Electron transfer in pendant-group and molecularly doped polymers

Utilizing the molecular-ion model previously constructed to describe photoemission and ultraviolet absorption in pendant-group polymers, we derive an expression for the probability of electron transfer between a molecular ion and a neutral molecular species embedded in a frequency-dependent dielectric medium described by the (nonlocal) longitudinal dielectric response function $\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$. The medium is taken to exhibit three branches of its longitudinal polarization excitation spectrum defined by $\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}))=0$: a low-frequency branch corresponding to intermolecular motions, an infrared branch corresponding to molecular vibrational modes, and a high-frequency branch corresponding to valence-electron excitations. In addition, the linear coupling of the electron to the intramolecular modes of the initial and final molecular ions is incorporated into the model. The electron-transfer probability is evaluated as a function of the spacing, $R$, between the molecular-ion sites and the energy difference between the intitial and final molecular-ion states. Utilizing parameters in $\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ typical of pendant-group polymers (e.g., polystyrene, polyvinylpyridine) or the matrices utilized in molecularly doped polymers films (e.g., polycarbonates), we find that the electron-transfer process is activated and that the low-frequency dielectric relaxations characteristic of these polymers create this activation. Explicit expressions for the activation energies are derived and evaluated numerically for poly(2-vinylypyridine): a material for which a model of $\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ is available in the literature. The valence-electron excitations do not influence the electron-transfer activation energies, but both the intramolecular and longitudinal-polarization vibrational modes increase these activation energies above the values predicted using the low-frequency relaxations alone. The energies, $\ensuremath{\hbar}{\ensuremath{\omega}}_{n}$, of many of these vibrational modes are, however, larger than thermal energies, $\mathrm{kT}$. Consequently, the predicted electron-transfer activation energies are smaller than those given by traditional semiclassical models of electron transfer. Moreover, these activation energies also depend explicitly on the spacing, $R$, between the two sites. This $R$ dependence is evaluated for both classical and quantum-mechanical models of the change densities on the molecular ions. Our analysis predicts, therefore, the complete spacing and temperature dependence of the electron-transfer probabilities as functions of the intramolecular molecular-ion vibrational frequencies and electron-vibration coupling constants, and of the frequency and spatial dependence of the dielectric response of the medium in which these ions are embedded. This prediction permits the identification of scaling laws relating both the activation energies and electron-transfer prefactors to molecular and dielectric observables: an identification which provides valuable guidance in the molecular design of efficient electronic transport media.