A new approximation to the Bethe-Goldstone wave function is proposed, constructed from the two-nucleon potential as follows: The eigenstates of the interacting two-nucleon system with energy eigenvalues less than a certain cutoff energy span a subspace $W$ of the Hilbert space representing the two-nucleon system. Our approximation to the Bethe-Goldstone wave function is the projection of the free two-nucleon wave function onto this subspace $W$. For a suitable choice of cutoff energy, this approximation has the qualitative healing properties expected of the exact Bethe-Goldstone wave function. Moreover, it permits an easy evaluation of reaction matrix elements. The approximation is also applied to the calculation of the reaction matrix elements in the $\mathrm{sd}$ shell of the harmonic-oscillator shell model. The nucleon-nucleon potential of Hamada and Johnston was used in our calculation. The resulting energy levels of ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}$ and ${}_{}{}^{18}{}_{}{}^{}\mathrm{F}$ are similar to those obtained by Kuo and Brown.

• Received 14 April 1969

DOI:https://doi.org/10.1103/PhysRev.186.963