Molecular-orbital theory

Pour liquid nitrogen between the poles of a strong magnet and it simply runs off; pour pale-blue liquid oxygen and it sticks, bridging the poles. Attraction to a magnet means unpaired electrons. Yet in the Lewis structure O=O every electron is paired. Here the Lewis picture fails, and we need a theory in which electrons belong to the whole molecule: molecular-orbital theory.

When two atoms approach, their atomic orbitals can combine in two ways (linear combination of atomic orbitals, LCAO). Added in phase, they build up electron density between the nuclei and drop below the original atomic orbitals in energy: a bonding orbital. Subtracted, they leave a node between the nuclei and rise in energy: an antibonding orbital (marked with a star). The number of molecular orbitals always equals the number of atomic orbitals combined.

Electrons fill molecular orbitals by the aufbau principle, Pauli exclusion and Hund’s rule, just as in atoms. Bonding electrons stabilise the molecule and antibonding electrons undo that stabilisation, which gives the bond order:

The simplest case: H₂’s two electrons both sit in the bonding $\sigma_{1s}$ orbital — bond order 1. He₂ has 4 electrons, two bonding and two antibonding: bond order 0, which is why helium is monatomic. Remove one electron and $\ce{He2+}$ has bond order ½ — and it really exists, detectable in helium discharges, with a dissociation energy of about 230 kJ/mol.

From Li₂ to Ne₂ the valence shell is 2s and 2p. The 2s orbitals give $\sigma_{2s}$ and $\sigma^*_{2s}$; of each atom’s three 2p orbitals, the pair along the bond axis gives $\sigma_{2p}$ and $\sigma^*_{2p}$, and the two perpendicular pairs give two $\pi_{2p}$ and two $\pi^*_{2p}$.

The subtle point is whether $\sigma_{2p}$ or $\pi_{2p}$ lies lower. For O₂ and F₂, $\sigma_{2p}$ is lower. But for B₂, C₂ and N₂ the 2s and 2p energies are close, and $\sigma_{2s}$ and $\sigma_{2p}$ (which share the same symmetry) push each other apart, lifting $\sigma_{2p}$ above $\pi_{2p}$. This is s–p mixing. Across the period the 2s–2p gap grows from 5.8 eV for B to 21.5 eV for F, the mixing weakens, and the order flips between N and O.

The same reasoning extends to heteronuclear diatomics, except that the two sets of atomic orbitals no longer match in energy. CO has 10 valence electrons, isoelectronic with N₂: bond order 3, bond length 113 pm, one of the strongest bonds known (D₀ about 1072 kJ/mol). NO has 11; the extra electron goes into $\pi^*$, giving bond order 2.5 and a paramagnetic radical.