Brueckner Theory in an Oscillator Basis. I. The Method of Reference Bethe-Goldstone Equations and Comparison of the Yale, Reid (Hard-Core), and Hamada-Johnston Interactions

Abstract

Arguments are given for preferring discrete localized excited single-particle states to plane waves in Brueckner theory for finite nuclei. Short-range internuclear repulsion requires transforming pair states to relative and c.m. coordinates, in which only constant and harmonic potentials separate. This series of papers develops a form of Brueckner theory in which single-particle states are expanded in harmonic-oscillator functions. This paper is limited to the single-oscillator-configuration (SOC) approximation. The reaction matrix is obtained in two steps: A reference matrix, involving the potential $V^R\left(r_1\right)=-C+\frac{1}{2}m{\omega }^2{r_1}^2$ and Eden and Emery's rather good approximate Pauli operator, is obtained by solving radial Bethe-Goldstone equations. The off-diagonal elements of the tensor interaction are treated exactly in coupled equations. Second, essentially the exact SOC matrix is calculated; by making the Pauli and spectral corrections on the reference two-body rather than the relative matrix elements the exact SOC Pauli operator can be used, and energies of low-lying excited states can be varied (e.g., to satisfy self-consistency conditions). The Pauli corrections are much smaller than when the Pauli effect is omitted entirely in the reference matrix. By proper choice of $C$, the spectral corrections also may be reduced. Three interactions with hard cores are compared under the same conditions. The gap between occupied and excited states is greater than $\frac{3}{2}$ the normal oscillator spacing $\hslash \omega$. Occupied-state energies and pair-creation matrix elements of $V^R$ are made nearly self-consistent. The three interactions are practically equivalent, as demonstrated in a calculation of the energetics of ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$. The most advantageous choice of excited-state spectrum remains the outstanding uncertainty, probably requiring careful evaluation of the Bethe three-body cluster for finite nuclei.