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I apologize in advance if this turns out to be a duplicate question.

As far as I can understand, if you bring two or more atoms together their wave functions begin to interfere and, since there cannot be two electrons in the same state, the states "split" into more states.

However, what I don't understand is how this splitting occurs: Lets say that I bring two Helium atoms, $1s^2$, together. We already have two electrons in the $n=1$ state. If I'm not mistaken that's all you can fit in there. So the states have to split. When the states split do some electrons take on another $n$ value? If so, is this $n$ value necessarily an integer? Or is $n$ limited to integers only for "free" atoms?

Could it be that for a large collection of molecules, this "band structure" one finds in textbooks is a result of a lot of electrons in states $n+\delta$, where $1>>\delta\in \mathbb{R}$?

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The answer is that when two (or more) atoms are together the whole system has changed and $n$ cannot be used in the same way as for a single atom.

I find a helpful way of thinking about this is interms of molecular orbitals.

Two atomic orbitals can be combined to form two molecular orbitals - one bonding and one antibonding. This is shown below for H$_2$ formation from two H atoms

enter image description here

Note that for Helium the molecular orbitals are very similar to those for hydrogen and as you point out with 4 electrons you have too many for just one orbital.

In the case of bands when many atoms are combined I find the following image helpful, which is the case for Graphite made from carbon.

enter image description here

The band structure in Carbon comes from 2p electrons on the individual carbon atoms which are aligned perpendicularly to the planes of the carbon sheets. The bonding of these 2p electrons can be seen on the left hand side in Benzene molecules with the hexagonal structure - 6 atomic orbitals contribute to make 3 bonding and 3 antibonding molecular orbitals. The bonding orbitals ($\pi$) are filled with 6 electrons (the arrows) and the antibonding levels ($\pi^*$) are empty. Now on the right handside of the diagram many 2p orbitals from many carbon atoms combine to form the bonding (mostly filled) and antibonding (mostly empty) bands of graphite.

Now you could look at the energies of the electron states and say that we have non-integer $n$ values, but I think it makes more sense to think of the electron states being spread over many many atoms and having different (integer) numbers of nodes in the wavefunctions over a larger number of atoms.

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