Bond-length distributions for ions bonded to oxygen: results for the transition metals and quantification of the factors underlying bond-length variation in inorganic solids

Bond-length distributions are examined for 63 transition metal ions bonded to O 2− in 147 configurations, for 7522 coordination polyhedra and 41 488 bond distances, providing baseline statistical knowledge of bond lengths for transition metals bonded to O 2− . A priori bond valences are calculated for 140 crystal structures containing 266 coordination polyhedra for 85 transition metal ion configurations with anomalous bond-length distributions. Two new indices, Δ topol and Δ cryst , are proposed to quantify bond-length variation arising from bond-topological and crystallographic effects in extended solids. Bond-topological mechanisms of bond-length variation are (1) non-local bond-topological asymmetry and (2) multiple-bond formation; crystallographic mechanisms are (3) electronic effects (with an inherent focus on coupled electronic vibrational degeneracy in this work) and (4) crystal-structure effects. The indices Δ topol and Δ cryst allow one to determine the primary cause(s) of bond-length variation for individual coordination polyhedra and ion configurations, quantify the distorting power of cations via electronic effects (by subtracting the bond-topological contribution to bond-length variation), set expectation limits regarding the extent to which functional properties linked to bond-length variation may be optimized in a given crystal structure (and inform how optimization may be achieved) and more. These indices further provide an equal footing for comparing bond-length variation and the distorting power of ions across ligand types, including resolution for heteroligand polyhedra. The observation of multiple bonds is found to be primarily driven by the bond-topological requirements of crystal structures in solids. However, sometimes multiple bonds are observed to form as a result of electronic effects ( e.g. the pseudo Jahn–Teller effect, PJTE); resolution of the origins of multiple-bond formation follows calculation of the Δ topol and Δ cryst indices on a structure-by-structure basis. Non-local bond-topological asymmetry is the most common cause of bond-length variation in transition metal oxides and oxysalts, followed closely by the PJTE. Non-local bond-topological asymmetry is further suggested to be the most widespread cause of bond-length variation in the solid state, with no a priori limitations with regard to ion identity. Overall, bond-length variations resulting from the PJTE are slightly larger than those resulting from non-local bond-topological asymmetry, comparable with those resulting from the strong JTE, and less than those induced by π-bond formation. From a comparison of a priori and observed bond valences for ∼150 coordination polyhedra in which the strong JTE or the PJTE is the main reason underlying bond-length variation, the JTE is found not to have a cooperative relation with the bond-topological requirements of crystal structures. The magnitude of bond-length variation caused by the PJTE decreases in the following order for octahedrally coordinated d 0 transition metal oxyanions: Os 8+ > Mo 6+ > W 6+ >> V 5+ > Nb 5+ > Ti 4+ > Ta 5+ > Hf 4+ > Zr 4+ > Re 7+ >> Y 3+ > Sc 3+ . Such ranking varies by coordination number; for [4] it is Re 7+ > Ti 4+ > V 5+ > W 6+ > Mo 6+ > Cr 6+ > Os 8+ >> Mn 7+ ; for [5] it is Os 8+ > Re 7+ > Mo 6+ > Ti 4+ > W 6+ > V 5+ > Nb 5+ . It is concluded that non-octahedral coordinations of d 0 ion configurations are likely to occur with bond-length variations that are similar in magnitude to their octahedral counterparts. However, smaller bond-length variations are expected from the PJTE for non- d 0 transition metal oxyanions.

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