Bond valence model structures

Typical Ni—L bond lengths have been extracted from the Cambridge Structure Database (CSD) and listed in tabular form.321 Also, Ni11—L bond lengths from the CSD have been analyzed by the BDBO technique, which is related to the bond valence model (BVM) where the total bond order is equal to the oxidation state of any atom.322 Selected mean Ni—L distances from the CSD source are collected in Table 2. 

In Chapter 2 it was shown that the Madelung field of a crystal is equivalent to a capacitive electric circuit which can be solved using a set of Kirchhoff equations. In Sections 3.1 and 3.2 it was shown that for unstrained structures the capacitances are all equal and that there is a simple relationship between the bond flux (or experimental bond valence) and the bond length. These ideas are brought together here in a summary of the three basic rules of the bond valence model, Rules 3.3, 3.4, and 3.5. 

However, we do not need to abandon the bond valence model for those few inorganic compounds which contain homoionic bonds since there are a number of ways of adapting the model depending on the nature of the structure. If the two cations or two anions that form the bond are equivalent by symmetry, as the two Hg cations are, for example, in the tetragonal crystals of Hg2Cl2 (65441, Fig. 3.4), the normal rules still apply. In this compound the two Hg ... 

The bond valence model may also be used to refine the structure since it is based on the same assumptions as the two-body potential method. The network equations (3.3) and (3.4), can be used to predict the theoretical bond valences as soon as the bond graph is known. From these one can determine the expected bond... 

O Keeffe (1991Z)) has used bond valences to model the coherent interface that occurs between the semiconductors Si and MSi2 with M = Ni or Co (27139). Although these systems contain Si-Si bonds and therefore do not obey the assumptions of the bond valence model (condition 3.2), the mathematical formalism of the model still works because of the high symmetry. As both Si-Si and Si-Ni bonds are found in NiSi2, the cubic structure is strained (cf. BaTiOs in Section 13.3.2) and this strain affects the structure of the interface. Of the six possible interfacial structures examined, the two with the lowest BSI eqn (12.1) are those that are believed to occur in NiSi2 and CoSi2 respectively, and in both cases the strain introduced at the interface is correctly predicted. 

The symmetry between cations and anions in the bond valence model can best be seen in the compounds of the alkali metals and alkaline earths where the cation valences are similar to those of the anions. Binary compounds such as NaCl (18189), CsCl (22173), and ZnO (67454) are invariant under the interchange of the cations and anions since both kinds of ions occupy equivalent sites. For compounds such as CaF2 (29008) which crystallizes with the fluorite structure, changing the signs of ions gives the antifluorite structure adopted by the alkali metal oxides such as Na20 (60435). Although the antifluorite... 

In the bond valence model quantum effects are treated classically by including them in the interatomic repulsion described by eqn (3.1) or (3.2). There are, however, a number of cases where quantum effects are directly responsible for deviations from the higher symmetry that would otherwise be expected. Such electronically distorted structures were discussed in Chapter 8. 

A closer comparison of bond valence and electron density models is not possible because of the different underlying assumptions of the models. The forces in the bond valence model act between structureless point atoms, but the forces in the electron density model are exerted by electrons on nuclei and vice versa. This basic difference makes it difficult to compare the two models in greater detail. They are best seen as complementary, the electron density model providing important information about the nature of the bonding between the atoms, the bond valence model providing a simple tool for predicting structure and properties, particularly in cases where the structure is complex. 

Finally, a chemical compound must satisfy the constraints of three-dimensional space. The physical and chemical properties of a solid or liquid are determined by the interplay between the constraints of chemistry described, for example, by the bond valence model, and the constraints of space. Ultimately it is the ability of a chemical structure to be mapped into three-dimensional space that determines whether or not it exists. 

For closest-packed oxides, a Pannetier-type cost function [58] is more robust and faster to evaluate than the lattice energy as defined earlier. Here, the bond valence model [59] is used to calculate the charge on the ions and the discrepancy with the expected value is used to measure the quality of the structure. With an additional term, the discrepancy in the expected and calculated coordination numbers, the cost function becomes... 

A number of theorems associated with the bond valence model are useful in the analysis of inorganic structures. Equation (2), known as the valence sum rule, is central to the model. Since Gauss theorem is necessarily obeyed by the bond fluxes that terminate at each atom, equation (2) is always... 

X-ray diffraction does not separate atoms that are Periodic Table neighbours well, as the scattering factors of these species are so similar. Thus, problems such as the distribution of Fe2+ and Fe3+ over the available sites in a crystal structure may be unresolved by conventional structure determination methods. The bond valence model is an empirical concept that correlates the strength of a chemical bond between two atoms and the length of the bond. Because crystal structure determinations yield accurate interatomic distances, precise values of the bond strength, called the experimental bond valence, can be derived. 

This chapter describes some of the ways in which bond valences can be used. These now extend well beyond checking newly determined structures. The chapter starts with a description of the correlation between bond valence and bond length and some of the routine uses to which the Valence Sum Rule can be put. Section 10.3 presents a formal description of the Bond Valence Model in terms of bond graphs, showing how they can be used to predict bond lengths. The use of the model to generate the bond graph is covered in Section 10.4 while Section 10.5 describes how the model accounts for many of the distortions found in atomic environments. Finally, Section 10.6 discusses some of the limitations of the current version of the model. 

Each structure is composed of double and single bonds but the average, shown in Figure 10.6g, corresponds exactly to the Bond Valence Model prediction for this ion. 

A number of transition metals also show electronically driven asymmetries in their coordination environments. Typical of these are and U which form one or two strong bonds to O to give the vanadyl (VO) " and uranyl (U02) complexes. Cu typically shows an elongation of two trans bonds in its octahedral coordination sphere [46]. Cu and Hg ", though not strictly transition metals, tend to form two strong colinear bonds. Other distortions tend to be weaker and are only expressed when other factors favor a distorted crystal structure. The distortion of the Ti environment in BaTiOi is primarily a steric effect (see Section 10.6.2), but the polarizability of the Tr cation probably contributes. Similar effects are found for other cations with a d° configuration [47]. Most of these effects are not well understood in detail but their presence must be taken into account when using the bond valence model. 

The Bond Valence Model has its roots in the ionic models of Pauling, but it can be equally well derived from the covalent models of Lewis. It thus spans the full range between ionic and covalent bonds without making any distinction between them. In the formal development of the model, a chemical structure is treated as a network of bonds in which each bond is associated with a valence that expresses its strength. The two network equations, viz the Valence Sum Rule (Equation 10.5) and the Equal Valence Rule (Equation 10.6), can be used to predict bond valences, and hence bond lengths, when the bond network is known. The influence of one part of the structure on another is transmitted through the network by application of the network equations. 

Taken together, the rules of the Bond Valence Model provide a powerful tool for modelling and understanding the structures of compounds containing acid-base bonds. 

Although there are still no reliable ways of predicting the structure, this has not been for want of trying. There are currently two approaches that involve the use of bond valences. In the first, bond valences are used to assess the quality of, and improve, randomly generated trial structures in the second the structure is assembled directly using the crystal chemical rules of the bond valence model. 

There is much more work to be done before the model can make quantitative predictions of all of these effects, but in its present form, the bond valence model does provide a means of understanding the structure and predicting the geometry that would be expected in the absence of extraneous effects. This can then be used to examine the relative importance of different types of strain, leading to a better appreciation of the roles of, and interconnections between, the different influences at work in inorganic solids. 

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