The symmetry group of a cyclic system in a crystal is defined. Its relation to the space group of an infinite crystal is cleared up. It is shown that all the irreducible representations of the cyclic system symmetry group can be found from the definite representations of the space group. The connection between Herring's and reduction group methods developed in the space group representation theory and the large unit cell method is established. A simple and efficient method of the symmetry-adapted atomic orbitals generation is proposed and illustrated by the examples for α-quartz and β-cristobalite crystals (space group D63 and O7h).



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Symmetry Groups of Cyclic Systems in Crystals

An interpretation of double coset decompositions of groups is given in terms of the construction of systems from similar disjoint subsystems. A natural labelling scheme is developed for the direct product states of systems of this type, providing an extension of the concept of unique labelling. An application of the decoupling transformation to finite direct sum systems is investigated.

Symmetry properties of direct sum systems

A method is developed for the construction of symmetry-adapted molecular functions which are consistent with a given set of 3-jm symbols (or coupling coefficients) No irreducible representation matrices or projection operators are required because all group theoretic basis information is contained in the 3-jm symbols for the molecular symmetry group The authors use this information to construct basis functions for the permutation representation of the molecule and then use point group 3-jm symbols to combine this permutation structure with a set of xyz displacements to give irreducible basis functions for the molecular vibrations The molecular SF6 is used as an example

http://iopscience.iop.org/article/10.1088/0022-3700/16/6/013/pdf

Symmetry-adapted functions: molecular vibrations

The problem of classification and determination of standard irreducible bases for induced representations of an arbitrary finite group is discussed. A general prescription for classification and determination of these bases for an arbitrary induced representation is proposed, and an extra independent prescription for the special case of primitive representations is given. Relations between appropriate transformation coefficients are investigated and, in particular, a matrix similar to the Racah recoupling matrix from theory of multiple coupling of angular momentum is introduced.

Classification and determination of irreducible bases for induced representations for chains of subgroups of finite groups

General methods of generating complete sets of states for l(<or=2n) electrons in s-states on n centres are described. It is shown that a unique labelling of these states can always be obtained using a fictitious Sn symmetry of the n centres. Explicit descriptions of eight, seven and four electrons on eight centres are derived and the problem of determining a unique labelling for these states on rigid eight-centre structures with cubic and lower symmetry is solved.

http://iopscience.iop.org/0305-4470/16/11/012/pdf/0305-4470_16_11_012.pdf

Unique labelling of many-electron states in multicentre systems

Earlier work on unique labelling schemes in Oh symmetry is extended to include 48-atom shells and the symmetry coordinates of shells subject to periodicity conditions. A new approach to labelling is introduced based on the concept of cyclic regions in a crystal. These are described by finite factor groups of the crystallographic space group which contain the isogonal group as a subgroup. Some examples are worked out in detail.

https://iopscience.iop.org/article/10.1088/0305-4470/15/11/015/pdf

K S Chan

Papers

Unique labelling of symmetry coordinates and cyclic regions in crystals