Constrained variation in Jastrow method at high density

A method is derived for constraining the correlation function in a Jastrow variational calculation which permits the truncation of the cluster expansion after two-body terms, and which permits exact minimization of the two-body cluster by functional variation. This method is compared with one previously proposed by Pandharipande and is found to be superior both theoretically and practically. The method is tested both on liquid ³He using the Lennard-Jones potential and on the model system of neutrons treated as Boltzmann particles (“homework” problem). Good agreement is found both with experiment and with other calculations involving the explicit evaluation of higher-order terms in the cluster expansion. The method is then applied to a more realistic model of a neutron gas up to a density of 4 neutrons per F³, and is found to give ground-state energies considerably lower than those of Pandharipande.     

Model energy calculations of nuclear matter in the Jastrow variational approach

We calculate the model energy of nuclear matter within the FIY expansion scheme. Numerical results are presented for two semi-realistic central potentials and a state independent correlation function which is suitably constrained for good cluster convergence.     

Long-range Jastrow correlations

We develop a promising many-body method to evaluate the equation of state for dense neutron matter and liquid helium. The ground state of the Fermi fluid is described by a conventional Jastrow ansatz. We admit the presence of short- and long-range correlations. Under this assumption we study the generating function which has been introduced by Wu and Feenberg. We employ a graphic formulation and develop the diagrammatic expansion of the generating function and the radial distribution function. If long-range correlations are assumed, the diagrams have singular parts. We give a proof that the total contribution of such diagrams to the generating function which contain two, three, and four correlation lines is of finite value. The same property is shown for a selected class of singular diagrams containing α correlation lines (α>4). To verify the cancellation phenomenon we introduce a two-body function which serves graphically as an insertion into selected singular diagrams. For the remaining classes of diagrams we need three-, four-, ?, n-body insertions. The result is cast into the form of a theorem. The cancellation rests on the exclusion principle and does not depend on the special shape of the correlation function. Finally, a generalized hypernetted-chain summation of diagrams which represent the radial distribution function is executed. The procedure includes exchange contributions and can be employed if short-and/or long-range correlations are present.     

On Jastrow wavefunctions containing attractive correlations

A quantum mechanical explanation is given in support of the form of Jastrow wavefunction introduced by De Michelis and Reatto. We propose a new, simpler wavefunction which has the correct long-range behaviour.     

Constrained variational calculations on nuclear matter with state dependent correlations

Constrained variational calculations on the binding energy of nuclear matter have been performed employing three potentials with differing mixtures of central and tensor components. We present numerical results on the energy expectation value in two- and three-body cluster approximation as function of the separation distance at which the central and tensor correlations heal smoothly.     

Constrained variational calculations of closed shell nuclei with the reid potential

Lowest-order constrained variational calculations with harmonic oscillator wave functions are carried out for ⁴He, ¹⁶O, and ⁴⁰Ca nuclei with the Reid potential. The results obtained with this simple method are in very close agreement with those obtained by renormalized Brueckner-Hartree-Fock theory with orthogonalized plane wave intermediate states. The A-dependence of the difference between the experimental and calculated binding energies for A = 2, 3, 4, 16, 40 and ∞ can be explained by a three-body cluster term coming either from a wrong off-energy-shell behavior of the Reid potential or a three-body force. The calculated radii of ¹⁶O and ⁴⁰Ca are ≈ 10% too small, indicating that the Reid potential may not be very realistic.     

Jastrow correlations and the hard sphere fermi gas

A diagrammatic expansion for the variational energy of a Fermi fluid using state-dependent correlations is described. It is used to obtain the leading order terms (up to (k_Fa)²) in the low density expansion of a hard sphere Fermi gas using both state-dependent and state-independent correlations.     

Brueckner theory and Jastrow approach for finite nuclei

The results of Jastrow variational calculations and of Brueckner theory in lowest order (Brueckner-Hartree-Fock) are compared. The comparison is made for the calculations of ground state properties like binding energy and charge-radius of light and medium weight, closed shell nuclei (⁴He, ¹⁶O, ⁴⁰Ca). For the nucleon-nucleon interaction rather simple forces are used (Brink-Boeker potential B1, Afnan-Tang potential S3). For all cases considered it turns out that the results of the two different methods are in fairly good agreement, with the binding energy calculated in the Brueckner-Hartree-Fock approximation being always slightly below the corresponding upper bounds calculated in the Jastrow variational approach. This good agreement between the two methods indicates, that for light and medium weight nuclei the Jastrow variational approach and the Brueck-ner-Hartree-Fock approximation can be considered as reasonable approximations to a complete solution of the many-body problem.     

On the variational approach to Jastrow correlations in nuclei

The variational equation determining the Jastrow correlation function is investigated with particular emphasis on the healing problem for both nuclear matter and finite nuclei. The consequences of several healing conditions are discussed. Furthermore, influences from the choice of the single particle basis and from long range correlations are studied and are found to be small in the short range region.     

Model energy — A study of the Jastrow correlation factor

We calculate the model energy of nuclear matter for two semi-realistic potentials with state-independent hard cores. Here we have included the delta-function contributions arising from the hard cores, which have been omitted in a previously published paper.     

Calculation of Landau's Fermi-liquid parameters in the extended Jastrow theory

In the framework of the unrenormalized and renormalized Jastrow theory of Fermi liquids we derive general expressions for the Landau parametersF _i ^s andF _i ^a (i=0, 1, 2, ). As an illustrative application we numerically calculate the Landau parameters in pure neutron matter starting from the simple two-body approximation for the energy expectation value in the unrenormalized version. The limitations of such preliminary estimations are discussed and necessary improvements are pointed out.     

Numerical comparison of reaction-matrix and Jastrow methods for nuclear matter

The ground-state energy per particle of symmetrical nuclear matter has been calculated as a function of Fermi wave number for two representative central potentials containing hard cores, by means of both reaction-matrix and Jastrow methods. The agreement of results from the two methods is remarkably good in the density region near equilibrium.     

Comparison of the structure of Jastrow and exact many fermion theories

The structure of Jastrow and exact many body theory (in expS-form) are compared. The results are: 1. The correct linked clusters can be computed in terms of Jastrow correlation functions; 2. especially the Bethe-Goldstone (or defect) wave functions are defined uniquely and differ from those used in former comparisons; 3. one should use the (Jastrow) model energy instead of the expectation value thereby avoiding violation of the Pauli principle and some problems with higher order terms.     

Comparison of the structure of Jastrow and exact many fermion theories

The structure of Jastrow and exact many body theory (in expS-form) are compared. The results are: 1. The correct linked clusters can be computed in terms of Jastrow correlation functions; 2. especially the Bethe-Goldstone (or defect) wave functions are defined uniquely and differ from those used in former comparisons; 3. one should use the (Jastrow) “model energy” instead of the expectation value thereby avoiding violation of the Pauli principle and some problems with higher order terms.     

Jastrow approach to nuclear matter without truncation of the cluster expansion

It is shown that an approximate expression in closed form can be obtained for the Jastrow trial energy per particle in nuclear matter. The undesirable truncation of the cluster expansion may thus be avoided.     

On the relation between Brueckner theory and Jastrow theory of nuclear matter

The two- and three-hole-line contributions to the ground state energy as calculated from Brueckner theory are derived from a cluster expansion followed by variation of the trial function. The implications of that derivation both for Brueckner theory and for Jastrow theory are worked out in detail. It is argued that the Jastrow theory is able to give simpler methods to calculate the ground state energy which may be of the same accuracy as current Brueckner calculations. It is shown that the single-particle potential of Brueckner theory is intimately related to a subsidiary condition used in the variation of the trial function. The main steps which have to be taken in a derivation of the general hole-line expansion from Jastrow theory are indicated. It is shown that the hole-line expansion is not a cluster expansion in the sense of Jastrow theory, and an interpretation is given of the “self-consistent choice” of the single-particle potential advocated in Brueckner theory.     

Nonconvergence of the cluster expansion for antisymmetry in the Jastrow many-body theory

The cluster-expansion treatment of antisymmetry, as used in most Jastrow-theoretic calculations to date for Fermion systems, is shown to be seriously inadequate. This is offered as an explanation for the recently reported discrepancy between Jastrow-theoretic and Brueckner-Bethe-Goldstone results for nuclear matter. This should also help to resolve some current difficulties in accounting for the properties of neutron stars.     

Effect of three-body cluster on the healing properties of the Jastrow Correlation function

A variational equation for the Jastrow Correlation function is derived from the energy functional expanded up to three-body cluster terms. The asymptotic behaviour of this nonlinear equation is studied. The solutions show a healing at least of the type cos(tαr)/r ². The influence of higher cluster contributions is studied. Finally, it is discussed, how one can reduce the many-body cluster contributions to healing conditions to be used in the two-body cluster treatment.     

Monte Carlo test of the convergence of cluster expansions in Jastrow correlated nuclei

By comparing the results of the variational Monte Carlo method with various orders of a multiplicative cluster expansion appropriate to finite nuclear systems, we draw conclusions about the quality of the latter.