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Character Tables Appendix 1 CHARACTER TABLES The character tables for the more common point groups are presented. The last two columns of each character table list the infrared and Raman activity of the particular species. If one or more of the components of the polarizability a xx , a xy ' etc., is listed in the row for a certain species in the last column, that species is Raman active. Similarly, if one or more of the translation components Tz, T y , and Tz is listed in the next-to-last column, the species is infrared active. The components of the change in the dipole moment (/hz, /hy' /hz) should be listed in this column as wen as the components of translation (Tz, Ty, Tz). However, to save space, since they always occur together (both are vectors and transform in the same way if the symmetry operation is carried out similarly), the former are omitted in the character tables in this appendix. Also listed in the next-to­ last column are the components for the rotational coordinates Rz ' Ry , Rz • The subscripts x, y, z indicate the direction in which the translation or dipole moment change occurs for a particular vibrational species. When the components of translation, change in dipole moment, or polarizability are degenerate, they are enclosed in parentheses. 1. Tbe Cs , Ci' and Cn Groups Activity C. E (J",y IR Raman A' 1 1 T"" Ty, Rz axx , Ct.V1J , azz , Ct.a;1I A" 1 -1 Tz, R"" Ry Ct.1IZ , a xz 201 202 Appendix 1 Activity Ci E i IR Raman Ag 1 1 R x , R y, R. eta;x, ety'Y' etzz A" 1 -1 Tx , T., T. Activity C. E C'• IR Raman A 1 1 T., R. etxa:, (ty'Y , Ctzz , ctx'Y B 1 -1 Tx , T., Rx , R. a yZ ' Ctcvz Activity C3 E C3 C'3 IR Raman A 1 1 1 Tz, Rz ct xx + l:t yy , l:tzz (Tx , Ty) E (axx - ayy , a XY )' (axz , a yz ) {~ :* } :*} (Rx , R y) e = e21'li/3, Activity C. E C. C. C34 IR Raman A 1 1 1 1 Tz, R z ct xx + (lyy, l:tzz B 1 -1 1 -1 ct xx - (lyy, (txy i -1 E (Tx , Ty)(Rx , R y) (axz , Ci yz ) {~ -i -1 -:} Character Tables 203 Activity Cs E Cs C's C'S C'5 IR Raman -- A 1 1 1 1 1 Tz, R z a xx + fXyU ' CXzz S s' s'* EI (Tx , Ty)(Rx , R y) (axz , a yz ) {~ s* c: 2* s' :*} s' s* s s'*} E. (axx - a yy , a XY) {~ c: 2* 0 e* o' Activity C. E C' cs C. C. C. • 5 IR Raman A 1 1 1 1 1 1 Tz, R z (Xxx + fX yy , CXzz B 1 -1 1 -1 1 -1 0 -0* -1 -0 EI (Tx , Ty)(Rx , R y) (axz , a yz ) {~ 0* -0 -1 -0* :*} -0* -0 -0* 1 -0 } E. (axx - a yy , a XY ) {~ -0 -0* 1 -0 -0* e = e2Jti/ 6 . 2. The Cnv Groups Activity c.v E C. G,,(xz) G.(yz) IR Raman - Al 1 1 1 1 Tz lXxx, fXlIy , cxzz A. 1 1 -1 -1 R z a",y BI 1 -1 1 -1 T"" R y a",z B. 1 -1 -1 1 Ty , R", avz 204 Appendix 1 Activity Cs• E 2Cs 3eT. IR Raman Al 1 1 1 T. axa: + ay'U, a zz Az 1 1 -1 R. E 2 -1 0 (T", Ty), (R", Ry) (a=-aw , axv )' (ay., a",.) Activity C4V E 2q C'• 2eT. 2eTa IR Raman Al 1 1 1 1 1 T. a= + ay., a •• A z 1 1 1 -1 -1 R. BI 1 -1 1 1 -1 a:tZ - a VlI B. 1 -1 1 -1 1 axv E 2 0 -2 0 0 (T", Ty), (R", R.) (ay., a".) Activity Cs• E 2Cs 2q SeT. IR Raman Al 1 1 1 1 T. aa:z + allll , an A. 1 1 1 -1 R. EI 2 2 cos 72° 2 cos 144° 0 (T", T.), (R", R y ) (a"., a •• ) E. 2 2 cos 144° 2 cos 72° 0 (a"" - a •• , a",.) Activity C,. E 2C, 2C. C. 3eT. 3 eTa IR Raman Al 1 1 1 1 1 1 T. a:vx + a ll'll' azz A. 1 1 1 1 -1 -1 R. BI 1 -1 1 -1 1 -1 B. 1 -1 1 -1 -1 1 EI 2 1 -1 -2 0 0 (T", T.), (R", R.) (a"., a •• ) E. 2 -1 -1 2 0 0 (a"" - a •• ), (axv) Character Tables 205 3. The Cnh Groups Activity c_ h E c_ i Gh IR Raman Ag 1 1 1 1 R. (Xxx, (XlIY , U zz , (XXY A" 1 1 -1 -1 T. Bg 1 -1 1 -1 Rx , R. a yZ ' a xz Bu 1 -1 -1 1 Tx , T. Activity Cgh E 2Cg Gh 2Sg IR Raman -- A' 1 1 1 1 R. axa; + all'Y' azz AU 1 1 -1 -1 T. E' 2 -1 2 -1 (Tx , Ty ) (arex - a.y , aXY ) EU 2 -1 -2 1 (R re , R.) (are.' avz) Activity C.h E C. C_ e: i S: Gh S. IR Raman - Ag 1 1 1 1 1 1 1 1 R. (Xxx+ay'Y' U zz Bg 1 -1 1 -1 1 -1 1 -1 lXxx -ay1I , a xv i -1 -i 1 i -1 Eg (Rx , R.) (ax .. ay.) {~ -i -1 i 1 -i -1 - !} Au 1 1 1 1 -1 -1 -1 -1 T. Bu 1 -1 1 -1 -1 1 -1 1 i -1 -i -1 -i 1 Eu (Tx , T.) {~ -i -1 i -1 i 1 - !} 206 Appendix 1 Activity C5h E 2C5 2q (jh 2S: 2S: IR Raman -- A' 1 1 1 1 1 1 Rz axa; + a'Y'Y' azz A" 1 1 1 -1 -1 -1 Tz E'I 2 2 cos 72° 2cos 144° 2 2 cos 72° 2 cos 144° (Tx , Ty) Eil I 2 2 cos 72° 2 cos 144° -2 -2 cos 72° -2 cos 144° (Rx , R y) (axz , ayz ) E'• 2 2 cos 144° 2 cos 72° 2 2 cos 144° 2 cos 72° (aXX-ayy, aXY ) Eil• 2 2 cos 144° 2 cos 72° -2 -2 cos 144° -2 cos 72° Activity C6h E 2C6 2q"",C. q"",C:' (jh 2S6 2Ss S. "'" i IR Raman -- Ag 1 1 1 1 1 1 1 1 Rz axx + ayy , azz Au 1 1 1 1 -1 -1 -1 -1 Tz Bg 1 -1 1 -1 -1 1 -1 1 B" 1 -1 1 -1 1 -1 1 -1 Elg 2 1 -1 -2 -2 -1 1 2 Rx , R y (tu;, a yZ E,,, 2 1 -1 -2 2 1 -1 -2 Tx , Ty E. u 2 -1 -1 2 2 -1 -1 2 a~-a'1l'1l' a'X1/ E." 2 -1 -1 2 -2 1 1 -2 4. The Dn Groups Activity D. E C.(z) C.(y) C.(x) IR Raman A 1 1 1 1 etza: , a'1l'1l' a zz BI 1 1 -1 -1 Tz, Rz a"" B. 1 -1 1 -1 Ty, Ry axz Bs 1 -1 -1 1 Tx• Rx ayZ Character Tables 207 Activity Da E 2C. 3C. IR Raman Al 1 1 1 ax:c + (11111' a zz A. 1 1 -1 Tz, R z E 2 -1 0 (Tx , Ty), (Rx , R y) (a~x - a yy , a OOY ) (ax.. a yz ) Activity D. E 2C. C.(= C!) 2C; 2C;' IR Raman Al 1 1 1 1 1 Cixa:: + all1l , Clzz A. 1 1 1 -1 -1 Tz, R. Bl 1 -1 1 1 -1 axx - a YII B. 1 -1 1 -1 1 aXl/ E 2 0 -2 0 0 (Tx , TI/)' (Rx ' R II) (a", .. all.) Activity D5 E 2C5 2q 5C. IR Raman Al 1 1 1 1 a",x + allll , a •• A. 1 1 1 -1 Tz, R z El 2 2 cos 72° 2 cos 144° 0 (Tx , Ty), (Rx , R II) (ax.. allz) E. 2 2 cos 144° 2 cos 72° 0 (am - al/Y ' aXl/) 208 Appendix 1 Activity D8 E 2Ce 2Ca C2 3C: 3C;' IR Raman Al 1 1 1 1 1 1 a .... + aulI ' a •• AI 1 1 1 1 -1 -1 TI, R. Bl 1 -1 1 -1 1 -1 BI 1 -1 1 -1 -1 1 El 2 1 -1 -2 0 0 (T"" Tu), (R." Ru) (a",., al/.) EI 2 -1 -1 2 0 0 «(X"'''' - (XlIV' a.",) 5. The Dnh Groups Activity DIA E q q Cf j 11"'11 11.,. I1v• IR Raman Ag 1 1 1 1 1 1 1 1 a=:,a1l'll' all A" 1 1 1 1 -1 -1 -1 -1 B1U 1 1 -1 -1 1 1 -1 -1 R. a.", Blu 1 1 -1 -1 -1 -1 1 1 Ta Blu 1 -1 1 -1 1 -1 1 -1 Rv a.,. B2u 1 -1 1 -1 -1 1 -1 1 Tu Bau 1 -1 -1 1 1 -1 -1 1 R., avl Bau 1 -1 -1 1 -1 1 1 -1 T", Character Tables 209 Activity DSh E 2Cs 3C. ah 2Ss 3av IR Raman A'1 1 1 1 1 1 1 aa:x + ay'Y' a zz A~' 1 1 1 -1 -1 -1 A'• 1 1 -1 1 1 -1 R.
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