Group Theoretical Data Research Articles

Thermal decomposition and vibrational spectroscopic aspects of pyridinium hexafluorophosphate (C5H5NHPF6)

Thermal decomposition and vibrational spectroscopic properties of pyridinium hexafluorophosphate (C5H5NHPF6) have been studied. The structure of the compound is better interpreted as having a cubic space group, based on Raman and infrared vibrational spectroscopy experiments and group theoretical correlation data between site symmetry species and the spectroscopic space group. The 13C NMR data shows three significant signals corresponding to the three chemical environments expected on the pyridinium ring i.e. γ, β and α carbons, suggesting that the position of the anion must be symmetrical with respect to the pyridinium ring’s C2v symmetry. The process of thermal decomposition of the compound using TGA methods was found to follow a contracting volume model. The activation energy associated with the thermal decomposition reaction of the compound is 108.5 kJ mol−1, while the pre exponential factor is 1.51 × 109 sec−1.

A Higher Frobenius–Schur Indicator Formula for Group-Theoretical Fusion Categories

Group-theoretical fusion categories are defined by data concerning finite groups and their cohomology: A finite group $G$ endowed with a three-cocycle $\omega$, and a subgroup $H\subset G$ endowed with a two-cochain whose coboundary is the restriction of $\omega$. The objects of the category are $G$-graded vector spaces with suitably twisted $H$-actions; the associativity of tensor products is controlled by $\omega$. Simple objects are parametrized in terms of projective representations of finite groups, namely of the stabilizers in $H$ of right $H$-cosets in $G$, with respect to two-cocycles defined by the initial data. We derive and study general formulas that express the higher Frobenius-Schur indicators of simple objects in a group-theoretical fusion category in terms of the group-theoretical and cohomological data defining the category and describing its simples.

Fusion rules of equivariantizations of fusion categories

We determine the fusion rules of the equivariantization of a fusion category \documentclass[12pt]{minimal}\begin{document}${\mathcal {C}}$\end{document}C under the action of a finite group G in terms of the fusion rules of \documentclass[12pt]{minimal}\begin{document}${\mathcal {C}}$\end{document}C and group-theoretical data associated to the group action. As an application we obtain a formula for the fusion rules in an equivariantization of a pointed fusion category in terms of group-theoretical data. This entails a description of the fusion rules in any braided group-theoretical fusion category.

Vibrational spectra of CaGa2O4, Ca2GeO4, CaIn2O4 and CaSnO3 prepared by electrospinning

CaGa2O4 nanofibers, Ca2GeO4 microfibers, CaIn2O4 nanorods, and CaSnO3 nanofibers were synthesized by using an electrospinning technique. Structures and morphologies of the as-synthesized oxides were characterized by X-ray diffraction and scanning electron microscopy, respectively. Raman and infrared spectra were also recorded and analyzed at room temperature. More significantly, nuclear site group analysis was carried out and the number of normal vibrational modes, Raman-active, and infrared-active optical phonon modes were obtained by theoretical calculation. Finally, vibrational assignments of the observed Raman peaks and infrared absorption bands were given based on the group theoretical analysis and experimental data from literature.

Intermediate-coupling phenomena in electron-impact excitation of Hg atoms

In my talk the influence of the intermediate-coupling scheme of Hg on various electron scattering observables such as spin exchange cross sections, spin up-down asymmetries or spin-resolved circular polarization is discussed with special emphasis to recent experimental results from our group and recent theoretical data.

Modular invariants for group-theoretical modular data. I

We classify indecomposable commutative separable (special Frobenius) algebras and their local modules in (untwisted) group-theoretical modular categories. This gives a description of modular invariants for group-theoretical modular data. As a bi-product we provide an answer to the question when (and in how many ways) two group-theoretical modular categories are equivalent as ribbon categories.

Covers of Klein surfaces

Abstract We consider ramified (Galois) covers of the upper half plane in the category of Klein surfaces. We study the connection between the group theoretical ramification data of the cover and its geometrical properties, such as the number of the connected components of the boundary and orientability of the surface.

Generation of Clebsch–Gordan coefficients for the point and double groups

The theory of the point and double groups has been widely used in quantum physics to understand the structure and dynamical properties of molecules and solids. In order to construct wave functions for such systems, one often needs the Clebsch–Gordan coefficients for the symmetry groups. Here, we present an extension of the Bethe program to support the calculation of the Clebsch–Gordan coefficients as applied, for instance, in crystal field theory. Apart from the generation of the Clebsch–Gordan coefficients, the program also provides a simple access to the group theoretical data for all frequently applied point and double groups. Program summary Title of program: Bethe Catalogue number:ADUH_v3_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUH_v3_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Reference in CPC to previous versions: Comput. Phys. Comm. 162 (2004) 124–142; Comput. Phys. Comm. 171 (2005) 119–132 Catalog identifiers of previous versions: ADUH, ADVU Does the new version supersede the old version?: Yes Licensing provisions: None Computers for which the program is designed: All computers with a license of the computer algebra package Maple [Maple is a registered trademark of Waterloo Maple Inc.] Installations: University of Kassel (Germany) Operating systems under which the program has been tested: Linux 8.1+ and Windows 2000 Programming language used: Maple 7 and 8 Memory required to execute with typical data: 10–30 MB No. of lines in distributed program, including test data, etc.: 11 024 No. of bytes in distributed program, including test data, etc.: 210 244 Distribution format:tar.gz Nature of the physical problem: The energy levels of atoms, placed into a crystal environment, can be classified by using group theory. In order to represent, for instance, the wave functions, which are associated with these atomic levels, one often requires the Clebsch–Gordan coefficients of the underlying symmetry group of the overall system. These coefficients arise in the coupling of the electronic wave functions (subsystems) and therefore help investigate the interaction between the many-electron atom and the external field of the crystal. Method of solution: In the framework of the Bethe program [K. Rykhlinskaya, S. Fritzsche, Comput. Phys Comm. 162 (2004) 124–142; K. Rykhlinskaya, S. Fritzsche, Comput. Phys Comm. (2005), in press], we previously defined data structures to deal with a large number of group parameters of the point and double groups. Among other parameters, here we also implemented the irreducible (matrix) representations of these groups which are utilized in the present extension of the program in order to generate the Clebsch–Gordan coefficients for the point and double groups. In practice, of course, these coefficients are obtained by means of a proper summation over the matrix elements of the irreducible representations of the group. Reasons for the new version: Extension of the program. Summary of revision: A number additional procedure have been created to generate the Clebsch–Gordan coefficients for the symmetry groups (Bethe_CG_matrix(), Bethe_CG_coefficient(), Bethe_group_direct_product(), etc.) Restrictions onto the complexity of the problem: The generation of the Clebsch–Gordan coefficients is supported for the cyclic and related groups C i , C s , C n , C n h , C n v , the dihedral groups D n , D n h , D n d , the improper cyclic groups S 2 n ( n ⩽ 10 ) , the cubic groups O, T, O h , T h , T d as well as the icosahedral groups I and I h . Both the point and the double groups are supported. Unusual features of the program: All commands of the Bethe program are available for interactive work. Apart from the generation of the Clebsch–Gordan coefficients, the program also provides a simple access to the group theoretical data for all the groups specified above. The notation of the symmetry operations and of the irreducible representations follows the compilation by Altmann and Herzig [S. Altmann, P. Herzig, Point-Group Theory Tables, Clarendon Press, Oxford, 1994]. For a quick reference to the program, a description of all user-relevant commands is given in the (user) manual Bethe-commands.pdf which is distributed together with the code. Typical running time: Although the program replies ‘promptly’ on most requests, the running time depends strongly on the particular task.

Generation of molecular symmetry orbitals for the point and double groups

Symmetry-adapted molecular basis functions are widely applied for the electronic structure computations of molecules and clusters. These functions are obtained by exploiting the symmetry of the system and often help to simplify the computations considerably. In order to facilitate their use in algebraic and numerical computations, here we provide a set of Maple procedures which generates these functions by means of projection operators, both within the nonrelativistic and relativistic theory. All commonly applied point and double groups are supported by the program including, in addition, the access to their group-theoretical data such as the symmetry operators, characters, or irreducible representations. Program summary Title of program: Bethe Catalogue identifier:ADVU Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADVU Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions:none Computer for which the program is designed: All computers with a license of the computer algebra package Maple (Maple is a registered trademark of Waterloo Maple Inc.) Installations:University of Kassel (Germany) Operating systems or monitors under which the program has been tested:Linux 8.1+ and Windows2000 Programming language used: Maple 7 and 8 Memory required to execute with typical data:10–30 MB No. of lines in distributed program, including test data, etc.:14 190 No. of bytes in distributed program, including test data, etc.:370 795 Distribution format:tar.gz Nature of the physical problem:Molecular and solid-state quantum computations can be simplified considerably if the symmetry of the systems with respect to the rotation and inversion of the coordinates is taken into account. To exploit such symmetries, however, symmetry-adapted basis functions need to be constructed instead of using—as usual—the atomic orbitals as the (one-particle) basis. These so-called symmetry orbitals are invariant with respect to the symmetry operations of the group and are different for the point and double groups, i.e. for nonrelativistic and relativistic computations. Method of solution:Projection operator techniques are applied to generate the symmetry-adapted orbital functions as a linear combinations of atomic orbitals. Restrictions onto the complexity of the problem:The generation of the symmetry orbitals is supported for the cyclic and related groups C i , C s , C n , C n h , C n v , the dihedral groups D n , D n h , D n d , the improper cyclic groups S 2 n ( n ⩽ 10 ) , the cubic groups O, T, O h , T h , T d as well as the icosahedral groups I and I h . In all these cases, the symmetry orbitals can be obtained for either the point or double groups by using a nonrelativistic or, respectively, relativistic framework for the computations. Unusual features of the program:All commands of the Bethe program are available for interactive work. Apart from the symmetry orbitals generation, the program also provides a simple access to the group theoretical data for the presently implemented groups from above. The notation of the symmetry operations and the irreducible representations follows the compilation by Altmann and Herzig [Point-Group Theory Tables, Clarendon Press, Oxford, 1994]. For a quick reference to the program, a description of all user-relevant commands is given in the (user) manual Bethe-commands.ps and is distributed together with the code. Typical running time:Although the program replies ‘promptly’ on most requests, the running time depends strongly on the particular task.

Use of group theory for the analysis of vibrational spectra

To facilitate the use of group theory in the analysis of vibrational spectra, a set of Maple procedures is provided generating the normal coordinates and determining the spectral activities of polyatomic molecules. Our program, called Bethe, is based on the frequently applied point groups and provides an interactive access to the group data as needed in physical chemistry and elsewhere. Owing to the demand of the users the normal coordinates of the molecules are provided either in terms of Cartesian or internal coordinates. Program summary Title of program: Bethe Catalogue number: ADUH Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUH Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions:None Computers for which the program is designed: All computers with a license of the computer algebra package Maple 1 1 Maple is a registered trademark of Waterloo Maple Inc. Installations:University of Kassel (Germany) Operating systems under which the program has been tested: Linux 8.1+ and Windows 2000 Program language used: Maple 7 and 8 Memory required to execute with typical data: 10–30 MB No. of lines in distributed program including test data, etc.:11 859 No. of bytes in distributed program including test data, etc.:312 229 Distribution format:tar.gz Nature of the physical problem: Interaction of the infrared light with the molecule can lead to the excitation of the molecular vibrations [1]. Analysis of such vibrations is performed by the point group theory and helps to interpret the molecular spectra. Method of solution: Point group theory is applied to determine the normal coordinates of symmetric molecules and to carry out a spectroscopic analysis of their vibrational modes. The direct product of the irreducible representations is utilized to obtain the selection rules for infrared and Raman spectroscopy. Restrictions onto the complexity of the problem: The computation of the normal coordinates is supported for rather a large number of symmetries; in the present version the group data are provided for the cyclic and related groups C i , C s , C n , C n h , C n v , the dihedral groups D n , D n h , D n d , the improper cyclic groups S 2 n ( n ⩽ 10 ) , the cubic groups O, T, O h , T h , T d and the icosahedral groups I, I h . Unusual features of the program: All commands of the Bethe program are available for interactive work. Apart from the analysis of the vibrational motion of molecules, we also provide the group theoretical data of all the presently implemented point and double groups. The notation of the symmetry operations and the irreducible representations follows the compilation by Altmann and Herzig [2]. For reference to the program, a brief description of all the available commands is given in the user manual Bethe-commands.ps and is distributed together with the code. Typical running time: Although the program replies ‘promptly’ on most requests, the running time depends strongly on the particular task.

Phases of supersymmetric D-branes on Kähler manifolds and the McKay correspondence

We study the topological zero mode sector of type II strings on a K\"ahler manifold $X$ in the presence of boundaries. We construct two finite bases, in a sense bosonic and fermionic, that generate the topological sector of the Hilbert space with boundaries. The fermionic basis localizes on compact submanifolds in $X$. A variation of the FI terms interpolates between the description of these ground states in terms of the ring of chiral fields at the boundary at small volume and helices of exceptional sheaves at large volume, respectively. The identification of the bosonic/fermionic basis with the dual bases for the non-compact/compact K-theory group on $X$ gives a natural explanation of the McKay correspondence in terms of a linear sigma model and suggests a simple generalization of McKay to singular resolutions. The construction provides also a very effective way to describe D-brane states on generic, compact Calabi--Yau manifolds and allows to recover detailed information on the moduli space, such as monodromies and analytic continuation matrices, from the group theoretical data of a simple orbifold.

Symmetry analysis and the atomic distortions of the phase transitions in CsZnPO4

Abstract The phase transitions in CsZnPO4 at approximately 583 K and 533 K: Pnma (I) → Pn21 a (II) → P21/a (III) are analyzed by group theoretical methods and structural data published for the three polymorphic forms. An order parameter treatment of the phase transformations is presented in terms of the Landau formalism and projection operator techniques. A collective tilting of the ZnO4- and PO4-tetrahedra is taken as the primary order parameter for both transitions. In the case of the I → II transformation, the order parameter transforms according to the irreducible representation Γ 4 − of Pnma. The rotational distortion pattern in phase III can be explained by onset of an order parameter transforming according to the zone boundary representation X2 of phase I. Other distortions (Zn-, P- and Cs-ion displacements, polarization or strain components) can couple to the primary order parameters. The microscopic distortion patterns for the possible displacements as well as the expected domains are listed.

Aesthetic field theory—Extended particles in four-dimensional space–time

We continue our study of complex aesthetic field theory using a set of group theoretical data introduced previously. Our more detailed study shows significant differences with the corresponding real theory. We find that our theory allows for a four-dimensional particle system, where the particle system is viewed as a bound state of a three-dimensional maximum and minimum. The magnitude of the field is large in this ’’confluence’’ region. This property persists in time. We can think of this as a form of nonattenuation. We never see a large magnitude ’’free’’ maximum or minimum (confinement). A problem with the particle system is that the confluence region extends to a greater degree in z for large ‖t‖ compared to the case at t = 0.