is to avoid the treatment of the unitary transformations with specific forms. Therefore, the calculations of the single-particle scalar densities are not trivial at all in the SRG methods.

Alternatively, the Foldy-Wouthuysen (FW) transformation, also called the Pryce-Tani-Foldy-Wouthuysen transformation [@Pryce1948Proc.RoyalSoc.A195.62--81; @Foldy1950Phys.Rev.78.29--36; @Tani1951; @Foldy1952Phys.Rev.87.688--693], which was initially formulated around 1950 to describe the relativistic spin-$1/2$ particles such as electrons in the non-relativistic limit, is also an elegant method for the non-relativistic expansion of the Dirac equation [@Greiner1990.277--290].

The FW non-relativistic expansions of the Dirac equation have been carried out up to some higher orders for the selected families of potentials with specific properties, see, e.g., Refs. [@Chen2010Phys.Rev.A82.012115; @Chen2014Phys.Rev.A90.012112]. Very recently, with the consideration of the strong scalar potential, we further applied the FW transformation to the general cases in the covariant DFT, and the corresponding expansion was performed up to the $1/M^4$ order [@Guo2019Chin.Phys.C43.114105]. Furthermore, we also investigated the difference between the results of the conventional SRG method and the corresponding ones of the FW transformation as well as the origin of such a difference in Ref. [@Guo2019Chin.Phys.C43.114105]. As the same as the conventional SRG method, the FW transformation presents a systematic way to derive the non-relativistic expansion of the Dirac 