character tables for some of the more common point groups

7/28/2019 Character Tables for Some of the More Common Point Groups

1/11

Appendix VIII: CharacterTables for Som e of the M oreCommon Point Groups

Character tables are given in this appendix for the following point groups: (1)the groups which correspond to the 32 crystal classes (see Table 8-14), (2)groups containing 5-fold axes that may be needed to describe the symmetriesof certain molecules or complex ions, and (3) the infinite groups Goov and&oQ\\ that are appropriate to linear structures.Some comment on the notation for the irreducible representations(symmetry species) is necessary, as certain variations will be found inthe Hterature. Molecular spectroscopists usually designate one-dimensionalsymmetry species by A or B, two-dimensional species by , and three-dimensional species by F or T. Some authors prefer lower-case letters tocapitals. The symmetry or antisymmetry of a given species with respect tothe generating rotation operation distinguishes A and B. Subscripts 1 and 2are used on A and B according to the symmetry with respect to rotationabout a C2 axis perpendicular to the generating axis. Symmetry with respectto inversion is indicated by a subscript g or u (see Section 4.4.2), while aprime or double prime identifies species that are, respectively, symmetric orantisymmetric with respect to a horizontal plane. Finally numerical subscriptsare used to distinguish various doubly and triply degenerate species.

For linear molecules or ions the symbols are usually those derived fromthe term symbols for the electronic states of diatomic and other linearmolecules. A capital Greek letter H, n , A , O, . . . is used, corresponding toA. = 0, 1, 2, 3 , . . . , where X is the quantum number for rotation about themolecular axis. For E species a superscript + or is added to indicate thesymmetry with respect to a plane that contains the molecular axis.The components of the translation and rotation vectors are given as T^.Ty, T^and Rx,Ry,Rz, respectively. The components of the polarizabiHty tensorappear as Unear combinations such as a^x + ccyy, etc., that have the symmetryof the indicated irreducible representation.


2/11

374 MATH EMAT ICS FOR CHEMISTRY AN D PHYSICSTable 1 Character tables for the cycHc groups, Gn{n = 2, 3, 4, 5, 6).

^ 2AB

^ 3AE

E11

1

C21- 1

E Cs1 11 s1 *

*^3

hOtyz^Otzx

(T; , r,), (/?,, /?,)

Otxy

s = exp(27Ti/3)01XX -\-otyy,a^^

{a^^ -ayy.a^y), (of ,, Otzx)

^AABE

E11I !

C41- 1

/

C211

- 1- 1

1- 1 Tz, Rz(T,,Ty),iRx,Ry)

OixX -\-CXyy,a,,a^^ -ayy,oi,,

((Xyz.Oi,^)

(?5AEl

E2

E

i i11Cs1e

^ 2

"-51

g 2 *

c|1

2 *

h 2 * 1 ^ 1

(T,, T,), (/?,, /?,)

= exp(2ni/5)

fc;c -ayy,axy)

(?6ABE,

E2

i !I i

C61- 1e

e

c.11

ee

Ci

- 1

^311 e

-8 ^ 6

11*1- * |

Tz,Rz(T Ty), (Rx, R,.)

= exp(27ii/6)01XX +ayy,azz

{Uyz -Olzx)

\Ol-XX Oiyy^ Oixy)


3/11

VIII. CHARACTER TABLES 375Table 2 Character tables for the dihedral groups, ^(n = 2, 3, 4, 5, 6).

&2 = VAB iB2B3

E1111

Ciiz)11- 1- 1

C2(y)1- 11- 1

Ciix)1- 1- 11

T.,R.rv,/?vTx, Rx

o(xx,ayy,a^.01XVOtzxOiyz

^ 3

A iA2E

E

112

2C 3

11- 1

3q1- 10 {Oixx -ayy,axy), {ay^â^x)

AA iA2B iB2E

E11112

2C411- 1

- 10

C21111- 2

2q1- 11

- 10

1C {1- 1- 110

7;,/?z

{Jx.Ty).{Rx.Ry)

aXX -\-GLyy.OL^z

CLXX - OiyyOlxx{ay.ÔL^x)

&5

A iA2E lE2

E1122

2C 5112 cos a2 co s 2oi

1C\112 cos lot2 co s oi

5C 21- 100

T.,R.(Tx,Tyi(Rx.Ry)

a = 72 oixx -\-ayy,a,,

(aya,x)(aXX -ayy.otxy)

^ 6

A iA2B,B2E lE2

E111122

2C 6

1 1- 1

2C 3

- 1 1

C211- 1- 1- 22

3C^1- 11- 100

3 ^ '1- 1- 1100

Tz, Rz

{Tx,T,)ARx,Ry)

otxx Ôtyy,azz

(ay^â^x)(aXX -ayy,axy)


4/11

376 MATH EMA TICS FOR CHEMISTRY AN D PHYSICSTable 3 Character tables for the groups 0nh(n = 2, 3, 4, 5, 6).

^ 2 / A gB i gB 2gB 3gA uB i B 2uB s u

= n E C2(Z)1 - 11 - 1

1 - 11 - 1

Ciiy)1

- 11

- 11

- 11

- 1

C2ix) i

- 1- 1

1 - 1 -- 1

1

cr(xy)

- 1- 1 1- 1

oixz)

- 1

- 1 1- 1


5/11

VIII. CHARACTER TABLES 377

'^6hA igA2gBigB2gEigE2gAiuA2uB2uEiuE2u

E11112211122

2C 6

- 1- 1- 1

- 1- 1

2C 3

- 1- 1

- 1- 1

Ci11

- 1- 1- 2

211- 1- 1- 22

3C ^1

- 11

- 100

- 11- 10

0

Table 33C^' / 253

1 1 1- 1 1 1- 1 1 - 1

1 1 - 10 2 10 2 - 11 1 1- 1 -1 -1- 1 - 1 11 - 1 10 -2 -10 - 2 1

(Continued).2 5 6 ^h ^^d âi;

1 1 1 11 1 - 1 - 11 - 1 1 - 11 - 1 - 1 1

- 1 - 2 0 0- 1 2 0 0 1 1 1 1- 1 - 1 1 1- 1 1 - 1 1- 1 1 1 - 11 2 0 0

1 - 2 0 0

Rz

(Rx,Ry)

Tz

(Tx^Ty)

Olxx -^0(yy,azz

(0(yz,0(zx)(axx -ayy,axy)

Table 4 Character tables for the groups tfn(n = 2,4, 6, 8).J 2 ^ (? i

AgAu

E11

/1

- 1Rx, Ry, RzTx.Ty^Tz

^x x 1 ^y y 1 ^zz' ^xyOiyz.Olzx

J^ 4ABE

E11i ;

^41- 1/ /

C 211- 1- 1

S^0 41- 1 RzTz(Tx,Ty),{Rx,Ry)

OtxX +Ctyy,OlzZotxx -ayy,axy

(ayz,azx)

-AAgEgAE

E

1111C 311s8^

C] I1 ] * ]s ]1 - ]8^ - ]8 -]

^6 ^^[ 1 1[ 8 *1[ e* J[ -1 -1

8 *1e* 8 j

Rz(Rx,Ry)Tz{Tx,Ty)

8 = exp(2ni/3)a XX +Olyy,ClzZ

(axx -ayy,axy), (ayz, 01 zx)

J'8ABE,E2E3

E 58

1 ;I II I

[ 1[ - 18

/[ 8

C 4

i1

I

sl C21 1- 1 1

-8 ^ - 18 1 / 1/ 1

8 1 * - 1

^ 81- 1

8I i8^8

C^^ 411 /- 1- 1 /

^^81- 1 * 1e 1- ; i4 * 1

8 =RzTz

(Tx.Ty)

(Rx.Ry)

exp(2jTi/S)otxx ôiyyâzz

(axx -ayy,axy)(ayz,azx)


6/11

378 MATHEMATICS FOR CHEMISTRY AND PHYSICSTable 5 Character tables for the groups 6'nh(n = 1, 2, 3,4 , 5 , 6).

^M/. ^ 6^.A '

E11

CTh1

Tz,Rx,Ryo(xx,oiyy,azz,oexy

^hhA gB gA uBu

E1111

Ci1

- 11 -

- 1 -

(^h1 11 - 11 - 11 1

RzRx,Ry

TzTx,Ty

axx.o(yY,azz,oexv0 f y z ' z x

^^'3hA'E '

A "E^'

E C31 1

[1 ^*1 11^ ^

r231

1 *

c /z '^31 11 e1 . *

- 1 - 1 1 - 1 - . *

5 ^^ 31:*l- 1:*l

Rz(Tx,Ty)

Tz(Rx,Ry)

= expilni/S)oixx +ayy,azz

{aXX -ayy,axy)

{ayz,azx)

6\Ah E C4 C2 Cl S. Oh 54AgBa

AuBu

{Rx.Ry)Tz

{Tx,Ty)

Oixx -\-ayy,azzOixx ~ 01 yy , CKxy

(oeyz,o(zx)

^hhA'

^ 2A^^^ 1E'f^ 2

E Cs1 1

{1 Uj 1 sî 1 2*

1 11;:.1 1 .21 1 . 2 *

C 2^ 51. 2 *. *e1. 2^ 2 *. *6:

*-51. 2 *^ 2 *1^ 2 *. 28*

Cs Or/j 5518^8. 2 *. 218^ -8 ^ 2 * -. 2 -

1

8^. 2^ 2 *

- 1-8^- . 2- ^ 2 *

"^51. 2. 2 *

- 1- . 2- . 2 *- . *f

^51^ 2 *. 28^

- 1- . 2 *- ^ 2 8- 8 *

^51;*l^ 2 * 1. 2 1

- 1:f 1- ^ 2 * 1- 1

Rz(Tz,Ty)

Tz(Rx,Ry)

8 = exp{2iTi/5)oixx -ôiyyâzz

{aXX -ayy,axy)

(ayz +azx)


7/11

VIII. CHARACTER TABLES 379Table 5 (Continued).

^'%h

EigE2gAuBuEluE2u

E C61 11 - 1

{ 1 UI I : ; *1 11 1

11 Uf 1 - *i l

ea11 ss-8 ^

11

-es- *

Ci

- 1

- 1

C2^ 311

11

- *

c | '1 1- 1 1 * 1e 1

- 1- e * 1

1 - 1- 1 - 1

* - 1e - 1

- - 1- e * - 1

5^^ 31- 1 *

- 1

1 *e

^6 or;, ^6 5311 -

- * - - *- 1 -- 1 *

* -

1 11 - 1 * \ e * e J * 1- e - e * j- 1 - 1- 1 1

f* \*e Je * e 1e * J

^z(/?x,/?.v)

T'z

iTx,Ty)

e = exp{27ti/6)otxx +>>>', "zz

(>'Z.ZJC)

( x x - a>7 , x > ' )

Table 6 Character tables for the groups 6\^{n = 2, 3, 4, 5, 6).^ '21;A iA2B iB2

1111

Ci11

- 1- 1

Ovixz)1

- 11

- 1

ov{yz)1

- 1- 1

1

TzRz

Tx,RyTy,Rx

oixxôiyyôizz0(xyotzxoiyz

E 2C3 3cryA iA2E

1 11 12 - 11

- 10TzRziTx,Ty),{Rx,Rv)

01XX -^ciyy^cizzidXX - oiyyâxy), {ayz,azx)

G\ÂiA2B iB2E

E11112

2C 411

- 1- 10

CiI111- 2

lo-o1

- 11

- 10

2CTJ1

- 1- 1

10

TxRz

{Tjc,Ty)ARx,Ry)

otxx +o(yy,azzotxx - OCyy

Olxy(ayz'Oizx)

(^'5vA iA2E lE2

E1122

2C 511

2 cos a2 cos la

ic l11

2 cos 2a2 cos a

5cry1

- 100

Tx{Tx,Ty)ARx,Ry)

a = 12o(xx âyy,azz

( > Z ' Z x )(o(xx -ayy,axy)

^''6vA iA2B iB2E lE2

E111122

2C 6

- 1- 1

2C 3

- 1- 1

C2111

- 1- 22

3o v1- 1

- 100

3or^1- 1

00 (7:v

TzRz

Ty), (R x Ry )

oixx +ayy,axy

(ayz,azx){axx -ayy,axy)


8/11

380 MATH EMA TICS FOR CHEMISTRY AN D PHYSICSTable 7 Character tables for the groups, 0nd(n = 2, 3, 4, 5, 6).

CJ)2d = 'I'dA iA 2B iB 2E

E 2 5 41 11 11 - 11 - 12 0

Ci IC'^1 11 - 11 11 - 1

- 2 0

lO d1

- 1- 1

10

TziT^,Ty),iRx,Ry)

(Xxx +(Xyy,azz

Otxy{ayz,azx)

^^3dA i gA 2gE gA i uA 2uEu

E112112

2 C 3

- 1

- 1

3 C 21

- 101

- 10

/112

- 1- 1- 2

2 5 6

- 1- 1- 1

3 a ^1

- 10

- 110

Rz(Rx,Ry)

Tz(Tx,Ty)

otxx +ayy,azz

{Olxx - Clyy.OL xy). (0(yz,0(zx)

^ 4 JA iA 2B lB 2E lE 2E 3

E1111222

2 5 811

- 1- 1V20

-V 2

2 C 411110- 20

2 5 |11

- 1- 1

-V 20V2

C 21111

- 22- 2

4 C ^1

- 11

- 1000

4or j1

- 1- 1

1000

RzTz

(Tx,Ty)(Rx,Ry)

otxx -\-Otyy,CtzZ

ioixx -Olyy,Olxy){ayz,otzx)

'^5dA l gA 2gE l gE 2gA i uA 2uE i uE2u

E11221122

2 C 511

2 C O 5 Q ;2 cos Jjoi

11

2 cos a2 co s 2oL

IC ]11

2 C05 2Qf2 COS a

12 C05 2Qf2 COS a

5 C 21

- 1001

- 100

/1122

- 1- 1- 2- 2

2^?011

2 cos a2 cos 2a

- 1- 1

2 cos a2 cos 2 a

2 5 i o11

2 C05 2Qr2 cos a

- 1- 1

2 cos 2af2 co j : a

5or^1

- 100

- 100

Rz(Rx,Ry)

Tz(Tx,Ty)

a = 12oixx -\-ayy,azz

{ctyz,oizx)\OtxX ~ Otyy , Olxy )

^ 6 JA iA 2B lB 2E lE 2E 3E 4E 5

E11122222

2 5 i 211

- 1V310

- 1-V 3

2 C 6

- 1- 2- 1

2 5 4111

- 10

- 2020

2 C 3

- 1- 12- 1- 1

2 5 ^ 211J

- 1-V 310

- 1V3

C 2111

- 22- 22

- 2

6 C ^1

- 1J

00000

6 a j1

- 11100000

RzTz

(Tx,Ty)

(Rx,Ry)

otxx ^ayy.otzz

{axx -ocyy,axy)

(otyz^otzx)


9/11

VIII. CHARACTER TABLES 381Table 8 Character tables for the cubic groups.

srAE

T = F

E1

i ;3

4C 31e^ *0

4 C j1e0

3C21I I

- 1 T , R

e = exp{2ni/?>)

(aj,y,ayz,azx)- Qf^.Ô

.5^/,AgEg

T g ^ F gAuEu

T u ^ F u

E1

i :31

i ;3

4C 31e *01

0

4Cf1e *e01 *e0

3C2

- 1

- 1

i1113

- 1- 1- 1- 3

4561 *0

- 1-

0

4^6^1e *0

- 1e

0

3a

I I- 1 1:|l

R

T

e = exp(27ri/3)otxx + >->. + a^ z

{aXX -ôiyy -lotzz,otxx - 01 yy){0lxy,0lyz,0lzx)

^ d

AiA2E

T i = F iT 2 ^ F 2

E11233

8C311

- 100

3C2112

- 1- 1

6^41

- 101

- 1

6or^1

- 10

- 11

RT

OixX + Ctyy + Uzzf c j c + a > 7 -lOLzz.Olxx -

iolxy.OLyzÔLzx)-Ctyy)

Oh

AigA2gEg

Tig = F igT2g = F2g

AiuA2uEu

T i u = F i uT2u = F2u

E

1123311233

8C3

11- 1

0011

- 100

3C2

112

- 1- 1

112

- 1- 1

6C 4

1- 101

- 11

- 101

- 1

6q1- 10

- 111

- 10

- 11

/11233

- 1- 1- 2- 3- 3

856

11- 1

00

- 1- 1

100

3a/ ,

112

- 1- 1- 1- 1- 2

11

654

1- 101

- 1- 1

10

- 11

6a^

1- 10

- 11

- 1101

- 1

R

T

otxx + OLyy + azz

{OLxX + Olyy - lotzz->01XX - Olyy){axy,0(yz,azx)


10/11

382 MATHEMATICS FOR CHEMISTRY AND PHYSICS

O H2

3

(0 o

ttq

^

++

1I

1+II I

< H

1+^1

IIIH

+

H

o -^ -^

(N -H o

(N -^ O

O X

b n'"t oo(N

- ^ Om CO( N

oCOr s i

>->o

mu< Nc s ^( N

u(NC*J

S""+=S ^+

_

-

---

,

^

T- H

--

(50 ^ H

'ICS 'ICN

1 ^ L ^

1 + ItN ' l (N

CO CO

1 1o o

1 ^ l ^1 +^ H ^ H'ICN 'l(N

1 ^ 1 ^+ 1

Irvi 'IfN

CO CO

DO 60tiT t CIII IIIDO 00 (NH H

o_

11

71

T f

o

^

-1

r^1

^

00a

^( N1>.+JJ

^1O

O

o

^1

o

o

m

00ffi

S Kb ^' JHS

1,_1

'-p1

1

7,

^

^

^

-

3r)+

-H | (N

CO

3tCIII3

H

1^-H | (NCO

3III(NH

o

1

^

-1

o

,_!

1

1

r t -

3o

1^

o

o

i n1,__ i

,__1

o

o

l O

3ff i


11/11

VIM. CHARACTER TABLES 383Table 10 Character tables for the infinite groups of hnear structures.

^ôouAi = i :+El = nE2 = AE3 = O

E11222

112 co^ 0

2 C05 2 0 2 cos 3 0

OOOTi;1

- 1000

RziT^,Ty)ARx,Ry)oizz -\-oiyy,azz

{ayz,azx)(axx -ayy,axy)

^ O O / I

^ g ^^ g "ngAg^^ûOuAu

E11221122

2r'^11

2 C05 02 cos 2 0

112 coj' 0

2 C05 2 0

OOO-y

1 - 1

001

- 100

i1122

- 1- 1- 2- 2

I1

2cos2 cos 2 0

- 1- 1

2 C05 02 COS 2(f)

00C21

- 100

- 1100

Rz(Rz^Ry)

Tz(Tx,Ty)

(Xxx +oeyy,azz(ciyzÔlzx)

(0(XX -0(yy,axy)

character tables for some of the more common point groups

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