Child,&Phillips:TablesforGroupTheory TablesforGroupTheory ByP.W.ATKINS,
M.S.CHILD,andC.S.G.PHILLIPS Thisprovidestheessentialtables(charactertables,directproducts,descentinsymmetryandsubgroups)requiredforthoseusinggrouptheory,togetherwithgeneralformulae,examples,andotherrelevantinformation. CharacterTables: 1TheGroupsC1,Cs,Ci
3 2TheGroupsCn(n=2,
3,…,8)
4 3TheGroupsDn(n=2,3,4,5,6)
6 4TheGroupsCnv(n=2,3,4,5,6)
7 5TheGroupsCnh(n=2,3,4,5,6)
8 6TheGroupsDnh(n=2,3,4,5,6) 10 7TheGroupsDnd(n=2,3,4,5,6) 12 8TheGroupsSn(n=4,6,8) 14 9TheCubicGroups: 15
T,Td,Th
O,Oh 10TheGroupsI,Ih 17 11TheGroupsC∞vandD∞h 18 12TheFullRotationGroup(SU2andR3) 19 DirectProducts: 1GeneralRules 20 2C2,C3,C6,D3,D6,C2v,C3v,C6v,C2h,C3h,C6h,D3h,D6h,D3d,S6 20 3D2,D2h 20 4C4,D4,C4v,C4h,D4h,D2d,S4 20 5C5,D5,C5v,C5h,D5h,D5d 21 6D4d,S8 21 7T,
O,Th,Oh,Td 21 8D6d 22 9I,Ih 22 10C∞v,D∞h 22 11TheFullRotationGroup(SU2andR3) 23 Theextendedrotationgroups(doublegroups): charactertablesanddirectproducttable 24 Descentinsymmetryandsubgroups 26 NotesandIllustrations: Generalformulae 29 Workedexamples 31 Examplesofbasesforsomerepresentations 35 Illustrativeexamplesofpointgroups: IShapes 37 IIMolecules 39 OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved.
1 Atkins,Child,&Phillips:TablesforGroupTheory CharacterTables Notes:
(1)Schönfliessymbolsaregivenforallpointgroups.Hermann–Mauginsymbolsaregivenforthe32crystaliographicpointgroups.
(2)InthegroupscontainingtheoperationC5thefollowingrelationsareuseful: η+=12(1+512)=1·61803L=−2cos144o η−=12(1−512)=−0·61803L=−2cos72o η+η+=1+η+ η−η−=1+η− η+η−=–
1 η++η−=
1 2cos72o+2cos144o=−
1 OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved.
2 Atkins,Child,&Phillips:TablesforGroupTheory
1.TheGroupsC1,Cs,Ci C1
E
(1)
A 1 Cs=Ch
E (m) A′
1 A″
1 Ci=S2E
(1) Ag
1 Au
1 σh
1 x,y,Rz –
1 z,Rx,Ry x2,y2,z2,xyyz,xz i
1 Rx,Ry,Rz x2,y2,z2, xy,xz,yz –
1 x,y,z OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved.
3 Atkins,Child,&Phillips:TablesforGroupTheory
2.TheGroupsCn(n=2,
3,…,8) C2
E C2
(2)
A 1
1 z,Rz x2,y2,z2,xy
B 1 –
1 x,y,Rx,Ry yz,xz C3EC3C32
(3) ε=exp(2πi/3)
A 11 1z,Rz x2+y2,z2 ⎧1εε*⎫
E ⎨⎩
1 ε* ε ⎬⎭ (x,y)(Rx,Ry) (x2–y2,2xy)(yz,xz) C4 EC4C2C43
(4)
A 1111z,Rz
B 1–11–
1 x2+y2,z2x2–y2,2xy ⎧⎪1i−1−i⎫⎪
E ⎨ ⎪⎩
1 −i −
1 ⎬ i⎪⎭ (x,y)(Rx,Ry) (yz,xz) C5 EC5C52C53C54
A 11111z,Rz ⎧1εε2ε*2ε*⎫ E1 ⎨⎩
1 ε* ε*
2 ε
2 ε ⎬⎭ (x,y)(Rx,Ry) ⎧1ε2ε*εε*2⎫ E2 ⎨⎩
1 ε*
2 ε ε* ε
2 ⎬⎭ ε=exp(2πi/5)x2+y2,z2(yz,xz) (x2–y2,2xy) C6E
(6)A1B1 ⎧1E1⎨ ⎩
1 ⎧1E2⎨ ⎩
1 C6C3 11–11 ε−ε*ε*−ε−ε*−ε−ε−ε* C2C32C65 11
1 –11–
1 −1−εε*⎫ −1−ε* ε ⎬⎭ 1−ε*−ε⎫ 1−ε −ε * ⎬⎭ z,Rz (x,y)(Rz,Ry) ε=exp(2πi/6)x2+y2,z2 (xy,yz)(x2–y2,2xy) OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved.
4 Atkins,Child,&Phillips:TablesforGroupTheory
2.TheGroupsCn(n=2,
3,…,8)(cont..) C7 EC7C72C73C74C75C76 ε=exp(2πi/7)
A 1111111z,Rz x2+y2,z2 ⎧1εε2ε3ε*3ε*2ε*⎫ E1 ⎨⎩
1 ε*ε*2ε*3ε
3 ε
2 ε ⎬⎭ (x,y)(
R,R) (xz,yz) xy ⎧1ε2ε*3ε*εε3ε*2⎫ E2 ⎨⎩
1 ε*
2 ε
3 ε ε* ε*
3 ε
2 ⎬⎭ (x2–y2,2xy) ⎧1ε3ε*ε2ε*2εε*3⎫ E3 ⎨⎩
1 ε*
3 ε ε*
2 ε
2 ε* ε
3 ⎬⎭ C8 EC8C4C2C43C83C85C87 ε=exp(2πi/8)
A 11111111z,Rz x2+y2,z2
B 1–1111–1–1–
1 ⎧1εi−1−i−ε*−εε*⎫(x,y) E1 ⎨⎩
1 ε* −i −
1 i −ε −ε* ε ⎬⎭ (Rx,Ry) (xz,yz) ⎧1i−11−1−ii−i⎫ E2 ⎨⎩
1 −i −11 −
1 i−i ⎬i⎭ (x2–y2,2xy) ⎧1−εi−1−iε*ε−ε*⎫ E3 ⎨⎩
1 −ε* −i −
1 iε ε* −ε ⎬⎭ OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved.
5 Atkins,Child,&Phillips:TablesforGroupTheory
3.TheGroupsDn(n=2,3,4,5,6) D2
E (222)
A 1 B1B
1 B2B
1 B3B
1 C2(z) C2(y) C2(x)
1 1
1 x2,y2,z2
1 –
1 –
1 z,Rz xy –
1 1 –
1 y,Ry xz –
1 –
1 1 x,Rx yz D3(32)A1A2E D4(422)A1A2B1BB2BE
E 2C3 3C2
1 1
1 1
1 –
1 2 –
1 0 z,Rz(x,y)(Rx,,Ry) x2+y2,z2(x2–y2,2xy)(xz,yz) E2C4C2(=C42)2C2'2C2"
1 1
1 1
1 x2+y2,z2
1 1
1 –
1 1 –1–1z,Rz
1 1–
1 x2–y2
1 –
1 1 –
1 1 xy
2 0 –
2 0 0(x,y)(Rx,Ry)(xz,yz) D5
E 2C5 2C52 5C2 A1
1 1
1 1 x2+y2,z2 A2
1 1
1 –
1 z,Rz E1
2 2cos72º2cos144°
0 (x,y)(Rx,Ry)(xz,yz) E2
2 2cos144º2cos72°
0 (x2–y2,2xy) D6(622) A1A2B1BB2BE1E2 E2C62C3C23C2′3C2′′ 11111–11–1212–
1 11111–11–111–1–1–1–20–120
1 x2+y2,z2 –1z,Rz –
1 1 0(x,y)(Rx,Ry)(xz,yz)
0 (x2–y2,2xy) OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved.
6 Atkins,Child,&Phillips:TablesforGroupTheory
4.TheGroupsCnν(n=2,3,4,5,6) C2ν
E (2mm) A1
1 A2
1 B1B
1 B2B
1 C2 σν(xz) σ′(yz) v
1 1
1 z x2,y2,z2
1 –
1 –
1 Rz xy –
1 1 –
1 x,Ry xz –
1 –
1 1 y,Rx yz C3ν(3m) A1A2E
E 2C33σν
1 1
1 1
1 –
1 2 –
1 0 zRz(x,y)(Rx,Ry) x2+y2,z2(x2–y2,2xy)(xz,yz) C4ν(4mm) A1A2B1BB2BE
E 2C4 C2
1 1
1 1
1 1
1 –
1 1
1 –
1 1
2 0 –
2 2σν2σd
1 1 –
1 –
1 1 –
1 –
1 1
0 0 zRz (x,y)(Rx,Ry) x2+y2,z2 x2–y2xy(xz,yz) C5ν E2C5 2C52 5σν A1
1 1
1 1z x2+y2,z2 A2
1 1
1 –1Rz E1 22cos72°2cos144° 0(x,y)(Rx,Ry)(xz,yz) E2 22cos144°2cos72°
0 (x2–y2,2xy) C6ν(6mm) A1A2B1BB2BE1E2 E2C62C3C2
1 1
1 1
1 1
1 1 1–
1 1–
1 1–
1 1–
1 21–1–
2 2–1–
1 2 3σν3σd
1 1z x2+y2,z2 –1–1Rz 1–
1 –
1 1
0 0(x,y)(Rx,Ry)(xz,yz)
0 0 (x2–y2,2xy) OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved.
7 Atkins,Child,&Phillips:TablesforGroupTheory
5.TheGroupsCnh(n=2,3,4,5,6) C2h
E C2
I (2/m) Ag
1 1
1 BgB
1 –
1 1 Au
1 1 –
1 BuB
1 –
1 –
1 σh
1 Rz x2,
y2,z2,xy –
1 Rx,Ryxz,yz –
1 z
1 x,y ()C3h EC3C32σhS3 S35
6 A' 11111 1Rz ⎧⎪1εε*1εε*⎫⎪ E' ⎨ ⎪⎩
1 ε* ε 1ε* ε ⎬⎪⎭ (x,y) A'' 111–1–1–
1 z ⎧⎪1εε*−1−ε−ε*⎫⎪ E'' ⎨ ⎪⎩
1 ε*ε −
1 −ε* −ε ⎬⎪⎭ (Rx,Ry) ε=exp(2πi/3)x2+y2,z2(x2–y2,2xy) (xz,yz) C4h(4/m)AgBgB Eg AuBuB Eu EC4C2C43i S43σhS4 11111 11
1 1–11–11–11–
1 ⎧⎪1i−1−i1i−1−i⎫⎪ ⎨ ⎪⎩
1 −i −
1 i1−i−
1 ⎬ i⎪⎭ 1111–1–1–1–
1 1–11–1–11–11 ⎧⎪1i−1−i−1−i1i⎫⎪ ⎨ ⎪⎩
1 −i −
1 i −
1 i 1−i⎬⎪⎭ Rz(Rx,Ry)z(x,y) x2+y2,z2(x2–y2,2xy) (xz,yz) OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved.
8 Atkins,Child,&Phillips:TablesforGroupTheory
5.TheGroupsCnh(n=2,3,4,5,6)(cont…) C5hEC5C52C53C54σhS5S57S53 S59 ε=exp(2πi/5) A′111111111
1 Rz x2+y2,z2 ⎧1εε2ε*2ε*1εε2ε*2ε*⎫ E′
1 ⎨ ⎩1ε*ε*2ε
2 ε 1ε* ε*
2 ε
2 ε ⎬⎭ (x,y) ⎧1ε2ε*εε*21ε2ε*εε*2⎫ E′
2 ⎨⎩
1 ε*
2 ε ε*ε21ε*2ε ε* ε
2 ⎬⎭ z (x2–y2,2xy) A′′11111–1–1–1–1–
1 ⎧1εε2ε*2ε*−1−ε−ε2−ε*2−ε*⎫ E1′′ ⎨⎩
1 ε* ε*
2 ε
2 ε −1−ε* −ε*
2 −ε
2 −ε ⎬⎭ (Rx,Ry) (xz,yz) ⎧1ε2ε*εε*2−1−ε2−ε*−ε−ε*2⎫ E′2′ ⎨⎩
1 ε*
2 ε ε*ε2−1−ε*2−ε −ε* −ε
2 ⎬⎭ C6h EC6C3C2C32C65i (6/m) S35S65σhS6S3 Ag 111111111111 BgB 1–11–11–11–11–11–1(Rx,Ry) ⎧1ε−ε*−1−εε*1ε−ε*−1−εε*⎫ E1g ⎨⎩
1 ε* −ε −1−ε*ε 1ε*−ε −
1 −ε* ε ⎬⎭ ε=exp(2πi/6) x2+y2,z2(xz,yz) ⎧1−ε*−ε1−ε*−ε1−ε*−ε1−ε*−ε⎫ E2g ⎨⎩
1 −ε −ε*1−ε −ε*1−ε −ε*1−ε −ε * ⎬⎭ (x2–y2,2xy) Au 111111–1–1–1–1–1–
1 Z BuB 1–11–11–1–11–11–11 ⎧1ε−ε*−1−εε*−1−εε*1ε−ε*⎫ E1u ⎨⎩
1 ε* −ε −1−ε*ε −1−ε*ε 1ε* −ε ⎬⎭ (x,y) ⎧1−ε*−ε1−ε*−ε−1ε*ε−1ε*ε⎫ E2u ⎨⎩
1 −ε −ε*1−ε −ε*−1ε ε*−1ε ε * ⎬⎭ OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved.
9 Atkins,Child,&Phillips:TablesforGroupTheory
6.TheGroupsDnh(n=2,3,4,5,6) D2h
E (mmm) Ag
1 BB1g
1 BB2g
1 BB3g
1 Au
1 BB1u
1 BB2u
1 BB3u
1 C2(z)
C2(y)C2(x)i
1 1 11
1 –
1 –
1 1 –
1 1 –
1 1 –
1 –
1 11
1 1 1–
1 1 –
1 –1–
1 –
1 1 –1–
1 –
1 –
1 1–
1 σ(xy) 11–1–1–1–111 σ(xz)σ(yz)
1 1 x2,y2,z2 –
1 –1Rzxy
1 –1Ryxz –
1 1Rxyz –
1 –
1 1 1z –
1 1y
1 –1x D3h
E (6)m2 2C33C2 σh2S33σv A1′
1 1 11
1 1 A′
2 1
1 –11 1–
1 E′ 2–
1 02 –
1 0 A1′′
1 1 1–
1 –1–
1 A′2′
1 1 –1–
1 –
1 1 E′′ 2–
1 0–
2 1
0 Rz(x,y) z(Rx,Ry) x2+y2,z2(x2–y2,2xy) (xy,yz) D4h(4/mmm) A1gA2gBB1gBB2gEgA1uA2uBB1uBB2uEu E2C4C22C2′2C2′′i2S4σh2σv2σd 111
1 111111 x2+y2,z2 111–1–1111–1–1Rz 1–11 1–11–111–
1 x2–y2 1–11–
1 11–11–11 xy 20–
2 0 020–200(Rx,Ry)(xz,yz) 111
1 1–1–1–1–1–
1 111–1–1–1–1–111Z 1–11 1–1–11–1–11 1–11–
1 1–11–11–
1 20–
2 0 0–20200(x,y) OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 10 Atkins,Child,&Phillips:TablesforGroupTheory
6.TheGroupsDnh(n=2,3,4,5,6)(cont…) D5h
E 2C5 2C25 5C2σh 2S5 2S53 5σv A′
1 1
1 1
1 1
1 1
1 x2+y2,z2 A′
2 1
1 1 –11
1 1 –
1 Rz E1′ 22cos72°2cos144°
0 22cos72° 2cos144°0(x,y) E′
2 22cos144°2cos72°
0 22cos144° 2cos72°
0 (x2–y2,2xy) A1′′
1 1
1 1 –
1 –
1 –
1 –
1 A′2′
1 1
1 –1–
1 –
1 –
1 1 z E1′′ 22cos72°2cos144°0–2–2cos72° –2cos144°0(Rx,Ry) (xy,yz) E′2′ 22cos144°2cos72° 0–2–2cos144°–2cos72°
0 D6h
E (6/mmm) A1g
1 A2g
1 BB1g
1 BB2g
1 E1g
2 E2g
2 A1u
1 A2u
1 BB1u
1 BB2u
1 E1u
2 E2u
2 2C6
2C3C23C2′3C2′′ 111111–11–1–11–11–1–2–1–12111111–11–1–11–11–1–2–1–12 11–1–1 1–1–11 000011–1–11–1–110000 i2S32S6σh3σd3σv 1111111111–1–1Rz1–11–11–11–11–1–1121–1–200(Rx–Ry)2–1–1200–1–1–1–1–1–1–1–1–1–111z–11–11–11–11–111–1–2–11200(x,y)–211–200 x2+y2,z2 (xz,yz)(x2–y2,2xy) OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 11 Atkins,Child,&Phillips:TablesforGroupTheory
7.TheGroupsDnd(n=2,3,4,5,6) D2d=Vd
E (42)m 2S4C2 2C2′ 2σd A1
1 1
1 1
1 x2+y2,z2 A2
1 1
1 –
1 –
1 Rz B1B 1–
1 1
1 –
1 x2–y2 B2B 1–
1 1 –
1 1z xy
E 2
0 –
2 0 0(x,y)(xz,yz) (Rx,Ry) D3d(3)mA1gA2gEg A1uA2uEu E2C33C2i 2S63σd
1 1 111
1 1–111 2–
1 02–
1 1
1 1–1–
1 1 1–1–1–
1 2–
1 0–21 1–1Rz 0(Rx,Ry) –11z0(x,y) x2+y2,z2 (x2–y2,2xy)(xz,yz) D4dE 2S8 2C4 2S83 C2 4C2′4σd A1
1 1
1 1
1 11 A2
1 1
1 1
1 –1–1Rz B1B
1 –
1 1 –
1 1 1–
1 B2B
1 –
1 1 –
1 1 –
1 1z E1
2 2 0–
2 –
2 00(x,y) E2
2 0–
2 E3 2–
2 0
0 2
2 –
2 00
0 0(Rx,Ry) x2+y2,z2 (x2–y2,2xy)(xz,yz) OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 12 Atkins,Child,&Phillips:TablesforGroupTheory
7.TheGroupsDnd(n=2,3,4,5,6)(cont..) D5dE 2C5 2C2 5C2i 2S130
5 2S10 5σd A1g1
1 1 11
1 A2g1
1 1 –11
1 E1g22cos72°2cos144°022cos72° 112cos144° 1–1Rz 0(Rx,Ry) E2g22cos144°2cos72° 022cos144°2cos72°
0 A1u1
1 1 1–
1 –
1 –
1 –
1 A2u1
1 1 –1–
1 –
1 –
1 1z E1u22cos72°2cos144°0–2–2cos72°–2cos144° 0(x,y) E2u22cos144°2cos72° 0–2–2cos144°–2cos72°
0 x2+y2,z2 (xy,yz)(x2–y2,2xy) D6d
E A11 A21 B1B
1 B2B
1 E12 E22E32E42E52 2S122C62S42C3
1 1
1 1
1 1
1 1 –
1 1–11 –
1 1–11 310–
1 1 –1–2–
1 0 –20
2 –1–12–
1 –310–
1 2S512 11–1–
1 –3 10–1
3 C26C2′6σd
1 1
1 x2+y2,z2 1–1–1Rz
1 1–
1 1–11z –20 0(x,y)
2 0 –20
2 0 –20
0 (x2–y2,2xy)
0 0 0(Rx,Ry)(xy,yz) OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 13 Atkins,Child,&Phillips:TablesforGroupTheory
8.TheGroupsSn(n=4,6,8) S4
E
(4) S4 C2 S43
A 11
1 1 Rz
B 1–
1 1 –
1 z x2+y2,z2(x2–y2,2xy) ⎧1i−1−i⎫
E ⎨⎩
1 −i −
1 ⎬(x,y)(Rx,Ry)(xz,yz)i⎭ S6E
(3) C3 C32i S65 S6 Ag1
1 1
1 1
1 Rz ⎧1ε ε*
1 ε ε*⎫ Eg⎨⎩
1 ε* ε
1 ε* ε ⎬⎭ (Rx,Ry) Au1
1 1 –1–
1 –
1 z ⎧1ε ε*
1 ε ε*⎫ Eu⎨⎩
1 ε* ε
1 ε* ε ⎬⎭ (x,y) ε=exp(2πi/3)x2+y2,z2(x2–y2,2xy)(xy,yz) S8
E S8 C4 S83 C2 S85 C43 S87 ε=exp(2πi/8)
A 111 11
1 1
1 Rz x2+y2,z2
B 1–11 –11 –
1 1 –
1 z ⎧1ε i−ε*−1−ε −iε*⎫ E1 ⎨⎩
1 ε* −i −ε −1−ε* i ε ⎬⎭ (x,y) ⎧1i−1−i1i−1−i⎫ E2 ⎨⎩
1 −i −
1 i1−i−
1 ⎬i⎭ (x2–y2,2xy) ⎧1−ε* −iε −1ε* i−ε⎫ E3 ⎨⎩
1 −ε iε*−1ε −i −ε * ⎬⎭ (Rx,Ry) (xy,yz) OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 14 Atkins,Child,&Phillips:TablesforGroupTheory
9.TheCubicGroups
T E (23) 4C34C323C2
A 111
1 ⎧1εε*1⎫
E ⎨⎩
1 ε* ε ⎬1⎭
T 3 00–
1 ε=exp(2πi/3) (x,y,z)(Rx,Ry,Rz) x2+y2+z2(3(x2–y2)2z2–x2–y2) (xy,xz,yz) Td(43m) A1A2ET1T2
E 8C3 3C2 6S46σd
1 1
1 1
1 x2+y2+z2
1 1
2 –
1 1 –1–
1 2
0 0 (2z2–x2–y2,3(x2–y2)
3 0 –
1 1–1(Rx,Ry,Rz)
3 0 –
1 –
1 1(x,y,z) (xy,xz,yz) Th
E (m3) Ag
1 ⎧
1 Eg ⎨ ⎩
1 Tg
3 Au
1 ⎧
1 Eu ⎨ ⎩
1 Tu
3 4C34C323C2i 4S64S623σd 11
1 1 111 ε ε*11 ε*ε11 ε ε*1⎫ ε* ε ⎬1⎭ 00–
1 3
0 0–
1 11 1–1–1–1–
1 ε ε* ε*ε 1−1−ε−ε*−1⎫1−1−ε*−ε−1⎬⎭ 00–1–300
1 ε=exp(2πi/3)x2+y2+z2(2z2–x2–y2, 3(x2–y2)(Rx,Ry,Rz)(xy,yz,xz) (x,y,z) O(432)A1A2E T1 T2
E 8C33C26C46C2′
1 1
1 1 2–
1 1
1 1 1–1–
1 2
0 0
3 0–
1 1–
1 3 0–1–
1 1 (x,y,z)(Rx,Ry,Rz) x2+y2+z2(2z2–x2–y2, 3(x2–y2)) (xy,xz,yz) OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 15 Atkins,Child,&Phillips:TablesforGroupTheory
9.TheCubicGroups(cont…) Oh(m3m) E8C36C26C43C2i(=C42) 6S48S63σh6σd A1g
1 1 11 A2g
1 1–1–
1 Eg 2–
1 00 11111111–111–1220–120 x2+y2+z2 (2z2–x2–y2,3(x2–y2)) T1g
3 0–11 –
1 3
1 0–1–1(Rx,Ry,Rz) T2g
3 0 1–
1 –13–10–11 (xy,xz,yz) A1u
1 1 11 1–1–1–1–1–
1 A2u
1 1–1–
1 1–11–1–11 Eu 2–
1 00 2–201–20 T1u
3 0–11 –1–3–1011(x,y,z) T2u
3 0 1–
1 –1–3101–
1 OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 16 Atkins,Child,&Phillips:TablesforGroupTheory 10.TheGroupsI,Ih
I E12C512C2 20C315C2
5 η± =
1 ⎛⎜
1 ±
5 12 ⎞⎟ 2⎝ ⎠
A 1 T1
3 T2
3 1
1 η+ η− η− η+
1 1 x2+y2+z2
0 –1(x,y,z) (Rx,Ry,Rz)
0 –
1 G
4 –1–
1 1
0 H
5 0
0 –
1 1 (2z2–x2–y2, 3(x2–y2) xy,yz,zx) IhE12C512C220C315C2i5 Ag1
1 T1g3η+ T2g3η− Gg4–
1 Hg5
0 1111 η− 0–13 η+ 0–13 –1104 0–115 Au1
1 T1u3η+ T2u3η− Gu4–
1 Hu5
0 111–
1 η− 0–1–
3 η+ 0–1–
3 –110–
4 0–11–
5 12S10 1η−η+–10 –1η−η+10 12S320S615σ10 111η+–1–1η−0–1–1100–11 –1–1–
1 η+ 01 η− 01 1–10 01–
1 (Rx,Ry,Rz)(x,y,z) η± =
1 ⎛⎜
1 ±
5 12 ⎞⎟ 2⎝ ⎠ x2+y2+z2 (2z2–x2–y2, 3(x2–y2)) (xy,yz,zx) OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 17 Atkins,Child,&Phillips:TablesforGroupTheory 11.TheGroupsC∞vandD∞h C∞v A1≡∑+A2≡∑–E1≡ΠE2≡ΔE3≡Φ …… EC2 2C∞φ 11
1 11
1 2–2222–
2 2cosφ2cos2φ2cos3φ … … … … … … …∞σv …
1 z x2+y2,z2 …–
1 Rz … 0(x,y)(Rx,Ry) (xz,yz) …
0 (x2–y2,2xy) …
0 …… …… D∞h
E 2C∞φ … Σ+g1
1 … Σ−g1
1 … ∏g
2 2cosφ… Δg 22cos2φ… … …… … Σu+
1 1 … Σu−
1 1 … ∏u
2 2cosφ… Δu 22cos2φ… … …… … ∞σvi11 –11 0202 ……1–1–1–10–20–2…… 2S∞φ
1 …∞C2 …
1 1 …–1Rz –2cosφ…2cos2φ… 0(Rx,Ry)
0 … … –
1 … –
1 … 2cosφ… –2cos2φ… … … …–1z 1 0(x,y)0… x2+y2,z2 (xz,yz)(x2–y2,2xy) OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 18 Atkins,Child,&Phillips:TablesforGroupTheory 12.TheFullRotationGroup(SU2andR3) ⎧⎪sin ⎛⎜ j +
1 ⎞⎟φ ⎪⎝2⎠ χ(j)(φ)=⎨sin1φ ⎪
2 ⎪ ⎩2j+
1 φ≠0φ=
0 Notation:RepresentationlabelledΓ(j)withj=0,1/2,1,3/2,…∞,forR3jisconfinedtointegralvalues(andwrittenlorL)andthelabelsS≡Γ
(0),P≡Γ
(1),D≡Γ
(2),etc.areused. OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 19 Atkins,Child,&Phillips:TablesforGroupTheory DirectProducts
1.Generalrules (a)ForpointgroupsinthelistsbelowthathaverepresentationsA,
B,E,Twithoutsubscripts,readA1=A2=A,etc. (b) gu g u g u g ′ ″ ′ ′ ″ ″ ′ (c)Squarebrackets[]areusedtoindicatetherepresentationspannedbytheantisymmetrizedproductofadegeneraterepresentationwithitself. Examples ForD3hE′×E′′ A′′
1 +A′′
2 +
E ForD6hE1g×E2g=2Bg+E1g.
2.ForC2,C3,C6,D3,D6,C2v,C3v,C6v,C2h,C3h,C6h,D3h,D6h,D3d,S6 A1 A2 B1B B2B A1 A1 A2 B1B B2B A2 A1 B2B B1B B1B A1 A2 B2B A1 E1 E2 E1
E1E1E2E2A1+[A2]+E2 E2E2E2E1E1B1B+B2+E1A1+[A2]+E2
3.ForD2,D2h
A B1 B2B B3B
A A B1 B2B B3B B1B
A B3 B2B B2B
A B1 B3B
A OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 20 Atkins,Child,&Phillips:TablesforGroupTheory
4.ForC4,D4,C4v,C4h,D4h,D2d,S4 A1 A2 B1B B2B
E A1 A1 A2 B1B B2B
E A2 A1 B2B B1B
E B1B A1 A2
E B2B A1
E E A1
+[A2]+B1+B2
5.ForC5,D5,C5v,C5h,D5h,D5d A1 A2 A1 A1 A2 A2 A1 E1 E2 E1E1E1A1+[A2]+E2 E2E2E2E1+E2A1+[A2]+E1
6.ForD4d,S8 A1 A2 B1B B2B E1 E2 E3 A1 A1 A2 B1B B2B E1 E2 E3 A2 A1 B2B B1B E1 E2 E3 B1B A1 A2 E3 E2 E1 B2B A1 E3 E2 E1 E1 A1
+[A2]+E2 E1+E2 B1B+B2+E2 E2 A1+[A2]+ E1+E3 B1B+B2 E3 A1+[A2]+E2
7.ForT,
O,Th,Oh,Td A1 A2 A1 A1 A2 A2 A1
E T1 T2 EEEA1+[A2]+
E T1T1T2T1+T2A1+E+[T1]+T2 T2T2T1T1+T2A2+E+T1+T2A1+E+[T1]+T2 OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 21
8.ForD6d A1A1A1A2B1BB2BE1 E2 E3 E4 E5 Atkins,Child,&Phillips:TablesforGroupTheory A2 B1B B2B E1 E2 E3 E4 E5 A2 B1B B2B E1 E2 E3 E4 E5 A1 B2B B1B E1 E2 E3 E4 E5 A1
A2 E5 E4 E3 E2 E1 A1 E5 E4 E3 E2 E1 A1+[A2]+E2 E1+E3 E2+E4 E3+E5 B1B+B2+E4 A1+[A2]+E4 E1+E5 B1B+B2+E2 E3+E5 A1+[A2]+B1B+B2 E1+E5 E2+E4 A1+[A2]+E4 E1+E3 A1+[A2]+E2
9.ForI,Ih
A A
A T1 T2G
H T1 T1A+[T1]+
H T2T2G+
H GGT2+G+
H A+[T2]+
H T1+G+
H A+[T1+T2]+G+
H HHT1+T2+G+
H T1+T2+G+HT1+T2+G+2H A1+[T1+T2+G]+G+2H 10.ForC∝v,D∝h Σ+ Σ+ Σ+ Σ– Π Δ: Σ– Π Δ Σ– Π Δ Σ+ Π Δ Σ++[Σ–] Π+Φ +Δ Σ++[Σ–]+Γ Notation Σ Π Δ Φ Γ … Λ=
0 1
2 3
4 … Λ1×Λ2=|Λ1–Λ2|+(Λ1+Λ2)Λ×Λ=Σ++[Σ–]+(2Λ). OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 22 Atkins,Child,&Phillips:TablesforGroupTheory 11.TheFullRotationGroup(SU2andR3) Γ(j)×Γ(j′)=Γ(j+j′)+Γ(j+j′–1)+…+Γ(|j–j′|)Γ(j)×Γ(j)=Γ(2j)+Γ(2j–2)+…+Γ
(0)+[Γ(2j–1)+…+Γ
(1)] OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 23 Atkins,Child,&Phillips:TablesforGroupTheory Extendedrotationgroups(doublegroups): Charactertablesanddirectproducttables D2*
E E1/2
2 R 2C2(z) 2C2(y) 2C2(x) –
2 0
0 0 D*
3 E
R 2C3 2C3R 3C2 3C2R E1/2
2 –
2 1 –
1 E3/2 ⎧
1 −
1 −
1 1 ⎨⎩
1 −
1 −
1 1
0 0 i −i⎫ −i ⎬i⎭ D4
E R 2C4 2C4R 2C2 4C′ 4C′′
2 2 E1/2
2 –
2 2 –
2 0
0 0 E3/2
2 –
2 –
2 2
0 0
0 D
*
6 E
R 2C62C6R 2C32C3R2C26C′ 6C′′
2 2 E1/2
2 –
2 3 –
3 1 –
1 0
0 0 E3/2
2 –
2 –
3 3 –
1 1
0 0
0 E5/2
2 –
2 0
0 –
2 2
0 0
0 T
*d ER8C38C3R6C26S46S4R12σd O*
E R 8C38C3R 6C2 6C4 6S4R 12C′
2 E1/2
2 –
2 1 –
1 0
2 –
2 0 E5/2
2 –
2 1 –
1 0 –
2 2
0 G3/2
4 –
4 –
1 1
0 0
0 0 E1/2
×E1/2=[A]+B1+B2+B3 E1/2 E3/2 E1/2 [A1]+A2+E2E E3/2 [A1]+A1+2A2 E1/2 E3/2 E1/2 [A1]+A2+EB1B+B2+
E E3/2 [A1]+A2+
E OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 24 Atkins,Child,&Phillips:TablesforGroupTheory E1/2 E1/2 [A1]+A2+E1 E3/2 E5/2 E3/2B1B+B2+E2[A1]+A2+E1 E5/2E1+E2E1+E2[A1]+A2+B1+B2 E1/2 E1/2 [A1]+T1 E5/2 G3/2 E5/2A2+T2[A1]+T1 E3/2E+T1+T2E+T1+T2[A1+E+T2]+A2+2T1+T2] Direct products of ordinary and extended representations for T* d and O* E1/2E5/2G3/2 A1E1/2E5/2G3/2 A2E5/2E1/2G3/2 EG3/2G3/2E1/2+E5/2+G3/2 T1E1/2+G3/2E5/2+G3/2E1/2+E5/2+2G3/2 T2E5/2+G3/2E1/2+G3/2E1/2+E5/2+2G3/2 OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 25 Atkins,Child,&Phillips:TablesforGroupTheory Descentinsymmetryandsubgroups Thefollowingtablesshowthecorrelationbetweentheirreduciblerepresentationsofagroupandthoseofsomeofitssubgroups.Inanumberofcasesmorethanonecorrelationexistsbetweengroups.InCstheσoftheheadingindicateswhichoftheplanesintheparentgroupesthesoleplaneofCs;inC2vitesmustbesetbyaconvention);wheretherearevariouspossibilitiesforthecorrelationofC2axesandσplanesinD4handD6hwiththeirsubgroups,thecolumnisheadedbythesymmetryoperationoftheparentgroupthatispreservedinthedescent. C2v C2 Cs Cs σ(zx) σ(yz) A1
A A′ A′ A2
A A" A" B1B
B A′ A′ B2B
B A" A" C3v C3 Cs A1
A A′ A2
A A"
E E A′+
A" C4v A1A2B1BB2BE [Othersubgroups:C4,C2,C6] C2vσvA1A2A1A2B1+B2 C2vσdA1A2A2A1B1+B2 D3h C3h C3v A′ A′ A1
1 A′ A′ A2
2 E' E'
E A′′ A" A2
1 A′′ A" A1
2 E" E"
E [Other
subgroups:D3,C3,C2] C2νσh→σνA1 B2B A1+B2A2 B1B A2+B1 CsσhA′A′ 2A'A" A" 2A" CsσνA′ A" A'+A"A"A′ A'+A" OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 26 Atkins,Child,&Phillips:TablesforGroupTheory D4h D2d C2′(→C2′) A1gA1 A2gA2 BB1g B1B BB2g B2B EgE A1uB1B A2uB2B BB1uA1 BB2uA2 EuE D2d C2′′ ( →
C ′
2 ) A1A2B2BB1BEB1BB2BA2A1E D2hC′
2 AgBB1gAgBB1gBB2g+B3gAuBB1uAuBB1uBB2u+B3u D2hC′′
2 AgBB1gBB1gAgBB2g+B3gAuBB1uBB1uAuBB2u+B3u Othersubgroups:D4,C4,S4,3C2h,3Cs,3C2,Ci,(2C2v) D2C′
2 AB1BAB1BB2B+B3AB1BAB1BB2B+B3 D2 C4hC4vC2v C2v C′′ C2,σv C2,σd
2 A AgA1A1 A1 B1B AgA2A2 A2 B1 BgBB1BA1 A2
A BgB2BA2 A1 B2B+B3EgEB1B+B2B1B+B2
A AuA2A2 A2 B1B AuA1A1 A1 B1 BuBB2BA2 A1
A BuB1BA1 A2 B2B+B3EuEB1B+B2B1B+B2 D6DC′′DC′ 3d2 3d2 A1gA1g A1g A2gA2g A2g BB1gA2g A1g BB2gA1g A2g E1gEg Eg E2gEg Eg A1uA1u A1g A2uA2u A2g BB1uA2u A1u BB2uA1u A2u E1uEu Eu E2uEu Eu D2h C6vC3vC2v C2v C2hC2h σh→σ(xy)σv→σ(yz) σvC′ C′′ C2C′
2 2
2 Ag A1A1A1 A1 AgAg BB1g A2A2B1B B1B Ag BgB BB2g B2B A2 A2 B2B BgB Ag BB3g B1B A1 B2B A2 BgB BgB BB2g+B3g E1E A2+B2A2+B22BgAg+Bg Ag+B1g E2E A1+B1A1+B12AgAg+Bg Au A2A2A2 A2 AuAu BB1u A1A1B2B B2B Au BuB BB2u B1B A1 B1B B1B BuB Au BB3u B2B A2 A1 A1 BuB BuB BB2u+B3u E1E A1+B1A1+B12BuAu+Bu Au+B1u E2E A2+B2A2+B22AuAu+Bu Othersubgroups:D6,2D3h,C6h,C6,C3h,2D3,S6,D2,C3,3C2,3Cg,Ci C2h C′′
2 AgBgBBgBAgAg+BgAg+BgAuBuBBuBAuAu+BuAu+Bu Td
T A1
A A2
A E
E T1
T T2
T D2dA1B1A1+B1A2+EB2B+
E Othersubgroups:S4,D2,C3,C2,Cs. C3vA1A2EA2+EA1+
E C2vA1A2A1+A2A2+B1+B2A1+B2+B1 OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 27 Atkins,Child,&Phillips:TablesforGroupTheory Oh
O Td Th D4h D3d A1g A1 A1 Ag A1g A1g A2g A2 A2 Ag BB1g A2g Eg
E E Eg A1g
+B1g Eg T1g T1 T1 Tg A2g+Eg A2g+Eg T2g T2 T2 Tg BB2g+Eg A1g+Eg A1u A1 A2 Au A1u A1u A2u A2 A1 Au BB1u BB1u Eu
E E Eu A1u+B1u Eu T1u T1 T2 Tu A2u+Eu A2u+Eu T2u T2 T1 Tu BB2u+Eu A1u+Eu Othersubgroups:
T,D4,D2d,C4h,C4v,2D2h,D3,C3v,S6,C4,S4,3C2v,2D2,2C2h,C3,2C2,S2,Cs R3
O D4 D3
S A1 A1 A1
P T1 A2+
E A2+
E D E+T2 A1+B1+B2+
E A1+2E
F A2+T1+T2 A2+B1+B2+2E A1+2A2+2E
G A1+E+T1+T2 2A1+A2+B1+B2+2E 2A1+A2+3E
H E+2T1+T2 A1+2A2+B1+B2+3E A1+2A2+4E OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 28 Atkins,Child,&Phillips:TablesforGroupTheory NotesandIllustrations GeneralFormulae (a)Notation h Γ(i) Χ(i)(R) D(i)μν (
R ) li theorder(thenumberofelements)ofthegroup. labelstheirreduciblerepresentation.thecharacteroftheoperationRinΓ(i). theμvelementoftherepresentativematrixoftheoperationRintheirreduciblerepresentationΓ(i). thedimensionofΓ(i).(thenumberofrowsorcolumnsinthematricesD(i)) (b)Formulae(i)Numberofirreduciblerepresentationsofagroup=numberofclasses. (ii)∑li2=hi ∑(iii) χ(i)(R)= D(i)μμ (
R ) μ (iv)Orthogonalityofrepresentations: ∑ D(i)μν (R)* D(i')μ'ν' ( R) = (h / li )δ δii' δμμ'νν' (δij=1ifi=jandδij=0ifi≠j (v)Orthogonalityofcharacters: ∑χ(i)(R)*χ(i)(R)=hδii'
R (vi)positionofadirectproduct,reductionofarepresentation:If Γ=∑aiΓ(i) i andthecharacteroftheoperationRinthereduciblerepresentationisχ(R),thenthecoefficientsataregivenby ai=(l/h)∑χ(i)(R)*χ(R).R OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 29 Atkins,Child,&Phillips:TablesforGroupTheory (vii)Projectionoperators:Theprojectionoperator P(i)=(li/h)∑χ(i)(R)*RR whenappliedtoafunctionf,generatesasumoffunctionsthatconstituteponentofabasisfortherepresentationΓ(i);inordertogeneratepletebasisP(i)mustbeappliedtolidistinctfunctionsf.Theresultingfunctionsmaybemademutuallyorthogonal.Whenli=1thefunctiongeneratedisabasisforΓ(i)withoutambiguity: P(i)f=f(i) (viii)Selectionrules:Iff(i)isamemberofthebasissetfortheirreduciblerepresentationΓ(i),f{k)amemberofthatforΓ(k),andΩˆ(j)anoperatorthatisabasisforΓ(j),thentheintegral ∫dτf(i)*Ωˆ(j)f(k) iszerounlessΓ(i)ursinthepositionofthedirectproductΓ(j)×Γ(k) s (ix)ThesymmetrizeddirectproductiswrittenΓ(i)×Γ(i),anditscharactersaregivenby s χ (i) (R)× χ (i) (R) = 12 χ (i) (R)
2 + 12 χ (i) (R2) a TheantisymmetrizeddirectproductiswrittenΓ(i)×Γ(i)anditscharactersaregivenby a χ (i) (R)× χ (i) (R) = 12 χ (i) (R)
2 + 12 χ (i) (R2) OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 30 Atkins,Child,&Phillips:TablesforGroupTheory Workedexamples
1.ToshowthattherepresentationΓbasedonthehydrogen1s-orbitalsinNH3(C3v)containsA1andE,andtogenerateappropriatesymmetrybinations. Atableinwhichsymmetryelementsinthesameclassaredistinguishedwillbeemployed: C3vA1A2ED(R) x(R)Rh1Rh2 E 112⎛100⎞⎜⎜010⎟⎟⎜⎝001⎟⎠3h1h2 C3+11 –1⎛001⎞⎜⎜100⎟⎟⎜⎝010⎟⎠ 0h2h3 C3–11 –1⎛010⎞⎜⎜001⎟⎟⎜⎝100⎟⎠ 0h3h1 σ1 1–1 0⎛100⎞⎜⎜001⎟⎟⎜⎝010⎟⎠ 1h1h3 σ2 1–1 0⎛001⎞⎜⎜010⎟⎟⎜⎝100⎟⎠ 1h3h2 σ3 1–1 0⎛010⎞⎜⎜100⎟⎟⎜⎝001⎟⎠ 1h2h1 TherepresentativematricesarederivedfromtheeffectoftheoperationRonthebasis(h1,h2, h3);seethefigurebelow.Forexample ⎛001⎞ ⎜ ⎟ C3+(h1,h2,h3)=(h2,h3,h1)=(h1,h2,h3)⎜100⎟ ⎜⎜⎝010⎟⎟⎠ ordingtothegeneralformula(b)(iii)thecharacterχ(R)isthesumofthediagonalelements ofD(R).Forexample,χ(σ2)=0+1+0=
1.ThepositionofΓfollowsfromtheformula (b)(vi): Γ=a1A1+a2A2+aEE where a1 = 16 {1×
3 + 2×1×
0 + 3×1×1} =
1 a2 = 16 {1×3+ 2×1×
0 + 3×1×(–1)} =
0 aE=16{2×3+2×(–1)×0+3×0×1}=
1 OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 31 Atkins,Child,&Phillips:TablesforGroupTheory Therefore Γ=A1+
E Symmetrybinationsaregeneratedbytheprojectionoperatorin(b)(vii).Usingthelasttworowsofthetable, φ(A1)=℘(A1)h1= 16 (1× h1 + 1× h2 + 1× h3 + 1× h1 +1×h3+1×h2)= 13 (h1 + h2+ h3) ⎧φ(E)=℘(E)h1= 26 (
2 × h1 –1× h2 –1× h3 + 0× h1 ⎪ ⎪ +0×h3+0×h2)= 13 (2h1 – h2 – h3) ⎪ ⎨ ⎪⎪ φ ′(
E ) = ℘ (E)h2 = 26 (
2 × h2 –1× h3 –1× h1 + 0× h3 ⎪⎩ +0×h2+0×h1)= 13 (– h1 + 2h2 – h3) φ(E)andφ'(E)provideavalidbasisfortheErepresentation,butthebinations
1 1 φa(E)=(1/6)2(2h1–h2–h3)=(3/2)2φ(E)
1 1 φb(E)=(1/2)2(h2–h3)=(1/2)2{φ(E)+2φ'(E)} wouldbeamoreusefulbasisinmostapplications.
2.Todeterminethesymmetriesofthestatesarisingfromtheelectronicconfigurationse2ande1t21foraplex(Td),andtodeterminethegrouptheoreticalselectionrulesforelectricdipoletransitionsbetweenthem. ThespatialsymmetriesoftherequiredstatesaregivenbythedirectproductsinTable7. E×E=A1+[A2]+
E E×T2=T1+T2 Combinationoftheelectronspinsyieldsbothsingletandtripletstates,butforthee2 configurationsomepossibilitiesareexcluded.Sincethetotal(spinandorbital)statemustbe antisymmetricunderelectroninterchange,theantisymmetrizedbination[A2]mustbeatriplet,andthebinationsA1andEaresinglets.Forthee1t21configurationtherearenoexclusions.Therequiredtermsaretherefore e2→1A1+3A2+1E e1t21→1T1+1T2+3T1+3T2 Theselectionrulesareobtainedfromformula(b)(viii).ForelectricdipoletransitionstheoperatorΩ(j)hasthesymmetryofavector(x,y,z),whichfromthecharactertableforTdtransformsasT2.Fromthetableofdirectproducts,Table7, A1×T2=T2 A2×T1=T2 E×T2=E×T1=T1+T2 AssumingthespinselectionruleΔS=0,theallowedtransitionsare e21A1↔e1t211T2 e23A2↔e1t213T1 e21E↔e1t211T1,1T2 OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 32 Atkins,Child,&Phillips:TablesforGroupTheory
3.TodeterminethesymmetriesofthevibrationsofatetrahedralmoleculeAB4,andtopredicttheappearanceofitsinfraredandRamanspectra. ThemoleculeisdepictedinthefigurebelowandthecharactertableforthepointgroupTdisgivenonpage15. Therepresentationsspannedbythevibrationalcoordinatesarebasedonthe5×3cartesian displacementslesstherepresentationsT1andT2,whichareountedforbytherotations(Rx,Ry, Rz)andthetranslations(x,y,z).Thestretchingvibrationsarethesubsetbasedonthe4bondsof themolecule.Foraparticularsymmetryoperation,onlyatoms(orbonds)thatremaininvariantcancontributetothecharacterofthecartesiandisplacementrepresentation,Γ(all)(orthestretchingrepresentation,Γ(stretch)). C3:Twoatomsinvariant,x,y,z,interchangedOnebondinvariant χ(all)(C3)=0χ(stretch)(C3)=
1 C2(z):Centralatominvariant;x,y,signreversed,zinvariantχ(all)(C3)=
0 Nobondsinvariant χ(stretch)(C2)=
0 S4(z):Centralatominvariant;x,y,interchanged,zsignreversedx(all)(S4)=–
1 Nobondsinvariant χ(stretch)(S4)=
0 σd(z):Threeatomsinvariant;x,y,interchanged,zinvariant x(all)(σd)=
3 Twobondsinvariant χ(stretch)(σd)=
2 ThecharactersoftherepresentationsΓ(all)andΓ(stretch)aretherefore Γ(all)Γ(stretch)
E 8C3 3C2 6S4 6σd 15
0 –
1 –
1 3 =A1+E+T1+3T2
4 1
0 0
2 =A1+T2 Γ(alI)andΓ(stretch)havebeenposedwiththehelpofformula(b)(vi)pareExample1).AllowingfortherotationsandtranslationscontainedinΓ(all)therearethereforefourfundamental vibrations,conventionallylabelledν1(A1),ν2(E),ν3(T2),andν4(T2).ν1andv2arestretchingandbendingvibrationsrespectively,ν3andν4involvebothstretchingandbendingmotions. OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 33 Atkins,Child,&Phillips:TablesforGroupTheory Theselectionrule(b)(viii)givesthespectroscopicpropertiesofthevibrations.Infraredactivityisinducedbythedipolemoment(avectorwithsymmetryT2,ordingtothecharactertableforTd)astheoperatorΩˆ(j)InthecaseoftheRamaneffect,Ωˆ(j)isponentofthepolarizabilitytensor(A1+E+T2).f(i)isthegroundvibrationalstate(A1),andf(k)istheexcitedstate(withthesamesymmetryasthevibrationinthecaseofthefundamental;asthedirectproductoftheappropriaterepresentationsinthecaseofanovertoneorbinationband).v1(A1)andv2(E)arethereforeRamanactiveandν3(T2)andν4(T2)areinfraredandRamanactive.Thefollowingovertonebinationbandsareallowedintheinfraredspectrum: ν1+ν
3,ν1+ν
4,ν2+ν
3,ν2+ν4,2ν
3,ν3+ν4,2ν
4 OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 34 Atkins,Child,&Phillips:TablesforGroupTheory Examplesofbasesforsomerepresentations Thecustomarybases—polarvector(e.g.translationx),axialvector(e.g.rotationRx),andtensor(e.g.xy)—aregiveninthecharactertables. Itmaybeofsomeassistancetoconsiderothertypesofbasesandafewexamplesaregivenhere. Base1 IrreducibleRepresentation A2inTd 2x(1)y
(2)–x(2)y(1)3Thenormalvibrationofanoctahedralmolecule representedby A2inC4v AlginOh Thethreeequivalentnormalvibrationsofan4octahedralmolecule,oneofwhichisrepresentedby T2uinOh OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 35 Atkins,Child,&Phillips:TablesforGroupTheory5Theπ-orbitalofthebenzenemoleculerepresentedby A2uinD6h6Theπ-orbitalofthebenzenemoleculerepresentedby BB2ginD6h Theπ-orbitalofthenaphthalenemolecule7representedby AuinD2h OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 36 Atkins,Child,&Phillips:TablesforGroupTheory IllustrativeExamplesofPointGroupsIShapes Thecharactertablesfor(a),Cn,areonpage4;for(b),Dn,onpage6;for(c),Cnv,onpage7;for(d),Cnh,onpage8;for(e),Dnh,onpage10;for(f),Dnd,onpage12;andfor(g),S2n,onpage14. OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 37 Atkins,Child,&Phillips:TablesforGroupTheory Cs Ci . Td Oh tetrahedronOh cubeIh octahedronIh dodecahedronR3 icosahedron sphere ThecharactertableforCsisonpage3,forCionpage3,forTdonpage15,forOhonpage16,forIhonpage17,andforR3onpage19. OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 38 Atkins,Child,&Phillips:TablesforGroupTheory IIMolecules Pointgroup Example C1 CHFClBr Cs BFClBr(planar),quinoline Ci meso-tartaricacid C2 H2O2,S2C12(skew) C2v H2O,HCHO,C6H5C1 C3v NH3(pyramidal),POC13 C4v SF5Cl,XeOF4 C2h trans-dichloroethylene C3h
H O
H BO
O Pagenumberforcharacter table333477788
H (inplanarconfiguration) D2h trans-PtX2Y2,C2H4 10 D3h BF3(planar),PC15(trigonalbipyramid),1:3:5–trichlorobenzene 10 D4h AuCl4–(squareplane) 10 D5h ruthenocene(pentagonalprism),IF7(pentagonalbipyramid) 11 D6h benzene 11 D2d CH2=C=CH2 12 D4d S8(puckeredring) 12 D5d ferrocene(pentagonalantiprism) 13 S4 tetraphenylmethane 14 Td CCl4 15 Oh SF6,FeF63– 16 IhBB12H122–17 C∞v HCN,COS 18 D∞h CO2,C2H2 18 R3 anyatom(sphere) 19 OXFORDHigherEducation ©OxfordUniversityPress,2006.Allrightsreserved. 39