Group Theoretical Methods and Applications to Molecules and Crystals

  title={Group Theoretical Methods and Applications to Molecules and Crystals},
  author={Shoon-Kyung Kim},
Preface List of symbols 1. Linear transformations 2. Theory of matrix transformations 3. Elements of abstract group theory 4. Unitary and orthogonal groups 5. The point groups of finite order 6. Theory of group representations 7. Construction of symmetry adapted linear combinations based on the correspondence theorem 8. Subduced and induced representations 9. Elements of continuous groups 10. The representations of the rotation group 11. Single- and double-valued representations of point groups… 
Applications of Symmetry and Group Theory for the Investigation of Molecular Vibrations
The application of symmetry and mathematical group theory is a powerful tool for investigating the vibrations of molecules. In this paper, we present an overview of the methods utilized. First we
Decomposition of Sohncke space groups into products of Bieberbach and symmorphic parts
Abstract Point groups consist of rotations, reflections, and roto-reflections and are foundational in crystallography. Symmorphic space groups are those that can be decomposed as a semi-direct
Crystal Vision-Applications of Point Groups in Computer Vision
The icosahedral group is used as an example since it is the largest finite subgroup of the 3D rotation group and its usage for data compression in applications where the processes are (on average) symmetrical with respect to these groups.
Symmetry projection, geometry and choice of the basis
A geometrical point of view of symmetry adapted projection to irreducible subspaces is presented. The projection is applied in two stages. The first step consists in projecting over subspaces
Unfolding of electronic structure through induced representations of space groups: Application to Fe-based superconductors
We revisit the problem that relevant parts of bandstructures for a given cell choice can reflect exact or approximate higher symmetries of subsystems in the cell and can therefore be significantly
Equivalent rotations associated with the permutation inversion group revisited: symmetry projection of the rovibrational functions of methane
In this work the analysis of the equivalent rotations from the permutation inversion group formalism is revisited. We emphasize that explicit knowledge of changes in the Euler angles are not required
Symmetries, Conserved Properties, Tensor Representations, and Irreducible Forms in Molecular Quantum Electrodynamics
In the wide realm of applications of quantum electrodynamics, a non-covariant formulation of theory is particularly well suited to describing the interactions of light with molecular matter, and a variety of symmetry principles are drawn out with reference to applications.
Quantum Numbers and the Eigenfunction Approach to Obtain Symmetry Adapted Functions for Discrete Symmetries
  • R. Lemus
  • Mathematics, Computer Science
  • 2012
It is shown that the irreducible representations (irreps) associated with the eigenfunctions are indeed a shorthand notation for the set of eigenvalues of the class operators (character table) and the need of a canonical chain of groups to establish a complete set of commuting operators is emphasized.
General approach for the construction of hybrid orbitals
A general approach to construct hybrid orbitals based on the projection of both the subspace of atomic orbitals and the subspace of hybrid orbitals is presented. It is shown that the projection must
Determining the Order of a Molecular Point Group
Conventional methods of identifying the point group of a molecule require skills in finding symmetry elements and operations. A useful aid to error checking is an ability to determine the order of a


A unified theory of the point groups. IV. The general corepresentations of the crystallographic and noncrystallographic Shubnikov point groups
A new system of classification for all the Shubnikov point groups Gs is presented, which is best suited for describing their isomorphisms as well as their construction. By the new classification and
A simple theory of the spinor representations of the complex orthogonal group O(d,C) in the d‐dimensional Euclidean space V(d) is presented via a basic lemma on involutional transformations and
A new method of constructing the symmetry‐adapted linear combinations based on the correspondence theorem and induced representations
The theory of induced representations is incorporated into the method of constructing symmetry‐adapted linear combinations for a symmetry group based on the correspondence theorem developed
A Simple Alternative to Double Groups
The representations of point groups appropriate to spin-12 particles is usually determined by consideration of a group of double the order of the point group under consideration, known as the double
On Spinors in n Dimensions
The matrix which enters in the charge conjugation transformation of the usual spinors in 4‐space is an invariant matrix and is skew symmetric. It is shown that there exists such an invariant matrix C
A unified theory of the point groups. III. Classification and basis functions of improper point groups
This paper introduces a new system of classification of improper point groups which is most effective for describing their general irreducible representations. The complete set of the general angular
Representation Theory for Nonunitary Groups
The representation theory for nonunitary groups is formulated following the same development used in the case of unitary groups, and the orthogonality relations for the corepresentation matrices are
Theory of Brillouin Zones and Symmetry Properties of Wave Functions in Crystals
It is well known that if the interaction between electrons 1 a metal is neglected, the energy spectrum has a zonal ructure. The problem of these “Brillouin zones” is eated here from the point of view
Symmetry and strain-induced effects in semiconductors