Coordination number prediction

Coordination numbers in different crystals depend on the sizes and shapes of the ions or atoms, their electronic structures, and, in some cases, on the temperature and pressure under which they were formed. An oversimplified and approximate approach to predicting coordination numbers uses the radius ratio, r+/r. Simple calculation from tables of ionic radii allows prediction of possible structnres by modeling the ions as hard spheres. For hard spheres, the ideal size for a smaller cation in an octahedral hole of an anion lattice is a radius of 0.414r. Calculations for other geometries result in the radius ratios and coordination number predictions shown in Table 7.1. 

On the assumption that the pairs of electrons in the valency shell of a bonded atom in a molecule are arranged in a definite way which depends on the number of electron pairs (coordination number), the geometrical arrangement or shape of molecules may be predicted. A multiple bond is regarded as equivalent to a single bond as far as molecular shape is concerned. 

As with other high coordination numbers, there seems to be little difference in energy between these structures. Factors such as the number of counter ions and the stereochemical requirements of chelating ligands are probably decisive and a priori arguments are unreliable in predicting... 

Since the repulsive forces are determined by the true sizes of ions, and not their crystal radii, the radius ratios to be used in this connection are the ratios of the univalent cation radii to univalent anion radii.12 Values of this ratio for small ions are given in Table II, together with predicted and observed coordination numbers, the agreement between which is excellent. 

Ion Radius ratio Predicted coordination number Observed coordination number Strength of electrostatic bonds... 

The dominant features which control the stoichiometry of transition-metal complexes relate to the relative sizes of the metal ions and the ligands, rather than the niceties of electronic configuration. You will recall that the structures of simple ionic solids may be predicted with reasonable accuracy on the basis of radius-ratio rules in which the relative ionic sizes of the cations and anions in the lattice determine the structure adopted. Similar effects are important in determining coordination numbers in transition-metal compounds. In short, it is possible to pack more small ligands than large ligands about a metal ion of a given size. 

There may, however, be a number of other reasons to pursue a predictive first principles theory of Mossbauer spectroscopy. For example, one may want to elucidate structure/spectroscopy correlations in the cleanest way. To this end one may construct in the computer a number of models with systematic variations in oxidation states, spin states, coordination numbers, and identity of hgands to name only a few chemical degrees of freedom. In such studies it is immaterial whether these molecules have been made or could be made what matters is that one can find out which structural details the Mossbauer parameters are most sensitive to. This can provide insight into the effects of geometry or covalency that are very difficult to obtain by any other means. 

Very low energy differences also result for different polyhedra with higher coordination numbers, including coordination number 7. In these cases the electron pair repulsion theory no longer allows reliable predictions. 

Table 1. The Measured and Calculated Maximum Coordination Number of CO, and the Predicted Structure for each of Co a...

A DFT study was performed in order to determine the structures of phenylcalcium hydride and its magnesium analog in donor solvents.391 A dimeric phenylcalcium hydride was found to be the most stable species in solution, but monomers or tetramers cannot be excluded at very low or very high concentrations. Hydride bridging is favored over phenyl bridging, and a coordination number of six is predicted to be dominant in solution. 

The way by which all the factors involved influence the course of a reaction varies from case to case, and prediction is largely empirical. For catalytic processes, the actual species acting as catalyst is often unknown because coordination number, type of ligands, stereochemistry of the complex, and formal charge are difficult to establish in the reaction medium. Often many species are present, and the most active may be the one having the lowest coordination number and being present in a concentration so low that it cannot be detected spectroscopically. Only kinetic studies can provide evidence for such species. 

For example, atoms of both the alkaline-earth family (ZAval = 2) and the chalcogen family (ZAval = 6) correspond to FAemp = 2, and their stoichiometric proportionality (or coordination number) to monovalent atoms is therefore commonly two (AH2, ALi2, AF2, etc.). It is a remarkable and characteristic feature of chemical periodicity that the empirical valency FAemp applies both to covalent and to ionic limits of bonding, so that, e.g., the monovalency of lithium (Vuemp = 1) correctly predicts the stoichiometry and coordination number of covalent (e.g., Li2), polar covalent (e.g., LiH), and extreme ionic (e.g., LiF) molecules. Following Musher,132 we can therefore describe hypervalency as referring to cases in which the apparent valency FA exceeds the normal empirical valency (3.184),... 

The geometry of the coordination compounds can be similarly predicted based on the coordination number of the central atom. Coordination numbers 2 and 3 are both relatively rare and give linear and planar or pyramidal geometries, respectively. The most important coordination numbers are 4, 5 and 6 with the latter being the most important one as nearly all cations form 6-coordinate complexes. Table 2.4 shows the geometries corresponding to the commonest coordination numbers in biological systems. 

An introductory example to this subject is the well-known diagrams developed by Darken and Gurry (1953) for solid solution prediction. In such a diagram (as shown in Fig. 2.14) all elements may be included. The two coordinates represent the atomic size, generally the radius corresponding to the coordination number (CN) 12, and the electronegativity of the elements. 

Fig. D. The pair correlation function g0H and the running coordination number oh predicted by the central force model...

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