π Molecular Orbitals of 1,3-Butadiene

Conjugated dienes have lower heats of hydrogenation than cumulated and isolated dienes, making them more stable. The enhanced stabilization of conjugated systems can be understood from their π molecular orbitals.

The simplest conjugated diene is 1,3-butadiene: a four-carbon system where each carbon is sp²-hybridized and has an unhybridized p orbital that contains an unpaired electron. According to molecular orbital theory, atomic orbitals combine to form molecular orbitals such that the number of molecular orbitals formed is equal to the number of involved atomic orbitals. A linear combination of the four atomic p orbitals in 1,3-butadiene gives rise to four molecular orbitals as shown below.

The bonding molecular orbitals, ψ₁ and ψ₂, are lower in energy relative to the atomic orbitals, whereas the antibonding molecular orbitals, ψ₃ and ψ₄, are higher in energy. The unpaired electrons are filled starting from the lowest energy molecular orbital. Since each orbital can accommodate a maximum of two electrons, the four unpaired electrons are distributed between ψ₁ and ψ₂. Based on the electron distribution, ψ₂ is the highest occupied molecular orbital (HOMO) and ψ₃ is the lowest unoccupied molecular orbital (LUMO).

Each molecular orbital is distinct, with the difference attributed to the phases of the four p orbitals. In general, an in-phase overlap between two atomic p orbitals forms a π bond. However, an out-of-phase overlap results in a node, a region of zero electron density with no bonding interaction between the two atoms.

In 1,3-butadiene, the lowest energy molecular orbital, ψ₁, is formed by an in-phase overlap of all four p orbitals forming a continuous π system. In ψ₂, the out-of-phase overlap between the two central carbons gives rise to a node. The number of nodes increases to two in ψ₃ and three in ψ₄. In summary, the energy of the molecular orbital increases as the number of nodes increases, whereas the bonding interaction decreases.