GROUP THEORY GUIDE

The Mathematica Group Theory Package GTPack 1.2

W. Hergert, R. M. Geilhufe

The Mathematica group theory package GTPack contains more than 200 additional modules dedicated to applications in solid state physics and photonics. It can be loaded by Needs["GroupTheory`"].

An introduction into group theory as well as examples using GTPack can be found in the book:  [1] W. Hergert, R. M. Geilhufe, Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica, Wiley-VCH, ISBN: 978-3-527-41133-7 (2018).

A short description of the package including examples and comments about the implementation of the package can be found in: [2] R. M. Geilhufe, W. Hergert, GTPack: A Mathematica group theory package for applications in solid-state physics and photonics, Frontiers in Physics, 6:86, 2018

The development of GTPack is an academic project. To support the development of GTPack we ask you to cite both references [1,2].

We acknowledge the help of Sebastian Schenk who significantly contributed to creating a Mathematica documentation for GTPack.

Version release date June 26, 2020

ReferenceReference

Auxiliary »

Quaternion operations

GTQAbs  ▪  GTQConjugate  ▪  GTQInverse  ▪  ...

Logical operators

GTEulerAnglesQ  ▪  GTQuaternionQ  ▪  GTSymbolQ

Graphic representation of point group relationships

GTPointGroups  ▪  GTGroupConnection  ▪  GTGroupHierarchy

Spherical Harmonics

GTCartesianSphericalHarmonicY  ▪  GTCartesianTesseralHarmonicY  ▪  ...

Input and Output operations

GTReadFromFile  ▪  GTWriteToFile  ▪  ...

Handling of Cluster data

GTClusterFilter  ▪  GTCompactStore  ▪  ...

Special matrices

GTSU2Matrix  ▪  GTDiracMatrix  ▪  ...

Presentation of data

GTBlueRed  ▪  GTSetTableColors  ▪  ...

Basic »

Classes

GTClasses  ▪  GTClassMult  ▪  GTClassMultTable

Characterization of groups and elements

GTGenerators  ▪  GTGetSubGroups  ▪  GTGroupOrder  ▪  GTMultTable  ▪  ...

Logical operators

GTAbelianQ  ▪  GTCyclicQ  ▪  GTGroupQ  ▪  ...

Symbols, matrices, quaternions and Euler angles

GTAllSymbols  ▪  GTGetEulerAngles  ▪  GTGetMatrix  ▪  ...

Coordinate transformation

GTTransformation  ▪  GTTransformationOperator

Crystal Structure »

Load and save and modify structures

GTLoadStructures  ▪  GTSaveStructures  ▪  ...

Plot and manipulate crystsl strucutre

GTClusterManipulate  ▪  GTPlotCluster  ▪  GTPlotStructure  ▪  ...

Point and space group information

GTBravaisLattice  ▪  GTCrystalData  ▪  GTCrystalSystem  ▪  ...

Carbon nanostructures

GTBuckyBall  ▪  GTTubeParameters  ▪  GTTubeStructure

Crystal Field Theory »

Qualitative Crystal Field Theory

GTCrystalFieldSplitting

Quantitative Crystal Field Theory

GTCrystalField  ▪  GTStevensOperator  ▪  GTBSTOperator  ▪  ...

Crystal field Parameters and Database

GTCrystalFieldParameter  ▪  GTStevensTheta  ▪  GTCFDatabaseInfo  ▪  ...

Electronic Structure »

Calculations of Band Structures

GTBandStructure  ▪  GTBands  ▪  GTBandsPlot  ▪  ...

Calculations of Density of States

GTDensityOfStates  ▪  GTDensityOfStatesPlot  ▪  GTFermiSurface  ▪  ...

Install »

Installation of symmetry elements

GTInstallAxis  ▪  GTReinstallAxes  ▪  GTWhichAxes

Installation of point groups

GTGroupFromGenerators  ▪  GTInstallAxis  ▪  GTInstallGroup  ▪  ...

Standard Representation

GTChangeRepresentation  ▪  GTWhichRepresentation

Lattice »

Cluster construction and analysis

GTCluster  ▪  GTShells  ▪  GTAdjacencyMatrix

Lattices in Real Space or Reciprocal Space

GTLatCluster  ▪  GTLatShells

Symmetry Analysis of Real Structure Clusters

GTShellVectorsQlp  ▪  GTGroupGlp  ▪  GTTransformToQlp

Reciprocal Lattice

GTReciprocalBasis

Brillouin Zones

GTBZPointMesh  ▪  GTBZPath  ▪  GTVoronoiCell  ▪  ...

Molecules »

Parameter Sets and Libraries for Molecules

GTMolChemicalData  ▪  GTMolDatabaseInfo  ▪  GTMolToCluster  ▪  ...

Representation Theory for Molecules

GTMolPermutationRep

Photonics »

Structure Factors and Fourier Transforms of inverse Permittivity

GTPhDielectric  ▪  GTPhFields  ▪  GTPhMasterEquation  ▪  ...

Visualization of permittivity distributions

GTPhPixelStructure  ▪  GTPhShowStructure

Master Equation

GTPhMaster  ▪  GTPhMasterObjects  ▪  ...

Symmetry Analysis of fields

GTPhSymmetryBands  ▪  GTPhSymmetryPoint  ▪  GTPhSymmetryField  ▪  ...

Connection to MPB

GTPhMPBBands  ▪  GTPhMPBFields

Pseudopotential »

Parameter Sets and Libraries

GTPwDatabaseInfo  ▪  GTPwDatabaseRetrieve  ▪  ...

Model-Pseudopotential Hamiltonians

GTPwHamiltonian  ▪  GTPwDielectricF  ▪  GTPwModelPotential

Plane Waves and Group Theory

GTPwEmptyLatticeIrep  ▪  GTPwSymmetrizePW

Representation Theory »

Characters

GTCharacterTable  ▪  GTSOCSplitting  ▪  ...

Projection Operators

GTCharProjectionOperator  ▪  GTProjectionOperator  ▪  ...

Auxiliary Representations

GTAngularMomentumChars  ▪  GTAngularMomentumRep  ▪  ...

Irreducible Representations

GTClebschGordanSum  ▪  GTGetIrep  ▪  GTIrep  ▪  ...

Direct product of representations

GTClebschGordanCoefficients  ▪  GTDirectProductChars  ▪  GTDirectProductRep

Representation Theory of Space Groups »

Characters and matrices

GTSGCharacterTable  ▪  GTSGGetIreps

Structure of Space Groups

GTSGLeftCosets  ▪  GTSGRightCosets  ▪  GTSGClasses  ▪  GTSGGetInvSubGroup  ▪  GTSGCosetRepresentative

Symmetry Elements

GTSGgmat  ▪  GTSGOrderOfElement

Simpack

GTTbParmExport  ▪  GTStrucExport  ▪  ...

TightBinding »

Construction of two-center tight-binding Hamiltonians

GTTbHamiltonian  ▪  GTTbHamiltonianElement  ▪  GTTbMatrixElement  ▪  ...

Construction of three-center tight-binding Hamiltonians

GTSymmetryBasisFunctions  ▪  GTTbSymbol3C  ▪  ...

Real Space Formulation

GTTbRealSpaceMatrix  ▪  GTTbHamiltonianRS  ▪  ...

Parameter Sets and Libraries for tight-binding Models

GTTbGetParameter  ▪  GTTbDatabaseInfo  ▪  ...

Wavefunctions

GTTbAtomicWaveFunction  ▪  ...

Symmetry Analysis of Band Structures

GTTbSymmetryBands  ▪  GTTbSymmetryPoint  ▪  ...

GTTbSymmetryBandStructure  ▪  GTTbSymmetryBands  ▪ 

Output of Hamiltonians as FORTRAN Modules

GTTbToFortran  ▪  GTTbToFortranList

Nanotubes

GTTbTubeBands

Wannier90

GTTbReadWannier90  ▪  GTTbWannier90Hamiltonian

Miscellaneous

GTHamiltonianPlot  ▪  GTHamiltonianList  ▪ 

Vibrations »

Vibrational Modes of Molecules

GTVibDisplacementRep  ▪  GTVibModeSymmetry  ▪  GTVibSpectroscopy

Vibrational Modes of Solids

GTVibDynamicalMatrix  ▪  GTVibLatticeModes  ▪  ...

Options »

General Options

GOColorScheme  ▪  GODecimals  ▪  ...

Basic Group Theory

GOClasses  ▪  GOGroupOrder  ▪  ...

Representation Theory

GOlmax  ▪  GONames  ▪  ...

Lattices, Crystals, K-space

GOBonds  ▪  GOBravais  ▪  ...

Band Structure

GOBands  ▪  GOEigenvectors  ▪  ...

Pseudopotential Theory

GOBondCharges

Photonics

GODCMethod  ▪  GOPhPol  ▪  ...

Tight Binding

GOTbBasis  ▪  GOTbEquivalence  ▪  ...

Molecules, Vibrations

GOMolPlot  ▪  GOSpecMode