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According to MO theory, are the 1s orbitals in $\ce{O2}$ too radially contracted to interfere with each other and thus cannot form bonding and antibonding orbitals, or do they still do that?

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It is always possible to define a bonding and antibonding orbital regardless of how small the overlap is. Because we solve the homonuclear diatomics in the symmetry of the molecule ($D_\mathrm{\infty h}$), the MOs we choose must respect that symmetry. It wouldn't matter if I put the oxygen atoms 10 light years apart, if I am treating them as part of the same quantum system, there are bonding and antibonding molecular orbitals. (Of course, at large distances bonding and antibonding orbital pairs becomes very near degenerate.)

That said, because oxygen has a doubly occupied 1s orbital, the corresponding sigma 1s bonding and antibonding orbitals will both be filled. Within MO theory we are always free to 'rotate' any two occupied MOs and the observable properties will all be the same.
So, there's no difference between saying that each oxygen has a non-bonding 1s orbital, and saying that there is a filled bonding and antibonding orbital. When both orbitals are filled you can't even ask the question of what the electrons are 'doing' down there, because the concept of a distinct orbital has no meaning.

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