The distortion theorem

Condition 3.1. The stoichiometry must obey the electroneutrality principle, namely that the sum of all the atomic valences (formal ionic charges), having regard to their sign, is zero. 

Condition 3.2. The bond graph must be bipartite as described in Section 3.5, i.e. all of the bonds must connect a cation, e.g. Na, to an anion, e.g. Cl.  

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. 

This rule is obeyed by unstrained structures and is equivalent to eqn (2.11), the law of conservation of energy, if the capacitances are all set equal. 

The non-linear shape of the bond-valence - bond-length correlation immediately leads to the Distortion Theorem [8] which states that  

This corollary has important implications that are discussed in Section 10.6.2. 

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Rule 3.6a (An alternative statement of the Distortion theorem). For any ion, lengthening some of its bonds and shortening others, keeping the average bond length the same, will always increase the valence sum. 

This statement of the distortion theorem can be demonstrated in the same way as the previous statement, but it leads to an interesting corollary ... 

The ligands of a lone-pair cation lie on the surface of a sphere. When TU is surrounded by weakly bonding oxyanions, it lies at the centre of the sphere, forming nine bonds of 297 pm each (O.llvu, Fig. 8.2(a)). When it bonds to strongly bonding anions, as in TI3BO3, TU moves about 70 pm away from its centre to form three primary bonds of 266 pm (0.33 vu), and six secondary bonds of 324-372 pm (Fig. 8.2(b)). In the process the radius of the coordination sphere increases from 297 to 322 pm in accordance with the distortion theorem (Rule 3.6). 

The second type of distortion arises from the difficulty of mapping the ideal bond lengths calculated using the network equations into real space, either because of constraints of crystal symmetry or because of non-bonding repulsions. For example, if an atom is placed in a cavity that is too large for it, its valence sum will be too small. The Distortion Theorem (Section 10.2.2) predicts that the valence sum will be increased if the environment of the atom is distorted to give bonds of different lengths. 

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. 

If an atom finds itself in a cavity that is too large for its bonds to adopt their ideal length, the bonds must be stretched. According to the distortion theorem (5), the environment of the atom will distort in such a way as to make the bond lengths unequal in order that the bond valence sums becomes equal to the atomic valences. As mentioned in Sect. 7.4.1, this contributes to the distortion around titanium(IV) in BaTiOs. In many cases, such distortions are found in compounds where electronic distortions are also expected, the two effects being mutually supportive. 

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