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 fivefold 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)