⚛️ Molecular Orbital Theory — Bond Order for Diatomic Molecules

MO theory treats the electrons in a molecule as occupying orbitals that are delocalized over the entire molecule, built by combining (a linear combination of) the atomic orbitals of every atom involved, rather than as localized bonding pairs and lone pairs drawn between two specific atoms. This gives a more physically accurate picture of bonding, and unlike a Lewis structure or VSEPR, it correctly predicts properties such as magnetism that depend on whether electrons are paired.

A bonding molecular orbital is lower in energy than the atomic orbitals that formed it and has constructive overlap, building up electron density between the two nuclei that holds them together and stabilizes the molecule. An antibonding orbital, marked with an asterisk such as σ*, is higher in energy than the parent atomic orbitals and has a node — a region of zero electron density — between the nuclei; electrons placed in it destabilize the molecule.

Bond order equals (electrons in bonding orbitals − electrons in antibonding orbitals) ÷ 2. It estimates how many net bonds hold the two atoms together: a bond order of 1 behaves like a single bond, 2 like a double bond, 3 like a triple bond, and 0 means there is no net bonding at all.

Filling O₂'s 12 valence electrons by Aufbau and Hund's rule leaves the two degenerate π*2p orbitals with one electron each, both unpaired — making O₂ paramagnetic and attracted into a magnetic field, which is why liquid oxygen visibly clings to a strong magnet. A simple Lewis structure for O₂ pairs every electron and predicts no magnetism at all, so O₂'s paramagnetism was a landmark, very public confirmation that MO theory captures real bonding physics that Lewis structures miss.

For the lighter, less electronegative period-2 elements from Li through N, the 2s and 2p atomic orbitals are close enough in energy to mix (s–p mixing), which pushes the σ2p molecular orbital above the π2p pair. Moving across the period to O, F, and Ne, increasing nuclear charge pulls the 2s orbital down and separates it further from 2p, so mixing becomes negligible and the textbook order — σ2p below π2p — is restored.

Both fill their bonding and antibonding orbitals equally: Be₂'s 4 electrons give σ2s²σ*2s², and Ne₂'s 16 electrons fill every bonding and antibonding orbital completely. In both cases the antibonding electrons exactly cancel the stabilization from the bonding electrons, giving a bond order of 0 — no net attraction remains to hold the two atoms together as a stable molecule.

Hund's rule says that when electrons fill a set of degenerate orbitals of equal energy, such as the two π2p or two π*2p orbitals here, they first occupy each orbital singly with parallel spin before any pairing occurs. This simulation applies Hund's rule automatically to the degenerate π pairs, which is exactly what produces O₂'s two unpaired π*2p electrons.

For period-2 diatomics, the 1s core orbitals on each atom are so much lower in energy, and so much smaller, than the valence 2s/2p orbitals that they barely overlap with the other atom's 1s orbital. They form essentially non-bonding core molecular orbitals that contribute equally to bonding and antibonding character, so — as a standard simplification — they are left out of the bond order calculation entirely.

MO theory underlies virtually all modern computational and quantum chemistry software used to predict molecular structure, reactivity, and spectra. It also extends far beyond simple diatomics — to conjugated π-electron systems, aromaticity in molecules like benzene, and the band theory that explains how semiconductors and metals conduct electricity.