中国科学技术大学：《量子化学 Quantum Chemistry》课程教学资源（参考书籍）《物质结构导论》Point Group Character Tables

A Point Group Character Tables Appendix A contains Point Group Character (Tables A.1-A.34)to be used throughout the chapters of this book.Pedagogic material to assist the reader in the use of these character tables can be found in Chap.3.The Schoenflies symmetry (Sect.3.9)and Hermann-Mauguin notations (Sect.3.10)for the point groups are also discussed in Chap.3. Some of the more novel listings in this appendix are the groups with five- fold symmetry C5,Csh,Csv,D5;D5d,Dsh,I,Ih.The cubic point group On in Table A.31 lists basis functions for all the irreducible representations of Oh and uses the standard solid state physics notation for the irreducible representations. Table A.1.Character table for group Ci(triclinic) C(1) E A 1 Table A.2.Character table for group Ci=S2(triclinic) S2(① E 2,y,22,ty.tz.yz Rz;Ry,R: Ag 1 1 ,y,2 u 1 -1 Table A.3.Character table for group Cih=S1 (monoclinic) Cih(m) E Oh x2,y2,22,xy R:,x,Y 1 x2,y2 R:,Ry:z A" 1 -1

A Point Group Character Tables Appendix A contains Point Group Character (Tables A.1–A.34) to be used throughout the chapters of this book. Pedagogic material to assist the reader in the use of these character tables can be found in Chap. 3. The Schoenflies symmetry (Sect. 3.9) and Hermann–Mauguin notations (Sect. 3.10) for the point groups are also discussed in Chap. 3. Some of the more novel listings in this appendix are the groups with five￾fold symmetry C5, C5h, C5v, D5, D5d, D5h, I, Ih. The cubic point group Oh in Table A.31 lists basis functions for all the irreducible representations of Oh and uses the standard solid state physics notation for the irreducible representations. Table A.1. Character table for group C1 (triclinic) C1 (1) E A 1 Table A.2. Character table for group Ci = S2 (triclinic) S2 (1) E i x2, y2, z2, xy, xz, yz Rx, Ry, Rz Ag 1 1 x, y, z Au 1 −1 Table A.3. Character table for group C1h = S1 (monoclinic) C1h(m) E σh x2, y2, z2, xy Rz, x, y A 1 1 xz, yz Rx, Ry, z A 1 −1

480 A Point Group Character Tables Table A.4.Character table for group C2(monoclinic) C2(2) E C2 x2,2,22,xy R:,2 A 1 (E,y) E2,y2 B 1 (Rz,Ry) -1 Table A.5.Character table for group C2(orthorhombic) C2.(2mm) E C2 Gv 0% 2,y2,2 A1 1 1 1 y R: A2 1 1 -1 -1 Tz Ry:x B1 -1 1 -1 y2 Rr,y B2 1 -1 -1 1 Table A.6.Character table for group Cah(monoclinic) C2h(2/m) E C2 Oh i x2,2,22,y R Ag 1 1 1 1 Au 1 1 -1 -1 E2,y2 R:;Ry Bg -1 -1 1 x,y Bu 1 -1 1 -1 Table A.7.Character table for group D2=V (orthorhombic) D2(222) E C防 C嘡 C 2,y2,22 A1 1 1 1 1 y R:,2 B1 1 1 -1 -1 Cz Ryy B2 1 -1 1 -1 yz Rx,工 B3 1 -1 -1 1 Table A.8.Character table for group Dad Va (tetragonal) D2a(42m) E C2 2S4 2C% 20a x2+y2,2 A 1 1 1 1 1 R: A2 1 1 1 -1 -1 x2-2 B1 1 1 -1 1 -1 xy B2 1 1 -1 -1 1 (xz,yz) (x,) E 2 -2 0 0 0 (Rz:Ry) D2h D2 i (mmm)(orthorhombic)

480 A Point Group Character Tables Table A.4. Character table for group C2 (monoclinic) C2 (2) E C2 x2, y2, z2, xy Rz, z A 1 1 xz, yz (x, y) (Rx, Ry) B 1 −1 Table A.5. Character table for group C2v (orthorhombic) C2v (2mm) E C2 σv σ v x2, y2, z2 z A1 1111 xy Rz A2 1 1 −1 −1 xz Ry, x B1 1 −1 1 −1 yz Rx, y B2 1 −1 −1 1 Table A.6. Character table for group C2h (monoclinic) C2h (2/m) E C2 σh i x2, y2, z2, xy Rz Ag 1111 z Au 1 1 −1 −1 xz, yz Rx, Ry Bg 1 −1 −1 1 x, y Bu 1 −1 1 −1 Table A.7. Character table for group D2 = V (orthorhombic) D2 (222) E Cz 2 Cy 2 Cx 2 x2, y2, z2 A1 1111 xy Rz, z B1 1 1 −1 −1 xz Ry, y B2 1 −1 1 −1 yz Rx, x B3 1 −1 −1 1 Table A.8. Character table for group D2d = Vd (tetragonal) D2d (42m) E C2 2S4 2C 2 2σd x2 + y2, z2 A1 11 1 1 1 Rz A2 11 1 −1 −1 x2 − y2 B1 1 1 −1 1 –1 xy z B2 1 1 −1 −1 1 (xz, yz) (x, y) (Rx, Ry) E 2 −20 00 D2h = D2 ⊗ i (mmm) (orthorhombic)

A Point Group Character Tables 481 Table A.9.Character table for group Cs(rhombohedral) C3(3) E Cs C x2+y2,22 R.,2 A 1 1 (xz,yz) (x,y） (x2-2,xy)」 (Rz:Ry) E 1 w2 W e2mi/3 Table A.10.Character table for group C3o(rhombohedral) Csv (3m) E 2C3 30u x2+y,22 A1 1 1 1 R: A2 1 1 -1 (x2-2,x)1】 (x, E 2 -1 0 (x2,y2) (R:;Ru) Table A.11.Character table for group Csh=S3(hexagonal) C3h=C3⑧h(⑥ E Cs c Oh Ss (σhC3) x2+y2,22 R: A 1 1 1 1 1 1 2 A 1 1 1 -1 -1 -1 (x2-2,x) E 1 3 1 w2 (x,y) 1 3 1 3 -1 -w2 (xz,yz) (Rz;Ry) 3 w? -1 -w2 W=e2mi/3 Table A.12.Character table for group Ds(rhombohedral) D3(32) E 2C3 3C2 x2+y2,22 A 1 1 1 R:,2 A2 1 1 -1 (xz,yz) (E,y) (x2-y2,xy) (R=:Ru) E 2 -1 0 Table A.13.Character table for group D3d(rhombohedral) Dsd=Ds⑧i(3m) E 2C3 3C2 2 2iCs 3iC x2+y2,22 Aig 1 1 1 1 1 R: A2g y 1 -1 1 1 -1 (xz,y2),(x2-y2,x (Rz,Ry) Eg 2 -1 0 2 -1 0 Alu 1 1 1 -1 -1 -1 A2u -1 -1 -1 1 (x,y) Eu -1 0 -2 1 0

A Point Group Character Tables 481 Table A.9. Character table for group C3 (rhombohedral) C3(3) E C3 C2 3 x2 + y2, z2 Rz, z A 111 (xz, yz) (x2 − y2, xy) " (x, y) (Rx, Ry) " E (1 1 ω ω2 ω2 ω ω = e2πi/3 Table A.10. Character table for group C3v (rhombohedral) C3v (3m) E 2C3 3σv x2 + y2, z2 z A1 1 11 Rz A2 1 1 –1 (x2 − y2, xy) (xz, yz) " (x, y) (Rx, Ry) " E 2 −1 0 Table A.11. Character table for group C3h = S3 (hexagonal) C3h = C3 ⊗ σh (6) E C3 C2 3 σh S3 (σhC2 3 ) x2 + y2, z2 Rz A 11 1 1 1 1 z A 11 1 −1 −1 −1 (x2 − y2, xy) (x, y) E ( 1 1 ω ω2 ω2 ω 1 1 ω ω2 ω2 ω (xz, yz) (Rx, Ry) E ( 1 1 ω ω2 ω2 ω −1 −1 −ω −ω2 −ω2 −ω ω = e2πi/3 Table A.12. Character table for group D3 (rhombohedral) D3 (32) E 2C3 3C 2 x2 + y2, z2 A1 111 Rz, z A2 1 1 −1 (xz, yz) (x2 − y2, xy) " (x, y) (Rx, Ry) " E 2 −1 0 Table A.13. Character table for group D3d (rhombohedral) D3d = D3 ⊗ i (3m) E 2C3 3C 2 i 2iC3 3iC 2 x2 + y2, z2 A1g 111 111 Rz A2g 1 1 −1 1 1 −1 (xz, yz),(x2 − y2, xy) (Rx, Ry) Eg 2 −1 0 2 −1 0 A1u 111 −1 −1 −1 z A2u 1 1 −1 −1 −1 1 (x, y) Eu 2 −1 0 −210

482 A Point Group Character Tables Table A.14.Character table for group Dsh(hexagonal) D3h=D3⑧oh(6m2) E Oh 2C3 2S3 3C4 30r x2+,22 A 1 1 1 1 1 1 R A的 1 1 1 1 -1 -1 1 -1 1 -1 1 -1 A 1 -1 -1 -1 (x2-y2,xy） (x,y) 2 2 -1 -1 0 0 (xz,yz) (R:,Ry) ⊙。 2 -2 -1 0 0 Table A.15.Character table for group Ca(tetragonal) C4(4) E C2 Ca c x2+y2,2 R:,2 1 1 1 1 x2-y2,xy B -1 -1 (x,y) (xz,yz) ~1 -i (Rz;Ry) 1 -i Table A.16.Character table for group C4(tetragonal) C4v (4mm) E C2 2C4 20v 20d x2+y2,22 A1 1 1 1 R: A2 1 1 -1 -1 x2-y2 B1 1 1 -1 1 -1 工y B2 1 -1 -1 1 (xz,92) (x,） E -2 0 0 (Rx,Rg） 0 C4h=Cai(4/m)(tetragonal) Table A.17.Character table for group S4 (tetragonal) S4(④ E C2 SA 5 x2+,22 R: 1 1 1 1 B 1 1 -1 -1 (xz,yz) (x,y) E 1 -1 -i (x2-y2,xg) (Rz,Ry) -1 i i Table A.18.Character table for group D(tetragonal) D4(422) E C2=C 2C4 2C% 2C x2+2,z2 A1 1 R=,z A2 1 -1 -1 x2-2 B1 1 -1 1 -1 ry B2 -1 -1 (x,y) (x2,y2)】 E -2 0 0 0 (Rz;Ry) Dah =D4 i(4/mmm)(tetragonal)

482 A Point Group Character Tables Table A.14. Character table for group D3h (hexagonal) D3h = D3 ⊗ σh (6m2) E σh 2C3 2S3 3C 2 3σv x2 + y2, z2 A 1 11 1 1 1 1 Rz A 2 11 1 1 −1 −1 A 1 1 −1 1 −1 1 −1 z A 2 1 −1 1 −1 −1 1 (x2 − y2, xy) (x, y) E 2 2 −1 −1 00 (xz, yz) (Rx, Ry) E 2 −2 −11 00 Table A.15. Character table for group C4 (tetragonal) C4 (4) E C2 C4 C3 4 x2 + y2, z2 Rz, z A 1 111 x2 − y2, xy B 1 1 −1 −1 (xz, yz) (x, y) (Rx, Ry) " E ( 1 1 −1 −1 i −i −i i Table A.16. Character table for group C4v (tetragonal) C4v (4mm) E C2 2C4 2σv 2σd x2 + y2, z2 z A1 11 1 11 Rz A2 11 1 −1 −1 x2 − y2 B1 1 1 −1 1 −1 xy B2 1 1 −1 −1 1 (xz, yz) (x, y) (Rx, Ry) " E 2 −2 000 C4h = C4 ⊗ i (4/m) (tetragonal) Table A.17. Character table for group S4 (tetragonal) S4 (4) E C2 S4 S3 4 x2 + y2, z2 Rz A 11 11 z B 1 1 −1 −1 (xz, yz) (x2 − y2, xy) " (x, y) (Rx, Ry) " E (1 1 −1 −1 i −i −i i Table A.18. Character table for group D4 (tetragonal) D4 (422) E C2 = C2 4 2C4 2C 2 2C 2 x2 + y2, z2 A1 1 111 1 Rz, z A2 1 11 −1 −1 x2 − y2 B1 1 1 −1 1 −1 xy B2 1 1 −1 −1 1 (xz, yz) (x, y) (Rx, Ry) " E 2 −200 0 D4h = D4 ⊗ i (4/mmm) (tetragonal)

A Point Group Character Tables 483 Table A.19.Character table for group C6(hexagonal) C6(6) E Ce C3 C2 C ci x2+y2,22 R:,z A 1 1 1 1 1 1 B -1 -1 1 -1 1 w3 ws (x2,y2) (x,) E w3 (Rx,Ry)」 1 wi w3 w (x2-y2,xy) 1 3 1 3 w=e2mi/6 Table A.20.Character table for group Cev(hexagonal) C6e (6mm) E C2 2C3 2C6 30d 30u x2+y2,22 A 1 1 1 1 1 R A2 1 1 1 1 -1 -1 B1 1 -1 1 -1 -1 1 B2 1 -1 1 -1 1 -1 (xz,yz) (,y） E 2 -2 -1 1 0 (R=,Ru) 0 (x2-2,x) E2 2 2 -1 -1 0 Ceh =C6i (6/m)(hexagonal); S6=C3⑧i(③)(rhombohedral) Table A.21.Character table for group D6 (hexagonal) D6(622) E C2 2C3 2C6 3C%3C x2+2,2 A 1 1 1 1 1 R=,2 A2 1 1 1 1 -1 -1 B1 1 -1 -1 -1 B2 1 -1 -1 -1 1 (xz,yz) (x,） E 2 -2 -1 1 0 0 (Rx,Rg） (x2-y2,xy) E2 2 2 -1 -1 0 0 Deh =D6 i(6/mmm)(hexagonal) Table A.22.Character table for group Cs (icosahedral) C5(5) E Cs C号 c C x2+y2,z2 R=,z A 1 1 1 1 1 (xz,yz) (x,） E (1 w3 (Rz,Ru) w w3 w (x2-y2,x) Ew w2 1 w w=e2ri/5.Note group Csh =Cs h=S10(10)

A Point Group Character Tables 483 Table A.19. Character table for group C6 (hexagonal) C6 (6) E C6 C3 C2 C2 3 C5 6 x2 + y2, z2 Rz, z A 111111 B 1 −1 1 −1 1 −1 (xz, yz) (x, y) (Rx, Ry) " E (1 1 ω ω5 ω2 ω4 ω3 ω3 ω4 ω2 ω5 ω (x2 − y2, xy) E (1 1 ω2 ω4 ω4 ω2 1 1 ω2 ω4 ω4 ω2 ω = e2πi/6 Table A.20. Character table for group C6v (hexagonal) C6v (6mm) E C2 2C3 2C6 3σd 3σv x2 + y2, z2 z A1 11 1 1 1 1 Rz A2 11 1 1 −1 −1 B1 1 −1 1 −1 −1 1 B2 1 −1 1 −1 1 –1 (xz, yz) (x, y) (Rx, Ry) " E1 2 −2 −1 100 (x2 − y2, xy) E2 2 2 −1 −100 C6h = C6 ⊗ i (6/m) (hexagonal); S6 = C3 ⊗ i (3) (rhombohedral) Table A.21. Character table for group D6 (hexagonal) D6 (622) E C2 2C3 2C6 3C 2 3C 2 x2 + y2, z2 A1 11 1 1 1 1 Rz, z A2 11 1 1 −1 −1 B1 1 −1 1 −1 1 −1 B2 1 −1 1 −1 −1 1 (xz, yz) (x, y) (Rx, Ry) " E1 2 −2 −110 0 (x2 − y2, xy) E2 2 2 −1 −10 0 D6h = D6 ⊗ i (6/mmm) (hexagonal) Table A.22. Character table for group C5 (icosahedral) C5 (5) E C5 C2 5 C3 5 C4 5 x2 + y2, z2 Rz, z A 11111 (xz, yz) (x, y) (Rx, Ry) " E ( 1 1 ω ω4 ω2 ω3 ω3 ω2 ω4 ω (x2 − y2, xy) E ( 1 1 ω2 ω3 ω4 ω ω ω4 ω3 ω2 ω = e2πi/5. Note group C5h = C5 ⊗ σh = S10(10)

484 A Point Group Character Tables Table A.23.Character table for group Cse (icosahedral) Csu(5m) E 2Cs 2C号5am x2+y2,22,z3,z(2+） A 1 1 1 1 R: A2 1 1 1 -1 z(x,),2(x,,(x2+y2)(x,) (x,) E 2 2 cosa 2cos 2a (Rz,Ry) 0 (x2-y2,xy),z2(x2-y2,xy) [z(x2-3y2),y(3z2-y2】 E2 2 2 cos 2a 2cos4a 0 a =2/5 =72.Note that T =(1+v5)/2 so that T =-2cos 2a =-2 cos 4/5 and T-1 2 cosa 2 cos 2/5 Table A.24.Character table for group Ds (icosahedral) D5(52) E 2C5 2C号 5C x2+y,22 1 义 1 1 R=,z A2 1 2 1 -1 (x,y） (xz,yz) E 2 2cos a 2cos 20 0 (Rx,Ry） (x2-2,x) E2 2 2cos 2a 2cos 4a 0 Table A.25. Character table for Dsd (icosahedral) Dsd E 2C5 2C号 5C2 2S10 2S10 50d (h=20) A1g +1 +1 +1 +1 +1 +1 +1 +1 (x2+y),z2 A2g +1 +1 +1 -1 +1 +1 +1 -1 Ra E1s +2 T-1 -T 0 +2 r-1 -T 0 z(x+iy,x-iy) E2g +2 一T T-1 0 +2 -T T-1 0 [(x+ig)2,(x-iy)] Alu +1 +1 +1 +1 -1 -1 -1 -1 A2u +1 +1 +1 -1 -1 -1 -1 +1 E +2 T-1 一T 0 -2 1-T +T 0 (x+iy,x-iy） E2u +2 一T T-1 0 -2 +T 1-T 0 Note:Dsd Ds i,ics Sio and ic=S10.Also iC=d Table A.26.Character table for Dsh (icosahedral) Dsh (102m) E 2C5 2C号5C2 Oh 2S5 2S%5a.(h=20) A +1 +1 +1+1+1 +1 +1+1x2+y2,22 A +1 +1 +1-1+1 +1 +1-1 R: E +2T-1 一T 0+2T-1 -T0(x,,(x22,y22, z(2+y2),(x2+2】 的 +2 -TT-1 0+2 -rT-1 0(x2-y,xy, y(3x2-y2),x(x2-3y2】 +1 +1 +1+1 -1 -1 -1-1 A码 +1 +1 +1-1-1-1 -1+1 2,23,z(x2+2) +2T-1 一T 0-2 1-T +7 0(Rz;Ry),(xz,yz) +2 -TT-1 0 -2 十T 1-r [xy2,2(x2-y] D5h=D5⑧h

484 A Point Group Character Tables Table A.23. Character table for group C5v (icosahedral) C5v (5m) E 2C5 2C2 5 5σv x2 + y2, z2, z3, z(x2 + y2) z A1 1 1 11 Rz A2 1 11 −1 z(x, y), z2(x, y),(x2 + y2)(x, y) (x, y) (Rx, Ry) " E1 2 2 cos α 2 cos 2α 0 (x2 − y2, xy), z(x2 − y2, xy), [x(x2 − 3y2), y(3x2 − y2)] E2 2 2 cos 2α 2 cos 4α 0 α = 2π/5 = 72◦. Note that τ = (1 + √5)/2 so that τ = −2 cos 2α = −2 cos 4π/5 and τ − 1 = 2 cos α = 2 cos 2π/5 Table A.24. Character table for group D5 (icosahedral) D5 (52) E 2C5 2C2 5 5C 2 x2 + y2, z2 A1 1 1 11 Rz, z A2 111 −1 (xz, yz) (x, y) (Rx, Ry) " E1 2 2cos α 2cos 2α 0 (x2 − y2, xy) E2 2 2cos 2α 2cos 4α 0 Table A.25. Character table for D5d (icosahedral) D5d E 2C5 2C2 5 5C 2 i 2S−1 10 2S10 5σd (h = 20) A1g +1 +1 +1 +1 +1 +1 +1 +1 (x2 + y2), z2 A2g +1 +1 +1 −1 +1 +1 +1 −1 Rz E1g +2 τ − 1 −τ 0 +2 τ − 1 −τ 0 z(x + iy, x − iy) E2g +2 −τ τ − 1 0 +2 −τ τ − 1 0 [(x + iy) 2,(x − iy) 2] A1u +1 +1 +1 +1 −1 −1 −1 −1 A2u +1 +1 +1 −1 −1 −1 −1 +1 z E1u +2 τ − 1 −τ 0 −2 1−τ +τ 0 (x + iy, x − iy) E2u +2 −τ τ − 1 0 −2 +τ 1−τ 0 Note: D5d = D5 ⊗ i, iC5 = S−1 10 and iC2 5 = S10. Also iC 2 = σd Table A.26. Character table for D5h (icosahedral) D5h (102m) E 2C5 2C2 5 5C 2 σh 2S5 2S3 5 5σv (h = 20) A 1 +1 +1 +1 +1 +1 +1 +1 +1 x2 + y2, z2 A 2 +1 +1 +1 −1 +1 +1 +1 −1 Rz E 1 +2 τ − 1 −τ 0 +2 τ − 1 −τ 0 (x, y), (xz2, yz2), [x(x2 + y2), y(x2 + y2)] E 2 +2 −τ τ − 1 0 +2 −τ τ − 1 0 (x2 − y2, xy), [y(3x2 − y2), x(x2 − 3y2)] A 1 +1 +1 +1 +1 −1 −1 −1 −1 A 2 +1 +1 +1 −1 −1 −1 −1 +1 z, z3, z(x2 + y2) E 1 +2 τ − 1 −τ 0 −2 1−τ +τ 0 (Rx, Ry), (xz, yz) E 2 +2 −τ τ − 1 0 −2 +τ 1−τ 0 [xyz, z(x2 − y2)] D5h = D5 ⊗ σh

A Point Group Character Tables 485 Table A.27.Character table for the icosahedral group I(icosahedral) I(532) E 12C5 12C号 20C3 15C2 (h=60) A +1 +1 +1 +1 +1 x2+y2+22 +3 +r 1-T 0 -1 (,y,z);(R=,Ry,R:) +3 1-T +1 0 -1 G +4 -1 -1 +1 0 2z2-x2-y2 x2-y2 +5 +1 ty tz yz Table A.28.Character table for Ih(icosahedral) E12C512C号20C315C2 i12S1012S10205615a (h=120) Ag +1 +1 +1 +1 +1 +1 +1 +1 +1+1x2+y2+22 Fis +3 +T1-T 0 -1+3 T 1-T 0-1 R::Ru:R: F +3 1-T 十T 0 -1 +31-T T 0-1 Gg +4 -1 -1 +1 0+4 -1 -1 +1 0 222-x2-y2 x2-y2 Hg +5 0 0 -1 +1 +5 0 0 -1 +1 Ty Tz yz Au +1 +1 +1 +1 +1 -1 -1 -1 -1-1 iu +3 +T1-7 0 -1 -3 -TT-1 0+1 (x,y,2) F2u +31-7 +T 0 -1 -3T-1 一T 0+1 Gu +4-1 -1 +1 -4+1 +1 -1 0 Hu+5 0 0 -1 +1-5 0 0 +1-1 T=(1+v5)/2.Note:Cs and Cs are in different classes,labeled 12Cs and 12C in the character table.Then iCs=So and iCs=Sio are in the classes labeled 12Sio and 12510,respectively.Also iC2=ov and In=Ii Table A.29.Character table for group T (cubic) T(23) E 3C2 4C3 4C3 x2+y2+22 A 1 1 1 1 (x2-2,322-r2) E 1 w2 11 1 (Rz;Ry;R:) (x,,2) T 3 -1 0 0 (yz,zx,xy) w=e2i/3;Th=Ti,(m3)(cubic)

A Point Group Character Tables 485 Table A.27. Character table for the icosahedral group I (icosahedral) I (532) E 12C5 12C2 5 20C3 15C2 (h = 60) A +1 +1 +1 +1 +1 x2 + y2 + z2 F1 +3 +τ 1−τ 0 −1 (x, y, z); (Rx, Ry, Rz) F2 +3 1−τ +τ 0 −1 G +4 −1 −1 +1 0 H +5 0 0 −1 +1 ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ 2z2 − x2 − y2 x2 − y2 xy xz yz Table A.28. Character table for Ih (icosahedral) Ih E 12C5 12C2 5 20C3 15C2 i 12S3 10 12S10 20S6 15σ (h = 120) Ag +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 x2 + y2 + z2 F1g +3 +τ 1−τ 0 −1 +3 τ 1 − τ 0 −1 Rx, Ry, Rz F2g +3 1−τ +τ 0 −1 +3 1 − τ τ 0 −1 Gg +4 −1 −1 +1 0 +4 −1 −1 +1 0 Hg +5 0 0 −1 +1 +5 0 0 −1 +1 ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ 2z2 − x2 − y2 x2 − y2 xy xz yz Au +1 +1 +1 +1 +1 −1 −1 −1 −1 −1 F1u +3 +τ 1−τ 0 −1 −3 −τ τ − 1 0 +1 (x, y, z) F2u +3 1−τ +τ 0 −1 −3 τ − 1 −τ 0 +1 Gu +4 −1 −1 +1 0 –4 +1 +1 −1 0 Hu +5 0 0 −1 +1 –5 0 0 +1 −1 τ = (1 + √5)/2. Note: C5 and C−1 5 are in different classes, labeled 12C5 and 12C2 5 in the character table. Then iC5 = S−1 10 and iC−1 5 = S10 are in the classes labeled 12S3 10 and 12S10, respectively. Also iC2 = σv and Ih = I ⊗ i Table A.29. Character table for group T (cubic) T (23) E 3C2 4C3 4C 3 x2 + y2 + z2 A 1111 (x2 − y2, 3z2 − r2) E (1 1 1 1 ω ω2 ω2 ω (Rx, Ry, Rz) (x, y, z) (yz, zx, xy)  T 3 −10 0 ω = e2πi/3; Th = T ⊗ i, (m3) (cubic)

486 A Point Group Character Tables Table A.30.Character table for group O(cubic) 0(432) E 8C3 3C2=3C 6C 6C4 (x2+y2+z2) A1 1 1 1 1 A2 1 1 -1 -1 (x2-y2,3z2-r2) E 2 -1 0 0 (R=;Ry;R:) T 3 0 -1 -1 1 (c,y,2) (ty,y2,2x) T2 0 -1 -1 Oh=O⑧i,(m3m)(cubic) Table A.31.Character table for the cubic group Oh(cubic) repr.basis functions E3C6C46C码8C3i3iC6iC46iC吗 8iCs A时 1 111111111 1 x(2-z2)+ A y(z2-x2)+ 11-1-111 1-1 -1 1 (z4(x2-y2) E+ ∫x2-2 222-x2-y2 2200-12 20 0 -1 T E,y,2 3-11-1 0-3 1-1 1 0 T (x2-y2). 3-1-1 1 0-3 1 1 -1 0 xyz[4(y2-22)+ A (22-x2)+ 11111-1 -1-1 -1 -1 z4(x2-y2】 A 工y2 11-1-11-1-111 -1 E xyz(x2-y2)... 2200-1-2-2 00 1 T xy(a2-y2)... 3-11-103-11-1 0 T xy,yz,zx 3-1-1103-1-11 0 tThe basis functions for T2 are z(2-y2),x(y2-22),y(22-22),for E-are xyz(x2-y2),xyz(322-s2)and for Tf are ry(r2-y2),yz(y2-22),zx(22-x2) Table A.32.Character table for group Ta(cubic)a Ta(④3m) E 8C3 3C2 6od 6S4 x2+y2+z2 A 1 1 1 1 1 A2 1 1 1 -1 -1 (x2-y2,322-r2) E 2 -1 2 0 0 (R=;Ry;R=) T 2 0 -1 -1 1 y22t,xy） (x,,2) T 3 -1 1 -1 a Note that (yz,z,ry)transforms as representation Ti

486 A Point Group Character Tables Table A.30. Character table for group O (cubic) O (432) E 8C3 3C2 = 3C2 4 6C 2 6C4 (x2 + y2 + z2) A1 1 1 111 A2 11 1 −1 −1 (x2 − y2, 3z2 − r2) E 2 −1 200 (Rx, Ry, Rz) (x, y, z) " T1 3 0 −1 −1 1 (xy, yz, zx) T2 3 0 −1 1 −1 Oh = O ⊗ i, (m3m) (cubic) Table A.31. Character table for the cubic group Oh (cubic)† repr. basis functions E 3C2 4 6C4 6C 2 8C3 i 3iC2 4 6iC4 6iC 2 8iC3 A+ 1 1 1 1 1 1 11 1 1 1 1 A+ 2 ⎧ ⎨ ⎩ x4(y2 − z2)+ y4(z2 − x2)+ z4(x2 − y2) 1 1 −1 −1 11 1 −1 −1 1 E+ ( x2 − y2 2z2 − x2 − y2 2200 −12 2 0 0 −1 T − 1 x, y, z 3 −1 1 −1 0 −3 1 −11 0 T − 2 z(x2 − y2)... 3 −1 −110 −311 −1 0 A− 1 ⎧ ⎨ ⎩ xyz[x4(y2 − z2)+ y4(z2 − x2)+ z4(x2 − y2)] 11111 −1 −1 −1 −1 −1 A− 2 xyz 1 1 −1 −1 1 −1 −111 −1 E− xyz(x2 − y2). . . 2 2 0 0 −1 −2 −200 1 T + 1 xy(x2 − y2). . . 3 −1 1 −1 03 −1 1 −1 0 T + 2 xy, yz, zx 3 −1 −1 1 03 −1 −11 0 † The basis functions for T − 2 are z(x2 − y2), x(y2 − z2), y(z2 − x2), for E− are xyz(x2 − y2), xyz(3z2 − s2) and for T + 1 are xy(x2 − y2), yz(y2 − z2), zx(z2 − x2) Table A.32. Character table for group Td (cubic)a Td (43m) E 8C3 3C2 6σd 6S4 x2 + y2 + z2 A1 1 1111 A2 111 −1 −1 (x2 − y2, 3z2 − r2) E 2 −1 200 (Rx, Ry, Rz) yz, zx, xy) " T1 3 0 −1 −1 1 (x, y, z) T2 3 0 −1 1 −1 a Note that (yz, zx, xy) transforms as representation T1

A Point Group Character Tables 487 Table A.33.Character table for group Coo Coo(com) E 2C6 o (x2+y2,z2) A1(∑+) 1 1 R: A2(∑-) 1 1 -1 (xz,yz) (x,) E(II) 2cos中 0 (Rz,Ry) (x2-2,x) E2(△) 2 2 cos26 0 ... ... Table A.34.Character table for group Doh Doch (oo/mm) E 2C. C i 2iCo C x2+2,2 A1g(对) 1 1 1 A1u(∑a 1 1-1 -1 -1 R= A2g(∑g) 1 -1 1 1 -1 2 A2u(∑Dt) -1 -1 -1 (xz,yz) (Rz:Ry) E1g(Ⅱg） 2 2coso 0 2 2cos 0 (x,y) E1u(Ⅱu） 2 2cosφ 0 -2 -2c0s中 0 (x2-y2,x E2g(△g) 2 2c0s20 0 2 2 cos 26 0 E2u(△u) 2 2c0s2φ 0 -2 -2 cos 20 0 ... ·..

A Point Group Character Tables 487 Table A.33. Character table for group C∞v C∞v (∞m) E 2Cφ σv (x2 + y2, z2) z A1(Σ+) 11 1 Rz A2(Σ−) 1 1 −1 (xz, yz) (x, y) (Rx, Ry) " E1(Π) 2 2 cos φ 0 (x2 − y2, xy) E2(Δ) 2 2 cos 2φ 0 . . . . . . . . . . . . Table A.34. Character table for group D∞h D∞h (∞/mm) E 2Cφ C 2 i 2iCφ iC 2 x2 + y2, z2 A1g(Σ+ g ) 1 111 1 1 A1u(Σ− u ) 1 11 −1 −1 −1 Rz A2g(Σ− g ) 1 1 −11 1 −1 z A2u(Σ+ u ) 1 1 −1 −1 −1 1 (xz, yz) (Rx, Ry) E1g(Πg) 2 2 cos φ 0 2 2 cos φ 0 (x, y) E1u(Πu) 2 2 cos φ 0 −2 −2 cos φ 0 (x2 − y2, xy) E2g(Δg) 2 2 cos 2φ 0 2 2 cos 2φ 0 E2u(Δu) 2 2 cos 2φ 0 −2 −2 cos 2φ 0 . . . . . . . . . . . . . . . . . . . .