Molecular Orbital Theory for Beginners

Lewis structures and hybridisation describe bonds as electron pairs belonging to two particular atoms. That works remarkably well, but it fails in some important cases. It can’t explain why oxygen gas sticks to a magnet, why He₂ doesn’t exist while He₂⁺ does, or why removing an electron from O₂ makes its bond stronger. Molecular orbital (MO) theory answers all of these. It treats electrons as belonging to the whole molecule, in orbitals that spread over all the atoms. This introduction builds the key ideas step by step.

From atomic to molecular orbitals

In an atom, electrons occupy atomic orbitals (1s, 2s, 2p…) centred on one nucleus (see atomic orbital shapes). In MO theory, when atoms come together, their atomic orbitals combine to form molecular orbitals that belong to the molecule as a whole.

Three rules govern this:

1. Number conserved: combining n atomic orbitals always gives n molecular orbitals. Two 1s orbitals give two MOs.
2. Similar energies and matching symmetry: orbitals combine effectively only if they have similar energies and point in compatible directions.
3. Filling rules: electrons fill MOs in order of increasing energy (the aufbau principle), a maximum of two per orbital with opposite spins (Pauli), and singly into orbitals of equal energy before pairing (Hund’s rule) — exactly as in atoms (see the electron configuration tool).

Bonding and antibonding orbitals

Electrons behave like waves. When two waves overlap, they can combine in phase (adding up) or out of phase (cancelling). The same happens with orbitals.

Take two hydrogen 1s orbitals:

• In-phase combination → a bonding orbital, labelled σ1s. The electron waves reinforce between the nuclei, building up electron density there. Electrons in this orbital attract both nuclei and hold them together. Its energy is lower than the original 1s orbitals.
• Out-of-phase combination → an antibonding orbital, labelled σ*1s (the star means antibonding). The waves cancel between the nuclei, leaving a node — a region of zero electron density — between them. Electrons here pull the nuclei apart rather than together. Its energy is higher than the original 1s orbitals.

Crucially, the antibonding orbital is raised in energy by slightly more than the bonding orbital is lowered. So filling both a bonding and its matching antibonding orbital gives a net destabilisation.

Bond order

The key number in MO theory is the bond order:

Bond order = ½ × (number of bonding electrons − number of antibonding electrons)

• Bond order 1 = single bond, 2 = double, 3 = triple.
• Bond order 0 means no net bonding — the molecule won’t exist as a stable species.
• Fractional bond orders (such as ½ or 2½) are possible, and they’re real.

Higher bond order generally means a shorter and stronger bond (see bond length and bond strength). For more worked calculations, see bond order from MO diagrams.

Example 1: H₂

Two hydrogen atoms bring 2 electrons. Both go into the low-energy σ1s bonding orbital, with opposite spins.

• Configuration: (σ1s)²
• Bond order = ½(2 − 0) = 1
• A stable single bond, matching the Lewis picture H–H.

Example 2: He₂ — why it doesn’t exist

Two helium atoms bring 4 electrons. Two fill σ1s; the other two must go into σ*1s.

• Configuration: (σ1s)²(σ*1s)²
• Bond order = ½(2 − 2) = 0
• The antibonding electrons cancel the bonding ones. He₂ doesn’t form under normal conditions — which is why helium is a monatomic gas.

Example 3: the ions H₂⁺ and He₂⁺

• H₂⁺ (1 electron): (σ1s)¹; bond order ½. It exists, with a weaker, longer bond than H₂.
• He₂⁺ (3 electrons): (σ1s)²(σ*1s)¹; bond order ½. This ion exists too and has been detected in gas discharges.

Lewis structures can’t describe a “half bond” at all; MO theory handles it naturally.

Period 2 diatomic molecules

For atoms from lithium to neon, the valence orbitals are 2s and 2p. The 2s orbitals combine to give σ2s and σ*2s. The 2p orbitals give more:

• The two 2p orbitals pointing along the bond axis overlap head-on to give σ2p (bonding) and σ*2p (antibonding).
• The two pairs of 2p orbitals perpendicular to the axis overlap sideways to give two π2p (bonding, equal in energy) and two π*2p (antibonding, equal in energy) orbitals.

That’s 8 valence MOs from 8 atomic orbitals (four from each atom).

The order of energy levels

For O₂, F₂ and Ne₂, the order (lowest to highest) is:

σ2s < σ*2s < σ2p < π2p < π*2p < σ*2p

For Li₂ to N₂, the 2s and 2p orbitals are close enough in energy to interact (called s–p mixing), which pushes σ2p above π2p:

σ2s < σ*2s < π2p < σ2p < π*2p < σ*2p

This swap doesn’t change the bond orders of these molecules, but it does affect some magnetic properties (for example, it explains why B₂ is paramagnetic).

Results for period 2

Molecule	Valence electrons	Bond order	Unpaired electrons	Magnetism	Bond energy (kJ mol⁻¹, approx.)
Li₂	2	1	0	Diamagnetic	105
Be₂	4	0	0	— (not stable)	—
B₂	6	1	2	Paramagnetic	290
C₂	8	2	0	Diamagnetic	600
N₂	10	3	0	Diamagnetic	945
O₂	12	2	2	Paramagnetic	498
F₂	14	1	0	Diamagnetic	158
Ne₂	16	0	0	— (not stable)	—

The bond energy rises and falls with the bond order — peaking at N₂ with its triple bond — just as MO theory predicts.

Magnetism: the killer evidence

• A substance with unpaired electrons is paramagnetic: it’s attracted into a magnetic field.
• A substance with all electrons paired is diamagnetic: it’s very weakly repelled.

The Lewis structure of O₂ (O=O, with two lone pairs on each atom) has all its electrons paired, predicting diamagnetism. But liquid oxygen poured between the poles of a strong magnet clings to them. MO theory explains why: O₂’s last two electrons go singly into the two equal-energy π*2p orbitals (Hund’s rule), leaving two unpaired electrons — while still giving a bond order of 2, consistent with a double bond. For the full explanation, see why is oxygen paramagnetic?.

Heteronuclear molecules: a glimpse

When the two atoms are different, such as in CO or HF, their atomic orbitals have different energies. The more electronegative atom’s orbitals sit lower. The bonding MOs then lie closer in energy to — and have more character from — the more electronegative atom, so the bonding electrons are concentrated on that atom. That’s MO theory’s description of a polar bond (see bond polarity). CO is isoelectronic with N₂ (both have 10 valence electrons) and also has a bond order of 3, which fits its very high bond enthalpy of about 1,077 kJ mol⁻¹.

MO theory vs valence bond theory

Feature	Valence bond theory (Lewis, hybridisation)	Molecular orbital theory
Where electrons are	In bonds between two atoms, or lone pairs on one atom	In orbitals spread over the whole molecule
Strengths	Simple; great for shapes and organic chemistry	Explains magnetism, fractional bond orders, spectra, excited states
Weaknesses	Fails for O₂ magnetism, He₂⁺, delocalisation needs resonance patches	Harder to picture for large molecules

The two approaches aren’t rivals so much as tools for different jobs. Chemists routinely use hybridisation to describe the σ framework of organic molecules and MO theory for π systems, colour and reactivity.

Common misconceptions

• “Antibonding orbitals are empty by definition.” They’re just higher-energy orbitals; they fill when there are enough electrons (as in O₂ and F₂).
• “Antibonding electrons have no effect.” They cancel bonding and can even outweigh it.
• “Bond order must be a whole number.” H₂⁺, He₂⁺ and O₂⁺ have fractional bond orders.
• “Electrons in a molecule belong to individual atoms.” In MO theory, they’re spread over the whole molecule.

Key takeaways

• Atomic orbitals combine to form molecular orbitals; n in → n out.
• Bonding MOs (lower energy, density between nuclei) hold atoms together; antibonding MOs (higher energy, node between nuclei) push them apart.
• Bond order = ½(bonding − antibonding electrons); zero means no stable molecule.
• MO theory explains He₂’s non-existence, fractional bonds and O₂’s paramagnetism.
• In period 2, s–p mixing puts π2p below σ2p for Li₂–N₂.