Continuum treatments of solvation

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... 

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. 

Since AGsqIv is expressed on a per-mole basis, the right sides of these equations should be multiplied by the Avogadro constant iV. ) Although energy is both a molecular and a macroscopic property, both entropy and free energy are macroscopic but not molecular properties. The dielectric continuum model uses the macroscopic property of the solvent, and the macroscopic treatment of the solvent allows us to find AG°oi ei, which is a contribution to a macroscopic property. The dielectric continuum treatment of solvation is a combined quantum-mechanical and statistical-mechanical treatment. 

Continuum models of solvation treat the solute microscopically, and the surrounding solvent macroscopically, according to the above principles. The simplest treatment is the Onsager (1936) model, where aspirin in solution would be modelled according to Figure 15.4. The solute is embedded in a spherical cavity, whose radius can be estimated by calculating the molecular volume. A dipole in the solute molecule induces polarization in the solvent continuum, which in turn interacts with the solute dipole, leading to stabilization. 

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. 

As in our simple treatment of solvation dynamics in Chapter 15, the solvent in Marcus theory is taken as a dielectric continuum characterized by a local dielectric function ( )). Thus, the relation between the source, D (electrostatic displacement) and the response, (electric field) is (cf. Eqs (15.1) and (15.2))... 

Explicit-Solvent versus Continuum-Solvent Methods. Theoretical treatments of solvation can be categorized as either explicit-solvent approaches, in which many individual solvent molecules are explicitly included, or as continuum-solvent methods, where the solvent molecules are replaced by a continuous dielectric (Section 15.22). 

C. J. Cramer and D. G. Truhlar, Development and biological applications of quantum mechanical continuum solvation models, in Quantitative Treatments of Solute/Solvent Interactions, P. Politzer and J. S. Murray, eds., Elsevier, Amsterdam (1994), pp. 9-54. [Theor. Comp. Chem. 2 9 (1994).]... 

Finally we address the issue of contributions. In our view it is unbalanced to concentrate on a converged treatment of electrostatics but to ignore other effects. As discussed in section 2.2, first-solvation-shell effects may be included in continuum models in terms of surface tensions. An alternative way to try to include some of them is by scaled particle theory and/or by some ab initio theory... 

The theoretical modeling of electron transfer reactions at the solution/metal interface is challenging because, in addition to the difficulties associated with the quantitative treatment of the water/metal surface and of the electric double layer discussed earlier, one now needs to consider the interactions of the electron with the metal surface and the solvated ions. Most theoretical treatments have focused on electron-metal coupling, while representing the solvent using the continuum dielectric media. In keeping with the scope of this review, we limit our discussion to subjects that have been adi essed in recent years using molecular dynamics computer simulations. 

TvaroSka, KoS r and Hricovini in this book). One way to account for the effect of solvent on conforxnation might be to represent the molecule without environmental influences, and then explicitly include the solvent or other environmental molecules in the calculation. While avoiding built-in influences of environment is a satisfying concept, it is difficult to obtain by experiment parameters that lack those influences. Several methods have been used to study solvation effects, including continuum descriptions (24) and the explicit treatment of solvent molecules in Monte Carlo and molecular dynamics simulation. 

Liquid/liquid partition constants within pharmaceutical chemistry have been of primary interest because of tlieir correlation with liquid/membrane partitioning behavior. A sufficiently fluid membrane may, in some sense, be regarded as a solvent. With such an outlook, tlie partitioning phenomenon may again be regarded as a liquid/liquid example, amenable to treatment with standard continuum techniques. Of course, accurate continuum solvation models typically rely on the availabihty of solvation free energies or bulk solvent properties in order to develop useful parameterizations, and such data may be sparse or non-existent for membranes. Some success, however, has been demonstrated for predicting such data either by intuitive or statistical analysis (see, for example. Chambers etal. 1999). 

The Marcus treatment uses a classical statistical mechanical approach to calculate the activation energy required to surmount the barrier. It assumes a weakly adiabatic electron transfer process and non-equilibrium dielectric polarization of the solvent (continuum) as the source of activation. This model also considers the vibrational contributions of the inner solvation sphere. The Hush treatment considers ion-dipole and ligand field concepts in the treatment of inner coordination sphere contributions to the energy of activation [55, 56]. 

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. 

An Evaluation of the Debye-Onsager Model. The simplest treatment for solvation dynamics is the Debye-Onsager model which we reviewed in Section II.A. It assumes that the solvent (i) is well modeled as a uniform dielectric continuum and (ii) has a single relaxation time (i.e., the solvent is a Debye solvent ) td (Eq. (18)). The model predicts that C(t) should be a single... 

C6 A. Klamt and V. Jonas, Treatment of the outlying charge in continuum solvation models, J. Chem. Phys., 105 (1996) 9972-9981. 

The solvent reaction field calculations involve several different aspects. We would like concentrate on the points required to make these models successful as well as on the facts that limit their accuracy. One of them is the shape of the molecular cavity, which can be modelled spherically or according to the real shape of the solute molecule. First, we discuss the papers in which spherical cavity models were applied. The studies utilizing the solute-shaped cavity models are collected the second group. Finally, the approaches employing explicit treatment of the first-solvation shell molecules combined with the continuum models are discussed. 

Still within a continuum solvation approach [22,41], a unified treatment of the local field problem has recently been formulated within PCM for (hyper)polarizabilities [47] and extended to several optical and spectroscopic properties, including IR, Raman, VCD and VROA spectra [8,9,11,12],... 

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