∑ ∑ Discrete Modeling of Homometric and Isovector Divisions

Crystallography Reports, Vol. 48, No. 2, 2003, pp. 167–170. Translated from Kristallografiya, Vol. 48, No. 2, 2003, pp. 199–202. Original Russian Text Copyright © 2003 by V. Rau, T. Rau. THEORY OF CRYSTAL STRUCTURES Discrete Modeling of Homometric and Isovector Divisions V. G. Rau and T. F. Rau Vladimir State Pedagogical University, Vladimir, Russia e-mail: [email protected] Received April 24, 2001 Abstract—As a further development of the method of discrete modeling of packings, the uniqueness of the division of the packing spaces into homometric (isovector) polyominoes is studied on the basis of the relation between the basic and vector systems of points. The particular criterion of the packing-space division based on the concept of the isovector nature of polyominoes with different numbers of points is formulated. © 2003 MAIK “Nauka/Interperiodica”. Earlier, we suggested a criterion of division of a (periodic basic system). In comparison with the nonpe- two-dimensional space into arbitrarily shaped polyomi- riodic basic system, the periodic basic system has some noes within the framework of the method of discrete additional interpoint distances associated with the modeling of molecules–polyominoes in the periodic translation vectors. The complete set of the interpoint space of a crystal lattice [1]. The criterion based on the distances rij = rj Ð ri = rk transforms the vector system concept of a packing multispace (PMS) of the Nth order of points in both nonperiodic and periodic basic sys- represented as a superposition of the sublattices with tems constructed on the lattice with the same basis vec- the sublattice index N is formulated as follows. A polyomino consisting of N points of the basic system does tors. not divide the space translationally if, upon the super- To the points of real crystal structures there corre- position of the vector system of this polyomino onto the spond certain physical quantities. If each point of the corresponding PMS, the points of the vector system basic and the vector systems is brought into correspon- occupy the sites at which the lines from the complete ρ dence with the “electron” density (ri) and the Patter- set of the packing spaces with the sublattice index N are r intersected. Accordingly, if at least one line not occu- son density P( k), then the unit-cell content of the peri- pied by the points of the vector system of polyominoes odic lattices can be written as is among the lines passing through the PMS sites, then the space can be divided translationally. Obviously, the N ρ() ρδ() number of various variants of the space division into r = ∑ i rr– i , (1) specifically shaped polyominoes depends on the num- i = 1 ber of straight lines of the PMS having no common 2 points with the vector system of polyominoes. N () δ() Below, we present the results of the study of the P r = ∑ Pk rr– k , (2) uniqueness of the division of packing spaces into k = 1 homometric (isovector) polyominoes based on the analysis of the relation between the basic and vector where systems of the points. A particular criterion of the division is formulated on the basis of the concept of isovec- 0, for rr≠ δ() i tor polyominoes with different numbers of points. rr– i = 1, for rr= . As is well known from the theory of X-ray structure i analysis (see, e.g., [2]), the basic system of points is understood as a set of points fixed by the ends of the Single-crystal X-ray diffraction data allow one to radius-vectors (r1, …, ri , …, rN) for each ith nucleus construct a continuous function of the Patterson density of the structure atoms set either in an arbitrary coordi- that is a self-convolution of the electron-density func- nate system (nonperiodic basic system) or in a coordi- tion. This can be demonstrated on the above models of nate system determined by the basis vectors of the lat- functions (1) and (2) by determining the inverted solu- tice of translations whose unit cell contains these atoms tion for the electron-density function by replacing each 1063-7745/03/4802-0167 $24.00 © 2003 MAIK “Nauka/Interperiodica” 168 V. G. RAU, T. F. RAU 2 (a) 8 5 3 7 1 4 ρÐ ρ 6 (b) II 62 I + IV III 12 48 18 15 13 52 73 41 25 14 37 51 31 81 84 Ðρ ᮏ Ðρ ρ ᮏ ρ 21 24 26 Fig. 1. Vector point system (b) as a self-convolution of the fragments of (a) the basic point system related by the center of inversion. vector rj by the vector Ðrj: characteristic of all the geometric methods [3]. The first one is associated with the fact that even at the initial N N stage of the search for the solution by constructing, e.g., ρρ⊗ρδ()⊗ ρ δ() = ∑ i rr– i ∑ j rr+ j a Wrinch hexagon, not any arbitrary six points can form i = 1 j = 1 the star of a two-dimensional simplex (triangle of the N N basic system). The point is that the operation of differ- = ∑ ∑ {}ρ ρ []δ()δrr– ⊗ ()rr+ ence, which acts onto a pair of vectors of the basic sys- i j i j (3) tem, is an element of the sixth-order cyclic group with i = 1 j = 1 the generating matrix denoted by R1 and called here a 2 N Wrinch matrix: = P δ()rr– ()– r ==P δ()rr– P()r , ∑ k i j ∑ k k k = 1 01 R1 = . where kij≡ . –11 Expression (3) is a particular case of the operation Then, of convolution of two functions ρ(r) and ϕ(r), ρ ⊗ ϕ = ρ (u)ϕ(r Ð u)du, where integration is taken over the r r ∫Ω R i = j ; Ω 1 domain of definition of functions . The geometric r j r j – ri sense of the convolution in the case of a discrete set of points can readily be understood from Fig. 1, where the 2 –11 r r – r basic system of points is represented by the function ρ R ==R , R i = j i ; 2 1 2 and the inversion-related function ρ , in the shape of a –01 r j –ri tetrahedron (Fig. 1a), whereas the vector system is con- 3 r –r structed in accordance with the equality R ==R –01 , R i = i ; 3 1 3 ()ρρρρ+ ⊗ ()+ 01– r j –r j (4) (5) = ρρ⊗ρρ()I + ⊗ ()II + ρ⊗ ρ()ρρIII + ⊗ ()IV . 4 r –r R ==R 01– , R i = j ; The result of the division of the vector system into 4 1 4 four fragments is shown in Fig. 1b, where centrosym- 11– r j ri – r j metric fragments (I) and (IV) coincide and form a Gali- 5 r r – r ulin cuboctahedron (for clarity, some points are identi- R ==R 11– , R i = i j ; fied). 5 1 5 10 r j ri The solution of the inverse problem of the construction of the basic system from the vector one was 6 r r obtained by searching for the fragment that single out R ==R 10 , R i = i . 6 1 6 the image: triads of peaks, multiple peaks and frag- 01 r j r j ments, “imaginary crosses,” Wrinch hexagons, etc. In this case, the approach of the geometric simplex sug- The corresponding characteristic matrix and equa- λ gested by Galiulin encounters two serious difficulties tion allow one to calculate the eigenvalues 1, 2. Indeed, CRYSTALLOGRAPHY REPORTS Vol. 48 No. 2 2003 DISCRETE MODELING OF HOMOMETRIC AND ISOVECTOR DIVISIONS 169 (a) (b) (c) ρϕρ ᮏ ϕ = σ Ð Ð ρ ᮏ –ϕ = ω ω ᮏ ω = σ ᮏ σ Fig. 2. Construction of homometric polyominoes by the convolution method: (a) initial polyominoes and their basic point systems, (b) convolution of the basic point systems and the corresponding homometric polyominoes, and (c) common vector point system of homometric polyominoes. 1011 0 1 2 3 4 5 6 789 11 0 3 4 5 6 789 10 11 0 1 2 3 4 5 6 7 8 9 11 0 1 2 3 6 7 8 9 1011 0 1 2 3 4 5 6 7 8 9 11 0 1 2 3 4 5 6 9 10 11 0 1 2 3 4 5 6 7 8 9 11 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 11 0 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 11 0 1 2 3 6 7 8 9 1011 0 1 2 3 4 5 6 7 8 9 11 0 1 2 3 4 5 6 9 10 11 0 1 2 3 4 5 6 7 8 9 11 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1011 0 1 2 3 4 5 6 7 8 9 11 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 11 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 11 9 10 11 0 1 2 3 4 5 6 7 8 9 11 0 1 2 3 4 5 6 7 8 9 4 5 6 7 8 9 Fig. 3. Division (on the left) and packing (on the right) of the packing space P 12 19 into homometric polyominoes.

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