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Point Groups (point + groups)
Selected AbstractsUnderstanding chemical shielding tensors using group theory, MO analysis, and modern density-functional theoryCONCEPTS IN MAGNETIC RESONANCE, Issue 2 2009Cory M. Widdifield Abstract In this article, the relationships between molecular symmetry, molecular electronic structure, and chemical shielding (CS) tensors are discussed. First, a brief background on the CS interaction and CS tensors is given. Then, the visualization of the three-dimensional nature of CS is described. A simple method for examining the relationship between molecular orbitals (MOs) and CS tensors, using point groups and direct products of irreducible representations of MOs and rotational operators, is outlined. A number of specific examples are discussed, involving CS tensors of different nuclei in molecules of different symmetries, including ethene (D2h), hydrogen fluoride (C,v), trifluorophosphine (C3v), and water (C2v). Finally, we review the application of this method to CS tensors in several interesting cases previously discussed in the literature, including acetylene (D,h), the PtX42, series of compounds (D4h) and the decamethylaluminocenium cation (D5d). © 2009 Wiley Periodicals, Inc. Concepts Magn Reson Part A 34A: 91,123, 2009. [source] Crystallographic nets and their quotient graphsCRYSTAL RESEARCH AND TECHNOLOGY, Issue 11 2004W. E. Klee Abstract Crystallographic nets are defined. It is shown how these nets can be generated from finite graphs which are called the quotient graphs of the nets. The procedure yields the topology of the nets, their crystallographic point groups and their lattice types. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] A path from Ih to C1 symmetry for C20 cage moleculeJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 12 2005Zhigang Wang Abstract The symmetry of the C20 cage is studied based on the intrinsical relationship among point groups (Bradley, C. J.; Cracknell, A. P. The Mathematical Theory of Symmetry in Solids; Claredon Press: Oxford, 1972). The structure of the C20 cage with Ih symmetry is constructed, as are eight other structures with subgroup symmetry. A path from Ih symmetry to C1 symmetry is obtained for the closed-shell electronic state, and the structure with D2h symmetry is the most stable on this path. Using the D2h structure the correlation energy correction is studied on the condition of restricted excitation space at the CCSD(T) level. We obtain curves on the relation between the orbital numbers and the total energy at the CCSD(T), CCSD, and MP2 level, respectively. The results of these curves obtained from MP2 and CCSD(T) methods have the same tendency, while the results of CCSD gradually diverge with an increase in orbital numbers. When the orbitals used in the calculation reach 460, the total energy is ,759.644 hartree at MP2 level and is ,759.721 hartree by the CCSD(T) method. From the calculation results, we find that a large basis set can improve the reliability of the MP2 method, and to restrict excitation space is necessary when using the CCSD(T) method. © 2005 Wiley Periodicals, Inc. J Comput Chem 12: 1279,1283, 2005 [source] Magnon energy gap and the magnetically structural symmetry in a three-layer ferrimagnetic superlatticePHYSICA STATUS SOLIDI (B) BASIC SOLID STATE PHYSICS, Issue 8 2006Rong-ke Qiu Abstract The magnon energy band in a ferrimagnetic superlattice with three layers in a unit cell is studied by employing retarded Green's functions and the spin-wave method. Two modulated energy gaps ,,13 and ,,23 are evaluated systematically, which exist in the magnon energy band along the Kx -direction perpendicular to the plane of the superlattice. It is revealed that the energy gap ,,13 has a direct relation with the symmetry among the spin quantum numbers and the interlayer exchange couplings, while the energy gap ,,23 relates to the symmetry among these spin quantum numbers only. These symmetries differ from the symmetry of crystallographic point groups. We define the magnetically structural symmetry that is dominated mainly by the magnetic parameters. The absence of the energy gap at a certain condition means that the system has a high magnetically structural symmetry. The magnetically structural symmetry of the superlattice, which is an intrinsic property, strongly affects the magnon energy band structure and thus the magnetic behaviors of the system. Furthermore, two complete bandgaps are observed to extend through the Brillouin zone (referred to as "magnonic crystal") in this superlattice system. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] The application of eigensymmetries of face forms to anomalous scattering and twinning by merohedry in X-ray diffractionACTA CRYSTALLOGRAPHICA SECTION A, Issue 3 2010H. Klapper The face form (crystal form) {hkl} which corresponds to an X-ray reflection hkl is considered. The eigensymmetry (inherent symmetry) of such a face form can be used to derive general results on the intensities of the corresponding X-ray reflections. Two cases are treated. (i) Non-centrosymmetric crystals exhibiting anomalous scattering: determination of reflections hkl for which Friedel's rule is strictly valid, i.e.I(hkl) = I() (Friedel pair, centric reflection), or violated, i.e.I(hkl) , I() (Bijvoet pair, acentric reflection). It is shown that those reflections hkl strictly obey Friedel's rule, for which the corresponding face form {hkl} is centrosymmetric. If the face form {hkl} is non-centrosymmetric, Friedel's rule is violated due to anomalous scattering. (ii) Crystals twinned by merohedry: determination of reflections hkl, the intensities of which are affected (or not affected) by the twinning. It is shown that the intensity is affected if the twin element is not a symmetry element of the eigensymmetry of the corresponding face form {hkl}. The intensity is not affected if the twin element belongs to the eigensymmetry of {hkl} (`affected' means that the intensities of the twin-related reflections are different for different twin domain states owing to differences either in geometric structure factors or in anomalous scattering or in both). A simple procedure is presented for the determination of these types of reflections from Tables 10.1.2.2 and 10.1.2.3 of International Tables for Crystallography, Vol. A [Hahn & Klapper (2002). International Tables for Crystallography, Vol. A, Part 10, edited by Th. Hahn, 5th ed. Dordrecht: Kluwer]. The application to crystal-structure determination of crystals twinned by merohedry (reciprocal space) and to X-ray diffraction topographic mapping of twin domains (direct space) is discussed. Relevant data and twinning relations for the 63 possible twin laws by merohedry in the 26 merohedral point groups are presented in Appendices A to D. [source] Molecular crystal global phase diagrams.ACTA CRYSTALLOGRAPHICA SECTION A, Issue 1 2010In the first part of this series [Keith et al. (2004). Cryst. Growth Des.4, 1009,1012; Mettes et al. (2004). Acta Cryst. A60, 621,636], a method was developed for constructing global phase diagrams (GPDs) for molecular crystals in which crystal structure is presented as a function of intermolecular potential parameters. In that work, a face-centered-cubic center-of-mass lattice was arbitrarily adopted as a reference state. In part two of the series, experimental crystal structures composed of tetrahedral point group molecules are classified to determine what fraction of structures are amenable to inclusion in the GPDs and the number of reference lattices necessary to span the observed structures. It is found that 60% of crystal structures composed of molecules with point-group symmetry are amenable and that eight reference lattices are sufficient to span the observed structures. Similar results are expected for other cubic point groups. [source] Elastic properties of two-dimensional quasicrystalsACTA CRYSTALLOGRAPHICA SECTION A, Issue 4 2008Hans Grimmer Quasicrystals (QC) with two-dimensional quasiperiodic and one-dimensional periodic structure are considered. Their symmetry can be described by embedding the three-dimensional physical space VE in a five-dimensional superspace V, which is the direct sum of VE and a two-dimensional internal space VI. A displacement v in V can be written as v = u + w, where u,VE and w ,VI. If the QC has a point group P in VE that is crystallographic, it is assumed that w and a vector u,,VE lying in the plane in which the crystal is quasiperiodic transform under equivalent representations of P, inequivalent ones if the point group is 5-, 8-, 10- or 12-gonal. From the Neumann principle follow restrictions on the form of the phonon, phason and phonon,phason coupling contributions to the elastic stiffness matrix that can be determined by combining the restrictions obtained for a set of elements generating the point group of interest. For the phonon part, the restrictions obtained for the generating elements do not depend on the system to which the point group belongs. This remains true for the phason and coupling parts in the case of crystallographic point groups but, in general, breaks down for the non-crystallographic ones. The form of the symmetric 12 × 12 matrix giving the phonon, phason and phonon,phason coupling contributions to the elastic stiffness is presented in graphic notation. [source] Higher-dimensional point groups in superspace crystallographyACTA CRYSTALLOGRAPHICA SECTION A, Issue 2 2008A. Janner Crystallographic puzzles not covered by the present crystallography, like integral indexing and crystallographic scaling of axial-symmetric biomacromolecules and icosahedral viral capsids and/or integral lattices, can possibly be explained by extending (n,d)-dimensional superspace crystallography to include finite subgroups of the higher-dimensional orthogonal group O(n) and not only those of O(d), as restricted by the physical dimension d. [source] Application of modern tensor calculus to engineered domain structures.ACTA CRYSTALLOGRAPHICA SECTION A, Issue 2 2006This article is a roadmap to a systematic calculation and tabulation of tensorial covariants for the point groups of material physics. The following are the essential steps in the described approach to tensor calculus. (i) An exact specification of the considered point groups by their embellished Hermann,Mauguin and Schoenflies symbols. (ii) Introduction of oriented Laue classes of magnetic point groups. (iii) An exact specification of matrix ireps (irreducible representations). (iv) Introduction of so-called typical (standard) bases and variables , typical invariants, relative invariants or components of the typical covariants. (v) Introduction of Clebsch,Gordan products of the typical variables. (vi) Calculation of tensorial covariants of ascending ranks with consecutive use of tables of Clebsch,Gordan products. (vii) Opechowski's magic relations between tensorial decompositions. These steps are illustrated for groups of the tetragonal oriented Laue class D4z, 4z2x2xy of magnetic point groups and for tensors up to fourth rank. [source] Groupoid of orientational variantsACTA CRYSTALLOGRAPHICA SECTION A, Issue 1 2006Cyril Cayron Daughter crystals in orientation relationship with a parent crystal are called variants. They can be created by a structural phase transition (Landau or reconstructive), by twinning or by precipitation. Internal and external classes of transformations defined from the point groups of the parent and daughter phases and from a transformation matrix allow the orientations of the distinct variants to be determined. These are algebraically identified with left cosets and their number is given by the Lagrange formula. A simple equation links the numbers of variants of the direct and inverse transitions. The equivalence classes on the transformations between variants are isomorphic to the double cosets (operators) and their number is given by the Burnside formula. The orientational variants and the operators constitute a groupoid whose composition table acts as a crystallographic signature of the transition. A general method that determines if two daughter variants can be inherited from more than one parent crystal is also described. A computer program has been written to calculate all these properties for any structural transition; some results are given for Burgers transitions and for martensitic transitions in steels. The complexity, irreversibility and entropy of fractal systems constituted by orientational variants generated by thermal cycling are briefly discussed. [source] On the real crystal octahedraACTA CRYSTALLOGRAPHICA SECTION A, Issue 6 2002Yury L. Voytekhovsky A real crystal octahedron is defined as any polyhedron bounded, at least, by some of four pairs of parallel planes being in a standard crystallographic orientation with arbitrary distances between them. All the combinatorially non-equivalent shapes (30 in total) are found and characterized by 2-subordination symbols, automorphism group orders and symmetry point groups. The results are discussed with respect to the diamond crystal morphology. [source] Symmetry relations of magnetic twin lawsACTA CRYSTALLOGRAPHICA SECTION A, Issue 6 2001J. Schlessman Symmetry relationships between two simultaneously observed domain states (domain pair) are used to determine physical properties that can distinguish between the observed domains. Here the tabulation of these symmetry relationships is extended from non-magnetic cases to magnetic cases, in terms of magnetic point groups, i.e. all possible magnetic symmetry groups and magnetic twinning groups of domain pairs are determined and tabulated. [source] C20 to C60 fullerenes: combinatorial types and symmetriesACTA CRYSTALLOGRAPHICA SECTION A, Issue 6 2001Yury L. Voytekhovsky A figure giving the point groups for all combinatorially non-isomorphic C20 to C60 fullerenes (5770 in common) is contributed. The fullerenes of 6 to 120 automorphism group orders (80 in common) are drawn in the Shlegel projections and characterized by the point groups. [source] Chiral molecules with polyhedral T, O, or I symmetry: Theoretical solution to a difficult problem in stereochemistryCHIRALITY, Issue 8 2008Sri Kamesh Narasimhan Abstract Ever since point groups of symmetry have been used to describe molecules after Van't Hoff and Le Bel proposed tetrahedral structures for carbon atoms in 1874, it remains difficult to design chiral molecules with polyhedral symmetry T, O, or I. Past theoretical and experimental studies have mainly accomplished molecular structures that have the conformations for satisfying the T symmetry. In this work, we present a general theoretical approach to construct rigid molecular structures that have permanently the symmetry of T, O, and I. This approach involves desymmetrizaton of the vertices or the edges of Platonic solid-shaped molecules with dissymmetric moieties. Using density functional theory (DFT) and assisted model building and energy refinement (AMBER) computational methods, the structure, the rigidity, and the symmetry of the molecule are confirmed by assessing the lowest energy conformation of the molecule, which is initially presented in a planar graph. This method successfully builds molecular structures that have the symmetry of T, O, and I. Interestingly, desymmetrization of the edges has a more stringent requirement of rigidity than desymmetrization of the vertices for affording the T, O, or I symmetry. Chirality, 2008. © 2008 Wiley-Liss, Inc. [source] | |||||||||||||