correlated wave functions for finite systems is developed which systematically neglects the contribution from three-body cluster terms. The approximation made leads to a derivation of a Bethe-Goldstone equation for finite systems which includes modified occupation probabilities and self-consistent potentials for occupied states. The absence of potentials in unoccupied states follows from the neglect of three-body cluster terms. It is also shown that a careful treatment of the Pauli principle and occupation probabilities allows the application of variational techniques.
The cluster expansion for the energy for a system with Bijl-Dingle-Jastrow (BDJ) correlation structure is derived in a new form. It is shown how the theory may be related to the Brueckner-Bethe-Goldstone theory with a particular choice for the correlation operator (or wave matrix) appearing in the BDJ wave function. The three-body cluster terms are discussed. It is shown that these clusters contain a limited class of dispersive terms which may be included in the analysis of the two-body clusters. The three-body clusters are also shown to contain terms which correct for violations of the Pauli principle in the correlation operator.
It is shown that starting with a correlated wave function for nuclear matter of a general form one may make direct correspondence to the conventional diagrammatic theory of Brueckner, Bethe, and Goldstone. In particular, in this work we show that the expression for the three-body-cluster energy is the same as that obtained by Bethe. Also, it is shown that the current approximation which neglects the potential for particle states in the diagrammatic approach corresponds to a neglect of a coupling between the two-body (Bethe-Goldstone) equation and the equation for the three-body-cluster wave function (Bethe-Faddeev). The theory presented here provides a clarification of the relation of the nnn-body-cluster wave functions and the structure of the wave function of the entire system.
The cluster variation method, which was originally devised for quantum-mechanical many-body systems at zero temperature, is extended to treat finite-temperature systems. The new method is based on a variational principle that the partition function is the maximum of a certain functional, APF (approximate partition function). This method includes the conventional cluster variation method at zero temperature as a limiting case. A procedure is given for calculating APF by cluster expansion for fermion systems. In this procedure, use is made of single-particle energies which should be determined self-consistently. The non-orthogonality problem which comes from the presence of correlations among particles is solved by the use of linear combinations of (generalized) Jastrow wave functions. Further development of this method is briefly discussed.
The usual perturbation treatment of the many-body problem is modified by eliminating the two-body interaction and introducing in its place a ttt matrix. The Watson, Brueckner, and Bethe (W.B.B.) treatment of the many-body problem is shown to be a special case of the modified perturbation treatment. The modified perturbation treatment involves no assumption except that of supposing the convergence of the perturbation expansion. Thus to each order in the expansion the results must be meaningful and no unphysical "unlinked cluster" terms can appear. Within the framework of the modified perturbation treatment it is easy to interpret the concepts of the Pauli principle for intermediate states and the self-consistent choice of the comparison potential which play such an important role in the W.B.B. treatment. This work is similar to that of J. Goldstone in that we relate the W.B.B. treatment to a perturbation treatment of the many-body problem. We differ from Goldstone in that we use the usual time-independent perturbation theory instead of the time-dependent one. The time-independent approach permits a direct comparison with the W.B.B. treatment and facilitates discussion of the Pauli principle for intermediate states and of the self-consistent potential.
We study many-body correlations in the ground state of a general quantum system of bosons or fermions by including an additional Jastrow function in our recently proposed variational coupled-cluster method. Our approach combines the advantages of state-dependent correlations in the coupled-cluster theory and of the strong, short-ranged correlations of the Jastrow function. We apply a generalized linked-cluster expansion for the Jastrow wave function and provide a detailed analysis for practical evaluation of the Hamiltonian expectation value as an energy functional of the Jastrow function and the bare density-distribution functions introduced and calculated in our earlier publications; a simple, first-order energy functional is derived and detailed formulas for the higher-order contributions are provided. Our energy functional does not suffer the divergence as most coupled-cluster calculations often do when applying to Hamiltonians with hardcore potentials. We also discuss possible applications of our technique, including applications to strongly correlated fermion systems.
Bethe's treatment of three-body clusters is extended to the entire linked-cluster perturbation series. The result is a completely rearranged expansion in terms of "compact clusters." The spatial correlations within these terms are all of quite short range. As a direct consequence, each of these new compact-cluster terms is proportional to ${\ensuremath{\kappa}}^{{h}^{\ensuremath{'}}\ensuremath{-}1}$, where ${h}^{\ensuremath{'}}$ is the number of momenta inside the Fermi sea which can be summed independently after allowing for momentum conservation. The "small parameter" of the expansion is $\ensuremath{\kappa}=\ensuremath{\rho}\ensuremath{\int}{|\ensuremath{\zeta}|}^{2}d\ensuremath{\tau}$, where $\ensuremath{\rho}$ is the ordinary nuclear density and $\ensuremath{\zeta}=\ensuremath{\varphi}\ensuremath{-}{\ensuremath{\psi}}_{\mathrm{BG}}$ is the "wound" in the correlated two-body (bethe-Goldstone) wave function. In nuclear matter this $\ensuremath{\kappa}$ is of order 10%. This result requires that the single-particle potentials be defined in terms of a certain subset of all the self-energy insertions. The allowed insertions are those which can be evaluated entirely on the energy shell, by means of a "generalized time-ordering" factorization. The totality of these insertions is said to constitute an "on-energy-shell mass operator" Mon{M}^{\mathrm{on}}Mon. The resulting occupied-state potentials are quantitatively very similar to those of most previous versions of nuclear-matter theory. However, the present intermediate-state potentials are small enough (being only of order +1 MeV) to be safely ignored in practical calculations. This simplification is offset by the need for a separate calculation of the three-body clusters. These single-particle potentials are "physically meaningful" in the sense that they lead to an optimum treatment of the short-range correlations. The bulk properties of nuclear systems are determined almost entirely by these short-range correlations. It is therefore proposed that similar potentials be used in a many-body theory of finite nuclei. True occupation numbers occur in a simple and natural way throughout the entire expansion. The formal and practical consequences of this feature are carefully examined. It is argued that this is significant for a theory of finite nuclei. Comparisons are made with several different "renormalized" formulations of quantum statistical mechanics, and also with the Green's-function theory of nuclear matter. The present formulation tends to artificially suppress the long-range correlations (e.g., pairing correlations) between particles near the Fermi surface. This defect can be eliminated by using a degenerate analog of the Goldstone expansion.