single-particle density of the $2d_{5/2}$ state calculated by the reconstituted FW transformation and the corresponding exact density. The choice of legends follows that in Fig. \[fig2\]. []{data-label="fig3"}](Fig3.eps){width="45.00000%"}

In order to examine the details, the differences between the single-particle density of the $2d_{5/2}$ state calculated by the reconstituted FW transformation and the corresponding exact density are shown in Fig. \[fig3\]. It can be seen clearly that the results without the corrections show dramatic differences from the exact ones, and the discrepancy $\rho_v - \rho_{v,{\rm exact}}$ reaches $6.6 \times 10^{-5}$ fm$^{-3}$ around the first peak of the wave function, which corresponds to a relative discrepancy $3.2\%$. In contrast, after considering the picture-change-error corrections, the discrepancies $\rho_v - \rho_{v,{\rm exact}}$ are smaller than $0.97 \times 10^{-5}$ fm$^{-3}$ in the whole region.

![(Color online) Neutron density of $^{208}$Pb calculated by the reconstituted FW transformation and the corresponding exact density. []{data-label="fig4"}](Fig4.eps){width="45.00000%"}

Since the non-relativistic expansion is regarded as a crucial candidate for the connection between the relativistic and non-relativistic DFT, it is meaningful to further examine the total density of neutron by summing up the single-particle densities of the lowest $126$ single-particle orbitals as shown in Fig. \[fig4\]. In the center area with $r<6$ fm, 