Dielectric solvation - Born - models

3 Determine the interaction contribution to the chemical potential implied by the equation of state Eq. (4.1). 

4 Consider the simple case where the radial distribution function in the fluid is zero for radii less than a cut-off value determined by the size of the hard core of the solute, and one beyond that value. Calculate the value of the parameter a appearing in the equation of state Eq. (4.1) for a potential of the form cr , where c is a constant and n is an integer. An example is the Lennard-Jones potential where = 6 for the long-ranged attractive interaction. What happens if n 37 Explain what happens physically to resolve this problem. See Widom (1963) for a discussion of the issue of thermodynamic consistency when constructing van der Waals and related approximations. 

5 Consider the several random variables Xj, j = 1. n. Suppose that these Xj are distributed according to a multi-variable gaussian distribution with means [xj) and covariances [8xj8x = [xjX ) — [xj) (x ). Show that 

Summation over repeated subscripts from 1. n is implied. Compare this result with Eqs. (4.11) and (4.12) (van Kampen, 1992). 

A virtue of the PDT approach is that it enables precise assessment of the differing consequences of intermolecular interactions of differing types. Here we use that feature to inquire into models of electrostatic interactions in biomolecular hydration. This topic provides a definite example of the issues discussed in the previous section the perturbative interactions, I of the previous section, are just the classic electrostatic interactions. 

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The Born model [11] provides a means of estimating the Gibbs energy of solvation for an ion in an infinitely dilute solution. It is based on a continuum description of the solvent as a uniform dielectric with a relative permittivity of The work of transferring the ion from vacuum to the dielectric medium is estimated on the basis of the following three-step process (a) the ion is reversibly discharged in vacuum (b) the discharged ion, which is assumed to be a sphere of radius, r, is... 

The Born model certainly overestimates the solvation term at static frequencies. Using the MSA to account for the effects of dielectric saturation, equation (7.8.32) can be rewritten as... 

Such difficulties with the simple Born model have stimulated attempts to develop more physically valid theoretical treatments of ion solvation, as discussed previously [121,128,131 — 134]. Relying again on the basic electrostatic approach, improved models have addressed the dielectric satura-... 

It should be stressed that the SMC parameterization procedure is very different from standard Born model analysis where an ion of assigned radius is embedded in a uniform continuum dielectric with bulk permittivity. Here, the ion is fully solvated by the model water, and the dielectric constant refiects the solvent s electronic properties. The structural component of dielectric reorganization is treated explicitly. 

Implicit solvation models developed for condensed phases represent the solvent by a continuous electric field, and are based on the Poisson equation, which is valid when a surrounding dielectric medium responds linearly to the charge distribution of the solute. The Poisson equation is actually a special case of the Poisson-Boltzmann (PB) equation PB electrostatics applies when electrolytes are present in solution, while the Poisson equation applies when no ions are present. Solving the Poisson equation for an arbitrary equation requires numerical methods, and many researchers have developed an alternative way to approximate the Poisson equation that can be solved analytically, known as the Generalized Born (GB) approach. The most common implicit models used for small molecules are the Conductor-like Screening Model (COSMO) [96,97], the Dielectric Polarized Continuum Model (DPCM) [98], the Conductor-like modification to the Polarized Continuum Model (CPCM) [99], the Integral Equation Formalism implementation of PCM (lEF-PCM) [100] PB models and the GB SMx models of Cramer and Truhlar [52,57,101,102]. The newest Miimesota solvation models are the SMD (universal Solvation Model based on solute electron Density [57]) and the SMLVE method, which combines the surface and volume polarization for electrostatic interactions model (SVPE) [103-105] with semiempirical terms that account for local electrostatics [106]. Further details on these methods can be found in Chapter 11 of reference 52. 

You can calculate the enthalpies of ion solvation from the vapor phase to water (dielectric constant D) from the Born model by using the Gibbs-Helmholtz Equation (13.41) ... 

The Born model for the solvation of a charged sphere in a uniform dielectric gives an estimate of the photoionization energy shift. 

The simplest approach to describing the interactions of metal cations dissolved in water with solvent molecules is the Born electrostatic model, which expresses solvation energy as a function of the dielectric constant of the solvent and, through transformation constants, of the ratio between the squared charge of the metal cation and its effective radius. This ratio, which is called the polarizing power of the cation (cf Millero, 1977), defines the strength of the electrostatic interaction in a solvation-hydrolysis process of the type... 

The upshot is that the Born theory of solvation fails because it regards the solvent as a continuous dielectric, whereas in fact solute ions (especially metal cations with z > 1) often interact in a specific manner with solvent molecules. In any event the molecular dielectric is obviously very lumpy on the scale of the ions themselves. The Born theory and other continuous dielectric models work reasonably well when metal ion solute species are treated as solvent complexes such as Cr(OH2)63+ rather than naked ions such as Cr3+, but the emerging approach to solvation phenomena is to simulate solvation dynamically at the molecular level using computer methods. 

Solvation Effects. Many previous accounts of the activity coefficients have considered the connections between the solvation of ions and deviations from the DH limiting-laws in a semi-empirical manner, e.g., the Robinson and Stokes equation (3). In the interpretation of results according to our model, the parameter a also relates to the physical reality of a solvated ion, and the effects of polarization on the interionic forces are closely related to the nature of this entity from an electrostatic viewpoint. Without recourse to specific numerical results, we briefly illustrate the usefulness of the model by defining a polarizable cosphere (or primary solvation shell) as that small region within which the solvent responds to the ionic field in nonlinear manner the solvent outside responds linearly through mild Born-type interactions, described adequately with the use of the dielectric constant of the pure solvent. (Our comments here refer largely to activity coefficients in aqueous solution, and we assume complete dissociation of the solute. The polarizability of cations in some solvents, e.g., DMF and acetonitrile, follows a different sequence, and there is probably some ion-association.)... 

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