Solvation time correlation function

From the above equation it appears convenient to characterize solvation dynamics by means of the solvation time correlation function C(t), defined asa)... 

Fig. 2.8 Experimentally obtained solvation time correlation function, S(t), for the solvation of coumarin 343 in water (taken from Ref. [19 a]).

The surrogate Hamiltonian is expressed in terms of renormalized solute-solvent interactions, a feature that leads to a simple and natural linear response description of the solvent dynamics in the vicinity of the solute. In addition to the measurable solvation time correlation function (tcf), we can also calculate observables needed to elucidate the detailed mechanism of solvation response, such as the evolution of the solvent polarization charge density around the solute. 

A recent develtyiment in the theoty ftx- the dynamics structure factor of molecular liquids, which employs the interaction-site modd, is outlined. The theory is applied for a d cription of the solvation dynamics associated with a photo-excitation of a molecule in polar liquid. Preliminary results of the solvation time correlation functions for an atomic molecule in a variety of solvents are presented. 

The solvation d)mamics is monitored by the decay of solvation time correlation function (TCP), S(f), defined as. 

Note that C t) is essentially the solvation time correlation function. Both C(t) and J co) are molecular properties and can be related to the dielectric properties of the solvent, in the long wavelength approximation (see [39] for a comprehensive discussion). In terms of the dielectric permittivity (o)) [7,21],... 

The solvation time correlation function is often equated to the auto time correlation function of energy fluctuation. This is usually termed C(t) to distinguish it from S(t). Thus, C(t) is defined as... 

A rather simple experimental teehnique involving measurement of the time-dependent fluorescence Stokes shift (TDFSS) after an initial exeitation has been applied to measure SD in a large number of liquids. TDFSS oceurs due to dipolar solvation of the excited probe and thus gives an estimate of the solvation timeseales. In an important paper, Jimenez et al. reported the results of SD of the exeited state of the dye coumarin 343 (C343) in liquid water [14]. Their result is shown in Figure 3.13. The initial part of the solvent response of water was found to be extremely fast (few tens of femtoseconds) and it constituted more than 60% of the total solvation energy relaxation. The subsequent relaxation was found to occur in the picosecond timescale. The decay of the solvation time correlation function, S t)y was fitted to a function of the following form... 

Figure 3.13. Comparison of solvation time correlation function S t) and C i) for dye C343 in water. The dashed line shows the experimental result (labeled as expt). The MD simulation result is labeled Aq. Also shown is a simulation for solvation of a neutral atomic solute with the Lennard-Jones parameters of the water oxygen atom (S°). The experimental data were fitted to Eq. (3.9) (using the constraint that the long-time spectrum matched the steady-state fluorescence spectrum) as a Gaussian component (fi equency 38.5 ps 48% of total amplitude) and a sum of two exponential components 126 (20%) and 880 (35%) fs. Adapted with permission from Nature, 369 (1994), 471. Copyright(1994) Nature Publishing Group.

We next need to find the frequency-dependent friction C(s) that we use in Eq. (3.16), to obtain the barrier frequency. This is obtained from the solvation time correlation function [22,23]. 

Let us summarize the steps quickly. First, we use the Marcus theory to obtain the reaction free-energy surface. Second, we adopt the Grote-Hynes theory to obtain the reaction rate. The latter needs frequency-dependent friction on the reactive motion, which is the solvent polarization. Third, we use the solvation time correlation function to obtain the frequency-dependent friction. 

Since the solvation time correlation function is known both from experiments and from computer simulations, we can easily carry out the above exercise. When this is done, the theory predicts a lack of, or weak, dependence of the electron transfer rate on solvent dynamics, for weakly adiabatic reactions the reason being the dominance of the ultrafast component in SD of water, so the solvent moves too fast to offer any retardation ... 

There have been two different interpretations of the slow dynamics observed in the SD of the lysozyme hydration layer. The first attributes the intermediate time-scales (30 0 ps) to slow water. Bagchi and co-workers employed the dynamic exchange model to relate the observed slow dynamics to the timescale of the fluctuation of water in the hydration layer [11]. In an alternative interpretation. Song et al. used the formulation developed by Song and Marcus that relates the solvation time correlation function to the DR of the medium. They attributed the... 

In a series of important studies, Zewail and co-workers examined the SD of exeited tryptophan as a natural probe in several proteins by using the TDFSS teehnique [13]. The advantage of using tryptophan as a probe was twofold. First, it was a natural probe, so the conformation of the protein was not disturbed and the solvation of the native state was explored. Second, one could study proteins where the tryptophan is partly or fully exposed to water, and so SD studies allowed one to directly probe the response of biological water. They found a slow component in the solvation time correlation function, which was in the range 20-40 ps. This was more than an order of magnitude slower than the bulk response. 

Figure 8.5. Solvation time correlation function for atryptophan probe in flie surface of the protein SC. The inset shows the same for dansyl-bonded SC where die probe is 6-7 A away from the surface. Adapted widi permission from J. Phys. Chem. B, 106 (2002), 12376. Copyright (2002) American Chemical Society.

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