Bases for the irreducible representations of O(3) symmetry adapted to a crystallographic point group

1987 ◽  
Vol 31 (3) ◽  
pp. 535-563 ◽  
Author(s):  
M. D. Gould ◽  
G. S. Chandler
Keyword(s):  
Point Group ◽  
Irreducible Representations

Schwingungsspektren und Kraftkonstanten der Hexacyano-Komplexe des Mangan(I), Technetium(I) und Rhenium(I) / Vibrational Spectra and Force Constants of the Hexacyano Complexes of Mangan(l), Technicium(I) and Rhenium(l)

1972 ◽  
Vol 27 (8-9) ◽  
pp. 1193-1196 ◽  
Author(s):  
W. Krasser ◽  
K. Schwochau
Keyword(s):  
Force Field ◽  
Raman Spectra ◽  
Vibrational Spectra ◽  
Point Group ◽  
Force Constants ◽  
Irreducible Representations ◽  
Infrared And Raman Spectra ◽  
Valence Force ◽  
Valence Force Field ◽  
Complex Salts

The infrared and Raman spectra of the complex salts K5[Mn(CN)6], K5[Tc(CN)6] and K5[Re(CN)s] have been recorded in the range from 4000 to 40 cm-1. All expected fundamental vibrations have been observed and could be assigned to the irreducible representations of the sym­metry point group Oh . The calculation of the force constants is based on the concept of the generalized valence force field. The low CN-valence force constants indicate the relatively strong Π-bonding character of the metal carbon bond, which is especially pronounced for K5[Tc(CN)6).

scholarly journals Characterizing the universal rigidity of generic tensegrities

2021 ◽  
Author(s):  
Ryoshun Oba ◽  
Shin-ichi Tanigawa
Keyword(s):  
Finite Groups ◽  
Stability Condition ◽  
Point Group ◽  
Group Symmetry ◽  
Irreducible Representations ◽  
Sufficient Condition ◽  
Point Group Symmetry ◽  
Block Diagonalization ◽  
Representations Of Finite Groups ◽  
Universal Rigidity

AbstractA tensegrity is a structure made from cables, struts, and stiff bars. A d-dimensional tensegrity is universally rigid if it is rigid in any dimension $$d'$$ d ′ with $$d'\ge d$$ d ′ ≥ d . The celebrated super stability condition due to Connelly gives a sufficient condition for a tensegrity to be universally rigid. Gortler and Thurston showed that super stability characterizes universal rigidity when the point configuration is generic and every member is a stiff bar. We extend this result in two directions. We first show that a generic universally rigid tensegrity is super stable. We then extend it to tensegrities with point group symmetry, and show that this characterization still holds as long as a tensegrity is generic modulo symmetry. Our strategy is based on the block-diagonalization technique for symmetric semidefinite programming problems, and our proof relies on the theory of real irreducible representations of finite groups.

scholarly journals On the planarity of the NSi3 skeleton in the trisilylamine molecule. A normal coordinate analysis involving complex symmetry coordinates

1979 ◽  
Vol 57 (14) ◽  
pp. 1779-1784 ◽  
Author(s):  
H. F. Shurvell ◽  
A. Dunham ◽  
S. J. Cyvin ◽  
J. Brunvoll
Keyword(s):  
Secular Equation ◽  
Point Group ◽  
Irreducible Representations ◽  
Complex Numbers ◽  
Normal Coordinate ◽  
Energy Distributions ◽  
Normal Vibrations ◽  
Interaction Terms ◽  
Symmetry Coordinates ◽  
Unusual Problem

Normal coordinate calculations have been carried out for the N(SiH3)3 molecule. A model with C3h symmetry was used, which is based on a planar NSi3 skeleton. The presence of complex numbers for the characters of degenerate irreducible representations of the point group C3h leads to an unusual problem when factoring the G and F matrices. A set of real degenerate symmetry coordinates for the E′ and E′′ species can be constructed, but these are not true symmetry coordinates under the C3h point group. However, they can be obtained from the genuine (complex) symmetry coordinates by a unitary transformation. For a degenerate species, this procedure leads to a and b blocks of the factored G and F matrices, which contain interaction terms. Consequently both blocks must be taken together when forming the secular equation, and subsequently the calculated frequencies appear as pairs of identical numbers.A valence force field that includes all reasonable interactions has been obtained and used to predict the wavenumbers of the 15N and d9 isotopic molecules. Details of the normal vibrations in the three molecules have been obtained from potential energy distributions. The results show reasonable agreement with the limited experimental data available for the isotopic molecules.

scholarly journals Color-Coding Point Group & Space Group Diagrams and other Mathematical Journeys

2014 ◽  
Vol 70 (a1) ◽  
pp. C1275-C1275
Author(s):  
Maureen Julian
Keyword(s):  
General Position ◽  
Point Group ◽  
Three Dimensional ◽  
Irreducible Representations ◽  
Color Coding ◽  
Space Group ◽  
Point Groups ◽  
Symmetry Operations

Color clarifies diagrams point group and space group diagrams. For example, consider the general position diagrams and the symbol diagrams. Symmetry operations can be represented by matrices whose determinant is either plus one or minus one. In the former case there is no change of handedness and in the latter case there is a change of handedness. The general position diagrams emphasis this information by color-coding. The symbol diagrams are a little more complicated and will be demonstrated. The second topic is a comparison of the thirty-two three-dimensional point groups with their corresponding 18 abstract mathematical groups. The corresponding trees will be explored. This discussion leads into the topic of irreducible representations.

scholarly journals Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes

2014 ◽  
Vol 66 (6) ◽  
pp. 1201-1224 ◽  
Author(s):  
Jeffrey D. Adler ◽  
Joshua M. Lansky
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Fixed Points ◽  
Point Group ◽  
Reductive Group ◽  
Conjugacy Classes ◽  
Irreducible Representations ◽  
Reductive Groups ◽  
A Finite Group ◽  
The Relationship

AbstractSuppose that is a connected reductive group defined over a field k, and ┌ is a finite group acting via k-automorphisms of satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of -fixed points in is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair (,┌) and consider any group G satisfying the axioms. If both and G are k-quasisplit, then we can consider their duals *and G*. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in G*(k) to the analogous set for *(k). If k is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of G(k) and (k), one obtains a mapping of such packets.

MODELING THE ELECTROMAGNETIC TRANSITIONS IN TETRAHEDRAL-SYMMETRIC NUCLEI

2010 ◽  
Vol 19 (04) ◽  
pp. 621-632 ◽  
Author(s):  
A. GÓŹDŹ ◽  
A. DOBROWOLSKI ◽  
J. DUDEK ◽  
K. MAZUREK
Keyword(s):  
Ground State ◽  
Transition Probabilities ◽  
Point Group ◽  
Irreducible Representations ◽  
Collective Excitations ◽  
Tetrahedral Symmetry ◽  
Electromagnetic Transitions ◽  
Model Combining

Collective excitations involving a hypothetical nuclear tetrahedral symmetry, the latter described with the help of the T d point group are discussed. A model combining the zero- and one-phonon excitations with the solutions of the "generalised collective rotor" of Ref.1 is constructed. The solutions belonging to the irreducible representations of the T d-group are constructed and used to calculate transition probabilities from hypothetical tetrahedral bands to the ground-state bands. An example of 156 Dy is discussed.

The Vibronic and Infrared Spectra of Europium Dibenzoylmethane Chelate with Piperidine

1979 ◽  
Vol 34 (6) ◽  
pp. 748-751
Author(s):  
S. Zachara
Keyword(s):  
Infrared Spectrum ◽  
Absorption Spectra ◽  
Infrared Spectra ◽  
Infrared Absorption ◽  
Point Group ◽  
Group Symmetry ◽  
Irreducible Representations ◽  
Vibronic Spectrum ◽  
Point Group Symmetry ◽  
Infrared Absorption Spectra

Abstract The vibronic spectra of europium dibenzoylmethane chelate with piperidine in the neighbour-hood of the 5D0 → 7F0 line at 293 K and 77 K and the infrared absorption spectra in the region of 20- 200 cm-1 at various temperatures have been studied. It is shown that all vibrational frequencies obtained from the vibronic spectrum at 77 K can be observed in the infrared spectrum at the same temperature. For the above frequencies we assign some vibrations as belonging to irreducible representations of S4 point group symmetry.

scholarly journals EXPLORING THE SPECTRUM OF QCD USING A SPACE-TIME LATTICE

2006 ◽  
Vol 21 (04) ◽  
pp. 843-846 ◽  
Author(s):  
COLIN MORNINGSTAR
Keyword(s):  
Point Group ◽  
Space Time ◽  
Irreducible Representations ◽  
Baryon Spectrum ◽  
Quantum Numbers ◽  
Spin Quantum ◽  
Pure Gauge Theory ◽  
Gluonic Excitations ◽  
Static Quark ◽  
The Continuum

Some past and ongoing explorations of the spectrum of QCD using Monte Carlo simulations on a space-time lattice are described. Glueball masses in the pure-gauge theory are reviewed, and the energies of gluonic excitations in the presence of a static quark-antiquark pair are discussed. Current efforts to compute the baryon spectrum using extended three-quark operators are also presented, emphasizing the need to use irreducible representations of the cubic point group to identify spin quantum numbers in the continuum limit.

Vibrational Analysis of Polymeric Hydrogen Fluoride: Cyclic (HF)6

1973 ◽  
Vol 28 (11) ◽  
pp. 1787-1793 ◽  
Author(s):  
S. J. Cyvin ◽  
V. Devarajan ◽  
J. Brunvoll ◽  
Ø . Ra
Keyword(s):  
Electron Diffraction ◽  
Hydrogen Fluoride ◽  
Secular Equation ◽  
Point Group ◽  
Vibrational Analysis ◽  
Irreducible Representations ◽  
Mean Amplitudes Of Vibration ◽  
Symmetry Coordinates ◽  
Vibrational Problem ◽  
Complex Characters

The molecular vibrations of cyclic (HF)6 are studied. The consequences of complex characters for irreducible representations of the appropriate point group (C6h) are elucidated. The secular equation of the vibrational problem is solved in terms of real coordinates. Those of the degenerate species were obtained by a transformation of the complex symmetry coordinates. The force constants were taken from solid HF data. Mean amplitudes of vibration were calculated and compared with electron diffraction results.

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