Location of atoms

Much of the experimental work in chemistry deals with predicting or inferring properties of objects from measurements that are only indirectly related to the properties. For example, spectroscopic methods do not provide a measure of molecular stmcture directly, but, rather, indirecdy as a result of the effect of the relative location of atoms on the electronic environment in the molecule. That is, stmctural information is inferred from frequency shifts, band intensities, and fine stmcture. Many other types of properties are also studied by this indirect observation, eg, reactivity, elasticity, and permeabiHty, for which a priori theoretical models are unknown, imperfect, or too compHcated for practical use. Also, it is often desirable to predict a property even though that property is actually measurable. Examples are predicting the performance of a mechanical part by means of nondestmctive testing (qv) methods and predicting the biological activity of a pharmaceutical before it is synthesized. 

Figure 1.4 (a) Close packing of atoms in a cubic structure, showing six in-plane neighbours for each atom (b) An expanded diagram of the packing of atoms above and below the plane. A above and A below represents the location of atoms in the hexagonal structure, and A above with B below, the face-centred cubic structure... 

The changes in reorientation of surface atoms were explained using the dynamic model of the crystal space lattice. It was assumed that during anodic polarization, when the oxidation of adsorbed water is taking place, atoms oscillate mainly in a direction perpendicular to the electrode surface. This process leads to periodic separation of atoms in the first surface layer. Thus, the location of atoms in different orientations is possible. It was stated that various techniques of electrode pretreatment used for... 

The technique of single crystal X-ray diffraction is quite powerful. In this technique an individual crystal is oriented so that each hkl plane may be examined separately. In this manner it becomes a simple matter to determine the unit cell parameters and symmetry elements associated with the crystal structure. Furthermore, it is also possible to record the intensity for each reflection from a given hkl plane and from this determine the location of atoms in the crystal, i.e. the crystal structure. While the data derived from single crystal X-ray diffraction are very valuable, the experiments are sometimes quite time consuming and so the technique is limited in its appeal as a day to day analytical tool. 

Like their contemporaries, none of the major protagonists in the early debates abont chemical stmctnre believed that it was possible (at that time, at least) to determine or dednce the physical locations of atoms in a molecnle or radical. In some ways, therefore, this clear distinction between the chemical and physical stmctnre of the molecnle may have been a way to temper criticism of the new ideas by their more conservative contemporaries, and thus render the concept more palatable to a wider andience of chemists. 

Location of atoms in relation to symmetry elements. Equiva-... 

Because the Patterson function contains no phases, it can be computed from any raw set of crystallographic data, but what does it tell us A contour map of p(x,y,z) displays areas of high density (peaks) at the locations of atoms. In contrast, a Patterson map, which is a contour map of P(u,v,w), displays peaks at locations corresponding to vectors between atoms. (This is a strange idea at first, but the following example will make it clearer.) Of course, there are more vectors between atoms than there are atoms, so a Patterson map is more complicated than an electron-density map. But if the structure is simple, like that of one or a few heavy atoms in the unit cell, the Patterson map may be simple enough to allow us to locate the atom(s). You can see now that the... 

If the phasing model is not isomorphous with the desired structure, the problem is more difficult. The phases of atomic structure factors, and hence of molecular structure factors, depend upon the location of atoms in the unit cell. In order to use a known protein as a phasing model, we must superimpose the structure of the model on the structure of the new protein in its unit cell and then calculate phases for the properly oriented model. In other words, we must find the position and orientation of the phasing model in the new unit cell that would give phases most like those of the new protein. Then we can calculate the structure factors of the properly positioned model and use the phases of these computed structure factors as initial estimates of the desired phases. 

Figure 1.10 Structural formulas of four hydrocarbons, each containing eight carbon atoms, that illustrate the structural diversity possible with organic compounds. Numbers used to denote locations of atoms for purposes of naming are shown on two of the compounds.

The most powerful technique for determining the structure of a chemical compound is x-ray crystallography. In this technique, a beam of x rays is focused on a crystal of a compound. The diffraction pattern produced enables chemists to determine the location of atoms within the crystals and hence deduce the molecular structure. It was Dorothy Hodgkin who pushed the limits of the technique to determine the structures of some biologically important molecules, including penicillin, vitamin B12, and insulin. 

The coordinates of atomic positions can be located and a molecular model derived, either by contouring the map, or by use of a peak search routine.. An example is provided in Figure 9.6. In the most favorable cases this map will also give an indication of the locations of atoms that were not included in the phasing model. Such a map can also be used to improve the accuracy of a preliminary model by adjusting the model to best fit the electron density. 

An increased precision in the location of atoms in the crystal lattice has shown that some irregularity in their position is very common. This in turn has led to an understanding of the way in which a crystal grows and of the changes brought about in a crystal lattice when it is bombarded by particles from without or suffers atomic disintegration within. It will be seen below that these may be matters of very practical concern. 

Now let us consider the formation of a molecule. For convenience we shall picture this as happening by the coming together of the individual atoms, although most molecules are not actually made this way. We make physical models of molecules out of wooden or plastic balls that represent the various atoms the location of holes or snap fasteners tells us how to put them together. In the same way, we shall make mental models of molecules out of mental atoms the location of atomic orbitals—some of them imaginary—will tell us how to put these together. 

All of these models can be used to show the relative locations of atoms and electrons in the phosphorus trihydride (phosphine) molecule. 

For simplicity, assume that the distribution of electron or nuclear density in the unit cell is discrete rather than continuous and is zero everywhere except for the locations of atoms, viewed as dimensionless points (see Figure 2.59, which illustrates that both electron and nuclear density decreases rapidly away from the centers of atoms). Then, the result of Eq. 2.138 is a set of peaks originating in (0, 0, 0) and ending at (m, v, w) with heights (more precisely with peak volumes since atoms are not dimensionless points), Hy, given as... 

Figure 2.61. Patterson functions calculated in the uOw plane using Eq. 2.136 and employing experimental x-ray (left) and neutron (right) powder diffraction data shown in Figure 2.58. The strongest peak in any Patterson function is always observed at (0,0, 0) beeause the origins of all vectors coincide with the origin of coordinates. Since in this particular example the real crystal structure contains an atom in (0, 0,0, see Figure 2.59), some of the peaks on the Patterson map correspond to the actual locations of atoms (i.e. = x- 0).The contour of...

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