Generation of the structures of molecular crystals with two molecules related by the center of inversion in a primitive unit cell

An algorithm for the generation of possible crystal structures with two inversion-related molecules of known shape in a primitive unit cell is proposed within the method of discrete packing modeling in molecular crystals. The algorithm is based on the replacement of molecules by polycubes (geometric figures composed of identical cubes) and looking through a finite number of all the possible periodic packings of these polycubes with a given coefficient of packing. A program package for personal computers is developed on the basis of the proposed algorithm and is approved by the example of several crystal structures that were determined earlier by X-ray diffraction.     

Generation of crystal structures of heteromolecular compounds by the method of discrete modeling of packings

Within the method of discrete modeling of packings, an algorithm of generation of possible crystal structures of heteromolecular compounds containing two or three molecules in the primitive unit cell, one of which has an arbitrary shape and the other (two others) has a shape close to spherical, is proposed. On the basis of this algorithm, a software package for personal computers is developed. This package has been approved for a number of compounds, investigated previously by X-ray diffraction analysis. The results of generation of structures of five compounds—four organic salts (with one or two spherical anions) and one solvate—are represented.     

Generation of molecular Bravais structures by the method of discrete modeling of packings

An algorithm for the generation of possible crystal structures consisting of molecules with the known shapes in the symmetry classes with one translationally independent molecule (the so-called molecular Bravais structures) has been proposed within the framework of the method of discrete modeling of packings in molecular structures. The algorithm was used to write a special complex of computer programs. Some examples of testing this complex on molecules with the structures determined earlier by the methods of X-ray diffraction analysis are also considered.     

Morphogenesis of crystal structures in the discrete modeling of packings

Within the framework of the discrete modeling of molecular packings, a metric approach to the investigation of the mechanisms of crystal formation is proposed. This method is based on the construction of a combination of polyhedra in a space in which the space division into polyhedra (or the periodic packing of polyhedra) is specified by the multistage addition to the initial “seeding” polyhedron set of this division (packing) of the adjacent polyhedra. Crystal growth is modeled using the constructions of bounding boxes in the division of the plane into polyominoes and the three-dimensional space, into polycubes. The formation of phenomenological polygons (polyhedra) in the growth of periodic structures is revealed and theoretically grounded.     

Coordination sequences and coordination waves in matter

A possible way of partitioning a space into polycubes (n-dimensional modifications of Golomb polyominoes, which are generally nonconvex) is used as a basic model of ordered matter structure. It is suggested that layer-by-layer growth of a structure, occurring along the geodetics of the digraph of a net defined by the local rules of bonding of polycubes, justifies the phenomenological laws of shaping (self-similarity during the growth, independence of the polyhedron shape on the “seed,” the symmetry of the growth polyhedron, etc.). Specific results of the analysis of number sequences of the increase in coordination circles for planar periodic partitions of model and real crystal structures, as well as the preliminary results of investigation of standing coordination topological waves, revealed for the first time in computer experiments, are reported.     

Algorithm and computer-program search for variants of polyomino packings in plane

An algorithm and a computer program that can be used in the search for all possible periodic packings of a specified set of polyominoes with a specified packing factor have been developed. The algorithm is based on the method of discrete modeling of molecular packings.     

Molecular structures and crystal packings of 2-styrylquinoxaline derivatives

The crystal and molecular structures of 2-styrylquinoxaline derivatives with different substituents in the styryl fragment are determined. The degree of planarity of the molecules studied varies in a very wide range, from 1.7° to 33.5°. In the ethylene fragment, the double bond is essentially localized. The bicycle-pedal disordering of the ethylene fragment is found in the crystals of the methoxy and oxyacetyl derivatives of 2-styrylquinoxaline. None of the packings contains packing motifs favorable for the photocycloaddition (PCA) reaction with single crystal retention. The crystal packings of these compounds and that of 2-(4-methylstyryl)quinoxaline are characterized by a stacking motif of the head-to-head type, which eliminates the possibility of PCA taking place with single crystal retention but is suitable for this reaction in polycrystalline films. The crystal packing of 2-(3,4-dimethoxystyryl)quinoxaline does not contain elements with stacking interactions.     

Systematic studies of general structural characteristics of organic molecular crystals and prediction of their structures

A strategy is developed for constructing noncentrosymmetric (polar) packings of crystals necessary for the manifestation of a number of important physical properties of crystals. The strategy is based on the analysis of the data obtained in the systematic studies of general structural characteristics of pairs of centro-and noncentrosymmetric polymorphic modifications of a number of organic molecular structures. An algorithm for prediction of such packings is designed on the basis of the potential function singled out by the method of automatic statistical search for information (Data Mining) in the Cambridge Structural Database.     

Specific features of the crystal packing that enable styryl dyes of the pyridine series to undergo the solid-phase [2 + 2] photocycloaddition including the process with single crystal retention

The crystal packing of a number of styryl dyes of the pyridine series is analyzed. The structures of three dyes and three [2 + 2] photocycloaddition (PCA) products, 1,2,3,4-tetrasubstituted cyclobutanes, obtained in single crystals are determined by X-ray diffraction. Stacks of planar organic cations are characteristic of styryl dye packings. The proceeding of the PCA reaction as a single crystal-to-single crystal transformation in the syn head-to-head stacks is in principle impossible. The syn head-to-tail stacking packings are favorable for the PCA reactions resulting in the centrosymmetric rctt isomers of cyclobutane. The stacking packings, in which molecules are related by the twofold axes (the anti arrangement of molecules), are also favorable for PCA in single crystals. In this case, the products are the rtct isomers of cyclobutane. The presence of the I^? counterions in a packing is a factor impeding the PCA reaction, because the secondary I…H-C bonds increase the rigidity of the crystal lattice. The conditions necessary for proceeding the PCA reactions in styryl dyes as single crystal-to-single crystal processes are as follows: (1) the stacks split into pairs of organic cations (dimers) with the d distances within 4.2 Å in a dimer and d exceeding 4.2 Å between the dimers; and (2) the dimers are surrounded by flexible shells consisting of anions, solvate molecules, or flexible moieties of the organic cations themselves.     

An algorithm and program of exhaustive search for possible tiling of a plane with polyominoes

Within the method of discrete modeling of packings in molecular crystals, an algorithm of the exhaustive search for all possible tiling variants of a plane with polyominoes with the given number of translationally independent cells has been developed on the basis of the packing code. The suggested algorithm was used to write the corresponding program for a personal computer.     

Mathematical simulation and X-ray diffraction investigation of the crystal structure of 1-phenyl-1-<Emphasis Type="Italic">tert</Emphasis>-butyl-3-methyl-1,3-dihydroisobenzofuran

An algorithm for using a priori generation of crystal structures by the discrete modeling method for the interpretation of data obtained from single-crystal X-ray diffraction experiments is considered. The crystal structure of 1-phenyl-1-tert-butyl-3-methyl-1,3-dihydroisobenzofuran is mathematically simulated using the discrete modeling of molecular packings and studied by X-ray diffraction. The simulation is performed for two isomers of the initial chemical compound that are possible from the viewpoint of the mechanism of the chemical reaction used in the synthesis of this compound. Appropriate models that can serve as starting models for solving and refining the crystal structure with the use of X-ray diffraction data are chosen from a complete set of calculated structural models in accordance with specific criteria. The structure is solved using a starting model calculated using the discrete modeling method and refined by the full-matrix least-squares procedure.     

Formation of crystal structure of cyanovinylbenzene derivatives: X-ray structure investigation of ethyl esters of 3,4-dimethoxy-, 3,4,5-trimethoxy-, and 4-fluorobenzylidenecyanoacetic acids

Ethyl esters of 3,4-dimethoxy-, 3,4,5-trimethoxy-, and 4-fluorobenzylidenecyanoacetic acids are investigated by X-ray diffraction analysis with the aim of elucidating the factors responsible for the formation of crystal structures of cyanovinylbenzene derivatives. The crystal data for other known derivatives of this series are analyzed. It is revealed that, in the absence of strong hydrogen bonding, the main structure-forming function in crystals is fulfilled by intermolecular contacts of two types, namely, staking contacts of parallel and antiparallel molecules. The staking contacts of antiparallel molecules are most frequently realized. It is assumed that their presence in the structure determines the prevailing formation of centrosymmetric packings in the compounds under consideration.     

Molecular structures and crystal packings of 2-styrylquinoline and its derivatives

The crystal and molecular structures of five styrylheterocycles of the quinoline series are studied. All molecules are planar. The double bond in the ethylene fragment is essentially localized. In the molecule of 2-(4-methylstyryl)quinoline, the ethylene fragment is disordered by the bicycle-pedal pattern. In four of the five compounds, the crystal packings do not contain stacking dimers prearranged for the [2+2] photocycloaddition (PCA) reaction. In the crystal of 2-(3-nitrostyryl)quinoline, pairs of crystallographically independent molecules form stacking dimers. In a dimer, the ethylene fragments have a twist orientation, which is incompatible with the PCA reaction. An attempt to initiate a temperature-dependent process of bicyclepedal isomerization in the crystal and, as a consequence, the PCA reaction by means of simultaneous irradiation and heating of a single crystal is unsuccessful.     

Structural regularities of helicoidally-like biopolymers in the framework of algebraic topology: II. α-Helix and DNA structures

The developed apparatus of the “structural application” of algebraic geometry and topology makes it possible to determine topologically stable helicoidally-like packings of polyhedra (clusters). A packing found is limited by a minimal surface with zero instability index; this surface is set by the Weierstrass representation and corresponds to the bifurcation point. The symmetries of the packings under consideration are determined by four-dimensional polyhedra (polytopes) from a closed sequence, which begins with diamondlike polytope {240}. One example of these packings is a packing of tetrahedra, which arises as a result of the multiplication of a peculiar starting aggregation of tetrahedra by a fractional 40/11 axis with an angle of helical rotation of 99°. The arrangement of atoms in particular positions of this starting aggregation allows one to obtain a model of the α-helix. This apparatus makes it possible to determine a priori the symmetry parameters of DNA double helices.     

Inverse problem in the layer-by-layer growth model

The problem of finding structural features of growing structures for which a specified growth shape is allowable is considered within the model of layer-by-layer growth of tilings or packings and the generalization of this model to connectivity graphs. In particular, the problem of revealing growth shapes that are impossible for periodic structures is solved.     

Search for possible structures of cubic approximants of icosahedral quasicrystals

Admissible structures of the approximants are obtained within the model of the dodecahedral local order (DLO) with due regard for the constraints imposed by the space symmetry. It is shown that the number of cubic approximants of icosahedral quasicrystals having a certain order and dodecahedral local order is finite and that the number of positions that can be occupied by the atoms from the unit cells of an approximant of each order is also finite. The corresponding estimates from above are made. In particular, it is shown that there exists only one structure for the 1/?1 approximant. It is also shown that it is possible to determine all the structures for any approximant of any order. The corresponding algorithm of the exhaustive search for these structures is suggested. The implementation of this algorithm provided the determination of all the structures of the 0/1 approximants and also some possible structures of the 0/1 and 1/1 approximants. The tables of possible approximant structures can be useful in the studies of new phases having approximant structures.     

A tiling approach to counting inherent structures in hard potential systems

The number of distinguishable inherent structures of a liquid is the key component to understanding the thermodynamics of glass formers. In the case of hard potential systems such as hard discs, spheres and ellipsoids, an inherent structure corresponds to a collectively jammed configuration. This work develops a tiling based approach to counting inherent structures that constructs packings by combining sets of elementary locally jammed structures but eliminates those final packings that either, do not tile space, or are not collectively jammed, through the use of tile incompatibility rules. The resulting theory contains a number of geometric quantities, such as the number of growth sites on a tile and the number of tile compatibilities that provide insight into the number of inherent structures in certain limits. We also show that these geometric quantities become quite simple in a system of highly confined hard discs.     

Tetragonal packings of spheres

Nine regular tetragonal packings of spheres are considered. These packings satisfy the following conditions: all the spheres have the same dimensions and environment, i.e., belong to one regular system of points with the coordination number of the packing being not less than six. The examples of the real crystal structures corresponding to seven of these packings are considered.     

Simulation of self-organization processes in crystal-forming systems: Supramolecular cyclic R6 cluster precursors and self-assembly of TeO2-TEL (Tellurite) and TeO2-PAR (Paratellurite) structures

The supramolecular chemistry of oxides of sp elements (SO₂, SeO₂, and TeO₂) is considered. The self-assembly of TeO₂-TEL (Tellurite) and TeO₂-PAR (Paratellurite) crystal structures is simulated. Methods of combinatorial and topological analysis (TOPOS program package) are applied which are based on constructing a basis 3D network of the structure in the form of a graph, the sites of which correspond to the positions of centroids of TeO₂ molecules and the edges characterize bonds between them. The topological type of the basis 2D network in the TeO₂-TEL structure corresponds to graphite (C-GRA), while in the TeO₂-PAR structure the basis network corresponds to the 3D diamond network (C-DIA). A nanocluster precursor of cyclic type (R6) composed of six covalently bound TeO₂ molecules (chair conformation) is established for both structures. The desymmetrization of the cyclic structure of the R6 cluster in TeO₂-PAR is related to the formation of Te-Te bonds with lengths of 3.824 and 4.062 Å. The symmetry and topology code of the processes of self-assembly of 3D structures from nanocluster precursors is completely reconstructed into the form “primary chain → microlayer → microframework.” In both structures R6 clusters form 2D packings with a coordination number of 6. The cluster self-assembly model explains the specific features of the morphogenesis of TeO₂-TEL and TeO₂-PAR (phases with low and high crystallization temperatures, respectively): platelike shape, perfect cleavage in the (110) plane, and preferred growth in the primar-chain direction [100] in the former case and growth in the direction of the primary [001] axis with the preferred formation of tetragonal prism faces (110) in the latter case.     

Prediction of molecular crystal structures using a genetic algorithm: Validation by GenMol™ on energetic compounds

This work deals with the problem of finding for a given industrial target the molecule conformation leading to the most efficient crystal polymorph and/or designing unknown crystal packing of new molecules. The originality of the method relies first on constraining the domain of the most probable crystals, before converging towards the predicted packing by a double genetic-refining optimization algorithm. Validation of this approach is discussed on the case study of 46 energetic crystal compounds, using the semi Ab Initio and Molecular Mechanics code GenMol™. This validation shows that all observed structures are retrieved in any circumstances with a relatively high number of good rankings.