GROUP THEORY GUIDE

The Mathematica Group Theory Package GTPack, V 1.1

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.

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  ▪  GTOperatorEquivalentT  ▪  GTStevensOperator  ▪  ...

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

Related TutorialsRelated Tutorials

    More tutorials can be found in: 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).