Intersecting spheres

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 line which intersects a circle at only one point is a tangent to the circle at that point. Every tangent is perpendicular to the radius drawn to the point of intersection. Spheres may have tangent lines or tangent planes. 

Intersecting spheres, radii a and 2, with orthogonal intersection... 

Table 1. Analysis of U(VI) coordination in some U compounds using the method of intersecting spheres...

In applying 7(r) and V(r) to molecular reactivity, we have found it expedient to compute them on an outer surface of the molecule, since this is what is encountered by an approaching reactant. Molecular surfaces have often been defined by means of intersecting spheres centred on the nuclei [75-78], having perhaps the respective van der Waals radii. We prefer to again follow Bader et al. [69] and use the 0.001 au contour of p(r). This has the advantage that it reflects features such as lone pairs and strained bonds, which are specific to the molecular charge distribution. The notations 7s(r) and V s(r) indicate that the properties have been calculated on the molecular surface. 

As stated earlier in this chapter, we feel that the most effective use of the electrostatic potential in relation to reactive behavior is in analyzing and predicting non-covalent interactions, with V(r) being computed on an outer surface of the molecule. This poses the question of how the latter should be defined, because there is no rigorous basis for it. A common approach has been to use a set of intersecting spheres centered on the individual nuclei, with van der Waals or other appropriate radii... 

Starting from a 3D placement, we can also calculate an approximate volume, the van der Waals volume. Each atom is replaced by a sphere of van der Waals radius centered at the atom s coordinates. Details for calculating the volume of intersecting spheres follow in Example 7.4. These volumes are shown in Figure 2.3. Again,... 

Laves P (1967) Space limitation on the geometry of the crystal stmctures of metals and intermetallic compounds. Phase transition in metals alloys. In Rudman PS, Stringer J, Jaffee RI (eds) Phase stability in metals and alloys. McGraw-HiU, New York Serezhkin VH, Mikhailov YuN, Buslaev YuA (1997) The method of intersecting spheres for determination of coordination numbers of atoms in crystal structures. Russ J Inotg Chem 42 1871-1910... 

In addition to the present shape parameterization [43], other two-center shape param-eterizations (mainly in connection with KS-LDA jellium calculations) have been used [81-83] in studies of metal-cluster fission. They can be grouped into two categories, namely, the two-intersected-spheres jellium [81, 84], and the variable-necking-in parame-terizations [82, 83]. In the latter group. Ref. [82] accounts for various necking-in situations by using the funny-hills parameterization [85], while Ref. [83] describes the necking-in... 

The two-intersected-spheres jellium has also been used for describing the fusion of two neutral magic clusters (see Ref. [86]). 

To calculate the Van der Waals and the occupied volume of die repeating unit, we used the quantum chemical method AMI to refine the structure of the monomer unit [20]. The model of the repeating unit is a set of intersecting spheres whose coordinates of centers coincide with the eoordi-nates of atoms and the radii are equal to the Van der Waals radii of the eorresponding atoms, as shown in Figure 3.2. 

A visualization tool, which gives some impression of the volume and the outer shape of a molecule, is the space-filling model of Corey, Pauling, and Koltun (CPK-model). > Herein, atoms are represented as intersecting spheres with radii roughly... 

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