scholarly journals CARTESIAN AXIS AND PLANE CONVENTIONS IN Cnv SYMMETRY GROUPS

Química Nova ◽  
2021 ◽  
Author(s):  
Lucas Dias ◽  
Roberto Faria
Keyword(s):  
Water Molecule ◽  
Point Group ◽  
Molecular Orbitals ◽  
Symmetry Groups ◽  
Long Time ◽  
Point Groups

In this work, we call the attention to the ambiguity found in the literature when labeling vibrations and molecular orbitals as B1 and B2 for molecules belonging to the C2v point group as, for example, the water molecule. A survey of several books and some articles shows that this ambiguity comes from a long time ago and persists today, being a source of misunderstanding and a waste of time for students and teachers. It means that, in the case of the point groups Cnv, Dn, and Dnh (n = 2, 4, 6), it is very important to draw students’ attention to this ambiguity that exists in the literature. It is unfortunate that the recommendation made by Mulliken, more than sixty years ago, to always place the water molecule in the yz plane, has not been followed.

scholarly journals Fundamental Domains for Rhombic Lattices with Dihedral Symmetry of Order 8

10.37236/7802 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Joseph Ray Clarence G. Damasco ◽  
Dirk Frettlöh ◽  
Manuel Joseph C. Loquias
Keyword(s):  
Symmetry Group ◽  
Fundamental Domain ◽  
Point Group ◽  
Symmetry Groups ◽  
Fundamental Domains ◽  
Open Case ◽  
Point Groups ◽  
Index 2 ◽  
Rhombic Lattice ◽  
Planar Lattices

We show by construction that every rhombic lattice $\Gamma$ in $\mathbb{R}^{2}$ has a fundamental domain whose symmetry group contains the point group of $\Gamma$ as a subgroup of index $2$. This solves the last open case of a question raised in a preprint by the authors on fundamental domains for planar lattices whose symmetry groups properly contain the point groups of the lattices.  

scholarly journals The determination of point groups from imprecise molecular geometries

2021 ◽  
Author(s):  
Peter J. Knowles
Keyword(s):  
Symmetry Breaking ◽  
Point Group ◽  
Simple Function ◽  
Coordinate Frame ◽  
New Approach ◽  
Measure Of Symmetry ◽  
Point Groups

AbstractWe present a new approach for the assignment of a point group to a molecule when the structure conforms only approximately to the symmetry. It proceeds by choosing a coordinate frame that minimises a measure of symmetry breaking that is computed efficiently as a simple function of the molecular coordinates and point group specification.

scholarly journals Point-group symmetry detection in three-dimensional charge density of biomolecules

2019 ◽  
Vol 36 (7) ◽  
pp. 2237-2243
Author(s):  
Cyril F Reboul ◽  
Simon Kiesewetter ◽  
Dominika Elmlund ◽  
Hans Elmlund
Keyword(s):  
Charge Density ◽  
Single Particle ◽  
Statistical Tests ◽  
Point Group ◽  
Three Dimensional ◽  
Group Symmetry ◽  
Point Group Symmetry ◽  
Density Maps ◽  
Point Groups ◽  
Density Map

Abstract Motivation No rigorous statistical tests for detecting point-group symmetry in three-dimensional (3D) charge density maps obtained by electron microscopy (EM) and related techniques have been developed. Results We propose a method for determining the point-group symmetry of 3D charge density maps obtained by EM and related techniques. Our ab initio algorithm does not depend on atomic coordinates but utilizes the density map directly. We validate the approach for a range of publicly available single-particle cryo-EM datasets. In straightforward cases, our method enables fully automated single-particle 3D reconstruction without having to input an arbitrarily selected point-group symmetry. When pseudo-symmetry is present, our method provides statistics quantifying the degree to which the 3D density agrees with the different point-groups tested. Availability and implementation The software is freely available at https://github.com/hael/SIMPLE3.0.

3D Braided Geometry Structures Based on Space Group

2010 ◽  
Vol 152-153 ◽  
pp. 1156-1161 ◽  
Author(s):  
Wen Suo Ma ◽  
Bin Qian Yang ◽  
Xiao Zhong Ren
Keyword(s):  
Group Theory ◽  
Symmetry Groups ◽  
The Novel ◽  
Geometrical Structures ◽  
Braided Group ◽  
Space Group ◽  
Space Groups ◽  
Geometry Structure ◽  
Point Groups ◽  
Space Point

3D braided group theory is dissertated. The analysis procedure is described from the existing braided geometry structure to the braided space group; 3D braided geometrical structures are finally described by means of group theory. Some of novel 3D braided structures are deduced from the braided space groups. By describing the 3D braided materials with braided space point and braided space groups, the braided space groups are not always the same as symmetry groups of crystallographic because novel lattices can be produced and the reflection operation cannot exist in braided space point groups. Braided point and space groups are theoretical basis for deriving the novel braided geometry structure.

Symmetry adapted functions for double point groups II. Cubic point groups

1970 ◽  
Vol 67 (3) ◽  
pp. 647-656 ◽  
Author(s):  
A. P. Cracknell ◽  
S. J. Joshua
Keyword(s):  
Double Point ◽  
Point Group ◽  
Basis Functions ◽  
Kronecker Products ◽  
Symmetry Adapted Functions ◽  
Point Groups

AbstractA method of deriving the basis functions of the double-valued representations of a point group by the reduction of Kronecker products is described. The method has been used to derive expressions for these bases for cubic point groups for which the results are tabulated.

Ferroelectric space groups

1986 ◽  
Vol 42 (1) ◽  
pp. 44-47 ◽  
Author(s):  
D. B. Litvin
Keyword(s):  
Electric Dipole ◽  
Space Time ◽  
Discrete Space ◽  
Symmetry Groups ◽  
Space Groups ◽  
Point Groups

The 440 ferroelectric space groups, viz the Heesch-Shubnikov (magnetic) space groups, which are symmetry groups of ferroelectric electric-dipole arrangements in crystals, are derived and tabulated. By considering automorphisms induced by the automorphisms of the discrete space-time group, we show that although ferroelectric, ferromagnetic and ferrocurrent point groups all number 31, the number of ferroelectric space groups differs from 275, which is that of both ferromagnetic and ferrocurrent space groups.

SINIF CEBİRİ YAKLAŞIMI ALTINDA 𝑫𝟐𝒅 ve 𝑪𝟑𝒊 NOKTA GRUPLARI

2021 ◽  
Vol 8 (17) ◽  
pp. 141-149
Author(s):  
Melike DEDE ◽  
Harun AKKUS
Keyword(s):  
Symmetry Groups ◽  
Character Tables ◽  
Secular Equations ◽  
Point Groups ◽  
Cayley Tables

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.

Crystallography of embedded particles in Al–Mg–Zn alloys. Symmetry analysis

2014 ◽  
Vol 47 (5) ◽  
pp. 1736-1748 ◽  
Author(s):  
Gunnar Thorkildsen ◽  
Helge B. Larsen
Keyword(s):  
Proper Subgroup ◽  
Point Group ◽  
Alloy System ◽  
Group Symmetry ◽  
Symmetry Principle ◽  
Generating Sets ◽  
The Matrix ◽  
Embedded Particles ◽  
Point Groups ◽  
Zn Alloys

Following a proper heat treatment, the alloy system Al–Mg–Zn shows a great wealth of precipitate particles forming coherent (η′ crystals) and incoherent (η crystals) boundaries with the Al matrix. Both the matrix crystal and the precipitate crystals are holohedral, as their point groups correspond to their metric symmetries (m\overline{3}m and 6/mmm). On the basis of published orientational relationships for a principal variant of every known precipitate family, the full sets of orientational variants are deduced by the concepts of intersection groups,Hβ, and variant generating sets,Vβ. The intergrowth symmetry principle has been visualized by stereographic projections. Special attention has been given to patterns of superimposed lattice nodes in reciprocal space and the implications of overlap in terms of observable reflections. It has been uncovered, by artificially reducing the point group symmetry of the coherent η′ phase, whereVβis a proper subgroup of the matrix holohedral, that twin laws for merohedry are revealed whenVβis factorized into a weak direct product.

Sign in / Sign up

Export Citation Format

Share Document