We know that Helium can have basically the same solution as the Hydrogen atom if there is only one electron. You get the same equation with a different factor on the potential and the basic solution is the same as the Hydrogen atom. But I'm trying to figure out why it can't work the other way around. If you have two hydrogen atoms side by side and one electron, then treating them as two potential wells you get the well-known pair of ground states, symmetric and antisymmetric, with the symmetric state just a little bit lower in energy than the antisymmetric. You can presumably fill this state with one electron, in which case each hydrogen atom should have half an electron. Now add a second electron. Shouldn't each of the two hydrogen atoms now be able to incorporate half of the second electron just the way the helium atom does? I can't see how the equations should be any different. So the two hydrogen atoms would end up sharing two half-electrons each. It's true that this solution is not the lowest energy state for the two hydrogen atoms. There is a lower energy state where each of them has exactly one whole electron, which is the familiar solution. (There is an even lower state where the two atoms come together and form a molecule, but in this situation we're sort of assuming for sake of argument that the atoms are kept apart.) So my quasi-helium states are not the lowest energy state, but they still seem like they should be a valid state that satisfies the relevant equations. It doesn't seem like this happens in practise because the corresponding spectral lines are not present as far as I know. But I can't see why not.