density of $^{208}$Pb calculated by the reconstituted FW transformation and the corresponding exact density.[]{data-label="fig9"}](Fig9.eps){width="45.00000%"}

The total neutron scalar density of $^{208}$Pb is shown in Fig. \[fig8\]. It is smaller than the total neutron density shown in Fig. \[fig4\] by $3\sim8\%$ in general. For the details, the differences between the neutron scalar densities calculated by the reconstituted FW transformation and the exact result are shown in Fig. \[fig9\]. The results with the relativistic corrections can well reproduce the exact values, with a relative discrepancy less than $1\%$. The same as the total density, the $1/{M^\ast}^3$-order corrections can further improve the agreement with the exact result in the nuclear surface region.

Summary and Perspectives {#sec:IV}
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With the inspiration of the reconstituted SRG method [@Guo2019Phys.Rev.C99.054324], the reconstituted FW transformation has been proposed.

In the covariant DFT, since the inclusion of the strong scalar potentials, the conventional FW transformation shows a quite slow convergence for the spectrum of the single-particle energies. By replacing the bare mass $M$ with the Dirac mass $M^\ast$ and defining the corresponding new operators, the contributions that come from the products of the terms that already appear in the lower orders and $\left(\frac{S}{M}\right)^n$ ($n=1, 2, \cdots$) are absorbed into 