  Staring from the Lagrangian density that foots on the meson-propagated
picture of nuclear force, the full Hamiltonian, that contains both mean field
and pairing contributions, is derived by quantizing the Dirac spinor field in
the Bogoliubov quasi-particle space, and the expectation with respect to the
Bogoliubov ground state gives the full energy functional. As an extension of
the D-RHF model, the degree of freedom associated with the $\rho$-tensor
($\rho$-T) coupling is implemented, and incorporating with the Bogoliubov
scheme the finite-range Gogny force D1S is utilized as the pairing force.
Moreover, qualitative analysis on the nature of the $\pi$-PV and $\rho$-T
couplings are presented for better understanding their enhancements on the
deformation effects.
  Space convergence related to the spherical DWS base is confirmed for the
D-RHFB model by taking light nucleus $^{24}$Mg and mid-heavy one $^{156}$Sm as
candidates. Compared to light nuclei, extraordinary more negative energy states
are necessitated to keep the expansion completeness on the spherical DWS base
for mid-heavy and heavy nuclei, due to the enhanced correlations between the
expansion components with large $\kappa$-quantity as indicated by the nature of
the $\pi$-PV and $\rho$-T couplings. Furthermore, because of the enhanced
deformation effects by the $\pi$-PV and $\rho$-T couplings, the RHF Lagrangian
PKA1 presents deeper bound ground state for $^{24}$Mg than the other selected
Lagrangians, in addition to predicting a fairly deep bound local minimum with
large oblate deformation.
