Mean bond-length variation in crystal structures: a bond-valence approach

The distortion theorem of the bond-valence theory predicts that the mean bond length 〈D〉 increases with increasing deviation of the individual bond lengths from their mean value according to the equation 〈D〉 = (D′ + ΔD), where D′ is the length found in a polyhedron having equivalent bonds and ΔD is the bond distortion. For a given atom, D′ is expected to be similar from one structure to another, whereas 〈D〉 should vary as a function of ΔD. However, in several crystal structures 〈D〉 significantly varies without any relevant contribution from ΔD. In accordance with bond-valence theory, 〈D〉 variation is described here by a new equation: 〈D〉 = (D_RU + ΔD_top + ΔD_iso + ΔD_aniso + ΔD_elec), where D_RU is a constant related to the type of cation and coordination environment, ΔD_top is the topological distortion related to the way the atoms are linked, ΔD_iso is an isotropic effect of compression (or stretching) in the bonds produced by steric strain and represents the same increase (or decrease) in all the bond lengths in the coordination sphere, ΔD_aniso is the distortion produced by compression and stretching of bonds in the same coordination sphere, ΔD_elec is the distortion produced by electronic effects. If present, ΔD_elec can be combined with ΔD_aniso because they lead to the same kind of distortions in line with the distortion theorem. Each D-index, in the new equation, corresponds to an algebraic expression containing experimental and theoretical bond valences. On the basis of this study, the ΔD index defined in bond valence theory is a result of both the bond topology and the distortion theorem (ΔD = ΔD_top + ΔD_aniso + ΔD_elec), and D′ is a result of the compression, or stretching, of bonds (D′ = D_RU + ΔD_iso). The deficiencies present in the bond-valence theory in explaining mean bond-length variations can therefore be overcome, and the observed variations of 〈D〉 in crystal structures can be described by a self-consistent model.