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 donoir (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 (ΔE) 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 ΔE domain and a superexponential decrease with increasing ΔE for large ΔE. The EGL, AFD(0) - A exp(-γΔE), holds for large values of ΔE over physically relevant ΔE domains (of ∼5000 cm^-1), where the theoretical parameters γ 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 ΔE) does not hold in the relevant ΔE 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 finit-order solvent correction to the EGL is AFD(0) ≃ à exp[-γ(ΔE - λs)] with à = A exp(γ²λsk_BT) where λs is the solvent reorganization energy, with the γ 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 λs vary in the range λs = 0.16 eV (for n-hexane) to λs = 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.