``Self-Consistent'' SRPA

As we expected from our discussion in section 3, the ``self-consistent'' choice of the single-particle propagator according eq.([*]) yields a stabilization of the SRPA equations. With this choice we obtain solutions of the non-linear SRPA equations for the small and large model-spaces for all interactions considered. Results for the correlation energies are displayed in table [*] considering the same approximations, interactions and model-spaces as in table [*].

The occupation numbers for the single-particle orbits nα depend on the approximation considered. In table [*] occupation numbers are listed, which were calculated for the OBE B potential using different approximations and model-spaces. Considerable deviations from the occupations of the IPM are observed. These deviations are slightly larger if one uses the interaction OBE A and slightly smaller for the interaction C. As the occupation numbers depend on the method considered, also the single-particle propagators defined in eq.([*]) will be different. Consequently for these self-consistent calculations also the contribution of the second order term will be different for the different approaches. In table [*] we give for each approximation the contribution of this second order term calculated with the self-consistent single-particle propagator and separately the sum of all higher order terms.

The stabilization of the correlation effects due to the self-consistent choice of the single-particle propagator is well documented by the contribution to the energy of second order in the residual interaction. For approximations which yield large correlation effects, like the SRPA, one obtains sizeable deviations of the occupation probabilities from the IPM (see table [*]) and contributions to the binding energy from the second order term, which are considerably smaller than the second order terms calculated without readjustment of the single-particle features (see table 1). For approaches like RPA, which yield weaker correlation effects, the quenching of the second order term due to the single-particle propagator is weaker. Therefore the more correlations are taken into account (SRPA as compared to RPA) the smaller the contribution of the second order term. Also the contributions of the terms of third and higher order in the interaction are reduced due to the self-consistent single-particle propagator. The propagator effect, however, is generally weaker than for the second order terms.

Figure: The occupation probabilities in 16O for the orbits in the 0s to 1s0d shell are represented in terms of columns. The results obtained by various approximations are distinguihed by the shading of the bars. All results were obtained for OBE potential C.
\begin{figure} \epsfxsize 12cm \begin{center} \makebox[12cm][l]{ \epsfbox{figure2.eps} } \end{center} \end{figure}
It is interesting to note that the sum of second order plus all higher order terms yields a correction to the binding energy, which is almost independent on the approximation considered. For the small model-space we obtain around 9 MeV, while the calculations in the large model-space yield around 19 MeV. This does not imply that correlations beyond RPA are negligible if the single-particle propagator is chosen in a self-consistent manner. Indeed the contribution to the binding energy originating from third and higher order terms is increasing going from RPA via pp-hh RPA to SRPA. This enhancement of the higher order terms is partly canceled by reduction of the second order terms as already discussed above. The importance of correlations beyond RPA is also reflected by the occupation probabilities displayed in table [*]. The deviations from the Hartree-Fock values observed in the SRPA are much larger than those obtained in RPA (see also Fig. 2). This is true in particular for the 2 shells close to the Fermi level (the 0p and the 1s0d shells). In this case the deviations obtained in SRPA are larger than the sum of the deviations produced by RPA and pp-hh RPA correlations.

The inspection of the correlation energies for the various model-spaces (9 MeV for the small and 19 MeV for the large model-space) may cast doubt on the convergence of the calculated correlation energy with respect to the size of the model-space. It should be noted, however, that the occupation probabilities obtained for the shells away from the Fermi level are not very sensitive to the approximation considered. The RPA and also the pp-hh RPA yields results for the 0s and 1p0f shells, which are very similar as those obtained in SRPA. This indicates that the interplay between the various correlations contained in SRPA is of dominant importance only for the shells close to the Fermi energy. Therefore the pp-hh RPA, which is the summation of all particle-particle and hole-hole ladders, should provide a good approximation for the correlations of these high energy particle-hole excitations.

One possible way to account for these pp-hh correlations would be the use of the self-consistent Green function method [6]. The Green function approach, if considered beyond the Hartree-Fock approximation, requires a very sophisticated self-consistency between the approximation used for the 2-particle Green function and the single-particle Green function to guarantee number conservation. The approximation for the single-particle propagator, which we are using here, is similar to the Green function approach (see discussion in 3.2). The present scheme, however, has the advantage that it always yields the correct particle number.

Employing the self-consistent single-particle Green functions defined in eq.([*]) one can calculate the expectation value for single-particle operators like the radius of the nucleon distribution. The radii obtained for the various approximations, using the OBE potentials A, B and C in the large model-space are listed in table [*] and compared to the result obtained in the Hartree-Fock approximation. As we discussed already in the beginning of this section, the interactions have been modified to guarantee a Hartree-Fock solution with a radius close to the experimental value [24]. Therefore we should only discuss the modifications in the calculated radius due to the correlation effects. One finds a small but non-negligible enhancement for the calculated radius. As to be expected also for this observable the effect of correlations is larger for the SRPA than for the 2 RPA approximations. Also for the calculation of the radii, the effects of correlations is slightly increasing going from potential A to C.

It is remarkable that the inclusion of SRPA correlation increases the binding energy but also the value for the calculated radius. This implies that calculated ground-state properties are moved off the ``Coester-band'' towards the experimental point. This effect is not very large but together with the features of the Dirac-BHF approach it may be sufficient to yield results for the ground-state of finite nuclei, which are in good agreement with the empirical data.