Equal valence 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. 

Rule 3.4 (Equal valence rule (eqn (3.4)). The sum of bond valences around any loop in the bond network, having regard to the direction of the bond, is zero. 

The theoretical bond valence, which is calculated using the equal valence rule (3.4), clearly gives a very poor estimate of both the flux and the experimental bond valence since in these compounds the valence is not equally distributed between the bonds. Nevertheless, it provides a useful reference against which to measure the strain introduced by the electronic asymmetry. ... 

The equal valence rule or loop rule equation (3) is less rigorously obeyed, and does not apply to the enviromnents of atoms with electronically driven anisotropies arising from, for example, lone electron pairs see Lone Pair, Electronic Structure of Main-group Compounds) or Jahn Teller distortions see Jahn-Teller Effect, Copper Inorganic Coordination Chemistry). 

The sign of the valence is determined by the direction in which the bond is traversed in completing the circuit. Equation 10.6 is equivalent to requiring that the atomic valence be distributed as uniformly as possible among the bonds that each atom forms [31]. For this reason. Equation 10.6 is called the Equal Valence Rule. 

While the network equations hold for the majority of inorganic compounds there are occasions when they fail. The Equal Valence Rule is not obeyed when the atomic environment is distorted by electronic effects. Such effects are found in two classes of cation main group cations in which the valence shell contains electron pairs that are not involved in chemical bonding, e.g. Sn and S , and transition elements in which the partially filled d shell can influence the coordination environment, e.g. the Jahn-Teller distortion typically found around the Cu " cations. The influence of electronic effects on the bond valences are discussed in Section 10.6.1. 

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. 

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. 

The model is restricted to compounds that display acid-base bonding and therefore it cannot be used to describe metallic or organic bonding. Within these limits, the Valence Sum Rule is found to be widely obeyed and has proved remarkably useful in the analysis of inorganic structures. The Equal Valence Rule is less universal and fails when the environment of an atom is distorted as a result of its internal electronic structure or as the result of the application of the Distortion Theorem in constrained structures. In some highly constrained structures the Valence Sum Rule may also be violated, indicating the presence of internal strain and flagging the possible existence of a complex crystal chemistry. 

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

If the selected space group of the starting structure has high symmetry, the number of free parameters will be smaller than the number of constraints and it is not possible for all the predicted distances to be realized. If the deviations are small, the structure may be stable, but if they are large, the structure will relax or, in extreme cases, be so unstable that it cannot be prepared. Relaxation may involve only a small adjustment to the bond lengths so that the valence sum rule continues to be obeyed (at the expense of the equal valence rule), or it may involve a reduction in the symmetry as is found in the case of CaCrF5 described above. If the symmetry is reduced, the number of free variables is increased and the atomic coordinates may be under-determined. In this case, the constraints on the sizes of non-bonding distances become important. 

In the above example of La2Ni04, not only the valence sum rule, but also the equal valence rule is clearly violated and the network equations cannot therefore be used to predict the bond lengths however, the network equations do give a set of ideal bond lengths, i.e. the lengths that the bonds would have if it were not for the constraints imposed by the translational symmetry of the crystal. This allows us to determine the sizes of the strains that such symmetry imposes, and allows us to understand why certain compounds undergo phase transitions as the temperature is reduced, and why they sometimes adopt unusual oxidation states and stoichiometries. 

This almost trivial example illustrates two important points, (a) If there are m cations and n anions there are m+n-1 independent valence sums (note that the first equation given above is the sum of the second and third), (b) We assume that the bonds from 0(1) will be equal in valence (length) and likewise that the bonds from 0(2) will be equal in valence (we have implicitly used an equal valence rule). 

From the connectivity matrix, one can see that there are 5 kinds of bonds (with vaiences indicated as a, p,Y,S and e). However with 5 kinds of atom there are oniy 5-1 = 4 independent bond vaience sums emd another condition is required to determine the five bond vaience parameters. This is provided by Brown s equal valence rule. This ruie may be stated as requiring that, subject to the bond valence constraint, bonds between like pairs of atoms have valences as nearly equal as possible. 

The equal valence rule translates into sums around circuits on the connectivity matrix. There are three four-circuits yielding three equations, of which only two eire independent. These are given here without weights for simplicity ... 

For a more complicated examples, including the use of six-circuits to express the equal valence rule, see Wagner O Keeffe (1988) and O Keeffe (1990a). 

The equal valence rule has some simple but far-reaching implications in crystal chemistry. We may infer from its general applicability that low energy configurations of solids are those in which valences are as nearly as equal as possible. Thus we might expect an important factor in the determination of coordination numbers to be the ability of a structure with given coordination numbers to accommodate equal valences. Some simple examples, taken froiti oxide emd nitride chemistry, illustrate this point... 

In oxide chemistry at least, there are three classes of cation that often have local environments that are in gross violation of the equal valence rule. These are ... 

These papers were mainly concerned with interpreting observed bond lengths in crystals. An important step was taken by Brown (1977) who first proposed a method for predicting bond valences (and hence lengths) to be expected in crystals. This work introduced the equal valence rule for the first time. 

In all cases of steric constraint, the observed bond distances violate the equal valence rule (14b) and, in some cases, the valence sum rule (14a) as well. A simple... 

AEevr Energy penalty for a deviation from the equal valence rule... 

In the course of the discussion of bond valence parameter determination we will also link the energy of an atom M in a given structural environment to deviations of its bond valence sum V A) from the absolute value of its oxidation state and to a (also bond valence-based) penalty function AFevr that penalizes deviations in the bond arrangement from the equal valence rule ... 

A different empirical approach to assess the coefficients Dq and g in (1) as well as a suitable functional form for the influence of the equal valence rule ( evr) have been derived in our earlier work [25, 26] from comparing the distance dependence of the bond valence sum mismatch with the distance dependence of interaction energies in various empirical interatomic potentials The variation of an individual bond valence can be straightforwardly translated into the variation of a Morse-type interaction potential... 

Moreover, it has to be kept in mind that bond valence maps (or more precisely, bond valence sum mismatch maps) use bond valence units scale rather than an energy scale, and attempts to link energy or probability density to the bond valence mismatch were difficult to achieve in a general transferable way as the calculation of effective bond valence mismatches according to Eq. 1 requires scaling between bond valence terms and cation-cation (or anion-anion) repulsions or (for open structures) to discriminate between sites of identical bond valence sum based on the degree to which the equal valence rule is fulfilled. 

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