A. 1 Character tables for some chemically important symmetry groups. E = exp(21ti/3) T., R, x2 + yz, z2 x2 - yz, xy (T., Ty), (R.

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1 Appendix A Character and correlation tables and multiplication properties of irreducible representations A. 1 Character tables for some chemically important symmetry groups E (Th C; E A' 1 T., Ty, R, xz, yz Ag R,, Ry, R, z2, xy A" 1-1 T., R., Ry yz, zx Au 1-1 T,, Ty, Tz x2,y2,z2 xy, zx, yz The Cn groups A 1 T., R., x2, y2, z2, xy B -1 T., Ty, R., Ry yz, zx E = exp(21ti/3) A E T., Rz { 11 EE* EE* } (T., Ty), (R., Ry) (x2 - y2, xy), (yz, zx) A B E Cs A { ~ _; =~ -:} T., R, x2 + yz, z2 x2 - yz, xy (T., Ty), (R., Ry) (yz, zx) E = exp(21ti/5) (T,, Ty), (R., Ry) (yz, zx) (x2 - y2, xy) 173

2 174 Group theory for chemists c6 E c6 c3 Cz Cl C65 e = exp(21ri/6) A 1 1 T., R. x2 + y2, z2 B E -e* -1 -e (T., Ty), E1 {~ : } (yz, zx) e* -e -1 -e* (R., Ry) -e* -e 1 -e* Ez -e} (x2 - y2, xy) {~ -e -e* 1 -e -e* c1 E c1 Cl Cl C1 4 C75 C1 6 e = exp(21ti17) A T., R. x2 + y2, z2 E E2 E3 e3 e2 e* (T., Ty) (yz, zx) E1 {~ e* Ez. e3 E3 E2 E } (R., Ry) E2 e3 e* E E3 e2 } Ez {~ e2 E3 (x2-/, xy) E e* e3 E2 E3 e* E2 e2 E e3 } E3 {~ e3 E e2 E2 e* E3 Cs E Cs c4 Cz Cl Cl Cs5 Cs7 e = exp(21ti/8) A T., Rz xz + y2, z2 B E -1 -i -e* -e (T., Ty), E1 {~ : } (yz, zx) e* -i -1 -e -e* (R,, Ry) i -i Ez {~ } (x2 - yl, xy) -i i i -e -1 -i e* E E3 -e*} {~ -e* -i -1 E e* -e The Dn groups Dz E Cz(z) Cz(y) Cz(x) A 1 1 x2,y2,z2 Bl T., R. xy Bz Ty, Ry zx B T., R, yz D3 E 2C3 3Cz AI 1 x2 + y2, z2 Az 1-1 T., R. E (T,, Ty), (R., Ry) (x2 - yl, xy), (yz, zx) D4 E 2C4 Cz(=C42) 2Cz' 2C{ AI x2+y2,z2 Az T., R. B x2 _ y2 Bz xy E (T,, Ty), (R., Ry) (yz, zx)

3 Appendix A 175 Ds E 2Cs cos no 2 cos cos cos no 0 T., R, (T., Ty), (R., Ry) (yz, zx) (x2 - yz, xy) (T., Ty), (R., Ry) (yz, zx) (xz - yz, xy) The Cnv groups zx yz A1 1 T, x2 + yz, z2 A R, E (T., Ty); (R., Ry) (x2 - y2, xy), (yz, zx) x2- y (T., Ty), (R., Ry) (yz, zx) xy E 2Cs cos no 2 cos cos cos no 0 T, R, (T., Ty), (R., Ry) (yz, zx) (x2 - yz, xy) (yz, zx) (r- y2, xy)

4 176 Group theory for chemists The Cnh groups xz, yz, zz, xy yz, zx ~ = exp(21ti/3) A' E' A" E" * R, T, x2 + yz, z2 (x2 -yz, xy) (yz, zx) c", A' A" E( Et E~u 1 {~ 1 1 {~ 1-1 -i i i i -1 -i -i a a E C, C 5 2 C 5 3 C 5 4 rr, 5 5 S ' _1 * e* e~* e1 e* E * E 2 } E 1 * E * * E c, c, c, c,' c,' I I -1 -I -I I -1 -e* -1 -e c.. I e* -e -1 -e* I -e I -e I I I -1 -I -I -I -e* -1 -e e* -1 -e c e* -e -1 -e* -I I -e* -e -I - * I - -e* I.. {: =: =:. I -e* -e I -e* -e 1 -e* {: =: =:. -1 R, x2+y2,z2 xz-yz,xy T, S ' ~ -1 - } ~ } (yz, zx) R, T, (R,, R,) E = exp(21ti/5) (x2 - y', xy) (yz, zx) e = exp(21ti/6) I -I -I -1 (R,. R,) (yz. zx) -I - - } (x I - - * 2 - y', xy) I T, I -1 I (T,. T,) I : } -I e* -1 '

5 Appendix A 177 The Dnh groups D2h E C2(z) C2(y) Cl(x) cr(xy) cr(xz) cr(yz) Ag x2, y2, z2 Big R. xy B2g Ry zx B3g R, yz Au Blu T. B2u Ty B3u T, D3h E 2C3 3C2!Th 2S3 3crv A/ x2 + y2, z2 A2' R. E' (T,, Ty) (x2 -y2,xy) At A{ T. E" (R., Ry) (yz, zx) D4h E 2C4 c2 2C/ 2C{ 2S4!Th 2crv 2crd A1g x2 + y2, z2 A2g R. B1g x2 _ y2 B2g xy Eg (R,, Ry) (yz, zx) A1u A2u T. Blu B2u Eu (T., Ty) Ds,, E 2C5 2C,2 sc, rr, 25, 25,' Srr,. A,' I I x2 + yl, z2 A,' I -I I -I R, E,' 2 2 cos 72" 2 cos 144" cos 72" 2 cos 144" 0 (T,, T,) E,' 2 2 cos 144" 2 cos 72" cos 144" 2 cos 72" 0 (x'- y', xy) A," I I I -I -I -I -I A{' I I -I -I -I -I I T, E," 2 2 cos 72" 2 cos 144" cos 72" -2 cos 144" 0 (R,, R,) (yz, zx) E," 2 2 cos cos 72" cos cos 72" 0 D6h E 2C6 2C3 c, 3C2' 3C," 2S3 2S6 rrh 3rrd 3rr, A,g x 2 +y2,z 2 A2g R, Btg B2g E, (R., R,) (yz, zx) E2g (x2 - y2, xy) Atu A2u T, Btu B2u Etu (T,, T,) E2u

6 178 Group theory for chemists The Dnd groups D2d E 2S4 Cz 2C2' 2ad AI xz + yz, zz Az Rz Bl xz -l Bz Tz xy E (T., Ty), (R., Ry) (yz, zx) D3d E 2C3 3Cz 2S6 3ad A1g xz + yz, zz Azg Rz Eg (R., Ry) (x2 - y2, xy), (yz, zx) Alu Azu Tz Eu (T., Ty) D4d E 2S 8 2C4 2S 8 3 Cz 4C/ 4ad AI xz + yz, z2 Az Rz Bl Bz Tz El 2 v2 0 -v (T., Ty) Ez (x 2 -yz, xy) E3 2 -v2 0 v (R., Ry) (yz, zx) Dsd E 2C 5 2C5' sc, i 2Sui rrd A,, x~+y:!.z2 A2g -I -I R, E,, 2 cos no 2 cos 144 () 2 2 cos no 2 cos 144 () (R,. R,) (yz. zx) E2g 2 cos cos no () 2 2 cos cos no 0 (x' - y', xy) A,, I I I I -I -I -I -I A2u I I -I -I -I -I I T, E,, 2 2 cos no 2 cos 144 () -2-2 cos no -2 cos (T,. T,) E2u 2 2 cos cos no cos cos no 0 D6d E 2S12 2C6 2S4 2C3 2S12 5 Cz 6Cz' 6ad AI xz + yz, z2 Az Rz Bl Bz Tz E1 2 \ \ (T., Ty) Ez (x 2 -yz, xy) E E Es 2 -v3 0-1 v (R., Ry) (yz, zx)

7 Appendix A 179 The S" groups s4 E 54 c2 543 A R, x2 + y2, z2 B T, x2 -y2,xy E {~ -i ~1-1 -a (T., Ty), (R., Ry) (yz, zx) s6 E c3 Cl e = exp(2'1ti/3) Ag 1 1 R, x2 + y2, z2 Eg {~ E e* E : } (R., Ry) {(x2 -y2, xy) e* E e* (yz, zx) Au T, Eu {~ E e* -1 -e -e*} e* E -1 -e* -e (T., Ty) Ss E 5s c4 5s3 c2 5s 5 Cl 5/ e = exp(hi/8) A R, x2 + y2, z2 B T, E1 {~ E -e* -1 -e -i : } { (T., Ty) E2 {~ e* -i -e -1 -e* i (R., Ry) -1 -i 1-1 -i } (x2 - y2, xy) -i i -1 i E3 {~ -e* -i E -1 e* -e } (yz, xz) -e e* -1 E -i -e* The cubic groups Td E 8C3 3C rrd AI x2+y2+z2 A E (2z2- x2 - y2, x2- y2) T (R., Ry, R,) T (T., Ty, T,) (xy, yz, zx) o, E sc, 6C 2 6C, 3C 2(=C}) i 6S, ss. 3u 11 6rrd A,, I I I I I I I I x2+i+z2 A2g I I -I -I I I -I I -I E, 2 -I () I 0 (2z'- x2 - y', x' - y 2) T,, 0 -I -I 3 I 0 -I -I (R,. R,. R,) T,, 0 I -I -I 3 -I 0 -I I (xy, yz. zx) A,u I I -I -I -I -I -I A2u I I -I -I I -I I -I -I Eu 2 -I () () I -2 () T,u 3 () I -I -3 -I 0 I (T,, T,. T,) T2u 3 () I -I -I I

8 180 Group theory for chemists The C.v and Dooh groups for linear molecules Coov E 2Coo<l> ooav AI= I+ 1 1 Tz xz + y2, z2 Az =I Rz E1 = n 2 2 cos I}> 0 (T., Ty), (R,, Ry) Ez =.i 2 2 cos 211> 0 (yz, zx) E3 = II> 2 2 cos 311> 0 (x2 - y2, xy) Dxh E 2Cx<l> 00(1 v 2Sx<l> ooc2 ~+ g I x2+y2,z2 ~ḡ I -I -I Rz ng 2 2 cos <I> cos<!> 0 (R,, Ry) (yz, zx) ag 2 2 cos 2<1> cos2. 0 (x 2 - y2, xy) ~u + -I -I -I Tz ~u- I -I -I -I I Ilu 2 2 cos <I> cos I}> 0 (T,, Ty) au 2 2 cos 2<1> cos 211> 0 A.2 Multiplication properties of irreducible representations General rules A X A = A, B X B = A, A x B = B, A x E = E, B x E = E, A x T = T; B x T = T; g X g = g, U X U = g, U X g = u; ' X ' = ', " X " = ', ' X " = "; A X E1 = Eb A X E2 = E2, B X E1 = E2, B X E2 = E1 Subscripts on A or 8 1 x 1 = 1, 2 x 2 = 1, 1 x 2 = 2. except for D2 and D2h, where 1 X 2 = 3, 2 x 3 = 1, 1 X 3 = 2 Doubly -degenerate representations E1 x E1 = Ez x Ez = A1 + Az + Ez E1 X Ez = B1 + Bz + E1

9 Appendix A 181 E X E = A1 + Az + B1 + Bz For groups in above lists that have symbols A, B or E without subscripts, read A1 = A2 = A, etc. Triply-degenerate representations: E X T1 = E X T2 = T1 + Tz T1 X T1 = Tz X T2 = A1 + E + T1 + T2 T1 X T2 = Az + E + T1 + Tz Linear molecules (Coov and Dooh): ~+ X ~+ = ~- X ~- = ~+; P x = ~- x n = n; n x n = ~+ + ~- + a a x a = ~+ + ~- + r nxa=n+<i> ~+ X ~- = ~- ~+ X a = ~- X A = a; etc. A.3 c4 A B E Dz A Bl Bz B3 D4 AI Az B1 Bz E Correlation tables for the species of a group and its subgroups Cz c6 c3 Cz A A A A A B A B 2B E1 E 2B Ez E 2A Cz Cz Cz D3 c3 Cz A A A AI A A A B B Az A B B A B E E A+B B B A Cz' Cz" c4 Cz Cz Cz Ds Cs Cz A A A A AI A A A A B B Az A B B A A B E1 E1 A+B B A B A Ez Ez A+B E 2B A+B A+B

10 182 Group theory for chemists C/ Cz" C2' Cz" D6 c6 D3 D3 Dz c3 Cz Cz Cz AI A AI AI A A A A A Az A Az Az B1 A A B B B1 B AI Az Bz A B A B Bz B Az AI B3 A B B A E1 E1 E E Bz + B3 E 2B A+B A+B Ez Ez E E A+ B1 E 2A A+B A+B <T(zx) <T(yz) Czv Cz Cs Cs C3v c3 Cs AI A A' A' AI A A' Az A A" A" Az A A" Bl B A' A" E E A' +A" Bz B A" A' av (J"d <Tv (J"d C4v c4 Czv Czv Cz Cs Cs AI A AI AI A A' A' Az A Az Az A A" A" Bl B AI Az A A' A" Bz B Az AI A A" A' E E B1 + Bz B1 + Bz 2B A' +A" A' +A" <Tv <Td <Tv~ <T(zx) <Tv (J"d Csv Cs Cs C6v c6 C3v C3v Czv c3 Cz Cs Cs AI A A' AI A AI AI AI A A A' A' Az A A" Az A Az Az Az A A A" A" E1 E1 A'+A" Bl B AI Az B1 A B A' A" Ez Ez A'+ A" Bz B Az AI Bz A B A" A' E1 E1 E E Bl + Bz E 2B A' +A" A' +A" Ez Ez E E A1 + Az E 2A A' +A" A' +A" c2h Cz Cs C; c3h c3 Cs Ag A A' Ag A' A A' Bg B A" Ag E' E 2A' Au A A" Au A" A A" Bu B A' Au E" E 2A" c4h c4 s4 Czh Cz Cs C; Csh Cs Cs Ag A A Ag A A' Ag A' A A' Bg B B Ag A A' Ag El' E1 2A' Eg E E 2Bg 2B 2A" 2A 8 E/ Ez 2A' Au A B Au A A" Au A" A A" Bu B A Au A A" Au E{ E1 2A" Eu E E 2Bu 2B 2A' 2Au Ez" Ez 2A"

11 c6h c6 c3h s6 c2h CJ Cz c, C; Ag A A' Ag Ag A A A' Ag Bg B A" Ag Bg A B A" Ag Etg Et E" Eg ZBg E ZB ZA" ZAg Ezg Ez E' Eg ZAg E ZA ZA' ZAg Au A A" Au Au A A A" Au Bu B A' Au Bu A A A' Au Etu Et E' E. ZBu E ZB ZA' ZAu Ezu Ez E" Eu ZA. E ZA ZA" ZA. Appendix A 183 C2(z) Cz(y) C2(x) C2(z) C2(y) C2(x) C2(z) Cz(y) Cz(x) a(xy) a(zx) a(yz) D2h D2 c2, c2, c2, C2h c2h c2h Cz Cz Cz C, c, C, Ag A A, A, A, Ag Ag Ag A A A A' A' A' B, 8 B, A2 B2 B, Ag Bg Bg A B B A' A" A" B2 8 B2 B, A2 Bz Bg Ag Bg B A B A" A' A" B3 8 BJ B2 B, Az Bg Bg Ag B B A A" A" A' Au A A2 Az Az Au Au Au A A A A" A" A" Btu B, A, B, Bz Au Bu Bu A B B A" A' A' Bzu Bz Bz A, B, Bu Au Bu B A B A' A" A' B3u BJ B, B2 A, Bu Bu Au B B A A' A' A" ah--+ av(zy) ah av D3h c3h DJ C3v C2v CJ Cz c, c, At' A' At At At A A A' A' Az' A' Az A2 Bz A B A' A" E' E' E E At+ Bz E A+B ZA' A'+ A" At A" At Az Az A A A" A" Az" A" Az At Bt A B A" A' E" E" E E Az + Bt E A+B ZA" A'+ A" Cz'--+ Cz' Cz"-+ Cz' Cz' Cz" D4h 04 Du Du C4v c4h D2h D2h c4 s4 At 8 At At At At Ag Ag Ag A A A2g Az A2 A2 Az Ag Bt 8 Btg A A Btg Bt Bt B2 Bt Bg Ag Bt 8 B B B2g B2 B2 Bt Bz Bg Bt 8 Ag B B Eg E E E E Eg Bzg + B3 8 B2g + B3 8 E E Atu At Bt Bt A2 Au Au Au A B A2u A2 Bz B2 At Au Btu Btu A B Btu Bt At Az Bz Bu A. Btu B A Bzu B2 A2 At Bt B. Btu Au B A Eu E E E E E. B2u + B3u B2u + B3u E E

12 184 Group theory for chemists Cz' Cz" C2, a. Cz, ad Cz' Cz" D4h Dz Dz Czv Czv Czv Czv (cont.) A,g A A A, A, A, A, Azg B, B, Az Az B, B, B,g A B, A, Az A, B, Bzg B, A Az A, B, A, Eg Bz + B3 Bz + B3 B, + Bz B, + Bz Az + Bz Az + Bz A,. A A Az Az Az Az Azu B, B, A, A, Bz Bz B,. A B, Az A, Az Bz Bzu B, A A, Az Bz Az E. B 2 + B 3 Bz + BJ B 1 + B 2 B 1 + B 2 A 1 + B 1 A 1 + B 1 Cz Cz' Cz" Cz Cz' Cz" ah a. ad D4h Czh Czh Czh Cz Cz Cz Cs Cs Cs C; (cont.) A,g Ag Ag Ag A A A A' A' A' Ag Azg Ag Bg Bg A B B A' A" A" Ag B,g Ag Ag Bg A A B A' A' A" Ag Bzg Ag Bg Ag A B A A' A" A' Ag Eg 2Bg Ag + Bg Ag + Bg 2B A+B A+B 2A" A'+ A" A' +A" 2Ag A,. Au A. Au A A A A" A" A" Au Azu Au B. B. A B B A" A' A' A. B,. Au Au B. A A B A" A" A' Au Bzu Au Bu Au A B A A" A' A" Au E. 2Bu Au+ B. Au+ Bu 2B A+B A+B 2A' A'+ A" A'+ A" 2Au ah -> a(zx) ah av Dsh Ds Csv Csh Cs Czv Cz Cs Cs A,' A, A, A' A A, A A' A' Az' Az Az A' A B, B A' A" E,' E, E, E,' E, A 1 + B 1 A+B 2A' A'+ A" Ez' Ez Ez Ez' Ez A 1 + B 1 A+B 2A' A'+A" At A, Az A" A Az A A" A" Az" Az A, A" A Bz B A" A' Et E, E, Et E, A2 + Bz A+B 2A" A' +A" Ez" Ez Ez Ez" Ez A2 + Bz A+B 2A" A'+ A"

13 Appendix A 185 " --> rr(xy) C;.' C-/ C{ C/ rr,. --> rr(yz) Cz' C," rr, fij Do, o. o,. DJil c., c.. o,d o,d o,. c. c,. o, o, c,,. c,,. s. o, A,, A, A,' A,' A, A, A,, A,, A, A A' A, A, A, A, A, A A2g A, A/ A,' A, A, A2g A2g B,, A A' A, A, A, A, Ag B, B,, B, A," A-/' B, B, A,, A,, B2g B A" A, A, A, A, Ag B, B,, B, Az" A," B, B, A,, A2g B,, B A" A, A, A, A, A, B, E,, E, E" E" E, E,, E, E, B 2, + B3, E, E" E E E E E, B2 + B, E,, E, E' E' E, E,, E, E, A,+ B1, E, E' E E E E E, A+ B 1 A," A, A," A," A, A" A," A," A" A A" A, A, A, A, A" A A2u A, A-t A-/' A, A" A2u Azu B," A A" A, A, A, A, A" B, B," B, A,' A / B, B" A," A," Bzu B A' A, A, A, A, A" B, Bzu B, Az' A,' B, B" A," A," B," B A' A, A, A, A, A" B, E," E, E' E' E, E," E" E" Bzu + BJu E, E' E E E E E" B2 + B 3 Ezu E, E" E" E, Ezu E" E" A"+ B1" E, E" E E E E E" A+ B1 Cz' Cz" Cz Cz' Cz" Cz D6h Czv Czv c2h Czh Czh CJ Cz (cont.) Alg AI AI Ag Ag Ag A A Azg B1 Bl Ag Bg Bg A A Big Az Bz Bg Ag Bg A B Bzg Bz Az Bg Bg Ag A B Elg Az + Bz Az + Bz 2B 8 Ag + Bg A 8 + Bg E 2B Ezg AI+ Bl AI +Bl 2Ag A 8 + B 8 Ag + B 8 E 2A A1u Az Az Au Au Au A A Azu Bz Bz Au Bu Bu A A Blu AI B1 Bu Au Bu A B Bzu Bl AI Bu Bu Au A B Elu AI+ Bl AI+ Bl 2Bu Au+ Bu Au+ Bu E 2B E2u A 2 + Bz Az + Bz 2Au Au+ Bu Au+ Bu E 2A Cz' Cz" ah ad av D6h Cz Cz c, C, c, C; (cont.) Alg A A A' A' A' Ag Az 8 B B A' A" A" Ag Big A B A" A' A" Ag Bzg B A A" A" A' Ag E1g A+B A+B 2A" A' +A" A' +A" 2Ag Ezg A+B A+B 2A' A'+ A" A'+ A" 2A 8 Alu A A A" A" A" Au Azu B B A" A' A' Au Blu A B A' A" A' Au Bzu B A A' A' A" Au Elu A+B A+B 2A' A' +A" A'+ A" 2Au Ezu A+B A+B 2A" A'+A" A'+ A" 2Au

14 186 Group theory for chemists C2--> C2(z) c2 C2' D2d s4 D2 C2v c2 c2 Cs A, A A A, A A A' A2 A B, A2 A B A" B, B A A2 A A A" B2 B B, A, A B A' E E B2 + B3 B1 + B2 2B A+B A'+ A" D3d D3 C3v s6 c3 c2h c2 Cs C; A,g A, A, Ag A Ag A A' Ag A2 8 A2 A2 Ag A Bg B A" Ag Eg E E Eg E A 8 + B 8 A+B A' +A" 2A 8 Alu A, A2 Au A Au A A" Au A2u A2 A, Au A Bu B A' Au Eu E E Eu E Au+ Bu A+B A' +A" 2Au c2 C2' D4d D4 C4v Ss c4 C2v c2 c2 Cs A, A, A, A A A, A A A' A2 A2 A2 A A A2 A B A" B, A, A2 B A A2 A A A" B2 A2 A, B A A, A B A' E, E E E, E B, + B2 2B A+B A'+ A" E2 B1 + B2 B1 + B2 E2 2B A,+ A2 2A A+B A'+ A" E3 E E E3 E B, + B2 2B A+B A' +A" Dsd Ds Csv Cs c2 Cs C; A,g A, A, A A A' Ag A2 8 A2 A2 A B A" Ag E,g E, E, E, A+B A'+ A" 2A 8 E2 8 E2 E2 E2 A+B A' +A" 2A 8 A1u A, A2 A A A" Au A2u A2 A, A B A' Au Elu E, E, E, A+B A'+A" 2Au E2u E2 E2 E2 A+B A' +A" 2Au D6d D6 C6v c6 D2d D3 C3v A, A, A, A A, A, A, A2 A2 A2 A A2 A2 A2 B, A, A2 A B, A, A2 B2 A2 A, A B2 A2 A, E, E, E, E, E E E E2 E2 E2 E2 B, + B2 E E E3 B1 + B2 B1 + B2 2B E A1 + A2 A,+ A2 E4 ~ E2 E2 A,+ A2 E E Es E, E, E, E E E

15 Appendix A 187 Cz Cz' D6d Dz Czv s4 c3 Cz Cz C, (cont.) At A At A A A A A' Az Bt Az A A A B A" B1 A Az B A A A A" Bz Bt At B A A B A' E1 Bz + B3 B1 + Bz E E 2B A+ B A'+ A" Ez A +B1 At+ Az 2B E 2A A+ B A'+ A" E3 Bz + B3 B1 + Bz E 2A 2B A+ B A'+ A" E4 A+ B1 A 1 + Az 2A E 2A A+ B A'+ A" Es Bz + B3 Bt + Bz E E 2B A+ B A'+ A" s4 Cz s6 c3 C; Ss c4 Cz A A Ag A Ag A A A B A Eg E 2A 8 B A A E 2B Au A Au Et E 2B Eu E 2Au Ez 2B 2A E3 E 2B Td D2d c3,. s. o, c2,. c, c, C, A, A, A, A A A, A A A' A, B, A, B A A, A A A" E A,+ B, E A+B 2A A1 + A 2 E 2A A'+ A" T, A2 + E A,+ E A+E B1 + B2 + B3 A2 + B, + B2 A+E A+ 2B A'+ 2A" T, B 2 + E A1 + E B+E B B3 A1 + B A+E A+ 2B 2A' +A" oh Td D4h D3d Atg At Atg Atg Az 8 Az Btg Azg Eg E A1 8 + Btg Eg Ttg T1 A 28 + Eg Az 8 + Eg Tzg Tz B 28 + E 8 Atg + E 8 Atu Az Atu Atu Azu At Btu Azu Eu E Alu +Btu Eu T1u Tz Azu + Eu Azu + Eu Tzu T1 Bzu + Eu Atu + Eu

16 Appendix B Some useful properties of square matrices B.l A proof Here, we will prove that carrying out a similarity transformation: CAC-1 = B does not alter the character of any matrix. We need to prove that x(a) = x(b). The proof is as follows: x(b) = x(cac- 1 ) but the multiplication of matrices is associative, which means that we can bracket off any pair in any way we choose. Therefore: CAC-1 = (CA)c-t = C(AC-1) (This property is also possessed by symmetry operations- see Chapter 2.) Therefore: but as: Therefore: and hence: x(b) = x[c(ac- 1 )] (see page 61) x(b) = x[(ac- 1 )C] = x[a(c-1c)] x(b) = x(a) This proves the desired result. The last step follows from the fact that e-tc = E where E is the unit matrix, which has a value of + 1 for every diagonal element and zero for all other elements. Multiplying A by E, therefore, does not change any of the diagonal elements of A, and hence does not alter x(a). B.2 Irreducible representations Here, we look at some of the important properties of the characters of irreducible representation matrices, and a derivation of the equation showing the number of times each irreducible representation is present in a reducible representation. 188

17 Appendix The sum of the squares of the characters of any irreducible representation equals the number of symmetry operations of the group (g): ~ Xi(R)f =g R where R is any symmetry operation of the group and Xi(R) is the character of that operation in the irreducible representation i. As an example, consider the characters of the A2 representation of C 3v (using the Mulliken symbol for the irreducible representation generated by the R 1 vector- see the character tables in Appendix A): [x( )]2 + [x(cj 1 )f + [x(cj 2 )f + 3[x(a.)f = (-1)2 + (-1)2 + (-1)2 = 6 (the total number of symmetry operations of C 3.) 2 The sum (over all R) of the products of the characters of any symmetry operation in different irreducible representations is zero; that is: ~ x,{r)xj(r) = o R where i and j are two different irreducible representations. Using the A2 and E representations of C 3 as examples: (1 X 2) + (1 X -1) + (1 X -1) + ( -1 X 0) + ( -1 X 0) + ( -1 X 0) = 0 The two equations from 1 and 2 may be summarised as: L x,{r)x.;(r) = gliii R where!iii is the Kronecker delta symbol, which has the value + 1 when i = j, but zero when i >F j. If we use this equation, together with the result of 8.1 (p.188), we can determine which irreducible representations contribute to any reducible one (this equation also limits the possible combinations of characters permissible in any irreducible representation). Let the character of a given operation in any representation be x(r). Let the character of the same operation in the irreducible representation q be Xq(R). By the result of 8.1, the following relationship holds: x(r) = ~ aqxq(r) where the sum is taken over all irreducible representations of the group and aq is the number of times that the irreducible representation q is present. Now multiply both sides by Xp(R), the character of R in the irreducible representation p, and sum both sides over all the symmetry operations of the group: q L x(r)xp(r) = ~ ~ aqxq(r)xp(r) R R q = ~ ~ aqxq(r)xp(r) q R (the order of summations is not significant). But: L aqxq(r)xp(r) = aqlipq R

18 190 Group theory for chemists and so this term vanishes unless p = q. Therefore, when this is summed over all q, the only term that does not vanish is: which equals a,g. Therefore: L Qp')(p(R)x,.(R) R L x(r)x,.(r) = a,g R By rearrangement, we can see that the number of times that the irreducible representation p appears in any reducible representation is: ap = (~) ~ x(r)x,.(r)

19 Appendix C Additional useful information Procedure for assignment to point groups 1 Determine whether the molecule is linear, or belongs to the highly symmetrical cubic or icosahedral point groups (Td, Oh or Ih) If not, proceed to step 2. 2 Find the proper rotation axis of highest order (C.). In the absence of such an axis, look for (a) a plane of symmetry (C 5), (b) a centre of symmetry (C;) or (c) no symmetry other than E (C 1). 3 If an axis c. is found, look for a set of n C2 axes perpendicular to it. If these are present, proceed to step 4. If they are absent, look for (a) a horizontal plane (C.h), (b) n vertical planes (C.v), (c) an S2 axis coincident with c. (S2n) or (d) no symmetry planes or other axes (C.). 4 If a c. axis and n perpendicular axes are present, check on the presence of (a) a horizontal plane (D.h), (b) n vertical planes and no horizontal plane (D.d) or (c) no symmetry planes or other axes (D.). The reduction formula: The proiection operator equation: [P']x = [ ~{Xp(R) R}} Contribution to character per unshifted atom R x(r) E +3-3 cr +1 Cz -1 C31, Ci 0 C41, Cl +1 C61, c6s +2 S31, S35-2 S41, S43-1 S61, s6s 0 191

20 Answers to exercises Chapter 1 1 (a) A c4 axis (coincident with the F-W-Cl direction); two Uv planes each one including the W and Cl atoms, the axial F and two trans equatorial F' atoms); and two ud planes (bisecting the angles between the uv planes). (b) A C 4 axis (passing through the Pt perpendicular to the molecular plane); two C 2' axes (along the Cl-Pt-Cl bond directions); two Cz" axes (bisecting the angle between the Cz' axes); a uh; two Uv planes (including the Cz' axes); two ud planes (including the Cz" axes); ani (at the Pt atom); and an S 4 (coincident with C 4). (c) A c3 axis (along the Si--c-N direction); and three Uv planes (each containing the Si, C, Nand one H). (d) A C 2 axis (through the Cl atom and the opposite C); and two uv planes (one in the molecular plane and one perpendicular to it, passing through the Cl atom). (e) A C2 axis (along the C=C=C direction); two Cz' axes (perpendicular to C 2, each at 45 to the planes defined by the CH 2 groups); two ud planes (bisecting the angles between the C 2 axes); and an S 4 (coincident with C 2). (f) Four C 3 axes (one along each Ni--c-o direction); three C 2 axes (each one bisecting two opposite edges of the tetrahedron defined by the 0 atoms); three S4 axes (coincident with the C 2 axes); and six ud planes (each one containing one edge of the tetrahedron). (g) A C 3 axis (perpendicular to the molecular plane); three C 2' axes (one passing along each C-D bond); a uh; three uv planes (each containing a Cz'); and an S 3 (coincident with C3). (h) A C3 axis (passing through the ring centre); three C 2' axes (each one passing through opposite B and N atoms); a uh; three uv planes (each containing one Cz'); and an S 3 (coincident with C 3). 2 (a) Cs 1, Cs 2, Cs3, C 5 4, C 5 5 = E 192 (b) c61, Cl = C3l, C63 = Cz 1, C64 = Cl, C6 5, C66 = E (c) S3 1, S3 2 = Cl, Sl = uh, S3 4 = C31, S3 5, S36 = E (d) Ss\ Ssz = Csz = C4\ Ss3, Ss4 = Cs4 = Czl, Sss, Ss6 = Cs6 = Cl, Sg7, S 8 8 = E

21 Answers to exercises (a) <Tv(3) (b) C3 1 (c) <Tv(l) (d) C3 1 4 (a) <Tv(l) (b) C 2'(1) (c) S3 5 (d) C 3 2 Chapter 2 l Remember that E = E- 1, <1(xz) = <T(xzt', <T(yz) = <T(yz)- 1 and C2 = C2-1. Work through all the possible similarity transformations; for example: EC2E = EEC2 = C2 c2c2c2 = C2E = c2 <1(xz)C2<1(xz) = <1(xz)<1(xz)C2 = EC2 = C2 (show that ~ and <1(xz) commute) <T(yz)C2<1(yz) = <T(yz)<T(yz)C2 = EC 2 = C 2 Therefore, c2 is in a class by itself. 2 (a) C 1 (b) Cs (c) Cs (d) C2v (e) C2v (f) C3v (g) C4v (h) D4h (i) D2h G) Td (k) D3h (I) Dsh (m) oh (n) Dooh (o) Coov Chapter 3 l (a) [ 10 7] (b) Not possible (c) [ ~ ~]

22 194 Group theory for chemists 2 (a) [ ] (b) Not possible (c) [ -2-4] (d) (e) [ 0 10] [~ ~] (f) [ ~ ~] 3 [0-1][01]= [~ n =E Chapter4 1 (a) D4h E 2C4 Cz 2C2' 2Cz" 2S4 ah mv 2ad (b) The R, vector is: z / t/ - _,/ / F Xe F D4h E 2C4 Cz 2C2' 2Cz" 2S4 fih mv 2ad C4v E Ci Cz Cl [~ ~] [ ~ -~] [ -1 0] 0-1 [ -~ ~]

23 Answers to exercises 195 C4v O"v O"v (J"d (J"d (cont.) [1 OJ [-1 OJ [~ ~J [ 0-1 J (a) C4v E Ci C2 Cl O"v O"v (J"d (J"d (b) These orbitals will give the same representation as the T. and Ty vectors (see Exercise 2). Chapter 5 1 c = [ ~ ~; ~~] therefore x( C) = 13 2 D = [ _: ~ : ] therefore x(d) = 13 B=CAC- 1 = [~ -~J [: :J [_~ ~J 3 4 = [ 5-4 J therefore x(a) = x(b) = (a) F + 2(-1) (-1)2 + 2(1)2 = 8 (b) (1 X 2) + 2( -1 X 0) + (1 X -2) + 2(1 X 0) + 2( -1 X 0) = 0 (c) (-1)2 + 3(0? (-1)2 + 3(W = 12 (d) (2 X 1) + 2( -1 X 1) + 3(0 X -1) + ( -2 X 1) + 2(1 X 1) + 3(0 X -1) = 0 (a) 2Atg + Atu + Eu (b) A2 + Bt + E2 (c) 2At + Tt + T2 Chapter 6 1 The irreducible components of f 3N in each case are as follows: (a) 3At + A2 + 4E (b) SAt + A2 + 2Bt + B2 + 6E (c) At' + Az' + 3E' + 2Az" + E" (d) 2Atg + 2A2g + 2B2g + 2Etg + 4E2g + 2A2u + 2Btu + 2B2u + 4Etu + 2E2u (e) SA 8 + 4B 8 + 4Au + SBu (f) 2At + 2E + 2Tt + ST2

24 196 Group theory for chemists 2 Czv: 2At + A2 + Bt + Bz C4v: At+ Bt + B 2 + E D3h: At'+ E' + E" TJ: E + Tz 3 C3v: ~X~ C3v E 2C3 3crv =At+ E D3h: ~ ~X CD D3h E 2C3 3C 2 <Th 2S3 3<Tv =At'+ E' 4 D3h: 8 (+) ~~X y 8 t.-- 0 D3h E 2C3 3C 2 <Th 2S3 3<Tv = Az" + E" Chapter 7 l 0 '-r2 ~ r, ~NI----'----<

25 Answers to exercises 197 SALCs: (ltv'3) (r 1 + r2 + r3) (A 1 ') (l/v6) (2r 1 - r2 - r3) (E') (l/v2) (r2 - r3) 2 r P == Az" + E" SALCs: (1/V3) ( 1r1 + 1r2 + 1r3) (Az") (11\16) (21T 1-1r 2-1r3) (E") (l/v2) ( 1r2-1r3) 3 The procedure is as for the other examples. The only problem is defining the positions of the symmetry operations. Td (see Figure 7.8): C3 1 (1), C3 2 (1), etc.: C 3 axis along r~> etc; C 2 (12), etc.: C2 axis bisecting the angle between r 1 and r2, etc.; S 4 1(12), Sl(12), etc.: S 4 axis coincident with analogous C 2; ud(l2), etc.: plane containing r 1 and r2 etc. Oh (see Figure 7.9): C3\123), Cl(123), etc.: C 3 axes forming equal angles with rb r2 and r3, etc.; C2(12), etc.: C 2 axis bisecting angle between r 1 and r2, etc.; C4 1 (1), Cl(l), C 2(1), etc.: C 4 and C 2 axis along r~> etc.; S 4 1(12), Sl(12), etc.: S4 axis bisecting angle between r 1 and r2, etc.; S 6 1(123), S6 5 (123), etc.: S6 axis coincident with C 3; uh(1234), etc.: uh containing r~> r2, r3 and r4, etc.; ud(l2), etc.: ud bisecting angle between r 1 and r2, etc. Chapter 8 1 (a) 2A1 (R, pol.; i.r.) + 2E (R, depol.; i.r.) (b) 4A1 (R, pol.; i.r.) + 2B 1 (R, depol.) + B 2 (R, depol.) + 4E (R, depol.; i.r.) (c) A 1' (R, pol.)+ 2E' (R, depol.; i.r.) + Az" (i.r.) (d) 2A1g (R, pol.) + A 2g (inactive) + 2B 2g (inactive) + E 1g (R, depol.) + 4E2g (R, depol.) + A2u (i.r.) + 2B 1u (inactive) + 2Bzu (inactive) + 3E 1u (i.r.) + 2E 2u (inactive) (e) 4Ag (R, pol.) + 2Bg (R, depol.) + 3Au (i.r.) + 3Bu (i.r.) (f) 2A1 (R, pol.)+ 2E (R, depol.) + T 1 (inactive) + 4T 2 (R, depol.; i.r.). 2 v 1: 935 cm- 1 : Cl-0 symmetric stretch (A 1) v 2: 462 cm-1: deformation (E) v 3 : cm- 1 : Cl-0 stretch (T 2) v4 : 628/625 cm- 1 : deformation (T 2) 3 B-F stretches: A 1' (R, pol.) + E' (R, depol.; i.r.) In-plane deformations: E' (R, depol.; i.r.) [A 1' is redundant] Out-of-plane deformations: A2" (i.r.) [E" is redundant- no E" in fvib]

26 198 Group theory for chemists v 1 : 888 cm- 1 : B-F symmetric stretch (A 1') v 2 : 692 cm- 1 : out-of-plane deformation (Az") v 3: 1453/1454 cm- 1 :B-F stretch (E') v 4 : 481/479 cm- 1 : in-plane deformation (E') 4 fvib = 2Ai' + 3E' + 2Az" + E" Cleq~~ Cleq/1 Clax P--Cieq Clax P-o stretches: A1' (R, pol.) + Az" (i.r.) P-cleq stretches: A1' (R, pol.) + E' (R, depol.; i.r.) Cleq-P-Oeq deformations: E' (R, depol.; i.r.) [A 1' is redundant) Cl -P-cleq deformations: E' (R, depol.; i.r.) + Az" (i.r.) + E" (R, depol.) [A 1' is redundant) v 1 : em=~}: P--CI, P--cleq stretches (A 1') v 2 em v 3 : 444 cm-1: P--el stretch (Az") v 4: 299 cm- 1 : Cl -P--cleq deformation (Az'') Vs: 579/581 cm- 1 : P--cleq stretch (E') v 6: 279/277 cm-1: Cl.. -P--cleq deformation (E') v 1: 98 cm- 1 : Cleq-P--cleq deformation (E') v 8 : 261 cm- 1 : Cl -P--cleq deformation (E") Chapter 9 1 B-F stretches: (A 1') (11V3) (r 1 + r 2 + r3) (E') (1/V6) (2r1 - r 2 - r3) (1/V2) (r 2 - r 3) In-plane deformations: (E') (11V6) (2a 1 - a 2 - a3) (11V2) (az - a3) Out-of-plane deformations: (Az") (1/V3) (~ 1 + ~2 + ~ 3 ) 2 The band can be assigned to 2v 3, the first overtone of the E' B-F stretch (2 x 1453 = 2906). To determine the symmetry species of this overtone, use the equation on p For a first overtone, this is: xz(r) = i [ x(r)x(r) + x(r 2 ) J

27 Answers to exercises 199 For D3h, we have: R E c3i Cl Cz Cz Cz (Th s3i s3s CTv CTv CTv Rz E c3z c3i E E E E c3z c3i E E E x(r) x(r 2 ) x 2(R) The resulting representation is reducible to A1' + E'. Hence, the first overtone of the E' B-F stretch of BF3 has symmetry species A1' and E'. 3 D3h~ C3v AI'~ AI E' ~E Az"~ AI Czv AI AI+ Bz BI E" ~E Az + BI Therefore, BF 2Cl (C 2v) has stretches 2A1 + B 2 (NB: BF 2 stretches A1 + B 2; BCl stretch A1; in-plane bends A1 + B 2; out-of-plane bend B1.) PCl 4F (axial F; C 3v) has stretches 3A1 + E (that is, A1 P-F, P-Cl.., P-cleq; E PCh(eq)); 'equatorial' bends E; 'axial' bends 2A1 + E. PCl 4F (equatorial F; C2v) has stretches 3A1 + B1 + B 2 (that is, P-F A1; PCl 2(eq) A1 + Bz; PClz(ax) A1 + B1); 'equatorial' bends 2A1 + B 2 ; 'axial' bends A1 + A 2 + 2B1 + B 2 Chapter 10 (b) 5s, 4dxz: alg 5dx2-y2: big Spx, Spy: eu

28 200 Group theory for chemists (c) a1g: (d) blg: ell: (1/2) ( <1>1 + <f>z + <!>3 + <1>4) (1/2) (<!>J - <l>z + <!>3- <1>4) (1/v'2) ( <1>1 - <!>3); (11v'2) ( <l>z - <1>4) Xe 4F 2 'IT-bonding (in-plane): f('it, in-plane) = a 2g + b 2g + ell

29 Answers to exercises 201 1r-bonding (out-of-plane): f( '1T, out-of-plane) == a 2 u + b 2 u + eg SALCs: f(1r, in-plane): a 2g: (112) ('1TJ + '1Tz + '1T3 + '1T4) bzg: (112) ( 1T1 - '1Tz + '1T3 - '1T4) eu: (11v'2) (1r 1-1r3); (1/v'2) ('1Tz- '1T4) f(1r, out-of-plane): a2u: (112) (1r 5 + '1T6 + '1T7 + '1Ts) bzu: (112) ('lts - '1T6 + '1T7 - '1Tg) eg: (11v'2) (1r 5-1r 7); (11v'2) (1r 6-1r 8 ) Possible 'IT-bonding interactions with: Sd,y (b 2g) (in-plane) Spz (azu) and Sdy., Sdzx (eg) (out-of-plane) 3 f B == ag + bzg + blu + b3u [ H == ag + b3u )-, X v

30 202 Group theory for chemists SALCs (un-normalised): ag: <!>1 + <l>z + <!>3 + <1>4; H1 + Hz b3u: <!>1 - <l>z + <!>3 - <1>4; H1 - Hz bzg: <1>1 - <l>z - <!>3 + <1>4 b1u: <!>1 + <l>z - <1>3 - <1>4 Hence, bonding and anti-bonding combinations are formed for a and b 3 g u, the b2g and b1u combinations are non-bonding. Qualitative molecular orbital energy level diagram: a; ~---, I, I bju I /---, \ I / \ \ I / \ \ I I I I \ \ I / \ \ I I \ \ It/ ',, lj '\ ~,, It \\ 1/,, ----t ~ \' ====,, = b1u. b2g \\ ~ ~ ~ ~ ~ ~,,, \ ~\ \,,, 4... = ' \ // ' \... / ' \ / / ' \ / / ' '... / ' \ / ' \ / / \ b3u // ' / \ / ag 2B 2H There are four electrons to be accommodated in the BHzB unit; that is, the two bonding molecular orbitals are filled. 4 C 2v: Central C: b1 f(1r- terminal C atoms): az + b~> thus b 1 forms bonding and anti-bonding combinations, and az is non-bonding.

31 Answers to exercises 203 Molecular orbital energy level diagram: ---bi Number of 'IT-electrons = 3; ground state (b 1 f(a 2 ) 1 5 C(CH2h Central C Terminal C atoms (f('it)) " I.e. a2 i.e. az" + e" SALCs for terminal C atoms: az": (llv'3) ( 'ITz + 'ITJ + '1'1"4) e": (llv'6) (2'1Tz - 'ITJ - '1'1"4) (11v'2) (,.3 - '1'1"4)

32 204 Group theory for chemists Bonding and anti-bonding az" molecular orbitals formed, and e" non-bonding. a2* / ' / ' / ' / ' / ' / ' / ', / ' / ' / ' / ' / ' ' / / e"= ~==== c ' / ' / ' / ' / ' / ' / ' / ' / ' / ' / ' / ' / ''----/ a2 C6H6 fp(1t) == azu + bzg + e1g + ezu: </J4 SALCs: a2u: (1/V6) (<h + <l>z + <l>3 + <1>4 + <l>s + <l>6) b2g: (11V6) (<1>1 - <l>z + <l>3 - <1>4 + <l>s - <l>6) e1g: (11V12) (2<1>1 + <l>z - <j>3-2<1>4 - <l>s + <1>6); (112) (<l>z + <j>3 - <l>s - <l>6) ezu: (11V12) (2<1>1 - <l>z - <j>3 + 2<1>4 - <l>s - <l>6); (112) ( <l>z - <j>3 + <l>s - <l>6) a2u strongly bonding; e 1g weakly bonding; e 2u weakly anti-bonding; b2g strongly anti-bonding.

33 Answers to exercises 205 The six 1r-electrons fill all of the bonding molecular orbitals. Chapter 11 1 Ground electronic states: BH3: (at'?(e')2: 1 A1' NH 3: (a1?(e) 2 : 1 A1 Excited states: BH3: (a 1') 2(e')3(az") 1 : 1 E" or 3E" (at')2(e')3(at*')2: IE' or 3E' (a1'?(e')3(e*') 1 : 1 A 1' + 1 A2' + 1 E' or 3A 1' + 3 Az' + 3E' NH3: (a 1) 2(e)3(e*) 1 : 1 A A2 + 1 E or 3A 1 + 3A2 + 3E (a 1) 2(e) 3 (a 1*) 1 : 1 E or 3 E All singlet ~ triplet transitions are spin forbidden. Singlet ~ singlet transitions are spin allowed. The symmetry selection rule gives the following results: BH3: tat' ~ IE" forbidden ~ IE' allowed ~ 1At' forbidden ~ taz' forbidden NH3:!At ~ tat allowed 2 Ground state of allyl radical: Possible 1r* excited states: ~ IA2 forbidden ~ IE allowed (bt)2( a2) 1 : 2 A2 (b 1) 1(a 2) 1(b 1*) 1 : 2 A 2 or 4 A 2 (bt)2(bt *)I: 2BI

34 206 Group theory for chemists The 'IT~ 'IT* transitions are 2A2 ~ 2A2 (allowed), 2A2 ~ 2B 1 or 2A2 ~ 4 A2 (forbidden). 3 C03 2 -: Ground state (az")2(e") 4 : 1 A/;n ~'IT* excited state (az")2(e")3(az"*) 1 : 1E' or 3E'. Then~ 'IT* transition 1 A 1' ~ 3E' is spin forbidden; 1A 1' ~ 1E' is spin and symmetry allowed (dipole moment components are E' and Az"). 4 (tzg)2(eg)i (tzg)2: ~Jg, lajg, leg, ITzg (eg)i: 2Eg Possible symmetries for (t2g)2(eg)1: 3Tlg X 2Eg = ~lg + ~2g lalg X 2Eg = 2Eg leg X 2Eg = 2Alg + 2A2g + 2Eg 1T2g X 2Eg = 2Tlg + 2T2g Ground state 4 A2g, hence the only spin-allowed transitions are 4 A2g ~ ~Ig or 4 T2g. (Note that both are g ~ g and so are symmetry forbidden.) Chapter 12 1 The ground state of the allyl anion, CH2CHCH2-, is (see Figures and 12.11) ('1T)2('1T 0 ) 2. Conrotatory motion gives a cyclic product with the electronic configuration (a)2(p)2, while disrotatory motion gives the electronic configuration (a)2(a*)2. Thus, the thermal cyclisation of CH2CHCH2- is predicted to proceed by a conrotatory mechanism. For a photochemical reaction, the allyl anion will have the electronic configuration ( '1T)2( 'lf 0 ) 1 ( 'lf*)1. Conrotatory motion gives the product in the state (a)1(p)2(a*)1, disrotatory (a)2(p) 1 (a*)1, and so the latter is preferred. 2 Photochemically excited C2H 4 molecules will be in the electronic state ('1T)1('1T*) 1. This (see Figure 12.15) would give the cyclobutane in the ground state, and so it will be a symmetry-allowed process.

35 Bibliography Mathematical background R L Flurry, Symmetry Groups, Prentice-Hall (Englewood Cliffs), E P Wigner, Group Theory, Academic Press (New York), 1959 (the definitive treatment). General applications (including theory) B E Douglas and C A Hollingsworth, Symmetry in Bonding and Spectra, Academic Press (Orlando), L H Hall, Group Theory and Symmetry in Chemistry, McGraw-Hill (New York), D S Schonland, Molecular Symmetry, D. Van Nostrand & Co. Ltd. (London), Spectroscopy (both theory and practice) C N Banwell, Fundamentals of Modern Spectroscopy, (3rd edn.), McGraw-Hill (London), J C Decius and R M Hexter, Molecular Vibrations in Crystals, McGraw-Hill (New York), J M Hollas, Modern Spectroscopy, John Wiley & Sons (Chichester), D A Long, Raman Spectroscopy, McGraw-Hill (New York), B P Straughan and S Walker (eds.), Spectroscopy, (3 vols.), Chapman & Hall (London), E B Wilson, Jr., J C Decius and PC Cross, Molecular Vibrations, McGraw-Hill (New York), 1955 (the classical treatment of the applications of group theory to vibrational spectroscopy). L A Woodward, Introduction to the Theory of Molecular Vibrations and Vibrational Spectroscopy, Oxford University Press (Oxford),

36 208 Group theory for chemists Molecular orbitals In addition to the general texts listed on the previous page, see also: J N Murrell, SF A Kettle and J M Tedder, Valence Theory, (2nd edn.), John Wiley & Sons (Chichester), Orbital symmetry and chemical reactions R E Lehr and A P Marchand, Orbital Symmetry, Academic Press (New York), R B Woodward and R Hoffmann, The Conservation of Orbital Symmetry, Verlag Chemie (Weinheim), 1971.

37 Index Abelian group 18 Accidental degeneracy 115 Activities Infrared 96-8 Raman Allyl cation, cyclisation of Alternating axis of symmetry 9 Ammonia molecular orbitals 128 SALCs for vibrational modes 86 Angle deformation vectors 102 for a planar molecule in plane 106, 112 out of plane 106, 112 Anharmonic vibrations 113 Answers to exercises Antibonding molecular orbitals Associative multiplication 18, 188 AX 2 molecules, molecular orbitals AX 3 molecules, molecular orbitals A~ molecules, molecular orbitals 129 Axis of symmetry 2, 5 Benzene electronic spectrum 15~ orbital symmetries (Benzene )chromium tricarbonyl, solid state vibrational spectrum 121 Beryllium hydride, molecular orbitals Bibliography Bond vectors 53, 73, 81, 102 Bonding molecular orbitals Borane, BH 3, molecular orbitals 128 Butadiene Diels-Alder addition electrocyclic reactions electronic spectrum T molecular orbitals in C 1 point group, definition 22 ci point group character table 173 definition 23 en axis, definition 3 en point group character tables definition 23 Cnh point group character tables 176 definition 24 Cnv point group character tables 175, 180 definition 24 C, point group character table 173 definition 23 Carbonate ion, 1T molecular orbitals in Cartesian coordinate vectors 65, 93 Centre of symmetry 6 Character tables 58, Characters of atomic orbital wave function representation of matrices 57 per unshifted atom in f 3N useful properties of 60-3, Chemical reactions, orbital symmetry and

38 210 Group theory for chemists Class of group elements 20 of symmetry operations 20, 58 Combination bands 113 symmetries of Commutative multiplication 14 Conjugate elements 20 Conrotatory mechanisms Conservation of orbital symmetry Correlation splitting 121 Correlation tables 116-8, Coupling of vibrational modes 109 Cubic point groups 27 Cycloaddition reactions Cyclobutadiene, 'Tl' molecular orbitals in (Cyclobutadiene )iron tricarbonyl, 'Tl' molecular orbitals in Cyclobutane Cyclobutene, electrocyclic reactions Cyclopropyl cation d-orbitals participation in molecular orbital formation symmetry properties of 76 d 1 complexes, electronic spectra d 2 complexes, electronic spectra d 5 complexes, electronic spectra 160 Dn point group character tables definition 25 Dnd point group character tables 178 definition 25 Dnh point group character tables 177, 180 definition 26 Decreasing symmetry Degeneracy of irreducible representations 84 of orbitals 49 of vibrational modes 72 Degrees of freedom 91 Diagonal elements of a matrix 34 Diels-Alder reactions Difference bands 114 Dihedral symmetry plane 4 Dipole moment, molecular 96, 150 Direct-product representations 94, 99 Disrotatory mechanisms E, definition of 3 Electrocyclic reactions Electron spin 153 selection rule 153 Electronic spectroscopy group theory in selection rules 150, 153 Ethene, dimerisation of Factor group splitting 121 Fermi resonance 115 Force constants 92 Functions as bases for group representations 46 Fundamental transitions 96 r3n Cartesian coordinate vector representation 65, 93 examples of 72 Gases, vibrational spectra of Gerade 60 Group Abelian 18 basic properties of 17 infinite 27, 118 point - see 'Point group' space 20 High-spin transition metal complexes Horizontal symmetry plane 4 i, definition of 7 lh point group, definition 28 Identity 3, 18 Improper rotation 9. Induced dipole moment 98 Infinite groups 22, 118 Infrared spectroscopy 92 Inverse group element 18 matrix 37, 54 symmetry operations 14, 20 Inversion 6 Irreducible representations 53, 61-2 important properties of characters of multiplication properties of symbols for 58-9 Kronecker delta 189

39 Index 211 Linear molecules 27, 59, 91, , 180 molecular orbitals for Low-spin transition metal complexes Matrices 34 addition of 35 characters of 57 equality of 35 inverse 37 multiplication of 35, 36 null 34 subtraction of 35 transformation 43 unit 34, 36 Methane molecular orbitals for 129 SALCs for vibrational modes 88 Molecular orbitals, symmetries of 77-80, Mulliken symbols for irreducible representations 59 Multiplication commutative 14 non-commutative 14 of group elements 17 of matrices 35 of symmetry operations 13, 19 Mutual exclusion rule 100, 108 Nitrite ion, 1T molecular orbitals in Non-commutative multiplication 14, 36 Non-crossing rule 165 Normalisation of SALCs 84 oh point group character table 179 definition 28 Orbital symmetry and chemical reactions Orbitals as bases for group representations 47, Order of a group 19 Orthogonality of SALCs 87 Osmium tetroxide, solid state vibrational spectrum 120 Overtones 113 symmetries of p-orbitals, symmetry properties of 75 1r-bonding in transition metal complexes molecular orbitals symmetries of molecular orbitals Pauli principle 151 Phosphorus oxychloride, vibrational analysis 94, 100, 104 Photochemical cyclisation reactions Plane of symmetry 4 Point group cubic 27 examples of general definition 20 systematic classification 22, 30 Polarisability tensor 98 Polarisation of Raman-scattered light 93, 101, Projection operators 81-2, 111-3, 123 PtCll- SALCs for vibrational modes 84, 111 vibrational analysis 94, 100, 106 Rh(3) point group, definition 28 Raman spectroscopy 92 Reducible representations 52, formula for reduction of 62, 190 Redundant coordinates 105 Reflection, as symmetry operation 4 Representations of groups 39 direct product 94, 99 irreducible 52 reducible 52 Rotation as a symmetry operation 2 vectors 40, 58, 94 Rotational degrees of freedom 91 Rotational fine structure 120 Rotation-reflection as a symmetry operation 9 axis 9 Ruthenium tetroxide, vibrational analysis 94, 100 s-orbitals, symmetry properties of 49 a, definition 4 a-bonding in transition metal complexes , 134 symmetries of molecular orbitals 77, Sn axis, definition of 9

40 212 Group theory for chemists S 2n point group character tables 179 definition of 25 Schonflies symbols for point groups 22,59 Selection rules electronic spectra 150, 153 infrared spectra 96, Raman spectra 98, spin 153 Similarity transformation 20, 53, 61 Site symmetry 120 Solids, vibrational spectra of Space group 20 Spin multiplicity 153 Spin selection rule t53 Subgroup 19, 116 Sulphur dioxide SALCs for vibrational modes 82 vibrational analysis 91, 100, 102 Sulphur hexafluoride molecular orbitals for 133 SALCs for vibrational modes 89 Symmetry axis of 2, 5 element, general definition of 2 Symmetry-adapted linear combinations (SALCs) 81-90, 111-3, 12~8 for degenerate representations 85, 87 normalisation of 82 orthogonality of 87 Symmetry operations as group elements 19 classes of 20 general definition of 1 identity 3 inversion 6 reflection 4 rotation 2 rotation-reflection 9 successive 10 Symmetry plane 4 Symmetry species 60 Td point group character table 179 definition of 27 Thermal cyclisation reactions Transformation matrices 43, 66 Transition metal complexes electronic spectra of molecular orbitals for , 134, Transition moment 96 Transition probability 96 Translation vectors 39, 58, 94 Translational degrees of freedom 91 Ungerade 60 Unshifted atoms, in calculating f3n 68 Vectors as bases for group representations 39 basis 82 bond 53, 81, 102 generating 82 rotation 40, 58, 94 translation 39, 58, 94 vibration 93, 102 Vertical symmetry plane 4 Vibration vectors 93 exact forms of Vibrational analysis 100, 118 examples of Vibrational degrees of freedom 91 Vibrational modes 91 as bases for group representations 93 classification of 102 Vibrational spectra 92 of gases of linear molecules of solids Water molecule electronic spectrum molecular orbitals Wave functions as bases for group representations 47-50, Woodward-Hoffman rules