Dielectric response continuum approximation

It is worth pointing out that the solvation response functions for realistic models of SD chromophores resemble much more closely the TCFs that represent the low- longitudinal dielectric response of the solvent than the corresponding transverse response [18,67], This is true for the continuum approximations described in the previous section as well as for the results of MD simulations in which the solvent response is treated at the molecular level. 

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. 

Jepsen and Friedman found, however, that for microscopic impurities, (2.34a) and (2.34b)—in contrast to (2.34c)—no longer appeared to be satisfied beyond the lowest order iny in the low-density approximation they were considering, which left open the asymptotic form the microscopic results would have. Equation 2.33 reveals that only if the Onsager approximation (2.30d) were satisfied in the molecular solvent would (2.33) and (2.34) be the same. The reason for this will become clear in our discussion of the y->0 limit below, where we show that only in the Onsager continuum limit, in which (2.30d) becomes exact, is the dielectric response to each solvent dipole that of a vacuum in a macroscopic sphere surrounding the solvent dipole. Thus only in the Onsager continuum limit are the assumptions satisfied under which one can identify each solvent particle as a macrosphere within which 6= 1, and so assure the identity of the full set of ratios in (2.33) to (2.34). 

An accurate description of the aqueous environment is essential for atom-level biomolec-ular simulations, but may become very expensive computationally. An imphcit model replaces the discrete water molecules by an infinite continuum medium with some of the dielectric and hydrophobic properties of water. The continuum implicit solvent models have several advantages over the explicit water representation, especially in molecular dynamics simulations (e.g., they are often less expensive, and generally scale better on parallel machines they correspond to instantaneous solvent dielectric response the continuum model corresponds to solvation in an infinite volume of solvent, there are no artifacts of periodic boundary conditions estimating free energies of solvated structares is much more straightforward than with explicit water models). Despite the fact that the methodology represents an approximation at a fundamental level, it has in many cases been successful in calculating various macromolecular properties (Case et al. 2005). 

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 +... 

We present and analyze the most important simplified free energy methods, emphasizing their connection to more-rigorous methods and the underlying theoretical framework. The simplified methods can all be superficially defined by their use of just one or two simulations to compare two systems, as opposed to many simulations along a complete connecting pathway. More importantly, the use of just one or two simulations implies a common approximation of a near-linear response of the system to a perturbation. Another important theme for simplified methods is the use, in many cases, of an implicit description of solvent usually a continuum dielectric model, often supplemented by a simple description of hydrophobic effects [11]. 

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],... 

When the change in the solute-solvent interactions results mainly from changes in the solute charge distribution, one can employ the theory of electric polarization to formulate the dynamic response of the system. This formulation involves the nonlocal dielectric susceptibility m(r, r, i) of the solution. While this first step might lead to either the molecular or the continuum theory of solvation, in the continuum approach (r, r, t) is related approximately to the pure solvent susceptibility (r, r, t) in the portions of... 

In the usual implementation of the continuum theories of SD, one assumes that the surrounding solvent is sufficiently weakly perturbed by the presence of the solute that the system response to the solute electronic transition is well approximated by the dielectric susceptibility of the pure solvent. Further, one usually assumes that the contributions of solute motion to SD can be neglected. As shown in Section 3.4.3, continuum theories can be quite successful in predicting the solvation response in highly polar liquid solvents. It is worth examining the reasons for their success in greater detail and discussing their likely limitations. 

If one accepts the continuum, linear response dielectric approximation for the solvent, then the Poisson equation of classical electrostatics provides an exact formalism for computing the electrostatic potential (r) produced by a molecular charge distribution p(r). The screening effects of salt can be added at this level via an approximate mean-field treatment, resulting in the so-called Poisson-Boltzmann (PB) equation [13]. In general, this is a second order non-linear partial differential equation, but its simpler linearized form is often used in biomolecular applications ... 

A continuum solvent replaces explicit atomic details with a bulk, mean-field response. It is possible to demonstrate from statistical mechanics that an implicit solvent potential of mean force (PMF) exists, which preserves exactly the solute thermodynamic properties obtained from explicit solvent. It is possible to formulate a perfect implicit solvent in principle, but in practice approximations are necessary to achieve efficiency. This remains an active area of research.An implicit solvent PMF can be formulated via a thermodynamic cycle that discharges the solute in vapor, grows the uncharged (apolar) solute into a solvent W pgi fX), and finally recharges the solute within a continuum dielectric Weiec(X)... 

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