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

Bond-length distributions are examined for 63 transition-metal ions bonded to O2- in 147 configurations, for 7522 coordination polyhedra and 41,488 bond distances, providing baseline statistical knowledge of bond lengths for transi-tion metals bonded to O2-. A priori bond valences are calculated for 140 crystal structures containing 266 coordination poly-hedra 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] multi-ple-bond formation; crystallographic mechanisms are [3] electronic effects (with inherent focus on coupled electronic-vibra-tional degeneracy in this work), and [4] crystal-structure effects. The Δ𝑡𝑜𝑝𝑜𝑙 and Δ𝑐𝑟𝑦𝑠𝑡 indices allow one to determine the primary cause(s) of bond-length variation for individual coordination polyhedra and ion configurations, quantify the dis-torting power of cations via electronic effects (by subtracting the bond-topological contribution to bond-length variation), set expectation limits regarding the extent to which functional properties linked to bond-length variations may be optimized in a given crystal structure (and inform how optimization may be achieved), and more. We find the observation of multiple bonds to be primarily driven by the bond-topological requirements of crystal structures in solids. However, we sometimes observe multiple bonds to form as a result of electronic effects (e.g. the pseudo Jahn-Teller effect); 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 pseudo Jahn-Teller effect (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 asym-metry, comparable to those resulting from the strong JTE, and less than those induced by π-bond formation. From a compar-ison 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 Jahn-Teller effect is found not to have a symbiotic relation with the bond-topo-logical requirements of crystal structures. The magnitude of bond-length variations caused by the PJTE decreases in the fol-lowing order for octahedrally coordinated d0 transition metals 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+. We conclude 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.

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Bond-Length Distributions for Ions Bonded to Oxygen: Results for the Transition Metals and Quantification of the Factors Underlying Bond-Length Variation in Inorganic Solids

10.26434/chemrxiv.11605698.v1 ◽

2020 ◽

Author(s):

Olivier Charles Gagné ◽

Frank Christopher Hawthorne

Keyword(s):

Transition Metal ◽

Bond Length ◽

A Priori ◽

Bond Formation ◽

Length Variation ◽

Electronic Effects ◽

Coordination Polyhedra ◽

Jahn Teller ◽

Jahn Teller Effect ◽

Non Local

Bond-length distributions are examined for 63 transition-metal ions bonded to O2- in 147 configurations, for 7522 coordination polyhedra and 41,488 bond distances, providing baseline statistical knowledge of bond lengths for transi-tion metals bonded to O2-. A priori bond valences are calculated for 140 crystal structures containing 266 coordination poly-hedra 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] multi-ple-bond formation; crystallographic mechanisms are [3] electronic effects (with inherent focus on coupled electronic-vibra-tional degeneracy in this work), and [4] crystal-structure effects. The Δ𝑡𝑜𝑝𝑜𝑙 and Δ𝑐𝑟𝑦𝑠𝑡 indices allow one to determine the primary cause(s) of bond-length variation for individual coordination polyhedra and ion configurations, quantify the dis-torting power of cations via electronic effects (by subtracting the bond-topological contribution to bond-length variation), set expectation limits regarding the extent to which functional properties linked to bond-length variations may be optimized in a given crystal structure (and inform how optimization may be achieved), and more. We find the observation of multiple bonds to be primarily driven by the bond-topological requirements of crystal structures in solids. However, we sometimes observe multiple bonds to form as a result of electronic effects (e.g. the pseudo Jahn-Teller effect); 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 pseudo Jahn-Teller effect (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 asym-metry, comparable to those resulting from the strong JTE, and less than those induced by π-bond formation. From a compar-ison 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 Jahn-Teller effect is found not to have a symbiotic relation with the bond-topo-logical requirements of crystal structures. The magnitude of bond-length variations caused by the PJTE decreases in the fol-lowing order for octahedrally coordinated d0 transition metals 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+. We conclude 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.

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Bond-Length Distributions for Ions Bonded to Oxygen: Results for the Transition Metals and Quantification of the Factors Underlying Bond-Length Variation in Inorganic Solids

10.26434/chemrxiv.11605698 ◽

2020 ◽

Author(s):

Olivier Charles Gagné ◽

Frank Christopher Hawthorne

Keyword(s):

Transition Metal ◽

Bond Length ◽

A Priori ◽

Bond Formation ◽

Length Variation ◽

Electronic Effects ◽

Coordination Polyhedra ◽

Jahn Teller ◽

Jahn Teller Effect ◽

Non Local

Bond-length distributions are examined for 63 transition-metal ions bonded to O2- in 147 configurations, for 7522 coordination polyhedra and 41,488 bond distances, providing baseline statistical knowledge of bond lengths for transi-tion metals bonded to O2-. A priori bond valences are calculated for 140 crystal structures containing 266 coordination poly-hedra 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] multi-ple-bond formation; crystallographic mechanisms are [3] electronic effects (with inherent focus on coupled electronic-vibra-tional degeneracy in this work), and [4] crystal-structure effects. The Δ𝑡𝑜𝑝𝑜𝑙 and Δ𝑐𝑟𝑦𝑠𝑡 indices allow one to determine the primary cause(s) of bond-length variation for individual coordination polyhedra and ion configurations, quantify the dis-torting power of cations via electronic effects (by subtracting the bond-topological contribution to bond-length variation), set expectation limits regarding the extent to which functional properties linked to bond-length variations may be optimized in a given crystal structure (and inform how optimization may be achieved), and more. We find the observation of multiple bonds to be primarily driven by the bond-topological requirements of crystal structures in solids. However, we sometimes observe multiple bonds to form as a result of electronic effects (e.g. the pseudo Jahn-Teller effect); 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 pseudo Jahn-Teller effect (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 asym-metry, comparable to those resulting from the strong JTE, and less than those induced by π-bond formation. From a compar-ison 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 Jahn-Teller effect is found not to have a symbiotic relation with the bond-topo-logical requirements of crystal structures. The magnitude of bond-length variations caused by the PJTE decreases in the fol-lowing order for octahedrally coordinated d0 transition metals 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+. We conclude 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.

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Bond-length distributions for ions bonded to oxygen: metalloids and post-transition metals

Acta Crystallographica Section B Structural Science Crystal Engineering and Materials ◽

10.1107/s2052520617017437 ◽

2018 ◽

Vol 74 (1) ◽

pp. 63-78 ◽

Cited By ~ 23

Author(s):

Olivier Charles Gagné ◽

Frank Christopher Hawthorne

Keyword(s):

Transition Metal ◽

Metal Ions ◽

Bond Length ◽

Coordination Number ◽

Lone Pair ◽

Transition Metal Ions ◽

Length Variation ◽

Coordination Polyhedra ◽

Bond Lengths ◽

Lone Pair Electrons

Bond-length distributions have been examined for 33 configurations of the metalloid ions and 56 configurations of the post-transition metal ions bonded to oxygen, for 5279 coordination polyhedra and 21 761 bond distances for the metalloid ions, and 1821 coordination polyhedra and 10 723 bond distances for the post-transition metal ions. For the metalloid and post-transition elements with lone-pair electrons, the more common oxidation state between n versus n+2 is n for Sn, Te, Tl, Pb and Bi and n+2 for As and Sb. There is no correlation between bond-valence sum and coordination number for cations with stereoactive lone-pair electrons when including secondary bonds, and both intermediate states of lone-pair stereoactivity and inert lone pairs may occur for any coordination number > [4]. Variations in mean bond length are ∼0.06–0.09 Å for strongly bonded oxyanions of metalloid and post-transition metal ions, and ∼0.1–0.3 Å for ions showing lone-pair stereoactivity. Bond-length distortion is confirmed to be a leading cause of variation in mean bond lengths for ions with stereoactive lone-pair electrons. For strongly bonded cations (i.e. oxyanions), the causes of mean bond-length variation are unclear; the most plausible cause of mean bond-length variation for these ions is the effect of structure type, i.e. stress resulting from the inability of a structure to adopt its characteristic a priori bond lengths.

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Survey of Bond Lengths in the Solid State for Oxides: Results and Applications

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273314088962 ◽

2014 ◽

Vol 70 (a1) ◽

pp. C1103-C1103

Author(s):

Olivier Gagne ◽

Frank Hawthorne

Keyword(s):

Solid State ◽

Bond Length ◽

Crystal Structures ◽

A Priori ◽

Bond Valence ◽

Inorganic Crystal Structure Database ◽

Ionic Radii ◽

Coordination Polyhedra ◽

Coordination Numbers ◽

Bond Lengths

A complete survey of bond lengths from the Inorganic Crystal Structure Database (ICSD) is presented for all atoms of the Periodic Table of Elements, bonded to oxygen and in different oxidation states and coordination numbers. From over 135,000 crystal structures, a total of 33,343 coordination polyhedra and 188,462 bond distances were collected after passing a rigorous filtering process. One hundred thirty-six (136) ions in four hundred seventy-three (473) different configurations (coordination numbers) resulted. First, the bondlength distributions are visually inspected. This leads to (1) the observation and visual interpretation of known phenomena (e.g. Jahn-Teller effect), and (2) the isolation of new phenomena, as trends that are less obvious in smaller case-studies become more noticeable. Next, different applications of the data are investigated. The completeness of the survey allows the reassessment of important parameters of the solid state: ionic radii, and bond-valence parameters. Of the 473 ionic radii derived in this study, 329 revisions are made to Shannon's list of radii [1] (of which 176 were estimates), and 144 new ionic radii are derived. Next, a systematic evaluation of all bond-valence parameters published to date is done for oxides. Furthermore, using a new method of derivation, 136 new pairs of bond-valence parameters are obtained. In comparison to the previous-best published bond-valence parameters, an overall average decrease in the r.m.s.d. to the valence-sum rule of 20.7% (12.6% when weighted) is observed for the 33,343 coordination polyhedra, using the new parameters. New equations to describe the bond-length to bond-valence relation are also investigated. From an optimization between the experimental and a priori bond-valences of 54 carefully-selected crystal structures, roughly 20 relatively simple equations were selected for testing. Following a rigorous evaluation, the current exponential equation was found to be a viable choice in describing the relation. Finally, bond-length and bond-valence ranges are assigned to the 473 configurations of the atoms. Whereas the bondlength ranges are a useful aid in structure refinement, the assignment of a bond-valence range to ions allows a priori analysis of site occupancy in crystal structures.

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On the Crystal Chemistry of Inorganic Nitrides: Crystal-Chemical Parameters, Bonding Behavior, and Opportunities in the Exploration of Their Compositional Space

10.26434/chemrxiv.11626974.v4 ◽

2020 ◽

Author(s):

Olivier Charles Gagné

Keyword(s):

Multiple Bond ◽

A Priori ◽

Lone Pair ◽

Length Variation ◽

Electronic Effects ◽

Metal Centers ◽

Redox Active ◽

Jahn Teller ◽

Ion Substitution ◽

Compositional Space

The scarcity of nitrogen in Earth’s crust, combined with challenging synthesis, have made inorganic nitrides a relatively unexplored class of compounds compared to their naturally-abundant oxide counterparts. To facilitate exploration of their compositional space via a priori modeling, and to help a posteriori structure verification not limited to inferring the oxidation state of redox-active cations, we derive a suite of bond-valence parameters and Lewis-acid strength values for 76 cations observed bonding to N3-, and further outline a baseline statistical knowledge of bond lengths for these compounds. Examination of structural and electronic effects responsible for the functional properties and anomalous bonding behavior of inorganic nitrides shows that many mechanisms of bond-length variation ubiquitous to oxide and oxysalt compounds (e.g., lone-pair stereoactivity, the Jahn-Teller and pseudo Jahn-Teller effects) are similarly pervasive in inorganic nitrides, and are occasionally observed to result in greater distortion magnitude than their oxide counterparts. We identify promising functional units for exploring uncharted chemical spaces of inorganic nitrides, e.g. multiple-bond metal centers with promise regarding the development of a post-Haber-Bosch process proceeding at milder reaction conditions, and promote an atomistic understanding of chemical bonding in nitrides relevant to such pursuits as the development of a model of ion substitution in solids, a problem of great relevance to semiconductor doping whose resolve would fast-track the development of compound solar cells, battery materials, electronics, and more.

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On the Crystal Chemistry of Inorganic Nitrides: Crystal-Chemical Parameters, Bonding Behavior, and Opportunities in the Exploration of Their Compositional Space

10.26434/chemrxiv.11626974 ◽

2020 ◽

Author(s):

Olivier Charles Gagné

Keyword(s):

Multiple Bond ◽

A Priori ◽

Lone Pair ◽

Length Variation ◽

Electronic Effects ◽

Metal Centers ◽

Redox Active ◽

Jahn Teller ◽

Ion Substitution ◽

Compositional Space

The scarcity of nitrogen in Earth’s crust, combined with challenging synthesis, have made inorganic nitrides a relatively unexplored class of compounds compared to their naturally-abundant oxide counterparts. To facilitate exploration of their compositional space via a priori modeling, and to help a posteriori structure verification not limited to inferring the oxidation state of redox-active cations, we derive a suite of bond-valence parameters and Lewis-acid strength values for 76 cations observed bonding to N3-, and further outline a baseline statistical knowledge of bond lengths for these compounds. Examination of structural and electronic effects responsible for the functional properties and anomalous bonding behavior of inorganic nitrides shows that many mechanisms of bond-length variation ubiquitous to oxide and oxysalt compounds (e.g., lone-pair stereoactivity, the Jahn-Teller and pseudo Jahn-Teller effects) are similarly pervasive in inorganic nitrides, and are occasionally observed to result in greater distortion magnitude than their oxide counterparts. We identify promising functional units for exploring uncharted chemical spaces of inorganic nitrides, e.g. multiple-bond metal centers with promise regarding the development of a post-Haber-Bosch process proceeding at milder reaction conditions, and promote an atomistic understanding of chemical bonding in nitrides relevant to such pursuits as the development of a model of ion substitution in solids, a problem of great relevance to semiconductor doping whose resolve would fast-track the development of compound solar cells, battery materials, electronics, and more.

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On the Crystal Chemistry of Inorganic Nitrides: Crystal-Chemical Parameters and Opportunities in the Exploration of Their Compositional Space

10.26434/chemrxiv.11626974.v2 ◽

2020 ◽

Author(s):

Olivier Charles Gagné

Keyword(s):

Functional Properties ◽

A Priori ◽

Lone Pair ◽

Length Variation ◽

Electronic Effects ◽

Statistical Knowledge ◽

Redox Active ◽

Multiply Bonded ◽

Jahn Teller ◽

Compositional Space

The scarcity of nitrogen in Earth’s crust, combined with challenging synthesis, have made inorganic nitrides a relatively-unexplored class of compounds compared to their naturally-abundant oxide counterparts. To facilitate exploration of their compositional space via a priori modeling, and to help a posteriori structure verification not limited to inferring the oxidation state of redox-active cations, we derive a suite of bond-valence parameters and Lewis-acid strength values for 76 cations observed bonding to N3-, and further outline a baseline statistical knowledge of bond lengths for these compounds. We examine structural and electronic effects responsible for the functional properties and anomalous bonding behavior of inorganic nitrides, and identify promising venues for exploring uncharted compositional spaces beyond the reach of high-throughput computational methods. We find that many mechanisms of bond-length variation ubiquitous to oxide and oxysalt compounds (e.g., lone-pair stereoactivity, the Jahn-Teller and pseudo Jahn-Teller effects) are similarly pervasive in inorganic nitrides, and are occasionally observed to result in greater distortion magnitude than their oxide counterparts. We identify inorganic nitrides with multiply-bonded metal ions as a promising venue in heterogeneous catalysis, e.g. in the development of a post-Haber-Bosch process proceeding at milder reaction conditions, thus representing further opportunity in the thriving exploration of the functional properties of this emerging class of materials.

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Crystal structures of two erbium(III) complexes with 4-aminobenzoic acid and 4-chloro-3-nitrobenzoic acid

Acta Crystallographica Section E Crystallographic Communications ◽

10.1107/s2056989015020319 ◽

2015 ◽

Vol 71 (12) ◽

pp. 1457-1461 ◽

Cited By ~ 2

Author(s):

Graham Smith ◽

Daniel E. Lynch

Keyword(s):

Hydrogen Bonding ◽

Bond Length ◽

Dimethyl Sulfoxide ◽

Crystal Structures ◽

Water Molecules ◽

Aminobenzoic Acid ◽

Coordination Polyhedra ◽

Nitrobenzoic Acid ◽

The Third ◽

Length Range

The crystal structures of two erbium(III) complexes with 4-aminobenzoic acid (4-ABAH), namely bis(μ2-4-aminobenzoato-κ2O:O′)bis[bis(4-aminobenzoato-κ2O,O′)diaquaerbium(III)] dihydrate, [Er2(C7H6NO2)6(H2O)4]·2H2O, (I), and 4-chloro-3-nitrobenzoic acid (CLNBAH), namely poly[hexakis(μ2-4-chloro-3-nitrobenzoato-κ2O:O′)bis(dimethyl sulfoxide-κO)dierbium(III)], [Er2(C7H3ClNO4)6(C2H6OS)2]n, (II), have been determined. In the structure of solvatomorphic compound (I), the symmetry-related irregular ErO8coordination polyhedra in the discrete centrosymmetric dinuclear complex comprise two monodentate water molecules and six carboxylate O-atom donors, four from two bidentate carboxylateO,O′-chelate groups and two from the bis-monodentateO:O′-bridging group of the third 4-ABA anion. The Er—O bond-length range is 2.232 (3)–2.478 (3) Å and the Er...Er separation in the dinuclear complex unit is 4.7527 (4) Å. One of the coordinating water molecules is involved in an intra-unit O—H...O hydrogen-bonding association with an inversion-related carboxylate O-atom acceptor. In contrast, the anhydrous compound (II) is polymeric, based on centrosymmetric dinuclear repeat units comprising ErO7coordination polyhedra which involve four O-atom donors from two bidentateO:O′-bridging carboxylate groups, one O-atom donor from the monodentate dimethyl sulfoxide ligand and two O-atom donors from the third bridging CLNBA anion. The latter provides the inter-unit link in the one-dimensional coordination polymer extending along [100]. The Er—O bond-length range in (II) is 2.239 (6)–2.348 (6) Å and the Er...Er separation within the dinuclear unit is 4.4620 (6) Å. In the crystal of (I), extensive inter-dimer O—H...O and N—H...O hydrogen-bonding interactions involving both the coordinating water molecules and the solvent water molecules, as well as the amine groups of the 4-ABA anions, give an overall three-dimensional network structure. Within this structure are also weak π–π ring interactions between two of the coordinating ligands [ring-centroid separations = 3.676 (3) and 3.711 (2) Å]. With (II), only weak intra-polymer C—H...O, C—H...Cl and C—H...S interactions are present.

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

Acta Crystallographica Section B Structural Science Crystal Engineering and Materials ◽

10.1107/s2052520614011470 ◽

2014 ◽

Vol 70 (4) ◽

pp. 697-704 ◽

Cited By ~ 5

Author(s):

Ferdinando Bosi

Keyword(s):

Bond Length ◽

Crystal Structures ◽

Coordination Sphere ◽

Distortion Theorem ◽

Bond Valence ◽

Algebraic Expression ◽

Length Variation ◽

Bond Lengths ◽

Valence Theory ◽

New Equation

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

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