the integro-differential equations Eq. (3) can be formally rewritten as equivalent differential ones, where V$_{+}$ ≡ V D + + Y$_{G}$ , V − ≡ V D + X$_{F}$ , and In the above expressions, Σ$_{S}$ represents the scalar potential from the Hartree terms, Σ$_{0}$ is the time component of the vector potential, which contains the contributions from the Hartree terms and the rearrangement terms induced by the density-dependence of the meson-nucleon couplings [21], and X$_{G}$ , X$_{F}$ , Y$_{G}$ , Y$_{F}$ are the effective local potentials from the Fock terms. The equations Eq. (5) then can be solved self-consistently with the same numerical method as in RMF [27]. From the radial Dirac equation Eq. (5), the Schr¨ odinger-type equation for the dominant component F ( r ) can be obtained as, where V$_{CB}$ = κ (1 − κ ) r 2 and V$_{SOP}$ correspond to the centrifugal barrier (CB) and spin-orbit potential (SOP), respectively. In the above equation, the Hartree and Fock terms for V$_{1}$ , V$_{SOP}$ and V read as One may note that the denominator V$_{+}$ − E contains a state dependent potential Y$_{G}$ . How- ever, as the quantity Y$_{G}$ is around a few MeV and is negligible in comparison with V$_{+}$ − E which is of the order of 1 GeV, the Eq. (7) is accurate enough to estimate the Hartree and Fock contributions. Similar argument also holds for the time component of the vector potential Σ$_{0}$ which contains the rearrangement term from Fock channels. 4 V D =Σ$_{0}$ − Σ$_{S}$ − M, V E = X$_{F}$ + 1 V$_{+}$ − E ( Y$_{F}$ V ′ + V$_{+}$ − E − Y ′ F − X$_{G}$Y$_{F}$ ) . (8c) V D SOP = κ r V D + ′ V$_{+}$ − E , V E SOP = κ r ( Y ′ G V$_{+}$ − E − X$_{G}$ − Y$_{F}$ ) , (8b) V D 1 = − V D + ′ V$_{+}$ − E , V E 1 = X$_{G}$ − Y$_{F}$ − Y ′ G V$_{+}$ − E , (8a) 1 V$_{+}$ − E { F ′′ + ( V D 1 + V E 1 ) F ′ + [ V$_{CB}$ + V D SOP + V E SOP ] F } + V $^{D}$F + V $^{E}$F = EF, (7) [ d dr − κ r − Y$_{F}$ ( r ) ] F ( r ) − [ V$_{+}$ ( r ) − E ] G ( r ) = 0 , (5a) [ d dr + κ r + X$_{G}$ ( r ) ] G ( r ) + [ V − ( r ) − E ] F ( r ) = 0 , (5b) V D + ≡ M + Σ$_{S}$ + Σ$_{0}$ , V D ≡ Σ$_{0}$ − Σ$_{S}$ − M. (6)