Valences sum rule

The experimental uncertainties in the measured bond lengths ensure that the sum of experimental bond valences around any particular ion will never exactly equal the atomic valence and, as shown in later chapters, there are cases where this discrepancy gives important information about the crystal chemistry, but the valence sum rule, which states that the sum of experimental bond valences around each atom is equal to the atomic valence (F, in eqn (3.3)), is much better obeyed than Pauling s second rule. 

Equations (3.3) and (3.4) have become known respectively as the valence sum rule and the loop, or equal valence, rule, and are known collectively as the network equations. Equation (3.4) represents the condition that each atom distributes its valence equally among its bonds subject to the constraints of eqn (3.3) as shown in the appendix to Brown (1992a). The two network equations provide sufficient constraints to determine all the bond valences, given a knowledge of the bond graph and the valences of the atoms. The solutions of the network equations are called the theoretical bond valences and are designated by the lower case letter 5. Methods for solving the network equations are described in Appendix 3. ... 

The valence sum rule is not, in general, sufficient to determine the distribution of the valence among the various bonds, but the principle of maximum symmetry suggests that the distribution will be the most symmetric one that is consistent with the valence sum rule. The condition that makes the bond valences most nearly equal is the loop, or equal valence rule. 

The bonding in the hydrazinium and trifluoroacetate ions can also be described in a similar way. Since each atom of the N-N or C-C bond contributes a different number of electrons (valence) to the bond, one can show the net valence transfer by means of an arrow as shown in Fig. 3.3(b). The valence sum rule is obeyed by this graph but at the expense of ignoring the electron pairs that provide the primary bond between the two N atoms. As in the case of dmso, the bond valence of the N-N bond in Fig. 3.3(b) shows only the net electron transfer, not the total number of electron pairs that contribute to the bond. The bond valence does not, therefore, correlate with the bond length. 

Typically lattice-induced strain results in the bonds around one cation being stretched and the bonds around another cation being compressed as found in BaRuOs (10253) by Santoro et al. (1999, 2000). When this happens, the valence sum rule will be violated around the cations in question but the valence still distributes itself as uniformly as possible among the bonds, so that the experimental bond valences determined from the bond lengths remain as close as possible to the theoretical bond valences. For this reason the BSI is typically smaller than the GII for lattice-induced strains, though the opposite is true for compounds with electronically induced strain where the valence sum rule remains well obeyed. 

Lattice-induced strains are characterized by large values of the GII because the environments around some atoms are stretched and around other atoms are compressed but, since the valence is still distributed as uniformly as possible among the bonds, the BSI remains small. This contrast with the electronically driven distortions discussed in Chapter 8 where the GII is small (the valence sum rule is obeyed) but the BSI is necessarily large. 

B (or N) can only be determined with a precision of around 10 per cent even if care is taken (Tytko 1999), but the observation that B lies between 32 and 42 pm for many bonds means that it is convenient to fix its value at 37 pm for all bond types (Brown and Altermatt 1985). Any error that this assumption introduces is usually negligible for most bonds to provided that the appropriate value of i o is used. Using the same value of B for all bond types makes the determination of i o simpler since only one parameter now needs to be fitted. Combining eqn (3.1) with the valence sum rule and rearranging yields eqn (A1.3) ... 

Naskar et al. (1997) were interested in using bond valences to determine oxidation states around transition-metal cations, particularly those with negative or zero formal oxidation states. Since these numbers cannot, in principle, be reached by the standard equations, they proposed to create a fictional positive oxidation state by arbitrarily adding 4.0 to the actual oxidation state. They proposed to write the valence sum rule in the form of eqn (A 1.10) ... 

The network equations constitute a set of A a 1 valence sum rule equations (eqn (3.3)) and A b Xa+1 loop equations (eqn (3.4)) where the network contains atoms and A b bonds. Alternatively one can use the equivalent Kirchhoff equations (2.7) and (2.11). One can readily write down equations of type 3.3 but one of these is redundant since the sum of all atomic valences in the crystal must be zero. There are many more than Ab — Aa + 1 possible loops in most bond graphs, but only Ab —Aa+ 1 are independent. Equations (3.3) and (3.4) thus constitute a set of Ab equations which is exactly the number needed to solve for the Ab unknown bond valences,. s. 

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... 

Burdett and McLarnan (1984) went on to test the valence sum rule by assessing a further prediction implied in the rule, namely that anions of the same electronegativity should be found in sites of equal bond sum. This was confirmed by calculating that, when hypothetical anions with equal parameters midway between O and N are used, the most stable structure is the W-Pmc2, ) type, where all the sites are of equal bond-strength sum. Also investigated was the Zachariasen-Baur extension of the second rule (Baur, 1970), stating that oversaturated anions should... 

A closely related problem is that of the usage of the bond valence sum rule in structural chemistry. The bond valence may be defined as s = A.log(r/ro) or s = B(r/ro) where A, N and ro are parameters for the given pair of atoms forming the bond (length r)... 

Pauling s Electrostatic Valence Principle, or the Valence Sum Rule, as it is called in the modified form used here (see Section 10.2.3), provides a useful check on the correctness of a structure and can draw attention to problems in its chemical description. Many workers now use the Valence Sum Rule routinely as a check on the correctness of a structure determination. 

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. 

If the average length of the bonds formed by an atom is constrained by the structure to be greater than the length that satisfies the Valence Sum Rule, the environment of the atom will tend to distort in such a way as to make the bonds of different lengths. ... 

The bond valence represents a measure of the strength of a bond that is independent of the atomic size. Unlike bond lengths, bond valences can be used to compare the relative importance of bonds between different atoms even when the atoms are of very different sizes. The most widely used property of bond valences is the Valence Sum Rule which can be stated as ... 

The correctness of a structure determination and its interpretation is supported by the Valence Sum Rule holding around all atoms. A significant discrepancy between an atomic valence and the sum of the experimental bond valences usually means that the model proposed for the structure is faulty. Problems could arise from overlooking a bond or an atom or from the use of an incorrect space group. 

The Electroneutrality Rule and other chemical considerations usually lead to an unambiguous assignment of the atomic valences. If the number of bonds is less than the number of atoms, atomic valences, combined with the Valence Sum Rule expressed mathematically by Equation 10.5, are sufficient to assign the bond valences uniquely... 

Although by their nature, cations with electronically distorted environments do not obey the Equal Valence Rule, they are found to obey the Valence Sum Rule, indicating that the atomic valence is still shared between the bonds, albeit unequally. 

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