  \textbf{Background:} The initial observation of a negative-parity state in
proximity to the ground state in the 1950s marked the advent of extensive
research into octupole deformed nuclei. Since then, the physics of octupole
deformed nuclei has consistently held a special interest within the field of
nuclear physics. In the present era, with the advent of sophisticated
radioactive ion beam (RIB) facilities and advanced detectors, coupled with the
remarkable capabilities of high-performance computing, extensive and intensive
explorations are being conducted from both experimental and theoretical
perspectives to elucidate the physics of octupole deformed nuclei.
\textbf{Results:} This work establishes the OD-RHF model, which provides a
reliable tool for studying octupole nuclei over a fairly wide range. The
reliability of the newly developed OD-RHF model is illustrated by taking the
octupole nucleus $^{144}$Ba as an example. Furthermore, the octupole
deformation effects in $^{144}$Ba is verified by using the RHF Lagrangians
PKO$i$ ($i=1,2,3$) and the RMF one DD-ME2. The intrusion of the neutron
$1i_{13/2}$ and proton $1h_{11/2}$ components is demonstrated to play an
essential role in determining the notable octupole deformation of $^{144}$Ba
using PKO$i$ ($i=1,2,3$) and DD-ME2. It is indicated that the Fock terms play
an important role in stabilizing the octupole deformation. More specifically,
due to the repulsive tensor coupling between the intrude components and the
core of $^{144}$Ba, the tensor force component carried by the $\pi$-PV
coupling, that contributes only via the Fock terms, plays an opposing role in
the formation of the octupole deformation of $^{144}$Ba.
