Molecular Orbital Theory — Bond Order and Magnetic Behaviour
Valence bond theory draws O₂ with a double bond and every electron paired, and predicts that liquid oxygen should be diamagnetic. Pour liquid oxygen between the poles of a magnet and it sticks. That single experiment is why molecular orbital theory replaced the simple Lewis picture for diatomic molecules, and it is still the most-asked question in the topic. This article builds the MO diagrams for the second-row homonuclear diatomics, counts every electron, and derives every bond order.
Bond order — the one formula
Only valence electrons matter for the second row: the 1s core electrons fill σ1s and σ*1s, which cancel exactly and can be ignored. The four 2p orbitals on each atom combine to give one σ (from the two pz orbitals pointing along the bond) and two degenerate π orbitals (from px and py), plus their antibonding partners.
Two orderings, and the reason for the switch
Everything depends on whether σ2pz lies below or above the π2p pair. Both orderings are real, and the crossover happens between nitrogen and oxygen.
| Applies to | Energy order (lowest first) |
|---|---|
| Li₂, Be₂, B₂, C₂, N₂ (Z ≤ 7) | σ2s < σ*2s < π2px = π2py < σ2pz < π*2px = π*2py < σ*2pz |
| O₂, F₂, Ne₂ (Z ≥ 8) | σ2s < σ*2s < σ2pz < π2px = π2py < π*2px = π*2py < σ*2pz |
The cause is s–p mixing. The σ2s and σ2pz orbitals have the same symmetry, so they interact: the lower one is pushed down and the upper one is pushed up. The strength of that interaction depends on the 2s–2p energy gap in the free atom, which widens steadily across the period as the effective nuclear charge grows. In B, C and N the gap is small, mixing is strong, and σ2pz is pushed above the π pair. By oxygen the gap is large enough that mixing is weak and the "natural" order σ before π is restored.
Crucially, this is not a bookkeeping convenience — it is confirmed by magnetism. B₂ is observed to be paramagnetic, which is only possible if the two valence p electrons occupy the degenerate π pair singly. C₂ is observed to be diamagnetic, which is only possible if all four of its p-block electrons pair up in that same degenerate π level. Put σ2p first and you predict the opposite for both.
The full electron count for the second-row diatomics
| Molecule | Valence e⁻ | Configuration | Bonding | Antibonding | Bond order | Unpaired | Magnetism |
|---|---|---|---|---|---|---|---|
| B₂ | 6 | σ2s² σ*2s² π2p² | 4 | 2 | 1 | 2 | Paramagnetic |
| C₂ | 8 | σ2s² σ*2s² π2p⁴ | 6 | 2 | 2 | 0 | Diamagnetic |
| N₂ | 10 | σ2s² σ*2s² π2p⁴ σ2p² | 8 | 2 | 3 | 0 | Diamagnetic |
| O₂ | 12 | σ2s² σ*2s² σ2p² π2p⁴ π*2p² | 8 | 4 | 2 | 2 | Paramagnetic |
| F₂ | 14 | σ2s² σ*2s² σ2p² π2p⁴ π*2p⁴ | 8 | 6 | 1 | 0 | Diamagnetic |
| Ne₂ | 16 | σ2s² σ*2s² σ2p² π2p⁴ π*2p⁴ σ*2p² | 8 | 8 | 0 | 0 | Does not exist |
Worked example 1 — dioxygen, step by step. Oxygen has 6 valence electrons per atom, so O₂ has 12. Using the Z ≥ 8 ordering:
σ2s takes 2 · σ*2s takes 2 · σ2pz takes 2 · π2px and π2py take 4 · that is 10, leaving 2 electrons.
The remaining 2 go into the degenerate pair π*2px and π*2py. By Hund's rule they occupy different orbitals with parallel spins: π*2px¹ π*2py¹.
Bonding electrons = 2 (σ2p) + 4 (π2p) + 2 (σ2s) = 8. Antibonding = 2 (σ*2s) + 2 (π*2p) = 4.
Bond order = ½(8 − 4) = 2 · unpaired electrons = 2 · paramagnetic.
The ground state is a triplet, ³Σg⁻. Two unpaired electrons give a spin-only moment of √(2 × 4) = 2.83 BM, which matches experiment. Molecular orbital theory gets both the double bond and the magnetism right; the Lewis structure gets only the first.
Worked example 2 — why C₂ has no σ bond at all. C₂ has 8 valence electrons and uses the Z ≤ 7 ordering: σ2s² σ*2s² π2p⁴.
Bonding = 2 + 4 = 6 · antibonding = 2 · bond order = ½(6 − 2) = 2.
But look at which orbitals are occupied. The σ2pz level is empty. C₂ in the gas phase therefore has a double bond made of two π bonds and no net σ bond — a result that has no valence-bond equivalent, and a standard discussion question. It also explains why C₂ is highly reactive: its unusual bonding is far from the strong σ + π arrangement carbon prefers.
Bond order against bond length and bond energy
The check that a bond order is right is that it tracks the measured bond length and dissociation energy in the correct direction.
| Molecule | Bond order | Bond length (pm) | Bond dissociation energy (kJ/mol) |
|---|---|---|---|
| B₂ | 1 | 159 | ≈ 290 |
| C₂ | 2 | 124 | ≈ 600 |
| N₂ | 3 | 110 | 945 |
| O₂ | 2 | 121 | 498 |
| F₂ | 1 | 142 | 159 |
Higher bond order means shorter and stronger, every time. N₂ has the highest bond order of the set and the strongest bond in the whole of small-molecule chemistry — which is precisely why nitrogen fixation is so difficult and why so many explosives release their energy by forming N₂.
The oxygen ion series — the cleanest test of the model
Add or remove electrons from O₂ and you change only the population of the π* level. Since those are antibonding, removing an electron strengthens the bond.
| Species | Valence e⁻ | π* occupancy | Bond order | Unpaired | Bond length (pm) |
|---|---|---|---|---|---|
| O₂⁺ (dioxygenyl) | 11 | π*¹ | ½(8 − 3) = 2.5 | 1 | 112 |
| O₂ | 12 | π*² | ½(8 − 4) = 2.0 | 2 | 121 |
| O₂⁻ (superoxide) | 13 | π*³ | ½(8 − 5) = 1.5 | 1 | 128 |
| O₂²⁻ (peroxide) | 14 | π*⁴ | ½(8 − 6) = 1.0 | 0 | 149 |
Worked example 3 — order these by bond length. The bond order sequence is O₂⁺ (2.5) > O₂ (2.0) > O₂⁻ (1.5) > O₂²⁻ (1.0), so the bond length sequence must be the reverse: O₂⁺ < O₂ < O₂⁻ < O₂²⁻, which the measured values confirm.
Magnetism: only peroxide, with a filled π* level, is diamagnetic. Superoxide has one unpaired electron (μ = √3 = 1.73 BM) — this is the species involved in biological oxidative stress, and the reason superoxide dismutase exists.
Extending to heteronuclear diatomics
For NO, CO and CN⁻ the same counting works, using the Z ≤ 7 ordering because the average effective nuclear charge is low enough for s–p mixing to matter. The orbitals are no longer symmetric — the bonding MOs sit closer in character to the more electronegative atom — but the bond order arithmetic is unchanged.
| Species | Valence e⁻ | Bond order | Unpaired | Note |
|---|---|---|---|---|
| CO | 10 | 3 | 0 | Isoelectronic with N₂; HOMO is essentially a carbon lone pair, which is why CO binds metals through carbon |
| CN⁻ | 10 | 3 | 0 | Also isoelectronic with N₂; strong-field ligand |
| NO | 11 | 2.5 | 1 | One electron in π*; paramagnetic odd-electron molecule |
| NO⁺ | 10 | 3 | 0 | Losing the π* electron raises the bond order and shortens the bond |
NO is the reason the "molecules have even numbers of electrons" intuition fails. It has 11 valence electrons, one of them in a π* orbital, and it is stable enough to be a signalling molecule in mammals. Because the electron it loses is antibonding, NO ionises unusually easily and NO⁺ has a stronger, shorter bond than NO — a textbook counter-intuitive result.
The errors that appear most often
- Using the σ-below-π ordering for N₂ or below. This is the single most common mistake and it flips the predicted magnetism of both B₂ and C₂.
- Forgetting Hund's rule in the π* level. Two electrons in a degenerate pair go singly with parallel spins — pair them and you wrongly call O₂ diamagnetic.
- Counting core electrons. They cancel; include them and every bond order still comes out right, but only if you count both σ1s and σ*1s. Safer to use valence electrons only.
- Assuming removing an electron always weakens a bond. If the electron came from an antibonding orbital, the bond gets stronger — O₂⁺ and NO⁺ both demonstrate this.
- Treating bond order as necessarily a whole number. Odd-electron species have half-integral bond orders, and that is physically meaningful.
- Reading "paramagnetic" as "has electrons". It means at least one unpaired electron.
How to attack any MO question in four steps
- Count the total valence electrons, adjusting for charge.
- Choose the ordering: π below σ for Z ≤ 7 on both atoms, σ below π for O, F and beyond.
- Fill the levels bottom-up, obeying the Pauli principle and Hund's rule in every degenerate pair.
- Apply bond order = ½(Nbonding − Nantibonding) and count unpaired electrons for the magnetism.
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