MO theory for diatomic molecules

Lewis structures give you bond count. VSEPR gives you geometry. Neither predicts that O₂ is paramagnetic, that B₂ has a single bond while C₂ has a double, or why N₂’s bond dissociation energy is 945 kJ/mol and O₂’s is only 498. Molecular orbital theory answers all three. This guide builds the diatomic MO diagram from scratch, applies it to Li₂ through F₂, then extends to heteronuclear cases like CO and HF.

Prerequisites: atomic orbital shapes and the aufbau principle. If crystal field theory reads comfortably, this will too — it is the same “orbital energy diagram, fill electrons, read off the physics” workflow, one step earlier in the periodic table.

Two atomic orbitals in, two molecular orbitals out

MO theory’s central postulate: atomic orbitals on adjacent atoms combine linearly, and every N atomic orbitals in yields N molecular orbitals out. For two 1s orbitals on two hydrogen atoms:

$\psi_{\sigma} = \frac{1}{\sqrt{2}}(1s_A + 1s_B) \quad \text{(bonding)}$

$\psi_{\sigma^*} = \frac{1}{\sqrt{2}}(1s_A - 1s_B) \quad \text{(antibonding)}$

The bonding combination piles electron density between the nuclei; the antibonding combination has a node there. Bonding sits below the atomic energy, antibonding sits above by about the same amount. Fill 2 electrons: both go into σ, no electrons in σ*, giving H₂ with bond order 1.

Bond order is the diagnostic:

$\text{Bond order} = \frac{1}{2}(\text{bonding electrons} - \text{antibonding electrons})$

Positive → bond exists. Zero → no bond. Higher → shorter and stronger bond.

The first-row diatomic diagram

For homonuclear diatomics of period 2 (Li₂ to F₂), each atom contributes 1s, 2s, 2p_x, 2p_y, 2p_z — ten atomic orbitals total, producing ten MOs. The 1s combinations sit far below the valence set and are usually drawn as a “core” pair (σ_1s, σ*_1s) with cancelling contributions, so most diagrams start at 2s:

Standard ordering (O₂, F₂, Ne₂):

$\sigma_{2s} < \sigma^*_{2s} < \sigma_{2p_z} < (\pi_{2p_x}, \pi_{2p_y}) < (\pi^*_{2p_x}, \pi^*_{2p_y}) < \sigma^*_{2p_z}$

Reordered (Li₂, Be₂, B₂, C₂, N₂): for the earlier row, σ_2p_z sits above the π_2p pair because of s-p mixing (see below). So π MOs fill before the σ_2p MO.

The switchover happens between N₂ and O₂. Committing this ordering to muscle memory is the single largest payoff in inorganic MO work.

Why the ordering switches

The 2s and 2p_z orbitals on the same atom transform under the same symmetry as one another along the internuclear axis (both are σ under D_∞h). When their energy gap is small, they mix — the “bonding” σ_2s combination picks up σ_2p_z character (and vice versa) — pushing σ_2p_z up. The gap is small on the left of the row (2s and 2p are close in energy) and grows across the row as Z_eff pulls the 2s down faster than the 2p. By O and F, 2s-2p mixing is negligible and σ_2p_z drops back below the π pair.

Diagnostic: photoelectron spectroscopy shows the σ_2p ionisation of N₂ at lower binding energy than either π ionisation, confirming σ sits above π. In O₂, the ordering has reversed (Miessler 5e §5.2.3).

Worked example — bond order of the O₂ series

The paramagnetism of O₂ — the flagship prediction

Ground-state O₂ has 12 valence electrons. Filling with the standard ordering: σ_2s (2), σ_2s (2), σ_2p (2), π_2p (4), π_2p (2 — one in each degenerate π* by Hund’s rule). Two unpaired electrons. Paramagnetic.

Lewis structure of O₂ shows a double bond with two lone pairs on each oxygen — every electron paired, no unpaired spins. Lewis predicts diamagnetic. Reality: liquid O₂ is famously attracted to a magnet, sticking between the poles as it pours between them. MO theory got this right in 1932; Lewis structures cannot get it right at all. This is the standard-issue “why MO theory is worth learning” demonstration.

The full first-row lineup

Notice bond order peaks at N₂ and falls symmetrically. Every homonuclear diatomic that has been isolated (in the gas phase or a matrix) sits on this curve.

The two “no bond” cases are instructive. Be₂ has bond order zero on paper: σ_2s(2) plus σ*_2s(2) cancel. In reality, weak van der Waals bonding gives a “molecule” with dissociation energy of about 10 kJ/mol and a much longer bond than a covalent single bond. MO theory says “no covalent bond,” and that is exactly what the experimental values report. Ne₂ is a purer version of the same story.

B₂ — the second flagship prediction

B₂ has 6 valence electrons. In the reordered diagram (which applies for B₂), the filling is σ_2s(2), σ*_2s(2), π_2p(2 — one in each degenerate π by Hund). Bond order = ½(4 − 2) = 1. Two unpaired electrons: paramagnetic.

If the standard ordering applied, the last two electrons would go into σ_2p, and B₂ would be diamagnetic with bond order 1. The observed paramagnetism of B₂ (gas-phase ESR, 1970s) is direct evidence for the reordered π-below-σ arrangement in the early second row. This is one of the few clean experimental discriminators between the two diagrams.

Heteronuclear diatomics — CO and HF

When the two atoms are different, three things change:

1. Atomic energies are unequal. The more electronegative atom has lower AO energies (larger Z_eff at the same principal quantum number).
2. MO composition is polarised. Bonding MOs sit lower and have more character from the more electronegative atom; antibonding MOs sit higher and have more character from the less electronegative atom.
3. The g/u parity labels disappear. There is no inversion centre. Use symmetry labels appropriate to C_∞v (σ, π, δ without the g/u subscript).

CO is isoelectronic with N₂ (14 electrons total, 10 valence). Its MO diagram is a skewed N₂ diagram: 2s AOs of C sit above 2s AOs of O, and 2p AOs of C sit above 2p AOs of O, but the same ten MOs are filled in nearly the same order. Bond order 3, diamagnetic. The HOMO of CO is a σ MO with strong lone-pair character on carbon — which is why CO binds transition metals through carbon, not oxygen (see 18-electron rule).

HF has just 2 valence AOs available for π-symmetric MOs (2p_x and 2p_y of F, no matching orbital on H) so those become non-bonding lone pairs on F. The bond is one σ, and F has two non-bonding lone-pair MOs in π symmetry plus one non-bonding 2s. HF is the simplest case of “non-bonding MOs from symmetry mismatch.”

Common pitfalls

Ignoring the σ-π reorder before N₂. Predicting diamagnetic B₂ or C₂ paramagnetic is the classic error. Memorise “N₂ is the last one where σ_2p sits above π_2p.”

Counting core electrons in bond order. The 1s electrons are almost non-bonding — their MO combinations cancel to bond order zero at that level. Include them mechanically if the diagram shows them, or work with valence electrons only and trust the convention.

Applying homonuclear diagrams to heteronuclear cases without polarisation. CO is not “N₂ but drawn with different letters.” The MOs are polarised, and that polarisation is what gives CO its enormous dipole moment (0.11 D — small, and with the unusual polarity C⁻-O⁺, because the σ HOMO lone pair sits on carbon).

Confusing bond order with formal charge. Bond order comes from MO occupancy; formal charge comes from Lewis-structure electron counting. They can disagree — CO by Lewis structure has a triple bond with formal charges +C, −O; by MO theory, it has bond order 3 but with the electron density polarised toward oxygen everywhere except the σ HOMO.

Practice

Where this leads

The MO diagram is a starting-block for the whole of inorganic bonding theory:

• Polyatomic MO theory — walsh diagrams for H₂O, BeH₂, and NH₃; group theory does the bookkeeping.
• Ligand field theory — the MO version of crystal field theory, where ligand-metal σ and π bonding tunes the d-orbital splitting.
• Solid-state band structure — extend the diatomic diagram to a chain, a plane, a lattice, and the discrete MOs become continuous bands.

Every one of them starts with two AOs combining into two MOs and counting electrons.