scholarly journals Classification of atomic-scale multipoles under crystallographic point groups and application to linear response tensors

2018 ◽  
Vol 98 (16) ◽  
Author(s):  
Satoru Hayami ◽  
Megumi Yatsushiro ◽  
Yuki Yanagi ◽  
Hiroaki Kusunose
Keyword(s):  
Linear Response ◽  
Atomic Scale ◽  
Point Groups

Molecular Rovibronic Symmetries in Electric and Magnetic Fields

1975 ◽  
Vol 53 (19) ◽  
pp. 2210-2220 ◽  
Author(s):  
James K. G. Watson
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Electric Field ◽  
Symmetry Groups ◽  
Nuclear Spins ◽  
Symmetry Classification ◽  
Electric And Magnetic Fields ◽  
Field Symmetry ◽  
Point Groups

The structures of the symmetry groups for the rovibronic levels of a molecule in a homogeneous electric or magnetic field are derived, and the symmetry classification of the levels in terms of the representations and corepresentations of these groups is discussed. Specific results are given for molecules of the point groups C3, C2v, C3v, D2d, and Td in an electric field. Symmetry in combined electric and magnetic fields and the inclusion of nuclear spins are considered briefly.

The crystallographic point groups as semi-direct products

1963 ◽  
Vol 255 (1054) ◽  
pp. 216-240 ◽  
Keyword(s):  
Direct Product ◽  
Invariant Subgroup ◽  
Projection Operators ◽  
Direct Products ◽  
Finite Rotations ◽  
Space Groups ◽  
Symmetry Adapted Functions ◽  
Point Groups ◽  
Definition Of

This paper aims at providing a systematic treatment of the crystallographic point groups. Some well-known properties of them, in terms of the theory of the poles of finite rotations, are first discussed, so as to provide a simple way for recognizing their invariant subgroups. A definition of the semi-direct product is then given, and it is shown that all crystallographic point groups can be expressed as a semi-direct product of one of their invariant subgroups by a cyclic subgroup. Many useful relations between point groups can be obtained by exploiting the properties of the triple and mixed triple semi-direct products, which are defined. Much of the rest of the paper is devoted to the theory of the representations of semi-direct products. The treatment here parallels that given by Seitz (1936) for the reduction of space groups in terms of the representations of its invariant subgroups (the translation groups). The latter, however, are always Abelian and this is not always the case for point groups. The full treatment of the general case, such as given by McIntosh (1958), is laborious and it is shown that, if the emphasis is placed on the bases of the representations, rather than the representations themselves, it is possible to achieve the reduction of the point groups by a method hardly more involved than that required when the invariant subgroup is Abelian. It is also shown that, just as for space groups, the representations of the invariant subgroups can be denoted and visualized by means of a vector, which allows a very rapid classification of the representations, very much as the k vector as used by Bouckaert, Smoluchowski & Wigner (1936) allows the formalism of the Seitz method for space groups to be carried out in a graphical fashion. One of the major consequences of this work is that it affords a substantial simplification in the use of the symmetrizing and projection operators that are required to obtain symmetry-adapted functions: a very systematic alternative to the method given by Melvin (1956) is therefore provided. In the last section of the paper all the techniques discussed are applied in detail, as an example, to the cubic groups. The projection operators are used to obtain symmetry-adapted spherical harmonics for these groups. The paper might be found useful as an introduction to the methods for the reduction of space groups.

Magnetic modes compatible with the symmetry of crystals

2021 ◽  
Vol 77 (4) ◽  
Author(s):  
Piotr Fabrykiewicz ◽  
Radosław Przeniosło ◽  
Izabela Sosnowska
Keyword(s):  
Space Group ◽  
Ferromagnetic Component ◽  
Magnetic Mode ◽  
Magnetic Symmetry ◽  
Point Groups

A classification of magnetic point groups is presented which gives an answer to the question: which magnetic groups can describe a given magnetic mode? There are 32 categories of magnetic point groups which describe 64 unique magnetic modes: 16 with a ferromagnetic component and 48 without. This classification focused on magnetic modes is helpful for finding the magnetic space group which can describe the magnetic symmetry of the material.

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