Apply LCAO to benzene and linear polyenes

A conjugated system contains a continuous row or ring of aligned p orbitals. Instead of forming separate π bonds between fixed pairs of atoms, these p orbitals combine into molecular orbitals extending across the conjugated framework.

GCE A-Level H3 Chemistry 9813-2027

On this page

Learning objectives

Apply LCAO to benzene and linear polyenes

Extend overlap across several atoms

Linear polyenes such as buta-1,3-diene and cyclic systems such as benzene follow the same counting rule: N contributing p orbitals form N π molecular orbitals.

Nodes, phases and delocalisation

The lowest-energy π molecular orbital has all neighbouring p orbitals in phase and no node between adjacent atoms. Higher π orbitals contain progressively more phase changes and nodes.

A node is a region where the wavefunction is zero. More nodes mean more antibonding interaction and therefore higher molecular-orbital energy.

Delocalisation means that a π molecular orbital and its electrons extend over more than two atoms; it does not mean that electrons move randomly between isolated double bonds.

Buta-1,3-diene and benzene

Four aligned p orbitals in buta-1,3-diene form four π molecular orbitals. Their phase patterns contain zero, one, two and three nodes respectively. The four π electrons occupy the two lowest orbitals, so the second is the HOMO and the third is the LUMO.

Six aligned p orbitals in benzene form six π molecular orbitals. The qualitative energy pattern contains one lowest bonding level, a degenerate bonding pair, a degenerate antibonding pair and one highest antibonding level.

Benzene's six π electrons fill the three bonding orbitals. Quantitative coefficients and mathematical derivation of the energies are not required here; focus on number of orbitals, phase, nodes, degeneracy and occupancy.

Worked example: build the butadiene set

Label the four carbon p orbitals C1 to C4. Draw ψ1 with the same phase on all four centres, then introduce one extra phase change for ψ2, two for ψ3 and three for ψ4.

Place two electrons in ψ1 and two in ψ2. The occupied pair are bonding overall; ψ2 is the HOMO and ψ3 is the LUMO.

Try this

How many π molecular orbitals and how many occupied π levels are present in ground-state buta-1,3-diene?

Check your answer

Four p orbitals form four π molecular orbitals. Four π electrons fill the two lowest levels, two electrons per level.

Construct the benzene pattern

Start with six p orbitals, so require six π molecular orbitals. Arrange them as 1 + 2 + 2 + 1 levels from lowest to highest, showing the equal-energy pairs on the same horizontal line.

Fill six electrons from the bottom. The lowest level and the degenerate bonding pair become occupied; the degenerate antibonding pair is the LUMO level.

Try this

Identify the HOMO and LUMO of benzene in the qualitative six-level pattern.

Check your answer

The HOMO is the occupied degenerate bonding pair. The LUMO is the next, unoccupied degenerate antibonding pair.

Transfer to a longer linear polyene

Hexa-1,3,5-triene contains six aligned p orbitals and six π electrons. It therefore has six π molecular orbitals and three occupied levels in its ground state.

The shapes are not copied from benzene because the molecular geometry and boundary conditions differ, even though the orbital and electron counts match.

Try this

For hexa-1,3,5-triene, state the number of π molecular orbitals, occupied levels and π electrons.

Check your answer

There are six π molecular orbitals, three occupied π levels and six π electrons.

Common mistake: pairing atoms into separate double bonds

Drawing three isolated π bonds for benzene misses the molecular-orbital model required here. All six p orbitals contribute to every benzene π molecular orbital, although the coefficients and nodes differ.

Do not count p-orbital lobes or electron pairs to determine the number of molecular orbitals. Count the contributing p atomic orbitals.

Try this

Correct: ‘Benzene has three π molecular orbitals because it has three π electron pairs.’

Check your answer

Benzene has six π molecular orbitals because six p atomic orbitals combine. Its six π electrons occupy the three bonding orbitals.

Check your understanding

You should be able to construct qualitative orbital sets for a linear polyene and benzene, order them by node count, and fill the electrons correctly.

Next, combine orbital construction with complete energy-level diagrams for H₂, O₂ and F₂, including HOMO, LUMO, bond order and magnetism.

Try this

What single rule links the number of atoms in a conjugated p system to the number of π molecular orbitals?

Check your answer

N aligned contributing p atomic orbitals form N π molecular orbitals.