Causal mechanisms underlying bond-length variation in inorganic solids are resolved, and the extent to which these mechanisms individually result in bond-length variation is quantified for transition metals with anomalous bond-length distributions.Bond-length distributions are examined for transition metals 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 O2−. 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, Δtopol and Δcryst, 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  and (4) crystal-structure effects. The indices Δtopol and Δcryst 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 , 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 ; resolution of the origins of multiple-bond formation follows calculation of the Δtopol and Δcryst 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 d0 transition metal oxyanions: Os8+ > Mo6+ > W6+ >> V5+ > Nb5+ > Ti4+ > Ta5+ > Hf4+ > Zr4+ > Re7+ >> Y3+ > Sc3+. Such ranking varies by coordination number; for [4] it is Re7+ > Ti4+ > V5+ > W6+ > Mo6+ > Cr6+ > Os8+ >> Mn7+; for [5] it is Os8+ > Re7+ > Mo6+ > Ti4+ > W6+ > V5+ > Nb5+. It is concluded that non-octahedral coordinations of d0 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-d0 transition metal oxyanions.Bond-length distributions are examined for 63 transition metal ions bonded to O For instance, the wide array of metastable oxidation states characteristic of transition metals facilitates electron-transfer reactions central to catalysis  in the late 1970s. Recently, Waroquiers et al. , using 177 446 bond lengths extracted from 9210 crystal structures refined since 1975; these data cover most ions of the periodic table and the coordination environments in which they occur in inorganic compounds. Here, we report bond-length data for 63 transition metal ions bonded to O2.2−; (II) to resolve the causal mechanisms underlying bond-length variation for transition metals bonded to O2−; (III) to quantify the extent to which causal mechanisms result in bond-length variation for those transition metal configurations with anomalous bond-length distributions. We split this article into three parts in accord with these objectives.There are three objectives in this work: (I) to provide a comprehensive description of bond-length variations for transition metals bonded to OIntroduction; essentially, the resolution and quantification of anomalous bonding behaviour will facilitate the targeted design of materials whose functional properties are linked to asymmetric coordination environments, and will further facilitate optimization of these properties within the constraints of physically realistic crystal structures.The information we derive in (I) will provide crystallographers in a broad sense with more comprehensive and accurate bond-length data than are currently achievable via addition of ionic radii. These data are useful for refining and interpreting new crystal structures (particularly Rietveld refinements), modelling crystal structures and assessing the validity of computational studies, without which much effort is wasted on unrealistic atomic arrangements  on the grand mean bond length (and its standard deviation), skewness and kurtosis for the alkali and alkaline earth metal ions bonded to O2− in the first article of this series  and kurtosis (±0.6) are obtained for sample sizes greater than ∼115 coordination polyhedra. Table 2Fig. 23.2.2−, transition metals have an average range of bond lengths of 0.475 Å for ion configurations with sample sizes greater than ten coordination polyhedra (n = 74 ion configurations). As a function of electronic configuration, the average bond-length ranges are 0.492 (d0), 0.736 (d1), 0.399 (d2), 0.221 (d3), 0.505 (d4), 0.391 (d5), 0.338 (d6), 0.414 (d7), 0.245 (d8), 0.646 (d9) and 0.585 Å (d10). For octahedral coordination (n = 33 ion configurations), these numbers are 0.585 (d0), 0.788 (d1), 0.399 (d2), 0.221 (d3), 0.755 (d4), 0.517 (d5), 0.264 (d6), 0.614 (d7), 0.589 (d8), 0.893 (d9), 0.756 Å (d10). The wide variation in these bond-length ranges demonstrates inconsistent bonding behaviour as a function of electronic configuration and coordination number. This anomalous behaviour is, to some extent, expected from the Jahn–Teller effect (JTE); bond-length variations attributable to the JTE will be discussed in detail below, along with other significant causes of bond-length variation. The largest bond-length ranges are for [6]V4+ , [6]Mo6+ , [7]Hg2+ , [6]V5+ , [6]W6+ , [6]Hg2+  and [6]Cu2+ , and it is notable that all these ion configurations show multimodal distributions.When bonded to OFig. S3, and those with adequate sample size , while that of ions exhibiting the classic JTE in octahedral coordination (weak or strong) is 0.097 Å. For the latter group, the largest mean bond-length ranges are observed for [6]Cu2+ 0.174, [6]V4+ 0.103, [6]Mn3+ 0.094 and [6]Mo5+ 0.070 Å. For the d0 transition metals, the largest ranges  are [6]Nb5+ 0.102, [6]W6+ 0.100, [6]Ta5+ 0.100, [8]Y3+ 0.096, [6]Ti4+ 0.094, [6]Mo6+ 0.091 and [6]Zr4+ 0.082 Å.Mean bond-length distributions are given in 2− (0.085 Å) is typical of ions not showing electronic and/or crystal-structure effects, and is due to their high Lewis acidity  to moderately distorted (10–20 × 10−3) to highly distorted (>20 × 10−3). There is a strong correlation between bond-length distortion and mean bond length for moderately distorted ion configurations (10–20 × 10−3) or higher. In previous articles in this series, we found that the correlation between bond-length distortion and mean bond length is strong for ion configurations with values of distortion >20 × 10−3 for metalloids and post-transition metal ions bonded to O2− , citing the distortion theorem  bond lengths within the constraints of space group and translational symmetry is the leading cause of mean bond-length variation in crystals. Below, we expand on their analysis to investigate the underlying causal mechanisms of bond-length variation in transition metal oxide and oxysalt crystals.Gagné & Hawthorne 2017b examined4.Fig. S1). We further summarize the theoretical underpinnings of these mechanisms.Transition metal oxides and oxysalts are ideally suited for examining bond-length variations in solids as they are highly susceptible to two important mechanisms underlying bond-length variation: (i) coupled electronic vibrational degeneracy (leading to the JTE) and (ii) formation of multiple (π) bonds. In recent years, the non-local bond-topological asymmetry of coordination environments (sometimes referred to as asymmetry in the bond network) has been proposed as an additional mechanism underlying bond-length variation in crystal structures  non-local bond-topological effects, (2) multiple-bond formation, (3) electronic effects  and (4) crystal-structure effects. As shown below, mechanisms (1)–(3) may each lead to large bond-length variations and/or multi-modal bond-length distributions. However, these effects rarely occur in isolation, and their combination is often what leads to wide variations in bond lengths. It is worth noting that these mechanisms may be present in a limited number of crystal structures, or in all crystal structures in which the given ion configuration occurs. As a result, the shape of a bond-length distribution depends on the relative importance/magnitude of the effect(s) sampled. Thus, we focus our attention below on quantifiable features that may be recognized within the constituent data of these bond-length distributions.In Table 44.1.The connection between bond topology and bond-length variation was first demonstrated by Kunz & Brown 1995. Unfortu4.1.1.valence-sum rule, which states that the sum of the directed bond valences around an ion is equal to its oxidation state , and (ii) the path rule, which states that the sum of the directed bond valences along any path of bonds in a structure is zero when the path begins and ends on symmetrically equivalent ions :s is the bond valence for a bond of length R, and R0 and B are the bond-valence parameters of the ion pair. The bond-valence parameters are constants that are typically derived empirically; large sets of bond-valence parameters include those of Gagné & Hawthorne , and where the sum is taken over the j bonds involving ion i. For structure verification, deviation up to ∼6–7% is deemed acceptable for any given site of a crystal structure  to remove doubtful data from our dataset; similarly, equation (2)pymatgen  bond valences intrinsic to a crystal structure. A priori bond valences depend only on the formal valences (oxidation states) of the ions at the sites in the structure and the connectivity of chemical bonds in the structure (i.e. the bond topology). Similarly, a priori bond lengths are defined as the ideal  bond lengths intrinsic to a given site assignment for a crystal structure, and are obtained via the conversion of a priori bond valences using equation (1)A priori bond lengths further depend on the identity of the site occupants, due to their dependence on the bond-valence parameters R0 and B in equation (1)Of greatest relevance to this work is that the bond-topological underpinnings of the bond-valence model allow prediction of the Pnma perovskite crystal structure, with two crystallographically distinct cation (A and B) and anion (X1 and X2) sites, Ae.g. GdB3+Mn3+O3 R0 = 1.988 and B = 0.433 v.u. for Gd3+, and R0 = 1.823 and B = 0.247 v.u. for Mn3+; Gagné & Hawthorne, 2015For example, let us consider the B Table 5. Denotin al. 2018]. Thus, 2−.As we show below, crystal structures often have intrinsic requirements for an uneven distribution of bond valences (and thus bond lengths). Next, we investigate the inner workings of this phenomenon and the extent to which it results in bond-length variation for transition metals bonded to O4.1.2.T–P stability ranges, rate of crystallization, external field(s) etc. Where such constraints are absent or have negligible effect, a crystal structure is observed with the lowest possible number of crystallographically distinct sites, i.e. equal to the number of distinct elements in the compound . In this configuration, cations and anions distribute their a priori  bond valences evenly Cr6+ readily polymerizes into corner-sharing dimers . Ideally, the Cr6+O4 tetrahedron would have four bonds 1.5 v.u. in strength, but this is prohibited by the bond-valence constraints of the bridging [2]O2−. As a result, Cr6+O4 dimers form three bonds 5/3 v.u. (1.608 Å) in strength and one bridging bond 1 v.u. (1.799 Å) in strength, thus resulting in a bimodal distribution of bond lengths for corner-sharing Cr6+O4 dimers. While this mechanism may seem to be different from that of non-local bond-topological asymmetry (see above), the difference is a matter of interpretation; bridging and non-bridging bonds are necessarily bond-topologically inequivalent. As such, we sometimes use the terms ‘bond-topological asymmetry’ and ‘bond-topological effects’ interchangeably in this article, although we tend to use the latter for more general discussion.Bond-length variation driven by bond-topological asymmetry can be plainly illustrated via localized bonding interactions in strongly bonded units, for which bond-length variations result from the inter-connectedness of the bond-valence constraints of cations and anions with regard to the valence-sum rule V5+, [6]W6+, [6]Mo6+, and sometimes [6]Ti4+, [6]Nb5+ and [6]Ta5+, and (iii) additional (non-bridging) bonds made by the bridging anion(s) etc. In other words, these bond-valence constraints will experience further variability based on the exact bond topology of the crystal structure. Fortunately, one does not have to keep track of all such variables, unless one wishes to rationalize the exact pattern of a priori bond valences, whose simple calculation is otherwise sufficient for all intents and purposes. Thus for oxides, polymerization of strongly bonded units invariably leads to bond-length variation when OS/CN > 1 for [2]O2−br, > 2/3 for [3]O2−bretc. This condition is necessary but not sufficient, as some combinations of charge and coordination number cannot result in polymerization of the strongly bonded unit, e.g. for [4]Os8+.One may list all possible bond-valence patterns arising from various polyhedra, polyhedral connectivity and degree of polymerization to realize the richness of bond-valence constraints in the solid state and the scale of bond-length variation that arises solely from bond-topological constraints. For example, we enumerate possible bond-valence patterns arising from various corner-sharing tetrahedra for +5 and +6 oxidation states in Figs. 83Nb5+O4 . This split is further accentuated by displacement of the Nb5+ ion toward the face made by the O1 ions as a result of a pseudo Jahn–Teller effect, resulting in an effective split in the bond lengths of 1.858–2.130 Å .We illustrate this concept with Li4.1.4.cat is greater than 〈BV〉br an. In those cases, the bond-valence constraints on bridging anions induce an increase in the bond valences at the non-bridging bonds, leading to considerable bond-length variation. The inverse situation, where 〈BV〉cat is smaller than 〈BV〉br an, also results in a clear-cut bond-length variation driven by bond-topological asymmetry, this time in the form of a decrease in the bond valences at the non-bridging bonds. This phenomenon may lead to bond-length variability of similar magnitude; although the bond-valence variability is typically less, the cations involved are of lower Lewis acidity . The resulting dispersion of bond lengths for W5+ is 1.833–2.005 Å.A simple illustration of this effect is seen in PWs Table 8, comparea priori bond valences for Fe3+ in aegirine, NaFe3+Si2O6 , while the two non-bridging O2 sites bond only to Si4+ and Na+  in holosymmetric coordination, whereas bonds to O1 decrease to 4 × 0.45 v.u. The predicted dispersion of bond lengths is 1.950–2.053 Å (observed 1.939–2.113 Å), with Δtopol = 0.067 and Δcryst = 0.038 v.u.To show this effect for a cation of lower Lewis acidity, we calculate the + Table 9. To acco4.2.et al., 2003Many molecular features of coordination complexes are preserved as they are incorporated into crystal structures, and their electronic properties are often more important than the ensuing steric constraints of the crystal structure  of the transition metal. However, only the tg2 orbital has the appropriate symmetry and spatial overlap to mix with that of the transition metal, and the other three orbitals remain non-bonded. The MO levels are filled with ligand electrons up to tg2 , at which point transition metal d electrons begin to fill levels starting from tg2* in a way that progressively negates the favourable π interaction, i.e. a tg2 state of lower energy in comparison with non-bonding. For this reason, the most favourable π-bonding interactions for octahedrally coordinated transition metal oxyanions  involve transition metals with few to no d electrons; this is well supported by our data, as we will see below. Following favourable π-type interaction, the complex is described as forming ‘multiple bonds’ to one or more of its ligands  molecular fragments. Additional bonding schemes, topol. The continuous (non-integer) nature of bond valences, combined with the overlapping effects of non-local bond-topological asymmetry and multiple-bond formation, renders a clear-cut divide of whether or not multiple-bond formation is the main reason for bond-length variation difficult to achieve. Our analysis suggests that multiple-bond formation may be the main reason underlying bond-length variation where (i) Δtopol > Δcryst, (ii) BVmax > 1.25 MBV  > 1.75 v.u. and (iii) the formation of π bonds is not an inherent result of the polymerization of strongly bonded units, in which case we consider non-local bond-topological asymmetry as the main cause of bond-length variation. For (ii), the condition BVmax > 1.25 MBV is implemented so as not to mistakenly classify those ion configurations that form strong π bonds by necessity (e.g.[4]Re7+) as due to multiple-bond formation. For example, we calculated the a priori bond valences for seven [6]V4+ polyhedra in six crystal structures to find 〈Δtopol〉 = 0.207 and 〈Δcryst〉 = 0.145 v.u. For this ion configuration, BVmax is frequently >1.75 v.u., higher than 1.25 × 2/3 v.u. [V4+O6]8− units are often observed as monomers. As the formation of strong π bonds is not an inherent product of polymerization, we conclude that the formation of π bonds is the main factor underlying bond-length variation for this ion configuration when bonded to O2−. Following similar logic, we find bond-length variation to result from bond-topological asymmetry for [Cr6+O4]2−, under the special case ‘Polymerization of strongly bonded units where 〈BV〉cat is greater than 〈BV〉br an’, as the formation of the strong π bonds is indeed caused by polymerization of the [Cr6+O4]2− unit. When Δcryst > Δtopol, multiple-bond formation is always a result of the pseudo Jahn–Teller effect (below) for d0 transition metals.As discussed in Section 4.1.24.3.for a given crystal structure, i.e. for which the effect of non-local bond-topological asymmetry on bond lengths is disregarded as a quantifiable and predictable constant. However, the extent to which electronic effects affect bond lengths in solids has yet to be quantified on a large scale, hindering the rapid identification of these phenomena in crystal structures and clouding the extent to which bond-length variations may be expected from these effects within the constraints of physically realistic crystal structures .In recent decades, significant developments in electronic structure theory have reduced the problem of molecular engineering and materials design to increasingly quantitative calculations of the electronic structure of both known and hypothetical compounds. It is expected that electronic effects represent the dominant force underlying bond-length variation s and anion p orbitals, leading to a high-energy antibonding state which, via structure distortion, may interact with empty cation p orbitals to form a localized electronic state where the lone pair resides  are typically only relevant to hetereo-ligand coordination centres and are not considered here.The two most common types of electronic effects present in inorganic solids are (i) lone-pair stereoactivity and (ii) coupled electronic vibrational degeneracy. Lone-pair stereoactivity results from strong interaction between cation 4.3.1.et al., 1937The Jahn–Teller effect is a mechanism of symmetry breaking in molecules and solids, and results from strong electron–vibrational (vibronic) and electron–phonon interactions in molecules and crystals, respectively  is most succinctly described via crystal-field theory (CFT). Fig. 11Dq values from Bersuker, 2010d electronic states split into triply degenerate tg2  and doubly degenerate eg  electronic states, with the t2 orbital higher in energy , and ‘weak’ otherwise. For an octahedral crystal field, degeneracy is strong for electron configurations HS d4, LS d7 and d9; those prone to weak JTE are d1, d2, LS d4, LS d5, HS d6 and HS d7 . For a tetrahedral field, the largest distortions (i.e. bond-length variations) are expected for configurations HS d3, HS d4, d8 and d9, and so on and so forth for every coordination geometry. For example, the strong JTE is observed for [6]Mn3+ (d4) in Gd3+Mn3+O3  bond valences 4 × 0.471 (2 × 0.194 and 2 × 0.700) and 2 × 0.559 (0.612) v.u. for Mn3+, and Δtopol = 0.039 and Δcryst = 0.186 v.u. As a side note, the wide discrepancy seen between the a priori and observed bond valences in this example emphasizes the inability of a priori bond valences to model the Jahn–Teller distortion (more on this below).In the classic description of the JTE, degenerate y Fig. 11. The occtopol and Δcryst indices, we identify three ion configurations from Table 4[6]Mn3+, [6]Cr2+ and [6]Cu2+) and none whose main cause is the weak JTE. We observe these effects as minor contributors to bond-length variation for [6]Co2+ (JTEs) and [6]V3+ and [6]Mo5+ (JTEw). These will be discussed in Section 5.3After examining our dataset in detail for the strong and weak JTE  in various crystal fields, and comparing the magnitude of the effect with that of distortion of bond-topological origin via the ΔNear-degenerate electronic states. The pseudo Jahn–Teller effect (PJTE) results from the vibronic mixing of two (or more) near-degenerate electronic states under nuclear displacement  the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), must be small, and there must be a distortion mode that has the same symmetry as the HOMO to LUMO transition (the energy gap is a function of ligand identity)  and the LUMO |3yzd〉 (from Ti4+) of the system. It can then be seen that favourable vibronic mixing results in a positive overlap integral upon displacement of the Ti4+ ion along the y axis (i.e. ‘off-centring’), resulting in an energetically favoured lower-symmetry configuration. For example, Mo6+ displaces toward a corner in Cs[Mo6+2O3(PO4)2]  bond valences 0.724 (0.702), 0.731 (0.541), 2 × 0.768 (0.640 and 0.730), 0.772 (0.678) and 1.237 (1.988) v.u., and Δtopol = 0.134 and Δcryst = 0.204 v.u. In this example, Mo6+ displaces toward O1 along the O6—Mo1—O1 axis, resulting in strong/weak bonds of 1.988/0.541 v.u.uker 2006]. We plod0 electronic configuration, although it should be noted that its occurrence is not limited to this configuration , moderate  and weak (Zr4+ and Hf4+) distorters  may further result in a variability in bond lengths up to the point of phase transition . Here, we group these effects under the designation of ‘crystal-structure effects’. These effects do not lead to multi-modality of the bond-length distributions, and they cause bond-length variations of significantly lower magnitude than those of mechanisms (1)–(3). As such, quantifying bond-length variations due to crystal-structure effects can only be done in the absence of other crystallographic mechanisms of bond-length variation for a given polyhedron  which otherwise overwhelm the Δcryst effect. Moreover, bond-length variation caused by these mechanisms is expected to fall semi-randomly about the mean bond length in a way that does not significantly alter the shape of bond-length distributions.The constraints of long-range order and periodicity have important implications with regard to bond distances. Variations in external conditions .Some polyhedra are inherently unable to adopt their configuration of highest symmetry as a result of the imperfect projection of their 4.4.2.d0 transition metals under the PJTE. We observe only one ion configuration in Table 45+–Ru5+ interactions for [6]Ru5+), discussed in Section 5.5Sometimes called steric effects, factors such as metal–metal and anion–anion repulsion and nearby stereoactive lone-pair electrons have been shown to influence bond-length variations and polyhedral distortion  may be skipped on first reading, until the reader requires further detail on specific ion configurations.In this section, we identify the principal and minor causal mechanism(s) underlying bond-length variation for the 52 ion configurations listed in Table 4a priori bond valences and indices Δtopol and Δcryst for polyhedra with anomalous bond-length dispersion; (ii) calculate 〈Δtopol〉 and 〈Δcryst〉 values to determine whether the bond-length distribution is primarily irregular as a result of bond-topological or crystallographic effects; (iii) identify the main and minor causes of bond-length variation based on frequency of observation and magnitude; (iv) describe the effect of causal mechanisms on the shape and range of the bond-length distribution; and (v) compare main and minor causes of bond-length variation with similar ion configurations where pertinent, and/or other relevant information.Our discussion of ion configurations follows a consistent form throughout: we (i) calculate 5.1.[6]Sc3+ [Fig. 1a)] has a subtly bimodal distribution of bond lengths. The main distribution peaks at 2.09 Å (0.50 v.u.) and the other maximum is at 2.12 Å (0.47 v.u.), and this occurs in conjunction with a hidden peak at 2.07 Å (0.53 v.u.). For example, Na3Sc2(PO4)3  bond valences 3 × 0.519 (0.555) and 3 × 0.481 (0.477) v.u., with Δtopol = 0.019 and Δcryst = 0.020 v.u. The tail at shorter bond lengths is longer than expected; constituent data include those of LiSc(SiO3)2  bond valences 4 × 0.45 (2 × 0.491 and 2 × 0.383) and 2 × 0.6 (0.607) v.u., with Δtopol = 0.067 and Δcryst = 0.038 v.u. Thus, bond-length variations for this ion configuration mainly result from non-local bond-topological asymmetry, and to a lesser extent from the pseudo Jahn–Teller effect.[6]V3+ [Fig. 1e)] occurs as monomers, oligomers, chains, sheets and frameworks. Despite the proclivity of the [V3+O6]9− unit for polymerization, its bond-length distribution is rather regular and does not cover an overly large range of bond lengths that is typical of other ions exhibiting this behaviour . There is, however, a subtle second maximum in Fig. 1e) at 2.06 Å (0.44 v.u.). This maximum arises from structures in which [V3+O6]9− polymerizes into oligomers, chains, sheets and frameworks, all of which result in weaker bonds 0.4–0.5 v.u. (versus 0.55–0.65 v.u. for the strongest bonds). This for example agrees with a priori (observed) bond valences calculated for distorted polyhedra of edge-sharing chains in LiV3+(Si2O6) 2  for this ion configuration. Average values of Δtopol and Δcryst for five polyhedra in four structures are 0.064 and 0.040 v.u., respectively.[4]V5+ [Fig. 1h)] has a somewhat symmetrical distribution of bond lengths when bonded to O2−. The [V5+O4]3− unit polymerizes into various linear oligomers, chains and rings, with important implications regarding Obr as the mean bond valence for this ion configuration is >1 v.u. [special case (i) of non-local bond-topological asymmetry]. Symmetrization of the [4]V5+—O2− bond-length distribution is well illustrated from the bond-length pattern that results from V4O12 and V6O18 rings in K3CaV5O15  bond valences for 13 coordination polyhedra in six structures; the mean values of Δtopol and Δcryst of 0.122 and 0.099 v.u. indicate that, despite a high value for Δcryst (largely attributed to the PJTE), non-local bond-topological asymmetry is the main reason underlying bond-length variation for this ion configuration. We do not find coordination polyhedra in our data where multiple-bond formation is the principal driver of bond-length variation; the strong π bonds are a product of polymerization and displacement of the cation away from the centre of the polyhedron via the PJTE. In KCu2+5V5+3O13 , 1.121 (O8) and 2 × 1.364  v.u. In this tetrahedron, V5+ moves off-centre toward O8 and away from O1, resulting in observed bond valences of 1.448 (O8) and 0.738 v.u. (O1) for a particularly strong case of PJTE; Δtopol = 0.081 and Δcryst = 0.123 v.u. In Cu2+2V5+2O7  bond valences are 0.980 (0.908), 1.274 (1.279), 1.288 (1.205) and 1.291 (1.329) v.u. for V1, with Δtopol = 0.135 and Δcryst = 0.039 v.u., and 0.976 (1.054), 1.271 (1.166), 1.288 (1.205) and 1.466 (1.478) v.u. for V2, with Δtopol = 0.137 and Δcryst = 0.069 v.u.[6]Cr3+ [Fig. 1l)] has a somewhat regular distribution of bond lengths, with two anomalies: (i) a spike of bond lengths at 1.99 Å (v.u.) and (ii) a relatively long tail at shorter bond lengths. For (i), the bond distances originate from a study of the Cr23+O4 solid-solution series in spinels, whereby 11 structures were refined with holosymmetric [Cr3+O6]9− units, thus providing 66 distances of 1.99 Å for our dataset 6  bond valences between 0.409–0.639 (0.383–0.617) v.u., with Δtopol = 0.074 and Δcryst = 0.038 v.u. for Cr1, and 0.415–0.715 (0.396–0.643) v.u., with Δtopol = 0.089 and Δcryst = 0.061 v.u. for Cr2, thus showing the non-local bond-topological asymmetry as the origin of bond-length variation. For a total of seven coordination polyhedra for which we calculated Δtopol and Δcryst, the mean values for these indices are 0.077 and 0.056 v.u., respectively. With a d3 electronic configuration, Cr3+ is not susceptible to the JTE, suggesting that crystal-structure effects are a significant cause of bond-length variation for this ion configuration.[4]Cr6+ [Fig. 1m)] has a multimodal distribution of bond lengths where bonded to O2−. [Cr6+O4]2− units polymerize as linear oligomers (primarily dimers), in addition to forming very strong π bonds in monomers. As discussed for [4]V5+ above, polymerization of the [Cr6+O4]2− unit inherently results in varying bond lengths whereby the mean bond-valence for this ion configuration is >1 v.u. [special case (i) of non-local bond-topological asymmetry]. Thus, the maxima in Fig. 1m) result from the superposition of the bond-length constraints of dimers, 1 v.u. (1.799 Å) and 3 × (5/3) v.u. (1.608 Å), on the normal distribution of bond lengths for monomers . We calculated the a priori (observed) bond valences for eight coordination polyhedra from three structures; average values of Δtopol and Δcryst are 0.127 and 0.085 v.u., respectively, confirming that the primary reason for bond-length variation is non-local bond-topological asymmetry [special case (i)] for this ion configuration, closely followed by the PJTE which presumably accounts for most of Δcryst. The structure of Tl3+2(Cr6+O4)3  bond valences are 1.2 (1.053), 2 × 1.4 (1.533 and 1.617) and 2 (1.843) v.u. for Cr1, and 4 × 1.5 (2 × 1.477 and 2 × 1.639) v.u. for Cr2. These values give Δtopol = 0.250 and Δcryst = 0.164 for Cr1, and Δtopol = 0 and Δcryst = 0.081 for Cr2. Multiple-bond formation appears to be the main reason underlying bond-length variation in a small number of structures, for example in (NH4)Fe3+(Cr6+O4)2  by Cr6+ must result from proper multiple-bond formation.[4]Fe3+ [Fig. 1u)] has a unimodal distribution with anomalously low kurtosis, with a maximum at 1.85 Å . [Fe3+O4]5− units polymerize into various oligomers, chains and frameworks via corner and/or edge-sharing, thus creating a wide range of bond-length constraints around that of 0.75 v.u. for a regular tetrahedron. These constraints effectively ‘flatten’ the shape of the distribution, similar to what happens in [4]V5+. With a mean bond valence of 0.75 v.u., [4]Fe3+ sometimes redistributes its bond valences to satisfy Obr requirements . For example, Fe—O—Fe bonds adjust to 1.764 Å (1.004 v.u.) (and 3 × 1. 884 Å) for Fe3 in K(Fe3+11O17)  (above) and NaZnFe3+2(PO4)3  and not to crystal-structure effects.[6]Co2+ [Fig. 1z)] has a regular bond-length distribution when bonded to O2−, although with a very long tail at longer bond lengths. This ion is susceptible to the strong JTE when in the low-spin state (d7) and to the weak JTE when in the high-spin state. Co2+ is usually presumed to be HS ], in contrast to the classic bimodal distribution observed for ions under the effect of the strong JTE. Of the ten longest bonds observed for this ion configuration (with one bond length >2.35 Å), spin is only reported for one coordination polyhedron, in Co2+3(BPO7) . Similar bond-length patterns occur in other structures with no report of spin state, e.g. in SrCo2+2(As5+O4)2   bond valences for five coordination polyhedra from four crystal structures, namely YCo2+(BO2)5 2  2  in a low-spin state.[6]Zn2+ [Fig. 1ai)] has a regular distribution of bond lengths when bonded to O2−, with a very long tail at longer bond lengths. We calculated the a priori bond valences for five polyhedra from five crystal structures, and obtained average Δtopol and Δcryst values of 0.056 and 0.050 v.u., respectively. In NaZnFe3+2(PO4)3  bond valences are 2 × 0.252 (0.168 and 0.274), 0.263 (0.212), 0.266 (0.412) 0.416 (0.433) and 0.550 (0.558) v.u., with Δtopol = 0.100 and Δcryst = 0.055 v.u. In Mn2+Zn2Ta5+2O8  bond valences are 3 × 0.256  and 3 × 0.410  v.u., with Δtopol = 0.077 and Δcryst = 0.129 v.u. The large Δcryst value is attributable to the PJTE for this polyhedron, as is the case for Zn4 in Zn3(Se4+O3)·3H2O Fig. S15  bond valences gives Δtopol = 0.086 and Δcryst = 0.079 v.u. Thus, despite the strong distorting effect of the PJTE, bond-length variations primarily result from non-local bond-topological asymmetry for this ion configuration . This configuration is the only one for Y3+ where Δcryst > Δtopol; bond-length variations in coordinations [6] to [9] primarily result from the PJTE.[6]Mo4+ [cj)Fig. S1 and 2.03 Å (0.62 v.u.). [Mo6+O4]2− units occur as monomers and corner-sharing dimers. The monomers are more or less regular. In corner-sharing dimers, the [Mo6+O4]2− units adjust their bond valences to 1 × 1 v.u. at Obr and 3 × 0.66 v.u. for the three non-bridging bonds. This rearrangement, a result of non-local bond-topological asymmetry [special case (ii)], results in a bimodal distribution of bond lengths. For example, a priori (observed) bond valences in Pb2+2[Mo4+2O(PO4)2(P2O7)] ] forms a unimodal bond-length distribution with a very wide dispersion of bond lengths when bonded to O2−, with the majority of MoO4 tetrahedra occurring as monomers. We calculated the a priori bond valences for eight polyhedra from five crystal structures, and obtained average Δtopol and Δcryst values of 0.104 and 0.087 v.u., respectively. Although 〈Δcryst〉 is much smaller than for [5]Mo6+ and [6]Mo6+, it shows that a considerable amount of bond-length variation is due to the PJTE for this ion configuration. Nonetheless, the effect of bond-topological asymmetry is stronger in most cases. One convincing example is for the structure of Zr(Mo6+O4)2  bond valences 3 × 1.333  and 2 (1.918) v.u., with Δtopol = 0.250 and Δcryst = 0.109 v.u.[6]Cd2+ [Fig. 1bd)] forms a unimodal bond-length distribution with a few anomalously long bonds. Calculation of a priori (observed) bond valences for two structures gave average values of Δtopol and Δcryst of 0.044 and 0.040 v.u., respectively. In Cd3Te6+O6  bond valences are 4 × 0.314 (2 × 0.324 and 2 × 0.409) and 2 × 0.373 (0.296) v.u., with Δtopol = 0.026 and Δcryst = 0.061 v.u. Presumably, the contribution to Δcryst is due to the PJTE, which is not uncommon in d10 transition metal complexes . Anomalously long bonds, e.g. 2.754 Å in Cd8(OH)12(SO4)2(H2O)  ] forms what appears to be a multi-modal distribution of bond lengths made entirely of [Re7+O4]− monomeric units, with a main maximum at 1.72 Å (1.73 v.u.) and other maxima at 1.69 (1.86) and 1.75 (1.61 v.u.) Å. We calculated the a priori (observed) bond valences for two coordination polyhedra in (V4+O)(Re7+O4)2 Fig. S1[(HO)2Te6+O4]2](H2O)16  is unlikely to result in such a marked variation (>0.25 Å).[6]Hg2+ [Fig. 1bq)] has a very messy distribution of bond lengths when bonded to O2−. This is partly due to the difficulty of defining a coordination polyhedron for this ion configuration across different structure types. Hg2+ typically forms two relatively short bonds (and a series of longer bonds for coordination numbers [3] to [7]), although this is not always true for [6]. For example, [Hg2+(H2O)6](Cl7+O4)2  in Hg2+(PO3)2  bond valences are 4 × 0.286  and 2 × 0.429 (0.439 and 0.543) v.u., with Δtopol = 0.063 and Δcryst = 0.066 v.u. In this structure, the Hg2+ ion presumably moves off-centre as a result of the PJTE ] forms a bimodal distribution of bond lengths with maxima at 1.61 (1.58) and 1.96 Å (0.60 v.u.), integrating for one and four bonds, respectively. [5]V4+ forms one (vanadyl) π bond for all but one structure in our dataset, that of NaV6O11  with bond valences 3 × 0.975 (1.785) and 2 × 0.423 v.u. (2.089 Å). We calculated a priori (observed) bond valences for two polyhedra from two crystal structures. In V3+2[V4+O(P2O7)2] 2]  bond valences are 2 × 0.648 (0.506 and 0.542), 2 × 0.690 (0.579 and 0.613) and 1.325 (1.818) v.u., with Δtopol = 0.210 and Δcryst = 0.186 v.u. The high value of Δtopol is primarily driven by π-bond formation, but presumably also by non-local bond-topological asymmetry (their respective contributions cannot be resolved). The very high value of Δcryst is somewhat suspicious. While the [5]V4+ ion (d1) is not susceptible to the JTE in a square-pyramidal coordination, some of the bond-length variation due to crystallographic effects is probably due to the PJTE, whereby the V4+ ion typically moves off-centre toward the O2− ion of the vanadyl bond. In this particular structure, however, a more important factor is at play. We may trace the value of Δcryst to a calculated a priori bond valence of 1.325 v.u. for the π bond; this value is very low because the O2− ion involved in the π bond is predicted to form a bond of 0.675 v.u. to Pb2+ (observed 0.167 v.u.). These values are not those observed in practice. Instead, the [V4+O5]6− units form corrugated (and presumably highly strained) layers, and the high value of Δcryst results from structural incommensuration.[6]V4+ [Fig. 1g)] forms a trimodal distribution of bond lengths, with maxima around 1.61 (1.58), 2.00 (0.54) and 2.25 Å (0.27 v.u.). [V4+O6]8− units occur as monomers, oligomers, chains, sheets and frameworks where bonded to O2−. By and large, these polyhedra adopt a [1+4+1] octahedral coordination, described by Schindler and co-workers as the number of vanadyl, equatorial and trans bonds of the polyhedron (listed in order), where the trans bond is the weakest bond formed  bond valences for four polyhedra from as many crystal structures. In (V4+O)(Re7+O4)2 (H2O)0.5 2(PO4)2  bond valences are 4 × 0.5  and 2 × 1 (0.388 and 1.624) v.u., with Δtopol = 0.222 and Δcryst = 0.219 v.u. In this structure, the large value of Δtopol results from non-local bond-topological asymmetry alone; however, the V4+ ion moves off-centre in the direction of the strong bond, resulting in one much stronger and one much weaker bond than predicted. It is unclear whether this phenomenon results from the weak JTE, the PJTE or a combination of both   bond valences 4 × 0.641 (0.675) and 2 × 0.718 (0.563) v.u., with Δtopol = 0.034 and Δcryst = 0.074 v.u.[5]V5+ [Fig. 1i)] forms a bimodal distribution of bond lengths with maxima at about 1.64 (1.50) and 1.92 Å (0.73 v.u.), and two superimposed peaks at about 1.89 (0.79) and 1.99 Å (0.61 v.u.). [V5+O5]5− units occur primarily in a square-pyramidal coordination  but also as triangular-bipyramidal and intermediate coordinations. They occur as monomers, oligomers, chains, sheets, clusters and frameworks, sharing corners and/or edges. We calculated a priori (observed) bond valences for three polyhedra in as many crystal structures. In K4(Cu2+V5+5O15Cl)  and 2 (1.873) v.u., with Δtopol = 0.400 and Δcryst = 0.073 v.u. In V5+AlMo6+O7  , 0.875 (0.827), 1.188 (1.190) and 1.438 (1.703) v.u., with Δtopol = 0.250 and Δcryst = 0.112 v.u. In this structure, bond-length variation associated with the strong π bond results from non-local bond-topological asymmetry and the PJTE.[6]V5+ [Fig. 1j)] forms a very messy distribution of bond lengths. We primarily observe the [V5+O6]7− units as clusters, but also as monomers, dimers, chains, sheets and frameworks, sharing corners and/or edges. Of the 293 coordination polyhedra in our dataset, 251 originate from edge-sharing clusters and 223 from decavanadate clusters. As we show below, cluster-type polyanions have specific bond-valence constraints for each of their crystallographically distinct polyhedra, which further vary as a function of the symmetry of the structure. These constraints result in a very messy  appearance for Fig. 1j), with added variability resulting from a suite of other effects. To elucidate the occurrence, prevalence and magnitude of these effects, we calculated a priori (observed) bond valences for 11 polyhedra from six crystal structures containing this ion configuration, including a structure containing the decavanadate cluster. First, we show the effect of π-bond formation in V5+2Se4+2O9  bond valences 2 × 0.4 (0.176/0.230 and 0.685/0.651), 2 × 0.6 (0.549/0.683 and 0.685/0.746), 1 (1.121/0.934) and 2 (1.743/1.756) v.u. for V1/V2, with Δtopol = 0.444/0.444 and Δcryst = 0.171/0.160 v.u. In addition, a significant amount of bond-length variation seems to be caused by non-local bond-topological asymmetry and the PJTE in this structure. The strong distorting effect of the PJTE is shown in Hg2+(V5+2O6)  bond valences 3 × 0.667 , 2 × 0.917 (0.516 and 0.846) and 1.167 (1.621) v.u., with Δtopol = 0.167 and Δcryst = 0.306 v.u. In hummerite, KMg(V5+5O14)(H2O)8 , the a priori (observed) bond valences are 2 × 0.553 (0.433 an 0.449), 0.836 (0.758), 0.845 (0.710), 1.102 (1.350) and 1.111 (1.282) v.u. for V1, with Δtopol = 0.187 and Δcryst = 0.143 v.u., 0.280 (0.319), 0.564 (0.624), 0.573 (0.590), 0.870 (0.922), 1.203 (0.919) and 1.510 (1.638) v.u. for V2, with Δtopol = 0.361 and Δcryst = 0.097 v.u., 0.316 (0.329), 0.600 (0.608), 0.609 (0.573), 0.920 (0.891), 0.988 (0.986) and 1.568 (1.643) v.u. for V3, with Δtopol = 0.325 and Δcryst = 0.027 v.u., 0.132 (0.273), 0.682 (0.513), 0.804 (0.752), 0.876 (0.917), 1.055 (0.901) and 1.451 (1.668) v.u. for V4, with Δtopol = 0.294 and Δcryst = 0.129 v.u., and 0.166 (0.246), 0.725 (0.557), 0.756 (0.831), 0.770 (0.798), 0.910 (0.901), 1.673 (1.672) v.u. for V5, with Δtopol = 0.305 and Δcryst = 0.060 v.u. Thus, bond lengths for [V5+O6]7− in the decavanadate cluster largely vary as a result of bond-topological asymmetry, including the formation of moderately strong π bonds (〈Δtopol〉 = 0.295 v.u.), and to a more modest extent as a result of the PJTE (〈Δcryst〉 = 0.091 v.u.). The average values of Δtopol and Δcryst are 0.290 and 0.173 v.u. for the 13 polyhedra considered for this ion configuration.[5]Mo5+ [ck)Fig. S1(] occurs in two structures in our dataset. In Pb2+(Mo5+O)10(P2O7) ] forms a bimodal distribution of bond lengths, with maxima at 1.68 (1.95) and 2.04 Å (0.62 v.u.). This ion configuration is characterized by the formation of a strong double bond (1.953 v.u. on average), sometimes followed by polymerization into corner-sharing or edge-sharing dimers. We calculated a priori (observed) bond valences for two polyhedra in as many crystal structures. In Ba(Mo5+2P4O16) , 2 × 0.604 (0.651 and 0.659), 0.639 (0.676) and 2 (2.079) v.u., with Δtopol = 0.389 and Δcryst = 0.075 v.u., showing the strong distorting effect of π-bond formation for this ion configuration. In Pr3Mo5+O7  bond valences are 2 × 0.746 (0.663 and 0.810) and 4 × 0.877  v.u., with Δtopol = 0.058 and Δcryst = 0.159 v.u. Thus no π bond is predicted for this structure, and bond-length variation is primarily driven by crystallographic factors, most likely the weak JTE (d1 electronic configuration).[5]Mo6+ [Fig. 1au)] forms what appears to be a unimodal bond-length distribution, despite the formation of a strong π bond in all of its structures (1.738 v.u. on average). The somewhat unimodal appearance is due to a continuous series of observed bond valences whereby [Mo6+O5]4− units form multiple strong bonds (e.g. 1.552 v.u. on average for the second shortest bond). [Mo6+O5]4− units occur as monomers with square-pyramidal and triangular-bipyramidal shape, sometimes polymerizing into chains, sheets and frameworks with Mo6+O6 octahedra. We calculated a priori (observed) bond valences for three polyhedra from as many crystal structures. In Cs2(Mo6+3O10) 2]  bond valences are 0.915 (0.850), 1.026 (0.801), 1.078 (0.735) and 2 × 1.491 (1.735 and 1.891) v.u., with Δtopol = 0.232 and Δcryst = 0.256 v.u. In Cs(Np5+O2)(Mo6+O4) ] forms a clear bimodal distribution, typical of that observed for octahedrally coordinated cations under the influence of the strong JTE. We calculated a priori (observed) bond valences for two of the seven structures in our dataset containing this ion configuration (the two anhydrous structures). In Cr2+3Cr3+4(PO4)6  bond valences are 4 × ∼0.25 (0.08–0.45) and 2 × ∼0.5 (0.48–0.58) v.u., with Δtopol = 0.170 and Δcryst = 0.118 v.u. In Cr2+3Cr3+4(PO4)6 ; the other nine polyhedra for this ion configuration clearly follow the [4+2] coordination expected of a ‘z-out’ Jahn–Teller distortion. We list the JTE as the main factor underlying bond-length variation for this ion configuration, although it is closely followed by the effect of non-local bond-topological asymmetry.[6]Mn3+ [Fig. 1q)] forms a bimodal distribution of bond lengths, typical of the strong JTE. However, inspection of the data making up the distribution reveals that of the 82 coordination polyhedra, only 39 have a distinct [4+2] (‘z-out’) coordination, while 17 have a [2+2+2] coordination, six have a [2+4] (‘z-in’) and two are regular (18 are ambiguous). A [4+2] coordination is easily explained by the strong JTE, for example in KMn3+(Se6+O4)2   bond valences 2 × 0.404 (0.155 and 0.230), 2 × 0.480 (0.506 and 0.874) and 2 × 0.616 (0.806 and 0.846) v.u., with Δtopol = 0.077 and Δcryst = 0.211 v.u. Regular polyhedra are probably due to disordered JT distortion, as shown by high anisotropic displacement parameters in Mn2+3Mn3+2(SiO4)3 ] forms a smooth bimodal distribution of bond lengths. Inspection of the data making up the distribution shows that of the 365 coordination polyhedra, 269 have distinct [4+2] (‘z-out’) coordination, 49 have a [2+2+2] coordination, ten have a [2+4] (‘z-in’) coordination and seven are regular (30 are ambiguous). [4+2] coordination follows classic (strong) JTE arguments, for example in Cu2+2V5+2O7  2(OH)2  bond valences 2 × 0.459 (0.54) and 4 × 0.270 (2 × 0.246 and 2 × 0.253) v.u., with Δtopol = 0.084 and Δcryst = 0.039. It is interesting that Δtopol > Δcryst for two of the three coordination polyhedra above. While this may be accidental, this result shows two things: (i) bond-topological effects are an important driver of bond-length variation for this ion configuration, and (ii) some structures appear to accommodate the bond-valence constraints of JTEs ions . We further calculated the a priori (observed) bond valences for a regular octahedron in Ba3Cu2+(Sb5+2O9) .5.4.[4]Ti4+ Ti4+ [Fig. 1c)] forms a peculiar bond-length distribution with a maximum at 1.96 Å (0.66 v.u.) and a very long tail at shorter bond lengths. The shape of the distribution results mainly from the formation of a strongly bonded axial ligand in square-pyramidal geometry, relative to the equatorial ligands. [Ti4+O5]6− units range in shape from square-pyramidal to distorted square-pyramidal to triangular-bipyramidal, with the strongest bond decreasing in strength along that series. In Cs2(Ti4+O)(P2O7)  bond valences 4 × 0.739  and 1.044 (1.66) v.u., with Δtopol = 0.098 and Δcryst = 0.217 v.u. In La3Ti4+O4Cl5  bond valences 2 × 0.625 (0.509), 2 × 0.915 (1.013 and 1.031) and 0.920 (1.019) v.u., with Δtopol = 0.140 and Δcryst = 0.109 v.u. These examples show that the strongest cause underlying bond-length variation for this ion configuration may vary between non-local bond-topological asymmetry and the PJTE as a function of polyhedral shape. Overall, the PJTE results in larger bond-length variations than bond-topological asymmetry, with mean Δtopol and Δcryst values of 0.119 and 0.163 v.u., respectively. For the square-pyramidal units, there is sometimes  a possibility of a sixth bond to complete the octahedron. However, these distances range from 2.70–3.77 Å in different structures, and we consider them far too long for inclusion as bonds.[6]Ti4+ [Fig. 1d)] forms a multi-modal bond-length distribution with subtle maxima arising from different effects. Primarily, [Ti4+O6]8− units form oligomers, chains, rings, sheets, clusters and frameworks, both as strictly corner-sharing or edge-sharing units, and sometimes as a mixture of both, resulting in a wide range of bond-valence constraints. Adding to the intrinsic bond-valence constraints is the off-centring of Ti4+ due to the PJTE, resulting in a very large range of observed bond lengths (1.648–2.474 Å). The near-uniqueness of the bonding environment of each and every TiO6 octahedron in our dataset somewhat detracts from the usefulness of rationalizing the shape of its compound bond-length distribution. Some of the noteworthy features of Fig. 1d) include a small maximum between 1.7–1.8 Å with a maximum at 1.74 Å, a strong maximum at ∼1.95 Å and a subtle maximum at 2.10 Å (0.44 v.u.). These maxima originate from the strongest and weakest bonds of corner-sharing chains, for which we typically observe one bond of 1.25–1.35 v.u. (1.72–1.74 Å) and five bonds of gradually decreasing strength in the ∼0.40–0.70 v.u. (1.94–2.13 Å) range, with an average bond length of 2.10 Å for the longest bond. A representative example is that of (Cs0.07Rb0.95)(Ti4+O)(As5+O4)  bond valences of 2 × 0.363 (0.397 and 0.673), 0.570 (0.631), 0.856 (0.525), 0.883 (0.474) and 0.964 (1.318) v.u. for Ti1, with Δtopol = 0.235 and Δcryst = 0.250, and 0.379 (0.522), 0.449 (0.577), 0.603 (1.218), 2 × 0.786 (0.615 and 0.654) and 0.997 (0.418) v.u. for Ti2, with Δtopol = 0.190 and Δcryst = 0.295 v.u. We may compare these numbers with the a priori (observed) bond valences of sheets of edge-sharing octahedra in Sm3+(Ti4+O3Cl)  bond valences for sheets of edge- and corner-sharing octahedra of [Ti4+O6]8− in La2(Ti4+2SiO9) , 2 × 0.625 (0.791), 2 × 0.685 (0.554) and 0.731 (0.871) v.u. for Ti2, with Δtopol = 0.055 and Δcryst = 0.162 v.u. Altogether, we get average values of Δtopol and Δcryst of 0.111 and 0.163 v.u., respectively, confirming that the PJTE has a larger effect on bond-length variation than non-local bond-topological asymmetry, overall.[7]Ti4+ [j)Fig. S13]4[Ti4+O(CO3)3](H2O)2  and making a partial double bond (1.50 v.u.) to a seventh O2− ion (in turn forming two hydrogen bonds to NH2 groups). The positions of the H atoms were not refined, and we cannot calculate the a priori bond valences for this structure. There is, however, significant displacement of Ti4+ toward O1 and away from O9 , suggesting the PJTE as the main reason underlying bond-length variation for this ion configuration.[6]Y3+ [Fig. 1aj)] has a somewhat regular distribution of bond lengths, with a main maximum at 2.27 Å (0.49 v.u.) and a more subtle maximum at 2.22 Å (0.55 v.u.). This latter maximum does not occur as a result of polymerization constraints; it either results from non-local bond-topological asymmetry, or because of slight (but consistent) off-centring of the central cation via the PJTE, resulting in some bonds in the range 2.20–2.24 Å. We calculated the a priori (observed) bond valences for olivine-structured NaY(GeO4) ] forms a regular distribution of bond lengths, with a ‘wide’ maximum, as also observed for [7]Y3+ and [9]Y3+. We calculated a priori (observed) bond valences for three coordination polyhedra from two polymorphs of Y2Ba2Cu2+Pt4+O8 [ICSD 63103 ] forms a very wide bond-length distribution with high kurtosis. A sharp maximum at 2.39 Å (0.36 v.u.) accounts for the majority of polyhedra where Y3+ makes eight equal bonds (i.e. does not move off-centre). A slightly distorted variant of a regular polyhedron can be seen in paraniite-(Y), Ca2Y(As5+O4)(W6+O4)2  bond valences of 4 × 0.317 (0.422) and 0.433 (0.362) v.u., with Δtopol = 0.072 and Δcryst = 0.131 v.u. A priori (observed) bond valences are much more scattered in Y3Re7+O8  v.u., with Δtopol = 0.085 and Δcryst = 0.071 v.u. for Y2. In KY(W6+O4)  bond valences are 2 × 0.230 (0.495), 4 × 0.400 (2 × 0.174 and 2 × 0.427) and 2 × 0.470 (0.475) v.u., with Δtopol = 0.058 and Δcryst = 0.088 v.u. The mean values of Δtopol and Δcryst are 0.082 and 0.093 v.u. over those structures, showing the PJTE and non-local bond-topological asymmetry to be approximately equally responsible for bond-length variations for this ion configuration.[9]Y3+ [Fig. 1am)] forms a regular distribution of bond lengths with a wide maximum, as observed for other coordination numbers of this cation. A priori (observed) bond valences in Y3Re7+O8 ] forms a regular distribution of bond lengths when bonded to O2−. We calculated a priori (observed) bond valences for five coordination polyhedra in four structures: Na4Zr2(SiO4)3 ZrB(Al8.82Ti4+0.18O18) 2 2FO3 Fig. S1, two of the O2− ions (O3 and O17) bond to 2 × [6]Nb5+ and 1 × [4]Nb5+, while the other two (O21) bond only to 2 × [6]Nb5+. This creates a strong bond-topological mismatch between the bond-valence requirement of O2− (2 v.u.) and an even distribution of bond valences for the NbO6 and NbO4 polyhedra. Thus, the [Nb5+O4]3− unit makes two bridging bonds via O21 of 1.573 v.u. (1.742 Å), while the two bridging bonds made via O3 and O17 are weaker at 0.961 v.u. (1.926 Å). Although we could not calculate the a priori bond valences of this structure, we assume that the PJTE has a similarly strong effect on bond-length variability as bond-topological asymmetry for this ion configuration.[5]Nb5+ [ce)Fig. S1 bond valences for two of these structures: for Na5Nb5+O5 ] forms a symmetric distribution of bond lengths with a maximum at 1.98 Å (5/6 v.u.). As we have seen above for [6]Ti4+ and [4]V5+, a symmetric distribution typically results from an ion configuration with a strong proclivity for polymerization. [Nb5+O6]7− units polymerize into dimers, chains, rings, clusters, sheets and frameworks, via corners, edges, faces and combinations of these. All of these polymerization circumstances lead to different a priori bond valences (and thus bond lengths) for the constituent crystal structures; the resulting distribution of these a priori bond lengths approaches that of a Gaussian distribution, in contrast to the positively skewed shape of simpler bond-length distributions driven by a two-body Morse potential. For example, monomeric [Nb5+O6]7− units are evenly distributed between [1+4+1] coordination and regular octahedra in our dataset. The off-centring of the Nb5+ ion via the PJTE, with no preferential off-centring direction . To compare the magnitude of bond-length variability caused by bond-topological asymmetry and the PJTE, we calculated the a priori (observed) bond valences for 15 polyhedra in 12 structures containing [6]Nb5+. In La(Nb5+5O14) (P2O7)O , they are 2 × 0.805 (0.993), 2 × 0.837 (0.824) and 2 × 0.859 (0.869) v.u. for Nb1, with Δtopol = 0.019 and Δcryst = 0.071 v.u., and 0.261 (0.462), 2 × 0.782 (0.714), 2 × 0.831 (0.837) and 1.515 (1.753) v.u. for Nb2, with Δtopol = 0.227 and Δcryst = 0.098 v.u. In Li(Nb5+U6+O6) Fig. S1  bond valences are 2 × 0.591 (0.240 and 0.849), 0.688 (0.590), 2 × 0.739 (0.497 and 1.037) and 2 × 0.826 (0.865 and 0.888) v.u. for Nb1, with Δtopol = 0.078 and Δcryst = 0.192 v.u., and 0.545 (0.506), 4 × 0.614 (2 × 0.613 and 2 × 0.617) and 2 × 1 (0.582 and 1.470) v.u., with Δtopol = 0.163 and Δcryst = 0.133 v.u. Average values for Δtopol and Δcryst are 0.121 and 0.163, respectively.[6]Mo6+ [Fig. 1av)] forms a peculiar trimodal distribution of bond lengths when bonded to O2−, with maxima at 1.71 (1.74), 1.92 (0.95) and 2.30 Å (0.32 v.u). The [Mo6+O6]6− units occur as monomers, dimers and other oligomers, chains, rings, sheets, clusters and frameworks, sharing corners, edges and/or faces. The near-uniqueness of the bonding environment of each and every Mo6+O6 octahedron in our dataset renders the elucidation of all bond-valence constraints leading to the observed bond-length distribution very lengthy and tedious. Here, we will simply calculate a priori (observed) bond valences for various modes of polymerization (from ten coordination polyhedra and eight structures) to determine whether bond-topological asymmetry, or the PJTE, is the main reason underlying bond-length variation for this ion configuration. Some of the results are given below, ordered in increasing degree of polymerization of the [Mo6+O6]6− unit. In K2Mo6+O2(I5+O3)4  bond valences are 2 × 0.951 (1.740), 2 × 1.001 (0.384) and 2 × 1.048 (0.862) v.u., with Δtopol = 0.033 and Δcryst = 0.531 v.u.; this is the largest value we calculate for our entire dataset. In Na2Mo6+3Te4+3O16   ] forms a regular distribution of bond lengths. We calculated a priori (observed) bond valences for three polyhedra in two structures. In Pb2+HfO3 ]. Similar to [6]Ti4+, [6]Nb5+ and other ions with similarly symmetrical bond-length distributions, [Ta5+O6]7− units are very susceptible to polymerization. [Ta5+O6]7− octahedra occur as monomers, dimers, chains, sheets, clusters and frameworks, either sharing corners, edges and/or faces. We calculated a priori (observed) bond valences for six coordination polyhedra in as many crystal structures. In Na2Ca3Ta5+2O9  and 2 × 1 (0.762 and 1.029) v.u. for chains of corner-sharing octahedra in CsTa5+(B2O5)  and 2 × 1 (1.167) v.u. for chains of edge-sharing octahedra in Yb(Ta5+O4) ] forms what appears to be a bimodal distribution of bond lengths when bonded to O2−, with a main maximum at 2.01 Å (0.76 v.u.) and a secondary maximum at 2.43 Å (0.22 v.u.). Pentagonal bipyramids typically polymerize into sheet and framework structures with TaO6 octahedra. We calculated the a priori (observed) bond valences for one polyhedron, in DyTa5+7O19 ] has a regular distribution of bond lengths. We calculated the a priori (observed) bond valences for two polyhedra in as many structures. In KNa3(W6+O5) 3 ] has a peculiar trimodal distribution of bond lengths when bonded to O2−, with two maxima at 1.74 (1.65) and 1.92 Å (0.97 v.u.) and a third, very broad, maximum around 2.18 Å (0.45 v.u.). The [W6+O6]6− units occur as monomers, chains, rings, sheets, clusters and frameworks, sharing corners and/or edges. The shape of the bond-length distribution is influenced by the fact that the majority of our dataset (roughly 2/3) involve clusters of more or less similar bond-valence constraints. The strongest and weakest bonds of these clusters are 1.69 (1.73) and 0.38 v.u. (2.24 Å) on average, with significant variability in between, presumably as a result of bond-topological constraints. The bonds in chains and rings tend to split into three groups, with mean bond valences of 2 × 1.5, 2 × 1 and 2 × 0.5 v.u. The main maximum at 1.92 Å (0.97 v.u.) represents the mean bond valence  for this ion configuration, although the [W6+O6]6− unit is regular in only a few structures. We calculated the a priori (observed) bond valences for five polyhedra from four crystal structures. In K2Ni2+[W6+O2(PO4)2] 2   bond valences are 6 × 1  v.u., with Δtopol = 0 and Δcryst = 0.319 v.u. Altogether, the average values of Δtopol and Δcryst are 0.117 and 0.269 v.u. for these structures, showing that the PJTE is the principal cause of bond-length variation for this ion configuration, with bond-topological asymmetry having a lesser role. Although many polyhedra have bond valences >1.75 v.u., these strong bonds result either from polymerization or the PJTE.[5]Re7+ [dv)Fig. S1(] forms a regular but slightly negatively skewed distribution of bond lengths when bonded to O2−. The largest bond-length variation is in Ba10(Re7+O5)6Br2  bond valences of 2 × 1.391 (1.263), 2 × 1.402 (1.178) and 1.414 (1.882) v.u., with Δtopol = 0.007 and Δcryst = 0.235 v.u. These values clearly show the PJTE to be the main cause of bond-length variation for this ion configuration.[6]Re7+ [Fig. 1bm)] forms a regular distribution of bond lengths when bonded to O2−. We calculated a priori (observed) bond valences for two coordination polyhedra from as many crystal structures. In Pr3(Re7+O8) Fig. S1(] occurs in two structures in our dataset. In Rb[Os8+2O8(OH)]  bond valences are 1.235 (0.624), 2 × 1.647 (1.735 and 1.739) and 2 × 1.735 (1.693 and 2.140) v.u., with Δtopol = 0.146 and Δcryst = 0.248 v.u. Thus the PJTE is the main cause of bond-length variation for this ion configuration.[6]Os8+ [ed)Fig. S12]  bond valences are 2 × 1.033 (0.607), 2 × 1.383 (1.756) and 2 × 1.583 (1.743) v.u., with Δtopol = 0.200 and Δcryst = 0.320 v.u. While the PJTE is the main driver of bond-length variation for this ion configuration, bond-topological asymmetry also contributes a significant amount.5.4.1.d0 electronic configurations in various coordination numbers , listed below are the three non-d0 non-octahedrally coordinated configurations of our dataset for which the PJTE is the main cause of bond-length variation.A variety of structural problems have been resolved via the pseudo Jahn–Teller effect for transition metals with non-[5]Cr2+ [r)Fig. S1(] occurs in two structures in our dataset. In Cr2+(HPO3)(H2O)2   bond valences are 0.349 (0.369), 0.412 (0.514) and 3 × 0.413  v.u., with Δtopol = 0.020 and Δcryst = 0.077 v.u. The effect of the PJTE on bond-length variation seems to be weaker than in Cr2+(HPO3)(H2O)2 but is clearly present. The four shortest bonded distances for these two polyhedra range between 1.994–2.110 Å, and the fifth distance is between 2.361–2.426 Å. The sixth shortest interatomic distance is between 2.964–3.3.053 Å, far too long to be a bond.[5]Co2+ [Fig. 1y)] forms a unimodal distribution with anomalously long bond lengths. There are five bonds making up the tail at longer bond lengths, representing the five (out of 15) polyhedra where [Co2+O5]8− units appear to distort as a result of the PJTE. In BaCo2+2(Si2O7)  bond valences are 0.362 (0.397), 2 × 0.379 (0.178 and 0.442), 0.432 (0.448) and 0.449 (0.483) v.u., with Δtopol = 0.032 and Δcryst = 0.070 v.u., showing that the PJTE is the main driver of bond-length variation for this ion configuration. The four shortest bonds in these five polyhedra range between 1.946–2.045 Å, and the fifth distance is between 2.345–2.574 Å. The sixth shortest interatomic distance is between 2.898–3.444 Å, far too long to be considered as a bond.[5]Cu2+ [Fig. 1ae)] forms a unimodal distribution of bond lengths when bonded to O2−, with a very long and flat tail for longer bonds. This ion configuration is very common (218 coordination polyhedra), making it an excellent candidate for probing the magnitude of the PJTE for [5]-coordinated cations. We calculated a priori (observed) bond valences for 13 polyhedra from nine crystal structures. In Cu2+4O(PO4)2 , 0.293 (0.482), 2 × 0.391 (0.253), 0.408 (0.517) and 0.517 (0.427) v.u., with Δtopol = 0.050 and Δcryst = 0.133 v.u. for Cu2 , and 0.295 (0.159), 0.384 (0.458), 0.393 (0.446), 0.409 (0.418) and 0.519 (0.564) v.u., with Δtopol = 0.051 and Δcryst = 0.063 v.u. for Cu3 (square pyramid). In Cu2+5O2(PO4)2 , and 2 × 0.277 (0.203 and 0.261), 0.444 (0.502) and 2 × 0.501 (0.486 and 0.505) v.u., with Δtopol = 0.099 and Δcryst = 0.033 v.u. for Cu3 . Although there is a weak correlation in which the weakest bond of the polyhedron becomes progressively stronger from a square-pyramidal to a triangular-bipyramidal configuration, there is no correlation between Δtopol and Δcryst as a function of polyhedron shape. We obtain average values of Δtopol and Δcryst of 0.061 and 0.084 v.u., with ranges of 0.016–0.099 and 0.033–0.133 v.u., respectively, for the 13 polyhedra. Thus, it seems that the long tail results from a mixture of continuous off-centring of Cu2+ via the PJTE, and non-local bond-topological asymmetry. One more structure worth discussing is that of PbCu2+(Cu2+Te6+O7)  bond valences in this structure are 0.265 (0.060), 0.372 (0.342), 2 × 0.399 (0.498) and 0.565 (0.590) v.u., with Δtopol = 0.066 and Δcryst = 0.091 v.u. Thus, the weak bond in this structure results from a mixture of non-local bond-topological asymmetry  and the PJTE.5.5.[6]Ru5+ [Fig. 1aw)] occurs as monomers, chains of corner- and edge-sharing octahedra and face-sharing oligomers. Face-sharing octahedra have Obr bond valences 3 × 0.53–0.78 v.u., with Obr bonding to other cations. Non-bridging bonds vary between 0.83–1.17 v.u. in these structures. For example, a priori (observed) bond valences are 3 × 0.786 (0.667) v.u. for bridging and 3 × 0.881 (1.075) v.u. for non-bridging bonds for Ru5+ in Ba3Ca(Ru5+2O9)  bond valences are 3 × 0.767 (0.633) and 3 × 0.900 (1.047) v.u., with Δtopol = 0.067 and Δcryst = 0.140 v.u. The reason for the large values of Δcryst is unclear, but they may be due to slight Ru5+–Ru5+ interactions between the dimers ; Ru5+ is JT-inactive with a d3 electronic configuration. Otherwise, monomers and chains of corner-sharing octahedra have regular polyhedra with a mean bond length of 1.957 Å (5/6 v.u.).6.6.1.Of the 52 ion configurations of Table 4topol and Δcryst values is given in Fig. 14a priori bond valences were calculated (Table S2); 〈Δtopol〉 = 0.102 and 〈Δcryst〉 = 0.113 v.u. For the 235 coordination polyhedra with Δtopol and/or Δcryst > 0.05 v.u., 〈Δtopol〉 = 0.113 and 〈Δcryst〉 = 0.123 v.u. Next, we wish to calculate the average magnitude of these indices between the main two factors identified in transition metal oxyanions: non-local bond-topological effects and the PJTE. We remove the polyhedra of this dataset in which π bonding is the main cause of bond-length variation, i.e. those for [5]V4+, [6]V4+, [5]V5+, [6]V5+, [5]Mo5+, [6]Mo5+ and [5]Mo6+, to get 〈Δtopol〉 = 0.091 and 〈Δcryst〉 = 0.120 v.u. . Similarly, if we remove those coordination polyhedra where the main factor underlying bond-length variation is the strong or weak JTE , 〈Δtopol〉 = 0.093 and 〈Δcryst〉 = 0.120 v.u. for the remaining 195 polyhedra . From here, it is not possible to remove the component of Δcryst which is due to crystal-structure effects, as those effects are both widespread and structure dependent. While Δcryst may be interpreted as mainly driven by the PJTE in this subset, a non-negligible component is due to crystal-structure effects. Thus, we are left comparing 〈Δtopol〉 = 0.091 and 〈Δcryst〉 = 0.120 v.u. for 195 polyhedra. In terms of frequency, Δtopol and Δcryst are >0.05 v.u. for 141 and 171 polyhedra, respectively, while Δtopol > Δcryst for 78 of those polyhedra; however, we emphasize that this subset of data is biased toward ion configurations with a d0 electronic configuration, to address other questions in this work. In terms of magnitude, eight of the 15 widest ranges of observed bond lengths in Table 4The distribution of observed Δa priori limitations with regard to ion identity.Thus we conclude the following: (i) non-local bond-topological asymmetry is the most frequently encountered cause of bond-length variation in transition metal oxides and oxysalts, closely followed by the PJTE; and (ii) 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. We further suggest non-local bond-topological asymmetry to be the most widespread cause of spontaneous distortion (and bond-length variation) in the solid state, with no 6.2.a priori bond valences in crystal structures with d0 transition metals 6  bond valences 2 × 0.568 (0.484), 2 × 0.174 (0.199) and 2 × 0.259 (0.244) v.u. Thus, this polyhedron strongly deviates from regularity (6 × 2/3 v.u.) in a way that accommodates the JT distortion; Δtopol = 0.156 and Δcryst = 0.041 v.u. However, in Gd(Mn3+O3)  bond valences for Mn3+ are 4 × 0.471 (2 × 0.194 and 2 × 0.700) and 2 × 0.559 (0.612) v.u. for Mn3+. Thus, deviation from regularity (6 × 0.5 v.u.) is not captured by the bond topology; Δtopol = 0.039 and Δcryst = 0.186 v.u. Another such example is that of Cu2+2Co2+O(B2O5)  bond valences of 3 × 0.304 , 2 × 0.309 (0.099 and 0.453) and 0.470 (0.540) v.u. for [6]Cu2+, with Δtopol = 0.046 and Δcryst = 0.161 v.u.For the strong JTE, strong correlation is observed for 5+5O14)(H2O)8  bond valences 0.316 (0.329), 0.600 (0.608), 0.609 (0.573), 0.920 (0.891), 0.988 (0.986) and 1.568 (1.643) v.u., with Δtopol = 0.325 and Δcryst = 0.027 v.u. We also showed the case for CaNb5+2(P4O13)(P2O7)O . However, the number of cases in which the JT distortion functions independent of (or against) bond-topological requirements is overwhelmingly large. In painite, (Ca0.81Na0.19)ZrB(Al8.82Ti4+0.18O18)  bond valences are 6 × 0.667 (3 × 0.187 and 3 × 0.599) v.u., with Δtopol = 0 and Δcryst = 0.274 v.u. In K2Mo6+O2(I5+O3)4  bond valences for [6]Mo6+ are 2 × 0.951 (1.740), 2 × 1.001 (0.384) and 2 × 1.048 (0.862) v.u., with Δtopol = 0.033 and Δcryst = 0.531 v.u. In WO3  bond valences are 6 × 1  v.u. for [6]W6+, with Δtopol = 0 and Δcryst = 0.319 v.u.For the PJTE, we find few structures where bond-topological and crystallographic effects appear to work in cooperation. Nonetheless, the decavanadate cluster represents a very good example of apparent cooperation. In hummerite, KMg on a qualitative scale, e.g. strong for Mo6+ and V5+, moderate for W6+, Ti4+, Nb5+ and Ta5+, and weak for Zr4+ and Hf4+ , Mo6+ , W6+ , V5+ , Nb5+ , Ti4+ , Ta5+ , Hf4+ , Zr4+ , Re7+ , Y3+  and Sc3+ . These 〈Δcryst〉 values are not set in stone; they may vary slightly as more a priori bond valences are calculated for these ion configurations.Octahedrally coordinated ions with d0 ions. For [4]-coordination , we have the following order: Re7+ 0.116 (n = 2), Ti4+ 0.107 (n = 1), V5+ 0.099 (n = 13), W6+ 0.098 (n = 2), Mo6+ 0.087 (n = 8), Cr6+ 0.085 (n = 8), Os8+ 0.077 (n = 1) and Mn7+ 0.028 (n = 1). For [5]-coordination, we observe Os8+ 0.248 (n = 1), Re7+ 0.235 (n = 1), Mo6+ 0.197 (n = 3), Ti4+ 0.163 (n = 2), W6+ 0.127 (n = 2), V5+ 0.121 (n = 3) and Nb5+ 0.103 (n = 2).Moreover, we find the magnitude of bond-length variations to vary as a function of coordination number for 6.4.The inseparability of electronic and nuclear coordinates in Jahn–Teller systems makes JT-active compounds unsuited to density functional theory (DFT), whose foundation rests on the Born–Oppenheimer approximation 4 , followed by [7]-coordination with 〈Δcryst〉 = 0.115 v.u. (n = 6), [9]-coordination with 〈Δcryst〉 = 0.110 v.u. (n = 1), [4]-coordination with 〈Δcryst〉 = 0.092 v.u. (n = 36), [8]-coordination with 〈Δcryst〉 = 0.087 v.u. (n = 7) and [10]-coordination with 〈Δcryst〉 = 0.079 v.u. (n = 1). Considering that crystal structures containing octahedrally coordinated d0 ions have been the subject of intense scrutiny and targeted syntheses , [5]Co2+ (d7), [5]Cu2+ (d9), and [6]Zn2+ and [6]Hg2+ (d10). The PJTE for [5]-coordinated square-pyramidal complexes has been demonstrated for [5]Cu2+ and several ligands  complexes. While the bond-length distributions of [6]Zn2+ and [6]Cd2+ are regular, the range of bond lengths for [6]Zn2+ is much larger than that for [6]Cd2+ (0.868 versus 0.591 Å), suggesting that the PJTE is stronger for Zn2+ than it is for Cd2+ when bonded to O2−. However, both these ions  have been shown to exhibit the PJTE in S2− structures of various d10 transition metal ions  for [5]Cu2+, 0.077 v.u. for [5]Cr2+ (n = 2), 0.07 v.u. for [5]Co2+ (n = 1), 0.050 v.u. for [6]Zn2+ (n = 5) and 0.038 v.u. for [6]Hg2+ (n = 2). Thus, it seems that the distorting power of non-d0 ion configurations is slightly less than that of their d0 counterparts, although the sample size is far from significant.For the five ion configurations with non-6.5.e.g. the ionic radius of S2− is larger than that of O2− by roughly 0.5 Å; Shannon, 1976e.g. NbO6 versus NbO4S2 polyhedra. For example, in Sm3+3Nb5+O4S3  and two bonds to S2− (2.586 and 2.719 Å). Is this polyhedron distorted, or is the bond-length variation simply due to ligand size differences? Can we quantitively resolve the distortion power of this PJTE-active cation in this configuration?Comparing the magnitude of bond-length variations across ligand type is clouded by ligand size differences . Thus, we calculate Δtopol = 0.049 and Δcryst = 0.148 v.u. for Nb5+, indicating that the distorting power of the PJTE is approximately the same here as it is for the average NbO6 polyhedron . As such, the introduction of S2− seemingly has little effect on the distorting power of Nb5+, a result whose generalization may bear important consequences for the design of hetero-ligand materials.A great advantage of the Δ6.6.topol and Δcryst indices allows one to pinpoint the causal mechanism(s) underlying material properties linked to bond-length variations, and to determine if, and how, optimization of these properties may be done. Understanding the extent to which these mechanisms materialize into bond-length variations is further crucial to maximize the harnessing of these effects within the constraints of physically realistic crystal structures.Calculation of the Δtopol >> Δcryst, as said property would arise primarily from the bond-topological underpinnings of the crystal structure. Along those lines, optimization via compositional variation would have maximum potential where the coordination unit responsible for the functional property has Δtopol = 0. For example, Δtopol = 0.000 and Δcryst = 0.017 v.u for the B site of the Pnma perovskite ACaBTi4+O3  or coordination number . Such compositional variation analyses may be complemented by DFT calculations, where constrained to values outlined in Tables 1S2. Going back to Gd3+Mn3+O3, with Δcryst = 0.186 v.u., consulting Table S2 informs us that little optimization of the functional properties resulting from Mn3+ seems possible; Table S2 lists three values of Δcryst in as many crystal structures for [6]Mn3+: 0.186, 0.211 and 0.211 v.u. In this instance, increasing Δcryst would necessarily follow from substitution of AGd3+, whereby a small amount of bond-length variation via mechanism (4)  may trickle down to the Δcryst index of BMn3+.Values given in this work may be used to set expectation limits regarding the maximum values of Δi.e. Δtopol > Δcryst), optimization is complicated by the fact that spontaneous distortion of crystal structures via non-local bond-topological asymmetry is a static emergent phenomenon which can only be predicted from a priori knowledge of ion connectivity. As a result, properties arising from non-local bond-topological asymmetry are less tunable, for they require subtle and less predictable changes in ion connectivity. These changes may occur either at the site of interest  or at other sites in the structure, where subtle changes in coordination number via homovalent and/or (multi-site) heterovalent substitution may have an effect on the a priori bond valences of the crystal structure which carries through to the Δtopol index of the  site of interest. While these substitutions can be modelled with relative ease, a significant constraint is that such fine tuning must not result in the crystallization of a different structure type. As a result, the optimization of functional properties linked to non-local bond-topological asymmetry is significantly more challenging than that for crystallographic effects.In the reverse scenario, where the functional property is linked to non-local bond-topological asymmetry  and 3 × 1.190 (1.169) v.u., with Δtopol = 0.090 and Δcryst = 0.059 v.u for V1, and 1.220 (1.647), 2 × 0.980 (0.938) and 2 × 0.910 (0.738) v.u., with Δtopol = 0.088 and Δcryst = 0.171 v.u for V2. For V1, non-local bond-topological asymmetry is mainly responsible for variation away from a regular tetrahedron with four bonds of 1.25 v.u.; for V2, the PJTE is the principal reason for variation away from observing five bonds of 1 v.u. More importantly, both mechanisms cause variation in the same spatial direction within both their polyhedron and the unit cell; both strongest bonds (V1—O1 and V2—O2) point in the same direction, along the c axis. As a result, the effect of Δtopol and Δcryst is entirely additive, and K3V5+5O14 is observed with a series of marked functional properties, i.e. second-harmonic generation, piezoelectricity and polarization .topol and Δcryst indices for the optimization of functional properties tied to bond-length variations. Calculation of these indices allows identification of the causal mechanism(s) upon which optimization should be focused, while the magnitude of these values is used to gauge qualitatively the extent to which these values  may be maximized. Along those lines, examination of the relation between Δtopol and Δcryst and the magnitude of various functional properties linked to bond-length variation, and the optimization of these properties, seems warranted.Perhaps one of the most promising opportunities resulting from this work regards the strategic use of the newly proposed Δ10.1107/S2052252520005928/lt5028sup1.pdfAdditional figures and tables. DOI: Click here for additional data file.10.1107/S2052252520005928/lt5028sup2.zipRaw data bond-length files. DOI:


Scheltema NM, Gentile A, Lucion F, et al. Global respiratory syncytial virus-associated mortality in young children (RSV GOLD): a retrospective case series. Lancet Glob Health 5: e984–91. http://dx.doi.org/10.1016/S2214-109X(17)30344-32017; —The following sentence in the Findings section of the Summary should have read: Median age for RSV-related deaths was 5·0 months (IQR 2·3–11·0) in low-income or lower middle-income countries, 4·0 months (2·0–10·0) in upper middle-income countries, and 7·0 months (3·6–16·8) in high-income countries.” This correction has been made to the online version as of Oct 3, 2017.


The global burden of hepatitis B virus (HBV) infection in terms of morbidity and mortality is immense. Novel treatments that can induce a protective immune response are urgently needed to effectively control the HBV epidemic and eventually eradicate chronic HBV infection.We designed and evaluated an HBV therapeutic vaccine consisting of a novel Toll-like receptor 7 (TLR7) agonist T7-EA, an Alum adjuvant and a recombinant HBsAg protein. We used RNA-seq, ELISA and hTLR7/8 reporting assays to characterize T7-EA in vitro and real-time PCR to evaluate the tissue-retention characteristics in vivo. To evaluate the adjuvant potential, we administrated T7-EA intraperitoneally in a formulation with an Alum adjuvant and HBsAg in normal and HBV mice, then, we evaluated the HBsAg-specific immune responses by ELISA and Elispot assays.T7-EA acted as an hTLR7-specific agonist and induced a similar gene expression pattern as an unmodified TLR7 ligand when Raw 264.7 cells were exposed to T7-EA; however, T7-EA was more potent than the unmodified TLR7 ligand. In vivo studies showed that T7-EA had tissue-retaining activity with stimulating local cytokine and chemokine expression for up to 7 days. T7-EA could induce Th1-type immune responses, as evidenced by an increased HBsAg-specific IgG2a titer and a T-cell response in normal mice compared to mice received traditional Alum-adjuvant HBV vaccine. Importantly, T7-EA could break immune tolerance and induce persistent HBsAg-specific antibody and T-cell responses in an HBV mouse model.T7-EA might be a candidate adjuvant in a prophylactic and therapeutic HBV vaccine. Chronic hepatitis B virus (HBV) infection is a major global health problem that affects > 350 million people worldwide. The infection can eventually lead to hepatocellular carcinoma, liver failure, or cirrhosis, and causes ~ 1 million deaths every year ; therefoHBV is a DNA virus that is converted into a covalently closed circular DNA (cccDNA) in the host cell nucleus . LicenseMany different approaches have been investigated in preclinical models to overcome immune tolerance in chronic HBV infection. One option is to optimize HBV components in the vaccine, such as DNA or peptide vaccines, vector-based or cell-based vaccines, or multi-epitope therapeutic vaccine candidates that sufficiently cover different HBV genotypes have been developed . Anotherstaphylococcus aureus  359.2; found 359.0. methods . Raney N6-amino-9-(4-(aminomethyl)benzyl)-2-(2-methoxyethoxy)-9H-purin-8-ol (compound3): A suspension of compound 2  in concentrated hydrochloric acid (10 mL) was placed in a round-bottomed flask. The reaction was stirred at room temperature overnight. The mixture was adjusted to pH 9 with 2 N NaOH and further purified by precipitating and washing in water to obtain a white solid . ESI–MS (positive ion mode): calculated for C16H21N6O3 m/z [M + 1] 345.2; found 345.0.N-(4-((6-amino-8-hydroxy-2-(2-methoxyethoxy)-9H-purin-9-yl)methyl)benzyl)-2-phenoxy)acetamide : To a solution of ethacrynic acid , O-Benzotriazole-N,N,N’,N’-tetramethyl- uronium-hexafluorophosphate (HBTU)  dissolved in anhydrous DMF (8 mL) was added triethylamine (TEA)  and a catalytic amount of 4-dimethylaminopyridine. A solution of compound 3  in DMF (2 mL) was added dropwise to the reaction mixture and stirred at room temperature for 4 h. After reaction completion, the reaction content was diluted with water (100 mL), then filtered and washed with ice-cold water. The crude material was purified by flash chromatography (5% MeOH/CH2Cl2) to afford compound 4 (T7-EA) as a white solid . 1H-NMR  δ 9.97 , 8.54 , 7.34 , 7.25 , 7.21 , 7.10 , 6.48 , 6.07 , 5.55 , 4.83 , 4.79 , 4.30 , 4.28-4.22 , 3.62-3.55 , 3.27 , 2.38 , 1.09 . 13C-NMR  δ 195.59, 167.14, 160.31, 155.97, 152.71, 149.88, 149.64, 148.18, 144.42, 138.65, 136.28, 132.97, 129.86, 129.81, 127.98, 127.93, 121.73, 112.39, 98.80, 70.70, 68.40, 65.75, 58.53, 42.62, 42.19, 23.41, 12.84. ESI–MS (positive ion mode): calculated for C29H31Cl2N6O6 m/z [M + 1] 629.2; found 629.1.6 cells per well. Compounds were added to the cultures at a final concentration of 10 μM. After 24 h of incubation, total RNA was isolated and purified using an RNA Purification Kit . RNA-seq was performed on a BGI-SEQ 500 platform and analyzed by BGI .Raw264.7 macrophage cells were seeded in 6-well plates in dulbecco’s modified eagle medium (DMEM) with 10% of fetal bovine serum (FBS), at a density of 1 × 10HEK-Blue hTLR7 and hTLR8 reporter cells were purchased from InvivoGen . These cells stably express human TLR7 and TLR8, and a SEAP reporter that can detect TLR7 and TLR8 agonists through NF-κB signaling activation. The cells were maintained in selective DMEM growth medium with 10 μg/mL blasticidin and 100 μg/mL Zeocin™ . After incubation with different doses of SZU-101 (T7) and T7-EA, the cells were tested using a HEK-Blue detection kit, according to the manufacturer’s instructions. The TLR7/8 agonist R848 was used as a positive control. The induction of TLR7/8 activation was visualized and assessed by reading the OD at 620–655 nm.4) were incubated for 24 h with T7-EA at concentrations ranging from 0.01 to 10 μM, cell viability was detected with a Cell Counting Kit-8 (CCK8) .Human monocyte cell line THP-1 cells (0.5 × 106/mL) were incubated with various compounds for 18 h at 37 °C and 5% CO2, and then the culture supernatants were collected. The levels of cytokines  in the supernatants were determined by ELISA . The minimum cytokine detection level was 15 pg/mL.Human peripheral blood mononuclear cells (PBMCs) were isolated from human buffy coats obtained from 3 healthy volunteers, as described previously . PBMCs . The minimum IL-6 detection level was 15 pg/mL.Bone marrow-derived macrophages (BMDMs) and bone marrow-derived dendritic cells (BMDCs) were isolated from C57BL/6 mice and grown as previously described , 22. TheThe metabolic stability of the test compounds (1 µM test concentration) was determined in freshly isolated Sprague–Dawley rat hepatocytes or cryopreserved primary human hepatocytes. Incubations were performed using 100,000 cells/mL  in William’s E Medium at 37 °C in a constant temperature incubator shaker. Then, 100 µL aliquots were taken at 0 min (control), 15 min, 30 min, 1 h, 2 h and 4 h. The reactions were quenched with 300 µL acetonitrile containing 200 ng/mL tolbutamide, vortex-mixed, and centrifuged at 14,000 rpm for 20 min. The supernatants were separated and assessed by LC/MS/MS (SHIMADZU LCMS-8030). All incubations were performed in triplicate.Balb/c mice (n = 3) were injected in the gastrocnemius muscles with 35 nmol T7, T7-EA, R848 or vehicle  in a 50μL volume. After 1, 3 and 7 days of injection, the muscles were harvested and immediately frozen in liquid nitrogen and stored at − 80 °C. Total RNA was isolated from cells or tissues using TRIzol reagent . The RNA samples were further purified using a Qiagen RNeasy Protect Kit . Reverse transcription was carried out with 1 μg total RNA using SuperScript III . Serum HBV DNA was extracted from 100 uL of serum and measured following the manufacturer’s instructions . The cDNA and HBV DNA samples were run in duplicate and quantified by real-time PCR  using the following primer sets: HBV forward 5′CACATCAGGATTCCTAGGACC3′, reverse 5′GGTGAGTGATTGGAGGTTG3′; β-actin forward 5′ GGCTGTATTCCCCTCCATCG3′, reverse 5′ CCAGTTGGTAACAATGCCATGT3′; IL-4 forward 5′ATGGATGTGCCAAACGTCCT3′, reverse 5′AAGCACCTTGGAAGCCCTAC3′, CCL2 forward 5′TCAGCCAGATGCAGTTAACG3′, reverse 5′CTCTCTTGAGCTTGGTGACA3′; CCL4 forward 5′GCAACACCATGAAGCTCTGC3′, reverse 5′CCATTGGTGCTGAGAACCCT3′; TNF-α forward 5′CAAAATTCGAGTGACAAGCCTG3′, reverse 5′GAGATCCATGCCGTTGGC3′. β-actin was used as an internal control for sample loading and normalization. The data were analyzed using the comparative Ct method, where Ct is the cycle number at which the fluorescence first exceeds the threshold. The ΔCt values from each cell line were obtained by subtracting the Ct values for β-actin from the Ct values for the sample. One difference of Ct value represents a 2-fold difference in the level of mRNA. The mRNA level was expressed as a percentage with respect to the control.C57BL/6 mice were injected with the indicated amounts of recombinant virus  via the tail vein. The mice were then bled retro-orbitally at different time points to monitor HBV surface antigen (HBsAg), HBs Antibody (HBsAb) and HBV genomic DNA levels in the serum. HBsAg and HBsAb levels were measured using an ARCHITECT Detection Kit  following the manufacturer’s instructions.5 cells/well splenocytes were added in duplicate to 96-well plates coated with anti-mouse IFN-γ . Then, 20 μg/mL HBsAg protein was added per well and incubated for 18 h at 37 °C. The number of spot-forming cells was determined using an Axioplan 2 imaging system .Purified HBsAg antigen and Alum adjuvant were supplied by Kangtai Biological Products CO., Ltd. . For in vivo adjuvanticity studies, uninfected Balb/c and AAV/HBV-infected C57BL/6 mice were immunized intraperitoneally with 2 μg HBsAg mixed with 100 nmol T7, T7-EA or R848 with or without Alum adjuvant at day 0, 14 and 28. Mice immunized with normal saline or HBsAg mixed with vehicle served as controls. Serum was obtained at different time points for HBsAg-specific antibody detection, and splenocytes were isolated to detect the HBsAg-specific T-cell response by Elispot. Anti-HBsAg IgG antibody levels were measured by ELISA with the ARCHITECT Detection Kit. Anti-HBsAg IgG1 and IgG2a were measured as previously described . BrieflyMouse antibodies (Abs) against CD4 (GK1.5), Ly6C cells and respective isotype control were purchased from Bio X cell . Abs were injected 1 day before vaccine administration and repeated every 4 days to maintain cell depletion. Cell depletion was confirmed by flow cytometry on peripheral blood samples 2 days after Ab injection. HBsAg (2 μg) mixed with 100 nmol T7-EA and Alum adjuvant was used for immunization at day 0, and HBsAg mixed with Alum adjuvant alone was used as a control. The mice were sacrificed 7 days after immunization and splenocytes were isolated to detect the HBsAg-specific T-cell response by Elispot, as described above.P < 0.05. *P < 0.05, **P < 0.01. All experiments were performed at least three times with comparable results.All the data are expressed as the mean ± SD. Statistical analysis was performed with GraphPad Prism 5 software. Data were analyzed by one-way ANOVA or unpaired two-tailed t tests, and differences were considered statistically significant at 4) is outlined in Fig. 1), a TLR7 agonist intermediate previously described by our lab [2 was obtained via reduction of the 9-benzylnitrile group of intermediate 1. Demethylation of compound 2 with concentrated hydrochloric acid provided 8-hydroxyl analogue 3. T7-EA (4) was synthesized by coupling the TLR7 agonist intermediate 3, with ethacrynic acid (EA) activating by HBTU. The SZU-101 (T7) pharmacophore and the EA electrophilic α, β-unsaturated ketone moiety were kept.The synthetic route of T7-EA  in our RNA-seq dataset. We found similar TLR gene expression patterns between T7-EA and T7 treatments in Raw 264.7 cells: only six genes were differentially expressed with an absolute fold-change > 2. These six genes, Stat1, Ifnb1, Irf7, Cxcl10, Ccl5 and Mapk13  between Raw 264.7 cells treated with T7-EA and Raw 264.7 cells treated with T7  maturation. Thus, TLR7/8 agonists are able to drive T-cell-mediated and B-cell-mediated adaptive immune responses , 44. LocIn summary, we have generated a novel TLR7 agonist known as T7-EA, which has promising pharmacodynamic and pharmacokinetic characteristics. We propose that T7-EA might be used as an adjuvant in HBV therapeutic vaccines as it can break immune tolerance and induce high quality HBsAg-specific immune responses in normal and HBV mouse models. Further studies are now needed to confirm the potential of T7-EA as an adjuvant in HBV therapeutic vaccines in other models and in clinical practice. We believe that T7-EA is a candidate for application as an adjuvant in prophylactic and therapeutic hepatitis B vaccines.Additional file 1: Table S1. Complete list of DEGs in Raw 264.7 cells treated with T7-EA and T7. All genes had an absolut fold-change (FC) >2 and a false discovery rate (FDR) <0.05 in T7-EA vs T7 group, and passed a low-expression filter.Additional file 2: Table S2. Complete list of genes in the Toll-like receptor signaling pathway. Genes in Toll-like receptor signaling pathway were taken from PathCards (https://pathcards.genecards.org). Passed a low-expression filter.Additional file 3: Figure S1. T7-EA treatment suppresses ISGs expression compared to T7 treatment in Raw 264.7 cells. Raw 264.7 cells (1 × 106 ) were treated with 10 μM T7 or T7-EA for 24 h; then, total RNA was extracted and purified for RNA-seq. A list of ISGs were taken from Schoggins et al. (2011) Nature 472: 481-485. All genes had an absolute fold-change >2 and a false discovery rate <0.5 in the T7-EA vs T7 group, and passed a low-expression filter. ISGs, interferon-stimulated genes.Additional file 4: Table S3. Complete list of ISGs significantly downregulated by T7-EA compared to T7. ISGs were taken from Schoggins et al. (2011) Nature 472: 481-485. All genes had an absolute fold-change (FC) >2 and a false discovery rate (FDR) <0.05 in T7-EA vs T7 group, and passed a low-expression filter.Additional file 5: Figure. S2. Evaluation of T7-EA activity in primary immune cells and after tissue injection. (A) Human PBMCs (2 × 106/mL) were incubated for 18 h with T7-EA and T7 at concentrations ranging from 0.01 to 10 μM. IL-6 levels in the culture supernatants were determined by ELISA. (B) BMDCs (0.5 × 106/mL) derived from C57BL/6 mice were incubated for 22 h with T7-EA, T7 and R848 at concentrations ranging from 0.01 to 10 μM. The levels of IL-6 in the supernatants were measured by ELISA. (C-F) Balb/c mice (n=3) were injected in the gastrocnemius muscles with 35 nmol T7, T7-EA, R848 or vehicle  in a 50 μL volume. Then, 1, 3 and 7 days after injection, the muscles were harvested and RNA was isolated. IL4 and CCL4 expression at the injection site were determined by real-time PCR. A student’s t test was used for data analysis. The data represent the means ± SD of triplicates and are representative of three independent experiments. ** P<0.01. T7, SZU-101; BMDCs, bone marrow derived dendritic cellsAdditional file 6: Figure S4. Evaluation of adverse effect of T7-EA. (A) THP-1 cells (0.5 × 104) were incubated for 24 h with T7-EA at concentrations ranging from 0.01 to 10 μM. CCK8 was used to assess cell viability. (B) Groups of HBV mice (n=6 per group) were immunized intraperitoneally with HBV therapeutic vaccine consisting of T7-EA, Alum adjuvant and a recombinant HBsAg protein at day 0, 14 and 28. Normal saline and a traditional HBV vaccine were used as controls. On day 35, the mice were sacrificed and the spleen/body weight ratio was calculated.Additional file 7: Figure S3. The effects of vaccines on HBV DNA and HBsAg levels in an HBV mouse. Groups of HBV mice (n=6 per group) were immunized intraperitoneally with an HBV therapeutic vaccine consisting of T7-EA, Alum adjuvant and a recombinant HBsAg protein at day 0, 14 and 28. Normal saline and a traditional HBV vaccine were used as controls. (A) HBV DNA in the serum was detected by real-time PCR at different time points. The data are expressed as relative to HBV DNA copies at day 0 for each mouse. (B) HBsAg levels in the serum was detected by Elisa at different time points. The data are expressed as relative to HBsAg levels at day 0 for each mouse.Additional file 8: Table S4.In vitro hepatocyte metabolic stability of T7-EA.


Nature Communications 10.1038/s41467-020-18438-0, published online 25 September 2020.Correction to: 16Oy1−18Oy)3”, but in fact it should read “SrTi(16Ox1−18Ox)3”.The original version of this Article contained an error in the caption of Fig. 1, which incorrectly says “SrTi(


Scientific Reports 10.1038/s41598-019-39250-x, published online 28 February 2019Correction to: This Article contains numerical errors in Table 2, whereby some p-values are incorrectly published as the mean and standard deviation under the heading ‘Relapse 1wk-1 yr) Mean ± SD’. The correct Table 2 appears below as Table wk-1 yr M