scholarly journals Explicit expressions for totally symmetric spherical functions and symmetry-dependent properties of multipoles

2014 ◽  
Vol 470 (2171) ◽  
pp. 20140435
Author(s):  
Boris Zapol ◽  
Peter Zapol
Keyword(s):  
Point Group ◽  
Product Structure ◽  
Spherical Functions ◽  
Matrix Elements ◽  
Linear Combinations ◽  
Selection Rules ◽  
Charge Distributions ◽  
Symmetry Operators ◽  
Point Groups ◽  
Symmetry Operations

Closed expressions for matrix elements 〈 lm' | A (G)| lm 〉, where | lm 〉 are spherical functions and A (G) is the average of all symmetry operators of point group G, are derived for all point groups (PGs) and then used to obtain linear combinations of spherical functions that are totally symmetric under all symmetry operations of G. In the derivation, we exploit the product structure of the groups. The obtained expressions are used to explore properties of multipoles of symmetric charge distributions. We produce complete lists of selection rules for multipoles Q l and their moments Q lm , as well as of numbers of independent moments in a multipole, for any l and m and for all PGs. Periodicities and other trends in these properties are revealed.

scholarly journals Isomorphism and matrix representation of point groups

2019 ◽  
Vol 15 (1) ◽  
pp. 88-92
Author(s):  
Wan Heng Fong ◽  
Aqilahfarhana Abdul Rahman ◽  
Nor Haniza Sarmin
Keyword(s):  
Group Theory ◽  
Matrix Representation ◽  
Point Group ◽  
Multiplication Table ◽  
Matrix Representations ◽  
The Matrix ◽  
Point Groups ◽  
Complete Set ◽  
Symmetry Of Molecules ◽  
Symmetry Operations

In chemistry, point group is a type of group used to describe the symmetry of molecules. It is a collection of symmetry elements controlled by a form or shape which all go through one point in space, which consists of all symmetry operations that are possible for every molecule. Next, a set of number or matrices which assigns to the elements of a group and represents the multiplication of the elements is said to constitute representation of a group. Here, each individual matrix is called a representative that corresponds to the symmetry operations of point groups, and the complete set of matrices is called a matrix representation of the group. This research was aimed to relate the symmetry in point groups with group theory in mathematics using the concept of isomorphism, where elements of point groups and groups were mapped such that the isomorphism properties were fulfilled. Then, matrix representations of point groups were found based on the multiplication table where symmetry operations were represented by matrices. From this research, point groups of order less than eight were shown to be isomorphic with groups in group theory. In addition, the matrix representation corresponding to the symmetry operations of these point groups wasis presented. This research would hence connect the field of mathematics and chemistry, where the relation between groups in group theory and point groups in chemistry were shown.

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 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.

SYMMETRY AND ELECTRO-OPTICAL PROPERTIES OF NANOTUBES

2002 ◽  
Vol 01 (03n04) ◽  
pp. 313-325 ◽  
Author(s):  
M. DAMNJANOVIĆ ◽  
I. MILOŠEVIĆ ◽  
T. VUKOVIĆ ◽  
B. NIKOLIĆ ◽  
E. DOBARDŽIĆ
Keyword(s):  
Carbon Nanotubes ◽  
Optical Properties ◽  
Density Functional ◽  
Tight Binding ◽  
Optical Characteristics ◽  
Optical Transitions ◽  
Matrix Elements ◽  
Selection Rules ◽  
Single Wall Carbon ◽  
Quantum Numbers

The symmetry of single-wall carbon and inorganic tubes is reviewed. For the carbon nanotubes it is used to get the full set of quantum numbers, in the efficient precision (combined density functional and tight-binding methods) calculation of electronic bands and their complete assignation, to obtain the selection rules for optical transitions and the momenta matrix elements for the Bloch eigen-states. The optical characteristics are thoroughly found, and discussed.

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.

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