Numerical examples

In order to illustrate theoretical findings presented in Sec. 3, we carried out numerical calculations within the Skyrme-DFT method. We used the code HFBTHO [39] which is capable of handling spherical and axially deformed nuclei within the Lipkin-Nogami (LN) approximation followed by the PNP. This corresponds to the projection-after-variation (PAV) method of restoring the PN symmetry. By using a new version of HFBTHO, we also performed full variation-after-projection (VAP) calculations analogous to those of Ref. [23].

As illustrative examples, we study spherical and deformed configurations in $^{18}$O and in $^{32}$Mg calculated using the Skyrme functionals SIII [40] and SLy4 [41]. These two parametrizations differ in a significant way with respect to the PNP method. The density-dependent term of SIII contributes to the energy density as $(\rho_n+\rho_p)\rho_n\rho_p$. Therefore, both in the neutron and proton subsystems, the powers $p$ (Sec. 3.5) of the density dependence are equal to 2. Consequently, from the PNP perspective, the density-dependent term of SIII is not any different than the density-independent terms. On the contrary, the density-dependent term of SLy4 is proportional to $\left[\rho_n+\rho_p\right]^{1/6}$ and exemplifies the case of fractional-power dependence discussed in Sec. 3.6. The contact pairing force of the volume type (density-independent) was used in the particle-particle channel. All calculations have been performed in the spherical harmonic-oscillator basis of $N_0=6$ or 10 shells, for $^{18}$O or $^{32}$Mg, respectively.