Linear response theory of solvation

We see that in this case the relaxation is characterized by the time tl which can be very different from td For example in water s /s = 1/40 and while td = 8ps, TL is of the order of 0.2 ps  

The continuum dielectric theory used above is a linear response theory, as expressed by the linear relation between the perturbation T and the response , Eq. (15.1b). Thus, our treatment of solvation dynamics was done within a linear response framework. 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 (as opposed to continuum) level theories. Here we derive this general formalism. For simplicity we disregard the fast electronic response of the solvent and focus on the observed nuclear dielectric relaxation. 

The origin of the terms transverse and longitudinal dielectric relaxation times lies in the molecular theory of dielectric relaxation, where one finds that the decay of correlation functions involving transverse and longitudinal components of the induced polarization vector are characterized by different time constants. In a Debye fluid the relaxation times that characterize the transverse and longitudinal components of the polarization are tq and rL = (ee/esltD respectively. See, for example, P. Madden and D. Kivelson, J. Phys. Chem. 86, 4244 (1982). 

In linear response theory the solvation eneigies are proportional to the corresponding products q ), fi and g (VV4 ) where ( ) denotes the usual 

We now apply linear response theory to the relaxation that follows a sudden change in the external force, see Section 11.1.2. Focusing on the simple case where 

Our starting point (see Eqs (11.1)—(11.3)) is the classical Hamiltonian for the atomic motions 

In this case p (r) is the external force and (r) is the corresponding system response. Alternatively we may find it convenient to express the charge distribution in tenns of point moments (dipoles, quadrupoles, etc.) coupled to the coiTesponding local potential gradient tensors, for example, H will contain terms of the form fi V t and g VV (b , where fi and Q are point dipoles and quadrupoles respectively. 

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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 E, Eq. [4.3.2]. Linear response theory of solvation dynamics may be cast in a general form that does not... 

The dielectric response of a solvated protein to a perturbing charge, such as a redox electron or a titrating proton, is related to the equilibrium fluctuations of the unperturbed system through linear response theory [49, 50]. In the spirit of free energy... 

The formation and transport properties of a large polaron in DNA are discussed in detail by Conwell in a separate chapter of this volume. Further information about the competition of quantum charge delocalization and their localization due to solvation forces can be found in Sect. 10.1. In Sect. 10.1 we also compare a theoretical description of localization/delocalization processes with an approach used to study large polaron formation. Here we focus on the theoretical framework appropriate for analysis of the influence of solvent polarization on charge transport. A convenient method to treat this effect is based on the combination of a tight-binding model for electronic motion and linear response theory for polarization of the water surroundings. To be more specific, let us consider a sequence... 

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). ... 

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. 

The solvation TCP can be related by linear response theory to auto TCP of energy fluctuation. This is usually termed C(f) and defined as,... 

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 ... 

The time dependent solvation funetion S(t) is a directly observed quantity as well as a convenient tool for numerical simulation studies. The corresponding linear response approximation C(t) is also easily eomputed from numerical simulations, and can also be studied using suitable theoretical models. Computer simulations are very valuable both in exploring the validity of such theoretical calculations, as well as the validity of linear response theory itself (by comparing S(t) to C(t)). Furthermore they can be used for direct visualization of the solute and solvent motions that dominate the solvation process. Many such simulations were published in the past decade, using different models for solvents such as water, alcohols and acetonitrile. Two remarkable outcomes of these studies are first, the close qualitative similarity between the time evolution of solvation in different simple solvents, and second, the marked deviation from the simple exponential relaxation predicted by the Debye relaxation model (cf Eq. [4.3.18]). At least two distinct relaxation modes are... 

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. 

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