Molecular Orbital Theory of Homonuclear Diatomic Molecules

Valence bond theory, as this chapter's earlier section noted, cannot explain the paramagnetism of $\text{O}_2$, because it always predicts fully paired electrons. Molecular orbital theory (MOT), developed by Hund and Mulliken, resolves this by abandoning the idea that bonding electrons belong to a single, localised bond at all. Instead, MOT treats the electrons of a whole molecule as occupying molecular orbitals that extend over the entire molecule, formed mathematically by the linear combination of atomic orbitals (LCAO) — the atomic orbitals of the combining atoms are added together and subtracted from each other to generate an equal number of new molecular orbitals.

Combining two atomic orbitals always produces exactly two molecular orbitals:

• A bonding molecular orbital, formed by the constructive combination (addition) of the atomic orbitals, which concentrates electron density in the region between the two nuclei. Electrons placed here are lower in energy than they were in the separate atomic orbitals, and stabilise the molecule.
• An antibonding molecular orbital (marked with an asterisk, e.g. $\sigma^*$), formed by the destructive combination (subtraction) of the atomic orbitals, which places a node — a region of zero electron density — directly between the two nuclei. Electrons placed here are higher in energy than in the separate atomic orbitals, and destabilise the molecule.

Head-on combination of orbitals along the internuclear axis (as with two $s$ orbitals, or two $p$ orbitals pointed along the bond axis) produces $\sigma$ and $\sigma^*$ molecular orbitals; sideways combination of parallel $p$ orbitals produces doubly-degenerate $\pi$ and $\pi^*$ pairs. Electrons fill these molecular orbitals following the same three rules used for atomic orbitals: the Aufbau principle (lowest energy orbitals fill first), the Pauli exclusion principle (at most two electrons per orbital, with opposite spins), and Hund's rule (degenerate orbitals, such as the two $\pi$ or two $\pi^*$ orbitals, are singly occupied before any is doubly occupied).

For second-period homonuclear diatomics, the energy ordering of the valence molecular orbitals is:

Once the configuration is written, the bond order is calculated as

$\text{Bond order} = \tfrac{1}{2}\big[(\text{number of bonding electrons}) - (\text{number of antibonding electrons})\big]$ …