I know that $\ce{He2}$ (homonuclear diatomic helium) does not exist because its bond order is zero. Since the bond order of $\ce{He2+} = 1/2$, that means that the positive $\ce{He2+}$ ion exists, but how does the positive ion exist if the neutral molecule doesn't?


Although we know that the helium dimer has a tiny bond dissociation energy, we don't consider it a molecule, as the "bond" does not survive a molecular vibration, even rotation of the molecule is enough to cleave it.

From an answer by Lighthart on "Molecular orbital theory & predicting the stability of a molecule?" I took the liberty to borrow the MO schemes:

MO schemes helium dimers

In the scheme you can clearly see how we arrive at the bond order of zero for $\ce{He2}$. Both, the bonding $\sigma$ and the anti-bonding combination $\sigma^{*}$ are doubly occupied. Dihelium is only held together by weak van der Waals forces.
In the case of the dihelium cation $\ce{He2+}$, one electron is removed from the anti-bonding orbital. There are now more electrons in bonding than anti-bonding orbitals. This effectively results in a net energy loss or bond energy gain, and the overall bond order increases to 0.5. In a more phenomenological context, three negative charges are now in a field of four positive charges, this gives rise to Coulomb attraction.

Reference: Wikipedia article on dihelium and references therein.

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