Extrapolating from a homologous series of oligomers to the infinite-mer: it's a long long way to infinity

Abstract

The usual strategy for deducing the $\pi\mbox{--}\pi^\ast$ electronic energy (or optical bandgap) in a molecule with an "infinite" number of conjugated double bonds consists in fitting a function with some adjustable parameters to the relevant data for a set of homologous molecules with increasing number of repeat units ($N$), and assuming that, after its parameters have been optimized according to the least-squares criterion, the function can be extended indefinitely, and its output will coincide with, or come close to, the correct limit. Since more than ten homologues are seldom available, one might wonder whether extrapolation to the infinite-mer upon such slender basis is an instance of sound inductive reasoning or a mere leap of faith. The present article argues that the shape of the fitting function is an equally important criterion, and points out that the expressions proposed by Hirayama and by Meier and coworkers are flat functions of $1/N$, and that such functions are incongruent with the currently available evidence. Formulas derived by Davydov and by W. Kuhn are shown to be special cases of a new equation that outperforms both.