Why does the polygon vertex at the bottom correspond to the most bonding MO?

With the standard orientation, that vertex has the largest downward projection on the energy axis, matching the most negative (most bonding) Hückel eigenvalue 2 cos(0) = +2 in μ units, which becomes lowest E when β < 0.

Does 4k+2 always mean “aromatic”?

Only within the same one-electron, planar monocycle assumptions. Real molecules need strain, conjugation length, and reactivity considerations beyond this diagram.

Why not show MO coefficients on atoms?

The Frost construction is deliberately geometric; the separate Hückel π-MO page shows LCAO vectors and matrices for selected chains and rings.

Frost Circle & Aromaticity (4n+2)

Q: Does 4k+2 always mean “aromatic”?

Only within the same one-electron, planar monocycle assumptions. Real molecules need strain, conjugation length, and reactivity considerations beyond this diagram.

Q: Why not show MO coefficients on atoms?

The Frost construction is deliberately geometric; the separate Hückel π-MO page shows LCAO vectors and matrices for selected chains and rings.

Inscribed n-gon on Hückel π energies; π count vs 4k+2 / 4k; Aufbau filling.

Level (E ↑)	μ	g	e⁻
2.000	-2.0000	1	0
1.000	-1.0000	2	0
-1.000	1.0000	2	4
-2.000	2.0000	1	2

Yellow bar: energy axis; cyan: inscribed polygon; dashed vertical through center.

Live graphs

About this model

The Frost–Musulin circle is a mnemonic for cyclic Hückel π molecular orbitals: inscribe a regular n-gon in a circle with one vertex down; each vertex height (relative to the center line) matches μ_k = 2 cos(2πk/n), the eigenvalues of the ring adjacency matrix, so E_k = α + β μ_k with β < 0 places bonding orbitals lower on the diagram. Degeneracies appear when cos symmetry pairs k and n−k. Students fill π electrons from the bottom with two per spatial orbital (Aufbau within degenerate sets). The page compares the electron count to the textbook 4k+2 vs 4k Hückel electron-count rules for planar monocycles — a qualitative aromatic / anti-aromatic hint, not a substitute for full quantum chemistry (no σ framework, no Jahn–Teller distortion, no explicit electron correlation).

Who it's for: Organic chemistry alongside Hückel π-MO pages; introductory aromaticity before Frost–Dewar–MCBT refinements.

Key terms

Frost circle
Hückel theory
cyclic polyene
4n+2 rule
anti-aromaticity
degenerate π orbitals

How it works

The Frost circle mnemonic inscribes a regular n-gon in a circle (one vertex down) so vertex heights match cyclic Hückel π energies E = α + 2β cos(2πk/n) (same as eigenvalues of the ring adjacency). With β < 0, lower vertices are more bonding. Fill π electrons from the bottom (Aufbau, 2 per spatial MO; degeneracies from cos symmetry). Compare your count to the textbook 4k+2 aromatic vs 4k anti-aromatic electron counts for planar monocycles — still a one-electron cartoon (no σ framework, no Jahn–Teller, no correlation).

Key equations

μ_k = 2 cos(2πk/n) ⇒ E_k = α + β μ_k

Planar monocycle (Hückel): 4k+2 π often aromatic; 4k π often anti-aromatic

Frequently asked questions

Why does the polygon vertex at the bottom correspond to the most bonding MO?: With the standard orientation, that vertex has the largest downward projection on the energy axis, matching the most negative (most bonding) Hückel eigenvalue 2 cos(0) = +2 in μ units, which becomes lowest E when β < 0.
Does 4k+2 always mean “aromatic”?: Only within the same one-electron, planar monocycle assumptions. Real molecules need strain, conjugation length, and reactivity considerations beyond this diagram.
Why not show MO coefficients on atoms?: The Frost construction is deliberately geometric; the separate Hückel π-MO page shows LCAO vectors and matrices for selected chains and rings.

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About this model

How it works

Key equations

Frequently asked questions

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Live graphs

About this model

How it works

Key equations

Frequently asked questions