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# Tables for Group Theory

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Material para uso nas disciplinas de Simetria e Estrutura em Química Inorgânica e Fundamentos em Espectroscopia.

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### Tables for Group Theory

1. 1. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory Tables for Group Theory By P. W. ATKINS, M. S. CHILD, and C. S. G. PHILLIPSThis provides the essential tables (character tables, direct products, descent in symmetry andsubgroups) required for those using group theory, together with general formulae, examples,and other relevant information.Character Tables:1 The Groups C1, Cs, Ci 32 The Groups Cn (n = 2, 3, …, 8) 43 The Groups Dn (n = 2, 3, 4, 5, 6) 64 The Groups Cnv (n = 2, 3, 4, 5, 6) 75 The Groups Cnh (n = 2, 3, 4, 5, 6) 86 The Groups Dnh (n = 2, 3, 4, 5, 6) 107 The Groups Dnd (n = 2, 3, 4, 5, 6) 128 The Groups Sn (n = 4, 6, 8) 149 The Cubic Groups: 15 T, Td, Th O, Oh10 The Groups I, Ih 1711 The Groups C∞ v and D∞ h 1812 The Full Rotation Group (SU2 and R3) 19Direct Products:1 General Rules 202 C2, C3, C6, D3, D6, C2v, C3v, C6v, C2h, C3h, C6h, D3h, D6h, D3d, S6 203 D2, D2h 204 C4, D4, C4v, C4h, D4h, D2d, S4 205 C5, D5, C5v, C5h, D5h, D5d 216 D4d, S8 217 T, O, Th, Oh, Td 218 D6d 229 I, Ih 2210 C∞v, D∞h 2211 The Full Rotation Group (SU2 and R3) 23The extended rotation groups (double groups): character tables and direct product table 24Descent in symmetry and subgroups 26Notes and Illustrations:General formulae 29Worked examples 31Examples of bases for some representations 35Illustrative examples of point groups: I Shapes 37 II Molecules 39 OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 1
2. 2. Atkins & de Paula, Physical Chemistry 9e: Tables for Group TheoryCharacter TablesNotes:(1) Schönflies symbols are given for all point groups. Hermann–Maugin symbols are given forthe 32 crystaliographic point groups.(2) In the groups containing the operation C5 the following relations are useful: η + = 1 (1 + 5 2 ) = 1·61803L = −2 cos144o 1 2 η − = 1 (1 − 5 2 ) = −0·61803L = −2 cos 72o 1 2 η +η + = 1 + η + η −η − = 1 + η − η +η − = –1 η + +η − = 1 2 cos 72o + 2 cos144o = −1 OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 2
3. 3. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory1. The Groups C1, Cs, CiC1 E(1)A 1 Cs=Ch E σh (m) A′ 1 1 x, y, Rz x2, y2, z2, xy A″ 1 –1 z, Rx, Ry yz, xz Ci = S2 E i (1) Ag 1 1 Rx, Ry, Rz x2, y2, z2, xy, xz, yz Au 1 –1 x, y, z OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 3
4. 4. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory2. The Groups Cn (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 C3 E C3 C32 ε = exp (2πi/3) (3) A 1 1 1 z, Rz x2 + y2, z2 ⎧1 ε ε*⎫ E ⎨ ⎬ (x, y)(Rx, Ry) (x2 – y2, 2xy)(yz, xz) ⎩1 ε* ε⎭ C4 E C4 C2 3 C4 (4) A 1 1 1 1 z, Rz x2 + y2, z2 B 1 –1 1 –1 x2 – y2, 2xy ⎧1 ⎪ i −1 −i⎫ ⎪ E ⎨ ⎬ (x, y)(Rx, Ry) (yz, xz) ⎪1 ⎩ −i −1 i⎪ ⎭ C5 E C5 C52 C53 C54 ε = exp(2πi/5) A 1 1 1 1 1 z, Rz x2 + y2, z2 ⎧1 ε ε2 ε *2 ε*⎫ E1 ⎨ ⎬ (x,y)(Rx, Ry) (yz, xz) ⎩1 ε* ε *2 ε2 ε⎭ ⎧1 ε2 ε* ε ε *2 ⎫ E2 ⎨ ⎬ (x2 – y2, 2xy) ⎩1 ε *2 ε ε* ε2 ⎭ C6 E C6 C3 C2 C32 5 C6 ε = exp(2πi/6) (6) A 1 1 1 1 1 1 z, Rz x2 + y2, z2 B 1 –1 1 –1 1 –1 ⎧1 ε −ε * −1 −ε ε*⎫ (x, y) E1 ⎨ ⎬ (xy, yz) ⎩1 ε* −ε −1 −ε * ε⎭ (Rz, Ry) ⎧1 −ε * −ε 1 −ε * −ε ⎫ E2 ⎨ ⎬ (x2 – y2, 2xy) ⎩1 −ε −ε * 1 −ε −ε * ⎭ OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 4
5. 5. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory2. The Groups Cn (n = 2, 3,…,8) (cont..) C7 E C7 C72 C7 3 C74 C7 5 C7 6 ε = exp (2πi/7) A 1 1 1 1 1 1 1 z, Rz x2 + y2, z2 ⎧1 ε ε2 ε3 ε *3 ε *2 ε*⎫ ⎨ ⎬ (x, y) E1 ⎩1 ε* ε *2 ε *3 ε3 ε2 ε ⎭ (Rx, Ry) (xz, yz) ⎧1 ε2 ε *3 ε* ε ε3 ε *2 ⎫ ⎨ ⎬ (x2 – y2, 2xy) E2 ⎩1 ε *2 ε3 ε ε* ε *3 ε2 ⎭ ⎧1 ε3 ε* ε2 ε *2 ε ε *3 ⎫ ⎨ ⎬ E3 ⎩1 ε *3 ε ε *2 ε2 ε* ε3 ⎭ C8 E C8 C4 C2 C4 3 C83 C85 C87 ε = exp (2πi/8) A 1 1 1 1 1 1 1 1 z, Rz x2 + y2, z2 B 1 –1 1 1 1 –1 –1 –1 ⎧1 ε i −1 −i −ε * −ε ε*⎫ (x, y) E1 ⎨ ⎬ (xz, yz) ⎩1 ε* −i −1 i −ε −ε * ε ⎭ (Rx, Ry) ⎧1 i −1 1 −1 −i i −i ⎫ E2 ⎨ ⎬ (x2 – y2, 2xy) ⎩1 −i −1 1 −1 i −i i⎭ ⎧1 −ε i −1 −i ε* ε −ε * ⎫ E3 ⎨ ⎬ ⎩1 −ε * −i −1 i ε ε* −ε ⎭ OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 5
6. 6. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory3. The Groups Dn (n = 2, 3, 4, 5, 6)D2 E C2(z) C2(y) C2(x)(222)A 1 1 1 1 x2, y2, z2B1 B 1 1 –1 –1 z, Rz xyB2 B 1 –1 1 –1 y, Ry xzB3 B 1 –1 –1 1 x, Rx yzD3 E 2C3 3C2(32)A1 1 1 1 x2 + y2, z2A2 1 1 –1 z, RzE 2 –1 0 (x, y)(Rx,, Ry) (x2 – y2, 2xy) (xz, yz)D4 E 2C4 C2 (= C4 ) 2 2C2 2C2"(422)A1 1 1 1 1 1 x2 + y2, z2A2 1 1 1 –1 –1 z, RzB1 B 1 –1 1 1 –1 x2 – y2B2 B 1 –1 1 –1 1 xyE 2 0 –2 0 0 (x, y)(Rx, Ry) (xz, yz)D5 E 2C5 2C52 5C2A1 1 1 1 1 x2 + y2, z2A2 1 1 1 –1 z, RzE1 2 2 cos 72º 2 cos 144° 0 (x, y)(Rx, Ry) (xz, yz)E2 2 2 cos 144º 2 cos 72° 0 (x2 – y2, 2xy)D6 E 2C6 2C3 C2 ′ 3C2 ′′ 3C2(622)A1 1 1 1 1 1 1 x2 + y2, z2A2 1 1 1 1 –1 –1 z, RzB1 B 1 –1 1 –1 1 –1B2 B 1 –1 1 –1 –1 1E1 2 1 –1 –2 0 0 (x, y)(Rx, Ry) (xz, yz)E2 2 –1 –1 2 0 0 (x2 – y2, 2xy) OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 6
7. 7. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory4. The Groups Cnν (n = 2, 3, 4, 5, 6)C2ν E C2 σν(xz) σ′ (yz) v(2mm)A1 1 1 1 1 z x2, y2, z2A2 1 1 –1 –1 Rz xyB1 B 1 –1 1 –1 x, Ry xzB2 B 1 –1 –1 1 y, Rx yzC3ν E 2C3 3σν(3m)A1 1 1 1 z x2 + y2, z2A2 1 1 –1 RzE 2 –1 0 (x, y)(Rx, Ry) (x2 – y2, 2xy)(xz, yz)C4ν E 2C4 C2 2σν 2σd(4mm)A1 1 1 1 1 1 z x2 + y2, z2A2 1 1 1 –1 –1 RzB1 B 1 –1 1 1 –1 x2 – y2B2 B 1 –1 1 –1 1 xyE 2 0 –2 0 0 (x, y)(Rx, Ry) (xz, yz)C5ν E 2C5 2C52 5σνA1 1 1 1 1 z x2 + y2, z2A2 1 1 1 –1 RzE1 2 2 cos 72° 2 cos 144° 0 (x, y)(Rx, Ry) (xz, yz)E2 2 2 cos 144° 2 cos 72° 0 (x2 – y2, 2xy)C6ν E 2C6 2C3 C2 3σν 3σd(6mm)A1 1 1 1 1 1 1 z x2 + y2, z2A2 1 1 1 1 –1 –1 RzB1 B 1 –1 1 –1 1 –1B2 B 1 –1 1 –1 –1 1E1 2 1 –1 –2 0 0 (x, y)(Rx, Ry) (xz, yz)E2 2 –1 –1 2 0 0 (x2 – y2, 2xy) OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 7
8. 8. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory5. The Groups Cnh (n = 2, 3, 4, 5, 6)C2h E C2 I σh(2/m)Ag 1 1 1 1 Rz x2, y2, z2, xyBgB 1 –1 1 –1 Rx, Ry xz, yzAu 1 1 –1 –1 zBuB 1 –1 –1 1 x, yC3h E C3 C32 σh S3 S35 ε = exp (2πi/3)( 6)A 1 1 1 1 1 1 Rz x2 + y2, z2 ⎧1 ⎪ ε ε* 1 ε ε*⎫ ⎪E ⎨ ⎬ (x, y) (x2 – y2, 2xy) ⎪1 ⎩ ε* ε 1 ε* ε ⎪ ⎭A 1 1 1 –1 –1 –1 z ⎧1 ⎪ ε ε * −1 −ε −ε ⎫ ⎪ *E ⎨ ⎬ (Rx, Ry) (xz, yz) ⎪1 ⎩ ε* ε −1 −ε * −ε ⎪ ⎭C4h E C4 C2 C4 i 3 3 S4 σh S4(4/m)Ag 1 1 1 1 1 1 1 1 Rz x2 + y2, z2BgB 1 –1 1 –1 1 –1 1 –1 (x2 – y2, 2xy) ⎧1 ⎪ i −1 −i 1 i −1 −i⎫ ⎪Eg ⎨ ⎬ (Rx, Ry) (xz, yz) ⎪1 ⎩ −i −1 i 1 −i −1 i⎪ ⎭Au 1 1 1 1 –1 –1 –1 –1 zBuB 1 –1 1 –1 –1 1 –1 1 ⎧1 ⎪ i −1 −i −1 −i 1 i ⎫ ⎪Eu ⎨ ⎬ (x, y) ⎪1 ⎩ −i −1 i −1 i 1 −i⎪ ⎭ OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 8
9. 9. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory 5. The Groups Cnh (n = 2, 3, 4, 5, 6) (cont…) C5h E C5 C52 3 C5 C54 σh S5 S57 3 S5 9 S5 ε = exp(2πi/5) A′ 1 1 1 1 1 1 1 1 1 1 Rz x2+y2, z2 ⎧1 ε ε2 ε *2 ε* 1 ε ε2 ε *2 ε*⎫ ′ E1 ⎨ ⎬ (x, y) ⎩1 ε* ε *2 ε2 ε 1 ε* ε *2 ε2 ε ⎭ ⎧1 ε2 ε* ε ε *2 1 ε2 ε* ε ε *2 ⎫ E′ 2 ⎨ ⎬ z (x2 – y2, 2xy) ⎩1 ε *2 ε ε* ε2 1 ε *2 ε ε* ε2 ⎭ A′′ 1 1 1 1 1 –1 –1 –1 –1 –1 ⎧1 ε ε 2 ε *2 ε * −1 −ε −ε 2 −ε *2 −ε * ⎫ ′′ E1 ⎨ ⎬ (Rx, Ry) (xz, yz) ⎩1 ε* ε *2 ε2 ε −1 −ε * −ε *2 −ε 2 −ε ⎭ ⎧1 ε2 ε* ε ε *2 −1 −ε 2 −ε * −ε −ε *2 ⎫ E′′ 2 ⎨ ⎬ ⎩1 ε *2 ε ε* ε2 −1 −ε *2 −ε −ε * −ε 2 ⎭C6h E C6 C3 C2 C32 5 C6 i 5 S3 5 S6 σh S6 S3 ε = exp(2πi/6)(6/m)Ag 1 1 1 1 1 1 1 1 1 1 1 1 x2+y2, z2Bg B 1 –1 1 –1 1 –1 1 –1 1 –1 1 –1 (Rx, Ry) (xz, yz) ⎧1 ε −ε * −1 −ε ε * 1 ε −ε * −1 −ε ε ⎫ * ⎨ ⎬E1g ⎩1 ε* −ε −1 −ε * ε 1 ε* −ε −1 −ε * ε ⎭ ⎧1 −ε * −ε 1 −ε * −ε 1 −ε * −ε 1 −ε * −ε ⎫ ⎨ ⎬E2g ⎩1 −ε −ε * 1 −ε −ε * 1 −ε −ε * 1 −ε −ε * ⎭ (x2 – y2, 2xy)Au 1 1 1 1 1 1 –1 –1 –1 –1 –1 –1 ZBu B 1 –1 1 –1 1 –1 –1 1 –1 1 –1 1 ⎧1 ε −ε * −1 −ε ε* −1 −ε ε* 1 ε −ε * ⎫ ⎨ ⎬E1u ⎩1 ε* −ε −1 −ε * ε −1 −ε * ε 1 ε* −ε ⎭ (x, y) ⎧1 −ε * −ε 1 −ε * −ε −1 ε* ε −1 ε* ε ⎫ ⎨ ⎬E2u ⎩1 −ε −ε * 1 −ε −ε * −1 ε ε* −1 ε ε*⎭ OXFORD Higher Education © Oxford University Press, 2010. All rights reserved. 9
10. 10. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory6. The Groups Dnh (n = 2, 3, 4, 5, 6)D2h E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz)(mmm)Ag 1 1 1 1 1 1 1 1 x2, y2, z2B1g B 1 1 –1 –1 1 1 –1 –1 Rz xyB2g B 1 –1 1 –1 1 –1 1 –1 Ry xzB3g B 1 –1 –1 1 1 –1 –1 1 Rx yzAu 1 1 1 1 –1 –1 –1 –1B1u B 1 1 –1 –1 –1 –1 1 1 zB2u B 1 –1 1 –1 –1 1 –1 1 yB3u B 1 –1 –1 1 –1 1 1 –1 xD3h E 2C3 3C2 σh 2S3 3σv(6) m2 ′A1 1 1 1 1 1 1 x2 + y2, z2A′ 2 1 1 –1 1 1 –1 RzE′ 2 –1 0 2 –1 0 (x, y) (x2 – y2, 2xy) ′′A1 1 1 1 –1 –1 –1A′′ 2 1 1 –1 –1 –1 1 zE′′ 2 –1 0 –2 1 0 (Rx, Ry) (xy, yz)D4h E 2C4 C2 ′ 2C2 2C2′ ′ i 2S4 σh 2σv 2σd(4/mmm)A1g 1 1 1 1 1 1 1 1 1 1 x2 + y2, z2A2g 1 1 1 –1 –1 1 1 1 –1 –1 RzB1g B 1 –1 1 1 –1 1 –1 1 1 –1 x2 – y2B2g B 1 –1 1 –1 1 1 –1 1 –1 1 xyEg 2 0 –2 0 0 2 0 –2 0 0 (Rx, Ry) (xz, yz)A1u 1 1 1 1 1 –1 –1 –1 –1 –1A2u 1 1 1 –1 –1 –1 –1 –1 1 1 ZB1u B 1 –1 1 1 –1 –1 1 –1 –1 1B2u B 1 –1 1 –1 1 –1 1 –1 1 –1Eu 2 0 –2 0 0 –2 0 2 0 0 (x, y) OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 10
11. 11. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory6. The Groups Dnh (n = 2, 3, 4, 5, 6) (cont…) D5h E 2C5 2C52 5C2 σh 2S5 3 2S5 5σv ′ A1 1 1 1 1 1 1 1 1 x2 + y2, z2 A′ 2 1 1 1 –1 1 1 1 –1 Rz ′ E1 2 2 cos 72° 2 cos 144° 0 2 2 cos 72° 2 cos 144° 0 (x, y) E′ 2 2 2 cos 144° 2 cos 72° 0 2 2 cos 144° 2 cos 72° 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 2 2 cos 72° 2 cos 144° 0 –2 –2 cos 72° –2 cos 144° 0 (Rx, Ry) (xy, yz) E ′′ 2 2 2 cos 144° 2 cos 72° 0 –2 –2 cos 144° –2 cos 72° 0 D6h E 2C6 2C3 C2 3C′ 2 ′′ 3C2 i 2S3 2S6 σh 3σd 3σv (6/mmm) A1g 1 1 1 1 1 1 1 1 1 1 1 1 x2 + y2, z2 A2g 1 1 1 1 –1 –1 1 1 1 1 –1 –1 Rz B1g B 1 –1 1 –1 1 –1 1 –1 1 –1 1 –1 B2g B 1 –1 1 –1 –1 1 1 –1 1 –1 –1 1 E1g 2 1 –1 –2 0 0 2 1 –1 –2 0 0 (Rx – Ry) (xz, yz) E2g 2 –1 –1 2 0 0 2 –1 –1 2 0 0 (x2 – y2, 2xy) A1u 1 1 1 1 1 1 –1 –1 –1 –1 –1 –1 A2u 1 1 1 1 –1 –1 –1 –1 –1 –1 1 1 z B1u B 1 –1 1 –1 1 –1 –1 1 –1 1 –1 1 B2u B 1 –1 1 –1 –1 1 –1 1 –1 1 1 –1 E1u 2 1 –1 –2 0 0 –2 –1 1 2 0 0 (x, y) E2u 2 –1 –1 2 0 0 –2 1 1 –2 0 0 OXFORD Higher Education © Oxford University Press, 2010. All rights reserved. 11
12. 12. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory7. The Groups Dnd (n = 2, 3, 4, 5, 6)D2d = Vd E 2S4 C2 ′ 2C2 2σd( ) 42 mA1 1 1 1 1 1 x2 + y2, z2A2 1 1 1 –1 –1 RzB1 B 1 –1 1 1 –1 x2 – y2B2 B 1 –1 1 –1 1 z xyE 2 0 –2 0 0 (x, y) (xz, yz) (Rx, Ry)D3d E 2C3 3C2 i 2S6 3σd (3)mA1g 1 1 1 1 1 1 x2 + y2, z2A2g 1 1 –1 1 1 –1 RzEg 2 –1 0 2 –1 0 (Rx, Ry) (x2 – y2, 2xy) (xz, yz)A1u 1 1 1 –1 –1 –1A2u 1 1 –1 –1 –1 1 zEu 2 –1 0 –2 1 0 (x, y)D4d E 2S8 2C4 2S83 C2 ′ 4C2 4σdA1 1 1 1 1 1 1 1 x2 + y2, z2A2 1 1 1 1 1 –1 –1 RzB1 B 1 –1 1 –1 1 1 –1B2 B 1 –1 1 –1 1 –1 1 zE1 2 0 –2 0 0 (x, y) 2 – 2E2 2 0 –2 0 2 0 0 (x2 – y2, 2xy)E3 2 0 –2 0 0 (Rx, Ry) (xz, yz) – 2 2 OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 12
13. 13. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory7. The Groups Dnd (n = 2, 3, 4, 5, 6) (cont..) D5d E 2C5 2C52 5C2 i 3 2S10 2S10 5σd A1g 1 1 1 1 1 1 1 1 x2 + y2, z2 A2g 1 1 1 –1 1 1 1 –1 Rz E1g 2 2 cos 72° 2 cos 144° 0 2 2 cos 72° 2 cos 144° 0 (Rx, Ry) (xy, yz) E2g 2 2 cos 144° 2 cos 72° 0 2 2 cos 144° 2 cos 72° 0 (x2 – y2, 2xy) A1u 1 1 1 1 –1 –1 –1 –1 A2u 1 1 1 –1 –1 –1 –1 1 z E1u 2 2 cos 72° 2 cos 144° 0 –2 –2 cos 72° –2 cos 144° 0 (x, y) E2u 2 2 cos 144° 2 cos 72° 0 –2 –2 cos 144° –2 cos 72° 0 D6d E 2S12 2C6 2S4 2C3 5 2S12 C2 ′ 6C2 6σd A1 1 1 1 1 1 1 1 1 1 x2 + y2, z2 A2 1 1 1 1 1 1 1 –1 –1 Rz B1 B 1 –1 1 –1 1 –1 1 1 –1 B2 B 1 –1 1 –1 1 –1 1 –1 1 z E1 2 1 0 –1 –2 0 0 (x, y) 3 – 3 E2 2 1 –1 –2 –1 1 2 0 0 (x2 – y2, 2xy) E3 2 0 –2 0 2 0 –2 0 0 E4 2 –1 –1 2 –1 –1 2 0 0 E5 2 1 0 –1 –2 0 0 (Rx, Ry) (xy, yz) – 3 3 OXFORD Higher Education © Oxford University Press, 2010. All rights reserved. 13
14. 14. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory8. The Groups Sn (n = 4, 6, 8)S4 E S4 C2 3 S4(4)A 1 1 1 1 Rz x2 + y2, z2B 1 –1 1 –1 z (x2 – y2, 2xy) ⎧1 i −1 −i ⎫E ⎨ ⎬ (x, y) (Rx, Ry) (xz, yz) ⎩1 −i −1 i⎭S6 E C3 C32 i 5 S6 S6 ε = exp (2πi/3)(3)Ag 1 1 1 1 1 1 Rz x2 + y2, z2 ⎧1 ε ε* 1 ε ε*⎫Eg ⎨ ⎬ (Rx, Ry) (x2 – y2, 2xy) (xy, yz) ⎩1 ε* ε 1 ε* ε ⎭Au 1 1 1 –1 –1 –1 z ⎧1 ε ε * 1 ε ε ⎫ *Eu ⎨ ⎬ (x, y) ⎩1 ε* ε 1 ε* ε ⎭S8 E S8 C4 S83 C2 S85 3 C4 S87 ε = exp (2πi/8)A 1 1 1 1 1 1 1 1 Rz x2 + y2, z2B 1 –1 1 –1 1 –1 1 –1 z ⎧1 ε i −ε * −1 −ε −i ε*⎫ ⎨ ⎬ (x, y)E1 ⎩1 ε* −i −ε −1 −ε * i ε ⎭ ⎧1 i −1 −i 1 i −1 −i ⎫ ⎨ ⎬ (x2 – y2, 2xy)E2 ⎩1 −i −1 i 1 −i −1 i⎭ ⎧1 −ε * −i ε −1 ε* i −ε ⎫ ⎨ ⎬E3 ⎩1 −ε i ε* −1 ε −i −ε * ⎭ (Rx, Ry) (xy, yz) OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 14
15. 15. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory9. The Cubic GroupsT E 4C3 4C32 3C2 ε = exp (2πi/3)(23)A 1 1 1 1 x2 + y2 + z2 ⎧1 ε ε* 1⎫E ⎨ ⎬ ( 3 (x2 – y2)2z2 – x2 – y2) ⎩1 ε* ε 1⎭T 3 0 0 –1 (x, y, z) (xy, xz, yz) (Rx, Ry, Rz)Td E 8C3 3C2 6S4 6σd(43m)A1 1 1 1 1 1 x2 + y2 + z2A2 1 1 1 –1 –1E 2 –1 2 0 0 (2z2 – x2 – y2, 3 (x2 – y2)T1 3 0 –1 1 –1 (Rx, Ry, Rz)T2 3 0 –1 –1 1 (x, y, z) (xy, xz, yz)Th E 4C3 4C32 3C2 i 4S6 4S62 3σd ε = exp (2πi/3)(m3)Ag 1 1 1 1 1 1 1 1 x2 + y2 + z2 ⎧1 ε ε* 1 1 ε ε* 1⎫ (2z2 – x2 –y2,Eg ⎨ ⎬ 2 2 ⎩1 ε* ε 1 1 ε* ε 1⎭ 3 (x – y )Tg 3 0 0 –1 3 0 0 –1 (Rx, Ry, Rz) (xy, yz, xz)Au 1 1 1 1 –1 –1 –1 –1 ⎧1 ε ε* 1 −1 −ε −ε * −1⎫Eu ⎨ ⎬ ⎩1 ε* ε 1 −1 −ε* −ε −1⎭Tu 3 0 0 –1 –3 0 0 1 (x, y, z)O E 8C3 3C2 6C4 ′ 6C2(432)A1 1 1 1 1 1 x2 + y2 + z2A2 1 1 1 –1 –1E 2 –1 2 0 0 (2z2 – x2 – y2, 2 2 3 (x – y ))T1 3 0 –1 1 –1 (x, y, z) (Rx, Ry, Rz)T2 3 0 –1 –1 1 (xy, xz, yz) OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 15
16. 16. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory9. The Cubic Groups (cont…) Oh E 8C3 6C2 6C4 3C2 i 6S4 8S6 3σh 6σd (m3m) (= C42 ) A1g 1 1 1 1 1 1 1 1 1 1 x2 + y2 + z2 A2g 1 1 –1 –1 1 1 –1 1 1 –1 Eg 2 –1 0 0 2 2 0 –1 2 0 (2z2 – x2 –y2, 2 2 3 (x – y )) T1g 3 0 –1 1 –1 3 1 0 –1 –1 (Rx, Ry, Rz) T2g 3 0 1 –1 –1 3 –1 0 –1 1 (xy, xz, yz) A1u 1 1 1 1 1 –1 –1 –1 –1 –1 A2u 1 1 –1 –1 1 –1 1 –1 –1 1 Eu 2 –1 0 0 2 –2 0 1 –2 0 T1u 3 0 –1 1 –1 –3 –1 0 1 1 (x, y, z) T2u 3 0 1 –1 –1 –3 1 0 1 –1 OXFORD Higher Education © Oxford University Press, 2010. All rights reserved. 16
17. 17. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory 10. The Groups I, Ih I E 12C5 12C52 20C3 15C2 1⎛ 1 ⎞ η ± = ⎜1 ± 5 2 ⎟ 2 ⎝ ⎠ 2 2 2 A 1 1 1 1 1 x +y +z T1 3 η+ η− 0 –1 (x, y, z) (Rx, Ry, Rz) T2 3 η− η+ 0 –1 G 4 –1 –1 1 0 H 5 0 0 –1 1 (2z2 – x2 – y2, 3 (x2 – y2) xy, yz, zx)Ih E 12C5 12C52 20C3 15C2 i 12S10 3 12S10 20S6 15σ 1⎛ 1 ⎞ η ± = ⎜1 ± 5 2 ⎟ 2 ⎝ ⎠ 2 2 2Ag 1 1 1 1 1 1 1 1 1 1 x +y +zT1g 3 η+ η− 0 –1 3 η− η+ –1 –1 (Rx,Ry,Rz)T2g 3 η− η+ 0 –1 3 η+ η− 0 –1Gg 4 –1 –1 1 0 4 –1 –1 1 0Hg 5 0 0 –1 1 5 0 0 –1 1 (2z2 – x2 – y2, 3 (x2 – y2)) (xy, yz, zx)Au 1 1 1 1 1 –1 –1 –1 –1 –1T1u 3 η+ η− 0 –1 –3 η− η+ 0 1 (x, y, z)T2u 3 η− η+ 0 –1 –3 η+ η− 0 1Gu 4 –1 –1 1 0 –4 1 1 –1 0Hu 5 0 0 –1 1 –5 0 0 1 –1 OXFORD Higher Education © Oxford University Press, 2010. All rights reserved. 17
18. 18. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory11. The Groups C∞v and D∞h C∞v E C2 φ 2C∞ … ∞σvA1≡∑ + 1 1 1 … 1 z x2 + y2, z2A2≡∑– 1 1 1 … –1 RzE1≡Π 2 –2 2 cos φ … 0 (x,y) (Rx,Ry) (xz, yz)E2≡Δ 2 2 2 cos 2φ … 0 (x – y2, 2xy) 2E3≡Φ 2 –2 2 cos 3φ … 0 … … … … … … … … … … … …D∞h E φ 2C∞ … ∞σv i φ 2S∞ … ∞C2Σ + g 1 1 … 1 1 1 … 1 x2 + y2, z2Σ− g 1 1 … –1 1 1 … –1 Rz∏g 2 2 cos φ … 0 2 –2 cos φ … 0 (Rx, Ry) (xz, yz)Δg 2 2 cos 2φ … 0 2 2 cos 2φ … 0 (x2 – y2, 2xy)… … … … … … … … …Σ + u 1 1 … 1 –1 –1 … –1 z −Σu 1 1 … –1 –1 –1 … 1∏u 2 2 cos φ … 0 –2 2 cos φ … 0 (x,y)Δu 2 2 cos 2φ … 0 –2 –2 cos 2φ … 0… … … … … … … … … OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 18
19. 19. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory12. The Full Rotation Group (SU2 and R3) ⎧ ⎛ 1⎞ ⎪ sin ⎜ j + 2 ⎟ φ ⎪ ⎝ ⎠ φ≠0χ( j ) (φ) = ⎨ 1 ⎪ sin φ 2 ⎪ ⎩ 2 j +1 φ=0Notation : Representation labelled Γ(j) with j = 0,1/2, 1, 3/2,…∞, for R3 j is confined to integralvalues (and written l or L) and the labels S ≡ Γ(0), P ≡Γ(1), D ≡Γ(2), etc. are used. OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 19
20. 20. Atkins & de Paula, Physical Chemistry 9e: Tables for Group TheoryDirect Products1. General rules(a) For point groups in the lists below that have representations A, B, E, T without subscripts, read A1 = A2 = A, etc.(b) g u ′ ″ g g u ′ ′ ″ u g ″ ′(c) Square brackets [ ] are used to indicate the representation spanned by the antisymmetrized product of a degenerate representation with itself.ExamplesFor D3h E′ × E′′ A1 + A′′ + E ′′ 2For D6h E1g × E2g = 2Bg + E1g.2. For C2, C3, C6, D3, D6,C2v,C3v, C6v,C2h, C3h, C6h, D3h, D6h, D3d, S6 A1 A2 B1 B B2 B E1 E2 A1 A1 A2 B1 B B2 B E1 E2 A2 A1 B2 B B1 B E1 E2 B1 B A1 A2 E2 E1 B2 B A1 E2 E1 E1 A1 + [A2]+ E2 B1 + B2 + E1 B E2 A1 + [A2] + E23. For D2 , D2h A B1 B2 B B3 B A A B1 B2 B B3 B B1 B A B3 B2 B B2 B A B1 B3 B A OXFORD Higher Education © Oxford University Press, 2010. All rights reserved. 20
21. 21. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory4. For C4, D4, C4v, C4h, D4h, D2d, S4 A1 A2 B1B B2 B E A1 A1 A2 B1B B2 B E A2 A1 B2B B1 B E B1 B A1 A2 E B2 B A1 E E A1 + [A2] +B1 + B25. For C5, D5, C5v, C5h, D5h, D5d A1 A2 E1 E2 A1 A1 A2 E1 E2 A2 A1 E1 E2 E1 A1 + [A2] + E2 E 1 + E2 E2 A1 + [A2] + E16. For D4d, S8 A1 A2 B1 B B2B E1 E2 E3 A1 A1 A2 B1 B B2B E1 E2 E3 A2 A1 B2 B B1B E1 E2 E3 B1B A1 A2 E3 E2 E1 B2B A1 E3 E2 E1 E1 A1 + [A2] + E2 E 1 + E2 B1 + B2 + E2 B E2 A1 + [A2] + E1 + E3 B1 + B2 B E3 A1 + [A2] + E27. For T, O, Th, Oh, Td A1 A2 E T1 T2 A1 A1 A2 E T1 T2 A2 A1 E T2 T1 E A1 + [A2] + E T 1 + T2 T 1 + T2 T1 A1 + E + [T1] + T2 A2 + E + T1 + T2 T2 A1 + E + [T1] + T2 OXFORD Higher Education © Oxford University Press, 2010. All rights reserved. 21
22. 22. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory8. For D6d A1 A2 B1 B B2 B E1 E2 E3 E4 E5 A1 A1 A2 B1 B B2 B E1 E2 E3 E4 E5 A2 A1 B2 B B1 B E1 E2 E3 E4 E5 B1 B A1 A2 E5 E4 E3 E2 E1 B2 B A1 E5 E4 E3 E2 E1 E1 A1 + [A2] + E 1 + E3 E 2 + E4 E 3 + E5 B1 + B2 + B E2 E4 E2 A1 + [A2] E 1 + E5 B1 + B2 + B E 3 + E5 + E4 E2 E3 A1 + [A2] + E 1 + E5 E 2 + E4 B1 + B2 B E4 A1 + [A2] E 1 + E3 + E4 E5 A1 + [A2] + E29. For I, Ih A T1 T2 G H A A T1 T2 G H T1 A + [T1] + G+H T2 + G + H T1 +T2 + G + H H T2 A + [T2] + H T1 + G + H T1 +T2 + G + H G A + [T1 +T2] T1 +T2 + G + 2H +G+H H A1 + [T1 +T2 + G] + G + 2H10. For C∝v, D∝h Σ+ Σ– Π Δ + + – Σ Σ Σ Π Δ Σ– Σ+ Π Δ Π Σ + [Σ–] + Π+Φ +Δ 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). OXFORD Higher Education © Oxford University Press, 2010. All rights reserved. 22
23. 23. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory11. The Full Rotation Group (SU2 and R3) 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)] OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 23
24. 24. Atkins & de Paula, Physical Chemistry 9e: Tables for Group TheoryExtended rotation groups (double groups):Character tables and direct product tables *D2 E R 2C2(z) 2C2(y) 2C2(x)E1/2 2 –2 0 0 0D3* E R 2C3 2C3R 3C2 3C2RE1/2 2 –2 1 –1 0 0E3/2 ⎧1 −1 −1 1 i −i ⎫ ⎨ ⎬ ⎩1 −1 −1 1 −i i⎭D4 E R 2C4 2C4R 2C2 4C2′ 4C2′′E1/2 2 –2 2 – 2 0 0 0E3/2 2 –2 – 2 2 0 0 0D6* E R 2C6 2C6R 2C3 2C3R 2C2 6C2′ 6C2′′E1/2 2 –2 3 – 3 1 –1 0 0 0E3/2 2 –2 – 3 3 –1 1 0 0 0E5/2 2 –2 0 0 –2 2 0 0 0Td* E R 8C3 8C3R 6C2 6S4 6S4R 12σ dO * E R 8C3 8C3R 6C2 6C4 6S4R 12C2′E1/2 2 –2 1 –1 0 2 – 2 0E5/2 2 –2 1 –1 0 – 2 2 0G3/2 4 –4 –1 1 0 0 0 0E1/2 × E1/2 = [A] +B1 +B2 + B3 E1/2 E3/2E1/2 [A1] + A2 + E 2EE3/2 [A1] + A1 + 2A2 E1/2 E3/2E1/2 [A1] + A2 + E B1 + B2 + E BE3/2 [A1] + A2 + E OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 24
25. 25. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory E1/2 E3/2 E5/2E1/2 [A1] +A2 + E1 B1 + B2+ E2 B E 1 + E2E3/2 [A1] +A2 + E1 E 1 + E2E5/2 [A1] + A2 +B1 + B2 E1/2 E5/2 E3/2E1/2 [A1] + T1 A2 + T2 E + T 1 + T2E5/2 [A1] + T1 E + T 1 + T2G3/2 [A1 + E + T2] + A2 + 2T1 + T2]Direct products of ordinary and extended representations for Td* and O* A1 A2 E T1 T2E1/2 E1/2 E5/2 G3/2 E1/2 + G3/2 E5/2 + G3/2E5/2 E5/2 E1/2 G3/2 E5/2 + G3/2 E1/2 + G3/2G3/2 G3/2 G3/2 E1/2 + E5/2+ G3/2 E1/2 + E5/2+ 2G3/2 E1/2 + E5/2+ 2G3/2 OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 25
26. 26. Atkins & de Paula, Physical Chemistry 9e: Tables for Group TheoryDescent in symmetry and subgroupsThe following tables show the correlation between the irreducible representations of a groupand those of some of its subgroups. In a number of cases more than one correlation existsbetween groups. In Cs the σ of the heading indicates which of the planes in the parent groupbecomes the sole plane of Cs; in C2v it becomes must be set by a convention); where there arevarious possibilities for the correlation of C2 axes and σ planes in D4h and D6h with theirsubgroups, the column is headed by the symmetry operation of the parent group that ispreserved in the descent. C2v C2 Cs Cs σ(zx) σ(yz) A1 A A′ A′ A2 A A" A" B1 B B A′ A′ B2 B B A" A" C3v C3 Cs A1 A A′ A2 A A" E E A′ + A" C4v C2v C2v σv σd A1 A1 A1 A2 A2 A2 B1 B A1 A2 B2 B A2 A1 E B1 + B2 B1 + B2[Other subgroups: C4, C2, C6] C2ν Cs Cs D3h C3h C3v σh→σν σh σν ′ A1 A′ A1 A1 A′ A′ A′ 2 A′ A2 B2B A′ A" E E E A1 + B2 2A A + A" A1′′ A" A2 A2 A" A" A′′ 2 A" A1 B1B A" A′ E" E" E A2 + B1 2A" A + A"[Other subgroups: D3, C3, C2] OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 26
27. 27. Atkins & de Paula, Physical Chemistry 9e: Tables for Group TheoryD4h D2d D2d D2h D2h D2 D2 C4h C4v C2v C2v ′ ′ C2 (→ C2 ) ′′ ′ C2 (→ C2 ) ′ C2 ′′ C2 ′ C2 ′′ C2 C2, σv C2, σdA1g A1 A1 Ag Ag A A Ag A1 A1 A1A2g A2 A2 B1g B B1g B B1 B B1 B Ag A2 A2 A2B1g B B1B B2 B Ag B1g B A B1 BgB B1 B A1 A2B2g B B2B B1 B B1g B Ag B1 B A Bg B2 B A2 A1Eg E E B2g + B3g B B2g + B3g B B2 + B3 B B2 + B3 B Eg E B1 + B2 B B1 + B2 BA1u B1B B1 B Au Au A A Au A2 A2 A2A2u B2B B2 B B1u B B1u B B1 B B1 B Au A1 A1 A1B1u B A1 A2 Au B1u B A B1 BuB B2 B A2 A1B2u B A2 A1 B1u B Au B1 B A Bu B1 B A1 A2Eu E E B2u + B3u B B2u + B3u B B2 + B3 B B2 + B3 B Eu E B1 + B2 B B1 + B2 BOther subgroups:D4, C4, S4, 3C2h, 3Cs,3C2,Ci, (2C2v) D6 ′′ D3d C2 ′ D3d C2 D2h C6v C3v C2v C2v C2h C2h C2h σh → σ(xy) σv ′ C2 ′′ C2 C2 ′ C2 ′′ C2 σv → σ(yz) A1g A1g A1g Ag A1 A1 A1 A1 Ag Ag Ag A2g A2g A2g B1g B A2 A2 B1 B B1 B Ag Bg B Bg B B1g B A2g A1g B2g B B2 B A2 A2 B2 B Bg B Ag Bg B B2g B A1g A2g B3g B B1 B A1 B2 B A2 Bg B Bg B Ag E1g Eg Eg B2g + B3g B E1 E A2 + B2 A2 + B2 2Bg Ag + Bg Ag + Bg E2g Eg Eg Ag + B1g E2 E A1 + B1 A1 + B1 2Ag Ag + Bg Ag + Bg A1u A1u A1g Au A2 A2 A2 A2 Au Au Au A2u A2u A2g B1u B A1 A1 B2 B B2 B Au Bu B Bu B B1u B A2u A1u B2u B B1 B A1 B1 B B1 B Bu B Au Bu B B2u B A1u A2u B3u B B2 B A2 A1 A1 Bu B Bu B Au E1u Eu Eu B2u + B3u B E1 E A1 + B1 A1 + B1 2Bu Au + Bu Au + Bu E2u Eu Eu Au + B1u E2 E A2 + B2 A2 + B2 2Au Au + Bu Au + Bu Other subgroups: D6, 2D3h, C6h, C6, C3h, 2D3, S6, D2, C3, 3C2, 3Cg, Ci Td T D2d C3v C2v A1 A A1 A1 A1 A2 A B1 A2 A2 E E A1 + B1 E A1 +A2 T1 T A2 + E A2 + E A2 + B1 + B2 T2 T B2 + E B A1 + E A1 + B2 + B1 Other subgroups: S4, D2, C3, C2, Cs. OXFORD Higher Education © Oxford University Press, 2010. All rights reserved. 27
28. 28. Atkins & de Paula, Physical Chemistry 9e: Tables for Group TheoryOh O Td Th D4h D3dA1g A1 A1 Ag A1g A1gA2g A2 A2 Ag B1g B A2gEg E E Eg A1g + B1g EgT1g T1 T1 Tg A2g + Eg A2g + EgT2g T2 T2 Tg B2g + Eg B A1g + EgA1u A1 A2 Au A1u A1uA2u A2 A1 Au B1u B B1uBEu E E Eu A1u + B1u EuT1u T1 T2 Tu A2u + Eu A2u + EuT2u T2 T1 Tu B2u + Eu B A1u + EuOther subgroups: T, D4, D2d, C4h, C4v, 2D2h, D3, C3v, S6, C4, S4, 3C2v, 2D2, 2C2h, C3, 2C2, S2, CsR3 O D4 D3S A1 A1 A1P T1 A2 + E A2 + ED E + T2 A1 + B1 +B2 + E A1 + 2EF A2 + T1 +T2 A2+ B1 +B2 + 2E A1 + 2A2 + 2EG A1 + E + T1 + T2 2A1 + A2 +B1 + B2 + 2E 2A1+ A2 + 3EH E + 2T1 + T2 A1 + 2A2 + B1 + B2 + 3E A1 + 2A2 + 4E OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 28
29. 29. Atkins & de Paula, Physical Chemistry 9e: Tables for Group TheoryNotes and IllustrationsGeneral Formulae(a) Notationh the order (the number of elements) of the group.Γ(i) labels the irreducible representation.Χ(i)(R) the character of the operation R in Γ(i). Dμν) ( R ) (i the μv element of the representative matrix of the operation R in the irreducible representation Γ(i).l i the dimension of Γ(i).(the number of rows or columns in the matrices D(i))(b) Formulae(i) Number of irreducible representations of a group = number of classes. 2(ii) ∑ li = h i(iii) χ (i ) ( R) = ∑ Dμμ) ( R) (i μ(iv) Orthogonality of representations: ∑ Dμν) ( R)* Dμiν) ( R) = (h / li ) δ ii δ μμ δνν (i ( (δij=1 if i = j and δij = 0 if i ≠ j(v) Orthogonality of characters: ∑ χ (i ) ( R)* χ (i ) ( R) = h δ ii R(vi) Decomposition of a direct product, reduction of a representation: If Γ = ∑ ai Γ (i ) iand the character of the operation R in the reducible representation is χ(R), then the coefficients atare given by ai = (l / h)∑ χ (i ) ( R)* χ ( R). R OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 29
30. 30. Atkins & de Paula, Physical Chemistry 9e: Tables for Group Theory(vii) Projection operators: The projection operator (i ) = (li / h)∑ χ (i ) ( R)* R R when applied to a function f, generates a sum of functions that constitute a component of a basis for the representation Γ(i); in order to generate the complete basis P (i) must be applied to li distinct functions f. The resulting functions may be made mutually orthogonal. When li = 1 the function generated is a basis for Γ(i) without ambiguity: P (i ) f = f (i )(viii) Selection rules: If f(i) is a member of the basis set for the irreducible representation Γ(i), f{k) a member of that ˆ for Γ(k), and Ω (j) an operator that is a basis for Γ(j), then the integral ( i )* ˆ ( j ) ( k ) ∫ dτ f Ω f is zero unless Γ(i) occurs in the decomposition of the direct product Γ(j) × Γ(k) s(ix) The symmetrized direct product is written Γ ×Γ , and its characters are given by (i ) (i ) s χ ( i ) ( R) × χ ( i ) ( R) = 1 χ ( i ) ( R) 2 + 1 χ (i ) ( R 2 ) 2 2 a The antisymmetrized direct product is written Γ ×Γ and its characters are given by (i ) (i ) a χ ( i ) ( R) × χ ( i ) ( R) = 1 χ ( i ) ( R) 2 + 1 χ (i ) ( R 2 ) 2 2 OXFORD Higher Education© Oxford University Press, 2010. All rights reserved. 30