equation up to an arbitrary order. However, when the strong scalar potential is included, the convergence of the FW transformation also becomes rather slow. Therefore, inspired by the reconstituted SRG method, it is interesting and important to improve the FW transformation for a faster convergence.

In this paper, inspired by the reconstituted SRG method, we will propose the novel reconstituted FW transformation, where the non-relativistic expansion is performed in the powers of ${1}/{M^\ast}$. The results up to the $1/{M^\ast}^3$ order will be shown explicitly, and the difference between these results and the corresponding ones obtained by the reconstituted SRG method will be investigated. Furthermore, by starting generally from the field operators, the relativistic corrections to the single-particle densities will be investigated. These corrections are originated from the so-called picture-change error between the Schrödinger and Dirac pictures, which was firstly applied in quantum chemistry and it was called the DKH method [@Douglas1974Ann.Phys.N.Y.82.89--155; @Hess1986Phys.Rev.A33.3742--3748; @Seino2010J.Chem.Phys.132.164108; @Nakajima2012Chem.Rev.112.385--402]. In this work, such relativistic corrections will be applied to the single-particle vector and scalar densities with the inclusion of the higher orders in a general case of the covariant DFT for the first time.

This paper is organized as follows. The theoretical framework for the 