Continuum treatments

The continuum treatment of dispersion forces due to Lifshitz [19,20] provides the appropriate analysis of retardation through quantum field theory. More recent analyses are more tractable and are described in some detail in several references [1,3,12,21,22],... 

The continuum treatment of electrostatics can also model salt effects by generalizing the Poisson equation (12) to the Poisson-Boltzmann equation. The finite difference approach to solving Eq. (12) extends naturally to treating the Poisson-Boltzmann equation [21], and the boundary element method can be extended as well [19]. 

Simonson, T. Archontis, G. Karplus, M., Continuum treatment of long-range interactions in free energy calculations. Application to protein-ligand binding, J. Phys. Chem. B 1997, 101, 8349-8362... 

It was noted above that a continuum treatment of the solvent can be helpful, although representing certain solvent molecules explicitly may be necessary. The expressions for handling the free energy contributions in such hybrid models have been derived by Gilson et al.11... 

An early continuum treatment of solvation, associated with Born,17 comes out of the analysis of the electrostatic work involved in building up a charge Q on a conducting sphere of radius R in a medium with dielectric constant e. From Poisson s equation, it follows that the potential outside of the sphere is Q/eR. Thus the work of charging is the result of each additional element dq interacting with the charge q already present 87... 

Sections III. 1-III. 3 have described some basic discrete molecular and continuum treatments of solute-solvent interactions. There are many variants and refinements of these that have not been discussed, such as the use of effective dielectric constants66 or the implementation of dielectric screening.155 156 For ionic solutions, it is sometimes preferred to find the reaction field potential via the Poisson-Boltzmann rather than the Poisson equation,132 157 since the effects of the other ions can readily be incorporated into the former.158... 

Analysis of the experimental measurements of transient solvation (primarily C(r)) in terms of contemporary theoretical models has led to several conclusions [15,22-26,30-33,41], which are reviewed in detail in Section II. Continuum treatments are seen to fail in several cases, but are remarkably predictive considering the simplicity of the model. Qualitative features predicted by theories that go beyond the simple continuum model are borne out in experiment, although the agreement is qualitative at best. 

Transferrin model compounds and 44 related iron(III) crystal structures were used to modify the AMBER force field for subsequent studies of ferric transferrin. Eneigy minimization was conducted both in vacuo and, more interestingly, with the generalized Bom/surface area (GB/SA) continuum treatment described in Chapter 2, Section 2.712201. 

As a second example, we have determined the influence of solvation on the steric retardation of SN2 reactions of chloride with ethyl and neopentyl chlorides in water, which has recently been studied by Vayner and coworkers [91]. In their study solvent effects were examined by means of QM-MM Monte Carlo simulations as well as with the CPCM model. Solvation causes a large increase in the activation energies of these reactions, but has a very small differential effect on the ethyl and neopentyl substrates. Nevertheless, a quantitative difference was found between the stability of the transition states determined using discrete and continuum treatments of solvation, since the activation free energies for ethyl chloride and neopentyl chloride amount to 23.9 and 30.4kcalmoF1 according to MC-FEP simulations, but to 38.4 and 47.6 kcal moF1 from CPCM computations. 

The outer-shell intrinsic reorganization energy, AG, is usually treated on the basis of the dielectric continuum treatment. For one-electron electrochemical reactions, this yields [32]... 

Such considerations also provide a rationale of the inability of the dielectric continuum model, as conventionally expressed with a fixed frequency factor, to describe the solvent-dependent kinetics of some other outer-sphere reactions [45b, 95b]. As noted in Sect. 3.3.1, the influence of solvent dynamics upon vn should disappear for reactions having moderate or large inner-shell barriers, the frequency factor being determined by vis instead [eqns. (22) and (25)] this can account for the success of the conventional (fixed-frequency) dielectric-continuum treatment in describing solvent-dependent kinetics for some reactant systems [45],... 

The appropriate generalization of equation (33) to arbitrary geometries in multicomponent mixtures can be shown from kinetic theory [5] or from a reasonable continuum treatment to be... 

The Ru(NH3)6 system was also studied [106]. Again, considerable changes of experimental rate constants corrected for double-layer effects were observed in several solvents, which contrasted with small changes expected from the dielectric continuum treatment. Elzanowska et al. [88] studied the electrode reaction of... 

A recent study of Kiefer and Hynes49 used an EVB formulation, with a continuum treatment of the solvent, in an attempt to derive an LFER for PT reactions. Unfortunately, they assumed that a Marcus relation was never actually derived for PT reactions, apparently overlooking all the above works. Furthermore, their derivation ignored the crucial effect of Hy. Nevertheless, it is encouraging to see again a... 

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