Dielectric constant continuum theory

Consider an alchemical transformation of a particle in water, where the particle s charge is changed from 0 to i) (e.g., neon sodium q = ). Let the transformation be performed first with the particle in a spherical water droplet of radius R (formed of explicit water molecules), and let the droplet then be transferred into bulk continuum water. From dielectric continuum theory, the transfer free energy is just the Born free energy to transfer a spherical ion of charge q and radius R into a continuum with the dielectric constant e of water ... 

If we now transfer our two interacting particles from the vacuum (whose dielectric constant is unity by definition) to a hypothetical continuous isotropic medium of dielectric constant e > 1, the electrostatic attractive forces will be attenuated because of the medium s capability of separating charge. Quantitative theories of this effect tend to be approximate, in part because the medium is not a structureless continuum and also because the bulk dielectric constant may be an inappropriate measure on the molecular scale. Eurther discussion of the influence of dielectric constant is given in Section 8.3. 

Some authors plot log k or AG against 1/e rather than against the Kirkwood function. Since 1/e is nearly linearly related to (e — 1)/(2e + 1), within the assumptions of a theory in which the solvent is treated as a continuum this substitution of variable is not serious. Another approach is to interpret the solvent dependence of the Hammett reaction constant p on a dielectric constant function. ... 

The quantitative theory of ionic reactions, within the limitations of a continuum model of the solvent, is based on the Bom equation for the electrostatic free energy of transfer of an ion from a medium of e = 1 to the solvent of dielectric constant... 

The Born solvation equation is based on the difference in the energy needed to charge a sphere of radius r,- in a solvent of dielectric constant e, and in vacuum having a dielectric constant of unity. Thae are basic flaws in the concept of the Born solvation equation (5) on which the continuum theory of ET reactions is based. First, Bom Eq. (5) does not take into account the interaction of ions with a water solvent that has a dielectric constant of approximately 80 at room temperature. Hence, the Born solvation energy will have negligible contribution from solvents with high dielectric constants. Consequently, for solvents of high dielectric constant, Eq. (5) can be written as... 

These points indicate that the continuum theory expression of the free energy of activation, which is based on the Born solvation equation, has no relevance to the process of activation of ions in solution. The activation of ions in solution should involve the interaction energy with the solvent molecules, which depends on the structure of the ions, the solvent, and their orientation, and not on the Born charging energy in solvents of high dielectric constant (e.g., water). Consequently, the continuum theory of activation, which depends on the Born equation,fails to correlate (see Fig. 1) with experimental results. Inverse correlations were also found between the experimental values of the rate constant for an ET reaction in solvents having different dielectric constants with those computed from the continuum theory expression. Continuum theory also fails to explain the well-known Tafel linearity of current density at a metal electrode. ... 

Compared to all other intermolecular interactions, the Coulomb interaction is described by a simple law, i.e.. Equation 15.2. A theory for Coulombic interaction, therefore, uses the concepts and laws that have been developed in classical electrostatics. However, it is worth pointing out that the dielectric constant is a macroscopic property and it is therefore, in principle, not correct to describe the solvent as a dielectric continuum on the molecular level. Nevertheless, experience has shown that it is in fact a useful approximation. 

It would seem from our correlations for the alcohols that the continuum model results in a valid expression for the energy states of the electron since the approximations arrived at by Platzman and Franck (23) and by Davydov (8) give the rough magnitude of the binding energy correctly, and the latter shows a trend with dielectric constant in accord with experiment. However, the theory is still in a rather primitive state, and other physical properties of the liquid, in addition to its dielectric behavior, may have to be taken into account. 

Both, the Gouy-Chapman and Debye-Hiickel are continuum theories. They treat the solvent as a continuous medium with a certain dielectric constant, but they ignore the molecular nature of the liquid. Also the ions are not treated as individual point charges, but as a continuous charge distribution. For many applications this is sufficient and the predictions of continuum theory agree with experimental results. At the end of this chapter we discuss the limitations and problems of the continuum model. 

Additional complications can arise when two bodies, i.e. the tip and the sample, interact in liquid (Fig. 3c). The interaction energy of two macroscopic phases across a dielectric medium can be calculated based on the Lifshitz continuum theory. In contact, when the distance between the phases corresponds to the nonretarded regime, the Hamaker constant in Eq. (2) is approximated by ... 

Pratt and co-workers have proposed a quasichemical theory [118-122] in which the solvent is partitioned into inner-shell and outer-shell domains with the outer shell treated by a continuum electrostatic method. The cluster-continuum model, mixed discrete-continuum models, and the quasichemical theory are essentially three different names for the same approach to the problem [123], The quasichemical theory, the cluster-continuum model, other mixed discrete-continuum approaches, and the use of geometry-dependent atomic surface tensions provide different ways to account for the fact that the solvent does not retain its bulk properties right up to the solute-solvent boundary. Experience has shown that deviations from bulk behavior are mainly localized in the first solvation shell. Although these first-solvation-shell effects are sometimes classified into cavitation energy, dispersion, hydrophobic effects, hydrogen bonding, repulsion, and so forth, they clearly must also include the fact that the local dielectric constant (to the extent that such a quantity may even be defined) of the solvent is different near the solute than in the bulk (or near a different kind of solute or near a different part of the same solute). Furthermore... 

In terms of traditional Transition State Theory (TST) for solution reactions [40,41], in which e.g. the activation free energy AG can be estimated with equilibrium solvation dielectric continuum theories [42-46], the nonequilibrium or dynamical solvation aspects enter the prefactor of the rate constant k, or in terms of the ratio of k to its TST approximation kTST, k, the transmission coefficient, k and kTST are related by [41]... 

The solvent effects are often described within a semiempirical selfconsistent reaction field theory (SCRF)248. In this theory the free energy of solvation is obtained from a set of selfconsistent equations describing the interaction of the solute (denoted by S) with the solvent modeled by a polarizable continuum characterized by a dielectric constant e. In the SCRF formalism, as developed by Rivail and collaborators249- 250 the solute-solvent system is modeled by a polarizable continuum (characterized by a dielectric constant e) in which the solvent molecule is immersed within an ellipsoidal cavity251,252. The Hamiltonian describing the solute in the cavity is given by,... 

Solvent effects can significantly influence the function and reactivity of organic molecules.1 Because of the complexity and size of the molecular system, it presents a great challenge in theoretical chemistry to accurately calculate the rates for complex reactions in solution. Although continuum solvation models that treat the solvent as a structureless medium with a characteristic dielectric constant have been successfully used for studying solvent effects,2,3 these methods do not provide detailed information on specific intermolecular interactions. An alternative approach is to use statistical mechanical Monte Carlo and molecular dynamics simulation to model solute-solvent interactions explicitly.4 8 In this article, we review a combined quantum mechanical and molecular mechanical (QM/MM) method that couples molecular orbital and valence bond theories, called the MOVB method, to determine the free energy reaction profiles, or potentials of mean force (PMF), for chemical reactions in solution. We apply the combined QM-MOVB/MM method to... 

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