Date Published: August 01, 2016
Publisher: International Union of Crystallography
Author(s): Olivier Charles Gagné, Frank Christopher Hawthorne.
Bond-length distributions have been examined for 55 configurations of alkali-metal ions and 29 configurations of alkaline-earth-metal ions, for 4859 coordination polyhedra and 38 594 bond distances (alkali metals), and for 3038 coordination polyhedra and 24 487 bond distances (alkaline-earth metals).
Many crystal structures have been refined in the past 100 years, and a large amount of information concerning interatomic distances in the solid state is available. There are many studies of bond-length distributions for specific pairs of ions, notably for cations bonded to oxygen [e.g. Baur, 1971 ▸ (Si); Burns et al., 1997 ▸ (U); Hawthorne et al., 2000 ▸ (S); Schindler et al., 2000 ▸ (V); Hawthorne & Huminicki, 2002 ▸ (Be); Huminicki & Hawthorne, 2002 ▸ (P); Mills & Christy, 2013 ▸ (Te); Majzlan et al., 2014 ▸ (As)]. However, many of these studies have focused on subsets of the available information, both with regard to the number of ions and coordination numbers, and to the amount of data available for each example. We have examined the distribution of bond lengths for 135 ions bonded to oxygen in 462 configurations using 180 331 bond lengths extracted from 9367 refined crystal structures; these data involve most ions of the periodic table and all coordination numbers in which they occur. Here we report the bond-length distributions for 10 ions, the common alkali-metal ions (Li+, Na+, K+, Rb+ and Cs+) and alkaline-earth-metal ions (Be2+, Mg2+, Ca2+, Sr2+ and Ba2+) in all observed coordination numbers where bonded to O2− for a total of 63 081 bond lengths in 7897 polyhedra from 4258 refined crystal structures. An advantage of working with a large number of ion pairs and a large amount of data is that it allows examination of subtle differences between the shapes of various distributions (e.g. bond-length distributions, mean-bond-length distributions) for various configurations of ions, which reflect differences in their bonding behaviour. These differences typically arise from either structural and/or electronic effects, and are well known for extreme examples such as -coordinated Cu2+ and -, - and -coordinated U6+; however, more subtle deviations from unimodality could be expected for the bond-length distributions of other ion configurations that are involved in related electronic or structural effects. Our motivation for this work is twofold: (1) The factors that affect bond distances are of continuing interest to all who work on crystal structures and their properties, and a comprehensive analysis of all the data should lead to increased understanding of those factors. Here we give a preliminary examination of the alkali-metal ions and alkaline-earth-metal ions in all observed coordination numbers where bonded to O2−, and make our complete dataset available for future more detailed work. (2) A comprehensive knowledge of the observed variation in bond lengths is critically important in assessing the validity of computational results on possible atomic arrangements (e.g. Richardson, 2013 ▸) and identifying unusual stereochemical features in newly solved or refined crystal structures.
In the interest of clarity, we define certain terms that we use in the following text. We make no claims of generality; these are merely working definitions.
The DVD-ROM version of the ICSD with FindIt, Version 2010-2, was used for data collection for all ions bonded to oxygen. The collection of bond-length data was done on the basis of coordination polyhedra for all cations of the periodic table. A set of structures containing each ion pair of interest was accumulated for each cation. In these structure sets, only the coordination polyhedra of the cation of interest were evaluated. The bonds in each coordination polyhedron were calculated and individually examined to ensure that only suitable entries were included. Valid coordination polyhedra were not discarded due to problems elsewhere in the structure that have no effect on the coordination polyhedron of interest.
Bond-length distributions commonly resemble a positively skewed Gaussian distribution. The shape originates from the variation in Born repulsion and Coulomb attraction as a function of interatomic distance. Two useful statistical measures used to describe the shape of these distribution are skewness and kurtosis. Skewness is a measure of the asymmetry of the distribution about its mean, and can be positive (as in Fig. 1 ▸ for Na+) or negative. Kurtosis is a measure of the distribution of data between the peak and the tails of the distribution: a high kurtosis indicates that the distribution has a sharper maximum and larger tails, and a low kurtosis indicates that the distribution has a rounder maximum and smaller tails. Thus, important data that we derive from the bond-length distributions determined here are mean bond length, skewness and kurtosis. Deviations from this typical shape are frequent, and can be the result of structural and/or electronic effects that result in emergent bond-length constraints. Hence, we can gain insight into the reasons underlying the bonding behaviour of atoms from a visual inspection of their bond-length distributions, e.g. the familiar (4 + 2) bimodal distribution of bond lengths for octahedrally coordinated Cu2+ from the Jahn–Teller effect (Jahn & Teller, 1937 ▸), associated with the degenerate electronic ground state of a d9 metal in a holosymmetric octahedral field.
Our collection and filtering criteria resulted in a combined sample size of 38 594 bonds and 4859 coordination polyhedra. Table 1 ▸ gives the 55 observed configurations, the mean bond length and standard deviation, the minimum and maximum bond length (and range), the skewness and kurtosis (where justified by sample size), and the number of bonds and coordination polyhedra for the five common alkali-metal ions. Fig. S1 gives all the bond-length distributions for the alkali metals; those with adequate sample sizes (as discussed above) are shown in Fig. 5 ▸. An important issue is the reliability of the data at the limits of its distribution, i.e. at the lowest and highest observed coordination numbers for each ion, and below we examine the data at the lower and upper limits of these distributions.
Our collection and filtering criteria resulted in a combined sample size of 24 487 bonds and 3038 coordination polyhedra. Table 2 ▸ gives the 29 observed configurations, the mean bond length and standard deviation, the minimum and maximum bond length (and range), the skewness and kurtosis (where justified by sample size), and the number of bonds and coordination polyhedra for the five common alkaline-earth-metal ions. Fig. S2 gives all the bond-length distributions for the alkaline-earth metals; those with adequate sample sizes (as discussed above) are shown in Fig. 8 ▸. These ions are found in slightly more than half the number of configurations observed for the alkali metals (55), primarily because these ions are not observed in coordinations higher than , with the exception of Ba2+ (observed as  and ).
Figs. S3 and S4 give all the mean bond-length distributions for the alkali-metal and alkaline-earth-metal ions; those with adequate sample sizes (below) are shown in Figs. 14 ▸ and 15 ▸. Tables 4 ▸ and 5 ▸ give the grand mean bond length and standard deviation, the minimum and maximum mean bond length (and range), the skewness and kurtosis of each distribution (where justified by sample size) and the number of coordination polyhedra and coordination numbers for all configurations for the alkali and alkaline-earth metals. A minimum sample size was determined in the same way as above for Na+, less than which the values of skewness and kurtosis have little significance; this threshold was set to 100 coordination polyhedra and is relatively high due to the wide range of mean bond lengths observed for these families.
(1) We have examined the bond-length distributions for 55 configurations of alkali-metal ions and 29 configurations of alkaline-earth-metal ions, for 4859 coordination polyhedra and 38 594 bond distances (alkali metals) and for 3038 coordination polyhedra and 24 487 bond distances (alkaline-earth metals).