  Inspired by the reconstituted similarity renormalization group method, the
reconstituted Foldy-Wouthuysen (FW) transformation is proposed. Applied to the
Dirac equation in the covariant density functional theory, the reconstituted FW
transformation shows a fast convergence of the spectrum of the single-particle
energy. The single-particle densities and the single-particle scalar densities
obtained by this new method are also investigated. In particular, the
relativistic corrections to the densities from the picture-change error between
the Schr\"odinger and Dirac pictures are discussed in detail. Taking these
relativistic corrections into account, both the single-particle densities and
the single-particle scalar densities are almost identical to their exact
values.
