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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 

<img src='content_image/369468.jpg'>

<img src='content_image/369469.jpg'>

<img src='content_image/369470.jpg'>

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, 

<img src='content_image/369467.jpg'>

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 

<img src='content_image/369466.jpg'>

<img src='content_image/369465.jpg'>

<img src='content_image/369464.jpg'>

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. 

