Delocalised MOs and Conjugation
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You should be comfortable with the linear combination of atomic orbitals (LCAO) method for diatomics and the concept of bond order as defined earlier in this unit.
Why this matters
Conjugation explains why molecules like benzene and polyenes are more stable than their isolated double-bond counterparts.
In the previous lecture, we treated molecular orbitals (MOs) as localised interactions between two atoms. However, in larger systems such as 1,3-butadiene or benzene, the p-orbitals on adjacent atoms overlap to form a continuous network of delocalised orbitals spanning the entire molecule. This delocalisation lowers the total electronic energy, a phenomenon known as resonance energy. Understanding this is crucial for predicting reactivity, colour, and magnetic properties in organic and inorganic materials.
The Idea of Delocalisation
Imagine a series of atoms lined up, each contributing a p-orbital perpendicular to the chain. In a localised picture, electrons are trapped between specific pairs of atoms. In the delocalised picture, the wavefunctions of these p-orbitals overlap to create new molecular orbitals that extend over the whole chain. The electrons are no longer "owned" by a single bond but are spread out over the entire conjugated system. This spreading out allows the electrons to occupy lower energy states than they would in isolated double bonds, stabilising the molecule. The more atoms involved in this conjugation, the more the energy levels split, and the more stable the system becomes relative to a hypothetical localised structure.
Particle in a Box Approximation
To quantify the effect of conjugation length, we can model the delocalised electrons as particles confined to a one-dimensional box of length \(L\). The allowed energy levels for a particle of mass \(m\) in a box are given by the quantum number \(n\):
Here, \(h\) is Planck's constant. As the length of the conjugated system \(L\) increases, the energy gap \(\Delta E\) between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) decreases. This is why longer polyenes absorb light at longer wavelengths (lower energies), shifting from the ultraviolet into the visible region, which is why many long-chain conjugated molecules are coloured.
Bond Order in Conjugated Systems
We can extend the bond order concept to conjugated systems. For a linear polyene with \(N\) p-orbitals, there are \(N\) molecular orbitals. The bond order between any two adjacent atoms is the sum of the contributions from all occupied MOs. For a simple system like 1,3-butadiene, the delocalisation leads to a partial double-bond character in the central single bond. The bond order is no longer strictly 1 or 2 but an intermediate value, about 1.45 for the central bond of butadiene (a Hückel \(\pi\) bond order of 0.45 on top of the \(\sigma\) bond). This is consistent with experimental bond lengths, which are intermediate between typical C–C and C=C distances.
where \(c_{i,A}\) and \(c_{i,B}\) are the coefficients of the atomic orbitals on atoms A and B in the \(i\)-th molecular orbital. This summation captures the cumulative effect of all electrons in the delocalised system on the bond strength between specific atoms.
Alternant Hydrocarbons and Symmetry
For alternant conjugated hydrocarbons (those whose carbons can be starred and unstarred alternately, i.e. with no odd-membered rings), the MO energy levels are symmetric about the non-bonding energy level (zero energy in the Hückel model). This symmetry leads to a pairing of orbitals: for every bonding orbital at energy \(\alpha + k\beta\), there is an antibonding orbital at \(\alpha - k\beta\). This property simplifies the analysis of electron distribution and reactivity. In particular, the HOMO and LUMO sit symmetrically below and above \(\alpha\), which is why the same orbitals control both photochemistry and the long-wavelength absorption.
The result
Molecular orbital theory
Two atomic orbitals \(\phi_A,\phi_B\) combine linearly into a bonding molecular orbital \(\psi_+\propto\phi_A+\phi_B\) (in-phase, lower energy, electron density concentrated between the nuclei) and an antibonding orbital \(\psi_-\propto\phi_A-\phi_B\) (out-of-phase, higher energy, with a node between the nuclei). Filling these molecular orbitals by the same aufbau and Hund's-rule procedure used for atoms gives a bond order \(\text{BO}=\tfrac{n_{\text{bonding}}-n_{\text{antibonding}}}{2}\), and correctly predicts molecular magnetism — including the paramagnetism of \(\text{O}_2\), which the Lewis/valence-bond picture (this unit's earlier results) cannot explain.
A common misconception is that delocalisation always increases bond order uniformly across the molecule. In reality, delocalisation can lead to bond order alternation, where some bonds become stronger and others weaker, depending on the specific electronic structure and symmetry of the system. This is particularly evident in polycyclic aromatic hydrocarbons where different bonds have significantly different lengths and reactivities. C
Worked example
Problem. Model the four \(\pi\) electrons of 1,3-butadiene as particles in a one-dimensional box of length \(L = 0.55\ \mathrm{nm}\). With two electrons per level, the HOMO is \(n = 2\) and the LUMO \(n = 3\). Find \(\Delta E\) and the wavelength absorbed. Use \(m_e = 9.11\times10^{-31}\ \mathrm{kg}\), \(h = 6.626\times10^{-34}\ \mathrm{J\,s}\), \(c = 3.00\times10^8\ \mathrm{m\,s^{-1}}\).
- \(E_n = n^2h^2/(8m_eL^2)\), so \(\Delta E = (3^2 - 2^2)\,h^2/(8m_eL^2) = 5h^2/(8m_eL^2)\).
- \(8m_eL^2 = 8(9.11\times10^{-31})(0.55\times10^{-9})^2 = 2.205\times10^{-48}\ \mathrm{kg\,m^2}\); \(h^2 = 4.390\times10^{-67}\ \mathrm{J^2\,s^2}\).
- \(\Delta E = 5 \times 4.390\times10^{-67}/2.205\times10^{-48} = 9.96\times10^{-19}\ \mathrm{J}\).
- \(\lambda = hc/\Delta E = (6.626\times10^{-34})(3.00\times10^8)/(9.96\times10^{-19}) = 2.00\times10^{-7}\ \mathrm{m}\).
Units check. \(\mathrm{J^2\,s^2}/(\mathrm{kg\,m^2}) = \mathrm{J}\), and \(\mathrm{J\,s \cdot m\,s^{-1}}/\mathrm{J} = \mathrm{m}\). The measured absorption maximum of butadiene is 217 nm, so this crude model gets within about 10%.
Three quick tests
- How does the energy gap between HOMO and LUMO change as the length of a conjugated system increases?
Answer
It decreases, leading to absorption at longer wavelengths. - What is the bond order of the central bond in 1,3-butadiene in the delocalised model?
Answer
About 1.45 in Hückel theory: intermediate between single and double, and closer to single. - Why are alternant hydrocarbons symmetric in their MO energy levels?
Answer
Due to the pairing of bonding and antibonding orbitals around the non-bonding energy level.
Back to CU-102 →
Every lecture and every result in this unit, with prerequisites and threads.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.