Solvation dynamics in polar solvents and imidazolium ionic liquids: failure of linear response approximations

Large scale computer simulations of different fluorophore-solvent systems reveal when and why linear response theory applies to time-dependent fluorescence measurements.

: Logarithm of the Stokes shift relaxation function after excitation, S(t), after deexcitation, S R (t), and time correlation functions C g (t) and C e (t) in ground and excited state in acetonitrile ( Figure S3: Comparison of C(t) and S(t) using forcefields from Ref. [3] (left), this study (middle) or a mixed description (right), where each correlation function C(t) is obtained from a single short trajectory of 1 ns length.

Effect of sampling and forcefield
In Ref. [3] Kumar and Maroncelli find that the equilibrium and nonequilibrium simulation of C153 in 255 molecules of MeOH yields S(t) C(t). However, the small number of solvent molecules, as well as the short simulation time (1 ns for the equilibrium simulations) lead to an insufficient sampling of configurations. Fig. S3 shows 20 individual equilibrium simulations of 1 ns length in the ground and excited state, as well as the true nonequilibrium response. For the left panel, forcefields for C153 and MeOH, as well as the general simulation setup (rigid molecules, coarse-grained solvent, no polarizability) correspond to Ref. [3], where the black lines correspond to data taken directly from literature. It is evident that depending on the initial configuration of the system, different findings can be obtained, where some of the obtained correlation functions C(t) equal S(t), but others do not. Clearly, 1 ns of simulation time is not sufficient to sample all important configurations, so that repeating the simulation (as shown for 20 different initial coordinates) leads to different conclusions. A similar analysis, namely the use of short trajectories of 1 ns length to describe the equilibrium response, also leads to large uncertainties in C(t) for the force fields and setup used in the main article (middle panel of Fig. S3). Nevertheless, even the fastest equilibrium response does not equal S(t), in contrast to the simulation setup of Ref. [3]. The equilibrium and nonequilibrium results differ to a larger extent for the flexible C153 in atomistic, polarizable MeOH than for the rigid C153 in rigid, coarse-grained MeOH. To detect whether this increased discrepancy stems from the solute or solvent force field, we conducted also simulations of flexible C153 (parametrization used in this study) in the rigid, coarse-grained MeOH employed in Ref. [3] (right panel in Fig. S3). We find that a high-level solvent force field leads to larger differences between C(t) and S(t).

Higher-order correlation functions
According to Wick's theorem [4], higher-order correlation functions are multiples of the first-order correlation δ∆U (0)δ∆U (t) e for odd orders, and zero for even orders. C e (t) and its higher order factorizations are plotted in Fig. S4   t / ps

Partial correlation functions
as well as the true nonequilibrium contributions  Fig. S5 shows S A (t) and S C (t), where it is clear that the anions contribute more to S(t), especially for C153 as chromophore. The partial correlation functions, however only partly capture this behavior in the case of MQ and fail to describe the correct partial relaxation for C153. Thus, the use of LRT to predict individual contributions to the overall functions, should be carefully tested for each system in use, as it may lead to quantitatively wrong results in some cases.  Figure Fig. S4. The colored area corresponds to a 95% confidence interval.

Partial higher-order correlation functions
As explained above, the calculation of higher-order correlation functions and comparison to the first-order correlation function is often used to test the Gaussian statistics of a system. This is also possible for partial correlation functions, so that we calculated the higher-order correlation for C C (t) and C A (t), shown in Fig. S6   t / ps  Figure Fig. S4. The colored area corresponds to a 95% confidence interval.

Higher-order corrections to Gaussian statistics
The authors of Ref. [5] suggested to apply corrections to C(t) that result from deviations of Wick's theorem.
The results for MQ and C153 in 2-PrOH, where large deviations from Eqn. 1 were found, are shown in the main article. For all other solvents, where deviations from Wick's theorem are very small, of course the resulting corrections are very small, as depicted in Fig. S8. In Ref. [5], a small artificial solute was employed to test the usefulness of such corrections. We therefore also applied the corrections to formerly conducted simulations of artificial excitations in benzene [6], where we observed deviations from Eqn. 1. The corresponding excitations are shown in Fig. S9, where we chose the dipole and quadrupole increase and decrease, to test whether Eqn. 4 provides any improvement over C e (t). As depicted in Fig. S10, for an increase in dipole or quadrupole moment (analogous to what we find in C153), the corrections are somewhat helpful, although for the quadrupole increase the corrections moves the curve in the right direction, but too far. For a decrease in dipole or quadrupole moment (similar to MQ), no improvement was found for C corr (t) over C e (t). It is interesting that for MQ and C153 in 2-PrOH, we find similar results, were only for C153 (increasing dipole upon excitation) C corr (t) showed a better fit to S(t) than C e (t).   In contrast, the contributions from cations and anions change to some extent around MQ and C153. Fig. S12 shows the corresponding cation and anion coordination numbers, and their evolution with time after excitation. Note that the first shell, obtained via Voronoi tessellation, is quite large, so that more than the first peak in the radial distribution functions contributes to the first shell. The corresponding figures for both chromophores in . The colored area corresponds to a 95% confidence interval.