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 O and in
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
.
Therefore, both in the neutron and proton subsystems, the powers
(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
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
or 10 shells,
for
O or
Mg, respectively.