Sphere space filling

One of the conceptually simplest representations of a molecule, also used in the construction of hard-sphere space filling models, is obtained by centring spheres of suitable Van der Waals radii at the positions of the atomic nuclei. To convey a feeling of size the aggregate is enclosed in a generating ellipsoid that circumscribes the atom most remote from the centre to define the volume of the Van der Waals body [212]. 

The van der Waals surface (or the hard sphere model, also known as the scale model or the corresponding space-filling model) is the simplest representation of a molecular surface. It can be determined from the van dcr Waals radii of all... 

The space filling model developed by Corey, Pauling, and Koltun is also known as the CPK model, or scale model [197], It shows the relative volume (size) of different elements or of different parts of a molecule (Figure 2-123d). The model is based on spheres that represent the "electron cloud . These atomic spheres can be determined from the van der Waals radii (see Section 2.10.1), which indicate the most stable distance between two atoms (non-bonded nuclei). Since the spheres are all drawn to the same scale, the relative size of the overlapping electron clouds of the atoms becomes evident. The connectivities between atoms, the bonds, are not visualized because they are located beneath the atom spheres and are not visible in a non-transparent display (see Section 2.10). In contrast to other models, the CPK model makes it possible to visualize a first impression of the extent of a molecule. 

FIGURE 1 6 Molecular models of methane (CH4) (a) Framework (tube) models show the bonds connecting the atoms but not the atoms themselves (b) Ball and stick (ball and spoke) models show the atoms as balls and the bonds as rods (c) Space filling models portray overall molecular size the radius of each sphere approximates the van der Waals radius of the atom (d) An electrostatic potential map of methane... 

The ball and wire display is used for model building Although it is convenient for this purpose other model displays show three dimensional molecular structure more clearly and may be preferred The space filling display is unique m that it portrays a molecule as a set of atom centered spheres The individual sphere radii are taken from experi mental data and roughly correspond to the size of atomic electron clouds Thus the space filling display attempts to show how much space a molecule takes up... 

All pictorial representations of molecules are simplified versions of our current model of real molecules, which are quantum mechanical, probabilistic collections of atoms as both particles and waves. These are difficult to illustrate. Therefore we use different types of simplified representations, including space-filling models ball-and-stick models, where atoms are spheres and bonds are sticks and models that illustrate surface properties. The most detailed representation is the ball-and-stick model. However, a model of a protein structure where all atoms are displayed is confusing because of the sheer amount of information present (Figure 2.9a). 

The most novel model is the Space-Filling model. No bonds are shown. Rather, each atom is displayed as a colored sphere that represents the atom s approximate size, and the complete model indicates the molecule s... 

Space-Filling Models. For most of this century, chemists have tried to answer the size question by using a special set of molecular models known as space-filling or CPK models. The space-filling model of an atom is simply a sphere of fixed radius. A different radius is used for each element, and the radii are chosen to reproduce certain experimental observations, such as the compressibility of a gas, or the spacing between atoms in a crystal. 

The next most important aspect of a molecular compound is its shape. The pictorial representations of molecules that most accurately show their shapes are images based on computation or software that represents atoms by spheres of various sizes. An example is the space-filling model of an ethanol molecule shown in Fig. C.2a. The atoms are represented by colored spheres (they are not the actual colors of the atoms) that fit into one another. Another representation of the same molecule, called a ball-and-stick model, is shown in Fig. C.2b. Each ball represents the location of an atom, and the sticks represent the bonds. Although this kind of model does not represent the actual molecular shape as well as a space-filling model does, it shows bond lengths and angles more clearly. It is also easier to draw and interpret. 

If atoms are considered as hard spheres, the packing density can be expressed by the space filling SF of the spheres. It is ... 

The space filling in the body-centered cubic packing of spheres is less than in the closest packings, but the difference is moderate. The fraction of space filled amounts to ns/3 = 0.6802 or 68.02 %. The reduction of the coordination number from 12 to 8 seems to be more serious however, the difference is actually not so serious because in addition to the 8 directly adjacent spheres every sphere has 6 further neighbors that are only 15.5 % more distant (Fig. 14.3). The coordination number can be designated by 8 + 6. 

Corresponding to its inferior space filling, the body-centered cubic packing of spheres is less frequent among the element structures. None the less, 15 elements crystallize with this structure. As tungsten is one of them, the term tungsten type is sometimes used for this kind of packing. 

Although the space filling of the body-centered cubic sphere packing is somewhat inferior to that of a closest-packing, the CsCl type thus turns out to be excellently suited for compounds with a 1 1 composition. Due to the occupation of the positions 0,0,0 and with different kinds of atoms, the structure is not... 

The space filling in the MgCu2 type can be calculated with the aid of equation (14.1) (p. 150) the geometric relations follow from the bottom image in Fig. 15.4 the four Cu spheres form a row along the diagonal of length a fl, therefore r(Cu) = s/2a two Mg... 

To a first approximation, atoms in molecules may be regarded as hard spheres with a segment cut off in the bonding direction, as in the familiar space-filling models. The radius of the atom in a nonbonding direction is called the van der Waals radius. Half the distance between two atoms of the same kind in adjacent molecules at equilibrium is taken as the van der Waals radius (Figure 5.1). In assigning a fixed radius in this way, we assume that atoms... 

FIGURE 7.15 A space-filling model of spheres arranged in the simple cubic pattern. 

Figure 11.3 Arrangement of atoms in an ionic solid such as NaCl. (a) shows a cubic lattice with alternating Na+ and Cl- ions, (b) is a space-filling model of the same structure, in which the small spheres are Na+ ions, the larger Cl-. The structure is described as two interlocking face-centred cubic lattices of sodium and chlorine ions. 

Space-filling parameter (and curves). The space-filling parameter introduced by Laves (1956) and by Parthe (1961) is an index which may be useful in studying the relationships between atomic dimensions and structure. For a compound it is defined by the ratio between the volume of atoms, assumed as spheres of well-defined radii in a unit cell, and the volume of the unit cell itself. 

To calculate the space-filling value for a specific compound, one has to know the radii of the atoms and the lattice constant. Neither of these is needed for the construction of a space-filling curve of a crystal structure type it is sufficient to know the point positions of the atoms and the axial ratios. The curve is based on a hard sphere... 

Reduced dimensional parameters (strain parameters and near-neighbours diagrams) By comparing the space-filling theoretical curves and the actual values of intermetallic phases it has been observed that an incompressible sphere model of the atom gives only a rough description when discussing metallic structures. 

FlC. 64. The tightly associated domains (one shown light and the other dark) of elastase. Figures 64 through 66 use a space-filling representation with a sphere around each a-carbon position they were photographed from Richard Feldmann s molecular graphics display at the National Institutes of Health. 

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