I guess this is more of a chemistry question, but whatever. I think it's interesting.
Suppose you had two bare atomic nuclei. For concreteness, lets assume the nuclei are the same with atomic number $Z$. Lets bring in a single electron and focus on the ground states of the nuclei.
When the nuclei are far apart, the ground states are degenerate. When we bring the nuclei together, the ground state splits into the bonding and anti-bonding orbitals. Let $\Delta E$ represent some measure of the energy difference between the bonding and anti-bonding orbitals.
From intuition, I would expect $\Delta E$ to increase with decreasing internuclear distance $R$. What happens as $R$ shrinks to zero?
I expect the bonding orbital to become the ground state of an "atom" with charge $2Z$. Is that correct? More importantly, what happens to the anti-bonding orbital?
This isn't an exercise in the Born-Oppenheimer Approximation. I magically hold the nuclei at a distance $R$, so their repulsion doesn't matter. Also, electron-electron repulsion doesn't matter because I only introduce one electron.