Butadiene Pi Molecular Orbitals

Introduction

Butadiene contains four connected p orbitals and four pi electrons. It is the simplest open-chain system large enough to show two occupied pi MOs, two unoccupied pi MOs and a reduced gap relative to ethene in the uniform Hückel model. Its orbital phase patterns also illustrate a general rule: higher chain eigenfunctions have more nodes. The calculation provides a controlled way to discuss delocalisation beyond a single carbon-carbon double bond.

Core explanation

Number the planar conjugated carbons 1–2–3–4. Set the same diagonal α on each p site, β between adjacent sites 1–2, 2–3 and 3–4, and zero direct coupling otherwise. For a uniform open chain of N=4 sites, the Hückel energies are E k = α + 2β cos[kπ/5] for k=1,2,3,4. The cosine factors are approximately +0.809, +0.309, −0.309 and −0.809, so the coefficients of β are approximately +1.618, +0.618, −0.618 and −1.618.

With the usual β<0 convention, E₁=α+1.618β is lowest, E₂=α+0.618β is next, E₃=α−0.618β is third and E₄=α−1.618β is highest. Four pi electrons doubly occupy E₁ and E₂ in the closed-shell ground-state model. E₂ is the HOMO and E₃ is the LUMO. The gap is E₃−E₂ = −1.236β = 1.236 β . Ethene's analogous gap is 2 β , so the four-site chain has a smaller simple orbital gap if comparable α and β are assumed.

The corresponding normalised coefficients can be written c j^(k)=√(2/5)sin(jkπ/5) for site j=1,…,4. The k=1 orbital has the same phase on all four sites and no internal node. The k=2 orbital has a phase change between central sites 2 and 3. The k=3 and k=4 orbitals have progressively more phase changes. This is analogous to standing-wave patterns in a finite chain, though Hückel's boundary conditions and electron interactions are model-specific.

Electron delocalisation does not mean every C–C bond in butadiene becomes identical in a real molecule. The simple uniform-β model makes all nearest-neighbour couplings equal by assumption, even though the actual equilibrium structure shows bond-length alternation between terminal and central connections. More advanced calculations allow geometry-dependent overlap and electron repulsion. Hückel theory nevertheless captures the idea that the central single bond participates in a connected pi network and differs from an isolated alkane C–C bond.

The total occupied pi-orbital energy in this model is 2E₁+2E₂ = 4α + 2(1.618+0.618)β ≈ 4α+4.472β. Comparing it with two isolated ethene pi systems, each contributing 2α+2β, gives 4α+4β. Since β<0, the butadiene value is lower by about 0.472 β in this particular comparison. This is a model delocalisation energy and not a measured heat of hydrogenation or a complete molecular stabilisation energy.

Frontier-orbital coefficients can help frame reactivity. The HOMO and LUMO have nonuniform amplitudes and signs across the chain, which influence overlap with a reaction partner. But orbital coefficients alone do not predict a reaction product. Sterics, charge, geometry, transition-state interactions and solvent can change selectivity, so a frontier picture is one piece of a larger argument.

Step-by-step reasoning

Draw the four-site chain and use the open-chain formula with denominator N+1=5. Calculate the four cosine values, then sort energies using β<0. Place four pi electrons into the two lowest orbitals. Identify HOMO and LUMO, compute their difference and compare with a two-site model only under consistent parameter assumptions.

Visual explanation

Draw four p sites in a row and four horizontal MO levels. Mark the lowest phase pattern ++++; the HOMO ++−−; the LUMO +−−+ or an equivalent overall phase convention; and the highest alternating +−+−. Indicate increasing nodes upward. Put paired electrons on the bottom two levels and a bracket across the middle HOMO–LUMO gap.

Real-world analogy

A longer string supports more standing-wave patterns packed into a given frequency range than a very short string. A longer connected p chain likewise produces more orbital levels and often a narrower middle gap in the simple model. The analogy helps with trend but should not be used to equate optical excitation exactly with one-electron orbital spacing.

Real-world example

Conjugated dyes often absorb at longer wavelengths as the effective pi system grows. Butadiene versus ethene illustrates the first step in that trend: a four-site Hückel gap is smaller than a two-site gap when β is comparable. Real colour requires sufficiently low excitation energy and adequate transition intensity, plus environmental and substituent effects.

Why?

Why are the two middle levels closer together than ethene's pair? Four coupled p sites support four standing-wave-like eigenpatterns rather than only two. Their eigenvalues spread across a range set by β, placing two states near the centre α. The occupied/unoccupied boundary then lies between E₂ and E₃, which are closer than ethene's only two levels.

Common misconception

The numerical value 1.236 β is not a universal experimental absorption energy. It follows from equal nearest-neighbour β and zero overlap in a fixed four-site chain. Another error is saying conjugation makes every bond exactly equal; uniform matrix parameters are a modelling convenience, not proof of identical real bond lengths.

Worked example

Let β=−2.0 eV and leave α as an arbitrary reference. Then E₂=α+0.618(−2.0)=α−1.236 eV and E₃=α−0.618(−2.0)=α+1.236 eV. Their gap is 2.472 eV, equal to 1.236 β . With the same β , ethene's Hückel gap would be 4.0 eV. These are illustrative orbital-gap comparisons, not fitted molecular absorption bands.

Quick check

1. How many butadiene pi MOs are occupied in the simple four-electron closed-shell ground state? Answer: Two spatial MOs are doubly occupied; the other two are empty. 2. What is the Hückel HOMO–LUMO gap for the uniform four-site chain with β<0? Answer: Approximately 1.236 β .

Exam focus

State the open-chain formula and β sign before sorting. Count four MOs and four electrons separately, show occupied levels and distinguish an orbital gap from an optical transition energy. Qualify any delocalisation-energy comparison by its chosen reference and model assumptions.

Advanced insight

The open-chain coefficient formula encodes both nodes and terminal amplitudes. Because higher and lower eigenvectors are orthogonal, their phase patterns cannot all be chosen uniformly positive. In more advanced orbital-symmetry analysis, the sign of frontier coefficients becomes important for constructive overlap along a reaction path, but a complete reaction prediction still requires geometry and energetics.

Summary

Uniform Hückel butadiene has four pi levels at α plus β times approximately 1.618, 0.618, −0.618 and −1.618. Four pi electrons fill the two lowest, leaving a model HOMO–LUMO gap of 1.236 β . Delocalisation lowers the model occupied energy relative to two isolated ethene units, while real bond alternation and excitation energies lie beyond the simple assumptions.

Practice questions

1. If β is negative, which butadiene level is the HOMO: α+0.618β or α−0.618β? Answer: α+0.618β is lower and occupied as the second level; α−0.618β is the higher, empty LUMO. 2. Why should the butadiene and ethene gaps be compared using the same β convention and similar parameter values? Answer: Otherwise a numerical gap difference could reflect changed parameter choice rather than the effect of adding connected p sites to the model. 3. Does a four-site Hückel coefficient pattern determine butadiene's exact central C–C bond length? Answer: No. The simplest model fixes geometry and coupling parameters. Accurate bond length requires an energy calculation allowing nuclear geometry and more complete electronic effects.