Ethene in Hückel Theory

Introduction

Ethene is the smallest conjugated pi system that displays the essential Hückel construction. Each sp² carbon contributes one p orbital, so two local functions generate one lower and one higher pi molecular orbital. Solving this small matrix makes the roles of α, β, coefficient phase, electron filling and orbital gap explicit. It also shows the limits of equating a one-electron gap with an observed electronic transition.

Core explanation

Let φ₁ and φ₂ be p orbitals perpendicular to ethene's carbon framework. In the simplest Hückel model they are orthonormal, have equal diagonal energy α and nearest-neighbour coupling β. The Hamiltonian is H = [[α,β],[β,α]]. Its secular condition det(H−EI)=0 gives (α−E)²−β²=0, so the two energies are E₊=α+β and E₋=α−β. Here the plus or minus subscript describes the algebraic combination, not an independently assumed energy order.

With the usual β<0 convention, α+β is lower. Solving the coefficient equations for that root gives equal same-sign amplitudes, yielding ψ b=(φ₁+φ₂)/√2 if overlap is neglected. For α−β, coefficients have opposite signs, yielding ψ a=(φ₁−φ₂)/√2. The first increases pi density between carbon atoms and is bonding; the second has an additional node between them and is antibonding. Globally changing both coefficient signs does not alter an MO, but their relative sign does.

Ethene has two pi electrons, one from each carbon. In a closed-shell ground state both occupy the lower ψ b spatial orbital with opposite spins, while ψ a is empty. The total model pi-electron energy is 2(α+β)=2α+2β. Relative to two separated p electrons each assigned energy α, the Hückel pi stabilisation is 2β, a negative quantity under the chosen convention. This is a model comparison, not a measured C=C bond dissociation energy, because sigma bonding, nuclear repulsion, geometry and electron correlation are omitted.

The simple HOMO–LUMO gap is (α−β)−(α+β)=−2β=2 β . A naive photon wavelength from this gap is only a rough orbital-level estimate. A true vertical electronic excitation involves many-electron states, Coulomb and exchange effects, orbital relaxation and selection rules. The gap helps compare related conjugated systems but should not be announced as the exact UV absorption energy of ethene.

An approximate pi bond order can be calculated from occupied orbital coefficients. For two electrons in ψ b, the off-diagonal pi density contribution between sites is 2c₁c₂ = 2(1/√2)(1/√2)=1. Filling ψ a with two more electrons would contribute an equal negative amount and cancel net pi bond order in this simple index. The idea is that occupancy of bonding and antibonding MOs has opposing effects on the bond, not that a real four-electron ethene ion necessarily follows the unrelaxed model exactly.

The symmetry of the two combinations can be expressed under exchange of equivalent carbon sites: ψ b is symmetric and ψ a antisymmetric under a simple site swap. Full molecular point-group labels require the complete ethene geometry and all atoms, but the two-site sign test already captures the basic p-overlap physics. A coordinate or orbital phase convention can change which algebraic expression carries a minus sign without changing the physical bonding pattern.

Ethene's two-orbital calculation also prepares the chain cases. Adding a third p site gives a nonbonding middle energy in an allyl graph, while four sites give four distinct open-chain energies in butadiene. The defining step stays the same: solve an adjacency-based matrix, then fill electrons rather than forcing a Lewis bond pattern into the eigenvalue calculation.

Step-by-step reasoning

Write the two-by-two Hückel matrix and solve its determinant. State β<0 before sorting energies. Substitute each root into the coefficient equations, normalise the vectors and draw relative phases. Place two pi electrons in the lower orbital, then calculate total energy or gap only after confirming occupancy.

Visual explanation

Draw two adjacent p orbitals as lobes above and below the carbon plane. In one picture, same-phase lobes overlap across the C–C region, labelled ψ b and α+β. In the second, reverse one orbital's phase and mark a node between carbons, labelled ψ a and α−β. Put two electron arrows on the lower line and leave the upper line empty.

Real-world analogy

Two coupled oscillators can move in a same-direction pattern or an opposite-direction pattern with different energies. Ethene's two p orbitals similarly form symmetric and antisymmetric combinations. The analogy helps with mode count and phase but should not be treated as an actual mechanical model of electron density.

Real-world example

Ethene absorbs in the ultraviolet rather than visible region under ordinary conditions. Its relatively short two-site pi system has a larger simple orbital gap than many extended conjugated chains. Comparing such gaps helps explain why sufficiently extended conjugation can shift absorption toward longer wavelengths, although measured spectra require more sophisticated treatment.

Why?

Why does the same-phase combination lie lower for β<0? The Hamiltonian expectation includes the coupling term twice, giving α+β for the normalised symmetric state. A negative coupling reduces the energy. The opposite-phase state has α−β, so the same coupling raises it. This is the algebraic reflection of constructive versus destructive intersite interference.

Common misconception

The value 2 β is a model orbital gap, not automatically an experimental excitation energy. Likewise, the Hückel stabilisation 2β is not the entire energy of a carbon-carbon double bond. The model isolates a pi contribution with fixed simplified parameters and leaves out much of the molecule's physics.

Worked example

Take illustrative α=−8 eV and β=−2 eV. The bonding level is α+β=−10 eV and the antibonding level is α−β=−6 eV. Two pi electrons fill the −10 eV orbital, giving a simple pi-electron energy of −20 eV versus −16 eV for two uncoupled α sites. The orbital gap is 4 eV. These values illustrate the parameter algebra and are not a fitted prediction of ethene's measured spectrum.

Quick check

1. How many pi MOs does ethene's two-p-orbital Hückel basis give? Answer: Two, one lower bonding and one higher antibonding for β<0. 2. What is the model HOMO–LUMO gap under the usual negative-beta convention? Answer: −2β, equal to 2 β and therefore positive.

Exam focus

Show both eigenvalues and normalised coefficient patterns. State electron count and beta sign before labelling HOMO or LUMO. Separate one-electron orbital energies, total occupied-orbital sum and true many-electron excitation energy.

Advanced insight

If overlap S between the two local p orbitals is retained, the symmetric and antisymmetric energies become (α+β)/(1+S) and (α−β)/(1−S) under a simplified equal-parameter model. The familiar α±β result relies on S=0. This illustrates how Hückel's orthogonality assumption simplifies algebra while not literally eliminating physical orbital overlap.

Summary

Ethene's two-site Hückel matrix yields bonding ψ b=(φ₁+φ₂)/√2 at α+β and antibonding ψ a=(φ₁−φ₂)/√2 at α−β when β<0. Two pi electrons occupy the lower MO. The simple gap is 2 β and the model pi stabilisation is 2β relative to uncoupled sites, neither being a complete experimental energy prediction.

Practice questions

1. If β=−3 units, what are the two ethene levels relative to α and their gap? Answer: The levels are α−3 and α+3 units. The lower is bonding and the gap is 6 units. 2. Why does changing ψ b to −ψ b not turn a bonding MO into an antibonding one? Answer: Multiplying the entire wavefunction by −1 changes no relative phase or probability density. Antibonding requires reversing one site's phase relative to the other. 3. What extra information would be needed before predicting ethene's exact UV absorption from a Hückel gap? Answer: A many-electron excitation treatment, including electron interactions, geometry, state relaxation and transition selection rules, would be needed; the simple gap is only an approximate model quantity.