Section 9.5.2. Methodology

All results given in Table 9.5.1.1 are based on X-ray and neutron diffraction results retrieved from the September 1985 version of the CSD. Neutron diffraction data only were used to derive mean bond lengths involving H atoms. This version of the CSD contained results for 49 854 single-crystal diffraction studies of organo-carbon compounds; 10 324 of these satisfied the acceptance criteria listed below and were used in the averaging procedures:

All calculations were performed on the University of Cambridge IBM 3081 D using the programs BIBSER, CONNSER, RETRIEVE, GEOM78, and PLUTO78 (Allen et al., 1979). A stand-alone program was written to implement the selection criteria, whilst a new program (STATS) was written to perform the statistical calculations described below. It was also necessary to modify CONNSER to improve the precision with which it locates chemical substructures. In particular, the program was altered to permit the location of atoms with specified coordination numbers. This was essential in the case of carbon so that atoms with coordination numbers 2, 3 and 4 (equivalent to formal hybridization states sp¹, sp², sp³) could be distinguished easily and reliably. Considerable care was taken to ensure that the correct molecular fragment was located by GEOM78 in the generation of geometrical tabulations. This often involved the explicit specification of H atoms in fragments, and the extensive use of geometrical tests on valence and torsion angles. Considerable use was also made of chemical structural diagrams, which are available in the Cambridge in-house version of the CSD for some 81% of all entries. Chemical diagrams proved useful, for example, in identifying the various coordination environments commonly adopted by atoms such as As, B, P, etc.

The classification of bonds used in Table 9.5.1.1 is based on common functional groups, rings and ring systems, coordination spheres, etc. It is designed: (i) to appear logical, useful and reasonably self-explanatory to chemists, crystallographers, and others who may use the table; (ii) to permit a meaningful average value to be cited for each bond length. With reference to (ii), it was considered that a sample of bond lengths could be averaged meaningfully if: (a) the sample was unimodally distributed; (b) the sample standard deviation (σ) was reasonably small, ideally less than ca 0.02 Å; (c) there were no conspicuous outlying observations – those that occurred at > 4σ from the mean were automatically eliminated from the sample by STATS, other outliers were inspected carefully; (d) there were no compelling chemical reasons for further subdivision of the sample.

Where there are less than four independent observations of a given bond length, then each individual observation is given explicitly in the table. In all other cases, the following statistics were generated by the program STATS.

The statistics given in the final table correspond to distributions for which the automatic 4σ cut-off (see above) had been applied, and any manual removal of additional outliers (an infrequent operation) has been performed. In practice, a very small percentage of observations was excluded by these methods. The major effect of removing outliers is to improve the sample standard deviation, as shown in Fig. 9.5.2.1 in which a single observation is deleted.

Figure 9.5.2.1| top | pdf | Effect of the removal of outliers (contributors that are > 4σ from the mean) for the C—C bond in Car—C≡N fragments. Relevant statistics (see text) are:

The statistics chosen for tabulation effectively describe the distribution of bond lengths in each case. For a symmetrical, normal distribution: the mean (d) will be approximately equal to the median (m); the lower and upper quartiles will be approximately symmetric about the median: , and 95% of the observations may be expected to lie within ±2σ of the mean value. For a skewed distribution, d and m may differ appreciably and and will be asymmetric with respect to m. When a bond-length distribution is negatively skewed as in Fig. 9.5.2.2,i.e. very short values are more common than very long values, then it may be due to thermal-motion effects; the distances used to prepare the table were not corrected for thermal libration.

Figure 9.5.2.2| top | pdf | Skewed distribution of B—F bond lengths in ions: d = 1.365, m = 1.372, σ = 0.029, ql = 1.352, qu = 1.390 for 84 observations. Note that d ≠ m and that ql, qu are asymmetrically disposed about the mean d.

In a number of cases, the initial bond-length distribution was clearly bimodal, as in Fig. 9.5.2.3(a). All cases of bimodality were resolved on chemical grounds before inclusion in the table, on the basis of hybridization, conformation-dependent conjugation interactions, etc. For example, the histogram of Fig. 9.5.2.3(a) was resolved into the two discrete unimodal distributions of Figs. 9.5.2.3(b), (c), which correspond to planar N(sp²), pyramidal N(sp³), respectively. The mean valence angle at N was used as the discriminator, with a range of 108–114° for Nsp³ and 117.5° for Nsp².

Figure 9.5.2.3| top | pdf | Resolution of the bimodal distribution of C—N bond lengths in Car —N(Csp3)2 fragments: (a) complete distribution; (b) distribution for planar N, mean valence angle at N > 117.6°; (c) distribution for pyramidal N, mean valence angle at N in the range 108–114°.