Q1(a). π-MO energy levels: benzene, cyclobutadiene, cyclopentadienyl anion

Introduction. In Hückel theory each ring carbon gives one p orbital, and a planar ring of N atoms gives N π MOs with energies Eₖ = α + 2βcos(2π k/N), k = 0, 1, …, N-1 (α = energy of an isolated p orbital, β negative). The Frost circle gives the same pattern: inscribe the polygon, one vertex down, in a circle of radius 2β; each vertex marks one level. There is always one lowest level, and the rest come in degenerate pairs.

Figure: π-MO levels (energy upwards; β<0, so α+2β is the lowest). Dashed line: non-bonding level α.

1. Benzene (N=6): α+2β; α+β (two); α-β (two); α-2β. Six π electrons fill the three bonding MOs exactly. Total π energy 6α+8β, against 6α+6β for three isolated C=C: delocalisation energy 2β. A closed shell of 4n+2 electrons (n=1): aromatic.
2. Cyclobutadiene (N=4): α+2β; α (two, non-bonding); α-2β. After two electrons fill ψ₁, the remaining two must go one each into the degenerate pair (Hund's rule): square cyclobutadiene would be a diradical with no delocalisation gain (π energy 4α+4β, the same as two C=C). This is the mark of a 4n system: antiaromatic. The real molecule distorts to a rectangle, isolable only in an argon matrix near 10 K.
3. Cyclopentadienyl anion (N=5): α+2β; α+0.618β (two); α-1.618β (two). Six electrons fill all three bonding MOs: a closed shell, 4n+2 (n=1), aromatic. The same five MOs hold only five electrons in the radical and four in the cation (antiaromatic).
4. General rule read from the diagrams: a planar cyclic conjugated system is aromatic when its bonding MOs are exactly filled, which happens for 4n+2 π electrons; with 4n electrons two are left unpaired in a degenerate pair.

Conclusion. Benzene and C₅H₅⁻ have six π electrons in three bonding MOs (closed shell, aromatic). Cyclobutadiene puts two electrons into a pair of non-bonding MOs (open shell, antiaromatic).

Application. The same Frost-circle diagrams predict the stability of the cyclopropenyl cation and the tropylium ion (2π and 6π) and explain why cyclopentadiene is far more acidic than other hydrocarbons; ferrocene is built from two aromatic C₅H₅⁻ rings.