Interlocking spheres

Over the years, many workers have addressed the problem of choice of cavity and the reaction field. Tomasi s polarized continuum model (PCM) defines the cavity as a series of interlocking spheres. The isodensity PCM (IPCM) defines the cavity as an isodensity surface of the molecule. This isodensity surface is determined iteratively. The self-consistent isodensity polarized continuum model (SQ-PCM) gives a further refinement in that it allows for a full coupling between the cavity shape and the electron density. 

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. 

The most common way to define molecular cavities, however, is to use a set of interlocking spheres centred on the atoms constituting the molecular solute (Figure 1.2). Based on such a definition of the cavity, we can define different molecular surfaces ... 

Figure 1.2 Definitions of cavities based on interlocking spheres. In black (dashed) the spheres centred on atoms A and B, in red the SAS, in cyan the shared parts of VWS and SES. In green the concave part of SES. In blue the crevice part of VWS. In black (dotted) some positions of tangent solvent probes (see Colour Plate section).

Cavities based on interlocking spheres allow a simple and accurate calculation of tessellation elements, thanks to weight function methods. A question not solved yet is... 

How is the cavity defined Most methods now use some variation on the interlocking spheres approach, but the size of these spheres is very much method dependent. 

The conductor-like screening model (COSMO) approach replaces the dielectric medium with a conducting medium (basically a medium that effectively has an infinite dielectric constant). Interlocking spheres are used to generate the cavity. The conductor-like screening has been implemented as a PCM version, called CPCM.128,129... 

Enzymatic reactions will be considered in another chapter of this book. We shall only remark that continuum ASC models, and in particular the PCM, may be of some help in studying reactions involving large molecules. In the set of approximations we have considered no additional changes in the PCM formalism were introduced. In particular, the same definition of cavity in terms of interlocking spheres, its partition in tesserae resulting from the inscription of a pentakisdodecahedron in each sphere, and the self-polarization of apparent charges, have been used. 

A very interesting semiempirical expression for 7 / ofeq.(104) has been suggested by Still et al. (1990) for a cavity of molecular shape (interlocking spheres). Their expression contains several parameters to be determined, describing, inter alia, theradii afc, and the portion ofthe exposed area for each sphere, when the others are placed at the desired positions. 

Solvent effects were evaluated by different self consistent reaction field (SCRF) procedures i) the parametrisation AM1-SM4 for cyclohexane and AM1-SM2.1 for water, implemented on the AMSOL-V suite of programs,ii) the ab initio Pisa model (interlocking spheres) implemented in the Gamess (Rev,97) package and, with different options, in the Gaussian 94 package. 

By interlocking spheres centered on each nuclei and with the radius defined in terms of the respective van der Waals radii (Bondi [82]) multiplied by a factor 1.2, the van der Waals surface of the solute is constructed. The SE cavity is then defined as the contact surface of a probe sphere (with radius equal to the molecular radius of the solvent molecule) rolling on the solute van der Waals surface. 

One may use different types of molecular cavities and surfaces definitions (e.g. equipotential surfaces, equidensity surfaces, van der Waals surfaces). Among them there is a subset that shares a common trait they consider that a molecule may be represented as a set of rigid interlocking spheres. There are three such surfaces a)... 

The simple virtual charge model discussed by Constanciel and Tapia [6] has been developed into an extended generalized Born (EGB) approach. Different approximations have been proposed. Constanciel [40] has analyzed the theoretical basis used as foundations for empirical reaction field approximations through the continuum model to the surrounding medium. Artifacts in the EGB scheme have been clearly identified. The new approximate formulation proposed derives from an exact integral equation of classical electrostatics following a well defined procedure. It is shown there how the wavefunction of solvated species imbedded in cavities formed by interlocking sphere in a polarizable continuum can be computed. 

The cavity is a constitutive component in all the continuum methods [2, 3]. In the PCM model the cavity may be accurately modeled on the shape of the molecular solute M. The basic PCM cavity is defined as a set of interlocking spheres centered on the nuclei of atoms of M, with radii related to the corresponding atomic van der Waals radii. Actually, the PCM model uses several variants of the basic cavity, which may introduce additional spheres [4, 5], to take into account of the portions of space not occupied by the charge distribution of molecular solute but non accessible or excluded to the solvent molecules. 

In this thesis, the models that have been used for all the calculations are the widely known polarizable continuum model (PCM) [79], and the recently developed SMD model [80]. These two models define the cavity for the solute as the union of a series of interlocking spheres centered on the atoms and differ only in that the latter includes the radii and non-electrostatic terms as suggested by Truhlar and coworkers. Other variants of the PCM model are, for example, the Isoelectronic-PCM (IPCM), which uses a static isodensity surface for the cavity, and its improved version self-consistent isodensity-PCM (SCI-PCM) [81]. 

Figure 5.5 Radiative decay rate (in calculated for a coumarin-type molecule near a silver aggregate composed of three identical interlocking spheres placed in a linear configuration. Results are depicted by using a color scale. The white squares indicate the molecule positions for which calculations have been performed (Cartesian axes are in A). The molecular transition dipole moment is directed perpendicular to the metal surface. Details on model can be found in the original work. Reprinted with permission from Ref. [54]. Copyright [2004], American Institute of Physics.

The PCM and COSMO approaches use cavities of interlocking spheres to represent the solvent accessible surface of a given solute. PCM has been incorporated into several quantum chemistry packages, e.g., in GAUSSIAN94, GAMESS-UK,... 

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