variational calculations [15]. 

## V. SPHERICAL NUCLEI 

In this section, we will present the binding energies, two-neutron separation energies, single-particle levels, charge radii and neutron skin thicknesses for selected spherical isotopes and isotones in different mass regions in the covariant EDF with PC-PK1. The results will be compared with the corresponding data available as well as the predictions with DD-PC1, PC-F1, PC-LA, and NL3* sets. 

## A. Binding energy 

The binding energies for the Ca, Ni, Sn, and Pb isotopes are calculated with PC-PK1 and their deviations from the data [23] are shown in Fig. 2 in comparison with those with DD-PC1, PC-F1, PC-LA, and NL3*. The calculation with PC-PK1 reproduces the experi- mental binding energies within 1 MeV for the Ca isotopes and 2 MeV for both Sn and Pb isotopes. For Ni isotopes, although remarkable improvement is achieved by PC-PK1 in com- parison with results by the other interactions, there is still an underestimation of 2 . 4 − 5 . 2 MeV for the even-even 58 − $^{64}$Ni. Former investigations have shown that these nuclei are soft against deformation [39]. Therefore, the dynamic correlation energies gained by restoration of rotational symmetry and configuration mixing are expected to reduce these deviations. Similarly, the underestimations of the binding energies for neutron-deficient Pb isotopes can also be improved by configuration mixing, as demonstrated in the non-relativistic calcula- tions for the even-even 182 − $^{194}$Pb [40]. For the spherical Ca and Sn isotopes, the energies gained from the restoration of rotational symmetry and configuration mixing are expected to be much smaller as illustrated in the systematic beyond mean-field studies [41, 42]. There- fore, the inclusion of these energies will not change significantly the discrepancy between the mean-field results and the corresponding data for such spherical isotopes.. 

The binding energies for the isotonic chains are very important to examine the balance between the Coulomb field and the isovector channel of the Lagrangian density in Eq. (2). Here the binding energies for the N = 20, N = 50, N = 82, and N = 126 isotones are calculated with PC-PK1 and their deviations from the data [23] are shown in Fig. 3 in com- 

