What molecules other than N$_2$ have wrong orbital ordering when calculated using Hartree-Fock?

N$\_2$ is a canonical example of this aspect of the failure of HF method, for instance, see table 4.9 in "Modern Quantum Chemistry" by Szabo and Ostlund. Here it is shown that in HF, the HOMO of N$\_2$ is of $\pi$ symmetry, whereas correlated method and experiments show that its HOMO should have $\sigma$ symmetry, which becomes HOMO-1 in HF.


