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. \[fig6\].[]{data-label="fig7"}](Fig7.eps){width="45.00000%"}

It can be seen in Fig. \[fig6\] that the differences between the results without the relativistic corrections and the exact one are systematic. There are differences in the positions of peaks and nodes as well as the amplitudes. In contrast, the results with the relativistic corrections reproduce remarkably the behavior of the exact result, in particular the positions of peaks and nodes. The remaining discrepancies are rather small (notice the small scale used here).

As a result, the exact single-particle scalar density for the $2d_{5/2}$ state can be well reproduced, with a relative discrepancy less than $0.2\%$, by the reconstituted FW transformation including the relativistic corrections, as illustrated in Fig. \[fig7\].

Comparing the dashed lines to the others in Figs. \[fig6\] and \[fig7\], one can confirm the critical role of the relativistic corrections playing in the difference between the vector and scalar densities, because they both start from the $1/{M^\ast}^2$ order.

![(Color online) Neutron scalar density of $^{208}$Pb calculated by the reconstituted FW transformation and the corresponding exact density.[]{data-label="fig8"}](Fig8.eps){width="45.00000%"}

![(Color online) Difference between the neutron scalar 