ENERGY-GAP LAW FOR NONRADIATIVE AND RADIATIVE CHARGE-TRANSFER IN ISOLATED AND IN SOLVATED SUPERMOLECULES

In this paper we explore the foundations and some applications of the energy gap law (EGL) for nonradiative and radiative charge recombination from an ion pair state to the ground electronic state of isolated (solvent-free) and solvated donor (D)-acceptor (A) complexes and DBA bridged (B) supermolecules. The energy gap dependence of the averaged Franck-Condon density AFD(E), which is proportional to the microscopic electron-transfer (ET) rate, k(E), at the excess energy E, was calculated numerically (for a range of E) and by saddle point integration (for E = 0) for a displaced harmonic potential system. The intramolecular electron vibration coupling parameters were inferred from resonance Raman data and from ET emission line shapes. For isolated supermolecules an energy gap (Delta E) dependence of AFD(E) was derived, which for the electronic origin (E 0) is a multi-Poissonian, with a Gaussian dependence over a narrow, low Delta E domain and a superexponential decrease with increasing Delta E for large Delta E. The EGL, AFD(0) = A exp(-gamma Delta E), holds for large values of Delta E over physically relevant Delta E domains (of similar to 5000 cm(-1)), where the theoretical parameters gamma and A have to be extracted from numerical calculations using a complete set of nuclear frequencies and their coupling parameters. Approximate coarse graining of the coupling parameters over a small number of frequencies reveals that within a few-mode approximation it is important to segregate between medium- and high-frequency modes; the averaged single-mode approximation is inadequate, while the maximal mode representation (which is valid in the asymptotic limit of huge Delta E) does not hold in the relevant Delta E domain. The failure of the single-mode approximation forces us to utilize the exponential EGL as a useful empirical relation for the representation of ''exact'' theoretical results or of experimental data for isolated systems. Focusing on the EGL for solvated supermolecules, we have shown that the first-order solvent correction to the EGL is AFD(0) similar or equal to ($) over tilde A exp[-gamma(Delta E - lambda(s))] with ($) over tilde A = A exp(gamma(2) lambda(s)k(B)T) where lambda(s) is the solvent reorganization energy, with the gamma parameter being solvent invariant and determined by the intramolecular dynamics. The EGL for solvated DBA was successfully applied for the analysis of the nonradiative ET rates in the pyrene-substituted barrelene-based donor-acceptor supermolecule in a series of solvents, with the solvent-dependent energy gaps being varied in the range of 0.45 eV, while the lambda(s) vary in the range lambda(s) = 0.16 eV (for n-hexane) to lambda(s) = 0.36 eV (for acetonitrile). Finally, we have explored the isomorphism between the description of the nuclear Franck-Condon vibrational overlap for nonradiative and radiative ET processes. We predict an exponential EGL for the low-energy tails in the charge-transfer fluorescence spectra of isolated and solvated supermolecules.