the total neutron density lies between $0.08$ fm$^{-3}$ and $0.10$ fm$^{-3}$. As seen in the figure, all the results calculated by the reconstituted FW transformation are in good agreements with the exact result.

![(Color online) Difference between the neutron density of $^{208}$Pb calculated by the reconstituted FW transformation and the corresponding exact density.[]{data-label="fig5"}](Fig5.eps){width="45.00000%"}

In order to investigate the results in Fig. \[fig4\] in detail, the differences between each calculated result and the exact neutron density are shown in Fig. \[fig5\]. It can be seen that the results calculated with the consideration of the relativistic corrections show quite different behaviors from the one without corrections. In general, the relative discrepancy between the results with corrections and the exact one is within $0.5\%$. It is also interesting to point out that, compared with the result with the corrections up to the $1/{M^\ast}^2$ order, the neutron density is further improved by the $1/{M^\ast}^3$-order corrections in the nuclear surface region, since the $1/{M^\ast}^3$-order corrections are proportional to the derivatives of the potentials as shown in Eq. .

Single-particle and total scalar densities {#sec:IIIC}
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![(Color online) Difference between the single-particle density and the single-particle scalar density for the neutron $2d_{5/2}$ state. The solution of 