Symmetry Elements Research Articles

GENETIC SIGNIFICANCE OF VARIATIONS IN THE FACES OF THE SIMPLE FORMS OF TOPAZ CRYSTAL FROM CHAMBER PEGMATITES OF THE KOROSTEN PLUTON (UKRAINIAN SHIELD)

Topaz crystal morphology and habit distortion has been studied in various mineral-structural zones of chamber pegmatites of the Korosten pluton, which is located in north-western part of the Ukrainian Shield. It was assumed that the symmetry of the crystals obey the Curie principle. This means that only the symmetry elements common to the crystal and the medium in which it is formed will remain on real polyhedrons. The types of symmetry that contain the axes of infinite order are reduced to the following groups: 1) ∞L∞∞PC is a ball; 2) ∞L∞ is a ball filled with an optically active liquid; 3) L∞∞L2∞PПC is a cylinder; 4) L∞ПС is a rotating cylinder; 5) L∞∞P is a cone; 6) L∞∞L2 is a twisted cylinder; 7) L∞ is a rotating cone. Symmetry of the real fluid-dynamic situation of the mineral-forming medium of topaz-bearing parageneses often evolves in the following way: ∞L∞∞PC → L∞∞P → P. In this case, the flow of the mineral-forming fluid has the symmetry P. The resulting topaz crystals can have P symmetry if their symmetry plane coincides with the flow symmetry plane, otherwise they have no symmetry elements at all. In particular, it is shown for the first crystals that the upper faces grew faster, and their size is smaller than that of the lower ones. Growth was limited by the supply of the necessary fluid to the growing crystal faces. Hence, it follows that the fluid flow was in the direction from top to bottom. If the planes of symmetry of the fluid flow and of the polyhedron do not coincide, then visually triclinic crystals of the second type are formed. They are much more abundant than the ones of the first type. In addition to these two types, polyhedra with external symmetry L2 are found. It is difficult to imagine an environment with such symmetry because; it is unlikely that an attached crystal would grow between two fluid streams moving in opposite directions. Nevertheless, polyhedra flattened along the faces M {110} and less often along l {120} are frequent. That is, they grew in the environment in which the fluid flow moved in a direction parallel to the {110} faces (and less often {120}), in the direction from the smaller faces of a simple forms to the larger ones. The direction of fluid flow is more difficult to establish, with more or less the same development of the faces of the simple form of the topaz crystal.

Crystal structure, Hirshfeld surface analysis and DFT study of 2,2''-({[(1E,1'E)-(diselanedi-yl)bis-(2,1-phenyl-ene)]bis-(methane-ylyl-idene)}bis-(aza-neylyl-idene))bis-[3',6'-bis-(di-ethyl-amino)-4a',9a'-di-hydro-spiro-[isoindoline-1,9'-xanthen]-3-one

The title compound, C70H70N8O4Se2, is a spiro bicyclic diselenide, made up of two [SeC6H4CH=N-N(CO)C6H4(C)C6H3NEt2(O)C6H3NEt2] units related by a twofold crystallographic symmetry element bis-ecting the diselenide bond. The compound crystallizes in a non-centrosymmetric polar space group (tetra-gonal, P b2) and the structure was refined as an inversion twin. The two diethyl amine groups and their attached phenyl groups of the xanthene ring are disordered over two orientations, with occupancies of 0.664 (19)/0.336 (19) and 0.665 (11)/0.335 (11), respectively. The dihedral angles between the mean planes of the central isoindoline and the phenyl rings are 26.8 (2) and 2.5 (4)°, respectively. The mean plane of the central xanthene ring forms dihedral angles of 2.0 (5), 8.8 (5), 1.7 (5) and 7.9 (6)° with the peripheral phenyl rings. The isoindoline and xanthene rings subtend a dihedral angle of 89.8 (2)°. The mol-ecular conformation is stabilized by an intra-molecular C-H⋯O hydrogen bond generating an S(6) ring motif. In the crystal, mol-ecules are linked by C-H⋯O hydrogen bonds together with C-H⋯π (ring) inter-actions, forming a three-dimensional network. A Hirshfeld surface analysis of the crystal structure indicates that the most important contributions to the crystal packing are from H⋯H (68.1%), C⋯H/H⋯C (21.2%) and O⋯H/H⋯O (8.7%) contacts. The optimized structure calculated using density functional theory (DFT) at the B3LYP/6 - 31 G(d) level is compared with the experimentally determined mol-ecular structure in the solid state. The HOMO-LUMO behaviour was used to determine the energy gap and the mol-ecular electrostatic potential (MEP) of the compound was investigated.

Crystal structure of nyerereite: A possible messenger from the deep Earth

Abstract Carbonates in the system Na2CO3-CaCO3 are nowadays suggested as having a wide stability field at conditions of the mantle transition zone. Our structural analysis of nyerereite, which has limited stability fields at ambient conditions, and its similarities with already known carbonates that are stable at high-pressure conditions, allowed us to propose that nyerereite likely undergoes phase transitions at both high-pressure and high-temperature conditions. This supports the hypothesis that nyerereite takes part in carbon transportation from the mantle/deep crust toward the surface, with important implications for the deep carbon cycle associated with carbonatites. K-free nyerereite [Na2Ca(CO3)2] was synthesized both at hydrothermal conditions and from the melt. The structure of nyerereite was refined as a three-component twinned structure in the centrosymmetric Pbca space group with ratios of the three twinning components 0.221(3):0.287(3):0.492(3). Twinning at micro- and nano-level can introduce some minor structural deformations that influence the likely occurrence of the inversion center as one of the symmetry elements in the nyerereite structure. Based on the automated topological algorithms, we show that nyerereite has a unique crystal structure, not having analogs among the known structures, except for the structure with a similar composition K2Ca(CO3)2 fairchildite. A comparison between the centrosymmetric Pbca nyerereite structure and that of aragonite (CaCO3, Pmcn space group) reveals two main scenarios for the high-pressure form of Na2Ca(CO3)2: (1) poly-somatic relations as the interlayering of the high-pressure polymorph Na2CO3 and CaCO3-aragonite, and (2) high-pressure structure with ninefold-coordinated Na and Ca sites resembling that of aragonite. Our discussion heightens the interest in the high-pressure behavior of the nyerereite structure and strengthens the hypothesis about the possibility for nyerereite to be stable at high-pressure/high-temperature conditions.

Results of studying the azimuthal anisotropy of the paleozoic basement with vsp technique at the Bystrovka vibroseismic test site

At the Bystrovka vibroseismic test site (Novosibirsk region) 3-component refracted waves profiling was performed at three intersecting lines. Shear waves analysis made possible to detect anisotropy of the Paleozoic basement occurring at depth of about 10 m and to suggest symmetry elements of this medium along with their orientation. Compressional waves data were used to construct depth sections estimating head waves velocities. These velocities demonstrate significant variation in lines of different orientation. The results obtained agree with previously performed VSP.

Informational Measure of Symmetry vs. Voronoi Entropy and Continuous Measure of Entropy of the Penrose Tiling. Part II of the “Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling”

The notion of the informational measure of symmetry is introduced according to: Hsym(G)=−∑i=1kP(Gi)lnP(Gi), where P(Gi) is the probability of appearance of the symmetry operation Gi within the given 2D pattern. Hsym(G) is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as Hsym(D3)= 1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, Hsym=0. The informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. The informational measure of symmetry does not correlate with either the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns. Quantification of the “ordering” in 2D patterns performed solely with the Voronoi entropy is misleading and erroneous.

4-dimensional lattice models for the quantum range

Some finite subspace models L are presented for quantum structures which replace the use of countable infinite Hilbert space H dimensions. A maximal Boolean sublattice, called block, is 24, where its four atoms directly above 0εL, base vectors of H in 24 are drawn as four points on an interval. Blocks can overlap in one or two atoms. Different kinds of operators can map one block onto another and interpretations are given such that subspaces can carry on their base vector tuple real, complex or quaternionic numbers, energies, symmetries and generate coordinate lines. Describing states of physical systems is done using L and its applications for dynamical modelling. They don‘t need the infinte dimensional vectors of H. L has in the first model 11 blocks and 24 atoms (figure 1). They correspond to the 24 elements of the tetrahedral S4 symmetry. S4 arises from a spin-line rgb-graviton whirl operator with center at the tip of a tetrahedron and a nucleon triangle base with three quarks as vertices. The triangles factor group D3 of S4 is due to the CPT Klein normal subgroup Z2 x Z2 of S4 . It has a strong interaction SI rotor for the nucleons inner dynamics which is used for integrating functions, exchanging energies of nucleon with its environment and setting barycentrical coordinates in the triangle. At their intersection B as barycenter sets a Higgs boson or field the rescaled quark mass of a nucleon. Each factor class of one element from D3 assigns to it a color charge, a coordinate, an energy vector and a symmetry. Symmetries attached can be different according to interactions involved. Every atom of L has then a specific character with different properties.Three characters are added to octonian base vectors, listed by their indices as n = 0,1,…,7, and named for the atoms of L as na, nb, nc. The structure and element attributes of the finite subspace lattices L are desribed in many examples and models which technical constructed run macroscopically. Several models are described below. Example, the color charge whirl as rgb-graviton projection operator maps the block 2c3b5a6a to 0a1a2a3a. The symmetries change dimension from 3x3- to 2x2-matrices. From SU(3) are λ1 on 3b mapped to the SU(2) x-coordinate Pauli matrix σ1, from λ2 on 5a to σ2 y-coordinate and from λ3 on 6a to σ3 z-coordinate of real Euclidean space R³. The SU(3) matrices have complex w3 = z +ict, w2 = (iy,f), w1 = (x,m) coordinates. In figure 3 is shown how a rotation of two proton tetrahedrons for fusion changes the two linearly independent wj vectors to the 1-dimensional x,y,z base vectors. In deuteron then on one coordinate line sit with Cooper paire u-d-quarks at the ends the Heisenberg coupled energy or space vector rays 15 (x,m), m mass measured in kg, x in meter, 23 (iy,E(rot)), E(rot) rotational energy measured in Joule J, y in meter, 46 (ict,f), t time measured in seconds, f = 1/∆t frequency s inverse time interval measured in Hz. The six color charges are red r on +x as octonian coordinate 1, green g on +y as 2 , blue b on -z as 6, turquoise on -x as 5, magenta on -y as 3, yellow on +z as 4..

Estimation of parameters and thickness of a horizontal layer of arbitrary anisotropy from P-wave moveout

We present an inversion scheme to estimate anisotropy parameters and thickness of a homogeneous layer of arbitrary anisotropy from traveltimes of a reflected P wave. The scheme is based on the weak-anisotropy approximation of P-wave moveout formula. For the description of anisotropy, we use the so-called A-parameters, which represent an alternative to standard elastic moduli specifying anisotropic media, and allow specification of anisotropy of any symmetry, orientation and strength. The proposed scheme allows estimation of, at most, nine of fifteen A-parameters, which describe P-wave propagation in the weak-anisotropy approximation. Therefore, generally, the reconstruction of the P-wave phase-velocity surface from the estimated parameters is impossible. If we deal, however, with higher-symmetry anisotropy (transversely isotropic, orthorhombic) and if information about the orientation of its symmetry elements is available, reconstruction of the phase-velocity surface is possible. Synthetic tests indicate that the inversion scheme works well even in situations, in which the use of other commonly used moveout formulae leads to unsatisfactory results.

The ubiquitous buried water in the beta-trefoil architecture contributes to the folding nucleus and ~20% of the folding enthalpy.

The beta-trefoil protein architecture is characterized by three repeating "trefoil" motifs related by rotational symmetry and postulated to have evolved via gene duplication and fusion events. Despite this apparent structural symmetry, the primary and secondary structural elements typically exhibit pronounced asymmetric features. A survey of this family of proteins has revealed that among the most conserved symmetric structural elements is a ubiquitous buried solvent which participates in a bridging H-bond with three different beta-strands in each of the trefoil motifs. A computational analysis reported that these waters are likely associated with a substantial enthalpic contribution to overall stability. In this report, a Pro mutation is used to disrupt one of the water H-bond interactions to a main chain amide, and the effects upon stability and folding kinetics are determined. Combined with Ala mutations, the separate effects upon side chain truncation and H-bond deletion are analyzed in terms of stability and folding kinetics. The results show that these buried waters act to assemble a central folding nucleus, and are responsible for ~20% of the overall favorable enthalpy of folding.

The wavevector star channel and symmetry group.

The concepts of `wavevector star channel' and `wavevector star channel group' are newly defined, which allow the effective study of phase transitions considering directly the translational symmetry breaking in crystals. A method is suggested by which the wavevector star channels can be found using the image of the representation of the translational group. According to this method, the wavevector star channels are found for the 80 Lifschitz stars in the reciprocal lattice. The wavevector star channel group is defined as the set of symmetry elements of the parent phase which leave the star channel invariant, and the wavevector star channel groups with one, two, three and four arms are calculated. It is shown that the complicated symmetry changes in the pyroelectric crystal Pb1-xCaxTiO3 (PCT) can be described using the new five-component reducible order parameter transformed according to the representation of the wavevector star channel group, rather than the nine-component one based on the theory of the full irreducible representation of the space group.

POINT GROUPS 𝑫𝟐𝒅 and 𝑪𝟑𝒊38T UNDER CLASS SUM APPROACH

In this study, the point groups 𝐷2𝑑 and 𝐶3𝑖 which belong to tetragonal and trigonal crystal systems, respectively, are handled under the class sum approach. Symmetry groups were formed with symmetry elements that left these point groups unchanged and Cayley tables of related groups were obtained. Using these tables, the conjugates of the elements and the classes of the group were formed. Secular equations are written for each class sum obtained by the sum of the elements that make up the class. By solving these secular equations, the character vectors are obtained. Thus, the character tables were reconstructed with the calculated characters for both point groups under the class sum approach.

Cryo-EM reveals new species-specific proteins and symmetry elements in the Legionella pneumophila Dot/Icm T4SS.

Legionella pneumophila is an opportunistic pathogen that causes the potentially fatal pneumonia known as Legionnaires' disease. The pathology associated with infection depends on bacterial delivery of effector proteins into the host via the membrane spanning Dot/Icm type IV secretion system (T4SS). We have determined sub-3.0 Å resolution maps of the Dot/Icm T4SS core complex by single particle cryo-EM. The high-resolution structural analysis has allowed us to identify proteins encoded outside the Dot/Icm genetic locus that contribute to the core T4SS structure. We can also now define two distinct areas of symmetry mismatch, one that connects the C18 periplasmic ring (PR) and the C13 outer membrane cap (OMC) and one that connects the C13 OMC with a 16-fold symmetric dome. Unexpectedly, the connection between the PR and OMC is DotH, with five copies sandwiched between the OMC and PR to accommodate the symmetry mismatch. Finally, we observe multiple conformations in the reconstructions that indicate flexibility within the structure.

Guest-Mediated Hierarchical Self-Assembly of Dissymmetric Organic Cages to Form Supramolecular Ferroelectrics

Guest-Mediated Hierarchical Self-Assembly of Dissymmetric Organic Cages to Form Supramolecular Ferroelectrics

Ducks in space groups! Students grasping 3D arrangement of symmetry elements with hands-on models

Ducks in space groups! Students grasping 3D arrangement of symmetry elements with hands-on models

Symmetry and Quantum Features in Optical Vortices

Optical vortices are beams of laser light with screw symmetry in their wavefront. With a corresponding azimuthal dependence in optical phase, they convey orbital angular momentum, and their methods of production and applications have become one of the most rapidly accelerating areas in optical physics and technology. It has been established that the quantum nature of electromagnetic radiation extends to properties conveyed by each individual photon in such beams. It is therefore of interest to identify and characterize the symmetry aspects of the quantized fields of vortex radiation that relate to the beam and become manifest in its interactions with matter. Chirality is a prominent example of one such aspect; many other facets also invite attention. Fundamental CPT symmetry is satisfied throughout the field of optics, and it plays significantly into manifestations of chirality where spatial parity is broken; duality symmetry between electric and magnetic fields is also involved in the detailed representation. From more specific considerations of spatial inversion, amongst which it emerges that the topological charge has the character of a pseudoscalar, other elements of spatial symmetry, beyond simple parity inversion, prove to repay additional scrutiny. A photon-based perspective on these features enables regard to be given to the salient quantum operators, paying heed to quantum uncertainty limits of observables. The analysis supports a persistence in features of significance for the material interactions of vortex beams, which may indicate further scope for suitably tailored experimental design.

Synthesis of agelastatin A and derivatives premised on a hidden symmetry element leading to analogs displaying anticancer activity

Concise total syntheses of (±)-7-hydroxy debromo agelastatin A (AglA), (±)-AglA, and 11-nitro AglA are presented based on an identified pseudo-symmetry element. This synthetic strategy was developed based on a desire to improve solubility of this potent anticancer agent while also developing a synthetic strategy that would enable late-stage variation of the pyrrole moiety. A stability study of pyrrole-derived carbinolamines revealed critical substituent effects impacting the equilibrium between the cyclic carbinolamine and keto pyrrole forms. 7-Hydroxy AglA existed primarily in the ketopyrrole form however the des-bromo variant existed primarily in the cyclic carbinolamine form. Cytotoxicity assays revealed activity for a 13-nitro AglA derivative (∼14–63 μM) for breast cancer cells (MDA-MB-231 and MCF7) and a glioblastoma cell line (U87) while for 7-hydroxy des-bromo AglA, measurable activity was only observed against the glioblastoma cell line.

Cis-Bis(L-DOPA-κ2N,O)copper(II) monohydrate: synthesis, crystal structure, and approaches to the analysis of pseudosymmetry.

The crystal structure of the cis isomer of cis-bis(L-DOPA-κ2N,O)copper(II) monohydrate (L-DOPA is 3,4-dihydroxy-L-phenylalanine) (CuLD), [Cu(C9H10NO4)2]·H2O, is a singular example of a structurally characterized, homoleptic, crystalline metal L-DOPA complex. CuLD crystallizes in the space group P21, with Z' = 2. The two independent molecules are square planar, and are interconnected by a linear hydrogen-bonded chain containing 12 independent hydrogen bonds. The copper ions in both molecules have weak apical intermolecular Cu...O interactions [2.739 (2) and 2.973 (2) Å] with catechol -OH groups. A survey of the Cambridge Structural Database suggested that cis and trans isomers of Cu(NH2-C-CO2)2 amino acid complexes are equally likely to occur. 12 strong O-H...O and N-H...O hydrogen bonds stabilize an unusual linear arrangement of the Cu complexes. The Cu...Cu' distances along the chain are nearly equal [5.0739 (3) and 5.1107 (3) Å] and the Cu...Cu'...Cu angles are nearly linear [176.75 (1)°]. The MATCH procedure available in the Oxford University Crystals for Windows package was used to carry out a detailed analysis of the relationship between the two independent molecules. MATCH has some particular advantages in studying the details of pseudosymmetry, which include: (i) no atomic-order requirements; (ii) the pseudosymmetry matrix is readily available, which allows quick insight into the symmetry elements involved and their location; and (iii) the differences between molecular centroids, as well as between all atomic positions and torsion angles, are listed. A tutorial presentation is designed to attract new users to the technique. In the present case, a search for a pseudosymmetric relationship between the two independent molecules showed that they are related by a pseudo-42 axis along the crystallographic c direction. A detailed analysis shows that the pseudo-42 symmetry is disrupted by torsions about the CH2-C(ipso) bonds, and that there is no supergroup that can be used to describe the crystal structure.

An Exploration-Based Activity to Facilitate Students’ Construction of Molecular Symmetry Concepts

Molecular symmetry is a topic that requires mental manipulation of molecules when searching for symmetry elements and their corresponding symmetry operations. Such mental activities are difficult for students in a traditional classroom where only definitions and demonstrations are provided. Therefore, an exploration-based activity was developed to help university chemistry students construct molecular symmetry concepts. Since symmetry is a mathematical concept, Pirie and Kieren’s model of growth of mathematical understanding was used as the framework to design the activity. Each task allowed students to self-construct the meaning of symmetry by performing an operation on 2D figures and/or 3D manipulatives. The activity was designed for students without a background in molecular symmetry and conducted at two public universities. In class, the students went through stages of understanding that were consistent with those in the model. The results from the conceptual test confirmed that they were able to apply the learned concepts to identify the symmetry elements and symmetry operations of the given molecules.

Chromatographic separation of the interconverting enantiomers of imidazo- and triazole-fused benzodiazepines

The toolbox of medicinal chemists includes the 1,4-benzodiazepine scaffold as a “privileged scaffold” in drug discovery. Several biologically active small molecules containing a 1,4-benzodiazepine scaffold have been approved by the FDA for the treatment of various diseases, with most of them being used for their psychotropic effects. The therapeutic potential of 1,4-benzodiazepines has stimulated the interest of synthetic chemists in developing new synthetic strategies to a range of substituted analogues for biological evaluation. A structural variation of the classical benzodiazepine skeleton is observed e.g. in alprazolam, midazolam, and related benzodiazepines, which contain a 1,2,4-triazole or an imidazole ring fused to the benzodiazepine core. Irrespective of the presence of the fused heterocyclic ring, the seven-membered diazepine ring is far from planar, and its shape resembles a twist chair. Then, the unsymmetrical substitution pattern around the seven membered cycle renders these molecules chiral, as they lack any reflection-type symmetry element. However, chirality of this molecules is labile at room temperature, becausea simple ring flipping process converts one enantiomer into the other, and 1,4-benzodiazepines exist as a mixture of rapidly interconverting conformational enantiomers in solution at or near room temperature. Physical separation of the interconverting enantiomers of diazepam and of other related 1,4-benzodiazepin-2-ones can be accomplished by low temperature HPLC on chiral stationary phases (CSPs). If the HPLC column is cooled down to temperatures where the interconversion rate is sufficiently low, compared to the chromatographic separation rate, distinct separated peaks can be observed, provided the CSP is sufficiently enantioselctive. The apparent rate constants for the on-column enantiomerization and the corresponding free energy activation barriers were obtained by simulation of exchange-deformed HPLC profiles using a computer program based on the stochastic model. Here we report on the dynamic HPLC investigations carried out on a set of fused imidazo and triazolo-benzodiazepines (alprazolam, midazolam, triazolam and estazolam) The experimental dynamic chromatograms and the corresponding interconversion barriers reported in this paper show that the third fused heterocyclic ring increase the energy barrier by 2 kcal/mol.

Classification and Effects of Symmetry of Mechanical Structure and Its Application in Design

Symmetry widely exists in natural objects and man-made objects. Mechanical structures, as man-made objects, have the property of symmetry without exception. The existence of symmetry affects the function and performance of mechanical products. Therefore, on the basis of analyzing a large number of examples and referring to the Schoenflies symbol of crystal, the symmetry of mechanical structures is divided into point group symmetry and space group symmetry, and these two types are further subdivided according to the types and spatial positions of the symmetry elements. Then, the general effects of symmetry are summarized according to symmetry types and functions, and several symmetry rules for design are further refined. Finally, after defining the requirements of speed change and technology background, a multispeed device for bicycle shaft drive is proposed by applying symmetry knowledge comprehensively.

Electron Symmetry Breaking during Attosecond Charge Migration Induced by Laser Pulses: Point Group Analyses for Quantum Dynamics

Quantum simulations of the electron dynamics of oriented benzene and Mg-porphyrin driven by short (<10 fs) laser pulses yield electron symmetry breaking during attosecond charge migration. Nuclear motions are negligible on this time domain, i.e., the point group symmetries G = D6h and D4h of the nuclear scaffolds are conserved. At the same time, the symmetries of the one-electron densities are broken, however, to specific subgroups of G for the excited superposition states. These subgroups depend on the polarization and on the electric fields of the laser pulses. They can be determined either by inspection of the symmetry elements of the one-electron density which represents charge migration after the laser pulse, or by a new and more efficient group-theoretical approach. The results agree perfectly with each other. They suggest laser control of symmetry breaking. The choice of the target subgroup is restricted, however, by a new theorem, i.e., it must contain the symmetry group of the time-dependent electronic Hamiltonian of the oriented molecule interacting with the laser pulse(s). This theorem can also be applied to confirm or to falsify complementary suggestions of electron symmetry breaking by laser pulses.