What is the theoretical quantum ground state energy of helium hydride positive ion? What is the theoretical quantum ground state energy of helium hydride positive ion? 

I am speaking of the usual assumptions such as the approximate modeling of the two nuclei here - one being a proton, the other being 2 protons plus 2 neutrons - as two infinitely massive particles alone in the universe at some equilibrium distance, in the non-relativistic simple case alone considering no spin dependent terms etc.

So actually only the charge of the nuclei is relevant in this approximation unless the vibrational energy - perhaps modeled in first approximation as a simple harmonic oscillator with some force constant - is also taken into account - which would be the positive 'zero' point value of harmonic oscillator. 

I would assume that it would have to be lower (that is greater bonding magnitude or absolute value of the negative energy) than that of the neutral helium atom, but I am not sure. But one place on internet gives $\pu{3600 cm^-1}$ as a fundamental frequency (but of what?) which - if I am not mistaken comes - to $\pu{1.64E-2}$ Hartrees or atomic units - so that cannot be correct or the value i am looking for as the helium atom is about -2.904 Hartrees or atomic units. I suppose it would be almost impossible to practically devise any experiment to arrive at any experimental value? Though, if so, what is that value?
You could get a quick estimate of the ground state energy with paper and pencil by using Heitler-London method, since all you need to do is change the charge on the second nucleus.

I would assume the vibrational energy absorption energy quoted referred to the He-H bond vibration.