Interatomic vectors distribution

The resultant function, unfortunately, does not reveal the distribution of atoms in the unit cell directly but it represents the distribution of interatomic vectors, all of which begin in a common point - the origin of the unit cell. Thus, Puvw is often called the function of interatomic vectors and it is also known as the Patterson function of the F -Fourier series. The corresponding vector density distribution in the unit cell is known as the Patterson map. 

Thus, since a Patterson map contains peaks, which are related to the real distribution of atoms in the unit cell, it is possible to establish both the coordinates of atoms and their scattering power by analyzing coordinates and heights of Patterson peaks. Unfortunately, the analysis of the distribution of interatomic vector density fimction is sometimes easier said than done due to the presence of several complicating factors. 

An example of the distribution of the interatomic vectors density function in the uOw plane of CeRhGea is illustrated in Figure 2.61. When compared with the electron and nuclear density distributions Figure 2.59), there are many more peaks in the two Patterson maps. Similar to the results shown in Figure 2.59, both Patterson functions are nearly identical except for the distribution of peak intensities, which is expected due to the differences in the scattering ability of Ce, Rh and Ge using x-rays and neutrons. 

Table 6.11. The three-dimensional distribution of the interatomic vectors (the Patterson function) in the symmetrically independent part of the unit cell of CeRhGcs calculated using...

Patterson method has a high success rate when applied to intermetallic structures mainly because of the high resolution of the resultant three-dimensional distribution of interatomic vectors. The latter is due to noticeably greater minimal interatomic distances and lower atomic displacement parameters when compared to other classes of compounds. 

The mathematical function that must be interpreted in order to deduce the heavy atom coordinates, a puzzle really, is called a Patterson function or Patterson synthesis (Patterson, 1935). It has a form similar to the equation for electron density except that all phases are effectively zero. It yields, also in a similar manner, a three-dimensional density distribution. The peaks in this map, however, do not correspond to electron density centers but mark the interatomic vectors relating those centers. 

If the real unit cell contains only a very few atoms, as is often the case with an ionic crystal or salt, the Patterson map, calculated by including its diffraction intensities as coefficients in the Patterson synthesis, may be treated as a puzzle. The object is to contrive a distribution of atoms whose interatomic vectors yield the highest peaks in the Patterson map. This direct approach in fact provided the means by which many of the first small molecules and simple ionic crystals were solved. It is not practical for larger, more complicated structures. 

Hauptman, H., and Karle, J. 1952. Crystal structure determination by means of a statistical distribution of interatomic vectors. Acta Cryst. 5 48. 

MD methods allow us to follow the coordinates / , and velocities Vj of all atoms throughout the simulations, and insight into the local order can be found from the distributions of the bond (0 ) and dihedral angles iviju)- The pair distribution function (PDF) g r) is a spherically averaged distribution of interatomic vectors. 

The discovery of confinement resonances in the photoelectron angular distribution parameters from encaged atoms may shed light [36] on the origin of anomalously high values of the nondipole asymmetry parameters observed in diatomic molecules [62]. Following [36], consider photoionization of an inner subshell of the atom A in a diatomic molecule AB in the gas phase, i.e., with random orientation of the molecular axis relative to the polarization vector of the radiation. The atom B remains neutral in this process and is arbitrarily located on the sphere with its center at the nucleus of the atom A with radius equal to the interatomic distance in this molecule. To the lowest order, the effect of the atom B on the photoionization parameters can be approximated by the introduction of a spherically symmetric potential that represents the atom B smeared over... 

Table 3. Classes of interatomic lengths of vectors, central atom-nearest ligand atom with maximum frequencies and dispersions of empirical distributions...

The Patterson function is a map that indicates all the possible relationships (vectors) between atoms in a crystal structure. It was introduced by A. Lindo Patterson " in 1934, inspired by earlier work on radial distribution functions in liquids and powders. In crystals the directionality as well as the lengths of vectors between atoms (atomic distances) can be deduced. By contrast, in liquids and powders the geometric information that can be obtained is limited to interatomic distances, because in these the molecules are randomly oriented. While the use of the Patterson function revolutionized the determination of crystal structures of small molecules in the 1930s to 1950s, direct methods are now the most widely used methods for obtaining structures of small organic molecules. The Patterson function, however, continues to play an essential part in the determination of crystal structures of inorganic compounds and macromolecules. It is also very useful when the structure of a small molecule proves difficult to solve by direct methods. 

Figure 4 displays the electron density distribution in the molecular plane of anthracene and contrasts it with that of phenanthrene along with the associated gradient vector field of the latter. In this figure, one can see the curved bond path linking the nuclei of the two hydrogen atoms, H4 and H5, in phenanthrene as well as the zero-flux interatomic surface they share, features lacking in the map of anthracene. 

Moreover, the RDF vectorial descriptor is interpretable by using simple rules and, thus, it provides a possibility of —> reversible decoding. Besides information about distribution of interatomic distances in the entire molecule, the RDF vector provides further valuable information for example, about bond distances, ring types, planar and nonplanar systems, and atom types. This fact is a most valuable consideration for a computer-assisted code elucidation. 

The strength of the water-metal interaction together with the surface corrugation gives rise to much more drastic changes in water structure than the ones observed in computer simulations of water near smooth nonmetallic surfaces. Structure in the liquid state is usually characterized by pair correlation functions (PCFs). Because of the homogeneity and isotropy of the bulk liquid phase, they become simple radial distribution functions (RDFs), which do only depend on the distance between two atoms. Near an interface, the PCF depends not only on the interatomic distance but also on the position of, say the first, atom relative to the interface and the direction of the interatomic distance vector. Hence, considerable changes in the atom-atom PCFs can be expected close to the surface. 

---