Continuum Solvation Models

Continuum models60 consider the solvent as a uniform polarizable medium with a dielectric constant of e, and with the solute M placed in a suitably shaped hole in the medium (Figure 14.8).61

Figure 14.8 Reaction field model

Creation of a hole in the medium costs energy, i.e. this is a destabilization, while dispersion interactions between the solvent and solute add a stabilization (this is roughly the van der Waals energy between solvent and solute). In principle, there may also be a repulsive component, thus the dispersion term is sometimes denoted dispersion/ repulsion. The electric charge distribution of M will polarize the medium (induce charge moments), which in turn acts back on the molecule, thereby producing an electrostatic stabilization. The solvation (free) energy may thus be written as in eq. (14.49).

Reaction field models differ in five aspects:

(1) How the size and shape of the hole are defined.

(2) How the cavity/dispersion contribution is calculated.

(3) How the charge distribution of M is represented.

(4) How the solute M is described, either classical (force field) or quantum (semi-empirical or ab initio).

(5) How the dielectric medium is described.

The dielectric medium is normally taken to have a constant value of e, but may for some purposes also be taken to depend for example on the distance from M. For dynamical phenomena it can also be allowed to be frequency dependent,62 i.e. the response of the solvent is different for a "fast" reaction, such as an electronic transition, and a "slow" reaction, such as a molecular reorientation. It should be noted that e is the only parameter characterizing the solvent, and solvents having the same e value (such as acetone, e = 20.7, and 1-propanol, e = 20.1, or benzene, e = 2.28, and carbon tetrachloride, e = 2.24) are thus treated equally. The hydrogen bonding capability of 1-propanol compared with acetone will in reality most likely make a difference, and the solvent dynamics of an almost spherical CCl4 will be different from the planar benzene molecule.

The simplest shape for the hole is a sphere or an ellipsoid. This has the advantage that the electrostatic interaction between M and the dielectric medium may be calculated analytically. More realistic models employ molecular shaped holes, generated for example by interlocking spheres located on each nucleus. Taking the atomic radius as a suitable factor (a typical value is 1.2) times a van der Waals radius defines a van der Waals surface. Such as surface may have small "pockets" where no solvent molecules can enter and a more appropriate descriptor may be defined as the surface traced out by a spherical particle of a given radius (a typical radius of 1.4A to model a water molecule) rolling on the van der Waals surface. This is denoted the Solvent Accessible Surface (SAS) and is illustrated in Figure 14.9.

Since an SAS is computationally more expensive to generate than a van der Waals surface, and since the difference is often small, a van der Waals surface is often used in practice. Furthermore, a very small displacement of an atom may alter the SAS in a discontinuous fashion, as a cavity suddenly becomes too small to allow a solvent molecule to enter. Alternatively, the cavity may be calculated directly from the wave function, for example by taking a surface corresponding to an electron density of 0.001.63 It is generally found that the shape of the hole is importan, and that molecular shaped cavities are necessary to be able to obtain good agreement with experimental data (such as solvation energies). It should be emphasized, however, that reaction field

Figure 14.9 On a surface generated by overlapping van der Waals spheres there will be areas (hatched) that are inaccessible to a solvent molecule (dotted sphere)

models are incapable of modelling specific (short-range) solvation effects, i.e. those occurring within the first solvation sphere.

The energy required to create the cavity (entropy factors and loss of solvent-solvent van der Waals interactions), and the stabilization due to van der Waals interactions between the solute and solvent (which may also contain a small repulsive component), is usually assumed to be proportional to the surface area. The corresponding energy terms may be taken simply as being proportional to the total SAS area (a single proportionality constant), or parameterized by having a constant X specific for each atom type (analogous to van der Waals parameters in force field methods), with the X parameters being determined by fitting to experimental solvation data.

For solvent models where the cavity/dispersion interaction is parameterized by fitting to experimental solvation energies, the use of a few explicit solvent molecules for the first solvation sphere is not recommended, as the parameterization represents a best fit to experimental data without any explicit solvent present. The electrostatic component of eq. (14.49) can be described at several different levels of approximation, as discussed in the following sections.

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