Hückel Theory + Reorganization Energy = Marcus–Hush Theory: Breakdown of the 1/n Trend in π‑Conjugated Poly‑p‑phenylene Cation Radicals Is Explained
2017-01-03T00:00:00Z (GMT) by
Among the π-conjugated poly-p-phenylene wires, fluorene-based poly-p-phenylene (FPPn) wires have been extensively explored for their potential as charge-transfer materials in functional photovoltaic devices. Herein, we undertake a systematic study of the redox and optical properties of a set of FPPn (n = 2–16) wires. We find that, while their absorption maxima (νabs) follow a linear trend against cos[π/(n + 1)] up to the polymeric limit, redox potentials (Eox) show an abrupt breakdown from linearity beginning at n ∼ 8. These observations prompted the development of a generalized model to describe the unusual evolution of redox and optical properties of poly-p-phenylene wires. We show that the cos[π/(n + 1)], commonly expressed as 1/n, dependence of the properties of various π-conjugated wires has its origin in Hückel molecular orbital (HMO) theory, which however, fails to predict the evolution of the redox potentials of these wires, as the oxidation-induced structural/solvent reorganization is unaccounted for in the original formulation of HMO theory. Accordingly, aided by DFT calculations, we introduce here a modified HMO theory that incorporates the reorganization energy (Δα) and coupling (β) and show that the modified theory provides an accurate description of the oxidized FPPn wires, reproducing the breakdown in the linear cos[π/(n + 1)] trend. A comparison with the Marcus-based multistate model (MSM), where reorganization (λ) and coupling (Hab) are introduced by design with the aid of empirically adjusted parameters, further confirms that the structural/solvent reorganization limits hole delocalization to ∼8 p-phenylene units and leads to the breakdown in the linear evolution of the redox properties against cos[π/(n + 1)]. The predictive power of the modified HMO theory and MSM offer new tools for rational design of the next-generation, long-range charge-transfer materials for photovoltaics and molecular electronics applications.