considered in the present work.

Results {#Result}
=======

In the present work, as a reference for the correlation analyses performed below with the standard Skyrme interactions, we use the MSL0 parameter set [@Che10], which is obtained by using the following empirical values for the $9$ macroscopic quantities: $\rho _{0}=0.16$ fm$^{-3}$, $E_{0}(\rho
_{0})=-16$ MeV, $K_{0}=230$ MeV, $m_{s,0}^{\ast }=0.8m$, $m_{v,0}^{\ast
}=0.7m$, $E_{\text{\textrm{sym}}}({\rho _{0}})=30$ MeV, $L=60$ MeV, $G_{V}=5$ MeV$\cdot $fm$^{5}$, and $G_{S}=132$ MeV$\cdot $fm$^{5}$. And the spin-orbit coupling constant $W_{0}=133.3$ MeV $\cdot $fm$^{5}$ is used to fit the neutron $p_{1/2}-p_{3/2}$ splitting in $^{16}$O. It has been shown [Che10]{} that the MSL0 interaction can describe reasonably the binding energies and charge rms radii for a number of closed-shell or semi-closed-shell nuclei. We further find that the MSL0 parameter set predicts a value of $1.06$ MeV for the splitting of the neutron $3p$ shell in $^{208}$Pb, which reasonably describes the experimental value of $0.9$ MeV. It should be pointed out that the MSL0 is only used here as a reference for the correlation analyses. Using other Skyrme interactions obtained from fitting measured binding energies and charge rms radii of finite nuclei as in usual Skyrme parametrization will not change our conclusion.

As numerical examples, in 