Signed Kekulé structures and Hückel gap-closure thresholds in phenylenes

Naphthalene and biphenylene have comparable Hückel HOMO–LUMO gaps, yet no positive bond couplings close the gap of naphthalene, whereas coordinated changes of 18.9% close that of biphenylene. We define the gap-closure threshold as the smallest maximum relative change of existing bond couplings needed to close the gap. At equal reference couplings, it depends only on the signed Kekulé polynomial: it is maximal when all Kekulé structures share one sign, is fixed by a polynomial equation when one structure has the minority sign, and is otherwise bounded. Phenylenes with the same sequence of linear and angular annelations share the threshold, even when their gaps differ. For linear [N ]phenylenes it approaches π2/[4(N + 1)2]; angular [3]phenylene closes only by breaking mirror symmetry. In a bond-length model, the threshold ranks closure energies better than the relaxed gap does, and the predicted closing distortion is nearly the steepest-descent direction of the B3LYP Kohn–Sham gap (cosine 0.989).