Tables for
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 9.6, pp. 812-814

Section 9.6.2. Methodology

A. G. Orpen,a L. Brammer,b F. H. Allen,c D. G. Watsonc and R. Taylorc

aSchool of Chemistry, University of Bristol, Bristol BS8 1TS, England,bDepartment of Chemistry, University of Missouri–St Louis, 8001 Natural Bridge Road, St Louis, MO 63121-4499, USA, and cCambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, England

9.6.2. Methodology

| top | pdf | Selection of crystallographic data

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All results given in Table[link] 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 hydrogen atoms. This version of the CSD contained results for 49 854 single-crystal diffraction studies of organo-carbon compounds; 9802 of these satisfied the acceptance criteria listed below and were used in the averaging procedures:

  • (i) Structure contains a d- or f-block metal.

  • (ii) Atomic coordinates for the structure have been published and are available in the CSD.

  • (iii) Structure was determined from diffractometer data.

  • (iv) Structure does not contain unresolved numeric data errors from the original publication (such errors are usually typographical and are normally resolved by consultation with the authors).

  • (v) Only structures of higher precision were included on the basis that either (a) the crystallographic R factor was ≤ 0.07 and the reported mean estimated standard deviation (e.s.d.) of the C—C bond lengths was ≤ 0.030 Å (corresponds to AS flag = 1, 2 or 3 in the CSD), or (b) the crystallographic R factor ≤ 0.05 and the mean e.s.d. for C—C bonds was not available in the database (AS = 0 in the CSD).

  • (vi) Where the structure of a given compound had been determined more than once within the limits of (i)–(v), then only the most precise determination was used.

The structures used in Table[link] do not include compounds whose structure precludes them from the CSD (i.e. not containing `organic' carbon). In practice, structures including at least one C—H bond are taken to contain `organic' carbon. Thus, the entry for Cr—CO distances has a contribution from [NEt4][Cr(μ-H)(CO)10] but not from K[Cr(μ-H)(CO)10] or [Cr(CO)6]. Program system

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All calculations were performed on a University of Bristol VAX 11/750 computer. Programs BIBSER, CONNSER, RETRIEVE (Allen et al., 1979[link]) and GEOSTAT (Murray-Rust & Raftery, 1985a[link],b[link]), as locally modified, were used. A stand-alone program was written to implement the selection criteria, whilst a new program (STATS) was used for 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 state sp1, sp2, sp3) could be distinguished easily and reliably. Considerable care was taken to ensure that the correct molecular fragment was located by GEOSTAT in the generation of geometrical tabulations. Searches were conducted for all metals together and statistics for individual metal elements and subdivision of the entry for a given metal carried out subsequently. An important modification to GEOSTAT allowed for calculation of metal-atom coordination number with due allowance for multihapto ligands and μ2 ligands. Thus, η5-C5H5, η6-C6H6, and other η5 and η6 ligands were assigned to occupy 3 coordination sites, η3 and η4 ligands such as allyls and dienes to occupy 2 coordination sites, and η2 ligands such as alkenes 1 site, and so on. The approach taken in dealing with (μ2) bridging ligands was that when a metal–metal bond is bridged by one atom of a ligand [e.g. as in Cl, CO, OMe etc. as in (a), (b) below] then only the non-metal atom is counted as occupying a coordination site. For the relatively rare case of bridging polyhapto ligands (in which the bridging atoms are linked by direct bonds), the assignment follows logically, thus, μ222-alkyne, see (c) below, occupies one site on each metal. Bridging ligands that do not have one atom bonded to both metals [e.g. acetate in (d) below] contribute to metal coordination numbers as do terminal ligands. In examples (a)–(d) below, the metal atoms therefore have coordination numbers as follows: (a), Rh 4; (b), Fe 6; (c), Co 4; (d), Rh 6. For cases where coordination number is very difficult to assign, notably where a metal atom is bonded to more than one other metal atom as in metal cluster complexes, no assignment was attempted. [Scheme cbch9.6scheme1]

The non-location of hydrogen atoms presents major difficulties, both in the determination of coordination numbers for metal atoms, and for correct identification of ligands (e.g. to distinguish methoxide from methanol). Care was therefore taken to exclude cases where any ambiguity existed [e.g. no data taken for M—(OCH3) and M—O(H)CH3 distances when both are present in a structure in which hydrogen-atom positions were not reported]. Classification of bonds

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The classification of metal–ligand bonds in Table[link] is based on the ligating contacting atom. Thus, all metal–boron distances appear in sections 2.1–2.3 of Table[link], all metal–carbon distances in sections 3.1–3.22, and so on. Where intra-ligand interatomic distances (e.g. P—C distances in tertiary phosphines) are given in Table[link], they are averaged over all metals and precede the individual metal–ligand interatomic distances for that ligand.

Table[link] is designated: (i) to appear logical, useful, and reasonably self-explanatory to chemists, crystallographers, and others who may use it; (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 0.04 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. It should be noted that Table[link] is not intended to be complete in covering all possible ligands. The purpose of the table is to provide information on the interatomic distances for ligands of the greatest chemical importance, notably for those that are simple and/or common. Statistics

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

  • (i) The unweighted sample mean, d, where [d=\textstyle\sum\limits^n_{i=1}d_i/n]and [d_i] is the ith observation of the bond length in a total sample of n observations. Recent work (Taylor & Kennard, 1983[link], 1985[link], 1986[link]) has shown that the unweighted mean is an acceptable (even preferable) alternative to the weighted mean, where the ith observation is assigned a weight equal to 1/var(di). This is especially true where structures have been pre-screened on the basis of precision.

  • (ii) The sample median, m. This has the property that half of the observations in the sample exceed m, and half fall short of it.

  • (iii) The sample standard deviation, σ, where [\sigma=\left[\textstyle\sum\limits^n_{i=1}(d_i-d)^2/(n-1)\right]^{1/2}.]

  • (iv) The lower quartile for the sample, [q_l]. This has the property that 25% of the observations are less than [q_l] and 75% exceed it.

  • (v) The upper quartile for the sample, [q_u]. This has the property that 25% of the observations exceed [q_u] and 75% fall short of it.

  • (vi) The number (n) of observations in the sample.

The statistics given in Table[link] 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) had been performed. In practice, a very small percentage of observations were excluded by these methods. The major effect of removing outliers is to improve the sample standard deviation, as shown in Fig.[link] in which four (out of 366) observations are deleted.


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Effects of outlier removal and subdivision based on coordination number and oxidation state. Cu—Cl: (a) all data; (b) all data without outliers [> 4σ (sample) from mean]; (c) all data for which Cu is 4-coordinate, CuII.[\matrix{&d &m &\sigma &q_l &q_u &N \cr (a) &2.282 &2.255 &0.105 &2.233 &2.296 &366 \cr (b) &2.276 &2.254 &0.092 &2.232 &2.292 &362 \cr (c) &2.248 &2.246 &0.032 &2.233 &2.263 &153\cr}]

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 ([q_l,q_u]) will be approximately symmetric about the median [m-q_l\simeq q_u-m], 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 [q_l] and [q_u] will be asymmetric with respect to m. When a bond-length distribution is negatively skewed, 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.

In a number of cases, the initial bond-length distribution was clearly not unimodal as in Fig.[link]. Where possible, such distributions were resolved into their unimodal components (as in Fig.[link]) on chemical or structural criteria. The case illustrated in Fig.[link], for Cu—Cl bonds, is one of the most spectacular examples, owing to the dramatic consequences of oxidation state and coordination number (and Jahn–Teller effects) on the structures of copper complexes.


Allen, F. H., Bellard, S., Brice, M. D., Cartwright, B. A., Doubleday, A., Higgs, H., Hummelink, T., Hummelink-Peters, B. G., Kennard, O., Motherwell, W. D. S., Rodgers, J. R. & Watson, D. G. (1979). The Cambridge Crystallographic Data Centre: computer-based search, retrieval, analysis and display of information. Acta Cryst. B35, 2331–2339.
Murray-Rust, P. & Raftery, J. (1985a). J. Mol. Graphics, 3, 50–59.
Murray-Rust, P. & Raftery, J. (1985b). J. Mol. Graphics, 3, 60–68.
Taylor, R. & Kennard, O. (1983). The estimation of average molecular dimensions from crystallographic data. Acta Cryst. B39, 517–525.
Taylor, R. & Kennard, O. (1985). The estimation of average molecular dimensions. 2. Hypothesis testing with weighted and unweighted means. Acta Cryst. A41, 85–89.
Taylor, R. & Kennard, O. (1986). Cambridge Crystallographic Data Centre. 7. Estimating average molecular dimensions from the Cambridge Structural Database. J. Chem. Inf. Comput. Sci. 26, 28–32.

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