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. A"• 1 1 -1 -1 -1 1 T. E' 2 -1 0 2 -1 0 (T"" Ty) (a= - aw , a.,y)
Eil 2 -1 0 -2 1 0 (R"" Ry) (ay., a",.)
Activity E C' i a 2av D'h 2C. • 2C. 2C; 2S. h 2ad IR Raman
Alg 1 1 1 1 1 1 1 1 1 1 a=+ayy , a ..
AlU 1 1 1 1 1 -1 -1 -1 -1 -1
A •• 1 1 1 -1 -1 1 1 1 -1 -1 R.
A.u 1 1 1 -1 -1 -1 -1 -1 1 1 T.
Blg 1 -1 1 1 -1 1 -1 1 1 -1 aza: - a y'1l
Bl" 1 -1 1 1 -1 -1 1 -1 -1 1 B.g 1 -1 1 -1 1 1 -1 1 -1 1 a",y
B.u 1 -1 1 -1 1 -1 1 -1 1 -1
Eg 2 0 -2 0 0 2 0 -2 0 0 (R"" Ry) (ay., a",.)
Eu 2 0 -2 0 0 -2 0 2 0 0 (T"" Ty ) ....N o
Activity
DiA E 2Ca 2q 5C. (JA 2S5 2S: 5(J~ IR Raman
A'1 1 1 1 1 1 1 1 1 a" + at/v' azz A'• 1 1 1 -1 1 1 1 -1 R.
E'1 2 2cos 72° 2 cos 144° 0 2 2cos 72° 2 cos 144° 0 (T." T.) E'• 2 2 cos 144° 2 cos 72° 0 2 2 cos 144° 2 cos 72° 0 (a.,. - a •• , a.,.)
A"1 1 1 1 1 -1 -1 -1 -1 A"• 1 1 1 -1 -1 -1 -1 1 T. Eil 1 2 2cos 72° 2 cos 144° 0 -2 -2 cos 72° -2 cos 144° 0 (R." Rw) (a., .. all.) Eil• 2 2cos 144° 2 cos 72° 0 -2 -2 cos 144° -2 cos 72° 0
» 'tI 'tI (1) :::I c.. x· .... o :T 111.. 111o .... CD Activity ..
D8h E 2C. 2C. CI 3C: 3C;' i 2S8 2S. (JA 3(Jd 3(J~ ~ r:1' IR Raman iii I/l
Au 1 1 1 1 1 1 1 1 1 1 1 1 a.,.,+all!l,a ••
Azu 1 1 1 1 -1 -1 1 1 1 1 -1 -1 R. BIg 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
Bau 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1
Eu 2 1 -1 -2 0 0 2 1 -1 -2 0 0 (R~,R,,) (a,.., av.)
Eu 2 -1 -1 2 0 0 2 -1 -1 2 0 0 (au - all!l,a~) Al,. 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1
Az,. 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 T.
Bl ,. 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1
Bau 1 -1 1 -1 -1 1 -1 1 -1 1 1 -1
2 1 0 0 1 2 0 0 (T~,TI/) El " -1 -2 -2 -1
Ea" 2 -1 -1 2 0 0 -2 1 1 -2 0 0
~ ~ 212 Appendix 1
6. Tbe Dnd Groups
Activity D'd E 2S: C'• 2C. 2ad IR Raman
Al 1 1 1 1 1 Ci"" + Civv ' Ci .. A. 1 1 1 -1 -1 R.
BI 1 -1 1 1 -1 a u -a'UlI ..
B. 1 -1 1 -1 1 T. Ci"v E 2 0 -2 0 0 (T", T.), (R", R.) (Ci •• , Ci".)
Activity D'd E 2C. 3C. i 2S6 3ad IR Raman
AlU 1 1 1 1 1 1 Ci"" + Ciw ' Ci •• A•• 1 1 -1 1 1 -1 R.
Eu 2 -1 0 2 -1 0 (R", Rv) (Ci",,,-Cil/V ' Cixv)'(Ci"., Civ.)
Alu 1 1 1 -1 -1 -1
A.u 1 1 -1 -1 -1 1 T.
Eu 2 -1 0 -2 1 0 (T". Tv)
Activity D'd E 2S8 2C, 2S: C. 4C; 4ad IR Raman
Al 1 1 1 1 1 1 1 Ci"" + Civv • Ci •• A. 1 1 1 1 1 -1 -1 R. BI 1 -1 1 -1 1 1 -1 B. 1 -1 1 -1 1 -1 1 T.
EI 2 V2 0 -V2 -2 0 0 (T". TlI)
E. 2 0 -2 0 2 0 0 (Ci"" - Cil/v' Ci",v)
Es 2 -V2 0 V2 -2 0 0 (R". RlI) (Ci" •• Civ.) n ::T III., III .... CI).," ;t 0- CD Activity Ul
D 5d E 2C5 2q SC2 i 2S~0 2S,o Sad IR Raman
A'g 1 1 1 1 1 1 1 1 a xx + C1.yy ' C1.zz
A 2g 1 1 1 -1 1 1 1 -1 Rz
Elg 2 2 cos 72° 2 cos 1440 0 2 2 cos 72° 2 cos 144° 0 (Rx , Ry ) (ayz, a zx)
E2g 2 2 cos 1440 2 cos 720 0 2 2 cos 1440 2 cos 720 0 (a xx - a yy , a XY)
Alu 1 1 1 1 -1 -1 -1 -1
A 2u 1 1 1 -1 -1 -1 -1 1 Tz
E,u 2 2 cos 72° 2 cos 1440 0 -2 -2 cos 720 -2 cos 1440 0 (Tx,T y)
E2u 2 2 cos 144° 2 cos 72° 0 -2 -2cos 1440 -2 cos 720 0
------_ .. _-
...N w ...N "'"
Activity D. d E 2S12 2C. 2S. 2C3 2St. C. 6C; 6ad IR Raman
Al 1 1 1 1 1 1 1 1 1 axx + ayy ' azz
A. 1 1 1 1 1 1 1 -1 -1 Rz
Bl 1 -1 1 -1 1 -1 1 1 -1 I
B. 1 -1 1 -1 1 -1 1 -1 1 Tz
El 2 V3 1 0 -1 -V3 -2 0 0 (Tx , Ty)
E. 2 1 -1 -2 -1 1 2 0 0 (axx - a yy ' a XY ) Es 2 0 -2 0 2 0 -2 0 0 E. 2 -1 -1 2 -1 -1 2 0 0
E. 2 -V3 1 0 -1 V3 -2 0 0 (R x , R y) (ayz. azx)
'0> '0 CD :::I C. ;C' ... Character Tables 215
7. Tbe Sn Groups
Activity S4 E S4 c. S: IR Raman ---
A 1 1 1 1 Rz a xx , + a yy , azz
B 1 -1 1 -1 Tz a xx - a yy , a XY
i -1 (Tz, Ty ) E -:} (ayZ ' azz ) {~ -i -1 (Rz ' R y )
Activity S S6 E C. C8 i sg S6 IR Raman
Ag 1 1 1 1 1 1 Rz azz + ayy , a zz
e e* 1 e (azz - ayy , a ZY )' Eg (Rz , R y ) e* e e* :*} ' D 1 (a yZ azz) Au 1 1 1 -1 -1 -1 Tz e e'" -1 -e Eu -e*} (Tz, Ty) {~ e'" e -1 -e* -e
S6 = C. xi; N ~ Q)
Activity i 7 S8 E S8 C. S"8 c. S: C." S8 IR Raman
A 1 1 1 1 1 1 1 1 R. (oc",,,,+ OCyy , OC••)
B 1 -1 1 -1 1 -1 1 -1 T.
6 i -6* -1 -6 -i (T"" Ty ) EI :*} {~ 6* -i -6 -1 -6* i (Rx , Ry ) i -1 -i 1 i -1 E. - ~} (oc",,,,- OCyy , OC",y) {! -i -1 i 1 -i -1
-6* -i 6 -1 6* i } Es -6 (OCyz. OC,"') {! -6 i 6* -1 6 -i -6*
6 = e''''i/8, » 'tI 'tI CD ::l C. )C' ~ Character Tables 217
8. The Cubic Groups
Activity
T E 4Ca 4q 3C2 IR Raman
A 1 1 1 1 a xx + a yy + a zz c c* (axx + a yy - 2azz, E {! c* c a a xx - a yy ) (Tx , T y , Tz); F 3 0 0 -1 (Rx , R y , R z) a xy , a xz , a yz ~ 00
Activity T" E 4Cs 4q 3C. i 4S. 4S: 3a" IR Raman
Ag 1 1 1 1 1 1 1 1 u"'''' + U yy + U zz Au 1 1 1 1 -1 -1 -1 -1
e e* 1 1 e e* Eg (u",,,, + U yy - 2uzz, u"'''' - U yy) {: e* e 1 1 e* e :}
e e* 1 -1 -e* -e* Eu -I} {: e* e 1 -1 e* -e -1
Fg 3 0 0 -1 3 0 0 -1 (R"" Ry , Rz) (u",y, u",z, uyz) Fu 3 0 0 -1 -3 0 0 1 (T""T y , Tz)
8 = e21ti/3•
'0> '0 CD ::l CL )C' ~ Character Tables 219
Activity Tu. E 8Ca 3C! 6S, 6C1a IR Raman
Al 1 1 1 1 1 a...+all'l/+a••
AI 1 1 1 -1 -1
E 2 -1 2 0 0 (a... + all'l/ - 2a••• a ... - a1lll)
Fl 3 0 -1 1 -1 (Rm. Ru. R.)
Fs 3 0 -1 -1 1 (T",. TII • T.) (a"",. av •• a",.)
Activity
0 E 8C. 3C. 6C4 6C'; IR Raman --- Al 1 1 1 1 1 axx + a yy + a .. A. 1 1 1 -1 -1 (axx + a yy - 2a... E 2 -1 2 0 0 axx - a yy ) (Tx • Ty • Tz) FI 3 0 -1 1 -1 (Rx • R y • Rz) F. 3 0 -1 -1 1 (aXY ' ayz • azx ) N No
Activity
Ok E SC. 3C. 6C4 6C~ i SS. 3ak 6S4 6aa IR Raman
AlU 1 1 1 1 1 1 1 1 1 1 a=+allll+a ••
A 2g 1 1 1 -1 -1 1 1 1 -1 -1
Eg 2 -1 2 0 0 2 -1 2 0 0 (2a= - a",,,, - auv' a= -- allll)
F,g 3 0 -1 1 -1 3 0 -1 1 -I (Ra;, Rv, Rz)
F2g 3 0 -I -I 1 3 0 -I -I 1 (al A2u 1 1 1 -1 -I -I -I -I 1 1 Eu 2 -I 2 0 0 -2 1 -2 0 0 i F,u 3 0 -I 1 -I -3 0 1 -I 1 (T"" T", Tz) F2U 3 0 -I -1 1 -3 0 1 1 -I » "C "C CD ::I c.. x· ... Character Tables 221 9. The Coov and D ooh Groups Activity Coov E ... 2q~/rp ... qZI OOl1v IR Raman l:+ 1 1 1 1 T. Ci... + Cin , Ci •• l:- 1 1 1 -1 R. (T"" T,,) II 2 2 cos!p -2 0 (Ci.,. , l4J.) (R." Rv) LI 2 2 cos 2tp 2 0 (CL.,., - Cin , CiIW) 11' 2 2 cos 311' -2 0 N N N Activity D=" E ... 2C2n lrp ... qz, 00(1" i '" 2S2n lrp ••• (1/0 ooC. IR Raman E+q 1 1 1 1 1 1 1 1 a ... + fXw, a •• E+ 1 1 1 1 -1 -1 -1 -1 " Tz E-q I 1 1 -1 1 1 1 -1 R. E- I 1 1 -1 -1 -1 -1 1 " II q 2 2 cos P -2 0 2 -2 cos P -2 0 (R", Rv) (a"., av.) II" 2 2 cos P -2 0 -2 2 cos P 2 0 (T", Tv) LI. 2 2 cos 2p 2 0 2 2 cos 2p 2 0 (a... - a vv ' ag:v) LI" 2 2 cos 2p 2 0 -2 -2 cos 2p -2 0 P. 2 2 cos 3p -2 0 2 -2 cos 3p -2 0 p" 2 2 cos 3p -2 0 -2 2 cos 3p 2 0 » '0 '0 CD :::l c.. x' ... Appendix 2 DESCRIPTION OF SYMBOLISM USED IN THE INTERNATIONAL TABLES FOR X-RAY CRYSTALLOGRAP'HY The space group Pbca/D~Z taken from the International Tables for X-Ray Crystallography(l) is used to describe further characteristics of the space group (Fig. A2-1). Pbca No. 61 m m m Orthorhombic D2hI5 -0 0.- ~0 i-- 0+ 1·0 0+ c ---~----I---- ,+0 0+ 1+0 ~i rV(b }-b 0i- -0 0)- i- 0 I----i----I---- x(o) -0 Ol~ <- 0+ r+G) O· . Origin al T Number of positions, Co-ordinates of equivalent positions Wyckolf notation, Conditions limiting and point symmetry possible reflections General: 8 c I x,y,z; !+x,l-y,i; x,!+y,l-z; !-x,y,t+z; hk[: No conditions x,y,i; !-x,!+y,z; x,!-y,t+z; i+x,y,t-z. 0Ie1: k~2n hOl: [~2n hkO: h~2n hOO: (h~2n) 0k0: (k~2n) 00[: (l~2n) Special: as aboye, plus 4 b O,O,!; !. M; 0, t ,0; ! ,0,0. } hk[: h+k,k+l, (/+h)~2n 0,0,0; !,!,O; O,U; 1.0,t. " a Symmetry of special projections (001) pgm; a' ~aI2, b' ~b (IOO)pgm; b'~bI2, c'~c (010) pgm; c' ~c/2, a'-a Fig. A2-I. 223 224 Appendix 2 The abbreviated symmetry elements are given across the top of Fig. A2-1 at the left corner; Pbca is the Hermann-Mauguin designation, and D~~ is the Schoenflies notation. This is followed by the space group number (in this case No. 61 of a total of 230), followed by the "fulI" space group symbols (these include other symmetry elements if they are present), fol• lowed by the crystal class out of 32 (mmm) and the crystal system (out of seven)-orthorhombic. The symbol P21 /b21/c21/a indicates the unit cell is "primitive," that the symmetry element 21 may be associated with the crys• tallographic axes a, b, and c respectively. The two-fold screw axes are des• ignated as 21 and indicate that a one-half unit cell translation follows the normal two-fold (180°) rotation. The glide planes are all perpendicular to the respective crystallographic axes. The top left-hand diagram symbolizes the way in which the "asymmet• rie unit" is repeated with the unit cell (circles and plus or minus signs are used; the plus sign indicates lying above plane of the page and the negative sign indicates lying below the plane of the page). The upper leji-hand corner of the rectangle is the origin, the x axis extends down the page, and the y axis extends across the page (from left to right). The symbols in the right-hand diagram indicate wh ich symmetry ele• ments are present and their positions. The symbol I represents one two• fold screw axis perpendicular to both a-c and b-c planes located at x = 0, z = ! and z = 0, y = !, respectively. The two-fold axis x = !, z = 0 parallel to c and perpendicular to the a-b plane is denoted by the symbol §. The broken lines represent glide planes seen edge-on and the small circles represent centers of symmetry. Under the heading "conditions limiting possible reflections" the sys• tematically absent reflections for the space group are listed. The listing of "coordinates of equivalent positions" gives the allow• able atom positional parameters for sites of various symmetries within the space group. REFERENCE 1. N. F. M. Henry and K. Lonsdale (eds.), International Tables Jor X-Ray Crystal• lography, Vol. 1, The Kynoch Press, Birmingham, England (1965). Appendix 3 SITE SYMMETRIES FOR THE 230 SPACE GROUPS The site symmetries for the 230 space groups are given. The symmetries can be used directly for a Bravais primitive cell, and the space groups are numbered from 1 to 230 to make for easier reference. The sequence of num• be ring corresponds to that used in the International Tables Jor X-Ray Crystal• lography.(1) In using these tables, it should be noted that when the site sym• metry is Cp , Cpv , or Cs and p = 1, 2, 3, etc., the number of sites is infinite. Note: The site symmetry is described by several symbols. For the example Space group 25 pmm2 C~v: 4C2v (1); 4Cs(2); CI (4) the coefficient represents the nonequivalent sets of sites (in this case 4C2v sites) with one equivalent site per set shown in parentheses; an infinite num• ber of Cs sites, each having an occupation number of two equivalent atoms and there are four such nonequivalent sets; an infinite nu mb er of Cl sites with four equivalent sites per set. For further discussion on this subject see Couture(3) and Irish and Brooker. (4) Space grouplIl Site symmetriesl2l 1 Pl ClI C,(l) 2 pI 0 , 8Ci (l); C,(2) 3 P2 C'2 4C.(1); C, (2) 4 P2, C 2 C, (2) 5 B. or C. c'• 2C.(l); C,(2) 6 Pm Cls 2CsCl); C, (2) 7 Pb or Pe C's C, (2) 8 Bm or Cm C's Cs(l); C, (2) 9 Bb or Ce C'8 C, (2) 10 P2/m qh 8C.h(1); 4C.(2); 2C8 (2); C,(4) 225 226 Appendix 3 Appendix 3 (eontinued) Space group(1) Site symmetries(2) 11 P2,/m qh 4Ci (2); Cs(2); C,(4) 12 B2/m Of C2/m qh 4C,h(1); 2Ci (2); 2C,(2); Cs(2); Cl(4) 13 P2/b Of P2/e Cih 4Ci (2); 2C,(2); Cl(4) 14 P2l /b Of P2l /e qh 4Ci (2); Cl(4) 15 B2/b Of C2/e qh 4Ci (2); C,(2); Cl(4) 16 P222 Dl, 8D,(l); 12C,(2); Cl(4) 17 P222l D', 4C,(2); Cl(4) 18 P2l2l2 Da, 2C,(2); Cl(4) 4 19 P2l2l2l D 2 Cl (4) 20 C222l D5, 2C,(2); Cl(4) 21 C222 D6, 4D,(l); 7C,(2); Cl(4) 22 F222 D7, 4D,(l); 6C,(2); Cl(4) 23 1222 D,S 4D,(1); 6C,(2); Cl(4) D9 24 1212121 2 3C,(2); Cl(4) 25 Pmm2 qv 4C,v(l); 4Cs(2); Cl(4) 26 Pme2l qv 2Cs(2); Cl(4) 27 Pee2 qv 4C,(2); Cl(4) 28 Pma2 Civ 2C,(2); Cs(2); C(4) 29 Pea2l qv Cl (4) 30 Pne2 qv 2C,(2); Cl(4) 31 Pmn2l qv Cs(2); Cl(4) 32 Pba2 qv 2C,(2); Cl(4) 33 Pna2l qv Cl(4) ClO 34 Pnn2 2V 2C,(2); C,(4) Cu 35 Cmm2 'v 2C,v(1); C,(2); 2Cs(2); Cl(4) Cu 36 Cme2l 'v Cs(2); Cl(4) Cl3 37 Cee2 'v 3C,(2); Cl(4) Cu 38 Amm2 'v 2C,v(1); 3C.(2); Cl(4) C15 39 Abm2 2V 2C,(2); Cs(2); Cl(4) C16 40 Ama2 2V C,(2); Cs(2); Cl (4) 41 Aba2 Cl72V C,(2); Cl(4) CIS 42 Fmm2 'v C,v(1); C,(2); 2C.(2); Cl(4) 19 43 Fdd2 C'v C,(2); Cl(4) 44 Imm2 C'o'v 2C,v(l); 2Cs(2); Cl (4) C2l 45 Iba2 2V 2C2 (2); Cl(4) Site Symmetries for the 230 Space Groups 227 Appendix 3 (continued) Space group(Il Site symmetries(21 461ma2 C22 'v C2(2); Cs(2); CI(4) 47 Pmmm D~h 8D2h (1); 12C2v (2); 6Cs (4); CI(8) 48 Pnnn D~h 4D,(2); 2Ci (4); 6C2(4); CI(8) 49 Pccm D~h 4C2h (2); 4D2(2); 8C,(4); Cs(4); CI(8) 50 Pban D:h 4D,(2); 2Ci (4); 6C,(4); CI(8) 51 Pmma D~h 4C'h(2); 2C'v(2); 2C2(4); 3Cs(4); CI(8) 52 Pnna Dh 2Ci (4); 2C,(4); CI(S) 53 Pmna Dh 4C'h(2); 3C,(4); Cs(4); CI(8) 54 Pcca D~h 2Ci (4); 3C2(4); CI(8) 55 Pbam D~h 4C2h (2); 2C,(4); 2Cs(4); CI(8) 56 Pccn DIO,h 2Ci (4); 2C2(4); CI(8) 57 Pbcm D",h 2Ci (4); C,(4); Cs (4); CI(8) 58 Pnnm D'22h 4C2h (2); 2C2(4); Cs(4); CI(8) 59 Pmmn D132h 2C'v(2); 2Ci (4); 2Cs(4); CI(8) 60 Pbcn D14,h 2Ci (4); C,(4); CI(8) DI5 61 Pbca 2h 2Ci (4); CI(S) 62 Pnma DI6,h 2Ci (4); Cs(4); CI(8) 63 Cmcm D172h 2C'h(2); C'v(2); Ci (4); C,(4); 2Cs(4); CI(8) DIS 64 Cmca 2h 2C2h (2); Ci (4); 2C2(4); Cs(4); Ci (8) 65 Cmmm DI92h 4D'h(l); 2C2h (2); 6C2v (2); C2(4); 4Cs(4); CI(8) D20 66 Cccm 2h 2D2(2); 4C'h(2); 5C2(4); Cs(4); C,(8) 67 Cmma D21 'h 2D,(2); 4C'h(2); C 2v (2); 5C2(4); 2Cs(4); CI(8) 68 Ccca D 222h 2D2(2); 2Ci (4); 4C2(4); CI(S) 69 Fmmm D'32h 2D'h(1); 3C2h (2); D,(2); 3C2V (2); 3C2(4); 3C8 (4); CI(S) D2. 70 Fddd 2ft 2D2(2); 2Ci (4); 3C2(4); CI(8) D25 71 Immm 2ft 4D2ft (l); 6C2V (2); Ci (4); 3Cs(4); C,(8) 72 Ibam D'·,ft 2D2(2); 2C2h (2); Ci (4); 4C2(4); Cs(4); C,(8) 73 Ibca D272ft 2Ci (4); 3C2(4); CI(8) 741mma D's,ft 4C2h (2); C2v (2); 2C2(4); 2Cs(4); CI(8) 75 P4 Cl• 2C.(1); C2(2); CI(4) C2 76 P41 • CI(4) 77 P42 q 3C2(2); CI(4) 78 P43 C'• CI(4) 79 14 C•5 C.(l); C2(2); CI(4) 80 141 C·• C,(2); CI(4) 228 Appendix 3 Appendix 3 (continued) Space groupCl) Site symmetries(2 ) SI SI P4 • 4S.(1); 3C2(2); C1 (4) S2 S2 14 • 4S.(1); 2C2(2); C1 (4) S3 P4/m Ch 4C.h (l); 2C2h (2); 2C.(2); C2(4); 2Cs(4); CI(S) S4 P42/m C:h 4C2h (2); 2S.(2); 3C2(4); Cs(4); C1 (S) S5 P4/n C:h 2S.(2); C.(2); 2Ci (4); C2(4); C1 (S) S6 P42/n C:h 2S4(2); 2Ci (4); 2C2(4); C1 (S) 87 14/m qh 2C4h (1); C 2h (2); S.(2); C4(2); Ci (4); C2(4); Cs(4); C1 (S) SS 141/a C:h 2S.(2); 2Ci (4); C2(4); C1 (S) S9 P422 D•I 4D.(1); 2D2(2); 2C.(2); 7C2(4); CI(S) D2 90 P4212 • 2D2(2); C4(2); 3C2(4); CI(S) Da 91 P4122 • 3C2(4); CI(S) 92 P41212 D'• C2(4); C1 (S) D5 93 P4222 • 6D2(2); 9C2(4); C1 (S) 94 P42212 D •6 2D2(2); 4C2(4); CI(S) 95 P4322 D'• 3C2(4); C1 (S) D8 96 P43212 4 C2(4); C1 (S) 97 1422 D •9 2D4(1); 2D2(2); C.(2); 5C2(4); C1 (S) IO 9S 14122 D• 2D2(2); 4C2(4); C1 (S) 99 P4mm CL 2C4v(l); C 2v (2); 3Cs(4); C1 (S) 100 P4bm C:v C.(2); C 2v (2); Cs(4); C1 (S) 101 P42cm C:v 2C2V (2); C2(4); Cs(4); CI(S) 102 P42nm C:v C2v (2); C2(4); Cs(4); C1 (S) 103 P4cc C!v 2C.(2); C2(4); C1 (S) 104 P4nc C:v C.(2); C2(4); C1 (S) 105 P42mc C1v 3C2V (2); 2Cs(4); C1 (S) 106 P42bc C:v 2C2(4); CI(S) 10714mm C:v C.v(1); C2v (2); 2Cs(4); C1 (S) CIO lOS 14cm .v C.(2); C 2v (2); C.(4); C1 (S) 109141md Cu.V C2v (2); Cs( 4); Cl (S) 110 141cd Cl2.v C2(4); C1 (S) 111 P42m D~d 4D2d(1); 2D2(2); 2C2v (2); 5C2(4); Cs(4); C1 (S) 112 P42c D:d 4D2(2); 2S.(2); 7C2(4); C1 (S) 113 P42,m D~d 2S.(2); C2v (2); C 2 (4); Cs(4); C1 (S) 114 P421c D~d 2S.(2); 2C2(4); C1 (S) 115 P4m2 D~d 4D2d (I); 3C2V (2); 2C2(4); 2Cs(4); C1 (S) Site Symmetries for the 230 Space Groups 229 Appendix 3 (continued) Space group(1) Site symmetries(') 116 P4c2 D:d 2D.(2); 2S.(2); 5C,(4); Cl(S) 117 P4b2 D~ä 2S.(2); 2D.(2); 4C,(4); Cl(S) l1S P4n2 D~ä 2S.(2); 2D.(2); 4C,(4); Cl(S) 11914m2 D:ä 4D'd(1); 2C'v(2); 2C,(4); C.(4); Cl(S) DlO 120 14c2 'ä D.(2); 2S.(2); D,(2); 4C,(4); C, (8) 121 142m DU,ä 2D.ä(1); D,(2); S.(2); C'v(2); 3C.(4); C.(4); Cl(S) l2 122142d D'd 2S.(2); 2C,(4); Cl(S) 123 P4/mmm Dh 4D.,,(1); 2D.,,(2); 2C4V (2); 7C'v(4); 5C.(S); C, (16) 124 P4/mcc Dh D.(2); C.,,(2); D.(2); C.,,(2); C,,,(4); D,(4); 2C.(4); 4C,(S); C.(S); C, (16) 125 P4/nbm Dh 2D.(2); 2D.ä(2); 2C,,,(4); C.(4); C.v(4); 4C.(S); C.(S); C, (16) 126 P4/nnc Dh 2D.(2); D.(4); S.(4); C,(4); Ct(8); 4C.(S); Cl(S) 127 P4/mbm Dh 2C,,,(2); 2D,,,(2); C,(4); 3C'v(4); 3C.(S); C, (16) 12S P4/mnc D:" 2C,,,(2); C.h(4); D.(4); C.(4); 2C,(S); C.(S); C, (16) 129 P4/nmm Dh 2D.a(2); C.v(2); 2C,,,(4); C.v(4); 2C.(S); 2C.(S); Cl(16) 130 P4/ncc D!" D,(4); S,(4); C,(4); Ct(S); 2C,(S); Cl(16) 131 P4,/mmc Dh 4D.,,(2); 2D,a(2); 7C'v(4); C,(S); 3C.(S); C, (16) O 132 P42/mcm Dl.10 D.h(2); D.a(2); D'h(2); D 2d (2); D.(4); C2h (4); 4C2V (4); 3C,(S); 2C.(S); C,(16) 133 P4./nbc DU." 3D.(4); S,(4); Ct(S); 5C,(S); C, (16) 134 P4,/nnm D."" 2D.d(2); 2D.(4); 2C.h(4); C2v (4); 5C.(S); C.(S); C, (16) 135 P4./mbc Dl."3 C.,,(4); S,(4); C.,,(4); D,(4); 3C.(S); C.(S); C, (16) 136 P4./mnm Dl'." 2D,,,(2); C.,,(4); S,(4); 3C'v(4); C.(S); 2C.(8); C, (16) 137 P4./nmc D."l5 2D.a(2); 2C2V (4); C;(S); C.(8); C.(S); C, (16) 13S P4./ncm Dlß D.(4); S.(4); 2C2h (4); C'v(4); 3C,(8); C.(S); C, (16) Dl7." 13914/mmm ,10 2D.,,(I); D.,,(2); D'd(2); C'v(2); C,,,(4); 4C.v(4); C,(S); 3C.(S); C, (16) 140 14/mcm Dl."S D,(2); D 2d (2); C,,,(2); D'h(2); C,,,(4); C.(4); 2C.v(4); 2C.(S); 2C.(8); C, (16) 141 14,/amd Dl9." 2D.d(2); 2C,,,(4); C'v(4); 2C.(S); C.(S); C, (16) D20 142 14,/acd ,10 S,(4); D.(4); C;(8); 3C.(S); C, (16) 143 P3 Cl3 3C.(1); C, (3) 144 P3, C'• C, (3) 145 P3. C·3 C, (3) 230 Appendix 3 Appendix 3 (continued) Space group(1) Site symmetries(') 146 R3 C'3 C3(1); C,(3) 147 P3 qi 2Cgi (1); 2Cg(2); 2Ci (3); C,(6) 148 R3 Ci, 2C3i(1); Cs(2); 2Ci (3); C,(6) 149 P312 D'3 6Dg (1); 3Cg(2); 2C.(3); C,(6) D2 150 P321 3 2Dg (1); 2C3(2); 2C.(3); C,(6) D3 151 P3,12 3 2C.(3); C,(6) 152 P3,21 D'3 2C.(3); C,(6) 153 P3.12 D5S 2C.(3); C,(6) 154 P3.21 D·3 2C.(3); C,(6) 7 155 R32 D 3 2D.(1); C.(2); 2C.(3); C,(6) 156 P3m1 qv 3Csv (1); C.(3); C,(6) 157 P31m Civ C3V(1); Cs(2); C.(3); C,(6) 158 P3c1 Civ 3C3(2); C,(6) 159 P31c C:v 2C3(2); C,(6) 160 R3m qv C3v(1); C.(3); C,(6) 161 R3c qv C3(2); C,(6) 162 P31m D~rL 2DsrL (1); 2D.(2); C.v(2); 2C.h(3); C3 (4); 2C2(6); C.(6); C,(12) 163 P31c D~rL Ds(2); Csi(2); 2D3 (2); 2Cs(4); Ci (6); C.(6); C,(12) 164 P3m1 D:rL 2DsrL (1); 2C.v(2); 2C2,,(3); 2C.(6); C.(6); C,(12) 165 P3c1 D~rL Ds(2); C.i (2); 2Cs(4); Ci (6); C,(6); C.(12) 166 R3m D:rL 2DsrL (1); Csv(2); 2C.h(3); 2C.(6); C.(6); C,(12) 167 R3c D:rL D 3(2); Csi(2); Cs(4); C,(6); C.(6); C,(12) 168 P6 C'• C.(l); Cs(2); C2(3); C,(6) 169 P6, C'6 C,(6) 170 P6. CS• C,(6) 171 P6. q 2C2(3); C,(6) 172 P6. C·• 2C.(3); C,(6) 173 P6. q 2C3 (2); C,(6) 174 P6 q" 6C3,,(1); 3C3(2); 2C.(3); C,(6) 175 P6/m Ch 2C.h(1); 2C3,,(2); C.(2); 2C2,,(3); Cg(4); C.(6); 2C.(6);" C,(12) 176 P63/m C:h Csh(2); C3,(2); 2C3,,(2); 2Cg(4); Ci (6); C.(6); C,(12) 173 P622 D'• 2D.(1); 2D3(2); C.(2); 2D.(3); C.(4); 5C2(6); C,(12) 178 P6,22 D2• 2C.(6); C,(12) " Site Symmetries for the 230 Space Groups 231 Appendix 3 (continued) Space group(') Site symmetries(') 179 P6522 D3• 2C.(6); C,(12) 180 P6.22 D •4 4D.(3); 6C.(6); C,(l2) D5 181 P6422 6 4D2 (3); 6C2 (6); C,(12) 182 P6322 D·6 4D3(2); 2C3(4); 2C.(6); C,(12) 183 P6mm qv C.v(l); C3v (2); C.v(3); C.v(3); 2Cs(6); C,(l2) 184 P6cc qv C.(2); C3(4); C2(6); C, (l2) 185 P63cm C:v C3v (2); C3(4); Cs(6); C,(12) 186 P63mc C:v 2C3V (2); Cs(6); C,(12) 187 P6m2 D~h 6D3h (l); 3C3V (2); 2C,v(3); 3C.(6); C, (12) 188 P6c2 D~h D 3(2); C3h (2); D 3(2); C3h (2); D 3(2); C3h (2); 3C3(4); C 2(6); C.(6); C,(12) 189 P62m D:h 2Dah (1); 2C3h (2); C3v (2); 2C.v(3); C3(4); 3C.(6); C,(12) 190 P62c D!h D 3(2); 3C3h (2); 2C3(4); C.(6); Cs(6); C,(l2) 191 P6/mmm Dh 2D.h(1); 2D3h (2); C.v(2); 2D'h(3); C3v (4); 5C'v(6); 4C.(12); C,(l2) 192 P6/mcc D:h D.(2); C.h(2); D 3(4); Cah (4); C6 (4); D.(6); C 2h (6); C3(8); 3C,(l2); C.(l2); C,(24) 193 P63/mcm D:h D ah (2); D 3d (2); Cah (4); D 3(4); C.(4); C 2h (6); C.v(6); C3(8); C.(12); 2C2 (l2); C,(24) 194 P63/mmc Dih D ad(2); 3D3h (2); 2C3V (4); C2h(6); C.v(6); C 2(l2); 2C8 (12); C,(24) 195 P23 T' 2T(1); 2D2(3); C3(4); 4C.(6); C,(12) 196 F23 T' 4T(1); C3(4); 2C.(6); C,(12) 197 123 T3 T(l); D,(2); C3(4); 2C.(6); C,(12) 198 P2,3 T4 C3(4); C,(12) 199 12,3 T6 C3(4); C.(6); C,(12) 200 Pm3 Tl 2Th(1); 2D.h(3); 4C2V(6); C3(8); 2C.(12); C,(24) 201 Pn3 T'h T(2); 2C3i (4); D,(6); C3(8); 2C2 (12); C,(24) 202 Fm3 T~ 2Th(l); T(2); C.h(6); C.v(6); C3(8); C.(12); C.(12); C,(24) 203 Fd3 Tfi 2T(2); 2C3i (4); C3(8);C.(12); C,(24) 204 1m3 Tt Th(1); D'h(3); C3i (4); 2C'v(6); C3(8); C8 (12); C,(24) 205 Pa3 T~ 2C3i (4); C3(8); C, (24) 206Ia3 TX 2C3i (4); C3(8); C2(12); C,(24) 207 P432 0' 20(1); 2D4(3); 2C.(6); C3(8); 3C,(12); C,(24) 232 Appendix 3 Appendix 3 (continued) Space group(l) Site symmetries(2) 208 P4.32 0 2 T(2); 2D3(4); 3D.(6); Ca(8); 5C.(12); C1(24) 209 F432 0 3 20(1); T(2); D 2(6); C.(6); C3(8); 3C2(12); C1 (24) 210 F4132 O' 2T(2); 2Da(4); C3(8); 2C2(12); C1(24) 211 1432 0 5 0(1); D.(3); Da(4); D 2(6); C.(6); Ca(8); 3C2(12); CI (24) 212 P4332 0 6 2D3(4); C3(8); C2(12); C1(24) 213 P4132 O' 2D.(4); C3(8); C.(12); C1(24) 214 14132 0 8 2D3(4); 2D.(6); C3(8); 3C2(12); C1(24) 215 P43m Ta 2Ta(1); 2D2d(3); Ca.(4); 2C2.(6); C.(12); C.(12); C1(24) 216 F43m TJ 4Ta(1); C3.(4); 2C2.(6); C.(12); C1(24) 217 143m Tl Ta(1); D.a(3); C3.(4); 8.(6); C •• (6); C.(12); C.(12); C1(24) 218 P43n n T(2); D 2(6); 28.(6); C3(8); 3C.(12); C1(24) 219 F43c Tj 2T(2); 28.(6); C3(8); 2C2(12); C1(24) 220 143d T3 2S.(6); C3(8); C.(12); C1(24) 221 Pm3m 01 20h (1); 2D4h(3); 2C.v (6); Cav(8); 3C2v(12); 3C.(24); C1(48) 222 Pn3n O~ 0(2); D.(6); C31(8); S.(12); C.(12); Ca(16); 2C2(24); C1(48) 223 Pm3n O~ Th (2); D 2h(6); 2D.a(6); D.(8); 3C.v (12); C.(16); C2(24); C.(24); C1(48) 224 Pn3m ot Ta(2); 2D.a(4); D 2d(6); C •• (8); D 2(12); C ••(12); 3C2(24); C1(48) 225 Fm3m O~ 20h (1); Ta(2); D.h (6); C.v (6); Cav(8); 3C.v (12); 2Cs (24); C1(48) 226 Fd3c O~ 0(2);. Th (2); D.a(6); C.h(6); C.v (12); C.(12); C.(16); C.(24); C.(24); C1(48) 227 Fd3m 01. 2Td (2); 2Dad(4); Cav(8); C •• (12); C.(24); C.(24); C1(48) 228 Fd3c O~ T(4); Da(8); Cal (8); S.(12); C.(16); 2C.(24); C1(48) 2291m3m O~ Oh(l); D.h (3); D.d (4); D.d (6); C ••(6); Cav(8); 2C2.(12); C.(24); 2Cs (24); C,(48) 230 la3d 01° C.1(8); D.(8); D.(12); S.(12); Ca(16); 2C.(24); C1(48) Note the following equivalent nomenclatures: Ci == S•• C. == C1h • D2 == V. D2h ~ Vh • D2a == Vd • and C.! == S6. Site Symmetries for the 230 Space Groups 233 REFERENCES 1. N. F. M. Henry and K. Lonsdale (eds.), International Tables/or X-Ray Crystallography, Vol. 1, Kynoch Press, Birmingham, England (1965). 2. R. S. Halford, J. Chern. Phys., 14:8 (1946). 3. L. Couture, J. Chern. Phys., 15:153 (1947). 4. D. E. Irish and M. H. Brooker, Appl. Spec., 27:395 (1973), Appendix 4 CORRELATION TABLES The correlation tables which follow are heipful in determining the site group reiating to the moIecuIar point group. The tables give the correiations be• tween species of a group and a subgroup. We wish to express our gratitude to St. Martin's Press forpermission to reproduce these tables from the book by D. M. Adams,(!) and to W. G. Fateley et al. for the use of their compre• hensive tables.(2) C. C. C. Ca C_ Cl D. CZ CY C'" Da Ca C. ------• • • A A A A A A A A A A Al A A B A B A B A Bl A B B A. A B E 2B El E 2B 2A B. B A B E E A+B E. E 2A 2A Ba B B A C" C" C~ • C~ 2 D. D. D. C. C. C. C. D5 C5 C. Al A A A A A A Al A A A. Bl Bl A A B B A. A B Bl A Bl B A A B El El A+B B. Bl A B A B A E. E_ A+B E B. + Ba B. + Ba E 2B A+B A+B _ C'• C"• C' C"• D. C. Da Da D_ Ca C_ C_ C_ Al A Al Al A A A A A A. A A_ A_ Bl A A B B Bl B Al A_ B_ A B A B B. B A_ Al Ba A B B A E l El E E B_ + Ba E 2B A+B A+B E_ E_ E E A + Bl E 2A A+B A+B 235 236 Appendix 4 G(zx) G(Yz) C.v C. C. C. C3v C3 C. Al A A' A' Al A A' A. A A" ftt" A. A A" Bl B A' A" E E A' +A" B. B A" A' Gv Gd, Gv Gd, C.v C. C.v C.v C. C. C. Al A Al Al A A' A' A. A A. A. A A" A" Bl B Al A. A A' A" B. B A. Al A A" A' E E Bl + B. Bl + B. 2B" A' +A" A'+A" Csv Cs C. Al A A' A. A A" El El A'+A" E. E. A'+A" Gv Gd, Gv ->- G(ZX) Gv Gd, Cs• Cs C3v C3v C.v C3 C. C. C. Al A Al Al Al A A A' A' A. A A. A. A. A A A" A" Bl B Al A. Bl A B tA' A" B. B A. Al B. A B A" A' El El E E Bl + B. E 2B A'+A" A'+A" E. E. E E Al + A. E 2A A'+A" A'+A" C.h C. C. Ci C3h C3 C. Cl Ag A A' Ag A' A A' A Bg B A" Ag E' E 2A' 2A Au A A" Au A" A A" A Bu B A' Au E" E 2A" 2A Correlation Tables 237 C.h C. S. C.h C. C. Ci Cl C.h C. C. Cl Ag A A Ag A A' Ag A A' A A' A Bg B B Ag A A' Ag A Et' EI 2A' 2A Eg E E 2Bg 2B 2Au 2Ag 2A E.' E. 2A' 2A Au A B Au A AU Au A AU A AU A Bu B A Au A AU Au A E{' EI 2A u 2A Eu E E 2Bu 2B 2A' 2Au 2A EU• E. 2Au 2A C6h C. C3h S. C.h C3 C. Cs Ci Cl Ag A A' Ag Ag A A A' Ag A Bg B AU Ag Bg A B AU Ag A Elg EI EU Eg 2Bg E 2B 2A u 2Ag 2A E.g E. E' Eg 2A g E 2A 2A' 2A g 2A Au A AU Au Au A A AU Au A Bu B A' Au Bu A B A' Au A EIU EI E' Eu 2Bu E 2B 2A' 2Au 2A E.u E. EU Eu 2A u E 2A 2A u 2Au 2A C.(z) C.(y) C.(x) C.(z) C.(y) C.(x) D'h D. C." C." C." C.h C.h C.h Ag A Al Al Al Ag Ag Ag Blg BI A. B. BI Ag Bg Bg B.g B. BI A. B. Bg Ag Bg B3g B3 B. BI A. Bg Bg Ag Au A A. A. A. Au Au Au Bw BI Al BI B. Au Bu Bu B.u B. B. Al BI Bu Au Bu B3U B3 BI B. Al Bu Bu Au D'h C.(z) C.(y) C.(x) a(xy) a(zx) a(yz) (cont.) C. C. C. Cs Cs Cs Ci Ag A A A A' A' A' Ag Blg A B B A' AU AU Ag B.g B A B AU A' AU Ag B3g B B A AU AU A' Ag Au A A A AU AU AU Au Bw A B B AU A' A' Au B2U B A B A' AU A' Au B3U B B A A' A' AU Au 238 Appendix 4 ah --+ av(zy) ah av D3h Cah D3 C3V C2v C3 C2 Cs Cs A,' A' A, A, A, A A A' A' A2' A' A2 A2 B2 A B A' A" E' E' E E A, + B2 E A+B 2A' A' + A" A'{l A"1 A" A, A2 A2 A A A" A"• A" A2 A, B, A B A" A' E" E" E E A. + B, E A+B 2A" A' + A" -+ C' C" C~ --+ C~ C~' C~ 2 • D4h D4 D2d D2d C 4n C4h D2h D2h C4 S4 Alg Al A, Al Al Ag Ag Ag A A A2g A2 A2 A2 A2 Ag B,g Blg A A Blg BI BI B2 B, Bg Ag B,g B B B2g B2 B2 B, B2 Bg Blg Ag B B Eg E E E E Eg B2g + B3g B2g + Bag E E AlU A, BI BI A2 Au Au Au A B A2u A2 B2 B. A, Au Blu B1u, A B BlU B, Al A2 B. Bu Au BlU B A B2u B2 A2 A, B, Bu B,u Au B A Eu E E E E Eu Bou + B3u B2u + B3u E E D4h C'2 C"2 C 2 , crv C 2 , erd C'2 C"2 (cant.) D2 D. C2v C2V C2V C2v Arg A A Al Al Al A, A2g BI 8 1 A2 A2 BI BI B,g A B, A, A. Al 8 , B2g B, A A2 Al B, A, Eg B2 + B3 B2 + B, B, + B2 BI + B2 A2 + B2 A. + B2 AlU A A A2 A2 A. A2 Aou BI BI A, A, B2 B2 B,u A B, A2 A, A2 B2 B2u B, A Al A2 B2 A2 Eu B2 + B3 B2 + B, BI + B2 B, + B2 Al + BI Al + BI Correlation Tables 239 Cu D4h C. C~ • C. C'• C"• Gh G v Gd (cont.) C.h C.h C.h C. C. C, C. C. C. Ci Alg Ag Ag Ag A A A A' A' A' Ag A.g Ag Bg Bg A B B A' A" A" Ag Blg Ag Ag Bg A A B A' A' A" Ag B.g 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 AlU Au Au Au A A A A" A" A" Au A.u Au Bu Bu A B B A" A' A' Au BlU Au Au Bu A A B A" A" A' Au B,u Au Bu Au A B A A" A' A" Au Eu 2Bu Au + Bu Au + Bu 2B A+B A+B 2A' A'+A" A'+A" 2Au Gh -+ G(zx) Gh G v D'h D. C.. C.h C. C,v C. C. C. A'1 Al Al A' A Al A A' A' A'• A. A. A' A B, B A' A" E{ E, E, E'1 E, Al + B, A+B 2A' A'+A" E'• E. E, E'• E. Al + B, A+B 2A' A'+A" A"1 Al A. A" A A. A A" A" A"• A. Al A" A B. B A" A' E{' E, El E"1 E, A. + B. A+B 2A" A'+A" E"• E. E. E"• E. A. + B. A+B 2A" A' +A" G h -+ a(xy) C'• c"• C"• C'• Gv -+ G(YZ) D6h D. D3h D3h C.v C6h D3d Dad D'h A,g Al A'1 A~ Al Ag A,g Alg Ag A.g A. A'• A'• A. Ag A.g A.g B,g B, A" A" B A,g B,g 1 • B. g A.g B.g B'g B. A"• A{' B, Bg A,g A.g Bag E,g E, E" E" E, E,g Eg Eg B.g + Bag E.g E. E' E' E. E2g Eg Eg Ag + B,g AlU Al A{' AU1 A. Au Alu AlU Au AU A 2U A2 • A"• Al Au A,u A.u B,u BlU B, A~ A'• B, Bu A.u AlU B.u A' A' B B.u B. • 1 B. u AlU A.u Bau E,u E, E' E' E, E,u Eu Eu B2a + Bau E.u E. E" E" E2 E.u Eu Eu Au + BlU 240 Appendix 4 D6h C'2 C"2 (Jv (Jd (cant.) C6 C3h D3 D3 C3V C3V S6 D2 Aig A A' Al Al Al Al Ag A A2g A A' A2 A2 A2 A2 Ag BI Big B A" Al A2 A2 Al Ag B2 B2g B A" A2 Al Al A2 Ag B3 Eig EI E" E E E E Eg B2 -+- B3 E2g E2 E' E E E E Eg A -+-BI AlU A A" Al Al A, A2 All A A2u A A" A2 A2 Al Al Au BI BlU B A' Al A2 Al A2 Au B2 B2U B A' A2 Al A2 Al Au Bs Elu EI E' E E E E Eu B2 -+- B3 E2u E2 E" E E E E Eu A -+- BI D6h C2 C'2 C"2 C2 C'2 C"2 C2 (cant.) C2V C2V C2V C 2h C 2h C 2h C3 C2 Alg Al Al Al Ag Ag Ag A A A2g A2 BI BI Ag Bg Bg A A Big BI A2 B2 Bg Ag . Bg A B B2g B2 B2 A2 Bg Bg Ag A B Eig BI -+- B2 A2 -+- B2 A2 -+-B 2 2Bg Ag -+- Bg Ag -+- Bg E 2B E2g Al -+- A2 Al -+- BI Al -+-BI 2A g Ag -+- Bg Ag -+- Bg E 2A AlU A2 A2 A2 Au Au Au A A A2u Al BI B2 Au Bu Bu A A Bw B2 Al BI Bu Au Bu A B B2u BI B2 Al Bu Bu Au A B Elu B2 + BI Al + B2 Al -+- BI 2Bu Au + Bu Au -+- Bu E 2B E2U A2 -+- Al A2 +BI A2 + B2 2A u Au + Bu Au + Bu E 2A Correlation Tables 241 C' C" (Jh (Ja (Jv D'h • • (cant.) C. C. Cs Cs Cs Ci A,g A A A' A' A' Ag A2g B B A' A" A" Ag Btg A B A" A' A" Ag B2g B A A" A" A' Ag E,g A+B A+B 2A" A' +A" A' +A" 2A g E2g A+B A+B 2A' A' + A" A' + A" 2A g AlU A A A" A" A" Au A2U B B A" A' A' Au BlU A B A' A" A' Au B2U B A A' A' A" Au ElU A+B A+B 2A' A' +A" A' +A" 2A u E.u A+B A+B 2A" A' + A" A' +A" 2Au A' E' E' E' A~f A~' E~f E~' E" • I 2 a a C2 -+ C2(z) C2 C'2 D2d S. D2 C2V C2 C2 Cs A, A A A, A A A' A2 A BI A2 A B A" BI B A A2 A A A" B2 B BI Al A B A' E E B2 + Ba B, + B2 2B A+B A' +A" Dad Da C3V S, Ca C 2h C2 Cs Ci A,g 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" 2A g AlU A, 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" 2A u 242 Appendix 4 C. C'2 D'd D. C.V SB C. C2V C2 C2 Cs Al Al Al A A Al A A A' A2 A2 A_ A A A2 A B A" BI Al A2 B A A2 A A A" B2 A2 Al B A Al A B A' EI E E EI E BI + B2 2B A+B A' +A" E2 BI + B2 BI + B2 E2 2B Al + A2 2A A+B A' +A" Ea E E Ea E BI + B2 2B A+B A' + A" D5d D5 C5V C5 C2 Cs Ci Alg Al Al A A A' Ag A2g A2 A2 A B A" Ag Elg EI EI EI A+B A' +A" 2A g E2g E2 E2 E2 A+B A' +A" 2A g AlU Al A2 A A A" Au A2u A2 Al A B A' Au EIU EI EI EI A+B A' +A" 2A u E2U E2 E2 E2 A+B A' +A" 2Au D.d D. C.v C. D 2d Da Cav Al Al Al A Al Al Al A2 A2 A2 A A2 A2 A2 BI Al A2 A BI Al A2 B2 A2 Al A B2 A_ Al EI EI EI EI E E E E2 E2 E2 E2 B, + B2 E E Ea B, + B2 B, + B2 2B E A, + A2 Al + A2 E. E2 E2 E2 Al + A2 E E E5 E, E, E, E E E Correlation Tables 243 D6d C. C'• (cant.) D. C.v S. C3 C. C. Cs Al A Al A A A A A' A. Bl A. A A B A" Bl A A. B A A A A" B. BI Al B A A B A' EI B. + B3 BI + B. E E 2B A+B A' +A" E. A +BI Al + A. 2B E 2A A+B A' +A" E3 B. + B3 BI + B. E 2A 2B A+B A' +A" E. A + BI Al + A. 2A E 2A A+B A' +A" E5 B. + BI BI + B. E E 2B A+B A' +A" S. C. Cl S. C3 Ci Cl SB C. C. Cl A A A Ag A Ag A A A A A B A A Eg E 2Ag 2A B A A A E 2B 2A Au A Au A EI E 2B 2A Eu E 2Au 2A E. 2B 2A 2A E3 E 2B 2A T D. C3 C. A A A A A E 2A E 2A 2A F BI + B. + B3 A + E A + 2B 3A Th T D'h S. D. Ag A Ag Ag A Eg E 2Ag Eg 2A Fg F Blg + B.g + B3g Ag + Eg BI + B. + B3 Au A Au Au A Eu E 2Au Eu 2A Fu F Blu + B.u + B 3u Au + Eu BI + B. + Ba 244 Appendix 4 Th (cont.) C.v C.h Ca C. Cs Ci Cl Ag Al Ag A A A' Ag A Eg 2A l 2A. E 2A 2A' 2Ag 2A Fg A. + Bl + B. Ag + 2Bg A+E A +2B A' + 2A" 3Ag 3A Au A. Au A A A" Au A Eu 2A. 2Au E 2A 2A" 2Au 2A Fu Al + Bl + B. Au +2Bu A+E A + 2B 2A' +A" 3Au 3A Ta T D.a Cav S. D. C.v Al A Al Al A A Al As A Bl A. B A A. E E Al + Bl E A+B 2A Al + A. Fl F A. + E A. +E A+E Bl + B. + Ba A. + Bl + B. F. F B. +E Al + E B+E Bl + B. + Ba Bl + B. + Ba Ta (cont.) Ca C. Cs Al A A A' A. A A A" E E 2A A'+A" Fl A+E A +2B A' +2A" F. A+E A +2B 2A' +A" 3C. C., 2C~ 0 T D. Da C. D. D. Al A Al Al A A A A. A Bl A. B A Bl E E Al + Bl E A+B 2A A +Bl Fl F A. +E A. + E A+E Bl + B. + Ba Bl + B. + Ba F. F B.+E Al + E B+E Bl + B. + Ba A + B. + Ba Correlation Tables 245 0 (cont.) C3 C2 C2 Al A A A A 2 A A B E E 2A A+B Fl A+E A + 2B A + 2B F2 A+E A +2B 2A +B Oh* 0 Td Th T D 4h D3d D 4d C3V D 3 D3i == S6 A lg Al Al Ag A A lg A lg Al. Al Al Ag A 2g A 2 A 2 Ag A Blg A 2g B lg A 2 A 2 Ag Eg E E Eg E A lg + Blg Eg A lg + Blg E E Eg Flg Fl Fl Fg F A 2g + Eg A 2g + Eg A 2g + Eg A 2+E A 2+E Ag+Eg F2g F2 F, Fg F B2g +Eg A lg + Eg B'g + Eg Al +E Al+E Ag+Eg AlU Al A 2 Au A AlU AlU AlU A 2 Al Au A,u A, Al Au A B lU A 2u Blu Al A 2 Au Eu E E Eu E Alu +Blu Eu AlU +BlU E E Eu Flu Fl F2 Fu F A 2u +Eu A'u+Eu A'u+Eu Al +E A,+E Au+Eu F2U F2 Fl Fu F B 2u + Eu Alu + Eu B,u + Eu A 2+E Al +E Au+Eu °h C" Gd C;, Gh (cont.) C3 D 2d D'd C4V D 4 C4h S4 C4 A lg A Al Al Al Al Ag A A A'g A Bl B, Bl B l Bg B B Eg E Al + B l Al + B, Al + B l Al + Bl Ag + Bg A+B A+B Flg A+E A 2 + E A, + E A 2 + E A 2 + E Ag + Eg A+E A+E F'g A+E B2 + E Bl + E B2 + E B, + E Bg + Eg B+E B+ E Alu A Bl Bl A, Al Au B A A 2u A Al A 2 B, Bl Bu A B Eu E Al + Bl A, + Bl A, + B, Al + Bl Au + Bu A+B A+B Flu A+E B, + E B 2 +E Al + E A, + E Au + Eu B+E B+E F,u A+E A 2 + E Al + E B l + E B 2 +E Bu + Eu A+E B+E * To find correlation~ with smaller subgroups, carry out the correlation in two steps; for example, if the correlation of 0h with C2V is desired, use the table to pass from 0h to Td and then employ the table for Td to go on to C2V . 246 Appendix 4 °h 3C, C" 2C~ C" Gh C" Gd (cont.) D2h D'h C'v C.v A lg Ag Ag Al Al A.g Ag Blg Al A, Eg 2Ag Ag + Blg 2Al Al + A. Flg Blg + B'g + Bag Blg + B.g + Bag A. + Bl + B. A. + Bl + B. F.g Blg + B.g + Bag A lg + B.g + Bag A, + Bl + B. Al + Bl + B. Alu Au Au A. A. A.u Au B,u A. Al Eu 2Au Au + Blu 2A. Al + A. Flu Blu + B.u + Bau Blu + B.u + B3u Al + Bl + B. Al + Bl + B. F.u Blu + B.u + Bau Au + B.u + Bau Al + Bl + B. A. + Bl + B. °h C~, Gh 3C. C" 2C~ C" Gh C~, Gh (cont.) C.v D. D. C.h C.h Alg Al A A Ag Ag A,g Bl A Bl Ag Bg Eg Al + Bl 2A A + Bl 2Ag Ag + Bg Flg A. + Bl + B. Bl + B. + Ba Bl + B. + Ba Ag + 2Bg Ag + 2Bg F.g Al + A. + B. Bl + B. + Ba A+B.+B3 Ag + 2Bg 2Ag + Bg Alu A, A A Au Au A.u B. A Bl Au Bu Eu A. + B. 2A A + Bl 2Au Au + Bu Flu Al + Bl + B. Bl+B,+Ba Bl + B, + Ba Au + 2Bu Au + 2Bu F.u Al + A. + Bl Bl + B. + B3 A+B.+Ba Au + 2Bu 2Au + Bu °h Gh G,t C. C'• (cont.) Cs Cs C, C. Ci Cl Alg A' A' A A Ag A A.g A' A" A B Ag A Eg 2A' A' +A" 2A A+B 2Ag 2A Flg A' +A" A' + 2A" A +2B A + 2B 3Ag 3A F.g A' +2A" 2A' +A" A +2B 2A +B 3Ag 3A Alu A" A" A A Au A A.u A" A' A B Au A Eu 2A" A' +A" 2A A+B 2Au 2A Flu 2A' +A" 2A' +A" A + 2B A +2B 3Au 3A F.u 2A' +A" A' + 2A" A +2B 2A +B 3Au 3A Correlation Tables 247 lh l C. Ca C. Cl Ag A A A A A Au A A A A A Flg Fl A + El A+E A + 2B 3A Flu Fl A + El A+E A + 2B 3A F.g Fa A + Ea A+E A +2B 3A F.u Fa A + E. A+E A +2B 3A Glg Gl El + Ea 2A + E 2A + 2B 4A Glu Gl El + Ea 2A +E 2A + 2B 4A Hg H A+El+E. A +2E 3A + 2B 5A Hu H A + El + E. A + 2E 3A + 2B 5A REFERENCES 1. D. M. Adams, Metal-Ligand and Related Vibrations, St. Martin's Press, New York (1968). 2. W. G. Fateley, F. R. DolIish, N. T. McDevitt, and F. F. Bentley, lnfrared and Raman .Selection Rules for Molecular and Lattice Vibrations: The Correlation Method, Wiley-Interscience, New York (1972). Appendix 5 ELEMENTARV MATHEMATICS This appendix will attempt to present the fundamental definitions and theo• rems necessary for an understanding of group theory. The presentation will not be detailed, for the approach to group theory followed in this book will be empirical rather than mathematical. For a more detailed discussion of the subject see the texts by Margenau and Murphy(l) and others. (2-41 DEFINITION OF A GROUP The group has been defined in Sectionl-4B. In summary,asetofelements A, B, C, ... is said to be a group iffor every pair of elements (e.g., A and B) a binary operation exists that yields the product AB which belongs to the set; if the associative law holds for the combination of elements; if the set contains the identity element; and if there is an inverse for each element. FINITE AND IN FINITE GROUPS Groups containing a limited number of elements are called finite, while groups containing an unlimited number of elements are called infinite. The number of elements, g, in a finite group determines the order of the group. All of the groups that we will encounter will be finite groups of order g, with the exception of those for linear molecules, of which there are two (Coov , D ooh ). SUBGROUPS Inspection of a group will show that within the group there are smaller groups with the same operation. In the group Cav , which is of order 6, the following smaller groups will be found; E by itself; O'v' of order 2; and Ca, of order 3. If the order of the group is g, then the order of the subgroup, h, must be an integral divisor of g. 249 250 Appendix 5 CLASSES If A and B are elements of a group, then B-lAB will be equal to some element Y of the group. Thus Y = B-lAB (A5-l) Y is called the transform of A by B, or we say that A is conjugate to Y. The following are properties of conjugate elements: 1) every element is a conjugate to itself; 2) if A is conjugate to Y, then Y is conjugate to A; e.g., A = B-l YB; 3) if A is conjugate to Yand Y is conjugate to X, then A is conjugate to X and A, Y, and X belong to the same dass. A complete set of elements conjugate to each other is called a dass of the group. The method of arranging the elements of a group into dasses exhibits the structure under the relation of conjugation. The result is that the sym• metry of the molecule can be presented as a set of disjoint sets of geometric (symmetry) elements. For Cav the complete set of elements conjugate to each other is E, Ca, C~; O"v1, O"v" O"v3. For C4V : E, C4 , Cf; C2 - C:; O"v1, O"v.; O"d1, O"d'· DEFINITION OF A MATRIX A collection of real or complex quantities displayed in a table of rows and columns is called an array. The most familiar type of array is the de• terminant, which always has the same number of rows and columns, and is always a number. It can be written as An A l2 A 13 ••• A ln A 2l A 22 A 23 . .. A 2n A = I A I = A 3l A a2 A 33 . .. A 3n (A5-2) A matrix, on the other hand, is an array in which the number of rows and columns can differ. It is an element from a set of matrices with a specific (row-by-column) multiplication (unlike determinants, which have different multiplications in the sense that the determinant is a number and this number is invariant under the interchange of rows and columns). The ma- Elementary Mathematics 251 trix product is not a number. However, a matrix product can have a set of determinants of various orders. We may represent a matrix as Au Au A l3 ••• Alm A 21 A 22 A 23 ... A 2m A = [Ai,j] = A 31 A 32 A 33 ... A 3m (A5-3) Here n and m determine the order of the matrix, n giving the number of the rows and m the number of columns. When n = m, the matrix is called square. MULTIPLICATION OF MATRICES A matrix A having three rows and three columns is to be multiplied by a matrix B having three rows and two columns. The row elements of matrix Aare multiplied by the corresponding column elements of B. The following example will illustrate this operation: Au Al2 AI~ BU Blj A = [A 21 A 22 A 23 B = [ B 21 B22 (A5-4) A 31 A a2 Aa BaI Ba (3 X 3 matrix) (3 X 2 matrix) AUBll + A u B 21 + A 13Bal A u Bl2 + A l2B 22 + A laBa2] [ = A 21Bu + A 22B 21 + A 2aBai A 2lBl2 + A 22B 22 + A 2aB32 = C (A5-5) AalBu + A a2B 21 + AaaBal AalBu + A a2B22 + A aaBa2 Thus, AB = C. Here the product is a matrix of three rows and two columns. TRANSPOSE OF A MATRIX Consider the matrix (A5-6) 252 Appendix 5 Hs transpose is (A5-7) REPRESENTATION OF GROUPS The elements of a group, such as the symmetry operations of a mole• eule, can be represented by matrices. For true representations, the multi• plication of the numbers representing A and B of the group must, if AB = C, lead to the number which represents the element C. A set of numbers or matrices which can be assigned to the elements of a group and which can properly represent the multiplications of the elements of this group is said to constitute a representation of the group. This can be illustrated by considering the moleeule NH3 in the C3V point group (Fig. A5-1). The fol• lowing treatment is taken from an article by Ziomek. (2) The following internal coordinates can be written: LlD l change in bond distance XY1 LlD2 change in bond distance XY2 LlD3 change in bond distance XY3 Lla12 change in angle Y1XY2 LI a 13 change in angle Y 1XY 3 Lla23 change in angle Y2XY3 Z I I I I Fig. A5-1. The NH3 (XY3) molecule showing the x, y, z coordinates. Elementary Mathematics 253 If the ct (1200 clockwise) symmetry operation is carried out for the XYa molecule, the following shifts occur: let LlD l LI Da LlD2 LlDl LI Da LlD2 (A5-8) Llal2 Llala Llala Lla 23 Lla23 Lla12 If the resulting shifts are written for all the symmetry operations in the Cav point group, the following table is obtained: Cav I E ct Cä O'v1 O'v' O'v 3 LlD l LlDl LI Da LlD2 LlDl LlD3 LlD2 LlD2 LlD2 LlDl LI Da LlD3 LlD2 LlDl LlD3 LlD3 LlD2 LlDl LlD 2 LlDl LlD3 (A5-9) Llal2 Llal2 Llala Lla23 Llal3 Lla23 Lla12 Llal 3 Lla13 Lla23 Llal2 Lla12 Llal3 Lla23 Lla23 Lla23 Llal2 Llala Lla23 Llal2 Llal3 Each column can be considered a vector. If we take column ct, the vector rat whose components are given under ct is a transform ofthe vector whose components are under c 3v . Symbolically rat = D(Ct)r, where D(Ct) is a matrix used to transform r into rat' and in more detail this becomes LlD3 001 000 LlDl LlD l 100 0 0 0 LlD2 LlD 1 0 0 0 0 LlD 2 o 3 (A5-1O) Llal3 o 0 0 0 1 0 Lla12 Lla23 o 000 0 1 Llal3 Lla12 000 1 0 0 Lla23 This procedure can be repeated for each of the symmetry operations. A set of 6 x 6 matrices that is a group is obtained, and it is a six-dimensional representation. This set is displayed below in such a way that the element R of the group is given first, its corresponding D(R) second, the sum X(R) of the terms along the diagonal (called trace) third, and the value of X(R) last: 254 Appendix 5 1 0 0 0 0 0 o 1 0 0 0 0 001 000 E~ -D(E) 000 1 0 0 x(E) = 6 (A5-11) 000 0 1 0 00000 1 001 000 1 0 0 0 0 0 o 1 0 0 0 0 C:~ = D(C:) 000 0 1 0 x(C:) = 0 (A5-12) o 0 0 0 0 1 000 1 0 0 o 1 0 000 o 0 1 000 100 0 0 0 C3~ == D(Cä) o 0 0 0 0 1 x(Cä) = 0 (A5-13) 000 1 0 0 000 0 1 0 100 0 0 0 o 0 1 000 o 1 000 0 O'v1 ~ o 0 0 0 1 0 == D(O'v1 ) x(O'vl) = 2 (A5-14) 000 1 0 0 00000 1 o 0 1 000 o 1 0 0 0 0 1 0 0 0 0 0 O'v2 ~ o 0 0 0 0 1 == D(O'v2) x(O'v2) = 2 (A5-15) o 0 0 0 1 0 000 1 Ö 0 o 1 0 0 0 0 1 0 0 0 0 0 o 0 1 000 O'v· ~ 000 1 0 0 = D(O'v·) x(O'v.) = 2 (A5-16) o 0 0 0 0 1 o 0 0 0 1 0 Elementary Mathematics 255 The matrices given in (A5-11) to (A5-l6) are ofa special form when they are partitioned as 2 X 2 matrices. For example, I 001 I 000 100 I 000 010:000 c;t -+ --- -1- - -- (A5-l7) 000 010 ~[~ ~] 000 001 000 100 where 00 1] 0 0 0] 01 0] A = [100 0= [000 C = [001 (A5-l8) 010 000 100 Here the matrix representing C3 + is said to be in the reducible form. If a set of matrices can be presented in this form, it, too, is said to be in the reducible form. Since the set of matrices in (A5-11) to (A5-l6) is called a representation and since its matrices can be presented in reduced form, the representation is reducible. This statement implies that a transforma• tion (called similarity) can be employed on the original set to display the matrices in the reduced form. The representation (set of matrices) so treated is called a reducible representation. If no similarity transformation exists, the representation is said to be irreducible Another criterion is the following. If 2 ~ 1 x(R) 1 > g (A5-l9) R the representation is reducible, and if 2 ~ 1 x(R) 1 = g (A5-20) R the representation is irreducible. The trace of the 6 x6 matrices of Eqs. (A5-11) to (A5-l6) is the sum of the diagonal terms. Thus, the traces for the transformations of the dis- placement coordinates are The set of traces is called the character of the representation. It may be summarized as folIows: 256 Appendix 5 vib (NHa) 6 0 0 2 2 2 (A5-21) L I X(R)2 = 62 + 22 + 22 + 22 = 48 R Since 48 > 6, where g = 6 (the number of elements in a Cav group), the representation is reducible. This reducible representation can be decomposed into a sum of irreducible representations. For the purpose of decomposition the characters of the irreducible representations are required. These charac• ters are conveniently given in tabular form, the character table for C3V , for instance, being written as folIows: Cav E 2Ca 3av Al XAl(E) XAl(Ca) XAl(aV ) A. XAa(E) XA2(Ca) XA.(av) E XE(E) XE(Ca) XE (I1v) Since, from the character table for C3V , 2XA/E) 2XAJCa) 2XA l (O'V) 2 2 2 2XE(E) 2XE(C3 ) 2XE(O'V) 4 -20 add 2XA l (E)+2XE(E) 2XA l (C3 )+2XE(C3 ) 2XA l (O'V) + 2XE(O'v) 6 o 2 we may write PROBLEMS l. Multiply the following matrices: a) [H~] X [1] b) [~] X [D E Al Elementary Mathematics 257 c) [4 56] x [ n d) U] x [456] e) [1 2 3] x [40 -6-7 96 6] 7 5 8 -11 -8 f) [~;:] x [ n g) [1 21] [3 -4] 402 x _~ ~ h) [AD EB C] F x [JLM K] G H I NO i) [A B C] [J K] D E F x t~ Answers a) [AJ+BK+CL] DI+EK+FL GI +HK+IL b) [AD + AE + AK] BD+BE+BK CD+CE+CK c) [29] d) [ 128 1015 12]18 -4 -5 -6 e) [19 4 -12 -4] f) [~~] 258 Appendix 5 g) [~ -1~] h) [Al + BL + CN AK+ BM + CO] Dl + EL + FN DK+ EM + FO Gl + HL + IN GK+ HM + 10 i) [Al + BL + CN AK+ BM + CO] Dl + EL + FN DK+ EM + FO REFERENCES 1. H. Margenau and G. M. Murphy, The Mathematics oJ Physics and Chemistry, D. Van Nostrand Co., Ine., New York (1956). 2. J. S. Ziomek, "Group Theory" in: Progress in InJrared Spectroscopy, Vol. 1 (H. A. Szymanski, ed.) Plenum Press, New York (1962). 3. F. A. Cotton, Chemical Applications oJ Group Theory, Interscienee Publishers, New York (1963). 4. G. Stephenson, Mathematical Methods Jor Science Students, Longmans, London (1962). Appendix 6 THE gELEMENTS It has been mentioned in Chapter 4 that the element~ of the g matrix are. given in terms of the internal coordinates, the bond stretching, r, and the angle deformation, rp. Wilson, Decius, and Cross(l) have suggested a method by which the gelements can be described. Figure A6-1 shows the schematic representations of the g elements. The superscript always in• dicates the number of common atoms. Atoms common to both coordinates are indicated by double circles, and are always put in a horizontalline. The noncommon atoms are indicated along a 45° diagonal. When a single com- Table A6-1. g Elements-General Case g:. "'1 + "'2 g~ "'1 COS 'P g~tp - (lIS"'Z sin 'P g~"m (lU"'l sin 'P cos T X (l14(l1.] -,------"'1 sin 'PZ14 sin 'P815 259 260 Appendix 6 2 9 r r ~ g~q> ~ I g~CP(1) grr :> > g~CP (6) ~ g~'P~ g~CP (~) ~ ~ Fig. A6-1. Schematic representations of gelements for nonlinear molecules. [From Molecular Vibrations by E. B. Wilson, J. C. Decius, and P. C. Cross, McGraw-Hill, New York (1955), as modified in Introduction to Infrared and Raman Spectroscopy by N. B. Colthup, L. H. Daly, and S. E. Wiberley, Academic Press, New York (1975). Used by permission of McGraw-Hill Book Company and Academic Press.] The gEIements 261 Table A6-2. gEIements for ffJ = 109°28' g:. ftl + ft. g;" -tftl g~CP -iV2e'3ft. g~CP mon atom is the terminal atom in a bending vibration, the notation g;'I'(D is used; when it is the central atom in a bending vibration, the notation glcp Table A6-3. gEIements for ffJ = 120° ftl + ft. g;" -tftl g~CP -tfie•3ft• g~'I'm tfiel3ftl cos, g~'I' g3 cpcp ~ 3 2 g2 (Ö)~ CPcP 3 2 4 gl (2) cpcp2 ~ 3 2 4 5 Fig. A6-2. Schematic representations of g elements for linear molecules. where rp = 120°. For results inc1uding torsions, out-of-plane bending, and cyc1ic structures, see Decius. (2) Ferigle and Meister(3) have determined formulas for gelements for linear cases, and these are illustrated in Fig. A6-2. For the use of the formulas in Tables A6-1 and A6-3, it is necessary to define certain terms; f-l and e are the reciprocals of mass and bond distance, respectively. The formulas also involve an angle 1p, which is defined as ß ~a8ß~~ß8Y a~Y 'Paay I 2'3~,. 4 Fig. A6-3. Definitions of notation used in Appendix 6. The gEIements 263 the notation being defined by the top diagram in Fig. A6-3. The torsion angle T is defined by the two bottom diagrams in Fig. A6-3. REFERENCES 1. E. B. Wilson, J. C. Decius, and P. C. Cross, Molecular Vibrations, McGraw-Hill, New York (1955). 2. J. C. Decius, J. Chern. Phys., 16: 1025 (1948). 3. S. M. Ferigle and A. G. Meister, J. Chern. Phys., 19: 982 (1951). Appendix 7 GENERAL METHOD OF OBTAINING MOLECULAR SYMMETRY COORDINATES A general method for obtaining molecular symmetry coordinates has been developed by Nielsen and Berryman. (1) According to them, the prescription for determining symmetry coordinates is given as follows. 1. Choose a linear combination L k of the internal coordinates from a set of equivalent coordinates. 2. Then the sum (A7-1) will be a symmetry co ordinate belonging to the ath row of the irreducible representation Dj and its lj - 1 partners will be given by (A7-2) Here the subscript k labels the number of symmetry coordinates in the jth irreducible representations, Dj(R)a.a. and Dj(R)ßa. are matrix elements from the ath row and ath column and from the ßth row and ath column of the matrix Dj(R) for the jth representation, respectively, R represents an element of the group, and RLk represents the transform of L k by R. The order of the molecular symmetry group is ~. Consider the moleeule XYa (NH3), whose symmetry is given by C3V • Figure 7-1 illustrates the geometrie configuration and the set of internal coor• dinates chosen. According to their prescription, one requires the matrix elements for the irreducible representations of Cav , the chosen set of inter• nal coordinates, the transforms of the internal coordinates to form tbe transforms ofthe linear combinations ofinternal coordinates, the dimensions ofthe irreducible representations, and the order ofthe group. First, the cho• sen set of internal coordinates is Lldl , Lld2, Lld3, Llal2' Llal3' and Lla23. 265 266 Appendix 7 ,------I Fig. A7-1. Second, the following table gives the transforms of these internal coordi• nates: rk Rlrk R.rk R 3rlc R.rlc R 5Yk R 6rlc R ----- c3V E c+3 c-3 aVl av• aV3 --- .dd, .dd, .dd3 .dd• .dd, .dd3 .dd• Yk Rrlc .dd. .dd2 .dd, .dd3 .dd3 .dd. .dd, .dd3 .dd3 .dd• .dd, .dd2 .dd, .dd3 .da12 .dal2 .dal3 .da'3 .dal3 .da'3 .dal • .da'3 .da23 .da12 .dal3 .da'3 .dal • .dal3 .dal3 .dal3 .da'3 .dal • .dal2 .dal3 .da'3 where the legend is R where rk is the kth internal coordinate and Rrk is the transform of rk by R. Third, the se1ection rules state that there are two fundamental modes of mo• tion in the Al and two in the E irreducible representations. This implies that k will take on values 1 and 2 in Al species and 1 and 2 in E species. Therefore for s~" for Al one has St,.l and stocl; and for S~" for E one has (Sfcc, S~) and (S~, S~). Fourth, the matrices for the irreducible represen• tation of C3V are given by the table General Method of Obtaining Molecular Symmetry Coordinates 267 R, R 2 R 3 R. Rs R. j DJ(R,) DJ(R2 ) DJ(R3 ) DJ(R4 ) DJ(Rs) DJ(R.) E C+a c-a aVI aV2 aV3 A, A2 -1 -1 -1 0) (-t tv'3) ( -! tv'3) C 0) (-t -!v'3) (-t_ ;!v'3) E (~ 1 -!v'3 -! !v'3 -! 0 -1 -tv'3 t tv'3 For the Al representation first choose and Then S!i' = tL DA' (R)l1 RLl R Sd' = t[DAI(E)l1ELl + DA'(Ct)CtLl + DA1(Cil)CiL l + DAI(av)aVILl + DAI(av.)av.Ll + DAI(ava)avaLd = t(1 L1dl + 1 L1d3 + 1 L1d2 + 1 L1dl + 1 L1d3 + 1 L1d2 ) = t(L1dl + L1d2 + Lld3 ) Thus S~l = tCL1dl + L1d2 + L1d3) and one then chooses as one of the normalized symmetry coordinates of Al type. There is no part• ner to this co ordinate since the Al representation is nondegenerate and hence one dimensional. In a similar manner one obtains S1' = t L DA' (R)Ral2 = t(L1al2 + L1a13 + L1a 23) R and then the choice is 268 Appendix 7 For the E species one has (Sfl' SiD and (S~, S~). First consider LI = LId, 11 = 2, and h = 6. Then S~ = i[DE(E)nLl + DE(Ct)nLl + DE(C"i)L1 + DE(av)L1 + DE(aV2 )L1 + DE(O'va)L1] = tel Lld1 - t Lld3 - t Lld2 + 1 Lld1 - 1 Lld3 - t Lld2 ) S~ = te2 Lld1 - Lld2 - Lld3 ) Normalized, we have The first and only partner of S~ is Sf2 and is obtained from the following expression: which becomes S~ = i[DE(E)21ES~ + DE(Ct)21CtS[i + DE(C"i)21C"iS~ + DE(aVJ)aV1S~ + DE(aV2)aV2S~ + DE(ava)avaS~] = 30(Lld 2 - Lld3 ) Then S~ is taken to be Hence the normalized and orthogonal pair selected for the first set of de• generate symmetry coordinates is S:;' = (1/0)(2 Lld1 - Lld2 - LIds) S~ = (1/V2)(Lld2 - LIds) For (S~, S~) one takes L 2 = Aa12 and k = 2, and S~ = t L DE(R)nRL2 R becomes General Method of Obtaining Molecular Symmetry Coordinates 269 while E - ~ ,-- DJt(R) RSE S22 - 6 L., 21 21 R becomes Then for (S~, S~) to be symmetry coordinates one chooses the following orthogonal and normalized pair: s~ = (l/V6)(Lla12 + Lla13 - 2 Ll(23) S~ = (l/V2)(Lla12 - il(13) REFERENCE I. J. R. Nielsen and L. H. Berryman, J. ehern. Phys., 17:659 (1949). Appendix 8 CALCULATION OF THERMODYNAMIC FUNCTIONS FROM VIBRATIONAL• ROTATIONAL SPECTRA From the vibrational-rotational spectra of a moleeule, it is possible to determine the thermodynamic functions of the moleeule. Abrief discussion folIows. For a more detailed discussion, see the books by Herzberg(l) or Glasstone. (2) It can be shown that~he total energy in a moleeule i8 the sum of the translational energy and the internal energy, where the internal energy is the sum of the vibrational, rotational, and electronic energies. E tota1 = Etr + Eint (A8-I) Eint = E Yib + Erot + E e1ec (A8-2) therefore, E tota1 = Etr + E Yib + Erot + E e1ec (A8-3) The total energy of a moleeule is found to be related to the partition function Q as folIows: (A8-4) where gi is the total statistical weight (degeneracy), k is the Boltzmann constant, and T the absolute temperature, Therefore (A8-5) and Qelec i8 usually neglected since it is very smalI. Qtotal is related to all the thermodynamic functions as folIows. Enthalpy or heat content, (HO) HO - Eo = RT d In Qtot + R (A8-6) T dT 271 272 Appendix 8 where R the molar gas constant, equal to 1.987 cal/deg . mole, and Eo is the total energy at absolute zero for one mole of an ideal gas. Heat capacity at constant volume (C~) c~ = :2 [d2Qto~:~1/T)2 - ( dQto~:~l/T) fJ (A8-7) Entropy (SO) SO = RT d In Qtot + R In Q - R In N + R dT tot (A8-8) where N is the number of distinguishable particles in agas, and can be assumed to be the number of moleeules in a mole, or Avogadro's number. Free energy (GO) GO - Eo = _ R In Qtot (A8-9) T N Qtot is reiated to the other Q's as in Eq. (A8-5). Since Qrot and Qvib can be related to spectroscopic data, it follows that thermodynamic functions can be obtained from spectroscopic data. For example, for a linear moleeule, _ 8nHkT Q (A8-IO) rot - ah2 where a is the symmetry number and is equal to the number of equivalent orientations of the moleeule which can be obtained by rotation, h is Planck's constant, and I is the moment of inertia. The value of I can be obtained from the equation h (A8-11) I = -'4:-n-::2c-Ll-'v- where Llv is the spacing between absorptions in the rotation al fine structure of the moleeule, and cis the velocity of light (2.9978 x 1010 ern/sec). Thus, the Qrot contribution to aII the thermodynamic functions can be determined from spectroscopic data. For the Qvib contributions, assuming a harmonie osciIIator and no coupling between rotation and vibration, 1 Q - e-hvc/2kT -,-----;,---,-;-=- vib - 1 _ e-hvc/ kT (A8-12) where v is the frequency in cm-1 and Qvib can be determined for each fun- Thermodynamic Functions from Vibrational-Rotational Spectra 273 damental frequency (v) and summed. The Qvib contribution to each ther• modynamic function can then also be determined. The Qtr contribution can be calculated from known formulas and added to the rotational and vibrational contributions. The thermodynamic functions are thus determined as functions of the temperature. REFERENCES 1. G. Herzberg, Molecular Spectra and Molecular Structure, II, Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand, New York (1945). 2. S. Glasstone, Theoretical Chemistry, Van Nostrand, New York (1944). Appendix 9 DIAGRAMS OF NORMAL VIBRATIONS FOR COMMON POINT GROUPS Normal modes of vibration of bent ABI molecule-point group C1W • A 8 C o ~ E 0 E 0 o )' o ~ ( 0 Normal modes of vibration of linear ABC molecule-point group Cootl • 8 A 0 0-+ 111 t ; V2. } o 0 0 + 1I2b o ,. ( " 1I3 Normal modes of vibration of linear A8 or ABA molecule-point group Dooh • 275 276 Appendix 9 Normal modes of vibration of pyramidal ABa molecule-point group Ca.- Normal modes of vibration of planar ABs molecule-point group DSh - 1I2(E) Normal modes of vibration of tetrahedral AB, molecule-point group Ttl - Diagrams of Normal Vibrations for Common Point Groups 277 I I I I I I I BI V------ Normal modes of vibrations of square planar AB. molecule-point group D.ho Normal modes of vibrations of pyramidal AB. molecule-point group C,vo 278 Appendix 9 " , , , ." , 0::____+,'" ___ +,'"______r+'------'' Zl6(E') Normal modes of vibrations of trigonal bipyramidal AB. molecule-point group D3,.- Normal modes of vibrations of octahedral ABs molecule-point group 0,,_ Appendix 10 DERIVATION OF THE CHARACTERS NECESSARY FOR SELECTION RULES In the development of selection rules for isolated molecules (see Chapter 2), it is necessary to derive the character of the dipole moment to be used in determining the infrared activity, the character to represent the polarizability for the determination of Raman activity, and the character to determine the number of fundamentals belonging to each vibration type. The derivation of these characters follows. DERIVATION OF THE CHARACTER OF THE DIPOLE MOMENT XM{R) Consider a co ordinate system (x, y, z) and assume a molecule whose center of gravity is located at p (Fig. AlO-l). If one performs a proper op• eration along the z axis such as a clockwise rotation of angle e(C(;\;), x is displaced from the equilibrium position to x', y is displaced to y', and z remains invariant. The relationships between x and x', y and y', and z and z' are x' = x cos e + y sin e y' = -xsin e + ycos e (AlO-l) z' = z In matrix notation this becomes X' ] [X] [COS e sin e 0] [X] (AlO-2) [:: = C(;\; ~ = -~in e ~os e . ~ ~ The character of the matrix is the sum of the elements along the diagonals (also called the trace or spur) and is given by XM(R) = XM(C(;\;) = (1 + 2 cos e) (AlO-3) 279 280 Appendix 10 z z ;----y x x Fig. AIO-l. The same results are obtained if the rotation operation is performed coun• terclockwise (Ce). Equation (AlO-3) can be written for the general case as follows: (AlO-4) where flR is the number of atoms left unchanged by the operation. If one considers an improper operation such as a rotation-refiection (S), the x, y coordinates are again displaced to x', y' but in this operation the z co ordinate also changes (Fig. AIO-2). The relationships between the equilibrium positions and the dis placements can now be written x' = x cos e + y sin e y' = - x sin e + y cos e (AlO-5) z' =-z z z Cf"xy J-----y ----4.. ",,~-----y /. , ... / ---_ -y I ,/ x/ x x z' Fig. AIO-2. Derivation of the Characters Necessary for Selection Rules 281 Written in matrix notation, the expressions beeome (AI0-6) The eharaeters beeome XM(R) = XM(S) = (-1 + 2 eos (9) (AlO-7) For the general ease the eharaeter is written XM(R) = flR( -1 + 2 eos (9) (AI0-8) Thus the eharaeter for the dipole moment neeessary to ealculate the in• frared aetivity is (AI0-9) where the plus sign is used for proper operations and the minus sign for im• proper operations. DERIVATION OF THE CHARACTER OF THE PO LARIZABI LlTY X,,(R) For most molecules the polarizability a may be different in the x, y, and z direetions. For the general ease the following equations apply: fix = axxEx + ayxEy + axzEz fly = ayxEx + ayyEy + ayzEz (AlO-lO) flz = azxEx + azyEy + azzEz where E represents a veetor of the eleetronie field and fl is the indueed di• pole moment. Sinee the polarizability tensor is symmetrie, a xy = a yx ' a yz = a zy ' and a xz = a zx . Thus, there are six eomponents to the symmetrie tensor of polarizability. If a partieular operation sueh as Ca is performed, the eomponents of polarizability will undergo the following ehanges: 282 Appendix 10 The relationships between the equilibrium polarizabilities and the ehanged polarizabilities are given in matrix notation as follows: CXIXCYIXCZIX] [CX'yCY'yCz,y (AlO-ll) CX,zCy,zCz,z A B where CXIX denote the direetion eosine between the axes Xl and X, ete. (Note Bis the transpose of A.) For a rotation (proper operation) through an an• gle f), the relationship beeomes -sin f) 0] eos f) 0 o 1 (AIO-12) whieh ean be written a XX ayy a zz Ce axy axz ayz_ eos2 f) sin2 f) 0 2 sin f) eos f) 0 0 sin2 f) eos 2 g 0 -2 sin g eos f) 0 0 0 0 0 0 0 -sin f) eos f) sin g eos g 0 2 eos 2f) - 1 0 0 0 0 0 0 eos f) sin f) 0 0 0 0 -sin f) eos f) aXX ayy azz X (AIO-13 axy ayz ayz_ The eharaeter of this representation is (AIO-14) Derivation of the Characters Necessary for Selection Rules 283 Since cos 2(9 = 2 cos 2 (9 - 1 and tecos 2(9 + 1) = cos 2(9 substituting in Eq. (AIO-14), one obtains X",(C e ) = 2 + 2 cos (9 + 2 cos 2(9 (AlO-15) For an improper operation X",(S) = 2 - 2 cos (9 + 2 cos 2(9 (AlO-16) Thus, for proper and improper operations one obtains the character X",(R) = 2 ± 2 cos (9 + 2 cos 2(9 (AlO-17) This is the character to be used for the determination of the Raman activity. DERIVATION OF THE CHARACTER FOR THE DETERMINATION OF NUMBER OF FUNDAMENTALS XE: (R) The character previous1y developed for the dipole moment refers to 3n variables, where n is the number of atoms in the molecule. For characters applying to representations for 3n - 6 coordinates, the characters for the motions of translation and rotation must be subtracted. The n vectors giving the displacements of the nuclei are, by the laws of mechanics, equivalent to the resultant vector acting at the center of gravity of a molecule. The three components of this vector transform for the translation like that of the dipole moment. Therefore (AlO-18) For displacements involving rotation one must consider angular momentum changes for each coordinate. For a proper rotation such as a rotation about the z axis (Ce), the changes are represented by Ix' = Ix cos (9 + Iy sin (9 ly' = -Ix sin (9 + Iy cos (9 (AlO-19) Iz' = lz 284 Appendix 10 For an improper operation S Is" = -Ix cos (9 - Iy sin (9 Iy" = Ix sin (9 - Iy cos (9 (AlO-20) Iz" = Iz Therefore XR(R) = 1 + 2 cos (9 for the Ce operation, and XR(R) = 1 - 2 cos (9 for the S operation, and for 3n - 6 coordinates the character for the number of fundamentals XB(R) be comes (A 10-2 1) For a proper operation (Ce) XB(Ce) = .uR(l + 2 cos (9) - (1 + 2 cos (9) - (1 + 2 cos (9) (AlO-22) XB(Ce) = .uR(l + 2 cos (9) - 2(1 + 2 cos (9) XB(Ce) = (.uR - 2)(1 + 2 cos (9) (AlO-23) For an improper operation (S) XE(S) = .uR[( -1 + 2cos (9) - (-I + 2 cos (9) - (1 - 2 cos (9)] (AlO-24) (AlO-25) REFERENCE J. E. Rosenthai and G. M. Murphy, Rev. Mod. Phys., 8:317 (1936). Appendix 11 UPDATED BIBLIOGRAPHY In the time elapsed since the submission of the manuscript for this edition, many publications relevant to the subject matter have appeared in the open literature. The most important of these are Iisted below. 1. J. D. H. Donnay and G. TUfrell, Chem. Phys. 6:1 (1974). 2. G. TUffelI, Infrared and Raman Spectra of Crystals, Academic Press, London (1972). 3. L. L. Boyle, Spec. Acta, 28A:I347 (1972). 4. L. L. Boyle, Acta Cryst. 28A:I72 (1972). 5. L. L. Boyle, Spec. Acta, 28A:1355 (1972). 6. C. Hsu and M. Orchin, J. Chem. Ed. 51:725 (1974). 7. G. Davidson, Introductory Group Theory for Chemists, Elsevier, London (1971). 8. S. D. Ross, Inorganic Infrared and Raman Spectra, McGraw-Hill, New York (1972). 9. R. L. Carter, J. Chem. Ed. 48:297 (1971). 10. W. L. Jolly, The Synthesis and Characterization of lnorganic Compollnds, Prentice-Hall, Englewood Cliffs, N.J. (1970). 11. D. Steele, The Theory of Vibrational Spectroscopy, W. B. Saunders, Philadelphia (1971). 12. M. St. C. Flett, The Theoretical Basis of Infrared Spectroscopy, in An lntrodllction of Organic Compounds, F. Schein mann (ed.), Vol. I, Pergamon Press, Oxford (1970). 13. K. N. Rao and C. W. Mathews (eds.), Moleclilar Spectroscopy-Modern Research, Academic Press, New Y ork (1972). 14. J. A. Salthouse and M. J. Ware, Point Group Character Tables and Related Data, Cambridge University Press, London (1972). 15. G. W. Chantry, Slibmillimetre Spectroscopy, Academic Press, London (1971). 16. P. M. A. Sherwood, Vibrational Spectroscopy of Solids, Cambridge University Press, London (1972). 17. V. C. Farmer, "Site Group to Factor Group Correlation Tables" and "Symmetry and Crystal Vibrations," in The lnlrared Spectra 01 Minerals, The Mineralogical Society, London (1974), pp. 515-525, 51-67. 18. D. F. Irish and M. H. Brooker, Appl. Spec., 27:395 (1973). 19. W. G. Fateley, Appl. Spec., 27:305 (1973). 20. J. R. Ferraro, Appl. Spec., 29:354, 418 (1975). 21. D. M. Adams, Coord. Chern. Rev., 10: 183 (1973). 22. C. F. Shaw and L. G. Newbury, Can. J. Spec., 20:65 (1975). 22. M. H. Brooker, Appl. Spec., 29: 528 (1975). 285 INDEX ADAMS, D. M., 106, 108, 109,235,285 Calculations (Cant.) ALPERT, N. L., 32 thermodynamic functions, 271 ANTOIN, D. J., 100 CARTER, R. L., 285 Centered lattice, 30 BAGLIN. F. G., 100 CHANTRY, G. W., 285 BARROW, G. M., 32 Character tables, 18 BASILE, L. J., 179 definition, 18 BAUER, S. H., 199 representation, 19 BEGUN, G. H., 186, 187, 194, 199 linear molecules, 20 BELL, J. W., 109 nonlinear molecu1es, 19 BENTLEY, F. F., 83, 109,247 types BERRYMAN, L. H., 265, 269 Cs , Ci and Cn groups, 201 BERTIE, J. E., 109 ~,~,~,~,~,~,~,W~ BHAGAVANTAM, S., 58, 108 203 Bhagavantam and Venkatearayudu Cnv groups, 203 Method, 58,75, 77 C2v , Csv , C4v , C5v , C6v , 201-204 BOYLE, L. L., 285 Cnh groups, 205 BRADLEY, C. A., 132 C2h , C Sh , C4h , C5h , C6h , 204-20 BRAGEN, J., 108 D n groups, 206 BRINKLEY, S. R., 180 D 2 , D s , D 4 , D5 , D6 , 206-208 BRINTZINGER, H., 179 D nh groups, 208 BROADLEY, J. J., 199 D 2h , D Sh , D 4h , D 5h , D6h , 208-211 BROOKER, M. H., 108,225,233,285 D nd groups, 208 BURKE, T. G., 198 D 2d , D Sd , D4d , D 5d , D6d , 212-214 Sn groups, 215 Calculations S4, S6 , SB, 215-216 combinations or difference tones, 38 Cubic groups, 217 Coov ,54 T, Th , Td , 0, 0h, 217-220 Td ,33 Coov and D ooh groups, 221 fundamenta!s Coov ,221 Coov ,53 D ooh ,222 Td ,36 CHERNICK, C. L., 199 infrared modes CLAASSEN, H., 165,179, 180, 198, 199 Coov ,54 CLASE, H. J., 186, 187, 199 Td ,39 CLEVELAND, F. F., 33, 56, 108, 137, 165, overtones 179, 180 Coov ,55 COLTHUP, N. B., 32, 165, 179 Td ,40 Comparison of BV and HH Methods, 75 Raman modes Coordinates Coov ,54 interna!, 135, 151, 156, 157 Td ,39 norma!ization, 136, 139, 151, 157 287 288 Index Coordinates (Cant.) DICKINSON, R. G., 108 orthogonalization, 136, 140, 151, 157 Dipole moment component, 201 redundant, 158 JJ-x, /J-y, /Lz, 201 symmetry, 135, 151, 157 DODD, R. E., 199 calculation, 265-269 DOLLISH, F. R., 83, 109,247 transformation, 141, 152 DONNAY, J. D. H., 285 Correlation method, 108 DUNCAN, A. B. F., 197 Correlation tables, 235 DURIG, J. R., 100, 108 C4 , C6 , D2 , D 3 , D 4 , D 5 , D 6 , 235 C2v , C3v , C4v , C5v , C6v , C2h , C3h , 236 EDGELL, W. F., 197 C4h , C5h , C6h , D 2h , 237 Examples of H-H method, 77 D 3h , D 4h , 238 D4h , D5h , D6h , 239 Factor group, 30 D6h ,240 FANO, L., 198 D6h , D7h , C7v , D 2d , D3d , 241 FARMER, V. C., 285 D4d , D5d , D6d , 242 FATELEY, W. G., 83, 109,235,247,285 D6d , S4, S6 , Sg , T, , Th , 243 FERIGLE, S. M., 56, 108, 179,262,263 0, 0h, 243 FERRARO, J. R., 32, 109, 179,285 Oh,246 FLETCHER, W. H., 199 Ih ,247 FLETT, M. St. C., 285 COSTA, G., 180 FORNERIS, R., 179 COTTON, F. A., 32, 199,258 FREEMAN, S. K., 32 COUTURE, L., 225, 233 CRAWFORD, B. E., 180 GALLASSO, V., 180 CROSS, P. C., 132, 173, 179, 259, 260, GAUNT, J., 179, 197 263 g elements, 259 Crystallographic symbolism, 223 for HARADA, 1., 132 MARGENAU, H., 249, 258 HART, R. R., 130, 133 Matrix, 250 HAVIGHURST, R. J., 103, 108 definition, 250 HEATH, D. F., 131, 133, 165, 179 multiplication, 250 HEDBERG, K., 199 transpose, 250 HENRY, N. F. M., 108,224,233 types Hermann-Mauguin system, 5, 28, 224 F matrix (F), 125, 126, 128, 129, HERZBERG, G., 32, 56, 179, 191,271,273 132, 135, 136, 137, 141, 142, HESTER, R. E., 179 144-146, 149, 150, 154, 156, 162 HIRAISHI, J., 179 fmatrix (1),124,143-147,154,159 HOCHSTRASSER, R. M., 32 o matrix (0), 127, 128, 136, 137, HOFFMAN, C. J., 199 146-149, 155, 156, 164 HOLLAS, J. M., 32 g matrix (g), 127, 146, 147, 154, 155 HORNIG, D. F., 58, 70, 75, 77, 103, 108 U matrix (U), 117, 123, 135-137, Hsu, C., 285 142-148, 153, 154, 160, 161 U' matrix (U'), 124, 137, 143-148, IBERS, J. A., 192, 199 153, 163 Identity, 2 Mc DEVITT, N. T., 83, 109,247 Improper rotation, 36 MEISTER, A. G., 33, 56, 108, 137, 165, IRISH, D. E., 108, 225, 233, 285 179,262,263 Irreducible representations, 22 MENARY, J. W., 108 characters, 22 MILLER, D. J., 179 MITRA, S. S., 58, 87, 89, 108 JAFFE, H. H., 32 Moleeules JAMES, D. W., 108 A3 , 57, 58 JANZ, G. J., 179 At,59 JOLLY, W. L., 285 A,;,17 JONES, E. A., 198 AB, 57, 58 JONES, L. H., 32 AB2 , 9-10, 16, 57, 275 ABA,275 KEISER, W. E., 32 ABC, 57, 275 KENDALL, D. N., 32 AB. , 6-9, 57, 256 KHANNA, R. K., 109, 199 ABC., 57 KIM, H., 180 AB., 16,57,276 KOPELMAN, R., 109 AB5 , 18, 57, 278 LA BONVILLE, P., 109, 179 AB6 , 57, 278 LAu, K. K., 109 AB7 , 57, 158 LA VILLA, R. E., 199 A,;Bs , 11, 17, 18 LEONG, W. H., 108 AIF3 ,107 LIEHR, A. D., 198 (AICl3)2 , 106 LINNETT, J. W., 131, 133, 165, 179 AuC14,57 LIPPINCOTT, E. K., 192, 199 BF3 , 7,16 LONSDALE, K., 108, 224, 233 BCI3 , 13 LORD, R. C., 185, 186, 199 BF.. (solid, melt), 172 LYNCH, M. A., 199 B2 C4,198 B2 Hs,32 MAccoLL, A., 178, 179 BeFl-, 172 MALM, J. G., 198, 199 BrCN,57 MANN, D. E., 198 BrF5 , 57, 197, 198 MATHEws, C. W., 285 CC4, 33, 35, 36, 57, 188, 190, 191 MATHIESON, A., 108 CD., 173 290 Index Moleeules (Cant.) Moleeules (Cant.) CF" 172 NH.(s), 13 CH., 166, 173 NH.NO., 92-100 CHCI.,166 NOCI,57 CHCI = CHCI (trans), 13 NO", 57 CH2 = C = CH2, 13 N 2 0, 32, 57 CH.CI, 167 NI5N 140 16 , NI'NI5016, N15N15016, CH.CHO,13 NI'NI'0'6, 107 CH.SnCI., 197 N20" 107, 194-196 COCl2 ,57 N",57 CO2 , 13, 20, 57 NpF6,17 c.H.,13 Ni(CNW,32 C6R., 13,57 N,S" 192-194 CIF., 31, 57, 198 N,S,H.,192 CI", 5 NSF.,107 Cr20.,30 OsF6,169 CsTeF5,197 P,,57 CuF2,12 PH,I, 99-102 CuO, 107 PCl,;, 18,57, 107 cyc1ohexane, 13 PF5, 31, 198 cyc1opropane, 89-91 PdF6 ,169 DCH.,31 POF.,198 diacetylene, 107 POCI.,57 DOD,173 PtF6,170 D2 CH2 ,31 PtF~- , 170 D20, 143, 173 PtCll-, 13, 16 GeF" 172 PuF6, 171 H2 ,5 PuBr.,76 HCI,6 pyrazine, 32 HOD,173 ReF6,169 H20, 10, 16,23, 57, 135, 138, 139, 143, RhF6 ,168 146-149, 151, 166, 173 RuF6, 168 HgzCI2 , 103, 104 S8,57 HgS (a and ß), 108 SbCli- , 107 Hg (C == CC!)2, 107 SbSI,107 HgCI (C == CCI), 107 SF" 57, 188, 189, 197 H 2 0 2 ,13 SF6 ,13 H 2 S(s), 13 SiF" 172 IF5, 13, 32, 185-187 S02,57 IF7 , 32, 57,182-185 SOC12,57 IrF6,170 TeFs ,197 KMnO, , 96-99 TiF, (gas), 172 KzCrO,,107 TiS,30 K.[Fe(CN)6l,84-86 TcF6,168 LaCI. , 105, 106 UF6 , 135, 156, 171 monochlorodiacetylene, 107 WF6,169 MoF6, 168, 197 XeF" 188--192 NaNO., 58, 67, 68-71, 72-75 ZrS,107 naphthalene, 86-89 ZrSiO,,108 ND., 16,57 MUETTERTIES, E. L., 199 NH., 135, 150-152, 154, 156 MURPHY, G. M., 108,249,258,284 Index 291 MURRAY, M. J., 33, 56, 108 Potential force fields (Cant.) valence force field, 131 NAGARAJAN, G., 108 Primitive lattice, 28-30 NAKAGAWA, 1., 84, 108, 179 Product rule, 167 NAKAMOTO, K., 32, 165, 178, 1~9 Proper rotation, 36 NEWBURY, L. G., 285 NEWNHAM, R. E., 109 RAo, K. N., 285 NEWTON, D. C., 106, 108, 109 REDLICH, 0., 180 NIELSEN, J. R., 265, 269 Reduction formula, 34, 37, 38, 51, 53 Normal coordinate treatment, 135 REIMANN, C. W., 109 molecules ROBERTS, H. L., 199 C2v , 138 ROBERTSON, J. M., 108, 199 Cav , 150 ROSENTHAL, J. E., 108,284 Oh' 156 Ross, S.' D., 285 results, 165-167 Rotation axes, 3 several hexafluorides (Oh symmetry), Rotation-reflection axes, 4 168-171 Rotational components, 201 several tetrafluorides (Td symmetry), R x , R y , R z , 201 172 Rotation (R), 23 NUDELMAN, S., 87, 89, 108 SABATINI, A., 179 ORCHIN, M., 32, 285 SACCONI, L., 179 OVEREND, J., 132, 180 SALTHOUSE, J. A., 285 SARMA, A. C., 197 PALENIK, G. J., 96, 108 SCHACTSCHNElDER, J. H., 132, 178, 180 PARISEAU, J., 180 SCHATZ, P. N., 165, 179 PHILLIPS, W. D., 199 SCHERER, J. R., 132 PlOTROWSKI, E. A., 156, 180 SCHETTINO, V., 179 PISTORIUS, C. W. F. T., 179 SCHUMB, W. C., 199 Planes of symmetry, 4 Schoenflies system, 5, 28, 224 Ud, 4, 11, 16,22 Secular determinant and equation, 117, Uh, 4-7, 12, 16, 17,21,22 148, 149 U v , 4-7, 9-10, 16,22,23 Selection rules, 33 Point group, 12 Aa to AB type molecular, 57 definition, 12 derivation, 33 rules for ciassification, 14-16 linear mo1ecules, 50 selection rules, 33 nonlinear molecules, 33 types, 13 forbidden vibrations, 59-66 Polarizability components, 201 point group, ll'xx, ll'xy, etc., 201 Coov , 50 Potential energy functions D ooh ,55 H2 0,142 Td ,33 NHa , 153 for systems involving translation, 58 MX., 119 SHAW, C. F., 285 Potential force fields, 111 SHERWOOD, P. M. A., 285 central force field, 130 SHIMANOUCHI, T., 132, 179 general quadratic potential function, SHURVELL, H. F., 179 119 SINCLAIR, V. C., 108 generalized valence force field, 131 Site group, 31 local symmetry force field, 116, 131 SLOWINSKI, E. J., 199 orbital valence force field, 131 SMITH, D. W., 199 292 Index SNYDER, R. G., 132, 178, 180 Tables of site symmetries, 225 SONNESSA, A. J., 32 TANNER, K. N., 197 SOU DER, P. A., 180 TOBIN, M. C., 192, 199 Space group selection rules, 58 TOPPING, G., 178 examples, Translati.n (T), 23 cyclopropane, 89 Translation components, 201 Hg"CI2 ,103 Tx , Ty , Tz, 201 KMnO.,96 TURRELL, G., 285 K 3 [Fe(CN)6], 84 LaC13 ,104 Unambiguous choice of site symmetry, NaN03 , 67, 70 75 naphthalene, 87 Update bibliography, 285 Nli.N03 ,91 UREY, H. C., 132 PH.I,99 Species of vibrations, 21 VENKATARAYUDU, T., 58, 108 symbols (linear molecules), 21 VENKATESWARLU, K., 165, 179 symbols (nonlinear molecules), 21 Vibration (Q), 23 STAMMREICH, H., 179 STEELE, D., 285 WAIT, S. C., 179 STEPHENSON, C. V., 198 WALKER, A., 33 STEPHENSON, G., 258 WALKER, M., 108 STRAUGHAN, B. P., 197 WALL, M. C., 179 Sum rule, 173 WALTER, J. L., 108 Summary of Fand G matrices, 174 WARD, A. T., 179 Oh molecules, 174 WARD, C. H., 197 D'h molecule, 174 WARE, M. J., 285 Td molecule, 175 WAUGH, J. S., 199 D 3h(AB3 ) moleeule, 176 WEINSTOCK, B., 198 C3.(AB3) molecule, 176 WELLS, A. F., 27 C2V molecule, 177 WEYL, H., 32 SUNDARAM, S., 165, 179, 180 WHEATLEY, P. J., 24-28 SUSZEK, F., 180 WHITE, J. E., 32 Symmetry,1 WHITE, W. B., 109 center, 4 WIBERLEY, S. E., 32,165,179,260 classes of operations, 22 WILSON, E. B., 132, 135, 137, 148, 173, coordinate, 117, 135-137, 151, 157 179,180,259,260,263 definition, 1 WINSTON, H., 108 effects on normal coordinates of H2 0, WOODWARD, L. A., 179, 199 23 Wu, E., 180 elements in condensed state, 24 WYCKOFF, R. W. C., 75-77, 91, 93, 96, operations, 5, 22 99, 101, 108 planes, 4 science, 1 YERANOS, W. A., 133, 178, 180 space, 24 structural chemistry, 2 ZACHARIASEN, W. H., 32 Symmetry elements, 2 ZELDIN, H., 14, 32 SZYMANSKI, H. A., 32, 258 ZIOMEK, J. S., 32, 156, 180,258