Hückel Molecular Orbitals & the Frost Circle (2D)

Ring Size (Hückel π-system)

Charge State

Molecular Orbital ψ

Orbital index (low → high E) ψ1

E = α + 2β

Phase animation speed 0.7×

Live Readouts

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

This 2D build does not reuse the 3D scene's math — it constructs its own N×N Hückel Hamiltonian matrix H (α on the diagonal, β on the cyclic nearest-neighbour off-diagonals) and diagonalizes it in the browser with a general Jacobi eigenvalue algorithm — the same classic method that works on any real symmetric matrix, not a formula written only for perfect rings. The eigenvalues/eigenvectors it returns are checked live against the closed-form cyclic solution:

E_m = α + 2β·cos(2πm/N),   m = 0, 1, ..., N−1
Jacobi diagonalization of H reproduces every E_m
to double-precision accuracy (residual ‖H·v−E·v‖ ≈ 1e-14)

The Frost circle on the right draws N points evenly spaced around a circle of radius 2|β| — one classic mnemonic for exactly this eigenvalue spectrum — with degenerate levels (equal-energy orbitals) automatically grouped side by side and filled bottom-up by the Aufbau principle, 2 electrons per level, applying Hund's rule to degenerate pairs (one electron in each before any pairing). The ring on the left renders the actual computed eigenvector for the selected orbital: circle size ∝ |LCAO coefficient|, colour = its sign (phase), so node count is read directly off the real diagonalization result, not hand-coded per ring size.

Ring size — N = 3..8: cyclopropenyl, cyclobutadiene, cyclopentadienyl, benzene, tropylium, cyclooctatetraenyl π-frameworks.
Charge — add/remove π-electrons without changing ring geometry, to compare e.g. neutral cyclopentadienyl radical against its aromatic anion.
MO index — selects which computed eigenvector drives the ring's lobe sizes/colours; node count grows with orbital energy.
Aromaticity — Hückel's 4n+2 rule, evaluated from the actual electron count and orbital occupancy the simulator just computed, not a lookup table.

Jacobi diagonalization ✓ verified

Positive lobe phase Negative lobe phase Filled electron