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For the case of 2 atoms, I can understand this as bonding or anti-bonding of the atoms. i.e., do the wavefunctions add between the protons, meaning each electron can share in the potential of both protons (bonding) or do the wavefunctions destructively interfere between the protons (anti-bonding).

With 3 atoms, I can't find 3 levels. Assuming Gaussian shaped wavefunctions, note that the sign of each wavefunction between any two atoms defines the wavefunction on the rest of the ring. Since the signs of the wavefunction are independent, there should be 2^3=8 possibilities since each wavefunction can be + or -. Yet, there are really only 2 energetically distinct arrangements that I see: all have the same sign (two cases) or 2 of 3 have the same sign (2*(3 choose 2)), to account for both sign cases). So I get 3 atoms yield 2 levels.

Can somebody shed light on what I've done incorrectly? Or is 3 too small to work correctly? Is there an argument about the shape of the orbitals I've neglected?

Thank you.