Molecular orbital diagram generator

In molecular orbital theory, atomic orbitals of similar energy and matching symmetry on the two atoms combine into molecular orbitals that belong to the whole molecule. Two atomic orbitals give two molecular orbitals: an in-phase bonding orbital, lower in energy than the atomic orbitals, with electron density between the nuclei, and an out-of-phase antibonding orbital (marked *), higher in energy, with a node between the nuclei. For second-row diatomics the 2s orbitals give σ2s and σ*2s; the 2p orbitals pointing along the bond give σ2p and σ*2p; the two pairs of 2p orbitals perpendicular to it give two degenerate π2p and two degenerate π*2p orbitals.

Electrons are placed in the molecular orbitals in order of increasing energy (aufbau), at most two per orbital with opposite spins (Pauli), and singly with parallel spins in degenerate orbitals before any pairs up (Hund’s rule). The bond order is

$$\begin{array}{rl}& \text{bond order}=\\ & \frac{\begin{array}{c}\text{number of bonding electrons}\\ -\text{number of antibonding electrons}\end{array}}{2}\end{array}$$

A bond order above zero means the molecule is expected to be more stable than the separate atoms; a bond order of zero (He₂, Be₂, Ne₂) means it is not expected to exist as a stable molecule. A species with unpaired electrons is paramagnetic (attracted into a magnetic field); one with all electrons paired is diamagnetic. The 1s core orbitals hold equal numbers of bonding and antibonding electrons, so they never change the bond order and are often left out.

s–p mixing. For Li₂ to N₂ the 2s and 2p orbitals are close enough in energy to mix. Mixing lowers σ2s and raises σ2p, enough that σ2p ends up above the π2p orbitals. The order is then σ2s, σ*2s, π2p, σ2p, π*2p, σ*2p. For O₂, F₂ and Ne₂ the 2s–2p gap is larger, mixing is small, and σ2p lies below π2p: σ2s, σ*2s, σ2p, π2p, π*2p, σ*2p. The generator uses the mixed order for Li₂ to N₂, N₂⁺, N₂⁻, CO and CN⁻, and the unmixed order for O₂, F₂, Ne₂, their ions and NO. For NO, O₂ and the O₂ ions the choice does not change the bond order or the magnetism.

Heteronuclear molecules. The atomic orbitals of the more electronegative atom are drawn lower in energy, so its orbitals contribute more to the bonding orbitals. In HF the hydrogen 1s orbital combines only with the fluorine 2p orbital that points along the bond; the F 2s and the two other F 2p orbitals have no partner and stay nonbonding (n), so they do not count in the bond order.

Energies in the diagrams are schematic: they show the order of the levels, not measured values. Source: Flowers P, Theopold K, Langley R, Robinson WR et al. Chemistry 2e, section 8.4 Molecular Orbital Theory. OpenStax, Houston, 2019, openstax.org/books/chemistry-2e/pages/8-4-molecular-orbital-theory.

O₂, step by step. Each oxygen atom has 6 valence electrons ($2{\mathrm{s}}^{2}2{\mathrm{p}}^4$), so O₂ has 12 valence electrons. Oxygen uses the unmixed order, with σ2p below π2p.

1. σ2s takes 2 and σ*2s takes 2 (8 left).
2. σ2p takes 2 (6 left).
3. The two π2p orbitals take 4 (2 left).
4. The last 2 go into the two degenerate π*2p orbitals, one in each with parallel spins (Hund’s rule).

Configuration: $(\sigma 2\mathrm{s}{)}^2({\sigma }^{\ast }2\mathrm{s}{)}^2(\sigma 2\mathrm{p}{)}^2(\pi 2\mathrm{p}{)}^4({\pi }^{\ast }2\mathrm{p}{)}^2$. Bonding electrons: $2+2+4=8$; antibonding: $2+2=4$. $\text{Bond order}=(8-4)/2=\mathbf{2}$, a double bond, as in the Lewis structure. But the two π* electrons are unpaired, so O₂ is paramagnetic with 2 unpaired electrons, which the Lewis structure (all electrons paired) cannot explain: liquid oxygen is attracted to a magnet.

Along O₂⁺, O₂, O₂⁻, O₂²⁻ each added electron goes into a π* orbital, so the bond order falls from 2.5 to 2, 1.5 and 1, and the O–O bond gets longer and weaker.