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 recombinati on from an ion pair state to the ground electronic state of isolated ( solvent-free) and solvated donor (D)-acceptor (A) complexes and DBA br idged (B) supermolecules. The energy gap dependence of the averaged Fr anck-Condon density AFD(E), which is proportional to the microscopic e lectron-transfer (ET) rate, k(E), at the excess energy E, was calculat ed 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 Ra man data and from ET emission line shapes. For isolated supermolecules an energy gap (Delta E) dependence of AFD(E) was derived, which for t he electronic origin (E 0) is a multi-Poissonian, with a Gaussian depe ndence over a narrow, low Delta E domain and a superexponential decrea se 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 rel evant Delta E domains (of similar to 5000 cm(-1)), where the theoretic al parameters gamma and A have to be extracted from numerical calculat ions using a complete set of nuclear frequencies and their coupling pa rameters. Approximate coarse graining of the coupling parameters over a small number of frequencies reveals that within a few-mode approxima tion it is important to segregate between medium- and high-frequency m odes; 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 failu re of the single-mode approximation forces us to utilize the exponenti al EGL as a useful empirical relation for the representation of ''exac t'' theoretical results or of experimental data for isolated systems. Focusing on the EGL for solvated supermolecules, we have shown that th e 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 ti lde A = A exp(gamma(2) lambda(s)k(B)T) where lambda(s) is the solvent reorganization energy, with the gamma parameter being solvent invarian t and determined by the intramolecular dynamics. The EGL for solvated DBA was successfully applied for the analysis of the nonradiative ET r ates in the pyrene-substituted barrelene-based donor-acceptor supermol ecule 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 a cetonitrile). Finally, we have explored the isomorphism between the de scription of the nuclear Franck-Condon vibrational overlap for nonradi ative and radiative ET processes. We predict an exponential EGL for th e low-energy tails in the charge-transfer fluorescence spectra of isol ated and solvated supermolecules.