Fused spheres

Figure 10. Effect of wall-fluid and fluid-fluid attractions on the density profiles of fused-sphere chains for N = 8 and (a) pavcr3 — 0.6 and (b) pavcr3 — 1.4. ewf and ff represent dimensionless strengths of wall-fluid and fluid-fluid attractions, respectively. The symbols coorrespond to values of (ewf, ff) — (0,0) ( ), (0.2,0) ( ), (0.2,0.2) (o), (0,0.2) (a), (0,0.5) (o).

The density functional theories are also accurate for the density profiles of fused-sphere chains. Figures 4(a) and 4(b) compare the theory of Yethiraj [39] (which is a DFT with the Curtin-Ashcroft weighting function) to Monte Carlo simulations of fused-hard-sphere chains at hard walls for N = 4 and 16, respectively. For both chain lengths the theory is in quantitative agreement with the simulation results and appears to get more accurate as the chain length is increased. Similarly good results were also found by SCMC who compared... 

All of the above tests were for hard chains at surfaces. The only comparison between theory and simulation for various values of fluid-fluid and bulk fluid attractions is that done by Patra and Yethiraj (PY) [137], who presented a simple van der Waals DFT for polymers and compared to simulations of fused-sphere chains. In their theory, PY used the Yethiraj functional [39] for the hard-chain contribution to the free energy and a simple mean-field term for the attractive contribution. Their excess free energy functional is given by... 

An effective approach is to compute V(r) on an appropriately-defined molecular surface, because this is what is seen or felt by the other reactant. Such a surface is of course arbitrary, because there is no rigorous basis for it. A common procedure has been to use a set of fused spheres centered on the individual nuclei, with van der Waals or other suitable radii [34—37]. We prefer, however, to follow the suggestion of Bader et al. [38] and take the molecular surface to correspond to an outer contour of the electronic density. This has the advantage of reflecting features such... 

Figure 4.1 An illustration of a fused sphere Van der Waals surface (VDWS) of a molecule.

In particular, if the atomic radii are taken as some of the recommended values of the atomic Van der Waals radii, then one obtains a fused sphere Van der Waals surface (VDWS) of the molecule. Several different sets of atomic radii have been proposed [85-87,255], and the fused sphere molecular surface obtained depends on this choice. 

The 3D space requirements of most molecules can be represented to a good approximation by such Van der Waals surfaces. Fused sphere VDWS s are used extensively in molecular modeling, especially in the interpretation of biochemical processes and computer aided drug design. These approximate molecular surfaces are conceptually simple, their computation and graphical display on a computer screen take relatively short time, even for large biomolecules. 

In Figure 4.1 an example of a fused spheres Van der Waals surface is shown. This figure illustrates an important difference between a MIDCO and a VDWS at the seam of interpenetration of the spheres the latter surfaces are not differentiable. 

Shape Analysis of Fused Sphere Van der Waals Surfaces and Other Locally Nondifferentiable Molecular Surfaces... 

Fused sphere surfaces, such as fused sphere Van der Waals surfaces (VDWS ) are simple approximations to molecular contour surfaces. By specifying the locations of the centers and the radii of formal atomic spheres in a molecule, the fused sphere surface is fully defined as the envelope surface of the fused spheres and can be easily generated by a computer. Although fused sphere VDW surfaces are not capable of representing the fine details of molecular shape, such surfaces are very useful for an approximate shape representation. 

The sequence of seeing graphs for families of MlDCO s of the ethanol molecule has been used for shape characterization [347], and the method is equally applicable to fused sphere VDW surfaces, and to solvent accessible surfaces. 

The input data for the shape analysis methods are provided by well-established quantum chemical or empirical computational methods for the calculation of electronic charge distributions, electrostatic potentials, fused spheres Van der Waals surfaces, or protein backbones. The subsequent topological shape analysis is equally applicable to any existing molecule or to molecules which have not yet been synthesized. This is precisely where the predictive power of such shape analysis lies based on a detailed shape analysis, a prediction can be made on the expected activity of all molecules in the sequence and these methods can select the most promising candidates from a sequence of thousands of possible molecules. The actual expensive and time-consuming synthetic work and various chemical and biochemical tests of... 

Fortuitously, for most molecules, the MIDCO s G(a) of the chemically most important small density threshold values a are those where the deviations are small from the simple fused sphere model surfaces. The usual Van der Waals surfaces fall within this range. For a molecule containing N nuclei, these VDWS s are obtained as the envelope surfaces of N interpenetrating spheres... 

The close resemblance between such fused sphere models and MIDCO s is valid not only for a specified density threshold value a, such as a = 0.002, but for... 

Figure 7.1 Illustration of the principle of the Fused Spheres Guided Homotopy Method (FSGH), applied for the generation of dot representations of density scalable MIDCO surfaces for the water molecule. Three families of atomic spheres (thin lines) and their envelope surfaces (heavy lines) are shown in the upper part of the figure. In the lower part of the figure, the selected point sets on the innermost family of spheres are connected by interpolating lines to the exposed points (black dots) on the envelope surfaces of two enlarged families of spheres. Linear interpolation along the lines for two selected density values leads to two families of white dots, generating approximations of two MIDCO s (heavy lines in the lower figure).

However, this sequence of envelope surfaces of gradually enlarged fused spheres will not, in general, approximate the MIDCO s adequately for some practical applications in particular, at the seams of interpenetrating spheres this representation does not follow the corresponding MIDCO G(a) well. 

The related recent development of Density Scalable Atomic Spheres, DSAS [255] allows one to build VDWS-like fused sphere approximations of MIDCO s for any density threshold value a (with the exception of very high densities such as those in the immediate vicinity of the nuclei). The basis of this technique is a family of scaling functions [255] developed for generating radii r Ca) of formal atomic spheres along which the electronic density of atom A is any selected constant value a, within a chemically important range of electronic densities. These "density scalable" radius functions rA(a) have been determined [255] for all the atoms A commonly encountered in molecular modeling problems. 

The FSGH method (Fused Sphere Guided Homotopy method) [43]. This method has been designed for the construction of approximate, density scalable ("inflatable") isodensity contour surfaces and their dot representations (i.e., for continuous transformations between different isodensity surfaces of a given molecule). 

Density Scalable Atomic Sphere (DSAS) surfaces [255]. This technique generates radii for atomic spheres for any desired electron density at the surface. The method is used for inexpensive representations of MIDCO s of large molecules, in combination with the Fused Sphere Guided Homotopy method (FSGH) [43]. 

A PCM calculation was performed using the HF method at the DZPsp(df) level for the electrostatic and induction contributions, and the method of Amovilli and Mennucci (1997) for the dispersion and repulsion contributions. Using PCM, the solute molecule is embedded in the homogeneous dielectric solvent, with its shape described by fused spheres centered on the atomic nuclei. Tire radii of these spheres are... 

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