Representations Of Point Groups Research Articles

Hofstadter Topology with Real Space Invariants and Reentrant Projective Symmetries.

Adding magnetic flux to a band structure breaks Bloch's theorem by realizing a projective representation of the translation group. The resulting Hofstadter spectrum encodes the nonperturbative response of the bands to flux. Depending on their topology, adding flux can enforce a bulk gap closing (a Hofstadter semimetal) or boundary state pumping (a Hofstadter topological insulator). In this Letter, we present a real space classification of these Hofstadter phases. We give topological indices in terms of symmetry-protected real space invariants, which reveal the bulk and boundary responses of fragile topological states to flux. In fact, we find that the flux periodicity in tight-binding models causes the symmetries which are broken by the magnetic field to reenter at strong flux where they form projective point group representations. We completely classify the reentrant projective point groups and find that the Schur multipliers which define them are Arahanov-Bohm phases calculated along the bonds of the crystal. We find that a nontrivial Schur multiplier is enough to predict and protect the Hofstadter response with only zero-flux topology.

Elastic and Inelastic Cross Sections for Low-Energy Electron Collisions with ClF Molecule Using the R-Matrix Method

The ClF molecule belongs to an interhalogen family and is important in laser physics and condensed phase molecular dynamics. The elastic and excitation scattering cross sections are obtained in a fixed nuclei approximation using the UKRmol+ codes based on R-matrix formalism. The scattering calculations were performed in the static-exchange (SE), static-exchange-plus-polarisation (SEP), and close-coupling (CC) models. Three CC models with different target states were employed, namely, the 1-state, 5-states, and 12-states. In the CC model, the target states were represented by configuration interaction (CI) wavefunctions. A good agreement of dipole and quadrupole moments of the ground state was obtained with the experimental values, which indicates a good representation of the target modelling. The study predicted the existence of a shape resonance in the SE, SEP, and 5-states CC models. This resonance vanished in the 12-states CC model. The excitation cross sections from ground to the lowest two excited states were also reported. The elastic differential and momentum transfer cross sections were obtained in the 12-states CC models. The contribution of long-range interactions to elastic scattering was included via Born closure approach. The quantities like collision frequencies and rate coefficients were also presented over a wide range of electron temperatures. The ionization cross sections were computed using the binary-encounter-Bethe (BEB) model. The results were reported in C2v point group representation.

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.

Wave functions for high-symmetry, thin microstrip antennas, and two-dimensional quantum boxes

For a spinless quantum particle in a one-dimensional box or an electromagnetic wave in a one-dimensional cavity, the respective Dirichlet and Neumann boundary conditions both lead to non-degenerate wave functions. However, in two spatial dimensions, the symmetry of the box or microstrip antenna is an important feature that has often been overlooked in the literature. In the high-symmetry cases of a disk, square, or equilateral triangle, the wave functions for each of those two boundary conditions are grouped into two distinct classes, which are one- and two-dimensional representations of the respective point groups, $C_{\infty v}$, $C_{4v}$, and $C_{3v}$. Here we present visualizations of representative wave functions for both boundary conditions and both one- and two-dimensional representations of those point groups. For the one-dimensional representations, color contour plots of the wave functions are presented. For the two-dimensional representations, the infinite degeneracies are presented as common nodal points and/or lines, the patterns of which are invariant under all operations of the respective point group. The wave functions with the Neumann boundary conditions have important consequences for the coherent terahertz emission from the intrinsic Josephson junctions in the high-temperature superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$: the enhancement of the output power from electromagnetic cavity resonances is only strong for wave functions that are not degenerate.

Harmonic fingerprint of unconventional superconductivity in twisted bilayer graphene

Microscopic details such as interactions and Fermiology determine the structure of superconducting pairing beyond the spatial symmetry classification along irreducible point group representations. From the effective pairing vertex, the pairing wave function related to superconducting order unfolds in its orbital-resolved Fourier profile which we call the harmonic fingerprint (HFP). The HFP allows to formulate a concise connection between microsopic parameter changes and their impact on superconductivity. From a random phase approximation analysis of twisted bilayer graphene (TBG) involving $d+id$, $s_{\pm}$, and $f$-wave order, we find that nonlocal interactions, which unavoidably enter the low-energy electronic description of TBG, not only increase the weight of higher lattice harmonics but also have a significant effect on the orbital structure of these pairing states. For gapped unconventional superconducting order such as $s_{\pm}$ and $d+id$, a change in HPF induces enhanced gap anisotropies. Experimental implications to distinguish the different gaps and HPFs are also discussed.

Broken symmetry driven phase transitions from a topological semimetal to a gapped topological phase in SrAgAs

We show the occurence of Dirac, Triple point, Weyl semimetal and topological insulating phase in a single ternary compound using specific symmetry preserving perturbations. Based on {\it first principle} calculations, \textbf{\textit{k.p}} model and symmetry analysis, we show that alloying induced precise symmetry breaking in SrAgAs (space group P6$_3/mmc$) leads to tune various low energy excitonic phases transforming from Dirac to topological insulating via intermediate triple point and Weyl semimetal phase. We also consider the effect of external magnetic field, causing time reversal symmetry (TRS) breaking, and analyze the effect of TRS towards the realization of Weyl state. Importantly, in this material, the Fermi level lies extremely close to the nodal point with no extra Fermi pockets which further, make this compound as an ideal platform for topological study. The multi fold band degeneracies in these topological phases are analyzed based on point group representation theory. Topological insulating phase is further confirmed by calculating Z2 index. Furthermore, the topologically protected surface states and Fermi arcs are investigated in some detail.

Halide Perovskite-Derived Compounds Rb2TeX6 (X = Cl, Br, and I): Electronic Structure of the Ground and First Excited States.

Herein, we report a study of the electronic structure of the ground and first excited states of Rb2TeCl6, Rb2TeBr6, and Rb2TeI6 halide-perovskite-derived crystals. Using X-ray photoelectron spectroscopy (XPS) measurements and density functional theory and multiconfiguration self-consistent field (MCSCF) calculations, the experimental and theoretical XPS spectra of the valence region were obtained. In addition, the effects of the cations and halogen atoms on the electronic structure were determined, and the classification of the excited states in double point group representation was carried out. Furthermore, a possible reason for the luminescence quenching in an isostructural series of crystals containing the [TeI6]2- anions was determined.

Isomorphism and matrix representation of point groups

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.

Development of the Bethe Method for the Construction of Two-Valued Space Group Representations and Two-Valued Projective Representations of Point Groups

A procedure of calculation of two-valued space group representations and two-valued projective representations of point groups is considered. A method of construction of factor systems w2(r2, r1), which reflect the transformations of half-integer spin quantum wave functions and are required in order to find the two-valued irreducible projective representations of the point groups, is presented. This method is based on the introduction of an operation q, firstly used by Bethe, as an additional symmetry element. The pathway of introducing the relations, which permit to make a one-valued algebra of double groups and, particularly, their multiplication tables, is shown by the examples of the 222 (D2) and 32 (D3) groups. The construction of a standard factor-system w′(1)(r2, r1) of the projective class K1 for the group 222 on the base of the discussed relations is presented for the first time. The whole role and the possibilities of Bethe’s method and its modifications for the construction of two-valued representations of the point and space groups are discussed.

Symmetry and representation in a three dimensional space

Schoenflies point groups are presented in terms of spatial partitions and Laue classes based on abstract groups. A much simpler system using only a minimal set of generators for three dimensional groups is then presented in the same form. This simplified treatment allows group operations of a given Laue class to be correlated to a greatly simplified Mulliken-style notation for irreducible representations of that class. Transformation matrix representations of point groups in the simplified style can then be manipulated according to their position in the class or to their sub-groups in very clear procedures. Simple rules can replace the large numbers of tables necessary in the Schoenflies approach. Some applications of the method are described.

Crystal Prediction using the Point Groups: an Application of Group Theory

As a blueprint for designing a crystal, a crystal net should be a geometric object. So a crystal design regime might be based on an algorithm that generates crystal nets using geometry. The "Crystal Turtlebug" is such a program, generating crystal nets as graphs embedded in 3-space. McColm et al (2011) described a "Maple" version; it generated crystal nets of one or two kinds of vertices and one or two kinds of edges. A new "Python" version can generate crystal nets of any number of kinds of vertices and edges. The algorithm employs matrix representations of point groups. The "core" of the program takes a fragment of the prospective crystal (e.g. for a crystal of 3 kinds of atoms and 2 kinds of bonds, a "transversal" of 3 vertices and 2 edges). The core assigns point groups to the vertices and isometries to the edges. The core then repeatedly applies groups to isometries and vice versa to generate a unit cell. The computational problem is finding a transversal that produces a chemically plausible crystal net, as most crystal nets found are implausible. We follow a breadth-first search and enumerate some transversals, screen them, generate crystal nets from the screened transversals, and screen the results. This process can entail generating millions of transversals to obtain a handful of crystal nets for users to view manually. Although such a geometric approach is somewhat different from the more popular directions of contemporary research in crystal prediction, there is precedent in the work of A. F. Wells, A. Le Bail, C. Wilmer et al, and in particular M. Treacy et al (2004). In principle, the Crystal Turtlebug will generate every crystal net up to topological equivalence. In practice, there is an exponential explosion in the number of transversals, so the time spent enumerating crystal nets within fixed parameters explodes exponentially with the number of kinds of atoms and bonds (McColm (2012)).

Hans Bethe (1906–2005) and Ligand Field Theory

Hans Bethe (1906–2005) and Ligand Field Theory

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

Group theoretical methods and applications to molecules and crystals

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 12. Projective representations 13. 230 space groups 14. The representations of space groups 15. The unirreps of the space groups to energy bands and vibrational modes of crystals 16. Time reversal, antiunitary point groups and their core presentations 17. Antiunitary space groups and their core presentations Appendix. Character tables for the crystal point groups References.

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.

Time-reversal symmetry breaking in a model of staggered superconductivity in UPt 3

This paper addresses some experimental consequences of a Ginzburg-Landau model proposed recently for the superconducting phases of UPt 3. The model assumes finite center-of-mass momentum of the Cooper pairs and has been shown to be consistent with the observed H-T-P phase diagram. Within this framework, we analyze in more detail the low-temperature time-reversal symmetry breaking phases, and determine how they are connected to the observed muon spin rotation anomalies. The results are discussed in the light of differences to other phenomenological models proposed for UPt 3 which are based on point-group representations.

The method of symmetrized bosons with applications to vibrations of octahedral molecules

A new technique for constructing representations of point groups, called symmetrized boson representation is introduced. This method is particularly useful when describing vibrations of large molecules and for high overtones. The method is illustrated by applying it to stretching overtones of octahedral molecules, SF6, WF6, and UF6 with group Oh.

Symmetry generalisation of the Euler–Schläfli theorem for multi-shell polyhedra

The point-group representations spanned by the vertices, edge vectors, face circulations and cell centres of a wide variety of multi-shelled polyhedral structures are related in a simple equation that generalises the well known Euler and Schläfli theorems. The new theorem can be expected to have applications in spectroscopic and electronic structure theory of a large class of clusters and coordination polyhedra.

How to obtain easily the induced representations of point groups: the icosahedral point groups

With tables of subduced representations as a starting point and use of the Frobenius reciprocity theorem, a simple method to obtain induced representations is given. Tables are given of the 22 induced representations of 532 (I) and of the 84 induced representations of {\overline 5}{\overline 3}2/m (Ih).

Paramagnetically Limited Critical Field of d-Wave Superconductor

We investigate the paramagnetically limited critical fields Hp of a bulk type II superconductor in the d -wave state which belongs to one-dimensional representation of point group D 4 h . Effects of nonmagnetic impurities on Hp are also considered.