Converged values of second-order core-polarization diagrams with orthogonalized-plane-wave intermediate states

We carry out an extended study of the Vary-Sauer-Wong effect on the second-order core-polarization diagrams $G_{3\mathrm{p}1\mathrm{h}}$ and $G_{3\mathrm{p}1\mathrm{h}}^T$ in the effective interaction between two valence nucleons. As in Vary-Sauer-Wong, we first calculate $G_{3\mathrm{p}1\mathrm{h}}$ using a harmonic oscillator propagator for the intermediate particle states $p$, including particle-hole excitations with energies up to $22\hslash \omega$. Results are in close agreement with those of Vary-Sauer-Wong. We then calculate $G_{3\mathrm{p}1\mathrm{h}}^T$ where a free particle propagator for $p$ is used. This is obtained by including the $-U$, the oscillator one-body potential, insertions of all orders to $p$ of $G_{3\mathrm{p}1\mathrm{h}}$. Although the resulting matrix elements are generally smaller in magnitude than those of Vary-Sauer-Wong, the qualitative feature of the Vary-Sauer-Wong effect is clearly maintained. Namely there are strong cancellations between the contributions from the low and high energy $p$ states. This makes the net effect from the core polarization diagrams significantly weaker than from $G_{3\mathrm{p}1\mathrm{h}}$ calculated with $2\hslash \omega$ excitations alone. We also study some intermediate choices for the propagator of $p$, where the free particle propagator is used only for high energy valence states. The Brueckner reaction matrix elements in a mixed representation where one particle is in a harmonic oscillator state and the other in a plane wave state are needed in our calculations. By using the vector transformation brackets of Wong and Clement and of Balian and Brezin and the Tsai-Kuo treatment of the Pauli exclusion operator, we have developed a technique for accurately calculating these matrix elements.