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Physical Review C

Subsidiary Conditions on Nuclear Many-Body Theories

J. W. Clark, MANFRED L. RISTIG

Published May 1, 1972• 15 citations
Publisher

Abstract

In the Jastrow theory applied to low cluster order it is necessary to restrict the variation of the two-body correlation factor $f(r)$ to the domain of functions corresponding to acceptable convergence of the cluster expansion for the energy. We examine in detail two constraints which have received some attention: the "normalization condition" and the "orthogonality (or average Pauli) conditions." The former constraint is motivated by the requirement that the liquid structure factor $S(k)$ vanish at zero wave number; the latter, by the Pauli principle in the Fermi medium. It is shown that two-body Jastrow theory cannot obey both constraints exactly for a state-independent correlation factor $f\ensuremath{\not\equiv}1$, but that the requirement $S(0)=0$ is automatically satisfied in (two-body) Brueckner reaction-matrix theory as a trivial consequence of the fulfillment, by definition of this theory, of the Pauli principle in the medium.

DOI
10.1103/physrevc.5.1553

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In the Jastrow theory applied to low cluster order it is necessary to restrict the variation of the two-body correlation factor f(r)f(r)f(r) to the domain of functions corresponding to acceptable convergence of the cluster expansion for the energy. We examine in detail two constraints which have received some attention: the "normalization condition" and the "orthogonality (or average Pauli) conditions." The former constraint is motivated by the requirement that the liquid structure factor S(k)S(k)S(k) vanish at zero wave number; the latter, by the Pauli principle in the Fermi medium. It is shown that two-body Jastrow theory cannot obey both constraints exactly for a state-independent correlation factor f\ensuremath≢1f\ensuremath{\not\equiv}1f\ensuremath≡1, but that the requirement S(0)=0S(0)=0S(0)=0 is automatically satisfied in (two-body) Brueckner reaction-matrix theory as a trivial consequence of the fulfillment, by definition of this theory, of the Pauli principle in the medium.
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.
A nonperturbative derivation of the Brueckner approximation to the ground-state energy of nuclear matter (with standard choice of hole and particle energies) is given within the framework of the method of correlated basis functions. The structure of the factor-cluster (FIY) expansion of the expectation value of the Hamiltonian with respect to a trial function incorporating short-range correlations in a general fashion, is analyzed for a uniform extended medium of fermions. With a special choice of correlated wave function, the Brueckner approximation is extracted from the FIY expansion by selective summation of two-body combination terms to all orders in the smallness parameter $\ensuremath{\kappa}$ associated, in this case, with both Brueckner-Bethe-Goldstone and FIY energy expansions. A reexamination of the numerical comparison of conventional Brueckner and Jastrow methods carried out earlier for two simple models of nuclear matter leads to the conclusion that it is inadvisable, at least in a Jastrow calculation satisfying the average Pauli condition, to incorporate the $O(\ensuremath{\kappa})$ contribution of the analog of the dispersion correction of reaction-matrix theory without at the same time including the $O(\ensuremath{\kappa})$ contribution of the analog of the Bethe-Faddeev term. For the uniform extended medium for the Jastrow choice of wave function, there is a formal cancellation of these two three-body contributions, the scale of this cancellation being governed by the size of $\ensuremath{\kappa}$. In the numerical example considered, $\ensuremath{\kappa}$ is about 0.22 and the cancellation goes something like +14 MeV-15 MeV=-1 MeV. These findings, when juxtaposed with the standard Brueckner results for the simple models in question, suggest that a departure from the standard prescription of hole and particle energies in reaction-matrix theory may be necessary when $\ensuremath{\kappa}$ is sizeable-as in liquid 3He^{3}\mathrm{He}3He and the high-density neutron gas, as well as equilibrium nuclear matter described with some potentials.
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.
The difficulties which arise in the conventional development of the Jastrow method for fermions, from the failure to impose proper wave-function antisymmetry, are exhibited by an extension of the diagrammatic analysis of Pandharipande and Bethe. These problems are readily solved by the use of second quantization and a corresponding modification of the usual cluster expansion. A numerical example for a strong repulsive interaction shows that inaccurate treatment of the antisymmetrization can lead to very large errors in low-order estimates of the energy.NUCLEAR STRUCTURE variational calculations, nuclear matter, Jastrow method, cluster expansion, antisymmetrization.
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.
A linked-cluster expansion for the expectation values of operators between correlated wave functions of the Jastrow type is described. The expansion is based on the logarithm of the correlation function and leads to terms that are explicitly proportional to the particle number for infinite systems. The lowest-order terms for a weakly interacting system of spin-1/2 particles are calculated explicitly. From these, the Euler-Lagrange equations for the correlation functions have been calculated. The equation for the correlation function between particles of antiparallel spin can be solved by Fourier transform, and it is shown that for $r\ensuremath{\rightarrow}\ensuremath{\infty}$, the correlation function behaves like 1+Ar21+\frac{A}{{r}^{2}}1+r2A​. In the case of a system of particles interacting by hard-core repulsions, the logarithm of the correlation function does not exist inside the hard-core radius. It is then necessary to sum over ladder diagrams as in the derivation of the reaction matrix in many-body perturbation theory. This is carried out to provide a low-density approximation to the ground-state energy of the system and the Euler-Lagrange equation for the correlation function. It is argued that the correlation function again behaves like 1+Ar21+\frac{A}{{r}^{2}}1+r2A​ for large rrr. The expansion is applied to the problem of calculating the correlation energy of the electron gas. It is shown that a variational calculation based on the lowest-order terms in an expansion of the wave function leads to a divergent result (unbounded from below), but that a convergent result is obtained by summing over ring diagrams, which correspond to long-range polarization of the electron cloud. The result is similar to that of Gaskell obtained by a somewhat different method. It is shown analytically that at high densities the result behaves logarithmically, specifically as $0.0570\mathrm{ln}{r}_{s}\ensuremath{-}0.1324$ Ry. The lowest-order exchange contributions have also been calculated and found to contribute 0.040 Ry to the energy at densities of physical interest.