Hückel Theory + Reorganization Energy = Marcus–Hush Theory: Breakdown of the 1/<i>n</i> Trend in π‑Conjugated Poly‑<i>p</i>‑phenylene Cation Radicals Is Explained
2017-01-03T00:00:00Z (GMT) by
Among the π-conjugated poly-<i>p</i>-phenylene wires, fluorene-based poly-<i>p</i>-phenylene (<b>FPP</b><sub><i>n</i></sub>) 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 <b>FPP</b><sub><i>n</i></sub> (<i>n</i> = 2–16) wires. We find that, while their absorption maxima (ν<sub>abs</sub>) follow a linear trend against cos[π/(<i>n</i> + 1)] up to the polymeric limit, redox potentials (<i>E</i><sub>ox</sub>) show an abrupt breakdown from linearity beginning at <i>n</i> ∼ 8. These observations prompted the development of a generalized model to describe the unusual evolution of redox and optical properties of poly-<i>p</i>-phenylene wires. We show that the cos[π/(<i>n</i> + 1)], commonly expressed as 1/<i>n</i>, 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 <b>FPP</b><sub><i>n</i></sub> wires, reproducing the breakdown in the linear cos[π/(<i>n</i> + 1)] trend. A comparison with the Marcus-based multistate model (MSM), where reorganization (λ) and coupling (<i>H</i><sub>ab</sub>) are introduced by design with the aid of empirically adjusted parameters, further confirms that the structural/solvent reorganization limits hole delocalization to ∼8 <i>p</i>-phenylene units and leads to the breakdown in the linear evolution of the redox properties against cos[π/(<i>n</i> + 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.