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,.
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
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 S3 '0 C5h E C5 C C (J/, S5 S; W = exp (2n i/5) 5 5 C! 5 ~ '0 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. Character Tables of the More Common Symmetry Point Groups D34 E 2C3 3C; i 2S3 30"4 Alg I I I I I I xl + y2, Z2 Alu I I I -1 -1 -1 A2g 1 1 -1 1 1 -1 R :r A2u 1 I -1 -1 -1 1 Z Eg 2 -1 0 2 -1 0 R", Ry xl - y, xy; YZ, zx E. 2 -1 0 -2 I 0 x,y 3 D44 E 2Ss 2C4 2SS C2 4C; 40"4 AI 1 1 1 1 1 1 I x2 + l, r A2 I I I I I -I -I R. 8 1 I -I 1 -I I I -1 8 2 1 -1 1 -I I -I I z EI 2 ]/2 0 -]/2 -2 0 0 x,y E2 2 0 -2 0 2 0 0 xl - y, xy E3 2 -]/2 0 ]/2 -2 0 0 R", R, I Y=. zx 2 DS4 E 2Cs 2C5 5C~ i 2S:o 2SIO 50"4 cp = 21[/5 Alg I I 1 I I 1 I I xl + y. r Alu I I 1 1 -I -I -I -I A2g 1 I I -I I I I -I R. A2• I I I -I -1 -I -I I z EIl 2 a b 0 2 a b 0 R", R, y=.zx EIU 2 a b 0 -2 -a -b 0 x,y E2f 2 b a 0 2 b a 0 xl - y,xy E2u 2 b a 0 -2 -b -a 0 a = 2 cos cp = 2 cos 4cp = Ws - 1)/2 = 0,618034, b = 2 cos 2cp = -q/5 + 1)/2 = -1,618034. D64 E 2S12 2C6 2S4 2C3 2.Si2 C2 6C~ 60"4 AI I I I I I I I I I x2 + y,r A2 I I 1 1 1 I I -1 -1 R. 8 1 1 -1 I -1 I -1 1 1 -I 82 1 -1 I -1 I -1 I -1 I Z EI 2 0 1 0 -I -0" -2 0 0 x,y E2 2 1 -I -2 -I I 2 0 0 xl-y,xy E3 2 0 -2 0 2 0 -2 0 0 E4 2 -I -I 2 -I -1 2 0 0 ~,R, YZ,zx Es 2 -0 I 0 -1 0" -2 0 0 Appendix A. Character Tables of the More Common Symmetry Point Groups 291 7. Dnh Groups Dlh E Cl C2 C2 i a a a D2h = Vh '" Y X xy ZX yz A g 1 1 1 1 1 1 1 1 x2, yl, Z2 A u 1 1 1 1 -1 -1 -1 -1 Big 1 1 -1 -1 1 1 -1 -1 Rz xy Blu 1 1 -1 -1 -1 -1 1 1 Z B2g 1 -1 1 -1 1 -1 1 -1 R y zx B2u 1 -1 1 -1 -1 1 -1 1 Y B3g 1 -1 -1 1 1 -1 -1 1 Rx yz B3u 1 -1 -1 1 -1 1 1 -1 X D3h E 2C3 3C~ ah 2S3 3a v 2 A'I 1 1 1 1 1 1 x + l, Z2 A"I 1 1 1 -1 -1 -1 A'2 1 1 -1 1 1 -1 Rz A"2 1 1 -1 -1 -1 1 z E' 2 -1 0 2 -1 0 x,y ~-l,xy E" 2 -1 0 -2 1 0 Rx' Ry yz, zx D4h E 2C4 C2 2C~ 2C;' i 2S4 ah 2av 2ad Aig 1 1 1 1 1 1 1 1 1 1 ~ + /, Z2 Alu 1 1 1 1 1 -1 -1 -1 -1 -1 A, 1 1 1 -1 -1 1 1 1 -1 -1 R ~g z A2u 1 1 1 -1 -1 -1 -1 -1 1 1 Z Bill 1 -1 1 1 -1 1 -1 1 1 -1 .>? -l Bill 1 -1 1 1 -1 -1 1 -1 -1 1 B2g 1 -1 1 -1 1 1 -1 1 -1 1 xy B2u 1 -1 1 -1 1 -1 1 -1 1 -1 E 2 0 ~-2 0 0 2 0 -2 0 0 yz,zx 9 Rx ' R), Eu 2 0 -2 0 0 -2 0 2 0 0 x,y t..:> DSh E 2Cs 2C; 5C~ Gh 2Ss 2S~ 5Gv A'1 1 1 1 1 1 1 1 1 _~+r,Z2 AU 1 1 1 1 -1 -1 1 -1 -1 "0>• A' "0 2 I 1 1 -I 1 1 I -1 R z ~ AU ::s 2 1 1 1 -1 -1 -1 -1 I Z p.. E' ><- 1 2 a b 0 2 a b 0 x,y EU 2 a b 0 -2 -b 0 yz, ZX ?> 1 -a Rx ' Ry Ci ::r' E'2 2 b a 0 2 b a 0 xl -r,xy EU 2 b a 0 -2 -b -a 0 ~ 2 ~ ~'"'" .... a = 2 cos T E 4C3 4c; 3C2 W = exp (211: i/3) A 1 1 1 1 x2 + l + :2 1 w w2 1 }x2 + l -2z2 E{ 1 w* W*2 1 x2 _ y2 F 3 0 0 -1 x,y, z Rx."Rr• Rz X.1'.),:. :x Td E SC3 3C2 6S4 6ad AI 1 1 ~+y2+:2 A2 1 1 1 -I -I E 2 -I 2 0 0 x2 + .1'2 _ 2z2• x2 _ y2 F I 3 0 -I -I Rx ' Rr• R~ -1 F2 3 0 -I Ix. Y. Z xy,yz, :x --L TI, E 4C) 4C; 3C2 i 4S~ 4S6 3a. w = exp (211: i/3) A 1 1 1 1 1 1 1 1 I X2 +y2+:2 9 A 1 1 1 1 -1 -I -1 -1 u , I (J) ur 1 I (J) w2 1 }.~ +.1- 2; Eg { 1 w* W*2 1 1 w* W*2 1 x2 _)'2 2 1 (J) w2 1 -I -(') _w -I Eu { 1 w* W*2 1 -I -w* _W*2 -1 zx Fg 3 0 0 -1 3 0 0 -1 Rx' R)" R~ xy,)'z, -3 I x, y, z F u 3 0 0 -1 0 0 0 E 8C) 3C2 6C4 6C~ A, 1 I 1 1 1 x2 + l + Z2 A2 1 1 1 -I -1 E 2 -1 2 0 0 x2 + l- 2z2, x2 -l FI 3 0 -1 1 -1 x,y,z Rx' R)" R, F2 3 0 -1 -1 1 xy.yz,zx Oh E 8C3 6Cl 6C4 3Cl(= C42) i 6S4 8S6 30"k 60"d A ig 1 1 1 1 1 1 1 1 1 1 r+l+z2 A lg 1 1 -1 -1 1 1 -1 1 1 -1 Eg 2 -1 0 0 2 2 0 -1 2 0 r-I-2r.r-1 T ig 3 0 -1 1 -1 3 1 0 -1 -1 R.,Ry,R, T2g 3 0 1 -1 -1 3 -1 0 -1 1 Xz,yx,xy AI. 1 1 1 1 1 -1 -1 -1 -1 -1 Alu 1 1 -1 -1 1 -1 1 -1 -1 1 Eu 2 -1 0 0 2 -2 0 1 -2 0 TI. 3 0 -1 1 -1 -3 -1 0 1 1 x,y,z T2. 3 0 1 -1 -1 -3 1 0 1 -1 9. The Icosahedral Groups tv I E 12Cs 12C; 20C3 15C2 >• "C "C (t) A 1 1 1 1 ~+y2+Z2 ::l 0... F1 3 a b 0 -I x,y,z R.•• Ry, R: ><. F2 3 b a 0 -I >• G 4 -I 0 (J -I ::r- {~ + y2 _ 2z2 H 5 0 0 -1 ~ ~ x2 - y2, xy, yz, zx (t) -.... ~ g: (t) 2 3 CI> I h E 12Cs 12Cs 20C3 15C2 i 12S1O 12S10 20S6 15a o ..... Ag 1 1 1 1 1 1 1 1 1 1 r+y+? ::r- -(t) T lg 3 a b 0 -1 3 b a 0 -1 Rz,Ry,Rz ~o T2g 3 b a 0 -1 3 a b 0 -1 @ Gg 4 -1 -1 1 0 4 -1 -1 1 0 (J Hg 5 0 0 -1 1 5 0 0 -1 1 r+y-2?, ~ r-y. S xy,yz, zx o ::l 1 1 1 r.n Au 1 1 -1 -1 -1 -1 -1 '< T lu 3 a b 0 -1 -3 -b -a 0 1 x,y,z § T2u 3 b a 0 -1 -3 -a -b 0 1 (t) .... G u 4 -1 -1 1 0 -4 1 1 -1 0 '<- '"':j H u 5 0 0 -1 1 -5 0 0 1 -1 8. ~ Q .... a = (I + Vs)/2 = 1,618034, §l "C b = (I - Vs)/2 = -0,618034. CI> 10. Continuous Groups for Linear Moleeules :-> '"<:l '"<:l t-..:> Kerneis and Co-Kernels of Degenerate Irreducible Representations* The sym-ops preserved in the kernels and co-kernels are given in parenthe• sis when necessary to specify the alinement of the subgroup. They are labeled as in Appendix A. Table B.I: D n Groups Table B.2: C nv Groups Group irrep Kernel Co - Kernels Group irl'ep Kernel Co - Kerneis D 3 E CI C 2 C lv E CI C. D 4 E CI C2(C~), C2(C~') C 4v E CI C.( D s EI CI C 2 C Sv EI CI C. E2 CI C 2 Ez CI C. Ds EI CI C2(C~) , C2(C~') C Sv EI CI C.( Table B.3: C nh Groups Table B.4: DnhGroups Group irrep Kernel Co - Kerneis Group I irrep Kernel Co - Kerneis C 3h E' C. - D 3h E' C.( C 4h Eg Ci - D 4h Eg Ci C 2h (C;) , C 2h (C;'j Eu C. - Eu C.( C Sh E' C. - I D 5h E'I C.( - C Sh Eig Ci D Sh Eig Ci C 2h ( C;) , C 2h ( C;'j - E2g C Zh E2g C 2h (Cz) D 2h Elu C. - Elu C.( • Adapted from Table 1 of P. Murray-Rust, H.-B. Bürgi and J.D Dunitz, Acta Crystallographica A35, 703 (1979). Appendix B. Kerneis and Co-Kernels of Degenerate Irreducible Representations 297 Table B.5: 5 0n Groups Group irrep Kernel Co - Kernels 5, E Cl - 56 E, C, - Eu Cl - 58 EI Cl - Eo C 2 - E3 Cl - Table B.6: D nd Groups Table B.7: Axial and Cylindrical Groups Group irrep Kernel Co - Kernels Group irrep Kernel Co - Kerneis D 2d E Cl C 2 (C;) , C,(iTd) C=u II Cl C, b. Cl C, D 3d E, C, C 2h eie ...... Eu Cl Co, C, D'd EI Cl C2 (C;), C, D=h II9 C, C 2h ( Cz) E2 C 2 (C2 ) C2v , D 2 b. 9 C, C 2h ( Co) E3 Cl Co(C;), C, eie ...... D Sd El9 C, C 2h IIu Cl C ou ( Cz) E29 C, C Oh b. u Cl C zu ( Co) Elu Cl C 2 , C, eie ...... E2u Cl Co, C, D 6d EI Cl C 2 (C;), C, Eo C O(C2 ) C 2v 1 D 2 E3 C3 C 3u , D 3 E, 5, D 2d Es Cl C 2 (C;) , C, Table B.8: Cubic Groups Group irrep Kernel Co - Kernels T d E Do Du Tl Cl 5, , C 3 , C, Tz Cl C 211 , CJv , C, Oh E, D 2h (CI,CJ) D'h Eu D2(Q,Ci) D, , D 2d Tl, C, C'h, 56, C 2h (C2) Tz, C, DZh ( Ci, Ci) , D 3d , C2h ( Co) Tlu Cl C'u, C 3u , C 2u (CO) , C,(iTd), C,(iTh) T2u Cl DZd(C;,CZ)' D3 , Czv(Co), C2(C2 ), C,(iTh) Appendix C Group Correlation Tables* °h ° Td D4h C 4v C 2v D3 D2d Alg Al Al Alg Al Al Al Al A2g A2 A2 BIg BI A2 A2 BI Eg E E Alg + BIg Al + BI Al + A2 E Al + BI TIg Tl Tl A2g + Eg A2+E A2 + BI + B2 A2+E A2+E T2g T2 T2 B2g + Eg B2+E Al + BI + B2 Al +E B2+E Alu Al A2 Alu A2 A2 Al BI A2u A2 Al Blu B2 Al A2 Al Eu E E Alu + Blu A2 +B2 Al + A2 E Al + BI Tlu Tl T2 A2u + Eu Al +E Al + BI + B2 A2+E B2+E T2u T2 Tl Bh + Eu BI +E A2 + BI + B2 AI+E A2+E T d T Du C 2v 54 D2 C 2v C3 C2 Al A Al Al A A Al A A A2 A BI A2 B A A2 A A E E Al + BI E A+B 2A Al +A2 E 2A Tl T A2 +E A 2 +E A+E BI + B2 + B3 A2 + BI + B2 A+E A+2B T2 T B2 +E Al +E B+E BI + B2 + B3 Al + BI + B2 A+E A+2B q ---+ C~ C'2 C"2 C'2 D4h D4 D2d C 4v C 4h D2h D2h C4 54 D2 Alg Al Al Al Ag Ag Ag A A A A2g A2 A2 A2 Ag BIg B Ig A A BI Bl9 BI BI BI Bg Ag BIg B B A B 2g B2 B2 B2 Bg Bl9 Ag B B BI Eg E E E Eg B2g + B3g B2g + B3g E E B2 +B3 Alu Al BI A2 Au Au Au A B A A2u A2 B2 Al Au Blu Blu A B BI Blu BI Al B2 Bu Au Blu B A A B2u B2 A2 BI Bu Blu Au B A BI Eu E E E Eu B2u + B3u B2u + B3u E E B2 +B3 • Adapted from B.E. Douglas and C.A. Hollingsworth: Symmetry in Bonding and Spectra. An Introduction. Acadernic Press Inc., Orlando San Diego New York London Toronto Montreal Sydney Tokio 1985. Appendix 3, pp. 412-414 Appendix C. Group Correlation Tables 299 D 4h C2 , 17 v C2 , 17d C2 C'2 C"2 C2 C'2 17h 17v (cont. ) C2v C2v C2h C2h C2h C2 C2 C. Cs Ci Alg Al Al Ag Ag Ag A A A' A' Ag A2g A2 A, Ag Bg Bg A B A' A" Ag Blg AI A2 Ag Ag Bg A A 11' A' Ag B2g A2 AI Ag Bg Ag A B A' A" Ag E g BI + JJ 2 BI +B2 2Bg Ag +Bg Ag +Bg 2B A+B 2A" A'+A" 2Ag Alu A2 A2 Au Au Au A A A" A" Au A2u AI AI Au Bu Bu A B A" A' Au Blu A2 AI Au Au Bu A A A" A" Au B2u AI A2 Au Bu Au A B A" A' Au Eu BI +B2 BI + 8 2 213" Au+B" A"+B,, 2B A+B 2A' A'+A" 2A" C2 (z) C,(y) C2 (x) C2 (z) C2 (y) C2 (x) C2(z) C2 (y) C2 (x) 17(xy) 17(Yz) D 2h D2 C2" C2" C2" Clh C2h C2h C2 C2 C2 C. Cs Ag A AI Al AI 11g Ag Ag A A A A' A' Big BI A2 B2 BI Ag Bg Bg A B B A' A" B2g B2 BI A2 132 Bg Ag Bg B A B A" A" 8 39 B3 B2 131 A2 Bg Bg Ag B B A A" A' Au A A2 A2 A2 Au Au Au A A A A' A" Blu BI AI BI B2 Au Bu Bu A B B All A' B2u B2 B2 Al BI Bu Au Bu B A B A' A' B3" B3 BI B2 Al 13" Bu A" B B A A' A" C2 C'2 D4d D4 C 4v S8 C4 C 2v C2 C2 C s Al Al Al A A Al A A A' A 2 A 2 A2 A A A2 A B A" BI Al A 2 B A A2 A A A" B 2 A2 Al B A Al A B A' EI E E EI E BI +B2 2B A+B A'+A" E 2 BI +B2 BI + B 2 E2 2B Al + A2 2A A+B A'+A" E3 E E E3 E BI + B2 2B A+B A' + A" C2 ~ C2(z) C2 C~ D2d S4 D2 C 2v C2 C2 C s Al A A Al A A A' A 2 A BI A 2 A B A" BI B A A 2 A A A" B 2 B BI Al A B A' E E B 2 + B 3 BI + B 2 2B A+B A'+A" 300 Appendix C. Group Correlation Tables D3d D3 C3v 86 C3 C2h C2 Cs C·• Alg Al Al Ag A Ag A A' Ag A2g A2 A2 Ag A Bg B A" Ag Eg E E Eg E Ag +Bg A+B A' + A" 2Ag Alu Al A2 Au A Au A A" Au A2u A2 Al Au A Bu B A' Au Eu E E Eu E Au +Bu A+B A' + A" 2Au ah ~ av(zy) ah av D3h C3h D3 C3v C2v C3 C2 Cs Cs A'I A' Al Al Al A A A' A' A'2 A' A2 A2 B2 A B A' A" E' E' E E Al + B2 E A+B 2A' A'+A" A"I A" Al A2 A2 A A A" A" A"2 A" A2 Al BI A B A" A' Eil Eil E E A2+B I ·E A+B 2 A" A' + A" Subject Index acetone, triplet 237 Bader-Pearson-Salem approach 4, 255, 258 acetylene (C 2 H2) barbaralane, Cope rearrangement 193, -, [.. 2 +.. 2]-cycloaddition 250 196-197 -, configuration, ground-state 87 basis set, minimal 77, 198 -, cycloaddition to dioxygen 171 benzene, 7r-system 5-7 -, cyclodimerization 171 -, excited configurations 255-256 -, excited states 87 -, excited states 237, 255-256 -, faciality 5 -, faciality 5 -, spin states 87 -, photoisomerization 255-258 -, symmetry coordinates 83 -, - to benzvalene 256-257 acrylonitrile, cycloaddition to allene 150 -, - to Dewar benzene 258 alignment, angular momentum 218, 229 -, valence isomers 125 -, symmetry coordinates 102 benze ne-anthracene adducts, cycloreversion -, vibration al coordinates 97 179 allene, cycloaddition to acrylonitrile 150 benzene dimer, cycloreversion 178-181 -, - to hexachlorocyclopentadiene 150 benzvalene (BV) 255-257 "allowedness" 119-132 -, isomerization 122-125, 151, 161,260 -, passive 121 Berry pseudorotation 269-270 anasymmetrization 161-166, 191, bicyclo[4.2.0]octa-2,4,7-triene (BCO) 194-196,206-209, 253 -, cis-trans-isomerization 118-119 - rules 164, 200, 203, 254 -, interconversion with cycloocta- angular momentum 29-33, 66 tetraene 118-119 - coupling 262, 218, 222, 229, bicyclo[4.2.2]-deca-trans-3, cis-7 ,9- triene 237, 276-280 -, dimerization 143 -, orbital 31, 215 cis-bicyclo[6 .1. 0]nona-2,4 ,6-triene -, orientation 229 -, re arrangement 205 -, spin 31 bicyclobutane, isomerization to CYclobutene -, -, analogy with orbital angular 124 momentum 216-218 bicyclopentene, rearrangement 205 -, total 222 bimanes see 9,10-dioxabimanes anharmonic coupling 210 biradical, perfect 220 [10]annulene, interconversion -, selectivity 235 with 9,10-dihydronaphthalene 117 -, tetramethylene 143 antiaromaticity 5-10 bisnorcaradiene ([4.4.1]propellatetraene) antisymmetrization 30, 65, 218-221 -, isomerization to 1,6-methano- aromaticity 5-10 [lO]annulene 116 associative law 36 bond symmetry rule 111 avoided crossing 72, 118, 149 bond-bisection requirement 16, 91, 122, 187 axis convention 74, 94, 101, 123, 126,226 Born-Oppenheimer approximation 57 azide ion (N3"), molecular orbitals 169-170 boron atom, c1assical model 29-30 -, cycloaddition to ethylene 169 -, electron configuration 29-33 azulene, fluorescence 245-246 302 Subject Index s-cis-butadiene configuration correlation 18-19, 131 -, cyclization to cyclobutene 8-9, 110-111, - interaction (CI) 70, 94, 145, 149, 218 114-116, 135 - mixing 70 -, molecular orbitals 11 conrotation 5 -, rearrangement to bicyclobutane 119-121 coordinate -, symmetry coordinates 114, 119 -, cartesian tert-butyl cyanoketene (TBCK) -, -, irreducible representation 39 -, cycloaddition to trimethyl- -, deformation 82 siloxypropene 150, 154 -, intern al 101 -, reaction 91-93, 102-104 carbenes -, -, anharmonic cross-terms 209 -, [l.n}-paracyclophane 223-224 -, rotation 74 -, intersystem crossing 223 -, -, symmetry properties 40 -, singlet, dimerization 90-94 -, translation 74 -, -, cycloaddition to alkenes 135-139 -, vibration 75, 82 -, triplet, dimerization 90 -, -, symmetry properties 74 see also methylene Cope rearrangement 187-188, 192-197 carbon atom, electron configuration 30 -, computational studies 193 carbon dioxide (C0 2) 77 core, atomic 77 -, linearity 82 correlation, configuration 18-19, 131 carbon monoxide (CO) -, orbital 15-16 -, electron configurations 74 -, -, vs. correspondence 113 -, orbital correlation diagram 70 correlation diagram -, 'Ir --+ .,... transition 74 -, Mulliken 4, 63 catalysis, homogeneous 3 -, -, heteronuclear diatomic molecules 71 centrifugal distortion 74, 91 -, -, homonuclear diatomic molecules 63 character 39 -,orbital 11-17 -, of a matrix see trace -, -, cyclization of cis-1,3,5-hexatriene 113 - table 39 -, -, [.. 2. + .. 2.}-cycloaddition 20 charge density 32 -, -, [.. 2a + .. 2.}-cycloaddition 20 -, directional properties 33 -, -, [.. 4a + .. 2.}-cycloaddition 16 -, symmetry 39, 61 -, -, [.. 4. + .. 2.}-cycloaddition 16 charge distribution 30 -, -, dimerization of methylene 89 chemiluminescence 248 -, -, isomerization of cyclooctatetraene chirality 55 to bicyclooctene 119 Claisen re arrangement 193 -, state 68 closed shell 7, 29 -, Walsh see Walsh diagram -, symmetry 66 correlation line 16 closure 36 correlation table co-kernel 69, 83 -, D ooh to C oo• to C2• 73 col see saddle point -, Dooh to D2h 69 combinations, linear correspondence 111 - of vibrational coordinates 99-10 1 -, comparison with correlation 113 - of configurations 68 -, direct 113 - of orbital products 219 -, induced 111-113 -, orthogonal 33, 37, 41, 47 correspondence diagram 91, 110 complete set 76 -, addition of CX2 to ethylene 136 complex ions, symmetry 48 -, Berry pseudorotation 270 -, square-planar 96 -, Cope rearrangement 195 -, -, interconversion with tetrahedral 103 -, cyclization, butadiene 110, 115 -, -, vibrational coordinates 96-98, 100-101 -, -, cis-l,3,5-hexatriene 112 composite motion 138, 254 -, cycloaddition, [.. 4 +.. 2)163 see also vibration, complex -, -, azide ion to ethylene 169 concertedness 109, 125, 151, 192 -, -, ozone to ethylene 170 configuration see electron configuration -, cyclodimerization of ethylene 140 Subject Index 303 -, dimerization, cyclobutadiene (to -, diphenylketene to styrene 151 cubane) 173 -, -, to a-methylstyrene 152 -, -, cyclobutadiene (to tricyclo- -, 1,3-dipolar 168-171 octadiene) 175, 177 -, ethyl vinyl ether to dimethyl azo- -, -, ethylene, stepwise 144 dicarboxylate 152 -, -, methylene 89 -, ketene-alkene 149-157 -, -, silacyclopropene 184 -, ozone to ethy lene 170 -, fragmentation of 1,2-dioxetane 238 -, teri-butyl cyanoketene to trimethyl- -, homo-Diels-Alder reaction 167 siloxypropene 150, 154 -, interconversion, 9,10-dihydronaphthalene -, tetracyanoethylene to alkenes 147 and [10]annulene 117 -, [.. 2 +.. 2] 5-10, 12-13, 19-21, 273 -, -, butadiene and bicyclobutane 120 -, -, biradical143-146 -, isomerization, benzvalene to benzene 123 -, -, concerted 139, 147 -, -, cyclobutadiene 208, 210 -, -, frontier-allowed pathway 12 -, -, Dewar benzene to benzene 125, 236 -, -, off-orthogonal approach 13,20, 142 -, -, NiX~- 276 -, -, stepwise 145-157 -, -, prismane to benzene 125 -, -, Woodward-Hoffmann-Rules 119 -, 102 + acetylene -t dioxetane -, -, zwitterionic 143, 147-150 -t a-diketone 249 -, [.. 4 +.. 2]5-10, 15-19,161-167 -, photoisomerization -, -, alternative approaches 166 -, -, benzene to benzvalene 256 -, [.. 6 +.. 4], cyclopentadiene to -, -, benzene to Dewar benzene 256 tropone 168 -, naphthvalene to naphthalene 260 see also cyclodimerization -, photolysis of formaldehyde 252 cycloalkenes -, rearrangement, [1,5]- of cyclopenta- -, dimerization 181-185 diene 204 cyclobutadiene (CBD) -, - of disilene 190 -, antiaromaticity 8 -, -,[1,3]-sigmatropic 199 -, dimerization to cubane 172-174 -, thermolysis of diazomethane 229 -, -, to tricyclooctadiene 171-172, 175-177 -, - of 7-methylene-2,3-diaza- -, faciality 5 [2.2.1]bicyclohept-2-ene 233 -, fluxion al isomerization 207-210, 215, -, - of methylenepyrazoline 231-232 218-221 - vs. correlation diagram 184 -, ground state 9 -, zwitterion formation in -, molecular orbitals 7 [.. 2 +.. 2)-cycloaddition 148 -, photolysis 250-251 cross-term, vibrational 96-96 -, square (D4h) 219-221 cubane 54 -, states 221 -, interconversion with cyclooctatetraene cyclobutadiene dianion 8 127-130, 161 cyclobutane, conformation 19 -, -, analysis in global symmetry 128-129 -, photofragmentation 13 -, -, analysis in local symmetry 129-130 cyclobutene, ring opening 110, 135 cyclization cyclodimerization, acetylene 171 -, s-cis-butadiene 110-111, 114-116, 135 -, ethylene 139-141 -, hexatriene 111-114 -, silaethylene 141-142 -, polyene 5, 109-116 1,3-cyc1ohexadiene, decyclization 127 -, polyenyl anion 112 cyc1ohexene, fragmentation 13 -, polyenyl cation 112 -, cis and trans 161 -, Woodward-Hoffmann Rules 112 cis, trans-1 ,5-cyclooctadiene, dimerization cycloaddition 147 -,102 to acetylene 171,249 cyc100ctatetraene (COT) -, acrylonitrile to alle ne 150-151 -, interconversion -, azide ion to ethylene 169 -, - with bicyclo[4.2.0]octa-2,4,7-triene -, carbenes to ethylene 135-139 118-119 -, CCl2CF2 to 1,3-butadiene 147 -, - with cubane 127-130 -, dimethylketene to styrene 152, 155-156 -, tub-tub inversion 130 304 Subject Index cyclopentadiene, cycloaddition to tropone Diels-Alder reaction 3, 161-167 168 -, desymmetrization of approach 162 -, [1,5]-rearrangement 203-204 -, transition state 6 cyclopropane, isomerization 147 dichlorodifluoroethylene (CChCF 2) cyclopropene, dimerization 182-183 -, addition to 1,3-butadiene 147 cycloreversion, benzene dimer 178-181 difluoromethylene (CF2) 78 -, benzene-anthracene adducts 179 -, dimerization 90 -, tetraalkyl-1,2-dioxetane 171 9,10-dihydronaphthalene -, [,,2. +" 2.]13-14, 19-21 -, interconversion with [lO]annulene 117 -, [,,4 +,,2]14 a-diketone, trip let -, chemiluminescence 250 d orbitals 46-47 dimerization -, importance in transition met als 46 -, bicyclo[4. 2. 2]-deca-trans-3, cis-7 ,9-triene -, irreducible representations 47, 276-277 143 -, splitting 46, 276 -, cis, trans-1,5-cyclooctadiene 147 -, unimportance in silicon compounds 274 -, cyclobutadiene 171-177 decyclization of 1,3-cyclohexadiene 127 -, cyclomonoalkenes 181-185 degeneracy, accidental 37, 95 -, cyclopropene 182 -, essential 37 -, silacyclopropenes 183-185 -, formal 209 -, silaethylene 185 - splitting 31-34, 46-49,60,64,78, 123 dimesityl-1 ,2-di- t-butyl-disiladioxetane density of states 237 -, 1,2 -+ 1,3 isomerization 240 derydbergization 262 dimethylazodicarboxylate desymmetrization 32, 80-81, 90-93 -, cycloaddition to ethyl vinyl ether 152 - by aperturbation 43, 90 dimethy lketene - by an external field 31-45 -, cycloaddition to styrene 152, 155-156 - by chelation 277 dinitrogen (N 2), excited states 65-69 -, distortional 55, 103 -, ground-state configuration 65-67, 228 -, D2h to C 2v 42 -, molecular orbitals 65 -, D ooh to D 2h 66 -, orbital correlation diagram 70 -, Oh to D3d 272 9,10-dioxabimanes (1,5-diazabicyclo[3.3.0]- -, Oh to S6 272 octadienediones) 246-248 -, substitutional 52, 55, 103, 116, 141, -, dipole moment 247 149,233 1,2-dioxetane, fragmentation 226, 237, 240 -, superposition of transition states 196 -, isomerization to 1,3-dioxetane 240 - to a subgroup 42, 90 -, symmetry coordinates 239 - to co-kernel 54 1,2-dioxetene 171, 249 - to kernel131, 225 dioxygen (02) deuterium, "inductive effect" 156 -, cycloaddition to acetylene 171,249 Dewar benzene (DB) -, molecular orbitals 65 -, isomerization to benzene 122, 125-127, -, singlet eLl g ) 68, 248-250 226, 236 -, -, reactive component eAg) 248 -, -, spin-non-conservative 22, 125, 255 -, states 67-68 -, molecular orbitals 237 diphenylcarbene 90, 223-224 diatomic molecule, heteronuclear -, intersystem crossing 226 -, configurational symmetry 74 -, spin-orbit coupling 225 -, molecular orbitals 70-74 -, spin-vibronic coupling 225 -, state symmetry 74 diphenyldiazomethane, photolysis 90, 244 -, symmetry coordinates 76 -, thermolysis 230 diatomic molecule, homonuclear diphenylketene (DPK) -, configurational symmetry 65-66 -, cycloaddition to alkenes 150 -, molecular orbitals 57-63 -, - to a-methylstyrene 152 -, state symmetry 66-70 -, - to styrene 151 -, symmetry coordinates 74-76 diphenylmethylene see diphenylcarbene diazomethanes, thermolysis 226-230 Subject Index 305 1,3-dipole, allyl and propargyl alle ny I types faciality 5-10 169 field, dipolar 41-43 , 90 direct sum 76, 99 -, magnetic 29-33, 46, 60-61, 215 direct product, of groups 219 -, quadrupolar 33-34, 40-43, 46, -, of representations 40 61-63, 70, 78-79, 81, 90, 248 -, -, degenerate 66, 70 -, -, effect on lower states of O2 68 dis rotation 5 -, -, effect on orbital energy 34-36, 48 distortion fluorescence 243-247 - along reaction coordinate 103 -, selection rules 243 -, vibration al 93 - of dioxabimanes 246-247 dyotropic shift 188 "forbiddenness" 119-132 force constants, harmonie 95 formaldehyde (H 2CO) -, symmetry coordinates 253-255 electrocyclic reactions 109 -, photolysis 251-253 electron formyl fluoride (HFCO), fragment at ion 255 - configuration 10, 74 fragmentation, unimolecular - -, closed shell 7 -, -, Bader's analysis 93-94 - -, irreducible representation 65 -, -, non-RRKM kinetics 235 - -, open shell 7 free-energy of activation, additivity 234 - -, symmetry 65-70 fulvene 257 - correlation 31 - density see charge density geminals 221 - inter change 219 group, commutative (Abelian) 37, 44 - jump, time-dependence 59 -, double 216 - repulsion 9, 67, 75 -, non-commutative 46-54, 96 -, rotating 76 - postulates 36 - spin 10, 65, 215-240 -, simply subducible 66 - -, non-conservation 22 - -, symmetry 215-221 harmonie approximation 95 -, spinning -, breakdown 102 -, -, classical model 215 heptatrienes, isomerization 202 electrons, frontier 10-15, 17,93 hexachlorocyclopentadiene indistinguishability 115, 217 -, cycloaddition to allene 150 elementary reaction 146 1,5-hexadiene, re arrangement 187-188, elements (of a group) 35 192-197 entropy of activation 178-181,201,227, 275 cis-l,3,5-hexatriene, cyclization 8-9, ethyl vinyl ether, cycloaddition to 111-114, 116 dimethylazodicarboxylate 152 Highest Occupied Molecular Orbital ethylene (C2H4 ) (HOMO) 10-15 -, cycloaddition to azide ion 169 homo-Diels-Alder reaction 167 -, -, to ozone 170 homomerism 141 -, cyclodimerization 139-141 Hund's rule 7, 277 -, distortion 92 Hückel Molecular Orbital (HMO) -, excited state configuration 94 theory 7-10, 17,23 -, faciality 5 Hückel rule 6-7, 9 -, formation 88 Hückel system 6-7 -, fragmentation 93-94, 102 hybridization 37, 41-43, 45, 60, 70, 78, 150 -, ground-state configuration 88 -, sp3 d 269 -, molecular orbitals 11, 88 -, sp3d2 267-268 -, vibrational coordinates 98-100 hydrogen chloride (HCI) 76 exciplex, charge-transfer 21 hydrogen cyanide (HCN) 77 exci ted state reactions 243-265 hydrogen molecule ion (Ht) exclusion principle, Pauli 65, 218 -, hybrid orbitals 61 Eyring equation 227 -, molecular orbitals 57-62 306 Subject Index hydrogen peroxide (H 20 2 ) 83-87 -, inverse 150, 156 -, ground-state configuration 87 - on triplet lifetime 234-235 -, conformation 87 -, secondary hydrogen shift, [1,3]197-200 -, -, in allene-cycloaddition 235 -, [1,5] 200-204 -, -, in ketene cycloadditions 155-157 -, [1,7] 202 -, -, of the first and second kind 156 hydrogen sulfide (H 2S) 78 -, -, thermolysis of methylene- hypervalent compounds 267-268 pyrazoline 233-235 identity operation 36 Jablonski diagram 244 interconversion J ahn-Teller effect, second-order 23 -, [lO]annulene and 9,1O-dihydro- naphthalene 117 Kasha's rule 243-244, 257-258 -, benzvalene and benzene 124, 127, 161 -, exceptions 245-246, 258, 262 -, biradical conformers 145 kernel 42, 45, 83 -, cyclooctatetraene and cubane ketene cycloaddition 149-157 127-130, 161, 172 -, faciality 5 see also isomerization intermediate, unstable 103 linear combinations of bond orbitals internal conversion (IC) 243 (LCBOs) 128, 272 internuclear repulsion 62, 75 linear combinations of atomic orbitals -, neglect 64-65 (LCAOs) 59 intersystem crossing (ISC) 10, 221-226, 244 Lowest Occupied Molecular Orbital - in carbenes 223 (LUMO) 10-15 -, reactive (RISC) 222-226 -, vibronically induced 234 methane (CH4 ), molecular orbitals 263 interaction, HOMO-LUMO 11-15 92-93 -, photolysis 261-263 136, 144 " methylene (CH2) 78-80, 223 -, through-bond 87, 145 -, dimerization 88-90, 111 -, through-space 87, 145, 196 -, electron configuration 80, 228 -, intersystem crossing 226, 230, 263 -, 1r-c1 196-197 inverse (of an operation) 36 -, orbitals 88 isomerization -, singlet-triplet gap 230 -, benzvalene to benzene 122-125 4-methylene-1-pyrazoline (MP) -, bicyclobutane to cyclobutene 124 -, thermolysis 226-232 -, bisnorcaradiene to 1,6-methano[1O]- 7-methylene-2 ,3-diaza[2.2 .1]bicyclohept-2-ene annulene 116 (MDBH), fragmentation 233 -, cis-1,3,5-hexatriene to 1,3-hexa• methylenecyclobutane, formation 151 diene 116 methylenecyclopropane (MCP), formation -, cyclooctatetraene to bicyclo[4.2.0]- 230 octa-2,4,7-triene 117 a-methylstyrene -, cyclopropanes 147 -, cycloaddition to diphenylketene 152 -, Dewar benzene to benzene 122 mirror plane, virtual 91 -, -, spin-non-conservative 236-237 mixing parameter 70 -, disilenes 188 mode selectivity 255 -, non-adiabatic 235 molecular orbital (MO) approximation -, norcaradiene to cycloheptatriene 116 65,67, 256 -, prismane to benzene 122, 125 -, breakdown 131, 208, 220 -, prismane to Dewarbenzene 125 moment, magnetic 29-33 -, tricyclo[3.1.0.02,4)hexane (TCH) Möbius surface 8 to cyclohexadiene 182 Möbius system see non-Hückel system -, cis-trans bicyclo[4.2.0]octa- multiplication table (group) 36 2,4,7-triene (BCO) 118-119 see also interconversion naphthalene 9 isotope effect 3 -, fluorescence 258 Subject Index 307 naphthvalene (NV), fluorescence 258-260 see also molecular orbital -, photoelectron spectroscopy 259 orbitals -, reactive internal conversion 258-260 -, bond 78, 128 -, reactive inter system crossing 258-261 -, -, symmetry-adaptation 78 nickel(II) complexes -, CH-bonding, inclusion in correspondence -, cis-trans isomerization 277-278 diagram 115, 120, 260 -, distortional desymmetrization 103-104 -, corresponding 113 -, substitutional desymmetrization 52-54, -, degenerate 7 103-104 -, occupied, omission from correlation -, tetrahedral-to-planar isomerization diagram 131 275-278 -, virtual 138 -, vibrational coordinates 96-98, 100-102 -, zero-order 79 nitrogen molecule see dinitrogen (N 2) -, (1, röle in polyene cyclization 114 "N 0 mechanism" reaction 3-4 order (of a grou p) 35 nodal surface 73 orientation see alignment non-crossing rule, orbital 72-73, 79, 93 orthogonality 40, 43 -, state 92, 257 overlap density function 255 non-Hückel system 8 oxygen molecule see dioxygen (02 non-RRKM kinetics 235 ozone (03 ), cycloaddition to ethylene 170 norcaradiene ([4.1.0]hepta-2,4-diene) -, molecular orbitals 170 116-117 ozonolysis 170 -, isomerization to cycloheptatriene 116 -, "walk" rearrangement 204-207 penta-atomic molecule, non-linear normal modes 95-97 -, -, vibrational coordinates 96-98 normalization 40 s- cis-pentadiene, [1,5]-sigmatropic - of symmetry coordinates 100 rearrangement 197, 199-202 notation, group theoretical 23 perturbation -, Schönfliess 32, 39 - theory, first order 43-44 -, Woodward-Hoffmann (S,A) 23 -, second-order 139 -, symmetry-breaking 70 Orbital Correspondence Analysis in phase continuity rule 164 Maximum Symmetry (OCAMS) 23 phenanthrene 9 -, comparison with WH-LHA 45, 91-93, 109 phosphorescence 244 -, extension to spin-forbidden processes 222 - of dioxabimanes 247 a\l- cis-octatetraene phosphorus, a\lotropy 270-273 -, cyclization to cyclooctatriene 112 -, cubic (Ps), non-existence 270-273 open shell 7 -, white (tetrahedral) (P 4) orbital -, -, dimerization to Ps 270-273 -, characterization by subgroup 38 -, -, polymerization to red phosphorus 273 -, characterization by irreducible phosphorus pentachoride (PCIs) 267 representation 39 phosphorus pentahalides, pseudorotation -, frontier 11-15 269-270 -, group 13 photochemistry -, hybrid 37, 40, 78 -, Rydberg 261-263 -, hydrogen-like 30, 57 see also excited state reactions - mixing 42, 44 "photochemistry without light" 226, 236-240 -, moiety 13 photoelectron spectroscopy 237 - phase 37, 59 photoexcitation, selection rules 243 -, Rydberg 262 photoextrusion of silylene 273-274 - splitting 31-34, 42, 47-49, photofragmentation 93, 250-255 231, 259, 277 see also photolysis -, subjacent 14, 17, 93 photoisomerization -, - interaction 19, 42 -, benzene to benzvalene 256 -, superjacent 14, 93 -, - to Dewar benzene 256 - symmetry conservation 3 -, -, spin-non-conservative 258-261 308 Subject Index photolysis Rules see Woodward-Hoffmann Rules -, formaldehyde 252 Rydberg state 261-263 -, cyclobutadiene 250-251 -, symmetry properties 262 -, diphenyldiazomethane, 90, 244 -, methane, 261, 263 saddle point 103-104 see also photofragmentation -, second-order 104 polarization function, d-type 267 scalar product 40 polyatomic moleeule Schönfliess notation 39 -, vibrational coordinates 95-102 semibullvalene, rearrangement 193, 196-197 polyenyl anion, cyclization 112 sensitization, chemical 248-250 - cation, cyclization 112 silacyclopropenes, dimerization 183-185 prefulvene 257 silaethylene, cyclodimerization 141-142, 185 primitive change 146 silicon, analogy to carbon 267 prismane silicon hexafluoride dianion (SiF~-) 267-268 -, isomerization to benzene 122, silylene (SiH 2), dimerization 93 125-127,236 -, singlet-triplet gap 223, 225 -, - to Dewar benzene 125 similarity transformation 50 -, photochemie al formation 255 Singly Occupied Molecular Orbital prochirality 55 (SOMO) 10-15 projection operator 99 Skell hypothesis 230 propylene solvent effect 3 -, [1,3]-sigmatropic rearrangement 197-200 - on ketene cycloaddition 150, 152, 155 1-pyrazoline, thermolysis 230-233 spectroscopy, vibrational 97 spin see electron spin reactive intersystem crossing (RISC) 244 spin-non-conservative reaction see also spin-non-conservative reaction -, Arrhenius parameters 227, 275 rearrangement see also reactive intersystem crossing -, bridged hexadienes 196-197 spin-orbit coupling 222-226, 262 -, s-cis-butadiene to bicyclobutane 119-121 spin-vibronic coupling 224-226, 261 -, circumambulatory 202-206 square terms, vibrational 96 -, Claisen 193 state correlation 74 -, Cope 187-188, 192-197 - coupling 257 -, degenerate 187-210 - mixing 93-94 -, electrocyclic 5 stereochemistry -, "norcaradiene walk" 204-207 -, biradical [.. 2 +.. 2]-cycloaddition 143, 146 -, sigmatropic 5, 22, 109, 188, 192-206 -, zwitterionic [.. 2 +.. 2]-cycloaddition 149 -, -, [1,3]198-200 -, ketene cycloaddition 153-154 -, -, [1,5] 200-204 stereoselectivity -, -, [3,3] 192 - of photophysical processes 246-248 redundancy, removal 47 steric parameter (E.) 154 relativistic effects 31 styrene representation, degenerate 46, 220 -, cycloaddition to dimethylketene 152, -, identity 39, 44 155-156 -, irreducible (irrep) 39 -, -, to diphenylketene 151 -, -, degenerate 46 subgroup 38 -, -, 2- and 3-dimensional 50 -, commutative 78 -, reducible 41, 76 -, invariant 51-54 -, -, reduction of 99 -, non-invariant 52 -, regular 115, 165 -, trivial 83, 121, 200 -, totally symmetrie see representation, substituent effect 3 identity - on ketene cycloaddition 150, 155 retro-Diels-Alder reaction 13, 179 substitution, bimolecular ring-current, momentary 248, 262-263 nucleophilic (SN2) 190-192 rule, commutative 36 sud den polarization 273 -, multiplication 36 sulfur dioxide (S02) 77, 82 Subject Index 309 sulfur hexaßuoride (SF 6) 267-268 tetraatomic molecules 77 supergroup 197 -, deformation 83-88 superposition of transition structures tetrahedrane 54 (STS) 195, 197 tetramethyl-l,2-dioxetane (TMD) surface crossing 222 -, fragmentation 22, 237, 243 symmetry adaptation 60 thermolysis -, axial (e oov ) 32, 70, 81 -, diazoomethanes 226-230 - axis, artificial 124 -, diphenyldiazomethane 230 - c1assification, primitive 22 -, 4-methylene-l-pyrazoline 226-232 -, configurational 18, 41 -, I-pyrazoline 230-233 - coordinate, inducing 91 see also fragmentation - coordinates 95-97 topology, molecular 9 - -, approaching ethylene molecules 141 trace 50 - -, dimerization of CBD to TCOD 176 transition, radiationless 22 - -, formaldehyde 253-255 - metals, reactions 23, 54 - -, homonuclear diatomic molecule 74-75 - -, importance of d-orbitals 46, 267 - -, linear HXXH molecule 84 - state 102-104 - -, linear tetraatomic molecule 83-84 - -, antiaromatic 6, 10 - -, linear XYX molecule 81 - -, aromatic 6 - -, non-linear XYX molecule 82 - -, Hückel 8-9 - -, superposition 76 - -, Möbius 8-9 -, cubic (Oh) 129 - -, open shell 9 -, cylindrical (D ooh ) 32, 58, 70 -- theory 227 - element 15-16, 32 - structure (TS) 188 - -, diagnostic 19-21 transmission coefficient 227 - -, discriminating 22 -, non-constancy 235 - -, distinction from symmetry operation 3 triatomic molecule -, essential 142 -, symmetry coordinates 80-83 -, global vs. local 109, 116-119 -, Walsh diagram 79 -, octahedral (0 h) 48 tricyclo[3.1.0.02,4]hexane (TCH) - operations (sym-ops) 16 -, fragmentation to cyclopropene 182 - -, effect on Cartesian coordinates 35 -, isomerization to cyclohexadiene 182 - -, multiplication 36 tricyclo[4.2.0 .02,5]octa-3, 7-diene - -, "paired" degenerate 50 (TCOD) 171, 195 -, orbital 15 trimethylenemethane (TMM) 230 -, overall 215, 218-222 tripie product 70 - point group 35 triplet, interconversion of components 244 - reduction see desymmetrization -, labeling 225 -, rhomboidal (D2h) 207 -, lifetime 234-235 -, space 41, 92 trisilane, cyclic -, spherical (Kh) 29-30 -, -, photofragmentation 273-274 -, spin 92, 215-221 tropone -, square-planar (D4h) 48 -, cycloaddition to cyclopentadiene 168 -, state 41, 215 tunneling 227 -, tetrahedral (T d) 48 -, one-dimensional model 210 synchronicity 109, 125, 151, 192 "turnstile rotation" 269 tetraalkyl-l,2-dioxetane, cycloreversion 171 united atom 63, 70 tetraaryldisilenes 1,2-rearrangement 188-190 tetracyanoethylene (TCNE) valence bond theory 9 -, cycloaddition to alkenes 147 vibration, complex 97 tetr aß uoroethy lene -, degenerate 96 -, cycloaddition to ethylene 149 -, non-degenerate 95 vinyl ethers tetrahaloethylenes -, cycloaddition to ketenes 147, 150 -, cycloaddition to conjugated dienes 143 310 Subject Index "walk rearrangement" Woodward-Hoffmann - Longuet-Higgins• see rearrangement, circumambulatory Abrahamson (WH-LHA) correlation Walsh diagram 77 procedure 18-21, 45 -, bending of HXXH 86 -, comparison with OCAMS 91, 109 -, tetraatomic molecule 84-88 Woodward-Hoffmann Ru/es 4, 5, 9, 12, 19 -, HXH molecule 79 -, extension to transition metal catalysis 23 -, triatomic molecule 78-80 - for sigmatropic re arrangements 197,204 water (H 2 0) 77 - for Cope rearrangement 194 -, geometry 82 - for thermal [lj]-rearrangement 202 wave-function, space 218 -, spin 217 zwitterion cascade 233