The first identification of the argonium ion (ArH+) towards the Crab Nebula supernova remnant was proclaimed by the Herschel in the sub-millimeter and far-infrared domain. Very recently the discovery of the hydro-helium cation (HeH+) in the planetary nebula (NGC 7027) has been reported by using the SOFIA. The elemental abundance of neon is much higher than that of the argon. However, the presence of neonium ions (NeH+) is yet to be confirmed in space. Though the hydroxyl radicals (OH) are very abundant either in neutral or in the cationic form, hydroxyl cations of such noble gases (i.e., ArOH+, NeOH+, and HeOH+) are yet to be identified in space. Here, we employ a spectral synthesis code to examine the chemical evolution of the hydride and hydroxyl cations of the various isotopes of Ar, Ne, and He in the Crab Nebula filament and calculate their line emissivity and intrinsic line surface brightness. We successfully explain the observed surface brightness of two transitions of ArH+ (617 and 1234 GHz), one transition of OH+ (971 GHz), and one transition of H2 (2.12 micrometer). We also explain the observed surface brightness ratios between various molecular and atomic transitions. We find that our model reproduces the overall observed features when a hydrogen number density of ~10^4-10^6 cm^-3 and a cosmic-ray ionization rate per H2 of ~10^-11-10^-10 s^-1 are chosen. We discuss the possibility of detecting some hydride and hydroxyl cations in the Crab and diffuse cloud environment. Some transitions of these molecules are highlighted for future astronomical detection.
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