Effect of hydrogen bonding on coordinate bond. semi-coordination in werner-type copper(ii) complexes revisited

Crystallographic statistics in chemical physics

Nonrigid properties of monatomic entities manifest themselves in the whole geometry of their environs. Until now we studied the statistics of these geometries using radial distributions of interatomic vectors. The classification of surroundings of a certain central monatomic entity MZ+ can be based also on their symmetry. It has group — theoretical properties and it can be described by means of non — crystallographic symmetry point groups, of which there are infinitely many. There is, however, a finite amount of crystallographic point groups, viz. 2 one-dimensional, 10 two-dimensional and 32 three-dimensional point groups. The latest of them also describe the site symmetry of monatomic entity in the crystal structure and thus also the symmetry of its surroundings. Surroundings of monatomic entity means here, however, not only the near surroundings usually consisting of coordinating ligands, but the complete surroundings containing all other non coordinating entities of the structure. The symmetry of an ideal crystal structure — of discontinuum, is described by one of 230 space groups. The site symmetry of each monatomic entity of the structure thus has a certain connection with the relative positions of entities which may be classified based on the conception of lattice complexes.

Statistics of Crystallographic Sites of Central Monatomic Entities

The search for the factors causing distortions of surroundings of central entities is the subject of interest especially of chemists. It is the region of interatomic vector lengths, which are less than the most probable minimum value a 0′ = 4.25 A, where it is already difficult to suppose a fitting of some known statistical model. The non-rigid properties of central entities in this region, let us call it the coordination sphere of entity Mz+, apparently manifest themselves also in distortions of the geometry of arrangement of monatomic ligand entities. In agreement with the commonly used terms for describing crystal structures of compounds we will call monatomic entities of ligands “ligand atoms” resp. atoms. The number of bonds formed by the central entity is the coordination number. The bond lengths between the central entity and ligand atoms will be signed M-L.

Statistics of Bond Distances Between Central Monatomic Entities and Ligands

An ideal crystal is usually understood to be a regular array of ions or atoms. However, if we speak of positions of these particles with respect to their crystallographic origin, physical reality forces us to make some approximations. An experiment based on the diffraction of X-ray or neutron radiation by crystal makes it possible to obtain information on the positions of atoms or ions in the form of coordinates of local maxima of electron densities.

Statistics of Interatomic Distances

Table 2.1 presents empirical estimates of the first two central moments of interatomic vector lengths with the described properties. In this case terminal monatomic entities belong to a certain type. The dispersion of lengths of these vectors expressed by the second central moment m 2 shows the value of 7.383 A2 for the sample of vector lengths of Zn2+ → 0, this value being much higher than 5.192 A2 for the sample of vector lengths for Zn2+ → S. In the case of such great sample sizes the variance equality of these data sets can be tested using sample characteristics of $F = S_1^2/S_2^2:$

$${H_0}:\sigma _1^2 = \sigma _2^2against{H_1}:\sigma _1^2 \ne \sigma _2^2$$

We will use the estimate of the above characteristics F 1 = m2(Zn2 + → 0)/m 2(Zn2 + → S) = 1.422. The critical Fvalue on the significance level of α = 0.05 for v1 = n(Zn2 + → 0) — 1 and v2 = n(Zn2 + →S) — 1 is 1.011 [1]. Since F > Fcrit the hypothesis H 0 of the equality of dispersions can be rejected. The vector lengths of Co3+ → Cl and Co3+ → N also show a significant difference between their dispersions. In both cases of central monatomic entities one could explain the differences between the dispersions of interatomic vector lengths with different qualities of their terminal entities. The values of m 2 of the samples of vectors of Zn2+ → S and Zn2+ → N exhibit, however, only a small difference (0.139 A2). Similarly, the estimates of the second central moments of samples Cu2+ →Cl and Cu2+ → N are comparatively near.

Jan Ondráček

Papers

Crystallographic statistics in chemical physics

Statistics of Crystallographic Sites of Central Monatomic Entities

Statistics of Bond Distances Between Central Monatomic Entities and Ligands

Statistics of Interatomic Distances

Factors Influencing the Dispersion of Lengths of Interatomic Vectors