Time correlation functions solvation dynamics

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

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 nonequilibrium solvation function iS (Z), which is directly observable (e.g. by monitoring dynamic line shifts as in Fig. 15.2), is seen to be equal in the linear response approximation to the time correlation function, C(Z), of equilibrium fluctuations in the solvent response potential at the position of the solute ion. This provides a route for generalizing the continuum dielectric response theory of Section 15.2 and also a convenient numerical tool that we discuss further in the next section. 

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... 

Molecular dynamics simulations focused on the solvation dynamics of a hypothetical dimer probe molecule in [C2mim][PF6] and [C2mim][Cl] have been done by Kim and co-workers.The solvation dynamics were characterized by the time correlation function of the vertical energy difference of two solute states relevant to a charge shift. The vertical energy difference between an initial state i and a final state /, AEj f, is assumed to be comprised of only Coulombic terms. The time correlation function is computed as... 

It is clear that the ensemble averages in equation (2) can be estimated from molecular dynamics simulations. Calculations along these lines began shortly after simulations on proteins became feasible, but only recently have systematic solvated simulations of useful length become accessible. These can be directly compared to experimental data (as a check on the quality of the force fields and simulation methods), and can also be used to help decide which approximate models are most appropriate. It is customary to compute from the simulation a normalized time-correlation function. 

The equilibrium calculation of solvation dynamics involves computing the equilibrium time correlation function ... 

Investigation of water motion in AOT reverse micelles determining the solvent correlation function, C i), was first reported by Sarkar et al. [29]. They obtained time-resolved fluorescence measurements of C480 in an AOT reverse micellar solution with time resolution of > 50 ps and observed solvent relaxation rates with time constants ranging from 1.7 to 12 ns. They also attributed these dynamical changes to relaxation processes of water molecules in various environments of the water pool. In a similar study investigating the deuterium isotope effect on solvent motion in AOT reverse micelles. Das et al. [37] reported that the solvation dynamics of D2O is 1.5 times slower than H2O motion. 

Odelius and co-workers reported some time ago an important study involving a combined quantum chemistry and molecular dynamics (MD) simulation of the ZFS fluctuations in aqueous Ni(II) (128). The ab initio calculations for hexa-aquo Ni(II) complex were used to generate an expression for the ZFS as a function of the distortions of the idealized 7), symmetry of the complex along the normal modes of Eg and T2s symmetries. An MD simulation provided a 200 ps trajectory of motion of a system consisting of a Ni(II) ion and 255 water molecules, which was analyzed in terms of the structure and dynamics of the first solvation shell of the ion. The fluctuations of the structure could be converted in the time variation of the ZFS. The distribution of eigenvalues of ZFS tensor was found to be consistent with the rhombic, rather than axial, symmetry of the tensor, which prompted the development of the analytical theory mentioned above (89). The time-correlation... 

Fig. 2 shows the effect of creating hydrogen-bonding complexes between HPTA and oxygen-bases on the solvation correlation function of HPTA, C(t) [10]. Utilizing a pump-probe set-up described elsewhere [11], with 400 nm excitation, the dynamic stokes shift of HPTA was analyzed with about 50fs time-resolution. The hydrogen-bonded HPTA exhibited much faster dynamics than the solvation dynamics of the uncomplexed HPTA in pure DCM. 

TABLE 1 Experimentally Observed Solvation Dynamics at 298K Determined by the Correlation Function C(t) tj, 2 Relaxation Times, Average Relaxation Time (First Moment of C(t))... 

Solvation dynamics are measured using the more reliable energy relaxation method after a local perturbation [83-85], typically using a femtosecond-resolved fluorescence technique. Experimentally, the wavelength-resolved transients are obtained using the fluorescence upconversion method [85], The observed fluorescence dynamics, decay at the blue side and rise at the red side (Fig. 3a), reflecting typical solvation processes. The molecular mechanism is schematically shown in Fig. 5. Typically, by following the standard procedures [35], we can construct the femtosecond-resolved emission spectra (FRES, Stokes shifts with time) and then the correlation function (solvent response curve) ... 

The solvation dynamics of bulk water have been well studied. Jarzeba et al. [33] obtained a correlation function with 160 fs (33%) and 1.2 ps (67%), and Jimemez et al. [34] reported an initial Gaussian-type component (frequency 38.5 ps-1 25 fs in time width, 48%) and two exponential decays of 126 fs (20%) and 880 fs (35%). Using Eq. (6), the correlation function we obtained for bulk water, as shown in Fig. 4, is best fitted by double exponential decays integrated with an initial Gaussian-type contribution through a stretched mode c t) = + c2e tlZl, where for a pure Gaussian-type decay, / = 2. The... 

Figure 19 shows the typical fluorescence transients of TBE from more than 10 gated emission wavelengths from the blue to the red side. At the blue side of the emission maximum, all transients obtained from four Trp-probes in the cubic phase aqueous channels drastically slow down compared with that of tryptophan in bulk water. The transients show significant solvation dynamics that cover three orders of magnitude on time scales from sub-picosecond to a hundred picoseconds. These solvation dynamics can be represented by three distinct decay components The first component occurs in about one picosecond, the second decays in tens of picoseconds, and the third takes a hundred picoseconds. The constmcted hydration correlation functions are shown in Fig. 20a with anisotropy dynamics in Fig. 20b. Surprisingly, three similar time scales (0.56-1.431 ps, 9.2-15 ps, and 108-140 ps) are obtained for all four Trp-probes, but their relative amplitudes systematically change with the probe positions in the channel. Thus, for the four Trp-probes studied here, we observed a correlation between their local hydrophobicity and the relative contributions of the first and third components from Trp, melittin, TME to TBE, the first components have contributions of 40%, 35%, 26%, and 17%, and the third components vary from 32%, to 38%, 43%, and 53%, respectively. The...

Strikingly, the solvation dynamics for all mutants are nearly the same. All correlation functions can be best described by a double exponential decay with time constants of 0.67 ps with 68% of the total amplitude and 13.2 ps (32%) for D60, 0.47 ps (67%) and 12.7 (33%) for D60G, and 0.53 ps (69%) and 10.8 ps (31%) for D60N. Relative to SNase above, the solvation dynamics are fast, which reflects the neighboring hydrophobic environment. We also measured the anisotropy dynamics and, as shown in the inset of Fig. 33, the local structure is very rigid in the time window of 800 ps. This observation is consistent with the inflexible turn (-T30W31-) in the transition from the second /1-sheet and the second x-helix (Fig. 31). Thus, the three mutants, with a charged, polar, or hydrophobic reside around the probe (Fig. 34) but with the similar time scales of... 

Unfortunately, theories starting from the basic (or factual ) Hamiltonian, Eq. (1). unavoidably lead (without further approximation) to expressions for the dynamic solvation properties that require the knowledge of two-time many-point correlation functions. To avoid this difficulty we turn to an alternative description of the systeip, in terms of the surrogate (subscript E) Hamiltonians ... 

While there is no unique criterion for choosing 4 E, the selection must lead to an accurate theory of solvation dynamics without invoking two-time many-point correlation functions. We have found that this goal can be achieved with a new theory for the nonequilibrium distribution function in which the renormalized solute-solvent interactions enter linearly. In this theory and are chosen such that the renormalized linear response theory accurately describes the essential solute-solvent static correlations that rule the equilibrium solvation both at t = 0 (when solvent is in equilibrium with the initial charge distribution of the solute) and at 1 = oc (when the solvent has reached equilibrium with the new solute charge distribution). ... 

According to Ekj. (7), it is the dielectric dynamics of the homogeneous solvent, as expressed in C (fc, ), that is the source of the time dependence of the estimate Z t) of the solvation tcf. In the RDT approximation the effect of the solute-solvent interactions is carried by the static coupling function B (fc). This factorization (to a function of the homogeneous solvent dynamics times a function of the static solute-solvent structure) is a characteristic feature of the RDT theory. The renonnalized character of the coupling function allows us to bypass the two-time many-point correlation functions that would necessarily appear in a dynamical theory that explicitly addressed the inhomogeneity of the solvent in the neighborhood of the solute particle. 

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