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Physical Review

Λ10 Approximation for Self-Bound Many-Fermion Systems

Robert D. Puff, A.S. Reiner, L. Wilets

Published Sep 23, 1966• 23 citations
Publisher

Abstract

We investigate several approximations for self-bound many-fermion systems, each of which employs the Green's-function formalism together with a suitable factorization to close the otherwise infinite hierarchy of coupled integral equations. The particular approximations, which attempt to include the important two-body correlations for short-range forces, differ in the choice of propagator $\ensuremath{\Lambda}$ for two-particle intermediate states in the $T$ matrix used to calculate the two-body correlation function. The advantages and shortcomings of the choice ${\ensuremath{\Lambda}}_{00}\ensuremath{\sim}{{G}_{1}}^{0}{{G}_{1}}^{0}$ have been noted previously and are not treated here. We show that the choice ${\ensuremath{\Lambda}}_{10}\ensuremath{\sim}{{G}_{1}}^{0}{G}_{1}$, which appears in a "natural" factorization, is formulated in a manner which preserves much of the simplicity of the ${\ensuremath{\Lambda}}_{00}$ approximation. (Numerical calculations using the ${\ensuremath{\Lambda}}_{10}$ approximation are presented by Foster in the following paper.) The ground-state energies for three $\ensuremath{\Lambda}$ approximations (${\ensuremath{\Lambda}}_{00}$, ${\ensuremath{\Lambda}}_{10}$, and ${\ensuremath{\Lambda}}_{11}\ensuremath{\sim}{G}_{1}{G}_{1}$) are expanded through third order in the interaction strength to facilitate comparison with each other and with the Brueckner-Goldstone series. Numerical comparison of these third-order contributions for a square-well potential is presented. ${\ensuremath{\Lambda}}_{10}$ is in dramatically closer agreement with the full perturbation theory (for ${{k}_{F}}^{0}c\ensuremath{\lesssim}1$) than are the other $\ensuremath{\Lambda}$ approximations or Brueckner theory.

DOI
10.1103/physrev.149.778

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