values. In particular for $^{16}$O and $^{40}$Ca, both the experimental proton and neutron single- particle spectra are well reproduced by PC-PK1. For $^{132}$Sn and $^{208}$Pb, the empirical levels close to the Fermi surface are also reproduced well. Moreover, the spurious shells at Z = 58 ( $^{132}$Sn) and Z = 92 ( $^{208}$Pb) are found for all the effective interactions, which may be improved by the inclusion of ρ -tensor couplings [45]. 

## D. Charge radii and neutron skin thicknesses 

In Fig. 7, the charge radii for Sn and Pb isotopes predicted by PC-PK1 are shown in comparison with data [24, 25] and those by DD-PC1, PC-F1, PC-LA, and NL3*. It is seen that all the effective interactions reproduce the observed charge radii of Sn isotopes quite well (within 0.3%). For the Pb isotopes, the kink in the charge radii has been excellently reproduced by all the effective interactions. Quantitatively, the observed charge radii of Pb isotopes are reproduced by the calculations with DD-PC1, PC-F1, and NL3* within ∼ 0 . 3%, while the calculations with PC-LA within ∼ 0 . 5%. 

In Fig. 8, the neutron skin thicknesses for Sn isotopes and $^{208}$Pb predicted by PC-PK1 are shown in comparison with data [46, 47] and those by DD-PC1, PC-F1, PC-LA, and NL3*. For Sn isotopes, although PC-PK1 slightly overestimates the neutron skin thickness in comparison with DD-PC1, it nicely reproduces the isotopic trend. For $^{208}$Pb, all the interactions except DD-PC1 give similar neutron skin thicknesses which are larger than the data deduced from antiprotonic atoms [48], polarized proton scattering [49, 50], elastic pro- ton scattering [51], proton-nucleus elastic scattering [47], and agree within the experimental error bar with that from inelastic α scattering [52]. The slightly overestimated neutron skin thicknesses are due to the enhanced symmetry energies for nuclear matter shown in Table IV. In overall, the DD-PC1 parametrization provides better description of experimental charge radii and neutron skin thickness due to its smaller symmetry energy at saturation density. 

## VI. DEFORMED NUCLEI 

In this section, we will focus on the description of the binding energies and deformations for selected well-deformed even-even nuclei. In order to investigate the fission barrier, a constrained calculation is also carried out by taking $^{240}$Pu as an example. 

