Discussion
The secular determinant (where \(\alpha\), \(\beta\) and \(E\) have their usual meanings) is given by
Answer: A
The secular determinant is (A), the tridiagonal one with β between neighbouring carbons only.
The allyl radical CH2=CH-CH2 has three p orbitals in a chain, C1-C2-C3. In Hückel theory every diagonal element is α - E, an off-diagonal element is β for carbons that are bonded and 0 for carbons that are not. C1 and C2 are bonded, C2 and C3 are bonded, C1 and C3 are not:
\[ \begin{vmatrix} \alpha-E & \beta & 0 \\ \beta & \alpha-E & \beta \\ 0 & \beta & \alpha-E \end{vmatrix} = 0 \]
(B) and (C) each drop one of the two bonds, describing an isolated ethylene plus a lone p orbital. (D) has -β, which flips the sign convention without any physical basis in standard Hückel theory.
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