Solvation continuum models

Continuum models 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).  

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  

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, 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 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 CCh 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 1.4 A to model a water molecule) rolling on the van der Waals surface. This is denoted the Solvent Accessible Surface (SAS) and is illustrated in Rgure 14.9. 

The simplest continuum model includes the dielectric constant of the medium in evaluating electrostatic terms in molecular mechanics calculations. Recall that Eq. 3.1 (for simple electrostatic interactions) included a dielectric term (e). Such a scaling of electrostatic interactions by the solvent dielectric constant is in principle useful and is theoretically justifiable. Note that for molecules dissolved in a solvent, the charges q,) are partial charges associated with each atom of the molecule that must be obtained by some other method. In principle this is a viable strategy, but in practice it has little impact on calculations. 

More advanced continuum models are based on parameterized, atom-specific terms that scale with the exposed surface area. In a molecular mechanics based approach, the amount of atomic surface (the sphere defined by an atom s van der Waals radius) that is exposed to solvent is determined for each particular atom in a molecule. Then, an equation that includes parameters related to the tjrpe of atom and to the specific solvent calculates a solvation term. These terms are summed over all atoms in the molecule. Such approaches blend into the molecular mechanics method quite naturally, without an overly burdensome increase in computation time. 

An especially interesting model, termed the generalized Born model, has been developed primarily for water as a solvent. We will describe it briefly here, because it nicely illustrates in a quantitative way some of the topics we have discussed in this chapter. The approach is a parameterized method that produces Gsoiv, the solvation free energy for a molecule or ion. First, Gsoiv is divided into three terms (Eq. 3.32). 

- is a parameter for each atom type (in the spirit of molecular mechanics) and SA is the solvent accessible surface area for atom i. 

What about Gp i for an ion in water We need to consider two types of interactions. The first is the interaction between solute ions, which should be modeled by Coulomb s law. The other is the interaction of an ion with the solvent, and this can be modeled by the Born equation, as mentioned in Section 3.2.2. These two equations are, to some extent, of a similar form, and so can be combined to give Eq. 3.34. 

In contrast to explicit solvation models, continuum models [110-113] only consider nonspecific long-range effects, whereas specific short-range contributions to 

Methods for evaluating the effect of a solvent may broadly be divided into two types those describing the individual solvent molecules, as discussed in Section 16.1, and those which treat the solvent as a continuous medium. Combinations are also possible, for example by explicitly considering the first solvation sphere and treating the rest by a continuum model. Each of these may be subdivided according to whether they use a classical or quantum mechanical description.  

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. 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. It is 

SIMULATIONS, TIME-DEPENDENT METHODS AND SOLVATION MODELS 

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Cramer C J and Truhlar D G 1996 Continuum solvation models Solvent Effects and Ohemical Reactivity ed O Tapia and J Bertran (Dordrecht Kluwer) pp 1-80... 

Cramer C J and Truhlar D G 1995. Continuum Solvation Models Classical and Quantum Mechanical Implementations. In Lipkowitz K B and D B Boyd (Editors) Reviews in Computational Chemistry Volume 6. New York, VCH Publishers, pp. 1-72. 

It is sometimes desirable to include the effect of the rest of the system, outside of the QM and MM regions. One way to do this is using periodic boundary conditions, as is done in liquid-state simulations. Some researchers have defined a potential that is intended to reproduce the effect of the bulk solvent. This solvent potential may be defined just for this type of calculation, or it may be a continuum solvation model as described in the next chapter. For solids, a set of point charges, called a Madelung potential, is often used. 

This method has been applied to derive a multitude of paths for the coil-to-helix transition in polyalanine using a continuum solvation model [36]. 

QuantlogP, developed by Quantum Pharmaceuticals, uses another quantum-chemical model to calculate the solvation energy. As in COSMO-RS, the authors do not explicitly consider water molecules but use a continuum solvation model. However, while the COSMO-RS model simpUfies solvation to interaction of molecular surfaces, the new vector-field model of polar Uquids accounts for short-range (H-bond formation) and long-range dipole-dipole interactions of target and solute molecules [40]. The application of QuantlogP to calculate log P for over 900 molecules resulted in an RMSE of 0.7 and a correlation coefficient r of 0.94 [41]. 

Although continuum solvation models do appear to reproduce the structural and spectroscopic properties of many molecules in solution, parameterization remains an issue in studies involving solvents other than water. In addition, the extension of these approaches to study proteins embedded in anisotropic environments, such as cell membranes, is clearly a difficult undertaking96. As a result, several theoretical studies have been undertaken to develop semi-empirical methods that can calculate the electronic properties of very large systems, such as proteins28,97 98. The principal problem in describing systems comprised of many basis functions is the method for solving the semi-empirical SCF equations ... 

How well can continuum solvation models distinguish changes in one or another of these solvent properties This is illustrated in Table 2, which compares solvation energies for three representative solutes in eight test solvents. Three of the test solvents are those shown in Table 1, one is water, and the other four were selected to provide useful comparisons on the basis of their solvent descriptors, which are shown in Table 3. Notice that all four solvents in Table 3 have no acidity, which makes them more suitable, in this respect, than 1-octanol or chloroform for modeling biomembranes. Table 2 shows that the SM5.2R model, with gas-phase geometries and semiempirical molecular orbital theory for the wave function, does very well indeed in reproducing all the trends in the data. 

C. J. Cramer and D. G. Truhlar, Continuum solvation models, in Solvent Effects and... 

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).]... 

As discussed in Section 2, one key assumption of reaction field models is that the polarization field of the solvent is fully equilibrated with the solute. Such a situation is most likely to occur when the solute is a long-lived, stable molecular structure, e g., the electronic ground state for some local minimum on a Bom-Oppenheimer potential energy surface. As a result, continuum solvation models... 

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