Bond Valence Methods

I.D. Brown

1 INTRODUCTION

Since the middle of the nineteenth century, chemists have been using the chemical bond model to predict the structure and properties of organic molecules and, surprisingly, the events that followed the discovery of the electron in 1898 made remarkably little difference to the way this model was formulated and applied. Quantum mechanics has provided insight into the nature of chemical bonding, but the bond model itself was essentially complete by the end of the nineteenth century. The only significant addition to the model in this century has been the determination of bond lengths and angles. In spite of the great advances in our understanding of the physics of the atom, the chemical bond model remains essentially empirical. The idea of a bond does not arise naturally from quantum mechanics and the only justification for the model lies in its success in predicting the existence, structure and properties of organic molecules.

Until recently it was generally thought that this model could not be used to describe the structure of inorganic solids because in inorganic materials the bonds clearly do not have integral bond orders. Consequently, in the early years of the twentieth century two other models were proposed. The ionic model was developed to describe inorganic salts in which the bonding electrons are incorporated into the valence shell of the anion, and the covalent model was developed for those compounds where the bonding electrons were more equally shared between the bonded atoms. While the early versions of these models have proved to be pedagogically useful, they were not able to give a quantitative prediction of structure except for the simplest compounds. Recently the ionic model has been transformed into the very successful interatomic potential model which is discussed in detail elsewhere in this book (see Chapter 3. and 7), but it has also become apparent that the chemical bond model itself, with only minor modifications, can also be adapted to inorganic compounds. It has shown itself as powerful in describing the structure and properties of inorganic solids as it has in describing those of organic molecules. This chapter will describe the modifications needed to adapt the model to inorganic solids and it will show how the model can be applied.

2 THE CHEMICAL BOND MODEL FOR INORGANIC SOLIDS

2.1 Differences between the organic and inorganic bond models

How must the rules of the chemical bond model be modified when used to describe inorganic solids? In both versions of the model, each atom is assumed to have an atomic valence which corresponds to the number of electrons (positive valence) or holes (negative valence) that are available in the valence shell of the neutral atom. In the original bond model the valence simply represented the number of bonds that the atom formed, i.e. its coordination number, but compounds were later discovered that contained double and triple bonds. In these compounds the coordination number was different from the atomic valence and each bond was associated with more than one unit of valence. However, in all cases the sum of the valences (or strengths) associated with the bonds around each atom was found to be equal to the valence of the atom. This is the principal rule of the chemical bond model and is known as the valence sum rule:

The term ‘bond valence’, which is used here for precision, is essentially synonymous with the more widely used but less well-defined term ‘bond strength’. The valence sum rule holds in both the organic and inorganic versions of the model.

The first important difference between the two versions of the model is that in inorganic compounds the bonds are not restricted to integral (or semi-integral) valences. The valence of each atom is distributed as uniformly as possible between the bonds that it forms (the equal valence rule discussed in more detail below) so that the valences of the bonds may have non-integral as well as integral values. In inorganic compounds, bonds with integral valences are the exception rather than the rule. Thus in crystals of NaCl, each of the six bonds formed by the Na atom has an equal share of its atomic valence of + 1, giving them each a bond valence of l/6 valence units (v.u.). Similarly, in the 42− ion each of the four bonds formed by S shares equally in its atomic valence of + 6, giving each a bond valence of 1.5 v.u.

The second difference between the two models is that in inorganic solids the sign of the atomic valence becomes important. It is therefore convenient to label atoms with positive valence as cations and those with negative valence as anions but without in any way implying that the model is restricted to those compounds traditionally thought of as ionic; the model works equally well for both ionic and covalent compounds and it is no longer necessary to distinguish between them. However, the model is mathematically restricted to compounds in which all the bonds have a cation at one end and an anion at the other. Although this appears to be a serious restriction, it is obeyed by almost all metal oxides and halides and those that do not conform can be treated by extensions to the model described by Brown (1992a).

A third difference lies in the way the structures are represented graphically. Organic molecules can be represented by a two-dimensional bond diagram which shows the topology of the molecule, i.e. the way in which the atoms are connected together. If carefully drawn, this diagram can be made to look like a projection of the three-dimensional structure of the molecule (Fig. 2.1).

Unfortunately, the corresponding bond diagram for inorganic compounds usually extends to infinity in one or more directions, making the same kind of simple two-dimensional representation impossible. However, for a single formula unit it is possible to draw a finite bond graph that retains the essential topological properties of the infinite bond network, though such a diagram does not give the same pictorial impression of the three-dimensional structure (Fig. 2.2). In order to use the bond model for inorganic solids it is necessary to know how to draw and interpret this bond graph.

2.2 Bond networks and bond graphs

Figure 2.2 illustrates how the finite bond graph is related to the infinite bond network. A single formula unit of the compound is isolated and extracted from the network. This necessarily involves breaking a number of bonds but these can be reconnected to other broken bonds within the formula unit in such a way that the nearest neighbour bonding topology is preserved. The resulting diagram differs in an important way from the traditional organic bond diagram, namely that the presence of two or more lines connecting a pair of atoms does not indicate a bond of double or triple strength; rather it indicates that each of the atoms is bonded to two or more spatially separated but chemically similar atoms. The bond graph for NaCl (Fig.2.2(b)) contains only two atoms, Na and Cl, but these are connected by 6 lines, indicating that each Na atom bonds to 6 different Cl atoms and vice versa. Similarly, the bond graph of SiO2 (Fig. 2.2(d)) contains three atoms, with Si forming four bonds to different O atoms and each of the two O atoms forming bonds to two different Si atoms. The number of lines terminating at an atom is its coordination number. For inorganic compounds it makes no sense to use single and double lines to represent the bond valence because in general the bonds do not have integral valences. Where necessary the bond valence can be shown by means of a number written alongside the bond.

Graphs which only have bonds linking cations to anions are called bipartite graphs, a property that can be indicated by arrows showing the bond directed from the anion to the cation. All the loops in a bipartite graph contain even numbers of bonds.

2.3 Prediction of bond valences

The valence sum rule and the equal valence rule can be expressed in mathematical form by equations (1) and (2), respectively (Brown, 1992a):

jSij=Vi

loopSij=0

where Sij is the valence of the bond between atoms i and j and Vi is the valence of atom i. The analogy is with KirchhofFs equations for the analysis of electrical networks where the electric current is assumed to distribute itself as equally as possible between the various branches of the network. The meaning of equation (2) is that the sum of bond valences around any loop in the network (taking account of the direction of the bonds) is zero and this condition is fulfilled when the valences are most uniformly distributed (Brown, 1992b, appendix). Equations (1) and (2) are called the network equations and are sufficient to determine...