For a butadiene molecule, the energy values of levels obtained by using HMO theory are : (1) α - 1.6β (2) α + 1.6β (3) α + 0.6β (4) α - 0.6β What is the sequence of levels in the order of increasing energy ?
(2), (3), (4), (1)
The whole question turns on one easily forgotten fact: β is negative.
In Hückel theory \(\alpha\) is the Coulomb integral (the energy of an electron in an isolated 2p orbital) and \(\beta\) is the resonance integral. Because bonding lowers the energy, \(\beta\) is a negative quantity, of the order of -75 kJ mol-1.
The consequence is that a term \(+n\beta\) lowers the energy while \(-n\beta\) raises it, and the larger the coefficient the greater the effect. So ordering from lowest to highest energy:
\(\alpha + 1.6\beta \;<\; \alpha + 0.6\beta \;<\; \alpha - 0.6\beta \;<\; \alpha - 1.6\beta\)
which corresponds to the labels (2), (3), (4), (1).
The pattern is exactly what is expected for a linear conjugated system. The four π molecular orbitals are arranged symmetrically about \(\alpha\): two bonding levels below it, at \(\alpha + 1.618\beta\) and \(\alpha + 0.618\beta\), and two antibonding levels the same distances above.
Butadiene's four π electrons fill the two bonding orbitals, so the HOMO is \(\alpha + 0.618\beta\) and the LUMO is \(\alpha - 0.618\beta\). The total π energy is \(2(\alpha + 1.618\beta) + 2(\alpha + 0.618\beta) = 4\alpha + 4.472\beta\), compared with \(4\alpha + 4\beta\) for two isolated ethene units — the difference of \(0.472\beta\) being the delocalisation energy that makes conjugation favourable.
Hence the order of increasing energy is (2), (3), (4), (1).
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