---
abstract: '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 Schrödinger 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.'
author:
- Yixin Guo
- Haozhao Liang
title: 'Non-relativistic expansion of Dirac equation with spherical scalar and vector potentials by reconstituted Foldy-Wouthuysen transformation'
---

Introduction
============

Since the 1970s, the density functional theory (DFT) has played a crucial role in investigating the ground-state and excited-state properties of atomic nuclei in a microscopic way [@Serot1986; @Reinhard1989Rep.Prog.Phys.52.439--514; @Ring1996Prog.Part.Nucl.Phys.37.193--263; @Bender2003Rev.Mod.Phys.75.121--180; @Vretenar2005Phys.Rep.409.101--259; @Meng2006Prog.Part.Nucl.Phys.57.470--563; @Liang2015Phys.Rep.570.1--84; @Meng2015J.Phys.GNucl.Part.Phys.42.093101; @Nakatsukasa2016Rev.Mod.Phys.88.045004; @Meng2016]. Both the non-relativistic and the relativistic DFT have attracted a lot of attention in nuclear physics, while there still remain open questions on the connection between these two frameworks [@Reinhard1989Rep.Prog.Phys.52.439--514; @Bender2003Rev.Mod.Phys.75.121--180; @Meng2016].

The non-relativistic 