∑ Information Patterns of Point Groups of Symmetry

Crystallography Reports, Vol. 46, No. 1, 2001, pp. 1–3. Translated from Kristallografiya, Vol. 46, No. 1, 2001, pp. 7–9. Original Russian Text Copyright © 2001 by Budnikov, Pitirimov, Soldatov, Chuprunov. CRYSTALLOGRAPHIC SYMMETRY Information Patterns of Point Groups of Symmetry N. S. Budnikov, A. V. Pitirimov, E. A. Soldatov, and E. V. Chuprunov Nizhni Novgorod State University, Nizhni Novgorod, Russia e-mail: [email protected] Received March 22, 1999 Abstract—An algorithm for calculating information entropy of regular point systems in crystallographic point groups has been described. The information patterns visualizing the information properties of the groups are described. © 2001 MAIK “Nauka/Interperiodica”. Traditionally, the structures of various materials are system of weighted points [2, 3]: described by geometric images with the aid of various (),, (),, polyhedra, spheres, sets of points, etc. Such a descrip- SfR=–∑ ij Zi Zj ln fRij Zi Zj .(2) tion makes the basis of the geometric method for repre- ij, senting the atomic structures of condensed media. Here, f(Rij, Zi, Zj) = fij is a nonnegative monotonic func- However, the geometric description is far from being tion of interpoint distances and point weights. The val- complete and can be used as a basis for studying the ues of this function can be interpreted as the probabili- relation between the structures of materials and their ties of certain (generally speaking, quite abstract) tran- physical properties only in some cases. In most sitions between the points i and j. The concrete form of instances, the geometric characteristics of atomic struc- this function depends on the physics of the problem to tures should be combined with the energy, entropy, and be solved. The simplest model of this kind was sug- other physical parameters of the respective atomic con- gested by analogy with the Newton gravitation law for- figurations. mulated within the framework of the Darwin–Fowler Earlier [1], we suggested the method for calculating method [1, 4]. the volume of information contained in the simplest (),, kZiZj geometric objects modeling various atomic configura- fRij Zi Zj = -------------2 . (3) tions. These objects were regular point systems of the Rij crystallographic point groups, i.e., the finite sets of This method for calculating the information entropy weighted points in a limited space region invariant with received the name of the gravitation model. respect to a certain symmetry point group. Below, we Since the volume of information extracted from the qualitatively analyze the information properties of reg- point configuration within the gravitation model ular point systems of all crystallographic point groups. depends only on the weights of the points forming the The information entropy of the point systems can set of interpoint distances, the information entropy of be calculated by the Shannon formula, which describes this configuration is determined by the vector system, the information entropy of a certain distribution which is a set of end points of the interpoint vectors ρ(x) [2] with the weights equal to the product of the weights of the points of the main system. S =.–ρ()x lnρ()x dx (1) In the X-ray analysis, the vector systems are used to ∫ represent the maxima of the interatomic (Patterson) ν function [5]. Thus, within the method based on this for- The system of weighted points can be well charac- mula, the information potential is higher the more terized by a set of point weights and a set of distances diverse the set of the weighted interpoint distances is. between these points. Therefore, the information prop- Isovector systems (i.e., the systems with the same vec- erties of the spatial configuration of a system of tor) are equally informative. The fact (well known to all weighted points can be characterized by a functional of those engaged in X-ray diffraction studies) is that while the set of interpoint distances and point weights, i.e., by singling out an isovector system from the main system, the information functional of the point system [1]. one usually obtains ambiguous/nonunique solutions reflecting the fact that the initial main systems and the Using the Shannon formula, one can derive the fol- corresponding crystal structures are equally informa- lowing equation for the information functional of the tive. 1063-7745/01/4601-0001 $21.00 © 2001 MAIK “Nauka/Interperiodica” 2 BUDNIKOV et al. nient to use a stereographic projection. The values of the information functional at the points of this projection can be reflected either by a half-tone pattern or by a three-dimensional graph. To calculate the information functional, we wrote a special package of programs in the ë++ language. These programs allow one to compute the information pattern of any point group as well as the minimum, the maximum, and the average functional values and the functional dispersion. It is also possible to examine in detail all the regions of the information pattern of interest and to store the pattern separately for further work. Using the above method, we obtained the stereographic projections of the information functionals for 32 crystallographic point groups. As an example, Fig. 1 shows the stereographic projection of the information functional for the regular point systems of the group D2d. Fig. 1. Information pattern for the point group D2d. Since within the framework of the gravitation model the entropy of the point system depends only on interpoint distances and point weights, the symmetry of the The configurations that can be transformed into one information pattern of a point group coincides with the another by the similarity operation provide the same symmetry of the corresponding vector system [5]. In volume of information, which is equivalent to the nor- particular, all the patterns are centrosymmetric. malization conditions imposed onto the distribution Each point group can be characterized by a number f(Rij, Zi, Zj) of symmetrically different and consists of different ( , , ) numbers of the elements of regular systems (orbits). In ∑ f Rij Zi Z j = 1. (4) , particular, this determines the number of various sim- i j ple forms which can form a polyhedron with the given It follows from the normalization condition that, within point symmetry. It would be expedient to consider var- the gravitation model, the system consisting of two ious regular point systems with the same symmetry but points contains the zero volume of information. To con- not invariant with respect to similarity. It is impossible sider the volume of information provided by one point to study such a difference within the framework of the is absolutely senseless. traditional crystallographic symmetry. Information pat- Geometric crystallography and crystal chemistry terns provide the characterization of each group, e.g., deal with the representations of crystal structure by sets by the minimum and the maximum information of regular point systems occupied by the atoms in the entropy, and, thus, the opportunity to study the possibil- symmetry groups. The regular point systems are char- ity of obtaining topologically and symmetrically differ- acteristic of symmetry groups; however, different regu- ent regular systems within the given group. lar point systems of one group can be of different orders Figure 2 represents the diagram showing the maxi- (have different numbers of points) and different sym- mum and the minimum information functionals for all metries [6, 7]. Therefore, the geometric properties of 32 crystallographic point groups. In this case, we the symmetry group can fully be characterized only by neglected the regular systems consisting of only one the whole set of possible regular systems. point, because, as was indicated above, the volume of The information properties of point groups can be information within the gravitation model for such sys- fully characterized on the basis of the information func- tems has no sense. The symmetry groups on the dia- tional whose values should be calculated for all the reg- gram are plotted along the abscissa in the order of an ular point systems given by the initial positions of the increase of the maximum information functionals. It is initial point. In order to represent the functional in the seen that the maximum amount of information is char- clear graphic form for all the point groups, it is conve- acteristic of the groups of the highest orders. The minimum functionals are zero for the groups of all the systems except for the cubic one. The most informative Information functional for cubic simple forms graph seems to be that of the maximum functional values. Thus, the maximum functional value can be con- Indices T Th Td O Oh sidered as the objective characteristic of the complexity (100) 2.68 2.68 2.68 2.68 2.68 of the corresponding symmetry group. (110) 4.06 4.06 4.06 4.06 4.06 Consider the volume information provided by indi- vidual regular point systems, All the regular point sys- (111) 1.79 3.25 1.79 3.25 3.25 tems of the crystallographic point groups can be CRYSTALLOGRAPHY REPORTS Vol. 46 No. 1 2001 INFORMATION PATTERNS OF POINT GROUPS OF SYMMETRY 3 H 7 6 5 4 3 2 1 0 C1 C2 C3 C2v D2 C3v C3h D3 D2h D2d C6v D3h D6 D4h Td O Ci Cs C2h C4 S4 C6 C3i C4v C4h D4 D3d C6h T D6h Th Oh Fig. 2. Diagram showing the minimum (rhombuses) and the maximum (squares) values of the information functional for regular point systems of the crystallographic point groups. Point groups are plotted along the abscissa; entropy is plotted along the ordinate. divided into general and special (lying on the symmetry correspondence with the points of the stereographic elements). It follows from the general symmetry con- projections of face normals and calculating the infor- siderations, in particular, from the Neumann principle, mational functionals for the regular point systems, one that the vector of the gradient of the function S at the can also study the informational properties of crystal points coinciding with the points of the special regular polyhedra.

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