**Date Published:** December 01, 2017

**Publisher:** International Union of Crystallography

**Author(s):** Olivier Charles Gagné, Frank Christopher Hawthorne.

http://doi.org/10.1107/S2052520617014548

**Abstract**

**Variations in mean bond length are examined in oxide and oxysalt crystals for 55 cation configurations bonded to O2−. Bond-length distortion is confirmed as a statistically significant causal factor of mean bond-length variation. The assignment of a coordination-based radius to O2− is found not to be supported by experimental data.**

**Partial Text**

In the 1960s and 1970s, a considerable amount of work was carried out on trying to understand the reasons underlying variations in mean bond length in crystals. This resulted from the improving precision of structure refinements which began showing variations in mean bond length that significantly exceeded experimental error. Several factors were examined as possible sources of this variation, and many studies were reported as ‘reasonably successful’ in correlating variation in mean bond length with one or more possible causal factors, e.g. variation in mean coordination number of the bonded anions, variation in mean electronegativity of the next-nearest-neighbour cations, dispersion of bond lengths about their mean value (distortion). However, these studies were typically limited to a single configuration of the oxidation state and coordination number of an ion, and often consisted of few data.

Persuasive examples have been reported describing correlations between mean bond length and bond-length distortion for ion configurations prone to large distortions (e.g. Brown & Shannon, 1973 ▸; Shannon, 1976 ▸) but the generality of this relation is not established for all ion configurations. Furthermore, the importance of other factors outlined above remains unclear. Here, we clarify these effects by using the results of a very large bond-length dispersion analysis carried out by the authors for 135 cations in 462 configurations, for a total of 180 331 bond lengths and 31 514 coordination polyhedra from 9367 refined crystal structures (described by Gagné & Hawthorne, 2015 ▸, 2016a ▸, 2017a ▸,b ▸; Gagné, 2017 ▸). Our data set is being released as it is published (e.g. Gagné & Hawthorne, 2016a ▸) with the hope of encouraging further detailed studies.

The following variables were systematically evaluated for all ion configurations as potential causal factors underlying mean bond-length variation: (1) bond-length distortion, (2) mean coordination number of the oxygen atoms bonded to the cation (previous references used the less accurate mean coordination number of all oxygen atoms of the structure), (3) mean electronegativity (〈χ〉) of the cations bonded to the oxygen atoms of the coordination polyhedron, (4) mean ionization energy (〈IE〉) of the cations bonded to the oxygen atoms of the coordination polyhedron (new).

The reporting of statistical significance for the correlations cited in the Introduction has been scarce. This is a cause for concern as some of the correlations reported as ‘significant’ have not been tested for a specific confidence level. Here, we have tested all correlations for statistical significance on an individual basis, and via multiple-regression analysis (Student t-test, 95% confidence level). In addition, we have not excluded different types of data (e.g. alkali-metal silicates from a consideration of Si4+—O bonds) from our analysis in order to avoid bias due to pre-conceived notions of causality.

In order to gauge the reliability of the correlations developed here, we have examined the variation of (1) p-values, and (2) R2 values as a function of sample size for SiO4, using data taken at random from our data set of 334 coordination polyhedra. For a large set of data, one expects p-values to be independent of sample size, such that a statistically significant correlation may be ascribed as such regardless of the number of data. As the number of data decreases, eventually the p-values and R2 values will become unreliable, and it is obviously important to know at what number of data the analysis begins to become unreliable.

A stepwise regression analysis based on t-tests (95% confidence level) was used to eliminate misleading correlations between the variables of this study. A step-by-step procedure of the stepwise regression analysis for [6]Na+ is shown in Table 1 ▸. When individually correlated to mean bond length, p-values for (1) bond-length distortion, (2) mean coordination of the bonded anions, (3) mean electronegativity (〈χ〉)and (4) mean ionization energy (〈IE〉) of the next-nearest-neighbour cations are 5.4 × 10−8, 1.9 × 10−3, 1.5 × 10−3, 0.26, respectively. The first step of the regression analysis shows that factoring in the variable with lowest p-value (italicized, i.e. distortion), leads to a R2 value of 0.24. The p-values of the other three variables update, and the mean electronegativity of the next-nearest-neighbour cation then gets factored in as the remaining variable with the lowest p-value below the threshold of 0.05, leading to an adjusted R2 value of 0.27. Interestingly, this then results in a p-value of 0.24 (above 0.05) for the mean coordination of the bonded anions. However, including the mean ionization energy of the next-nearest-neighbour cation in the next step (its p-value having dropped below 0.05) brings back its p-value below 0.05 for the final step. Thus all variables get included in the regression analysis, and are construed as significant. We note that running the regression for a 99% confidence level, the refinement would have stopped after step 2, and only bond-length distortion and the mean electronegativity of the next-nearest-neighbour cation would have been included as significant.

We selected samples for 55 ion configurations from our bond-length dispersion analysis, bonded solely to O2−, with a minimum size of ∼ 20 coordination polyhedra. Stepwise regression analysis was performed (based on p-values < 0.05) with mean bond length as the dependent variable and (1) bond-length distortion, (2) mean coordination of the bonded anions, (3) mean electronegativity and (4) mean ionization energy of the next-nearest-neighbour cations as independent variables. Results are listed in Table 2 ▸ for correlations significant at the 95% confidence level (at 99% shown in bold) in the form of R2 values, and adjusted R2 values for ion configurations for which two or more potential factors are statistically significant for a 95% confidence level. Results are arranged in decreasing order of R2. Of the 220 individual correlation trials (55 per variable), 77 are significant at the 95% confidence level and 62 at the 99% confidence level. Of the 77 correlations that are significant at the 99% confidence level, 42 involve distortion, 14 involve mean coordination of the bonded anions, 13 involve mean ionization energy of the next-nearest-neighbour cations, and eight involve the mean electronegativity of the next-nearest-neighbour cations. (1) Following a review of previous work on the variation of mean bond length in oxide and oxysalt crystals, we use 55 cation configurations bonded to O2− to analyze the relation between mean bond lengths and (a) bond-length distortion, (b) mean coordination number of the oxygen atoms bonded to the cation, (c) mean electronegativity of the next-nearest-neighbour cations, and (d) mean ionization energy of the cations bonded to the oxygen atoms of the coordination polyhedron via stepwise multiple regression analysis at the 95% confidence level. Source: http://doi.org/10.1107/S2052520617014548