expansion of the Dirac equation is considered to be a potential bridge for making the connection. By following this direction, the Dirac equation was reduced to some non-relativistic Schrödinger-like equations [@Reinhard1989Rep.Prog.Phys.52.439--514; @Bender2003Rev.Mod.Phys.75.121--180; @Meng2016]. However, on the one hand, the convergence of these methods is rather slow. On the other hand, which is more important, the effective Hamiltonian thus obtained is not Hermitian, since the upper- or lower-component wave functions as the solutions of the Schrödinger-like equations alone are not orthogonal to each other.

In 2012, the similarity renormalization group (SRG) [@Wegner1994Ann.Phys.Berl.506.77--91; @Bylev1998Phys.Lett.B428.329--333] was first applied by Guo [@Guo2012Phys.Rev.C85.021302] to perform the non-relativistic expansion of the single-nucleon Dirac equation for investigating the nuclear pseudospin symmetry [@Ginocchio1997Phys.Rev.Lett.78.436--439; @Meng1999Phys.Rev.C59.154--163; @Chen2003Chin.Phys.Lett.20.358--361; @Zhou2003Phys.Rev.Lett.91.262501; @Liang2013Phys.Rev.C87.014334; @Shen2013Phys.Rev.C88.024311; @Liang2015Phys.Rep.570.1--84]. The non-relativistic expansion is carried out in the powers of ${1}/{M}$, and the results up to the ${1}/{M^3}$ order were obtained in Ref. [@Guo2012Phys.Rev.C85.021302] (with $M$ the bare mass of the particle). It is remarkable that the non-relativistic reduced Hamiltonian thus obtained is Hermitian. In order to achieve the convergence of such a conventional SRG method, we recently showed in Ref. [@Guo2019Phys.Rev.C99.054324] that the ${1}/{M^4}$-order terms are needed. As a step further, by using the technique of resummation, which 