Symmetry Elements Research Articles

NMR discrimination in nonrigid prochiral solutes dissolved in chiral liquid crystals: symmetry considerations.

Enantiodiscrimination in the NMR spectra of flexible prochiral solutes dissolved in chiral liquid crystals (CLCs) is reviewed and compared with the analog phenomenon in such rigid solutes. In rigid prochiral solutes, the discrimination is brought about by the cancellation of improper symmetry elements upon dissolving in CLC within the frame of solute-solvent ordering mechanisms. If this reduction in symmetry renders the ordering of enantiotopic sites dissimilar, spectral discrimination may be observed. Symmetry considerations indicate that this is only possible for improper nonaxial groups lacking inversion symmetry. Nonrigid prochiral solutes consist of rapidly (on the NMR timescale) interconverting enantiomers, in which the racemization is accompanied by exchange of nonequivalent sites. These sites become, on the average, enantiotopically related, and in CLC, they exhibit spectral discrimination. The mechanism of the effect and the symmetry selection rules are different for the two cases. Specifically, the discrimination in flexible prochiral compounds results from the different ordering of the interchanging enantiomers in CLC. Using Altman's definition of average symmetry (Proc. R. Soc. A, 1967, 298, 184), selection rules for the phenomenon are derived. It follows that chiral discrimination in nonrigid prochiral solutes is much more abundant and can occur in all symmetry types except those possessing inversion. In particular, contrary to earlier thoughts, the effect can occur in compounds with axial symmetry. Illustrative examples of such studies with particular emphasis on compounds with average axial symmetry of the type D(3h), C(3v) and C(3h) are reviewed in this contribution.

Japanese Crystallography in Culture and Art

"We can find many seeds of crystallography in Japanese culture. Most of the family crests have symmetry elements such as rotation axes and mirror symmetry elements. Sekka-zue, a picture book of 86 kinds of crystals of snow, was made by Toshitura Doi, who is a feudal lord in Edo-period and he observed snow using a microscope in nineteenth century. In recent years, people enjoy to make crystal structures, polyhedrons, carbon nanotube, quasicrystal etc. by origami, the art of folding paper [1]. In the field of science, the Japanese crystallography has contributed to explore culture and art. An excellent example is unveiling the original color of Japanese painting "Red and White Plum Blossoms" by Korin Ogata [2]. Prof. Izumi Nakai (Tokyo University of Science) developed an X-ray fluorescence analyzer and an X-ray powder diffractometer designated to the investigation of cultural and art works and had succeeded in reproducing the silver-colored waves through computer graphics after X-ray analyses of crystals on the painting. The scientific approach by Prof. Nakai et al. unveiled the mystery of cultural heritage of ancient near east, ancient Egypt etc. and is being to contribute to insight into the history of human culture. [1] An event to enjoy making crystals by origami is under contemplation. [2] The symposium ""Crystallography which revives heritages"" was held on February 16, 2014 at Atami in Japan."

Twinning and Pseudosymmetry at High Pressures

Twinning is a common known problem in the study of crystal structures from single-crystal data and often related to a high degree of pseudosymmetry of the structure with respect to a higher symmetrical parent one. With the rising popularity of in situ single-crystal diffraction studies under high pressure, the occurrence of twinned structures is more and more frequently reported. In this contribution, we review the available information on merohedral and pseudomerohedral twinning as well as pseudosymmetry under high pressures [1]. For twinning by merohedry type I (inversion twinning), a reliable characterization of the twin domains and volume fractions is difficult and largely depends on the experimental conditions, i.e., on the number of measured Friedel pairs and the chosen wavelength. For twinning by merohedry (type II) and for twinning by pseudomerohedry, twin volume fractions could be reliably determined from high-pressure data for several cases. In none of these, a significant influence of hydrostatic pressure on the volume fractions of the individuals was observed. Pressure-induced twinning has also been observed for compounds which undergo first-order phase transitions. It is remarkable that the twinning operation in such cases is related to the loss of rotational symmetry elements of the higher symmetrical polymorph, although the high- and low-pressure phases are not in the group-subgroup relationship. The analysis of pseudosymmetry of several compounds as a function of pressure suggests that this parameter can be used to predict the (in)stability of compounds. In particular, a decrease in pseudosymmetry seems to be strongly correlated with the occurrence of first-order phase transitions in which the crystals break or amorphize.

New polymorphs of tiphenyltin(IV) halides

Triphenyltin(IV) halides, Ph3SnHal with Hal = Cl, Br, and I, are white, low melting solids soluble in many indifferent organic solvents like chloroform or toluene. In these solutions as well as in solid they consist of isolated, tetrahedral molecules which intermolecular interactions restricted to van-der Waals ones. These molecules are characterized through the three aromatic phenyl rings attached at a certain pitch angle around the shaft of the tin-halide bond giving them a propeller-like shape accompanied by molecular chirality as the normal vectors of the blades can be oriented clock-wise or anti-clock-wise with respect to the tin-halide bond. In the past, a lot of different polymorphs of triphenyltin(IV) halides with up to 24 molecules in the unit cell have been described, all crystallizing in centrosymmetric space groups, implying that both chiral molecules are present. From a structural point of view there are still several questions of general interest: (i) how are both enantiomers distributed in these polymorphs, (ii) how are the molecular dipole moments arranged, and (iii) how are the different polymorphs related to each other. During various studies on the syntheses and reactivity of triphenyltin(IV) halides we found four additional polymorphs, which constitute new structure types in case of δ-Ph3SnI (Z = 8, monoclinic, P21/c), and ε-Ph3SnI (Z = 12, monoclinic, P21/c) or complete the already existing α-Ph3SnCl structure type (Z = 24, rhomboedric, R-3) in case of β-Ph3SnBr and ζ-Ph3SnI. Analysing the new and already known polymorphs with respect to the three questions mentioned above points out (i) the influence of crystal symmetry elements (screw axes and glide planes) as well as (ii) the importance of the formation of supramolecular dimers composed of both enantiomers with an antiparallel orientation of the two molecular dipole moments, on the different structure types formed and their relation among each other.

Development of teaching materials to learn crystallographic symmetry

Authors have developed proper polyhedron models to enable people to learn the concept of three-dimensional symmetry. Touching and operating the symmetry elements of the proper polyhedron enables people to understand symmetry. In this study, authors made three-dimensional tessellation models. Certain polyhedra can be stacked in a regular periodic pattern to fill three-dimensional space completely. Figures show our models. The cube (Fig. (a)) is the only regular polyhedron to fill three-dimensional space completely. The cube is a Voronoi region of the simple cubic lattice (sc). The truncated octahedron (Fig. (b)) is the only Archimedean solid to fill three-dimensional space completely. The truncated octahedron is a Voronoi region of the body-centered cubic lattice (bcc). The rhombic dodecahedron (Fig. (c)) is the only Catalan solid (or Archimedean dual) to fill three-dimensional space completely. The rhombic dodecahedron is a Voronoi region of the face-centered cubic lattice (fcc). Figs. (a), (b), and (c) show three kinds of their aggregate respectively. In each of left-hand aggregate, there is a two-fold rotational axis along a vertical direction. In each of central aggregate, there is a three-fold rotational axis along a vertical direction. In each of right-hand aggregate, there is a four-fold rotational axis along a vertical direction. Fig. (d) is a nontrivial polyhedron to fill three-dimensional space completely. The external shape of the polyhedron was designed as a tree shape. We call such a model three-dimensional Escher shape (3DES) [1]. This can be stacked in a regular periodic pattern too.

Ultra short H-bond in potassium monobasic 1,4-butodioxybis(4'-benzoic) acid

Hydrogen bonds have been exhaustively investigated in order to understand their role in intermolecular interactions [1]. The importance of the hydrogen bonds follows from their particular role in the molecular association thanks to their variable strength and geometry. Indeed depending on the chemical nature of the donor/acceptor as well as on the physical parameters such as temperature and pressure, hydrogen bonds can give rise to directional and/or to hydrophilic like interactions which play a crucial role both in stabilizing static structures and in mediating dynamic biological processes. Among the hydrogen bonds the less common ones are the strongest (and shortest) ones, characterized by donor–acceptor interatomic distances shorter than 2.6 Å with D–H...A angle close to 180° [2]. Most of the short hydrogen bonds observed and reported so far are intramolecular interactions, nevertheless, both the bridging as well as the coordination of the metal ions by carboxylate groups and oxalate anions can provide the possibility of short interactions [3]. During the refinement of the structure of the potassium-monodeprotonated salt of 1,4-butodioxybis(4'-benzoic) acid it was observed a high electron density maxima over the mirror symmetry element. This electron density indicates a hydrogen atom doing an ultra short intermolecular hydrogen bond (2.4846(1) Å) between the deprotonated and non-deprotonated carboxylic acid groups. The seven fold coordinate potassium atom ties the organic moieties pushing the hydrogen atom closer to the deprotonated and negatively charged, carboxylate oxygen atom. The hydrogen atomic displacement parameter suggests, however, a possible positional disorder, which could be interpreted both as symmetrical O–H–O or asymmetrical O–H...O interaction.

Self-similarity and self-inversion of quasicrystals

The discovery of quasicrystals played a revolutionary role in the condensed matter science and forced to renounce the dogma of the classical crystallography that the regular filling of the space by identical blocks is reduced solely to the Fedorov space groups. It is shown that aperiodic crystals, apart from the similarity, exhibit the self-inversion property. In a broadened sense, the self-inversion implies the possible composition of the inversion with translations, rotations, and homothety, whereas pure reflection by itself in a circle can be absent as an independent symmetry element. It is demonstrated that the symmetry of aperiodic tilings is described by Schottky groups (which belong to a particular type of Kleinian groups generated by the linear fractional Mobius transformations); in the theory of aperiodic crystals, the Schottky groups play the same role that the Fedorov groups play in the theory of crystal lattices. The local matching rules for the Penrose fractal tiling are derived, the problem of choice of the fundamental region of the group of motions of a quasicrystal is discussed, and the relation between the symmetry of aperiodic tilings and the symmetry of constructive fractals is analyzed.

Potential functions of internal rotation in gauche-isomers of n-mononitroalkanes

Internal rotation in gauche-mononitroalkanes CH3(CH2) n NO2, n ≤ 7 is studied by means B3LYP/6-311++g(3df,3pd) and MP2/6-311++g(3df,3pd). Thirty six potential functions (V(φ)) are found for all rotations. It is shown that starting from a particular n value, the potential functions associated with certain molecular fragments change slightly during the lengthening of a hydrocarbon chain. Similar dependences V(φ) are presented in the form of generalized functions with averaged coefficients (V av(φ)) that are transferable and recommended for use in molecular simulations. The rotation of tops without symmetry elements is considered.

Numerical Analysis of Thermal Remediation in 3D Field-Scale Fractured Geologic Media.

Thermal methods are promising for remediating fractured geologic media contaminated with volatile organic compounds, and the success of this process depends on the coupled heat transfer, multiphase flow, and thermodynamics. This study analyzed field-scale removal of trichloroethylene (TCE) and heat transfer behavior in boiling fractured geologic media using the multiple interacting continua method. This method can resolve local gradients in the matrix and is less computationally demanding than alternative methods like discrete fracture-matrix models. A 2D axisymmetric model was used to simulate a single element of symmetry in a repeated pattern of extraction wells inside a large heated zone and evaluate effects of parameter sensitivity on contaminant recovery. The results showed that the removal of TCE increased with matrix permeability, and the removal rate was more sensitive to matrix permeability than any other parameter. Increasing fracture density promoted TCE removal, especially when the matrix permeability was low (e.g., <10(-17) m(2)). A 3D model was used to simulate an entire treatment zone and the surrounding groundwater in fractured material, with the interaction between them being considered. Boiling was initiated in the center of the upper part of the heated region and expanded toward the boundaries. This boiling process resulted in a large increase in the TCE removal rate and spread of TCE to the vadose zone and the peripheries of the heated zone. The incorporation of extraction wells helped control the contaminant from migrating to far regions. After 22 d, more than 99.3% of TCE mass was recovered in the simulation.

Modelling Development of a Thermal Gas/Oil Gravity-Drainage Process in an Extraheavy-Oil Fractured Reservoir

Summary Thermal gas/oil gravity drainage (T-GOGD) is an attractive enhanced-oil-recovery method applicable to naturally fractured reservoirs (NFRs). The process is applied successfully in Qarn Alam, a heavy-oil field in Oman. This paper presents a case study featuring dynamic-modelling optimization and uncertainty-analysis workflow for T-GOGD in a bitumen-bearing fractured reservoir on the basis of realistic 3D fracture characterization. In T-GOGD, the fractures are displaced to steam to provide a (matrix) gravity-drainage potential while heating the reservoir at the same time. One of the recovery mechanisms associated with T-GOGD is thermal expansion, which can result in high initial rates, but may cause plugging of the fracture system in the case of extraheavy oil (bitumen) if the expanded oil cools down before it is being produced. This situation requires short-distance well configurations and/or steam-stimulation cycles to establish communication. In an NFR, steam vapour occupies the fracture system while oil drains through the matrix, increasing the area for heat transfer with respect to the steam-chamber case; the process therefore differs significantly from steam-assisted gravity drainage, and a different production function applies. Shell’s in-house reservoir simulator MoReS with advanced dual-permeability capability is used to model development of T-GOGD in a bitumen reservoir employing 3D element-of-symmetry (EOS) models. A realistic fracture-characterization and -modelling process is described. The geometrical well configuration and operating schedule and strategy are optimized on an economic function for a deterministic subsurface realization. Using an uncertainty-analysis work flow, cumulative distribution functions of recovery and steam/oil ratio (SOR) are generated. Finally, robust optimization is explored by use of an economic objective function. The study concludes that (1) relatively large well spacing is feasible while injector/producer horizontal well offset is necessary to avoid steam channeling to the producer well, (2) live steamproduction control is a robust operating strategy, (3) performance is most sensitive to matrix permeability and oil viscosity, and (4) vertical fracture connectivity plays an important role in the process performance. T-GOGD has significant potential to develop the bitumen resource in naturally fractured carbonates.

Growth surface topography of kaolinite-group minerals and their simulation based on the regular sequence of enantiomorphic layers

The regularities of manifestation of polytypism in a number of kaolinite-group minerals (kaolinite, dickite, halloysite, and nacrite) have been investigated by transmission electron microscopy and vacuum decoration. Growth patterns of elementary layers with a thickness of 7 A, individual for each polytype, were observed on (001) faces of microcrystals. Specific features of the growth patterns of polytypes have been revealed by comparing them with simulated patterns constructed based on a packing of regularly alternating right- and left-handed (enantiomorphic) kaolinite layers. The new approach to the consideration of the polytypism of kaolinite minerals is substantiated by the absence of symmetry elements in the 7-A-thick layer, which determines their structure; the formation of enantiomorphic forms of kaolinite; the presence of grazing-reflection planes in the growth patterns; and the structure of polytypes with a two-layer period. Packing of enantiomorphic layers may yield eight structures, two of which correspond to the right- and left-handed forms of kaolinite, one is for dickite, two are for halloysite, and three are for nacrite. It is shown that the simulated and real growth patterns of these minerals are in good correspondence.

Ivan Shishov, the First Attempts of Melody Analysis in the Soviet Union

Describes the life and works of the Russian composer Ivan Shishov (1888-1947) and examines his article “On the Analysis Of Melodic Structure” (1927). The author argues that Shishov’s method was based on the numerical representation of melodic intervals and questing for the numerical and graphical elements of symmetry, which manifests in the interval structure of a melody.

Stereochemistry in subcomponent self-assembly.

CONSPECTUS: As Pasteur noted more than 150 years ago, asymmetry exists in matter at all organization levels. Biopolymers such as proteins or DNA adopt one-handed conformations, as a result of the chirality of their constituent building blocks. Even at the level of elementary particles, asymmetry exists due to parity violation in the weak nuclear force. While the origin of homochirality in living systems remains obscure, as does the possibility of its connection with broken symmetries at larger or smaller length scales, its centrality to biomolecular structure is clear: the single-handed forms of bio(macro)molecules interlock in ways that depend upon their handednesses. Dynamic artificial systems, such as helical polymers and other supramolecular structures, have provided a means to study the mechanisms of transmission and amplification of stereochemical information, which are key processes to understand in the context of the origins and functions of biological homochirality. Control over stereochemical information transfer in self-assembled systems will also be crucial for the development of new applications in chiral recognition and separation, asymmetric catalysis, and molecular devices. In this Account, we explore different aspects of stereochemistry encountered during the use of subcomponent self-assembly, whereby complex structures are prepared through the simultaneous formation of dynamic coordinative (N → metal) and covalent (N═C) bonds. This technique provides a useful method to study stereochemical information transfer processes within metal-organic assemblies, which may contain different combinations of fixed (carbon) and labile (metal) stereocenters. We start by discussing how simple subcomponents with fixed stereogenic centers can be incorporated in the organic ligands of mononuclear coordination complexes and communicate stereochemical information to the metal center, resulting in diastereomeric enrichment. Enantiopure subcomponents were then incorporated in self-assembly reactions to control the stereochemistry of increasingly complex architectures. This strategy has also allowed exploration of the degree to which stereochemical information is propagated through tetrahedral frameworks cooperatively, leading to the observation of stereochemical coupling across more than 2 nm between metal stereocenters and the enantioselective synthesis of a face-capped tetrahedron containing no carbon stereocenters via a stereochemical memory effect. Several studies on the communication of stereochemistry between the configurationally flexible metal centers in tetrahedral metal-organic cages have shed light on the factors governing this process, allowing the synthesis of an asymmetric cage, obtained in racemic form, in which all symmetry elements have been broken. Finally, we discuss how stereochemical diversity leads to structural complexity in the structures prepared through subcomponent self-assembly. Initial use of octahedral metal templates with facial stereochemistry in subcomponent self-assembly, which predictably gave rise to structures of tetrahedral symmetry, was extended to meridional metal centers. These lower-symmetry linkages have allowed the assembly of a series of increasingly intricate 3D architectures of varying functionality. The knowledge gained from investigating different aspects of the stereochemistry of metal-templated assemblies thus not only leads to new means of structural control but also opens pathways toward functions such as stereoselective guest binding and transformation.

Group Theory - a Tool to Analyse Infrared and Raman Spectra

Group theory is a technique which helps spectroscopists to establish theoretical calculation in vibrational frequencies of a molecule. Vibrational frequencies of a molecule can be determined by Infra red and Raman spectroscopy. In this article, different symmetry elements in group theory has been discussed. Point groups have also been discussed. Using the symmetry elements determination of dipole moments of molecule, and optical activity of molecule have been done.

Threefold Interpenetrated Cadmium(II) Organic Framework based on 3‐(5‐(Pyridin‐4‐yl)‐1H‐1,2,4‐triazol‐3‐yl)pyridine and 1,1′‐Biphenyl‐4,4′‐dicarboxylate

AbstractThe coordination polymer [Cd(HL)(BPDA)]n·nH2O (1) based on 3‐(5‐(pyridin‐4‐yl)‐1H‐1,2,4‐triazol‐3‐yl)pyridine (HL) and 1,1′‐biphenyl‐4,4′‐dicarboxylic acid (H2BPDA) was hydrothermally synthesized and characterized by single‐crystal X‐ray diffraction, XRPD, IR spectroscopy, and elemental analysis. The asymmetric unit of compound 1 consists of one crystallographically independent CdII ion, one HL ligand, “two half” BPDA2– ligand, and one lattice water molecule. There are two crystallographically different BDPA2– moieties, but each of them is placed in a symmetry element. The BPDA2– ligands and CdII ions connected to2D sheet, and the 2D sheets are connected to a 3D MOF by the HL ligands. Each dimeric CdII unit links six nearest neighbors through the BPDA2– and HL ligands, and the dimeric CdII units are centrosymmetric. Two parallel HL ligands connect two dimeric CdII units, generating long Cd2–HL–Cd2–HL rings, also centrosymmetric. So we can define the dimeric CdII unit as a 6‐connected node. Thus, the 3D structure can be described as a threefold parallel interpenetrated bsn beta‐Sn topology. In addition, the optical properties of compound 1 and the free ligand HL were investigated.

The Crystal Structure of Four Phosphorus-Containing Polyoxamacrocycles

The crystal structures of four phosphorus-containing polyoxamacrocycles 2,4, 6,11,13-Pentachloro-dibenzo[d,g][1,3,9,2]trioxaphosphacycloundecan-2-thione (1),2,4,9,14, 16-Pentachloro-dibenzo[g,j][1,3,6,12,2]tetraoxaphosphacyclotetra-decan-2-thione (2), 2,4, 10,16,18-Pentachloro-dibenzo[h,k][1,3,7,12,2]tetraoxa-phosphacyclohexadecan-2-thione (3) and 2,4,6,12,14,18,20,28,30—Nona-chloro-tetrabenzo[d,g,p,s] [1,3,9,13,15,21,2,14]hexaoxa diphosphacyclotetracosan-2,14-dithione (4) have been determined by X-ray diffraction. The result shows that compound 4 has a center of symmetry, and compounds 1, 2, and 3 have no elements of symmetry. The molecules of compounds 2 and 4 form inclusion complex with neutral molecule guest (CHCl3) which is outside of the phosphorus-containing macrocycle in the ratio of 1:1, and compounds 1 and 3 do not form inclusion complexes with CHCl3. The result obtained in this paper lays a foundation for further research on the complexing properties of phosphorus-containing polyoxamacrocycles. [Supplementary materials are available for this article. Go to the publisher's online edition of Phosphorus, Sulfer, and Silicon and the Related Elements for the following free supplemental files: Additional text, figures, and tables.]

Results of studying metamorphic rock samples (Outokumpu, Finland Borehole) by the acoustic polarization method

When studying the elastic properties of anisotropic solid media, it is necessary to determine the spatial orientation of the elastic symmetry elements. This is particularly important for randomly inhomogeneous elastically anisotropic media, such as rock. By applying the acoustic polarization method, the possibility of determining the direction of the symmetry elements of deep rock samples extracted from the Outokumpu, Finland borehole was demonstrated. The anisotropy indices of samples were obtained, calculated from the velocity values of longitudinal and transverse waves. Additional inelastic effects were revealed that are widespread in rock.

Notes of a protein crystallographer: the beauty of rose windows and the different meanings of symmetry

The magnificent rose windows of the Gothic cathedrals provide artistic illustrations of many of the planar point groups of symmetry. It is suggested that the presence of certain symmetry elements and their selection by the artists could have been motivated to convey specific meaning to the resulting compositions.

Capping Methodology in Cyclodextrin Chemistry: Use of a Symmetrical Diketone Reagent for Regiospecific Installation of Unsymmetrical Imine–Enamine and Imidazole Caps

Methylated α- and β-cyclodextrin skeletons were both equipped with an unsymmetrical N-(2-N-alkylaminoacenaphthenyl)alkylimine rigid handle. The capping reaction, which consists of condensing a diaminocyclodextrin with highly symmetrical acenaphthenequinone, was found to be regiospecific when starting from cyclodextrin-diamines without any symmetry element. All modified cyclooligosaccharides have intra-annular nitrogen donor atoms. They undergo further cyclization on oxidation, whether chemically with 2,3-dichloro-5,6-dicyano-1,4-benzoquinone or electrochemically, to give highly strained cyclodextrins capped with an unsymmetrical 1,2-disubstituted 1H-imidazole unit.

Tetra-kis(μ3-2-{[1,1-bis-(hy-droxy-meth-yl)-2-oxidoeth-yl]imino-meth-yl}-6-nitro-pheno-lato)tetra-copper(II).

The title cluster, [Cu4(C11H12N2O6)4], was obtained from the Cu0–FeCl2·4H2O–H4 L–Et3N–DMF reaction system (in air), where H4 L is 2-hy­droxy­methyl-2{[(2-hy­droxy-3-nitro­phen­yl)methyl­idene]amino}­propane-1,3-diol and DMF is di­methyl­formamide. The asymmetric unit consists of one Cu2+ ion and one dianionic ligand; a -4 symmetry element generates the cluster, which contains a {Cu4O4} cubane-like core. The metal ion has an elongated square-based pyramidal CuNO4 coordination geometry with the N atom in a basal site. An intra­molecular O—H⋯O hydrogen bond is observed. The solvent mol­ecules were found to be highly disordered and their contribution to the scattering was removed with the SQUEEZE procedure in PLATON [Spek (2009 ▶). Acta Cryst. D65, 148–155], which indicated a solvent cavity of volume 3131 Å3 containing approximately 749 electrons. These solvent molecules are not considered in the given chemical formula.