The concept of coordination number

In the description of crystal coordination formulae a coordination number was introduced, defined as the NNN of atoms X around the atom Y under consideration. 

More generally, in many cases of intermetallic compounds, unlike a high number of covalent compounds (compare for instance with the illustrative example of a carbon atom in the diamond structure), we cannot speak of bonds of an atom directed to (and saturated with) a well-defined group of atoms. 

As a consequence the coordination number (or ligancy) of a central atom which is easily obtained by enumerating the neighbours (and which, of course, has an important role in the description of the crystal structure and of its geometrical characteristics), very often is not clearly and immediately related to the bonding mechanism of the intermetallic phase. In other words, there are many cases in which the enumeration procedure alone gives only a very partial representation of the bonding. Moreover, there are numerous cases where the criteria for the enumeration 

As regards the second method of coordination evaluation, several schemes for the calculation of an effective coordination have been proposed. 

According to the Frank-Kasper definition, the coordination number is unambiguously 12 in the hexagonal close-packed metals and assumes the value 14 in a body-centred cubic metal. Generally in several complex metallic structures this definition yields reasonable values such as 14, even when the nearest-neighbour definition would give 1 or 2. 

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F. L. Carter, Quantifying the concept of coordination number. Acta Crystallogr. B 34 (1978) 2962. 

Alfred Werner was born in 1866 and died from arteriosclerosis in 1919, He started as an organic chemist and finished his chemical career in 1915 as one of the foremost inorganic chemists,. He won the Nobel Prize in 1913, During a period of two and a half decades he published 174 papers and supervised the work covered by 200 doctoral dissertations, Werner was the founder of coordination chemistry. He rejected the then prevailing concept formulated by Kekule) that the valence of an element is invariable and introduced instead the notion of principal and auxiliary valence. He also formulated the concept of coordination number, Werner used both the inductive and the deductive methods of reasoning. Most of his predictions on geometrical and optical isomerism were verified by experiment. 

The quantity Ncjh(Rm) may be referred to as the coordination number of particles, computed for the particular sphere of radius R. A choice of ocoordination number that conforms to the common usage of this concept. There exist other methods of defining the concept of coordination number, which are summarized and discussed by Pings (1968). 

The temperature dependence of the coordination number of water is also in sharp contrast to that of a simple fluid. According to computations made by Samoilov (1957), the coordination number of water increases slightly from 4.4 at 4°C to about 4.9 at 83°C. Whereas the coordination number of argon decreases substantially with rising temperature, Narten and Levy (1972), using more recent data on water, concluded that the coordination number of water is almost constant with temperature. [Note that differences in the definition of the concept of coordination number may exist. For a survey of the various definitions, the reader is referred to Pings (1968).]... 

Beck HP (1981) High-pressure polymorphism of Bal2- ZNaturforsch 36b 1255-1260 Carter EL (1978) Quantifying the concept of coordination number Acta Cryst B34 2962-2966 Batstmov SS (1983) On the meaning and calculation techniques of eflective coordination numbers. Kooidinatsionnaya Khimiya 9 867 (in Russian)... 

It should be pointed out here that the concept of coordination number is not a very clear one and various approaches to determine it have recently been proposed (Batsanov, 1977 Carter, 1978 O Keeffe, 1979). Jorgensen (1983, 1984a,b) has also discussed CN in relation to chemical bonding. 

Ho et al. (1990) have presented an approach to the description of independent kinetics that makes use of the method of coordinate transformation (Chou and Ho 1988), and which appears to overcome the paradox discussed in the previous paragraph. An alternate way of disposing of the difficulties associated with independent kinetics is intrinsic in the two-label formalism introduced by Aris (1989, 1991b), which has some more than purely formal basis (Prasad et al, 1986). The method of coordinate transformation can (perhaps in general) be reduced to the double-label formalism (Aris and Astarita, 1989a). Finally, the coordinate transformation approach is related to the concept of a number density function s(x), which is discussed in Section IV,B.5. 

Although examples exist, coordination numbers of 10 and above are relatively rare. Further, it seems that the concept of coordination geometry becomes less applicable. The reason is that, whilst idealized geometries can be identified, most real structures show distortions and there may be some arbitrariness about which of the ideal structures the distorted structure is derived from. Examples of idealised coordination geometries are given in Figs. 3.22 (coordination number 10), 3.23 (coordination number 11) and 3.24 (coordination number 12). The captions to these figures describe the construction of the polyhedra. 

In spite of the absence of periodicity, glasses exhibit, among other things, a specific volume, interatomic distances, coordination number, and local elastic modulus comparable to those of crystals. Therefore it has been considered natural to consider amorphous lattices as nearly periodic with the disorder treated as a perturbation, oftentimes in the form of defects, so such a study is not futile. This is indeed a sensible approach, as even the crystals themselves are rarely perfect, and many of their useful mechanical and other properties are determined by the existence and mobility of some sort of defects as well as by interaction between those defects. Nevertheless, a number of low-temperamre phenomena in glasses have persistently evaded a microscopic model-free description along those lines. A more radical revision of the concept of an elementary excitation on top of a unique ground state is necessary [3-5]. 

The concept of property space, which was coined to quanhtahvely describe the phenomena in social sciences [11, 12], has found many appUcahons in computational chemistry to characterize chemical space, i.e. the range in structure and properhes covered by a large collechon of different compounds [13]. The usual methods to approach a quantitahve descriphon of chemical space is first to calculate a number of molecular descriptors for each compound and then to use multivariate analyses such as principal component analysis (PCA) to build a multidimensional hyperspace where each compound is characterized by a single set of coordinates. 

If we multiply the probability density P(x, y, z) by the number of electrons N, then we obtain the electron density distribution or electron distribution, which is denoted by p(x, y, z), which is the probability of finding an electron in an element of volume dr. When integrated over all space, p(x, y, z) gives the total number of electrons in the system, as expected. The real importance of the concept of an electron density is clear when we consider that the wave function tp has no physical meaning and cannot be measured experimentally. This is particularly true for a system with /V electrons. The wave function of such a system is a function of 3N spatial coordinates. In other words, it is a multidimensional function and as such does not exist in real three-dimensional space. On the other hand, the electron density of any atom or molecule is a measurable function that has a clear interpretation and exists in real space. 

The concept of potential energy in mechanics is one example of a scalar field, defined by a simple number that represents a single function of space and time. Other examples include the displacement of a string or a membrane from equilibrium the density, pressure and temperature of a fluid electromagnetic, electrochemical, gravitational and chemical potentials. All of these fields have the property of invariance under a transformation of space coordinates. The numerical value of the field at a point is the same, no matter how or in what form the coordinates of the point are expressed. 

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