Average Observed Coordination Numbers

Effective Cation Radii of Some Metal Ions and Their Average Observed Coordination Numbers for Different Ligands in Structures of Small Molecule ... 

Cation coordination numbers using Rule 4.1 as a criterion have been determined for some 14000 cation environments (Brown 1988a). Some cations, such as are known with only one coordination number (4 in this case), but others, such as Cs+, can be found with every coordination number between 3 and 12. Whatever the total range, the frequency distribution of the coordination number for a given cation usually peaks close to the average. It is therefore convenient to take the average observed coordination number as a characteristic chemical property of the cation. 

Fig. 6.3. Ideal (average observed) coordination number versus predicted coordination number for various cations. The predicted coordination numbers are calculated using the ratios of the Pauling (1960) six-coordinate radii by interpolating coordination numbers between the values given in Table 6.1. Most cations are represented by points, selected cations are shown by their chemical symbol and, where necessary, oxidation state.

Table 2.1 Selected cation and anion bonding strengths and Average Observed Coordination Numbers (AOCN) to oxygen (taken from Brown, 1988, 1992a).

Here, Nq is the average observed coordination number in a large number of oxide crystal structures [1], and the same formula can be used to calculate the Lewis acid strength (5a) of cations (cf. Eq. 7 in chapter Bond Valence Theory of this volume). 5a and 5b take into account both the valence and size of atoms, providing an expectation value for individual bond valences. According to the valence matching rule, the most stable stractures are formed when the 5a and 5b values of the cations and anions are similar (see Eq. 9 in chapter Bond Valence Theory of this volume). 

Because of the interatomic distances overestimate mentioned before, we observe coordination numbers from AIMD that are slightly larger (by 0.4 in average) than the experimental ones. The values for Sb2Tc3 are in line with those of Caravati et al. [24], given that the cutoff distance was smaller in that study. 

If the value of N is known or can be inferred from other considerations, this value should be used in Eq. (6). If N is not known, a standard value, Nq, the average of the observed coordination numbers with oxygen ligands, can be used as a best guess instead, as shown in Eq. (7) [15]. The resulting value of So is known as the bonding strength of the atom, since, in the absence of any better information, the bonds formed by the atom are expected to have valences close to So-... 

In general one can say that those parts of the band that correspond to bonds which have been broken in order to create the surface are narrower. A similar effect can be expected for small particles the average coordination number of the atom decreases and the bands are narrower. This effect can be observed in photoemission experiments an example is shown in Fig. 3.18. 

At the same time, as the concentration decreases the exchange of water molecules by cooperative processes becomes easier and so significant fluctuations in the coordination numbers are observed, which are estimated to be about 1. At even lower concentrations, ion-water correlation patterns will become obscured and then undetectable. As an extrapolation, dynamic processes might contribute more and more to the description of the solution. The strong interaction between Li+ and OH2 (dH. i = 34 kcal/mole in the vapor phase 130>) may cause the cation-water complexes to remain quite well-defined tetrahydrates, on the average, despite all dynamic effects. 

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