Linear response approximations

The GLE can be derived by invoking the linear response approximation for the response of the solvent modes coupled to the motion of the reaction coordinate. 

This question can be explored within the linear-response approximation, which relates the response of the effective coordinate of the environment (e.g., the solvent or the protein) to the dipole, p, of the solute by (Ref. 11)... 

LD model, see Langevin dipoles model (LD) Linear free-energy relationships, see Free energy relationships, linear Linear response approximation, 92,215 London, see Heitler-London model Lysine, structure of, 110 Lysozyme, (hen egg white), 153-169,154. See also Oligosaccharide hydrolysis active site of, 157-159, 167-169, 181 calibration of EVB surfaces, 162,162-166, 166... 

In order to find the relation between Eq. (122) and the Marcus theory, we employ the linear response approximation. In this case, the free energies Fj )(i =1)2) for the donor and acceptor become a parabolic function of as... 

The cumulants [26] are simple functions of the moments of the probability distribution of 5V-.C2 = (V- V))2),C3 = (V- V)f),C4 = ((]/-(]/))4) 3C22,etc. Truncation of the expansion at order two corresponds to a linear-response approximation (see later), and is equivalent to assuming V is Gaussian (with zero moments and cumulants beyond order two). To this order, the mean and width of the distribution determine the free energy to higher orders, the detailed shape of the distribution contributes. 

We now turn to the problem of proton binding to proteins, an important area for simplified free energy methods. The linear response formalism earlier underlies most of the methods used today. It leads directly to one of the more useful practical methods, the so-called LRA, or linear response approximation method [59], presented here. 

The first version of the LIE method employed the linear response approximation to estimate the electrostatic part of the solvation/binding free energies. The linear response result for this component of the solvation... 

In order to actually examine the validity of the electrostatic linear response approximation for different types of solutes and solvents, Aqvist and Hansson13 investigated the relationships between electrostatic solvation... 

Warshel and coworkers have recently examined the LIE method and different versions of what they call the LRA (linear response approximation) method for the binding of a set of cyclic urea compounds to HIV protease.34 The key features of their LRA scheme is that both averages of Equation 2 are evaluated, thus requiring two extra simulations of the non-polar states (see above), that the ligand intramolecular electrostatic terms are included in the averages, and that the non-polar contribution is calculated with the PDLD method. Results of similar quality were reported with the different methods.34 However, it should be noted that the value Vi of the electrostatic coefficient was used in Ref. 34, which, as discussed above, has been shown... 

The fact that only intermolecular energies are needed for the binding estimate has often been interpreted in such a way that intramolecular relaxation/strain, entropy, receptor desolvation etc. are neglected. We have tried to illustrate here that, at least formally, this is not the case. Using ion binding to crown ethers as an example one can easily see that these effects are in principle embedded in the linear response approximation. Whether the approximations involved in the LIE type of equation are accurate enough is another matter. However, from the various reports on the method published so far it appears that most systems, irrespective of force fields, simulation... 

In Marcus original formulation of ET theory, the free energy curves Gj and Gj are assumed to be quadratic in x (linear response approximation). Using this assumption, Marcus derives the relationship between the activation free energy and the reaction free energy... 

Figure 2. MD simulation results for SD in response to electronic excitation of Cl 53 in room-temperature acetonitrile (left panel) and CO2 liquids. The solvent models and thermodynamic states are as in Ref. " and the solute model parameters are from Ref Nonequilibrium solvent response, S(f), and linear response approximations to it for the solute in the ground, Co(t), and excited, Q (f), electronic states are shown.

III. SOLVATION MECHANISMS WITHIN THE LINEAR RESPONSE APPROXIMATION... 

III. Solvation dynamics within the linear response approximation 213... 

The drawback of Eq. (14c) is that the integrands comprise a dynamic quantity p (f) and an induced distribution F( y). Both are perturbed by radiation field and therefore depend on its complex amplitude E. Because of that further calculations become cumbersome [18]. It is possible to overcome this drawback on the basis of a linear-response approximation. The field-induced difference 8p of the law of motion p (f) from the steady-state law p (f) is proportional to the field amplitude E. The same is supposed with respect to the difference F(y) — 1 of induced and homogeneous distributions (for the latter F = 1). A steady-state dipole s trajectory does not depend on phase y and therefore does not contribute (at F = 1) to the integral (14c). Then in a linear approximation we may represent the average7 of p (f)F(y) over y as a sum... 

In this chapter we consider the extension of continuum solvent models to nonlocal theories in the framework of the linear response approximation (LRA). Such an approximation is mainly applicable to electrostatic solute-solvent interactions, which usually obey the LRA with reasonable accuracy. The presentation is confined to this case. 

Continuum solvation models consider the solvent as a homogeneous, isotropic, linear dielectric medium [104], The solute is considered to occupy a cavity in this medium. The ability of a bulk dielectric medium to be polarized and hence to exert an electric field back on the solute (this field is called the reaction field) is determined by the dielectric constant. The dielectric constant depends on the frequency of the applied field, and for equilibrium solvation we use the static dielectric constant that corresponds to a slowly changing field. In order to obtain accurate results, the solute charge distribution should be optimized in the presence of the field (the reaction field) exerted back on the solute by the dielectric medium. This is usually done by a quantum mechanical molecular orbital calculation called a self-consistent reaction field (SCRF) calculation, which is iterative since the reaction field depends on the distortion of the solute wave function and vice versa. While the assumption of linear homogeneous response is adequate for the solvent molecules at distant positions, it is a poor representation for the solute-solvent interaction in the first solvation shell. In this case, the solute sees the atomic-scale charge distribution of the solvent molecules and polarizes nonlinearly and system specifically on an atomic scale (see Figure 3.9). More generally, one could say that the breakdown of the linear response approximation is connected with the fact that the liquid medium is structured [105],... 

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