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In all texts online, first the Born-oppenheimer approximation is used to split the wavefunction of a diatomic molecule (homonuclear considered only here) to that of a nuclear relative motion, and a electronic wavefunction. Then using the variational method, an electronic potential energy curve is drawn. Due to symmetry requirements when exchanging the two proton positions, the electronic wavefunction must have odd and even parity, which is done by building it as a linear superposition of the individual atom electronic wavefunction (for the ground state anyway), and so we get an anti-bonding orbital and a bonding-orbital of lower energy. Then, when including the vibrational energy of the bond (from the nuclear wavefunction), authors then taylor the bottom of the potential energy curve to show that energies follow that of a quantum harmonic oscillator. But, these seem to use only bonding-orbitals, is this still valid for anti-bonding orbitals? If not, are the vibrational states only numerically calculatable? How does the total energy ladder look like if have an anti-bonding orbital?

Furthermore, since the whole diatomic molecule will have some spin (either integer of half-integer I guess), then does this mean that the anti-bonding or bonding-orbitals correspond to different spin states of the whole atom? (such that if whole molecule were half-integer spinned, a bonding-orbital would have to correspond to anti-symmetric total spin wavefunction?)

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