Linear response theory of solvation dynamics

The continuum dielectric theory of solvation dynamics is a linear response theory, as expressed by the linear relation between the perturbation D and the response of E, Eq. [4.3.2]. Linear response theory of solvation dynamics may be cast in a general form that does not depend on the model used for the dielectric environment and can therefore be applied also in molecular theories. Let 

Alternatively we may find it convenient to express the charge distribution in terms of point moments (dipoles, quadrupoles, etc.) coupled to the corresponding local potential gradient tensors, e.g. H will contain terms of the form pVO and Q VV I where [1 and Q are point dipoles and quadrupoles respectively. 

In linear response theory the corresponding solvation energies are proportional to the corresponding products q I , and Q VV I where o denotes the usual observ- 

Going back to the general expressions [4.3.19] and [4.3.20], linear response theory relates non-equilibrium relaxation close to equilibrium to the dynamics of equilibrium fluctu- 

The left hand side of [4.3.23], normalized to 1 at t = 0, is a linear approximation to the solvation function 

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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). ... 

Approximate analytical theories of solvation dynamics are typically based on the linear response approximation and additional statistical mechanics or continuum electrostatic approximations to Cy(r). The continuum electrostatic approximation requires the frequency-dependent solvent dielectric response For example, the Debye model, for which e(a>) = + (cq - )/(l +... 

It should be emphasized that this description of solvation as a purely electrostatic process is greatly over-simplified. Short-range interactions exist as well, and the physical exclusion of the solvent from the space occupied by the solute must have its own dynamics. Still, for solvation of ions and dipolar molecules in polar solvents electrostatic solvent-solute and solvent-solvent interactions dominate, and disregarding short-range effects turns out to be a reasonable approximation. Of main concern should be the use of continuum electrostatics to describe a local molecular process and the fact that the tool chosen is a linear response theory. We will come to these points later. 

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. 

One example of the use of linear response theory has been that of Hwang et al. in their studies of an reaction in solution. > o In their work, based on the empirical valence bond (EVB) method discussed earlier, they defined their reaction coordinate Q as the electrostatic contribution to the energy gap between the two valence bond states that are coupled together to create the potential energy surface on which the reaction occurs. Thus, the solvent coordinate is zero at the point where both valence states are solvated equivalently (i.e., at the transition state). Hwang et al. studied the time dependence of this coordinate through both molecular dynamics simulations and through a linear response treatment ... 

This deviation from linearity shows itself also in the solvation dynamics. Figure 4.3.7 shows the linear response functions and the non-equilibrium solvation function, C(t) and S(t), respectively, computed as before, for the di-ether H(CH20CH2)2CH3 solvent. Details of this simulations are given in Ref. 1 lb. If linear response was a valid approximation all the lines in Figure 4.3.7 The two lines for C(t) that correspond to q=0 and q=l, and the two lines for S(t) for the processes q=0—K =l and the process q=l—X =0, would coalesce. The marked differences between these lines shows that linear response theory fails forfliis system. 

Such numerical simulations have played an important role in the development of our understanding of solvation dynamics. For example, they have provided the first indication that simple dielectric continuum models based on Debye and Debey-like dielectric relaxation theories are inadequate on the fast timescales that are experimentally accessible today. It is important to keep in mind that this failure of simple theories is not a failure of linear response theory. Once revised to describe reliably response on short time and length scales, e.g. by using the full k and (O dependent dielectric response function e(k,o , and sufficiently taking into account the solvent structure about the solute, linear response theory accounts for most observations of solvation dynamics in simple polar solvents. 

Numerical simulations have also been instrumental in elucidating the differences between simple and complex solvents in the way they dynamically respond to a newly created charge distribution. The importance of translational motions that change the composition or structure near the solute, the consequent early failure of linear response theory in such systems, and the possible involvement of solvent intramolecular motions in the solvation process were discovered in this way. 

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