Molecular-shaped cavity

Given the diversity of different SCRF models, and the fact that solvation energies in water may range from a few kcal/mol for say ethane to perhaps 100 kcal/mol for an ion, it is difficult to evaluate just how accurately continuum methods may in principle be able to represent solvation. It seems clear, however, that molecular shaped cavities must be employed, the electiostatic polarization needs a description either in terms of atomic charges or quite high-order multipoles, and cavity and dispersion terms must be included. Properly parameterized, such models appear to be able to give absolute values with an accuracy of a few kcal/mol." Molecular properties are in many cases also sensitive to the environment, but a detailed discussion of this is outside the scope of this book. ... 

A problem of molecular-shaped SCRF models is the absence of an analytical solution for the reaction field. One line of development was the search for an approximate expression for the dielectric interaction energy of a solute in a molecular-shaped cavity, without the need for explicit calculation of the solvent polarization. These models were summarized as generalized Born (GB) approximations [22,30]. The most popular of these models... 

There are two other main directions for the calculation of the electrostatic interaction between the solute and a surrounding dielectric continuum for molecular-shaped cavities. Both require intensive numerical calculations and are thus slower than GB methods. The first direction is the direct numerical solution of the Poisson equation for the volume polarization P(r) at a position r of the dielectric medium ... 

The efficient construction of proper and sufficiently accurate segmentations of a molecular-shaped cavity is an important technical aspect of apparent surface charge models, because it has a strong influence on both the accuracy and speed of the calculations. Before going into details, some common features will be discussed. 

Since the development of the Onsager model, there have been a number of elaborations on the model [4,5]. For example, the spherical cavity has been replaced by molecularly-shaped cavities. The state of the art within the field of solvent effects described by continuum solvent models is now implemented in, e.g., the Gaussian program package. 

There are currently three different approaches for carrying out ASC-PCM calculations [1,3]. In the original method, called dielectric D-PCM [18], the magnitude of the point charges is determined on the basis of the dielectric constant of the solvent. The second approach is C-PCM by Cossi and Barone [24], in which the surrounding medium is modelled as a conductor instead of a dielectric. The third, IEF-PCM method (Integral Equation Formalism) by Cances et al the most recently developed [16], uses a molecular-shaped cavity to define the boundary between solute and dielectric solvent. We have to mention also the COSMO method (COnductorlike Screening MOdel), a modification of the C-PCM method by Klamt and coworkers [26-28], In the latter part of the review we will restrict our discussion to the methods that actually are used to model solute-solvent interactions in NMR spectroscopy. 

The key differences between the PCM and the Onsager s model are that the PCM makes use of molecular-shaped cavities (instead of spherical cavities) and that in the PCM the solvent-solute interaction is not simply reduced to the dipole term. In addition, the PCM is a quantum mechanical approach, i.e. the solute is described by means of its electronic wavefunction. Similarly to classical approaches, the basis of the PCM approach to the local field relies on the assumption that the effective field experienced by the molecule in the cavity can be seen as the sum of a reaction field term and a cavity field term. The reaction field is connected to the response (polarization) of the dielectric to the solute charge distribution, whereas the cavity field depends on the polarization of the dielectric induced by the applied field once the cavity has been created. In the PCM, cavity field effects are accounted for by introducing the concept of effective molecular response properties, which directly describe the response of the molecular solutes to the Maxwell field in the liquid, both static E and dynamic E, [8,47,48] (see also the contribution by Cammi and Mennucci). 

For comparison, the results obtained using the Maier-Meier theory [4] are also shown this is a generalization of the Onsager model [13] to uniaxial media. The same dipole moment used for the calculations with the molecular shaped cavity was assumed, and the radius a was taken to be 3.9 A, a value derived from the density of the system. Improvement of the predictions, when the sphere is replaced by a molecular shaped... 

The molecule is often represented as a polarizable point dipole. A few attempts have been performed with finite size models, such as dielectric spheres [64], To the best of our knowledge, the first model that joined a quantum mechanical description of the molecule with a continuum description of the metal was that by Hilton and Oxtoby [72], They considered an hydrogen atom in front of a perfect conductor plate, and they calculated the static polarizability aeff to demonstrate that the effect of the image potential on aeff could not justify SERS enhancement. In recent years, PCM has been extended to systems composed of a molecule, a metal specimen and possibly a solvent or a matrix embedding the metal-molecule system in a molecularly shaped cavity [62,73-78], In particular, the molecule was treated at the Hartree-Fock, DFT or ZINDO level, while for the metal different models have been explored for SERS and luminescence calculations, metal aggregates composed of several spherical particles, characterized by the experimental frequency-dependent dielectric constant. For luminescence, the effects of the surface roughness and the nonlocal response of the metal (at the Lindhard level) for planar metal surfaces have been also explored. The calculation of static and dynamic electrostatic interactions between the molecule, the complex shaped metal body and the solvent or matrix was done by using a BEM coupled, in some versions of the model, with an IEF approach. 

The mutual polarization process between the solute and the polarizable medium is obtained by solving a system of two coupled equations, i.e., the QM Schrodinger equation for the solute in presence of the polarized dielectric, and the electrostatic Poisson equation for the dielectric medium in presence of the charge distribution (electrons and nuclei) of the solute. The solute occupies a molecular shaped cavity within the dielectric continuum, whose polarization is represented by an apparent surface charge (ASC) density spread on the cavity surface. The solute-solvent interaction is then represented by a QM operator, the solvent reaction potential operator, Va, corresponding to the electrostatic interaction of the solute electrons and nuclei with the ASC density of the solvent. 

This equation provides a way to estimate the molecular cavity volume for any system but the shape of the cavity has also to be defined. Constant coordinate cavities such as the sphere or the ellipsoid are obviously not appropriate for most systems and they have been almost definitively abandoned in favor of molecular-shaped cavities. The majority of current continuum methods, and MPE as well, use van der Waals-type molecular surfaces. Atomic radii are in general larger than standard Bondi radii so that the obtained surface is close to the so-called solvent-excluding surface [64,65], Consistent with the expression for the volume given above, the order of magnitude of atomic radii should be... 

Since Pierotti s theory was developed for solutes with spherical shape, its implementation to molecular-shaped cavities is performed by using the procedure proposed by Claverie (Eq. 4-7), [21] where the cavitation free energy of a given atom i is determined from the contribution of the isolated atom, AGcav,i, and a weighting factor, Wj, determined from the ratio between the surface of such an atom and the total surface of the sphere generated by that atom ... 

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 molecular shaped cavities, generated for example by interlockiiig 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 Figure 16.7. 

The Conductor-like Screening Model (COSMO) also uses molecular shaped cavities and represents the electrostatic potential by partial atomic charges (Klamt and Schuurmann [84]). COSMO was initially implemented for semiempirical methods but more recently was also used in conjunction to ab initio methods (Andzelm et al. [85]). 

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. 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... 

More sophisticated models employ molecular shaped cavities, but there is again no consensus on the exact procedure. The cavity is often defined based on van der Waals radii of the atoms in the molecule multiplied with an empirical scale factor. Alternatively, the molecular volume may be calculated directly from the electronic wave function, for example by using a contour surface corresponding to an electron density of 0.001. 

In constructing the practical Fock matrix two points should be clarified. One of them is the shape and size of the cavity, and the other is the description of the pM- The spatial character of an ideal cavity should be the inclusion of the whole charge distribution of the solutes and the exclusion of empty space where solvent can intrude. A molecular shaped cavity is feasible for fitting such a purpose, but expression of the interaction energy is in a complex form in general. In the case of a regularly shaped cavity, e.g., sphere, ellipsoid, and cylinder, the expression is usually given in analytical form. 

Polarizable Continuum Model (PCM) This method was developed by Tomasi s group in 1981 and many applications have been proposed [2]. The most distinctive feature of this method is to be able to treat a molecular shaped cavity. Applications not only to Hartree-Fock methods, but to UHF, MCSCF, MBPT, CASSCF, MR-SDCI and DFT etc. have been reported. They also proposed a extension to nonequilibrium solvation problems. The basic concept of their method is that the reaction potential may be described in terms of an apparent charge distribution on the cavity boundary s surface. The charge distributions a and the potential from them can be evaluated as... 

The chromophore has been treated at the Hartree-Fock or density functional theory level (see Chapter 4), in the determination of both its ground state and its properties. This model for the molecule represents a remarkable progress in the accuracy of the description of the molecular chromophore compared to polarizable point dipole model. The solvation effects have been described with the PCM. More in details, the solvent is described as a continuum dielectric which occupies all the space free from the metal specimen and the molecule is hosted in a molecular shaped cavity inside such dielectric. 

While for the exact dielectric problem only multipolar expansions are available, the COSMO approximation allows for the analytical calculation of the Green function in the case of a spherical cavity. Nevertheless, all implementations of COSMO are developed for the general case of a molecular shaped cavity, since any restriction to regular cavities such as spheres or ellipsoids is too severe a limitation. In this case the COSMO screening charges have to be calculated numerically. This requires a discretization of the cavity surface into... 

COSMO is a robust and efficient variant of the dielectric CSMs in QC and potentially even in MM calculations. Like other CSMs that use molecular-shaped cavities, COSMO gives good estimates of AGhydr for neutral and charged molecules if radii about 120% of the van der Waals (vdW)-radii are used for the cavity construction. Unfortunately the details of an optimal radii set depend on the level of theory for the solute, since the molecular potentials and even more the polarizabilities vary considerably between the different levels. A sound evaluation of an optimized set of radii for DMol/COSMO calculation has just been finished (see below). 

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