The Valence-Bond Method

the increment method allows the qualitative energetic assessment of new materials. When sodium nitride, NasN, was eventually synthesized for the first time [36], its properties pointed towards a metastable compound, and this was backed by quantum-mechanical total-energy calculations. Because the lattice parameter (4.73 A) of the ReOs-type sodium nitride corresponds to a molar volume of 63.7 cm /mol, one arrives at a huge experimental volume increment of 44.2 ctp /mol if (three times) the tabulated increment for Na+ (6.5 cm /mol) is subtracted. Thus, the NasN-derived nitride increment is more than twice the tabulated one (19 cm /mol) such that, within metastable NasN, the nitride ion is seemingly subject to extreme polarization, a clear sign of low stability. 

9) It is not easy to specify what chemists really mean by bond strength, a nonobservable quantity without a physical unit which the physicists will kindly allow. To measure the strength of a covalent bond, its dissociation energy or its force constant certainly proves useful. 

Formally, the bond-valence sum or, simply, the formal valence v, of a coordinated atom/ion i is equivalent to the sum of all bond-valences (or, bond strengths) Sjy to its j coordinating atoms/ions with interatomic/interionic distances Tij, that is, 

In the solid-state chemical community, bond-valence (BV) calculations have been tremendously useful, for example, when it comes to the localization of light atoms that do not show up very easily from X-ray density maps, especially when these include very heavy scatterers. Also, isoelectronic ions such as Ai3+/Si4+ 

There have been several compilations of bond-valence parameters ro (with B fixed at the above 0.37 A) for a multitude of anion-cation combinations, based on a very large number of well-resolved inorganic crystal structures. In addition, Brese and O Keeffe have included less common cation-anion combinations, and they have also predicted reference ro values for those cases where no experimental data were available. For convenience, we include their data [44] in Tables 1.3 and 1.4, so that they can be used for your own calculations. 

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Adams, St. and Swenson, J. (2000 ). Migration pathways in Ag-based superionic glasses and crystals investigated by the bond valence method. Phys. Rev. B 6, 054201. 

Brown, I. D. (1981). The bond valence method. An empirical approach to chemical structure and bonding. In M. O Keeffe and A. Navrotsky (eds). Structure and Bonding in Crystals, Vol. 2. New York Academic Press, pp. 1-52. 

Swenson, J. and Adams, St. (2001). The application of the bond valence method to reverse Monte Carlo produced structural models of superionic glasses. Phys. Rev. B64, 024204. 

The bonding in these polyatomic anions is delocalized, and for the diamagnetic species the bond valence method (see Section 13.4) can be used to rationalize the observed structures. This method is to be applied according to the following rules ... 

Wizard distances in crystals by the bond valence method, I. P. Orlov, K. A. Popov and V. S. Urusov, J. Struct. Chem., 1998, 39(4), 575 579 and Predicting bond lengths in inorganic crystals, I. D. Brown, Acta Crystallogr., Sect. B, 1977, 33, 1305 1310 calculations... 

We prefer an empirical approach to bonding as opposed to any complex theories of binding [4] or bonding [5, 6, 7], The bond valence method for determining the strength of bonds in crystals has developed from Pauling s concept of bond strength [8] and recently has been documented extensively [9, 10, 11]. 

Recently the bond valence method (see Section 7.8) has been applied successfully to the discrimination between various cation bonding sites in protein crystals (see Bibliography for the reference). 

Lists of atomic positions are not very helpful when a variety of structures have to be compared. This chapter describes attempts to explain and systematise the vast amount of structural data now available. The original aim of ways of comparing similar structures was to provide a set of empirical guidelines for use in the determination of a new crystal structure. With sophisticated computer methods now available, this approach is rarely employed, but one empirical rule, the bond-valence method, (see Section 7.8 below), is widely used in structural studies. 

Because the X-ray scattering factors of Mg and A1 are similar, it is not easy to assign the cations in the mineral spinel, MgAl204 to either octahedral or tetrahedral sites (see Section 7.8 for more information). The bond lengths around the tetrahedral and octahedral positions are given in the table. Use the bond valence method to determine whether the spinel is normal or inverse. The values of r0 are r0 (Mg2+) = 0.1693 m, r0 (Al3+)= 0.1651 nm, B = 0.037 nm, from... 

P. Muller, S. Kopke and G. M. Sheldrick, Is the bond-valence method able to identify metal atoms in protein structures Acta Crystallogr., D59, 32-37 (2003). 

Brown ID (1981) The bond-valence method An empirical approach to chemical stracture and bonding. In Structure and Bonding in Crystals. O Kieffe M, Navrotsl A (eds) Academic Press, p 1-30 Brrms RG (1993) Mineralogical Applicatiorrs of Crystal Field Theory (2nd edn). Cambridge Urriversity Press, Cambridge, UK... 

For some extensions of the Bond Valence method to thermodynamic measures of bond strength see (a) J. Ziolkowski, J. Solid State Chem., 1985, 57 269 (b) M. O Keeffe and J. A. Stuart, Inorg. Chem., 1983, 22, 177. 

There is yet another classical method used to predict these lattice parameters namely by the bond-valence method. In the [ZnS] type, the +2 bond-valence sum of Ca + results from four Ca +-0 bonds of bond valence 2/4 = 0.5. For the [NaCl] and [CsCl] types, there are six/eight corresponding bonds with bond valences of 2/6 = 0.3333 and of 2/8 = 0.125. Together with the bond-valence parameter of 1.967 A from Table 1.3 and using Equation (1.15), the interionic Ca +-0 distances are 2.22, 2.37, and 2.48 A which corresponds to (see above conversion) lattice parameters of 5.14, 4.75, and 2.86 A for the [ZnS], [NaCl], and [CsCl] types. 

Here the narrow prescription of Chapter 1 is widened to deal with more chemically complex phases, in which the materials may contain mixtures of A, B and X ions as well as chemical defects. In these cases, using an ionic model, it is only necessary that the nominal charges balance to obtain a viable perovskite composition. In many instances these ions are distributed at random over the available sites, but for some simple ratios they can order to form phases with double or triple perovskite-type unit cells. The distribution and valence of these ordered or partly ordered cations and anions are often not totally apparent from difEraction studies, and they are often clarified by use of the bond valence sums derived from experimentally determined bond distances. Information on the bond valence method is given in Appendix A for readers unfamiliar with it Point defects also become significant in these materials. The standard Kroger- fink notation, used for labelling these defects, is outlined in Appendix B. 

The utilisation of bond valence sums is widely used in these rather complex structures for either the allocation of an ion with a known nominal valence to the correct coordination polyhedron or the allocation of a charge state to an ion in a known coordination polyhedron. Using the bond valence method, the valence states of the Mn ions in i -Mn203 were determined to be Mn in A, Mn in A and Mnj Mn in B, giving an average valence of Mn for the phase, as necessitated by the overall formula. Similarly, bond valence sums for the perovskite LaCUjFe Oj allow the charge state at room temperature to be assigned La Cu Fe Ojj rather than La CUj fFe FCj )0,2. 

In the interest of economy, references are not provided to crystal structures used as examples where these are to be found in standard reference sources such as Structure Reports. On the other hand a fairly extensive bibliogr hy for applications of the bond valence method in crystal chemistry is provided. 

It should be remarked however, that the bond valence method is generally less suitable for metallic bonding than for covalenf and ionic bonds and some caution is required in the interpretation of metal-metal distances. Particularly for the moreelectropositive elements, it is often found that interatomic distances decrease substantially on forming a compound. To take a fairly typical and familiar example, the Mg-Mg distance in MgO is 2.97 A, in the elemental metal it is 3.20 A. in the former case it is generally conceded that there is not Mg-Mg bonding, but there must be in the latter. 

Bond valence sum constraints dictate that in this instance the coordination of B O will be irregular. Note the superiority of the bond valence method over a sum-of-radii method which would predict four equal B-0 bond lengths. 

The discussion so far has focussed on the calculation of valences (and bond lengths) in periodic crystals, but the bond valence method is equally plicable to aperiodic structures and is potentially very useful for predicting relaxations around defects in crystals (Brown, 1988) and at surfaces and interfaces (O Keeffe, 1991b). 

In 2.1 we showed how the bond valence method correctly accounted for the irregular coordination tetrahedron around B in high-pressure B2O3 and remarked that a sum-of-radii method would predict a regular tetrahedron. In fact most compilations of radii ascribe most, if not ail, of the variation of bond length with valence to a variation of cation radius with coordination number e.g. Shannon, 1976). Apart from the fact that one would not expect a cation "size (however defined) to vary significantly with coordination number, this approach can iead to some wrong conclusions as is now shown. 

Note that the bond valence method (using parameters from Brese O Keeffe, 1991) predicts close to the observed values, but the sums of ionic radii (using the radius for 8-coordinated La, Shannon, 1976) is substantially in error. The iast row of the table shows the sums of ionic radii assuming a coordination number of 6 for La in La-0 bonds (corresponding to a bond valence of 3/6 = 1/2) and a coordination number of 12 for La in the La-F bonds (corresponding to a bond valence of 3/12 = 1/4). Now the sums of radii are approximately correct, illustrating that it is bond valence rather than coordination number that determines bond length. 

A fairly obvious use of the bond valence method is to verify experimentally-determined crystal structures. In a good structure the bond valence sums should come close to the expected atom valences. However, particularly in compounds rich in large electropositive elements, sums less than anticipated may occur (see 3.3). 

Although the bond valence method has come into prominence only rather recently, it is worth recalling that it has a long history. Readers of the early papers cited here may care to speculate on why it has taken so long for the method to be generally accepted. 

O Keeffe, M. (1991b). Application of the Bond Valence Method to Si/NiSi2 Interfaces . J. Mats. Res. 6,2371-2374. 

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