Chemical Applications of Group TheorySome ReadingsChemical Application of Group TheoryF.A.CottonSymmetry through the Eyes of a ChemistI.Hargittai and M. HargittaiThe Most Beautiful Molecule - an Adventure in ChemistryH.Aldersey-WilliamsPerfect SymmetryJ.Baggott
Chemical Applications of Group Theory
The symmetry ofmolecules and solids isa verypowerful tool fondeveloping an understanding ofbonding and physical propertiesUsedtopredictthenatureof molecularorbitalsUsed topredictif electronicand vibration spectroscopictransitions can beobservedWe will cover the following material:Identification/classificationof symmetryelements and symmetry operationsAssignmentofpointgroups》 The point group of a moleculeuniquely and fully describes the molecules symmetryIdentifying polarity and chirality using point groupsIntroduction to what aCharacter Table"isAssigningsymmetrylabelsto“Symmetry adapted linear combinationororbitals"Assigning symmetrylabels to of vibrationmodesDeterminingtheIRandRamanactivityof vibrationalmodes We have learnt the point group theory of molecularsymmetry. We shall learn how to use this theory inour chemical research
• We have learnt the point group theory of molecular symmetry. We shall learn how to use this theory in our chemical research
1. Representation of groupsMatrix representationandreduciblerepresentation1.1Total Representation for C2vIndividuallyblockdiagonalizedmatricesEC2OxzOyz00000000000000000000000Reducedto1Dmatricesirreduciblerepresentation1-11-1x [1] [-1] [ 1] [-1]1-1y [ 1] [-1] [-1][ 1]1-1111z[1] [1] [1] [ 1]1-1Z
1. Representation of groups 1.1 Matrix representation and reducible representation
1.2 Reducing of representationsSuppose that we have a set of n-dimensional matrices, A, B,C, ... , which form a representation of a group. These n-Dmatrices themselves constitute a matrixgroupIf we make the same similarity transformation on each matrix,weobtainanewsetof matricesA'=IAF-l: B'= IBF-1:C'= CT-1This newset of matrices is also a representation of the groupIf A'is a blocked-factored matrix, then it is easy to prove thatB',C'...also areblocked-factored matrices[B,] [4][B,][4],B'=A-[4.] [B,] [4.][B.]A1,A2,A3...arenn2,n3...-Dsubmatriceswithn=n,+nz+n3
1.2 Reducing of representations • Suppose that we have a set of n-dimensional matrices, A, B, C, . , which form a representation of a group. These n-D matrices themselves constitute a matrix group. • If we make the same similarity transformation on each matrix, we obtain a new set of matrices, • This new set of matrices is also a representation of the group. • If A’ is a blocked-factored matrix, then it is easy to prove that B’,C’. also are blocked-factored matrices. A A B B ;C C . 1 1 1 ' ; ' ' ' , ' ,. 4 3 2 1 4 3 2 1 B B B B B A A A A A A1 ,A2 ,A3. are n1 ,n2 ,n3.-D submatrices with n= n1 + n2 + n3 +
Furthermore, it is also provable that the various sets ofsubmatrices[A1,B1,C....], {A2,B2,C2....], {A3,B3,C3...], {A4,B4,C4..-]are in themselves representations of the groupWe then call the set of matrices A,B,C, ... a reduciblerepresentationofthegroupIf it is not possible to find a similarity transformation to reducea representation in the above manner, the representation issaidtobe irreducibleThe irreducible representations of a group is of fundamentalimportance
• Furthermore, it is also provable that the various sets of submatrices {A1 ,B1 ,C1.}, {A2 ,B2 ,C2.}, {A3 ,B3 ,C3.}, {A4 ,B4 ,C4.}, are in themselves representations of the group. • We then call the set of matrices A,B,C, . a reducible representation of the group. • If it is not possible to find a similarity transformation to reduce a representation in the above manner, the representation is said to be irreducible. • The irreducible representations of a group is of fundamental importance
2.（CharacterTables ofPointGroupsExample-pointgroupC2vCharactertableECh=4ov(xz) oy(yz)+1x2, y2, z2ARzBasesBBX RLRVTop line:point groupsymmetryoperationsorder ofgroup,h, =number of symmetry operations
2. Character Tables of Point Groups Bases
2.1Constructionof Character TableTotal Representation for C2vIndividually block diagonalized matricesEC2OyzOxz000000000000000000000000Reducedto 1Dmatricesirreducible representation-1x[ 1] [-1] [ 1] [-1]-111T=1y[ 1] [-1] [-1] [1]-1-11=1111.川z [ 1] [1] [1] [ 1]1-1-1=1Z
2.1 Construction of Character Table
TranslationsMovementsof wholemolecule-representbyvectorsEoperationy(afteroperation)=ye.g-y vectorC2y'=-y (i.e. y'=-1 xy)ov(xz)y'=-yov(yz)y'=yall operationsz vector=Z29Eoperationx'=Xx vectorC2X'=-X一ov(xz)X'=Xov(yz)X'=-X
TranslationsConsidereffectof symmetryoperation onthevectorWrite+1fornochange,-1forreversalEov(xz)ov(yz)C2A1+1+1+1z vector+1B2+17yB1-1x+1ECCoy(xz) oy(yz)Labels A, etc.aresymmetryspecies;A+1+1+1+1theysummarisetheAeffects of symmetryBBoperations onthe+1+1工vectors.These translation vectors constitute a set of bases of C2y group
These translation vectors constitute a set of bases of C2v group
RotationsSimilarlyfor rotations of the moleculesEov(xz)ov(yz)A+1+1z vector-?y+7B,X+1+1RzA2+1+1-11RyB1+1店RxB2+1+1