Van der Waals cavity

We have given some highlights of a theory which combines the familiar multistate VB picture of a molecular system with a dielectric continuum model for the solvent which accounts for the solute s boundary effects — due to the presence of a van der Waals cavity which displays the solute s shape — and includes a quantum model for the electronic solvent polarization. 

An interesting alternative to van der Waals cavities is the use of isodensity or isopotential surfaces. Rivail et al. [70] demonstrated that for a given cavity volume, the electron isopotential surface is the one containing the largest electronic density, thus giving a physical meaning to this surface. Nevertheless, isodensity and isopotential cavities are computationally demanding, as they have to be recomputed at each SCF iteration, and are not quite used in practice. 

Previous studies have shown that there is a correlation between the enthalpy of hydration of alkanes and their accessible surface area [30,31] or related magnitudes. Moreover, relationships between the hydration numbers calculated from discrete simulations for hydrocarbons and both the free energy and enthalpy of hydration of these molecules have also been reported [32] and have been often used to evaluate solvation enthalpies. Analysis of our results, illustrates the existence of a linear relationship between A//n eie and the surface of the van der Waals cavity,. SVw, defined in MST computations for the calculation of the non-electrostatic contributions (Figure 4-1). In contrast, no relationship was found for the electrostatic component of the hydration enthalpy (A//eie data not shown). Clearly, in a first approximation, one can assume that the electrostatic interactions between solute and solvent can be decoupled from the interactions formed between uncharged solutes and solvent molecules. 

Figure 4-1. Representation of the change in the non-electrostatic component (kcal/mol) of the hydration enthalpy (A//n.eie) and entropy (-TAAn.eie) versus the surface (A) of the van der Waals cavity used in MST computations...

With the definition (112) one immediately arrives at the molecular (or van der Waals) cavity, characterized by the volume Vw and the surface iSw However, modellistic considerations suggest the use of other cavities, that may be derived from the van der Waals one by introducing further parameters related to the solvent molecules. 

The dicyclohexylamine/thiourea inclusion compound has been studied by CPMAS NMR. The results have shown that the guest (dicyclo-hexylamine) molecules are freely rotating and that the channels are perfect van der Waals cavities. 

A similar cooperation of hydrogen bonding and hydrophobic/van-der-Waals cavity packing may explain the preferential binding of alcohol from an aqueous alcohol solution [70]. 

The basic problem of these models is, of course, related to the oversimplified cavity shape, as evidenced in Fig. 17.5, which compares spherical and van der Waals cavities for a typical dye. A quite successful empirical recipe was suggested by Kawski et al. who showed that if the condition a/a = 1/2 is verified (a being the isotropic polarisability of the solute), the specific choice of the solute cavity shape (sphere or ellipsoid of revolution) affects negligibly the value of the excited state dipole moment. It is worth noting that such a condition occurs quite commonly [193-197] and for instance for phtalimide and stilbene derivatives the experimental values found for the radius and the polarisability are in a range from 0.41 to 0.62 [193], and from 0.53 to 0.68 [198, 199], respectively. 

Microscopic analyses of the van der Waals interaction have been made for many geometries, including, a spherical colloid in a cylindrical pore [14] and in a spherical cavity [15] and for flat plates with conical or spherical asperities [16,17]. 

Since taking simply ionic or van der Waals radii is too crude an approximation, one often rises basis-set-dependent ab initio atomic radii and constnicts the cavity from a set of intersecting spheres centred on the atoms [18, 19], An alternative approach, which is comparatively easy to implement, consists of rising an electrical eqnipotential surface to define the solnte-solvent interface shape [20],... 

A yet more realistic cavity shape is that obtained from the van der Waals radii of the atoms of the solute. This is the approach taken in the polarisable continuum method (PCM) [Miertus et al. 1981], which has been implemented in a variety of ab initio and semi-empirical quantu/rt mechanical programs. Due to the non-analytical nature of the cavity shapes in the PCM approach, it is necessary to calculate numerically. The cavity surface is divided... 

The PCM algorithm is as follows. First, the cavity siuface is determined from the van der Waals radii of the atoms. That fraction of each atom s van der Waals sphere which contributes to the cavity is then divided into a nmnber of small surface elements of calculable surface area. The simplest way to to this is to define a local polar coordinate frame at tlie centre of each atom s van der Waals sphere and to use fixed increments of AO and A(p to give rectangular surface elements (Figure 11.22). The surface can also be divided using tessellation methods [Paschual-Ahuir d al. 1987]. An initial value of the point charge for each surface element is then calculated from the electric field gradient due to the solute alone ... 

Tlic cavity and van der Waals contributions may also be modelled as separate terms. In som implementations an estimate of the cavity term may be obtained using scaled particle theor [Eierotti 1965 Claverie et al. 1978], which uses an equation of the form ... 

Molecular volumes are usually computed by a nonquantum mechanical method, which integrates the area inside a van der Waals or Connolly surface of some sort. Alternatively, molecular volume can be determined by choosing an isosurface of the electron density and determining the volume inside of that surface. Thus, one could find the isosurface that contains a certain percentage of the electron density. These properties are important due to their relationship to certain applications, such as determining whether a molecule will fit in the active site of an enzyme, predicting liquid densities, and determining the cavity size for solvation calculations. 

The secondary and tertiary structures of myoglobin and ribonuclease A illustrate the importance of packing in tertiary structures. Secondary structures pack closely to one another and also intercalate with (insert between) extended polypeptide chains. If the sum of the van der Waals volumes of a protein s constituent amino acids is divided by the volume occupied by the protein, packing densities of 0.72 to 0.77 are typically obtained. This means that, even with close packing, approximately 25% of the total volume of a protein is not occupied by protein atoms. Nearly all of this space is in the form of very small cavities. Cavities the size of water molecules or larger do occasionally occur, but they make up only a small fraction of the total protein volume. It is likely that such cavities provide flexibility for proteins and facilitate conformation changes and a wide range of protein dynamics (discussed later). 

Probably the most familiar of all clathrates are those formed by Ar, Kr and Xe with quinol, l,4-C6H4(OH)2, and with water. The former are obtained by crystallizing quinol from aqueous or other convenient solution in the presence of the noble gas at a pressure of 10-40 atm. The quinol crystallizes in the less-common -form, the lattice of which is held together by hydrogen bonds in such a way as to produce cavities in the ratio 1 cavity 3 molecules of quinol. Molecules of gas (G) are physically trapped in these cavities, there being only weak van der Waals interactions between... 

The Self-Consistent Reaction Field (SCRF) model considers the solvent as a uniform polarizable medium with a dielectric constant of s, with the solute M placed in a suitable shaped hole in the medium. Creation of a cavity 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). The electric charge distribution of M will furthermore 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... 

The simplest shape for the cavity is a sphere or possibly an ellipsoid. This has the advantage that the electrostatic interaction between M and the dielectric medium may be calculated analytically. More realistic models employ moleculai shaped cavities, generated for example by interlocking spheres located on each nuclei. Taking the atomic radius as a suitable factor (typical value is 1.2) times a van der Waals radius defines a van der Waals surface. Such a 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 rolling on the van der Waals surface. This is denoted the Solvent Accessible Surface (SAS) and illustrated in Figm e 16.7. 

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 coiTesponding to an electron density of 0.001. It is... 

The Polarizable Continuum Model (PCM) employs a van der Waals surface type cavity, a detailed description of the electrostatic potential, and parameterizes the cavity/ dispersion contributions based on the surface area. The COnductor-like Screening... 

On the other hand, the values of AH° and AS° for a-cyclodextrin-l-alkanol systems are significantly more negative than those for the corresponding P-cyclOdextrin systems. 1-Alkanols must fit closely into the cavity of a-cyclodextrin, so that the com-plexation is governed by van der Waals interaction rather than by hydrophobic interaction. 

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