ISSN 1070-3284, Russian Journal of Coordination Chemistry, 2008, Vol. 34, No. 10, pp. 723–728. © Pleiades Publishing, Ltd., 2008. Original Russian Text © V.N. Serezhkin, D.V. Pushkin, L.B. Serezhkina, 2008, published in Koordinatsionnaya Khimiya, 2008, Vol. 34, No. 10, pp. 733–738.
Maximum Filling Principle and Sublattice Characteristics for the Atoms of Period 3 Elements V. N. Serezhkin, D. V. Pushkin, and L. B. Serezhkina Samara State University, Samara, Russia E-mail:
[email protected] Received October 4, 2007
Abstract—The most important characteristics of the Voronoi–Dirichlet polyhedra (VDP) of A atoms (A = Na, Mg, Al, Si, P, S, Cl, or Ar) were determined. The sublattices contain chemically identical A atoms in the crystal structures of 232006 inorganic, coordination, and organoelement compounds. The VDP of A atoms have most often 14 faces, the Fedorov cuboctahedron being the most abundant VDP type. The effect of the nature of the A atoms on the characteristic A–A interatomic distances in the homoatomic sublattices of the crystal structures was considered. DOI: 10.1134/S1070328408100035
It is known [1–3] that the maximum space filling principle is in line with two complementary extreme models of the crystal structure, where atoms are approximated by spheres of a constant volume. In the classical model, atoms are regarded as hard spheres able only to contact one another and the structure is considered as some lattice type close packing of spheres in which the contact or coordination number (C.N.) of atoms is ideally (in particular, for the facecentered cubic lattice) equal to 12. In the stereoatomic model, atoms are modeled by soft (easily deformable or intersecting) spheres that tend to form a sparse coating; an ideal packing for a 3D space is arrangement of sphere centers in the points of a body-centered cubic lattice [4] with C.N. 14. Currently, the former model is generally accepted; in particular, this model underlies the known assumption of the crucial structure-forming role of heavy element atoms in the crystals of compounds and their preferred arrangement according to a close sphere packing (12-neighbor rule). However, in terms of the crystal structure as a sparse coating, a 14- rather than 12-neighbor rule should hold for heavy atoms. In order to verify experimentally the maximum filling principle and either 12- or 14-neighbor rule, we carry out systematic analysis of the most important geometrical and topological characteristics of sublattices comprising chemically identical atoms in all compounds with determined crystal structures. This analysis has been already carried out for sublattices comprising atoms of period 4–6 elements [1–3]. In view of this task, this study, which continues previous works [1–3], is devoted to crystal chemical analysis of compounds containing atoms of period 3 elements. The procedure based on characteristics of the Voronoi–Dirichlet polyhedra (VDP), all the notions and designations used
including the notation of the combinatorial-topological types (CTT) of the polyhedra were described in detail previously [1] and explained here only if necessary. The crystal chemical analysis covered all compounds containing atoms of period 3 elements (A atoms) whose crystal structures were present in databases [5, 6] by the beginning of 2007 and met two conditions, namely, the structure had been determined to an R-factor of 20 Å. The vertical dashed line in each diagram indicates the shortest r(A–A) distance in the crystal structure of the given element in the elemental form. Vol. 34
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nosilicon compounds with various compositions and structures. In chlorine-containing compounds dmax ≈ 3.8 Å corresponds to the contacts between the chlorine atoms either coordinated by the same metal atom (in particular, zinc in ä2ZnCl4 {66515}) or incorporated in neighboring molecules or complexes (for example, ë2Cl7P {HOCTOI} and C9H18Cl3O3Ti {ZUGZAC}). Note that a sharp maximum at ~2.9 Å in the Cl-sublattice (Fig. 1) is due to the contacts between chlorine atoms in −CHCl2, =CCl2, or CCl3 in the structure of numerous organochlorine compounds or ions (in particular, chloroform molecules, dichloroacetate ions, etc.). The pattern of the right-hand part of the column diagrams with r(Ä–Ä) ≥ D (Fig. 1a) almost does not depend on the nature of A. In all cases (except for Na atoms), the distributions of interatomic distances in the A-sublattices are close to normal distribution. Only in the Na-sublattices the number of Na–Na contacts decreases almost linearly with an increase in the interatomic distance. The available data indicate that the contacts with r(Ä–Ä) ≥ D correspond to A atoms that exist in the crystal structure as parts of independent neighboring molecules or complex ions and, most often, they are separated from each other by bulky polyatomic groups containing no A atoms. Thus, from the probabilistic standpoint, realization of some interatomic distance with r(A–A) ≥ D can be considered to be a random event independent of the nature of A element. This accounts for the similar patterns in the righthand parts of the column diagrams in Fig. 1a. The fundamental difference between the r(Ä–Ä) distributions in the two regions (Fig. 1a) can de described using the notion of the rank of VDP face (RF). It is known [8] that RF designates the minimum number of chemical bonds that connect two atoms (for example Q and R) whose polyhedra share a face. In general, three different cases are possible: (1) if RF = 1, the Q–R contact is a chemical bond. (2) If RF > 1, the Q and R atoms are not directly connected but are parts of the same molecule, chain, layer, or framework and, hence, it is possible to “walk” from Q to R along a finite set of chemical bonds the total number of which is equal to RF. In terms of classical crystal chemistry, faces with RF > 1 correspond to nonvalence contacts. (3) If RF = 0, this contact is equivalent to nonvalence but necessarily intermolecular contacts, as atoms Q and R are incorporated in different molecules (chains, layers, or nonintersecting frameworks) and, therefore, no bond path exists between them. In view of the foregoing, it can be stated that the r(Ä–Ä) ≥ D region in Fig. 1a represents mainly interatomic contacts with RF = 0, whereas the region with r(Ä–Ä) < D describes mainly contacts with RF ≥ 1. For example, in the Sisublattices the same distance of 4.23 Å may indicate nonvalence Si···Si contacts with RF = 2, 3, 4, 6, 8, 14, or 15. An example of compound with Si···Si interactions with RF = 15 is the zeolite Si24O48 · 2C6H4Cl2 {203221} (the Si(1)···Si(13) contacts).
The number of VDP faces in the A-sublattices varies from 4 to 44 and the average number of faces of a VDP (Nf, Table 1) is 12 to 15.3. The minimum Nf was found for argon where three of the five VDP have 12 faces and the other two, 8 and 16 faces. For other A atoms, 14-face polyhedra are most abundant in the A-sublattices; their fraction in the samples varies from ~22% for Cl to ~41% for Mg (Table 2). Dodecahedra were the second most abundant in the Mg-sublattice (~13%). In the A-sublattices with Na, Al, Si, P, S, and Cl, VDP with 16 faces were the second most abundant, whereas dodecahedra are the third (A = Na), fourth (Al), sixth (Al), and even seventh (P, S, Cl) in abundance, being encountered less frequently than the VDP with 13, 14, 15, 16, 17, and 18 faces (Table 2). Note that a similar situation was observed for s- and p elements of periods 4–6 [1–3]. In all cases, in A-sublattices of atoms of group 1 and 3–7 elements, the VDP with 12 faces are inferior in the frequency of occurrence not only to 14-face polyhedra but also to 16-face polyhedra. Only in sublattices comprising group 2 metals, dodecahedra are second most abundant being less frequently found than only VDP with 14 faces but always most frequently found than 16-face VDP. The number of vertices (Nv) at the VDP faces of A atoms ranges from 3 to 20. Most often, Nv = 4 or 6. The overall number of tetragonal and hexagonal faces is 50 to 87% of the total number of polyhedron faces, and the tetragon to hexagon ratio for VDP faces varies over a broad range from 1.03 (P) to 12 (Ar). Faces with Nv of 3 or 5 are also encountered rather often (on average, about 10 and 18% of the total number of faces, respectively). The existing diversity of Nf and Nv values gives rise to a large number of CTT of VDP existing in A-sublattices. The compounds under consideration have 201571 CTT of the VDP per 555702 sorts of A atoms, i.e., the same combinatorial type of the VDP should, on average, be found for three A atoms. However, the vast majority of CTT are represented by single examples, therefore, the VDP of several most abundant types are sharply distinguished against the background (Table 3). As previously [1–3], Table 3 includes the CTT of all VDP whose frequency of occurrence exceeded 3% of the sample size for at least one element. Note that in the sublattices we discuss, this threshold was crossed for the first time by the VDP with 16 faces described as the CTT [3246526284]. The Schlegel projection [9] of this polyhedron is shown in Fig. 2 and for the polyhedra of other CTT (Table 3), which are also typical of period 4−6 A-sublattices, these projections were reported [1]. As for elements of periods 4–6 [1–3], in the A-sublattices of period 3 elements, VDP with 14 faces representing Fedorov cuboctahedra with CTT [4668] are encountered most frequently (~7.3% of the total sample size). Some other 14-face polyhedra are also found rather frequently, in particular, those with CTT [445466] (Table 3). Meanwhile, the VDP with 12 faces shaped
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Table 2. Distribution (%) of the 555702 atoms of period 3 elements depending on the number of faces in their VDP in A-sublattices* The number of VDP faces
Na
4
**
5
0.1
6
Mg
Al
Si
P
0.2
**
**
0.7
0.2
0.1
**
1.0
3.2
1.9
0.2
0.1
0.1
0.1
7
0.5
0.3
0.2
0.2
**
**
**
8
3.9
6.0
2.8
0.7
0.3
0.3
0.4
9
1.0
0.3
0.8
1.1
0.3
0.3
0.4
10
2.4
1.9
3.8
1.6
0.7
1.1
1.3
11
2.1
1.9
1.9
1.8
1.5
2.4
2.5
12
10.0
12.8
8.7
6.3
4.9
6.8
6.1
13
5.5
4.9
5.5
8.3
8.0
8.8
8.4
14
28.5
40.9
35.3
27.8
29.1
24.7
21.9
15
6.1
5.4
10.5
15.1
16.0
14.5
14.0
16
15.7
10.7
13.1
15.7
18.0
16.6
16.7
17
7.4
3.5
5.6
8.8
9.4
9.7
10.4
18
8.1
4.0
5.4
6.2
6.7
7.2
8.2
19
3.1
1.5
1.9
2.8
2.8
3.7
4.4
20
2.9
1.2
1.4
1.6
1.5
2.1
2.6
21
0.7
0.2
0.3
0.7
0.6
1.0
1.2
22
0.4
0.3
0.3
0.4
0.3
0.5
0.7
23
0.2
0.1
0.1
0.1
0.1
0.2
0.3
24
0.2
**
0.1
0.1
**
0.1
0.1
25
**
0.1
**
**
**
**
0.1
26
**
0.1
**
**
**
**
**
**
**
**
**
**
**
**
**
**
**
**
27 28
**
**
29
**
**
30
0.1
S
Cl
Ar
**
20.0
60.0
20.0
31 32
0.1
**
**
**
* For atoms of each A element, the percentage of VDP polyhedra with the particular number of faces in the sample is indicated. The sample sizes are given in the third column of Table 1. For short, the only VDP with 34 faces found in P-sublattices (the P(3) atom in C52H80Br4F30Fe2KN4P13 {NIBJUD}) and the only VDP with 44 faces found in S-sublattices (the S(1) atom in C80H64O40S64SiW12 {JAPWIG}) are omitted. ** Only single examples with a contribution to the sample of
