Appendices Appendix A Character Tables* of the More Common Symmetry Point Groups 1. Cyclic Groups A C2 E C2 A 1 1 z R z xz., y2, Z2, xy B 1 -1 x,y Rx' R y YZ, zx 2 C3 E C3 C3 W = exp (2ni/3) A 1 1 1 Z Rz .~ + y2, Z2 1 W w2 xz. - y2, xy, x,y E{ 1 w* W*2 } } Rx' Ry } yz, zx 3 C4 E C4 C2 c4 2 A 1 1 1 1 z R z x + I, Z2 B 1 -1 1 -1 r -I,xy 1 i -1 -i E { x,y Rx' Ry YZ,zx 1 -i -1 i } } } • Adapted from 1. Gutman and O.E. Polansky: Mathematical Concepts in Organic Chemistry. Springer, Berlin Heidelberg New York London Paris Tokyo 1986. Appendix 5, pp. 184-199 284 Appendix A. Character Tables ofthe More Common Symmetry Point Groups Cs E Cs Cs2 Cs3 Cs4 W = exp (271ij5) I A 1 1 1 1 1 - i R% xl +.1, Z2 1 W w2 w3 w4 { } x,y } yz,zx E1 1 w* W*2 W*3 W*4 }R", Ry 1 w2 w4 W w3 } xl -.I, xy E2 { 1 W*2 W*4 w* W*3 2 S C6 E C6 C3 C2 C3 C6 W = exp (271ij6) A 1 1 1 1 1 1 Z R% X2 +.I,z2 B 1 -1 1 -1 1 -1 1 W w1 -1 -w _w2 E1 { 1 w* W*2 -1 -w* _W*2 } x,y } R", Ry } YZ,zx 1 w2 w4 1 w2 w4 Ez { 1 W*2 W*4 1 W*2 W*4 } xl -.I, xy 2. Cnv Groups C2v E C2 U V Ud zx yz Al 1 1 1 1 z xl, y2, Z2 A2 1 1 -1 -1 R% xy B1 1 -1 1 -1 x Ry zx B2 1 -1 -1 I Y R yz " I C3v E 2C3 3uv Al 1 1 1 z xl +.1, Z2 A2 1 I -1 R% -.I, xy E 2 -1 0 {.r x,y I R", Ry yz,zx Appendix A. Character Tables of the More Common Symmetry Point Groups 285 c4V E 2C4 C2 2a v 2ad AI 1 1 1 1 1 Z X2 +/,Z2 A2 1 1 1 -1 -1 Rz 2 8 1 1 -1 1 1 -1 x _/ 8 2 1 -1 1 -1 1 xy E 2 0 -2 0 0 x,y RX,R), YZ,ZX C5v E 2C5 2c; 5a,. <p = 2rr:5 AI 1 1 1 1 Z :xl + /, .::2 A2 1 1 1 -1 R z EI 2 a b 0 x,y Rx ' Ry yz, ZX E2 2 b a 0 :xl -/, xy I a = 2 cos <p = 2 cos 4<p = (0 - 1)/2 = 0.618034, b = 2cos2<p = -(0 + 1)/2 = -1.618034. C6v E 2C6 2C3 C2 3a v 3ad AI 1 1 1 1 1 1 Z :xl + /, Z2 A2 1 1 1 1 -1 -1 R z 8 1 1 -1 1 -1 1 -1 82 1 -1 1 -1 -1 1 EI 2 1 -1 -2 0 0 x,y Rx ' Ry y.::, zx E2 2 -1 -1 2 0 0 xZ-/,xy 3. Cnh Groups C1h E ah Clh = es xy A' 1 1 X,y R x2, y2, Z2, xy AU z 1 -1 z R x ' R y yz,zx 286 Appendix A. Character Tables ofthe More Common Symmetry Point Groups C2h E C2 i (jh 2 Ag 1 1 1 1 R z x , y2, Z2, xy A u 1 1 -1 -1 z 8g 1 -1 1 -1 R x ' Ry yz, zx 8 u 1 -1 -1 1 x,y 2 S2 C3h E C3 C3 (jh S3 3 A' 1 1 1 1 1 1 Rz :x? + f, r A" 1 1 1 -1 -1 -1 z 1 W w2 1 W w2 E' { } x,y }:x? - f, xy 1 w* W*2 1 w* W*2 1 W w2 -1 -w _w2 Eil { } yz, zx 1 w* W*2 -1 -w* _W*2 }Rx' Ry 3 S3 C4h E C4 C2 C4 i 4 (jh S4 Ag 1 1 1 1 1 1 1 1 R z :x? + f, r A u 1 1 1 I -1 -1 -1 -1 z 8 g 1 -1 1 -1 1 -1 1 -1 :x?-f,xy 8 u 1 -1 1 -1 -1 1 -1 1 1 i -1 -i 1 i -1 -i } YZ,zx Eg { 1 -i -1 i 1 -i -1 i } Rx ' Ry 1 i -1 -i -1 -i 1 i } x, Y Eu { I -i -1 i -1 i 1 -i > 2 3 '0 C5h E C5 C C (J/, S5 S; S3 W = exp (2n i/5) '0 5 5 C! 5 ~ <tl ------~- ::; X 2 +y2,z2 8: A' I I I >< AU ~I --I ~I ~I ~l > 2 3 4 2 3 4 (1 w w w w W w w w R, P"' E; { W*2 w*3 W*4 W*2 W*3 W*4 }:~yl w* I w* ~ ~ w w 2 w 3 w 4 ~I ~(I) ~·ul ~W3 ~W4 M­ <tl } YZ,::X ..., Ei' { w* W*2 W*3 (1)*4 ~I ~w* __ W*2 ~W*3 __ W*4 } Rx ' R y 3 w 2 w 4 W w 3 u/ w 4 w W 0"~ E2 { }X2~y2,XY (0 W*2 W*4 w* W*3 w*Z W*4 w* W*3 cn o w Z w 4 W w 3 ~I --ol ~W4 ~w ~u/ ...., E2' { w*Z W*4 W*3 ~I ~W*2 ~W*4 ~w* ~W*3 ;. w* <tl s;: o -------_. Iil (1 C6h E C C C C5 S2 S5 W = cxp (2n i/6) 6 3 2 C; 6 3 6 (J/, S'" S3 S~ o A ~ Z2 ::; 9 1 R z + jl, r:n A 1 ~I ~I -I ~I ~I --I Z " '<: 8 g ~I ~I ~l I ~l ~I I ~I S S <tl Bu ~l ~I ~I ~I ~l 1 -I M­ ..., w w 2 ~I ~W ~W2 1 (j) U)2 ~l ~w ~wz '<: E 10 { } Rx ' Rr } yz, ZX ""0 w* w*Z ~l ~w* ~W*2 w* (j)*Z ~l ~w* ~w*Z 9. ::; W w Z ~l ~w ~W2 ~l ~w ~W2 w w 2 M- } X, y CJ EI" { w'~ w*Z ~l ~w* ~w*Z ~l ~w* ~w*Z w* w*Z ..., w Z w 4 1 w 2 w 4 w 4 w 2 w 4 g 0./ 2 '0 EZg { } x ~ y2, xy W*2 W*4 W*2 W*4 1 W*2 W*4 W*2 W*4 '" w Z w 4 w 2 w 4 ~l ~wZ ~W4 ~l ~wZ ~W4 E2u { W*2 W*4 W*2 W*4 ~l ~W*2 ~W*4 ~l ~W*2 ~W*4 -l00"-' 288 Appendix A. Character Tables ofthe More Common Symmetry Point Groups 4. S2n Groups S2 E i S2 = Ci Ag 1 1 Rx ' Ry' R. x2,~,z2,xy,yz,zx A u 1 -1 x,y,z S3 S4 E S4 C2 4 A 1 1 1 1 R. _~ +~, Z2 B 1 -1 1 -1 z x2 _ y2, xy 1 i -1 -i } x,y } YZ,zx E{ 1 -i -1 i } Rx' Ry S6 E C3 C32 i S56 S6 I w = exp (2ni/3) A 1 1 1 1 1 1 R x2 + y2, z2 9 Au 1 1 1 -1 -1 -1 z 1 w ai 1 w w2 } x2 -/, xy, yz, zx Eg { 1 w* W*2 1 w* W*2 } Rx ' Ry I W w2 -1 -w -ai { }x,y Eu 1 w* W*2 -1 -w* _W*2 5. Dn Groups D2 E C2 C2 C2 D2 = V Z Y x A 1 1 1 1 x2, y2, Z2 -1 R xy BI 1 1 -1 z % B2 1 -1 1 -1 Y Ry zx B3 1 -1 -1 1 x R x YZ D3 E 2C3 3C2 AI 1 1 1 x2 + /, Z2 A2 1 1 -1 Z Rz 2 E 2 -1 0 x,y Rx' Ry x -/, xy, YZ, zx D4 E 2C~ C2 2C; 2C; AI 1 1 1 1 1 x2 + /, Z2 A2 1 1 1 -1 -1 Z R = 8 1 1 -1 1 1 -1 82 1 -I 1 -1 1 E 2 0 -2 0 0 x,y Rx ' R)' ~ -/, xy, yz, zx Ds E 2Cs 2C; 5C2 rp = 21[/5 AI 1 1 1 1 ~ +/,1 A2 1 1 1 -1 z Rz EI 2 a b 0 x,y Rx' R v yz, zx E2 2 b a 0 x2 -/, xy I a = 2 cos rp = 2 cos 4rp = (Vs - 1)/2 = 0.618034, b = 2 cos 2rp = -(Vs + 1)/2 = -1.618034. D6 E 2C6 2C3 C2 3C; 3C;' AI 1 1 1 1 1 1 x2 + /, Z2 A2 1 1 1 1 -1 -1 Rz 8 1 1 -1 1 -1 1 -1 8 2 1 -1 1 -1 -1 1 Z EI i 2 1 -1 -2 0 0 x,y Rx ' Ry yz, zx 2 Ez I 2 -1 -1 2 0 0 x _ y2, xy I 6. Dnd Groups D2d E 2S4 C2 2C; 2C;' Du = Vd AI 1 1 1 1 1 x 2 + /, Z2 A2 1 1 1 -1 -1 Rz 8 1 1 -1 1 1 -1 x2 -/ 8 2 1 -1 1 -1 1 z xy E 2 0 -2 0 0 x,y Rx' Ry yz, zx 290 Appendix A.