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 C 5 , C 5h , C 5v , D 5 , D 5d , D 5h , I , I h . The cubic point group O h in Table A.31 lists basis functions for all the irreducible representations of O h 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 x 2 ,y 2 ,z 2 , xy, xz, yz Rx,Ry ,Rz Ag 1 1 x,y,z Au 1 −1 Table A.3. Character table for group C 1h = S1 (monoclinic) C 1h (m) E σ h x 2 ,y 2 ,z 2 , xy Rz , x, y A 1 1 xz,yz Rx,Ry ,z A 1 −1
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A
Point Group Character Tables
Appendix A contains Point Group Character (Tables A.1–A.34) to be usedthroughout the chapters of this book. Pedagogic material to assist the readerin the use of these character tables can be found in Chap. 3. The Schoenfliessymmetry (Sect. 3.9) and Hermann–Mauguin notations (Sect. 3.10) for thepoint 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 groupOh in Table A.31 lists basis functions for all the irreducible representationsof Oh and uses the standard solid state physics notation for the irreduciblerepresentations.
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 1x, 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 1xz, 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, 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 1 1 1 1xy Rz A2 1 1 −1 −1xz Ry, x B1 1 −1 1 −1yz 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 1 1 1 1z Au 1 1 −1 −1
xz, yz Rx, Ry Bg 1 −1 −1 1x, y Bu 1 −1 1 −1
Table A.7. Character table for group D2 = V (orthorhombic)
D2 (222) E Cz2 Cy
2 Cx2
x2, y2, z2 A1 1 1 1 1xy Rz, z B1 1 1 −1 −1xz Ry, y B2 1 −1 1 −1yz 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 1 1 1 1 1Rz A2 1 1 1 −1 −1
x2 − y2 B1 1 1 −1 1 –1xy z B2 1 1 −1 −1 1
(xz, yz)(x, y)(Rx, Ry)
E 2 −2 0 0 0
D2h = D2 ⊗ i (mmm) (orthorhombic)
A Point Group Character Tables 481
Table A.9. Character table for group C3 (rhombohedral)
C3(3) E C3 C23
x2 + y2, z2 Rz, z A 1 1 1
(xz, yz)(x2 − y2, xy)
}(x, y)(Rx, Ry)
}E
{11
ωω2
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 1 1Rz 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 C23 σh S3 (σhC
23 )
x2 + y2, z2 Rz A′ 1 1 1 1 1 1z A′′ 1 1 1 −1 −1 −1
(x2 − y2, xy) (x, y) E′{
11
ωω2
ω2
ω11
ωω2
ω2
ω
(xz, yz) (Rx, Ry) E′′{
11
ωω2
ω2
ω−1−1
−ω−ω2
−ω2
−ωω = e2πi/3
Table A.12. Character table for group D3 (rhombohedral)
D3 (32) E 2C3 3C′2
x2 + y2, z2 A1 1 1 1Rz, 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)