Ab initio and DFT calculations were performed to investigate the structure, stability, and nature of chemical bonding of the F-Rg-BR₂ (R = F, OH, CN and CCH; Rg = Ar, Kr, Xe and Rn) molecules. The geometries are optimized for ground as well as transition states using the B3LYP-D3 and MP2 methods. It has been found that the F-Rg-B portion of F-Rg-BR₂ species is linear in the ground state but curved in the transition state. The NBO, AIM, ELF and EDA analyses suggest that the molecules can be expressed as F^-(Rg-BR₂)⁺ due to the covalent Rg-B bond and the ionic interaction between F and Rg. Calculations assert the metastable behavior of the F-Rg-BR₂ molecules, thermodynamic data shows that F-Rg-BR₂ can spontaneously dissociates into BFR₂ + Rg, the considerable energy barrier of this two-body dissociation channel calculated by the B3LYP-D3, MP2 and CCSD(T) methods affirms the kinetic stability of the F-Rg-BR₂ molecules. Thus F-Rg-BR₂ molecules are kinetically protected against the decomposition reaction and may be identified under cryogenic conditions in solid rare gas matrices or in the gas phase.