By following the conventional similarity renormalization group (SRG) expansion of the Dirac equation developed in [J.-Y. Guo, Phys. Rev. C 85, 021302 (2012)], we work out the analytic expression of the 1/M 4 order and verify the convergence of this method. As a step further, the reconstituted SRG method is proposed by using the re-summation technique. The speed of convergence of the reconstituted SRG becomes much faster than the conventional one, and the single-particle densities with the reconstituted SRG are also almost identical to the exact values.

By following the Foldy-Wouthuysen (FW) transformation of the Dirac equation, we work out the exact analytic expressions up to the 1/M 4 order for the general cases in the covariant density functional theory. These results are further compared with the corresponding ones derived from another novel non-relativistic expansion method, the similarity renormalization group (SRG).Based on that, the origin of the difference between the results obtained by the FW transformation and SRG method is explored. * haozhao.liang@riken.jp

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