The dynamics of unimolecular photo-triggered reactions can be strongly affected by the surrounding medium for which a large number of theoretical descriptions have been used in the past. An accurate description of these reactions requires knowing the potential energy surface and the friction felt by the reactants. Most of these theories start from the Langevin equation to derive the dynamics, but there are few examples comparing it with experiments. Here we explore the applicability of a Generalized Langevin Equation (GLE) with an arbitrary potential and a non-Markovian friction. To this end, we have performed broadband fluorescence measurements with sub-picosecond time resolution of a covalently linked organic electron donor-acceptor system in solvents of changing viscosity and dielectric permittivity. In order to establish the free energy surface (FES) of the reaction, we resort to stationary electronic spectroscopy. On the other hand, the dynamics of a non-reacting substance, Coumarin 153, provide the calibrating tool for the non-Markovian friction over the FES, which is assumed to be solute independent. A simpler and computationally faster approach uses the Generalized Smoluchowski Equation (GSE), which can be derived from the GLE for pure harmonic potentials. Both approaches reproduce the measurements in most of the solvents reasonably well. At long times, some differences arise from the errors inherited from the analysis of the stationary solvatochromism and at short times from the excess excitation energy. However, whenever the dynamics become slow, the GSE shows larger deviations than the GLE, the results of which always agree qualitatively with the measured dynamics, regardless of the solvent viscosity or dielectric properties. The method applied here can be used to predict the dynamics of any other reacting system, given the FES parameters and solvent dynamics are provided. Thus no fitting parameters enter the GLE simulations, within the applicability limits found for the model in this work.
The division by the mass of the second and third terms on the rhs of Eq. (1) is implicit in the definitions of the friction and the noise, respectively.
Usually defined as the energy difference between the lowest energy absorption maximum and the emission maximum.
Here we do not account for broadening due to electronic dephasing.
We also tested the log-normal function as the underlying distribution for convolution with the lineshape function but did not obtain significantly improved fits. Thus we opted for the distribution with less degrees of freedom, i.e., the Gaussian distribution.
For the Hamiltonian, Eq. (13), the reason for the gl elements to be zero is that the influence of these off-diagonal elements on the finally obtained values is minimal. This is most likely because of the large energy difference between these two states. For the dipole moment matrix, Eq. (14), the off-diagonal elements connecting the states l and c are considered to be zero because we have no observable providing us with information about them. If these elements would be much larger than zero, a transient absorption band should be observable, shifting from the midIR to the NIR as time increases.
This is of course tantamount to tacitly assuming that the energetic stabilization due to dispersion interactions is the same for both excited states.
The protic solvents (30-34) were not used for the solvatochromic fitting.
To the best of our knowledge, even in the case of the harmonic oscillator, the relationship between these two quantities is only defined via Laplace transforms.
For PeDMA we have used the shift of the first moment, m1, while for C153 we used the peak maximum of the log-normal function, .