A concealed difference between the structure-dependence of Dewar and topological resonance energy

In the 1970s, two Dewar-type resonance energies were put forward, one by Hess and Schaad (denoted here by DRE), the other by Gutman et al. and, independently, by Aihara (denoted here by TRE). Both were believed to be equivalent and, indeed, a good linear correlation between them could be established. We now show that there are some significant differences between the structure-dependencies of DRE and TRE. In particular, in the case of benzenoid molecules, DRE and TRE are found to be linearly related to ln ζ(0) and ln ζ(1), respectively, where ζ(x) is the Zhang-Zhang polynomial. Whereas ζ(0) is just the Kekulé structure count, ζ(1) is a novel structure-descriptor overlooked in the previous studies of resonance energy.