Hückel Molecular Orbitals & the Frost Circle

Ring Size (Hückel π-system)

Charge State

Molecular Orbital ψ

Orbital index (low → high E) ψ1

E = α + 2β

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π-electrons

Charge

Total Eπ

−6.00 β

HOMO–LUMO gap

1.00 |β|

Hückel 4n+2 verdict

Aromatic (6 = 4·1+2)

How it works

For a planar monocyclic ring of N sp²-hybridised carbons, each contributing one 2p_z orbital φ_k, the Hückel Hamiltonian keeps only same-atom (α) and nearest-neighbour (β) matrix elements, with cyclic boundary conditions. Solving the secular determinant analytically gives:

E_m = α + 2β·cos(2πm/N),   m = 0, 1, ..., N−1
c_k(m) = (1/√N)·e^(2πi·mk/N)     (real combinations: cos & sin)

β < 0, so m = 0 (all coefficients in phase, no nodes) is the most stable orbital; each further m adds one more pair of nodes around the ring, up to the fully alternating highest orbital. Levels come in degenerate cos/sin pairs except m = 0 and, for even N, m = N/2 — exactly the pattern the classic "Frost circle" mnemonic draws by inscribing a polygon vertex-down inside a circle of radius 2|β|.

Ring size — picks N = 3..8, matching cyclopropenyl, cyclobutadiene, cyclopentadienyl, benzene, tropylium (cycloheptatrienyl) and cyclooctatetraenyl π-frameworks.
Charge — adding/removing π-electrons (e.g. cyclopentadienyl anion, tropylium cation) without changing the ring geometry, so you can see the same orbital ladder filled differently.
MO index — selects which eigenvector's coefficients drive the p-orbital lobe sizes/colours on the 3D ring (red = positive lobe phase, blue = negative); watch the node count grow with orbital index.
Aromaticity — Hückel's rule: a fully conjugated, planar monocyclic system with 4n+2 π-electrons in closed-shell orbitals is aromatic (extra stabilisation); 4n electrons force an open-shell, geometry-distorting antiaromatic configuration (cf. Jahn–Teller); an odd electron count is a radical, neither.

This is the same math (real eigenvalues of a cyclic tridiagonal-with-corners matrix) used to rationalise benzene's extra 2β stabilisation over three isolated double bonds, and why cyclobutadiene — not cyclopentadiene or benzene — is the textbook antiaromatic outlier.

Positive lobe phase Negative lobe phase

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