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Physical Review

Nonsingular Formulation of the Brueckner Approximation for an Infinite Fermi System

Richard L. Becker

Published Aug 15, 1962• 8 citations
Publisher

Abstract

In the approximation neglecting any but single pair correlations, singularities of the reaction matrix render Brueckner's integral for the average energy per particle a singular integral. Several attempts to overcome this difficulty have been unsuccessful. By considering the infinite Fermi system to be a limit of finite systems, it is shown that the correct result merely involves replacing Brueckner's ordinary integral over diagonal reaction matrix elements by a principal value integral. In a finite system the level shift of a Bethe-Goldstone state differs from the diagonal reaction matrix element by a normalization factor which does not approach unity uniformly in the integration variable as the volume becomes infinite. In the neighborhood of a singularity the expression for the two-particle energy shift takes the form $\frac{\mathrm{cy}}{({y}^{2}+c{\mathcal{U}}^{\ensuremath{-}1})}$, where $y$ is the unperturbed energy measured from the singularity, $c$ is the square of a matrix element, and $\mathcal{U}$ is the quantization volume. Hence as $\mathcal{U}\ensuremath{\rightarrow}\ensuremath{\infty}$ the sum over the energy $y$ indeed approaches a principal value integral. An alternative derivation, employing a modified reaction matrix for which there is no difference between level shift and matrix element, leads to the same result. The general derivations are preceded by a soluble example.The connection of the Brueckner approximation with a phase-shift approximation for low-density systems is discussed. Some corrections to the higher order terms in existing derivations of the "separation method" expansion of the Brueckner reaction matrix are given.

DOI
10.1103/physrev.127.1328

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In the approximation neglecting any but single pair correlations, singularities of the reaction matrix render Brueckner's integral for the average energy per particle a singular integral. Several attempts to overcome this difficulty have been unsuccessful. By considering the infinite Fermi system to be a limit of finite systems, it is shown that the correct result merely involves replacing Brueckner's ordinary integral over diagonal reaction matrix elements by a principal value integral. In a finite system the level shift of a Bethe-Goldstone state differs from the diagonal reaction matrix element by a normalization factor which does not approach unity uniformly in the integration variable as the volume becomes infinite. In the neighborhood of a singularity the expression for the two-particle energy shift takes the form cy(y2+cU\ensuremath−1)\frac{\mathrm{cy}}{({y}^{2}+c{\mathcal{U}}^{\ensuremath{-}1})}(y2+cU\ensuremath−1)cy​, where yyy is the unperturbed energy measured from the singularity, ccc is the square of a matrix element, and U\mathcal{U}U is the quantization volume. Hence as U\ensuremath→\ensuremath∞\mathcal{U}\ensuremath{\rightarrow}\ensuremath{\infty}U\ensuremath→\ensuremath∞ the sum over the energy yyy indeed approaches a principal value integral. An alternative derivation, employing a modified reaction matrix for which there is no difference between level shift and matrix element, leads to the same result. The general derivations are preceded by a soluble example.The connection of the Brueckner approximation with a phase-shift approximation for low-density systems is discussed. Some corrections to the higher order terms in existing derivations of the "separation method" expansion of the Brueckner reaction matrix are given.
A closed set of coupled, nonlinear integral equations is obtained which completely describe a system of Fermi particles. The equations define the one and two-particle Green's functions which are completely analogous to those used in the field theory of fundamental particles. Special boundary conditions appropriate to the infinite medium of fermions are employed. Approximations to these equations are made which lead to a generalized Brueckner reaction-matrix formalism in the case of nuclear matter and which lead to an expression for the correlation energy in the case of the electron gas. The approximations for the system of nuclear matter are of three types: Ones which neglect the simultaneous interaction of three or more nucleons, the effective interactions of the vacant states below the Fermi sea, and the self-energy interactions with excited nucleons. The relationship between the reaction matrix and the expectation value of the actual potential energy is discussed. It is found that the energy levels given by the Brueckner formalism are nearly those given by a consideration of the actual energy, thus indicating that the shell-model energy levels for closed-shell-plus-one nuclei should be almost correct. A formula for the momentum distribution of nucleons in nuclear matter is obtained.
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.
The application of Brueckner's perturbation approach to interacting Fermion systems is prevented by the poles of the t-matrix which can be present for finite as well as infinite systems if the interaction is attractive. It is discussed that the singularity is removed if we take account of the contribution to the energy from diagrams involving two hole labels a number of times. These diagrams are not necessarily of ladder type and correspond to the possibility that a scattered pair of particles can go to the levels which they have left in the unperturbed state. Though the removal of poles makes one entertain a hope for the perturbation approach to finite systems, it is discussed that the same is not the case with infinite systems. It is pointed out that the difficulty in dealing with infinite systems is not caused by the pairs corresponding to the poles as is usually considered to be, but is due to pairs lying more closely to the Fermi surface.
Perturbation theoretic equations for several properties of the zero-temperature many-fermion normal system are rederived from general considerations. Though none of the equations are new, some of the derivations are, and taken together they form a brief and simple summary of many-fermion relations and a basis for further investigation of the basic perturbation description of the many-fermion normal system: the Brueckner-Goldstone expansion. A "change-of-parameter" technique is then developed and employed to investigate the possible use of the true momentum density instead of the unperturbed Fermi distribution in the Brueckner-Goldstone expansion and in the Brueckner KKK-matrix approximation. The result is a simpler, approximate perturbation series for the interaction energy, whose accuracy for nuclear matter is estimated to be approximately \ifmmode\pm\else\textpm\fi{}2 MeV. The new approximation is exact to the fourth order.
The general form of the energy of the ground state of a many-fermion system is shown to be exactly of the form proposed by Brueckner and Bethe, without approximation. In a variational treatment, if the trial wave function is picked containing only pair correlations, together with all possible unlinked pairs, it is described by a two-body excitation matrix 〈m1m2∣A∣p1p2〉〈{m}_{1}{m}_{2}|A|{p}_{1}{p}_{2}〉〈m1​m2​∣A∣p1​p2​〉. Variation of this matrix in the Ritz-Rayleigh principle yields a set of integral equations of the scattering type for the matrix AAA. Hole-state energies are given self-consistently in terms of the matrix AAA, but particle-state energies are Hartree-Fock energies. This may be corrected for by widely enlarging the class of terms admitted into the wave function. If the approximation is then made of omitting a class of terms, defined as cross-linked clusters in $〈\ensuremath{\psi}|H|\ensuremath{\psi}〉$, the particle-state energies are easily renormalized. Variation then leads to an infinite hierarchy of integral equations.
The method of superposition of configurations, which provides a general solution of the quantum-mechanical many-particle problem (for fermions), is reformulated so that equations may be compared with those used in the Brueckner theory. Important differences occur in application to finite or nonuniform systems for which the Hartree-Fock or Brueckner self-consistent orbitals are not plane waves. In such cases nonvanishing single-particle matrix elements occur which cannot be described by the Brueckner formalism based on a two-particle operator. Equations for an effective two-particle operator, equivalent to the variational equations of the method of superposition of configurations, are derived for a basis of Hartree-Fock orbitals. At the expense of making the orbital basis dependent upon the effective two-particle operator, the orbital basis can be determined by a condition which is essentially that of the Brueckner method. This condition removes a class of matrix elements which do not necessarily vanish in the Hartree-Fock basis although they would be neglected to second order in a perturbation calculation starting from the Hartree-Fock wave function. The equations for the effective two-particle operator are formally the same in both cases but lead to different operators because of the different choice of basis. In neither case can the equations be written in terms of products of such operators, a formalism assumed in the Brueckner theory. It is shown that the Brueckner condition is not equivalent to the condition which would determine the best possible orbital basis for the form of wave function implied by the use of an effective operator dependent on two particles only. The argument of the present paper is limited to systems with a finite number of particles, since the variational principle used here is not applicable without modification to systems with an infinite number of particles.