Why are eigenvalues of A shown as μ = (E−α)/β?

With uniform α on carbons, subtracting α and dividing by β diagonalizes the same eigenvectors as A; μ is dimensionless and β sets the energy scale.

Does this predict benzene bond lengths?

Not quantitatively. Bond-order hints from π occupation can be discussed, but geometry needs going beyond a minimal Hückel parameterization.

Why can math eigenvalues differ slightly from lecture chalkboard numbers?

Numerical diagonalization uses tolerances; degenerate pairs may appear as close splits. The qualitative level pattern and degeneracies match the analytic Hückel solution.

Hückel π-MO (Butadiene & Benzene)

Secular matrix H = αI + βA; eigen-energies and LCAO maps on the π skeleton.

Adjacency matrix A (H = αI + βA)

0	1	0	0	0	1
1	0	1	0	0	0
0	1	0	1	0	0
0	0	1	0	1	0
0	0	0	1	0	1
1	0	0	0	1	0

π MO energies (β < 0)

MO map (LCAO coefficients on carbons)

About this model

The Hückel molecular orbital model approximates π conjugation by a tight-binding Hamiltonian on one p_z-like orbital per conjugated carbon: H = α I + β A, where A is the graph adjacency matrix (β for bonded neighbors, zero otherwise). Diagonalizing H yields MO energies E_k = α + β μ_k with μ_k eigenvalues of A, and eigenvector components are LCAO coefficients on each carbon. Butadiene uses a four-site path graph; benzene uses a six-membered cycle, producing degenerate π levels. The page displays A, sorted MO energies for β < 0, and a schematic amplitude map for the selected MO. Electron count, Coulomb differences between atoms, σ–π separation, overlap non-orthogonality, and electron correlation are omitted — this is the textbook π-only introduction.

Who it's for: Organic chemistry alongside MO theory; complements homonuclear diatomic and crystal-field pages.

Key terms

Hückel theory
secular equation
LCAO
resonance integral
conjugation
degenerate orbitals

How it works

Simple Hückel π-electron model: H = α I + β A with A the carbon adjacency matrix (one per p_z basis). Eigenvalues give ε_k = (E_k − α)/β = μ_k (μ are eigenvalues of A). With β < 0, larger μ means lower π energy E = α + β μ. Eigenvectors are LCAO coefficients on each carbon for that MO. Butadiene is a path of four π centers; benzene is a six-cycle. Degeneracies appear for the ring. This is a one-electron, orthogonal-π cartoon — no σ framework, no electron–electron, no non-orthogonality.

Key equations

Secular: det(H − E I) = 0 , H_{ij} = α δ_{ij} + β A_{ij}

E_k = α + β μ_k (μ_k eigenvalue of A)

Frequently asked questions

Why are eigenvalues of A shown as μ = (E−α)/β?: With uniform α on carbons, subtracting α and dividing by β diagonalizes the same eigenvectors as A; μ is dimensionless and β sets the energy scale.
Does this predict benzene bond lengths?: Not quantitatively. Bond-order hints from π occupation can be discussed, but geometry needs going beyond a minimal Hückel parameterization.
Why can math eigenvalues differ slightly from lecture chalkboard numbers?: Numerical diagonalization uses tolerances; degenerate pairs may appear as close splits. The qualitative level pattern and degeneracies match the analytic Hückel solution.

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Hückel π-MO (Butadiene & Benzene)

About this model

How it works

Key equations

Frequently asked questions

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Hückel π-MO (Butadiene & Benzene)

About this model

How it works

Key equations

Frequently asked questions