  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.
