Defining Spheres

Figure 2-120. The center ofthe rolling probe sphere defines the solvent-accessible surface during movement of the probe over the van der Waals surface. Thus, the molecular surface is expanded by the radius of the solvent molecule,...

Figure 13.2 A typical Catalyst pharmacophore, where different colors indicate different chemical features and the spheres define tolerance spaces that each chemical feature would be allowed to occupy. See color plate.

Figure 4.8 The powder difTraction experiment, (a) Reciprocal space notation. The Ewald sphere is fixed, and the lattice is rotated about all angles about the origin. Only the rotations about [100] are shown in this two-dimensional section. Intersections with the Ewald sphere define the diffracting conditions, (b) The corresponding diffracted beams in real space...

The set of angles in Fig. 1(b) parametrizing the 3-sphere defines as follows a point on the 3-dimensional sphere [8] ... 

Thus, on the surface of the sphere defined by Eq. (C.l), r = m1v/3 = ca. The probability Pc that the atom is within the sphere is given by the integral over the isotropic Gaussian distribution ... 

Another convention expresses the electron density in terms of an effective radius such tliat exactly one electron would be contained within the sphere defined by diat radius were it to have the same density throughout as its center, i.e.. 

Diffusion into a sphere represents a three-dimensional situation thus we have to use the three-dimensional version of Fick s second law (Box 18.3, Eq. 1). However, as mentioned before, by replacing the Cartesian coordinates x,y,z by spherical coordinates the situation becomes one-dimensional again. Eq. 3 of Box 18.3 represents one special solution to a spherically symmetric diffusion provided that the diffusion coefficient is constant and does not depend on the direction along which diffusion takes place (isotropic diffusion). Note that diffusion into solids is not always isotropic, chiefly due to layering within the solid medium. The boundary conditions of the problem posed in Fig. 18.6 requires that C is held constant on the surface of the sphere defined by the radius ra. 

Figure 12.10. The xy plane (9 = nil) of the unit sphere. The section of the positive half-sphere defined by eq. (12.8.13) is shown by the shaded regions (which include the tails of the curved arrows but not their heads). Poles of the proper rotations C2a, C2 b, and C2care shown by filled digons and the poles of the improper rotations /C2mj m = d, e, f, are indicated by unfilled digons.

Differences resulting from nonisotropic electron distribution are significant only for H-X bond lengths X-rays see electrons rather than nuclei, and the simplest inference of a nuclear position is to place it at the center of a sphere whose surface is defined by the electron density around it. However, since a hydrogen atom has only one electron, for a bonded hydrogen there is relatively little electron density left over from covalent sharing to blanket the nucleus, and so the proton, unlike other nuclei, is not essentially at the center of an approximate sphere defined by its surrounding electron density ... 

Radius of sphere defining the effective atomic region, electron density being integrated in this sphere. 

Vi = Y so that Ytm is the value of the solid harmonic at points on the surface of the unit sphere defined by the coordinates 8 and (p, and hence Y is called a surface harmonic of degree l. Surface harmonics are orthogonal on the surface of the unit sphere and not at r = 0, as commonly assumed in the definition of atomic orbitals. 

The most general solution to the wave equation of a spherically confined particle is the Fourier transform of this Bessel function, i.e. the box function defined by ro- Such a wave function, which terminates at the ionization radius, has a uniform amplitude throughout the sphere, defined before (3.36)... 

In the same way that electronegativities determine the polarity of diatomic interactions, ionization radii should define the effective electronic charge clouds that interpenetrate to form diatomic molecules, as shown schematically in Figure 5.3. The overlap of two such spheres defines a lens of focal lengths fixed by the ionization radii, r and r2, at an interatomic distance d = x i + x2-... 

An electron in the valence state is confined to a sphere, defined by the ionization radius of the atom, and with electronic charge uniformly distributed. Such a charge density is correctly described by a wave function of constant amplitude within the sphere, and vanishing outside. The only parameter that differentiates between atoms of different type is the characteristic ionization radius, which is also a measure of the classical atomic property of electronegativity. 

Figure 4. Example of SPHGEN output. The spheres define the negative space for the...

Exo- and endo- define cluster regions lying outside and inside respectively, the sphere defined by the cluster core atoms... 

Thus we can describe the free hydrogen atom as having a heavy nucleus at the center of a mushy sphere defined by the space filled by the fasGmoving electron in its motion about the nucleus. This mushy sphere is about 2 A in diameter. 

Figure 8.1. The diffusion along IPMS by mobile ions in binary solid electrolytes, (a) Partition of the ln)ii]/-cfntrcd cubic (bcc) lattice into two primitwe cubic sub-lattices, separated by the P-surface. Average positions of the (mobile) silver cations, detected from X-ray measurements, in the solid electrolyte a-AgI are indicated by the smaller red spheres on the surface. The larger dark red spheres define the bcc lattice of the (frozen) iodine ions (only two occupied sites of the bcc array are shown), (b ) The bcc lattice can also be partitioned into two diamond sub-lattices by the D-surface. The curved net on the surface described trajectories of mobile Ag ions on the D-surface vertices of the net locate the average I" positions.

A set of thirty different descriptors [Stanton and Jurs, 1990] which combine shape and electronic information to characterize molecules and therefore encode features responsible for polar interactions between molecules. The molecule representation used for deriving CPSA descriptors views molecule atoms as hard spheres defined by the - van der Waals radius. The - solvent-accessible surface area SASA is used as the molecular surface area it is calculated using a sphere with a radius of 1.5 A to approximate the contact surface formed when a water molecule interacts with the considered molecule. Moreover, the contact surface where polar interactions can take place is characterized by a specific electronic distribution obtained by mapping atomic partial charges on the solvent-accessible surface. 

The method is based on the use of a - probe given by the excluded volume of two spheres with different radii and an identical centre corresponding to the barycentre of the molecule. The probe is layered like an onion, each layer being the excluded voliune. The first sphere, i.e. the component of the probe, is an atom (e.g. an iodine atom with van der Waals radius of 2.05 A, a carbon atom with van der Waals radius of 1.52 A, or a hydrogen atom with van der Waals radius of 1.08 A) whose volume defines the first layer, then 60 atoms of the same type (e.g. iodine atoms) construct the second sphere whose surface is like a fullerene and which shares the same centre as the first sphere. The excluded volume between the first and second spheres defines the second layer. In the same way the subsequent spheres and layers are also defined. 

It is worth noting that the radii of the spheres defining the cavity for the calculation of Gdis are not necessarily the same used for Ge (Bonacccorsi et al., 1990). 

The direct minimization procedure has been also used by Wang and Ford (1992) in an MNDO and AMI version of the program. These authors introduce an alternative partition of the spheres defining the cavity, which still starts from the pentakisdodecahedron. The program has been used to study SnI, and Sn 2 reactions as well as tautomeric equilibria (Ford and Wang, 1992). 

In the molten mixture, the Me + and M+ cations are in the first coordination sphere again surrounded by common X anions but have M+ and/or Me + cations in their second coordination sphere. Mixtures dilute in MeX2 have an excess of M+ cations. The divalent Me " " cations have higher field sfrengfh in comparison wifh fhe M+ cation. They fhus allracf more X anions in order fo supplemenf their coordination sphere to the maximum. This effort is compensated by the presence of M+ cations in the second coordination sphere. On the other hand, the probability of having two or more Me + cations in the second coordination sphere of M+ is small. Thus the first and second coordination spheres define the shape of the complex anion. 

For the charged atoms (Fig. 50), all of the distribution functions show features typical of charged-group solvation.112,113 There are four to five solvent molecules within the first solvation sphere, defined as extending to the first minimum ing(r) that is at - 3.5 A. This is indicative of tightly bound solvent around the charged group. 

The Gurney co-sphere defines a region around the ion which has solvent molecules whose structure has been modified by the field of the ion (see Section 10.16). Outside this region, the solvent has its macroscopic bulk structure. The diameter of the Gumey cosphere takes a value, R. The distance between the centre of ions with such co-spheres where the co-spheres just touch is also R (see Figure 12.4). This assumes that the ions are spherical, and the situation could well prove to be different if the ions are non-spherical. In addition, ion association was taken to be an integral part of the model and theory rather than an added-on factor used to explain deviations from a conductance equation based on the concept of complete dissociation. All ion association takes place within the Gurney CO-sphere. 

Position inferred froni supposing it to be at center of sphere defined by electron density around proton,... 

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