Irreducible Representations Of Point Groups Research Articles

Symmetry and Combinatorial Concepts for Cyclopolyarenes, Nanotubes and 2D-Sheets: Enumerations, Isomers, Structures Spectra & Properties

This review article highlights recent developments in symmetry, combinatorics, topology, entropy, chirality, spectroscopy and thermochemistry pertinent to 2D and 1D nanomaterials such as circumscribed-cyclopolyarenes and their heterocyclic analogs, carbon and heteronanotubes and heteronano wires, as well as tessellations of cyclopolyarenes, for example, kekulenes, septulenes and octulenes. We establish that the generalization of Sheehan’s modification of Pólya’s theorem to all irreducible representations of point groups yields robust generating functions for the enumeration of chiral, achiral, position isomers, NMR, multiple quantum NMR and ESR hyperfine patterns. We also show distance, degree and graph entropy based topological measures combined with techniques for distance degree vector sequences, edge and vertex partitions of nanomaterials yield robust and powerful techniques for thermochemistry, bond energies and spectroscopic computations of these species. We have demonstrated the existence of isentropic tessellations of kekulenes which were further studied using combinatorial, topological and spectral techniques. The combinatorial generating functions obtained not only enumerate the chiral and achiral isomers but also aid in the machine construction of various spectroscopic and ESR hyperfine patterns of the nanomaterials that were considered in this review. Combinatorial and topological tools can become an integral part of robust machine learning techniques for rapid computation of the combinatorial library of isomers and their properties of nanomaterials. Future applications to metal organic frameworks and fullerene polymers are pointed out.

Symmetrization of eigenfunctions of angular momentum in point groups

AbstractThe eigenfunctions |jm〉 of angular momentum can combine linearly to make basis functions of irreducible representations of point groups. We surmount the projection operator and find a new method to calculate the combination coefficients. It is proven that these coefficients are components of eigenvectors of some hermitian matrices, and that for all pure rotation point groups, the coefficients can be made real numbers by properly choosing the azimuth angles of symmetry elements of point groups in the coordinate system. We apply the coupling theory of angular momentum to obtain the general formulas of the basis functions of point groups. By use of our formulas, we have calculated the basis functions with half‐integers j from 1/2 to 13/2 of double‐valued irreducible representations for the icosahedral group. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 286–302, 2001

Modified Stereographic Projections of Point Groups and Diagrams of Their Irreducible Representations

Modified versions of the stereographic projections of the point groups of classical crystallography are presented. They show the consequences of symmetry operations rather than emphasizing the existence of symmetry elements. These projections may be used to give pictures of the irreducible representations of point groups and several examples are given.

On the relationship between crystal quantum numbers and irreducible representations of point groups

A classification of all point groups allows us to derive general character tables. The crystal quantum number description of eigenfunctions ¦JM〉 is related to the symmetry properties, and general relationships between them are given.

Irreducible representations of point groups

The method of obtaining the irreducible representations of crystallographic point groups is discussed. It employs the little-group technique and makes use of the solvability property. Octahedral group O is treated as an illustration.

Rotational displacements in crystals

The problem of random rotational displacement is solved and the theory of Brownian motion of molecules in crystals is developed. The probability distribution of orientations of a molecule is found as a decomposition into matrix elements of irreducible representations of point groups.

The symmetrized powers of group representations

AbstractTables of symmetrized powers of the irreducible representations of point groups are presented together with a derivation of the formula used to obtain them. It is shown how these tables may be applied to various quantum‐chemical problems.