E90 4 CDO CDO - Springer978-3-642-80021-4/1.pdf · B. Character Tables for Point Groups 367 Table...

Appendices

A. The Thirty-Two Crystallographic Point Groups

Table A.I. Stereograms of the 32 crystallographic point groups

Triclinic (Monoclinic) (1st setting) Tetragonal

00 CDO E90 1 2 4

00 E90 m(=2) 4

CDO CDO E8G I 21m 41m

Monoclinic (2nd setting) Orthorhombic

ffi@ EB8~ CDw x, 422

x, 2 222

(])CD~ CDEB~ E9@ x,

4mm x,

m 2mm

ffi@ 42m

x,

CDEB~ CDEB ffi@ 'X, ® ® X2

® ® X.

® ® ®®

21m mmm x, 4/mmm x,

A. The Thirty-Two Crystallographic Point Groups 361

Table A.I. (continued)

Trigonal Hexagonal Cubic

® GX3 00·· 0)xa ••• 00.' • • ° 0 :-,. ~ X21

• • • 0 •

3 6 23

eo.o'@®®GX' •.....• -3~' ° A ® ... .. /x2

. ° ® ® • •

Xl

3 6/-1n m3

®®~ ·0 .0

.' ~ 0 .

• o~ . 0 o • .

X2 . - 0

Xl Xl

32 622 432

oo·®®·@ . . ::. • • ~3

•• ••• . . x;/ X2.. • • • Xl X"

3m 6mm

®00@ &L.-~ /x

/ X2 o 0 Xl

6m2

1*---*--*100 0 ~):) ... :.::: . ' o : •••• -. l$. X2

• • X2 •• ••

X,

6/mmm m3m X,

362 Appendices

Table A.2. Schonflies and international symbols for the 32 crystallographic point groups

Crystal system

Cubic

Tetragonal

Orthorhombic

Hexagonal

Trigonal

Monoclinic

Triclinic

Schonflies symbol

T

D3 C3v

C3;(S6) C3

C2h

International symbol (abbreviated)

4-2 3-(m3m)

m m 432 43m 2 -- 3 (m3) m

23 422 ---(4/mmm) mmm

422 42m 4mm 4 - (4/m) m

4 4 222 ---(mmm) mmm

222 2mm 622 ---(6/mmm) mmm

622

6m2 6mm 6 -(6/m) m

6 6 - 2 -3-(3m)

m

32 3m :3 3 2 -(2/m) m

m 2 I 1

B. Character Tables for Point Groups 363

B. Character Tables for Point Groups

Characters and basis functions are tabulated below for the irreducible representations of point groups. For a proper rotation group and its direct-product group with Ci , only one of the two groups is considered. Both Mulliken and Bethe notations are written in parallel to denote the irreducible representations. For an explanation of the nomenclature, see Sects. 8.4 and 8.5. The representations below the broken line are double-valued. When barred and unbarred operations have different characters, they appear simultaneously in the same place. Spin functions a and f3 used in the basis functions are quantized along the z-axis. See Sect. 8.6 for transformation properties of the spin functions.

Complex conjugate representations are degenerate by virtue of time-reversal symmetry. The Mulliken notation regards such a pair of representations as a single (physically irreducible) representation. The other single-valued representations are all real and do not have additional degeneracy. Most doublevalued representations are pseudoreal, having no additional degeneracy. Real representations, which appear twice because of time-reversal symmetry, are noted in the tables. See Sects. 4.12, 13.1 and 13.2 for degeneracy due to timereversal symmetry.

For the symmorphic space groups O~(sc), O~(fcc) and O~(bcc), the labels of small representations [B.l] are given in the rightmost columns of the corresponding point groups. For convenience of reference, we summarize in Table B.l the point groups of k for high-symmetry points in the Brillouin zone.

Table B.l. Point group of k for high-symmetry points in the BZ

Ob Td 04b Old C4v °2b C2v °3d C3v Fig. 11.7

O~ rR MX ~T LZS A (a)

O~ r X w ~ LZS L A (b)

O~ rH p ~ N LOG AF (c)

B.1

Cub

ic P

oint

Gro

ups

Tab

le B

.2.

Oh

Oh

E

6C4

3Ci

6C;

8C3

I 6I

C4

30"h

A,.

r+ ,

1 1

1 1

1 1

1 1

A2•

r+

2 1

-1

1 -1

1

1 -1

1

Eg

r;

2 0

2 0

-1

2 0

2 T,

g r:

3

1 -1

-1

0

3 1

-1

T 2g

r;

3 -1

-1

1

0 3

-1

-1

A,u

r- ,

1 1

1 1

1 -1

-1

-1

A,

u r;

1

-1

1 -1

1

-1

1 -1

Eu

r;

2

0 2

0 -1

-2

0

-2

T,u

ri

3 1

-1

-1

0 -3

-1

1

T 2u

rs

3 -1

-1

1

0 -3

1

1 -_

.. _--

---

------

-----------

----

---

------

-------

----

----

---

----

E,/2

. r:

2

-2

)2

-)2

0

0 1

-1

2 -2

)2

-)

2

0

E 5/ 2•

r+

2

-2

-)2

)2

0

0 1

-1

2 -2

-)

2

)2

0 7

G3/2

g r:

4

-4

0 0

0 0

-1

1 4

-4

0 0

0

E,/2

u r-

2 -2

j2

. -)

2

0 0

1 -1

-2

2

-)2

)2

0

6

E 5/ 2u

r:;

-2

-2

-)2

)2

0

0 1

-1

-2

2 )2

-)

2

0

G3/ 2

u r;

4

-4

0 0

0 0

-1

1 -4

4

0 0

0

A,g:

r2,

X4

+ y4

+ Z4

-!r

4 A

,u: (A

lg) x

xyz

A

lu: x

yz

A2g

: X4 (

yl -

Zl) +

y4(Z

l -

X2) +

Z4(

X2

-y2

) E

u: {

xyzv

, -x

yzu

}

E.:

{u,

v},

u ==

2Zl

-X

l -

yl,

V ==

j3

(x2

-yl

) T,

u: {

X, y

, z}

T'

g: {

yZ(y

2 _

Zl),

ZX(Z

2 -

Xl)

, xy

(x2

-y2

)}

Tlu

: {X

(y2

_ zl

), Y

(Z2

_ X

l),

z(xl

_ y

2)}

Tl.

: {y

z, z

x, x

y}

r:: {

a, P

} n:

{xya

+ (y

z +

izx)

P,

-xyp

+ (y

z -

izx)

a}

r::

{vP,

-u

a,

up,

-va

}, {

(zx

+ iy

z)a

+ 2

ixyp

, -j

3(z

x +

iyz)

P, -j

3(z

x -

iyz)

a,

(zx

-iy

z)P

+ 2

ixya

} ri

: {z

a +

(X +

iy)P

, -z

P +

(X -

iy)a

}

60"d

8I

C3

1 1

-1

1 0

-1

-1

0 1

0 -1

-1

1

-1

0 1

1 0

-1

0 ---

-------

0 1

-1

0 1

-1

0 -1

1

0 -1

1

0 -1

1

0 1

-1

r;:

{xyz

a, x

yzP

} r;

: { -

j3(x

+ iy

)a,

2za

-(X

+ iy

)P,

2zP

+ (X

-iy

)a,

j3(x

-iy

)P}

Oh

=

0 X

C

i

r,R

,H

r,

r 2

r'2

r"5

r;s

r',

r;

r'l2

r'5

r 25

------

r+

6 r:;

r:

ri

r;

r;

Not

e th

at t

he a

bove

bas

is f

unct

ions

for

the

dou

ble-

valu

ed r

epre

sent

atio

ns a

re n

ot e

igen

func

tions

of

the

tota

l an

gula

r m

omen

tum

j = I +

s,

and

do n

ot d

iago

nali

ze t

he s

pin-

orbi

t in

tera

ctio

n. B

asis

fun

ctio

ns e

xpre

ssed

in

term

s of

the

eige

nfun

ctio

ns o

f j a

re g

iven

in

[B.2

].

..., 2:: I


Table B.3. T d

Td E 6IC4 3C2 6ud 8C3 Compatibility P with Oh

Al r l 1 1 1 1 1 AIg, A2u PI A2 r 2 1 -1 1 -1 1 A2g, Alu P2 E r3 2 0 2 0 -1 Eg, Eu P3 TI r 4 3 1 -1 -1 0 TIg, T2u P5 T2 r5 3 -1 -1 1 0 T2g, Tlu P4

-------- -------- ----------- ---- --- -------- ----------- ---

EI/2 r6 2 -2 ~ -~ 0 0 1 -1 r:,ri P6 E5/2 r7 2 -2 -J2 J2 0 0 1 -1 q,r.;- P7 G3/2 rs 4 -4 0 0 0 0 -1 1 r;,ri! Ps

Basis functions for T d may be readily obtained from those for Oh by noting that xyz is invariant in Td •

Table B.4. T

T E 3C2 4C3 4C/

A r l 1 1 1 1

r2 1 1 w w2

E r3 1 1 w2 W

T r 4 3 -1 0 0

------------ ----------- ----- ----------- -----------

EI/2 r5 2 -2 0 1 -1 -1 1

r6 2 -2 0 w -w _w2 w2

G3/2 r7 2 -2 0 w2 _w2 -w W

w = exp( -2ni/3), Th = T X Ci

B.2

Tet

rago

nal

Poi

nt G

roup

s

Tab

le B

.S.

D4

h

D4

h

A,.

r:

A

2•

r+ 2

B,.

r;

B

2•

r:

E.

r;

A,u

r!

A

2u

r;

B,u

r]

B

2u

ri

Eu

r- S

-----------

E'/2

. r:

E

3/ 2•

r:j

E'/2

U

ri

E3/ 2

u ri

A,.

: Z

2

A2.

: xy

(x2

_ y2

)

B,.

: x2

_ y2

B2.:

xy

E.:

{ -zx,

zy}

r::{I

X,p}

r:

j: {

XYIX

, xy

P}

D4

h =

D

4 X

C

i

E

2C4

C~

2C;

1 1

1 1

1 1

1 -1

1

-1

1 1

1 -1

1

-1

2 0

-2

0 1

1 1

1 1

1 1

-1

1 -1

1

1 1

-1

1 -1

2

0 -2

0

-----

------------

------1

----

2 -2

}i-

}i

2 -2

-J

i Ji

2

-2

Ji-

Ji

2 -2

-j

2

j2

A,u

: xy

z(x2

_ y

2)

A2u

: z

B,u

: xy

z B

2u: Z

(X2

_ y2

)

Eu:

{x,

y}

0 0 0 0

ri: {

ZIX, -

zP

}, {

(x +

iy)P

, (x

-iy

)lX}

ri: {

(x -

iy)P

, -

(x +

iy)lX

}

0 0 0 0

2C;

I 2

IC4

1 1

1 -1

1

1 -1

1

-1

1 1

-1

0 2

0 1

-1

-1

-1

-1

-1

-1

-1

1 1

-1

1 0

-2

0

------

----

----

---

----

----

-0

2 -2

}i-

}i

0 2

-2

-Ji

Ji

0 -2

2

-Ji

Ji

0 -2

2

j2-j

2

O"h

20

"v

1 1

1 -1

1

1 1

-1

-2

0 -1

-1

-1

1

-1

-1

-1

1 2

0

------ 1

----

--

0 0

0 0

0 0

0 0

20"d

1 -1

-1

1 0 -1

1 1 -1

0

------

0 0 0 0

M,X

X,

X4

X2

X3

Xs

X;

X~

X' 2

X~

X' 5

----

X:

X;

Xi

Xi

w ~ f [


Table B.6. D 2d

D 2d Basis E 2IC4 C2 2C~ 2ud W

Al r 1 Z2, xyz 1 1 1 1 1 WI A2 r 2 Z(X2 _ y2) 1 1 1 -1 -1 W2 Bl r3 x2 _ y2 1 -1 1 1 -1 W~ B2 r 4 z, xy 1 -1 1 -1 1 W~ E rs {x, y} 2 0 -2 0 0 W3 ----------- ---------- -------- ---------- ---- ---- ---- ------El/2 r6 {(X, p} 2 -2 Ji -Ji 0 0 0 W6 E3/2 r7 {z1X, zP} 2 -2 -J2 J2 0 0 0 W 7

When considering the group ofW, note that the C~ axis of the D 2d group is in the [110] direction in Fig. 11. 7b. Therefore, one has to read xy -+ x2 - y2 and x2 - y2 -+ xy in the aoove table.

Table B.7. C4v

C4v E 2C4 Ci 2uv 2ud Compatibility with T,~

D4h

Al r 1 1 1 1 1 1 A18, A2u ~1 A2 r 2 1 1 1 -1 -1 A28, A1u ~'1 Bl r3 1 -1 1 1 -1 B18, B2u ~2 B2 r 4 1 -1 1 -1 1 B28, B1u ~~ E rs 2 0 -2 0 0 E., Eu ~s ---------- ------ ---------- ---- --- ---- ------------ ----

El/2 r6 2 -2 Ji -h 0 0 0 r:,r6" ~6 E3/2 r7 2 -2 -J2 ;,n 0 0 0 r;,r7" ~7

C4 E C4 ci cl Basis

84 E IC4 C2 Iq Basis

A r 1 1 1 1 1 z xyz

B r 2 1 -1 1 -1 xy z

{r3 1 -i -1 i x + iy x - iy

E r 4 1 i -1 -i x - iy x + iy

----------- ------ -------- ----------------- -------- --------

rs 1 -1 p -p -i i -p* p* (X (X

E1/2 r6 1 -1 p* -p* i -i P P -p p

r7 1 -1 -p p -i i p* -p* (x - iy)P (x + iy)P

E3/2 rs 1 -1 -p* p* i -i p (x + iy)(X (x-iy)(X -p

p = exp( - ni/4), C4h = C4 X C j

368 Appendices

B.3 Orthorhombic Point Groups

Table 8.9. Dn

Dn Basis E Cl• Cly Cl" I CT. CTy CT" N

A. rt xl, yl, Zl 1 1 1 1 1 1 1 1 Nl Bll r; xy 1 1 -1 -1 1 1 -1 -1 Nl Bli rj xz 1 -1 1 -1 1 -1 1 -1 N4 B31 r; yz 1 -1 -1 1 1 ...;.1 -1 1 N3 A r 1 xyz 1 1 1 1 -1 -1 -1 -1 N2 u B1u ri z 1 1 -1 -1 -1 -1 1 1 N'1 B2u r3" y 1 -1 1 -1 -1 1 -1 1 N; B3u ri x 1 -1 -1 1 -1 1 1 -1 N. --------- -------- ----- ---- --- ---- ------ ---- --- ----

~

El/21 r; {a, II} 2 -2 0 0 0 2 -2 0 0 0 N; E1/lu r-, {za, - zll} 2 -2 0 0 0 -2 2 0 0 0 N;

Dn =Dl XCi In the group ofN, the X-, y-, and z-axes of the symmetry operations are directed zll [110], yll [001], x HIlO]. The names of the small representations obey the connectivity relations with E.

Table 8.10. C2V

C2v Basis E C2 CT, CT" Z

Al r 1 z, Xl, y2, Zl 1 1 1 1 ZI Al r l xy 1 1 -1 -1 Zl Bl r3 x,xz 1 -1 1 -1 Z3 B2 r 4 y, yz 1 -1 -1 1 Z4 --------- ------------ -------- -------- ------------ ----El/l r, {a, II} 2 -2 0 0 0 Z,

Table 8.11. 1:KUS

1:KUS Basis E C2 CTd IC~.

1:1 X + y, xy 1 1 1 1 1:2 z(x - y) 1 1 -1 -1 1:3 z 1 -1 1 -1 1:4 x-y 1 -1 -1 1 ------- ---------- --------- ------ ------ ------

1:, {a, II} 2 -2 0 0 0

B.6

Hex

agon

al P

oint

Gro

ups

Tab

le B

.12.

D3h

, C6v

, D

6

D3b

E

2I

C6

2C3

Uh

3C2y

3u

x

C6v

E

2C

6 2C

3 C

2 3u

y 3u

x C

6v

D3h

!:E

'

D6

E

2C6

2C3

C2

3C2y

3C

2x

Bas

is

Bas

is

Bas

is

A'

Al

rl

1 1

1 1

1 1

Z2

Z

y3 _

3x2

y I

A~

A2

r2

1 1

1 1

-1

-1

z x

3 -

3xy2

A~

BI

r3

1 -1

1

-1

+1

-1

y3 _

3x2

y x

3 _

3xy2

I i A~

B2

r 4

1 -1

1

-1

-1

1 x

3 _

3xy2

y3

_ 3

x2y

Z

0'

E"

EI

r6

2 1

-1

-2

0 0

{x,

y}

{x,

y}

{zx,

zy}

E

' E2

rs

2

-1

-1

2 0

0 {x

y, x

2 _

y2}

{xy,

x2

_ y2

} {x

, y}

... i --

----

----

----------

----

----

----

-------------

------

------

----

----

---

----

----

----

----

----

---

EI/2

r7

2

-2

A

-A

1 -1

0

0 0

{a, p

} ES

l2 r

s 2

-2

-j3

j3

1

-1

0 0

0 E3

/2

r9

2 -2

0

0 -2

2

0 0

0 {(

x +

iy)a

, (x

-iy

)P}

f ~--

..... ---

-------.-.--.~

------

D6b

=

D6

X C

; IN

Som

e au

thor

s [1

1.5-

-6]

inte

rcha

nge

rs a

nd r

6-

$

Tab

le B

.13.

C3

b,

C6

C3h

E

I C6

E

A'

A

r l

1

A"

B

r 2

1

r3 1

E'

E2

r 4

1

{rs

1 E

" E

I r6

1

----

----

----

----

---------

r7

1 -1

E

I/2

rs

1 -1

r9

1 -1

ES

/2

rio

1 -1

rll 1

-1

E3/2

r 1

2 1

-1

----------

-----

Q =

exp

( -

ni/6

), w

= Q

4 =

exp

( -

2ni/

3)

C6

h =

C

6 x

C;

IC6

C3

C6

C3

1 1

-1

1

w

w2

w2

w

-w

w2

_w

2 w

-------------

------------

Q

-Q

Q2

_Q

2

_Q

s

QS

_Q

4

Q4

-Q

Q

Q2

_Q

2

QS

_Q

s

_Q

4

Q4

i -i

-1

1 -i

i -1

1

------

---

----------

..., Cl

O"h

C2 3

Iq

Bas

is

C2

q q

1 1

1 Z2

,

-1

1 -1

z(

x ±

iy)3

1 W

w

2 (x

+ i

y)2

1 w

2 W

(x

-iy

)2

-1

W

_w

2 z(

x -

iy)

-1

w2

-w

z(x

+ i

y)

--------

----

----

----

-------------

----------

-i

i Q

4 _

Q4

QS

_

Qs

ex

i -I

_Q

2

Q2

-Q

Q

/3

i -i

Q4

_Q

4

_Q

s

QS

(x -

iy)2

/3

-i

i _

Q2

Q

2 Q

-Q

(x

+iy

)2ex

-i

i 1

-1

i -i

z(x

-iy

)/3

i -I

1 -1

-i

i z(

x +

iy)

ex

---

---_

._

---_

...

....

.. __

._-_

._--

-_.-


B.S Trigonal Point Groups

Table B.14. DJd

DJd Basis E 2CJ 3C~ I 2ICJ 3av L

Aig ri Z2 1 1 1 1 1 1 LI

A2g r+ 2 XIY2-YIX2 1 1 -1 1 1 -1 L2

Eg r+ J {zx, zy} 2 -1 0 2 -1 0 L3

Alu r;- 3x2y _ y3 1 1 1 -1 -1 '"" -1 L'l

A2u r-2 z 1 1 -1 -1 -1 1 L~

Eu r-3 {x, y} 2 -1 0 -2 1 0 L~

--------- ---------- ------ ----- ------ ------ ------ ----- --

EI/2g r: {IX, P} 2 -2 1 -1 0 2 -2 1 -1 0 L+ 6

r: (zx + izy)1X ± 1 -1 -1 1 i -i 1 -1 -1 1 i -i L: E3/2g r+ i(zx - izy) P 1 -1 -1 1 -i i 1 -1 -1 1 -i i L; 5

EI/2u r-6 {ZIX, zP} 2 -2 1 -1 0 -2 2 -1 1 0 Li

ri (x + iy)1X ± 1 -1 -1 1 i -i -1 1 1 -1 -i i Ls

E3/2u r- i(x - iylfJ 1 -1 -1 1 i -1 1 1 -1 i -i V 5 -) 4

Table 8.15. C3v

C3v E 2C3 3av F,A

Al r l 1 1 1 Al A2 r 2 1 1 -1 A2 E r3 2 -1 0 A3 ----------- -------- -------- -------- ------

EI/2 r6 2 -2 1 -1 0 A6

E3/2 r4 1 -1 -1 1 i -i A4

r3 1 -1 -1 1 -i i A3

372 Appendices

Table 8.16. C3

C3 E C3 C~ Basis

A r l 1 1 1 z

{r2 1 OJ OJ2 X + iy

E r3 1 OJ2 OJ X - iy

-------- ------------- ---------- ---------- -----------r4

1 -1 _OJ2 OJ2 OJ -OJ ex El/2 rs 1 -1 -OJ OJ OJ2 _OJ2 P B3/l r6 1 -1 -1 1 1 -1 (x + iy)ex

OJ = exp( - 21ri/3), C31( = S6) = C3 XCI' B3/l is a real representation. It appears in a pair because of time-reversal symmetry.

B.6 Monoclinic Point Groups

Clh( = c.) E t1

I Cl E Cl

A' A r l 1 1

A" B r l 1 -1 ----------- -------- ----------r3

1 -1 -i i EI/l r 4 1 -1 i -i

B. 7 Trielinic Point Groups

Table 8.18. C I

-1

Cj = CI XCI Bill is a real representation. It appears in a pair because of time-reversal symmetry.

B.8

Axi

al R

otat

ion

Gro

ups

Tab

le 8

.19.

Do

oh

Do

oh

B

asis

E

2C

(c/»

00.

(Iv

I 2I

C(c/»

CX

J C~

AIg

E

+

g Z

2

1 1

1 1

1 1

Alu

r u

1

1 -1

-1

-1

1

A2g

E

- g X

IY2

-YI

X 2

1 1

-1

1 1

-1

A2

u

E+ u

Z

l 1

1 -1

-1

-1

E

Ig

ITg

{z

x, z

y}

2 2e

osc/

> 0

2 2c

osc/

> 0

Elu

IT

u

{x,

y}

2 2e

osc/

> 0

-2

-2co

sc/>

0

E2g

.l

\g

{xy,

x2

_ y2

} 2

2cos

2c/>

0

2 2c

os2c

/>

0 E

2u

.l

\u

2 2c

os2c

/>

0 -2

-2

cos

2c/>

0

En

g

2 2c

osnc

/>

0 2

2cos

nc/>

0

En

u

2 2c

osnc

/>

0 -2

-2

cosn

c/>

0

---------

--------

-----------------

------------------------------------------~---------

EI/

2g

{ex, {

3} 2

2cos

c/>/2

0

2 2

cos #

2

0 E

I/2

u

{zex

, z{3

} 2

2cos

c/>/2

0

-2

-2co

sc/>

/2

0 E

3/ 2

g

{ex3

, f3

3}

2 2e

os 3c

/>/2

0 2

2 co

s 3c

/>/2

0 E

3/ 2

u

{(x

+ iy

)ex,

(x -

iy)f3

} 2

2cos

3c/>

/2 0

-2

-2co

s3c/

>/2

0

I

En

+ 1

/2g

2

2cos

(n +

!)c/>

0

2 2

cos

(n +

!)c/>

0

En

+ 1

/2u

2

2cos

(n +

!)c/>

0

-2

-2eo

s(n

+ !)c

/>

0 ------

Do

oh

= C

oov

X C

j

The

sym

bols

. E: '

E;;

, etc

. ar

e pr

efer

red

in l

inea

r m

olec

ules

.

!=' r l 0'

.... i ~ ~ W

-.l

W

Answers and Hints to the Exercises

Chapter 2

2.10 The four aces (SHDC) are rearranged as follows:

P(SHDC) = (DSCH), Q(DSCH) = (DCHS) .

The multiplication rule (2.11) gives

( 1 2 3 4) (1 2 3 4) (1 2 3 4) (1 3 4 2) (1 2 3 4) QP= 13423142 = 1342 3421 = 3421·

2.11 Putting Gi = E in (2.16), we obtain

f(E)f(G) = f(G) .

Since this relation holds for any f(G j ), the elementf(E) must be the unit element of the group C§'. Next put Gj = Gi- 1 , then

Since f(E) is the unit element,J(Gi- 1 ) is inverse to f(GJ

2.14 Consider some element e (other than the unit element E) of the group C§ of order g. Then we have em = E for some integer m, and the cyclic group {e, e2 , .•• , em - 1 , E} is a subgroup of C§. Now, C§ cannot have a proper subgroup because g is a prime number. Consequently, m must be equal to g.

2.15 Use the geometrical considerations explained in the text, or else use (2.23) and Table 2.3.

2.18

~1 ~2 ~3 ~4 ~5

~2 ~1 ~3 ~4 ~5 ~3 ~3 2~1 + 2~2 2~5 2~4 ~4 ~4 2~5 2~1 + 2~2 2~3

~5 ~5 2~4 2~3 2~1 + 2~2

2.19 If A is conjugate to B, then A - 1 is conjugate to B- 1• So, the elements inverse to the ones belonging to a class C(Jj will form a class C(Jj'. IfC(Ji i= C(Jj" then cb vanishes since C(JiC(Jj does not contain the unit element. IfC(Ji = C(Jj" products of the form GG- 1 appear hi times in the class product C(JiC(Jj.

Answers and Hints to the Exercises 375

2.20 From (2.26), GCCk = CCkG. Sum both sides over the elements G of the class CCj •

2.23 C4v = C2 + C2 C4 + C2 ux + C2 Ud •

2.24 For two elements K j and K j of the set f, we havef(KJ =f(Kj ) = E'. Then KjK j is an element of f, because

f(KjK) = f(Kj)f(K j ) = E' E' = E' ,

using the definition (2.16) of homomorphism. Furthermore, from Exercise 2.11,

indicating that K j- 1 is an element of f. Therefore, f is a subgroup of '§.

Next, for an arbitrary element G of '§,

f(GK jG- 1) =f(G)f(Kj)f(G- 1)

=f(G)E'f(G)-1 = E' ,

showing that GKjG- 1 belongs to f. Since the elements of the set GfG- 1 are distinct, GfG- 1 coincides with f as a set.

2.25 The group D3b has six classes: {E}, {C 3 , C;I}, {u 1 , U 2 , u3 }, rUb}' {C3Ub' C;I Uh }, {VI' V 2 , V3}'

Chapter 3

3.1 The eigenvalue A of an eigenvector v of the Hermitian operator A is given by A = (v, Av). Take the complex conjugate of this equation:

A* = (v, Av)* = (Av, v) ,

and use the hermiticity (3.45).

3.2 Because of the unitarity of A, we have in general (A v, Av) = (v, v). For an eigenvector v of A with eigenvalue A, we have (Av, Av) = A* A(V, v) so that IAI = 1.

If A is a symmetric unitary matrix, then A-I = A*. Take the complex conjugate of Av = AV, and show that

A - IV* = A * v* = A-I v* ,

which means Av* = AV*. Since v* is an eigenvector with the same eigenvalue as v, real vectors v + v* or i(v - v*) may be chosen as the eigenvector.

Chapter 4

4.1 For uyv to be a multiple of v, C2 has to be equal to C1 or -c1 • Neither choice can bring R(ot)v into a multiple of v, unless A = O.

376 Answers and Hints to the Exercises

4.3 Put X = r cos l/J, y = r sin l/J in polar coordinates. Then lz = - iiJ / iJl/J, and the right-hand side of (4.30) becomes f(r cos(l/J -IX), rsin(l/J -IX)).

4.4 You have only to verify the transformation for the generating elements C3

and cr 1 :

Cd~E) = (2fi - h, - ft)/.j6

= _~f~E) + f f~E) •

4.6 The matrix it commutes with any matrix of the representation if it commutes with the matrices for the generating elements. Show that the matrix it, which commutes with D(E)(C3 ) and D(E)(crd of(4.3), is necessarily a multiple of the unit matrix. For the three-dimensional representation (4.37),

satisfies (4.44).

4.9 Use (4.61) and (4.56) to evaluate the left-hand side.

4.10 Irreducibility of the direct-product representation is evident from

L L Ix(a x b)(AB)12 = L lia)(AW L IX(b)(BW A B A B

= g(d)g(PiJ) = g(d x PiJ) .

Furthermore, the number nr(d x PiJ) of direct-product representations constructed in this way is equal to the number nc(d x PiJ) of classes of the directproduct group d x PiJ, because

nr(d) = nc(d) , nr(aJ) = nc(PiJ) ,

nr(d x PiJ) = nr(d) nr(PiJ) , nc(d x PiJ) = nc(d) nc(PiJ) .

4.11

L [D(reg)(G)]ij[D(reg)(G')]jk = L b(Gi- 1 GGj ) b(Gj- 1 G' Gk )

j j

=b(Gi- 1 GG'Gk ) •

4.12 Use the result from Exercise 2.16.

4.13 The group C4v has five classes and hence five irreducible representations, whose dimensions can be obtained from 4 x 12 + 22 = 8. The characters may


be determined in the same way as mentioned in the text for the C3v group. Here, we show another method to determine the characters using the relation (5). Ifwe write Xi for hi X(a}(ti&'i), we have

The class constants may be found in the table of Exercise 2.18. For example, ti&' 2 ti&' 3 = ti&' 3 and ti&' 3 ti&' 3 = 2ti&' 1 + 2ti&' 2 give respectively X2X3 = daX3 and x~ = da(2da + 2X2), from which we find {X2 = da, X3 = ±2da} or {X2 = -da, X3 = O}. Proceeding in this way and using the orthogonality (2), complete'the character table.

4.15 The character X of the direct-product representation 151 x D2 is

Its sum over G is equal to g if Xl = X2 and vanishes otherwise. Invariance of IF 0 is verified as follows:

GIFo = L (Gtfr) (GcP) j

= L L L tfrkcPi 15kj (G) Dij(G) j k i

= L L tfrk cPi L Dij(G)Djk(G- l ) . k i j

= LLtfrkcPi[D(G)D(G-l)]ik . k i

Chapter 5

5.1 Evaluate WJ.i using (5.7, 8), to obtain

1 k

WJ.i = g ~ j~ hjj (P) X(J.}(Rj- 1 PRj) X(i)(P)*

On the right-hand side, you have only to sum over P = Rj SRj-l(S E J'f). Note that X(i}(p) = X(i)(S), which yields

WJ.i = ~ I x(J.}(S) X<i}(S)* , S

which proves the Frobenius reciprocity theorem.


5.2 The number of times the representation D().) appears in (D().) i ~) ! Ye is calculated using (5.8, 9):

~ I P)(S)* 1).)(S) = I Wi). W)'i = I W~i SEX i i

5.3 Sum (5.7) over P belonging to the class f(/i' and use (2.26).

5.4 Reduction of the subduced representations D(i)! Ye is obvious from Table 5.1. As for the induction D6 -+ D6h, the operation R2 in (5.7) is here the inversion I. Since IPI = P, we obtain

if PED6 ,

otherwise.

This character gives, for example, A i D6h = Alg + Alu . As for the induction C6 -+ D6, take R2 = c2y • Then C2y QjC2y = C6m will give

X().)(C6') = X().)(C6') + X().)(C6')* .

Ai D6 = Al + A2 , r5(Ed i D6 = E 1 , r 3 (E2) i D6 = E2

B i D6 = Bl + B2 , r 6(E1 ) i D6 = E1 , r 4(E2) i D6 = E2

5.5 From the text above this problem, subduction of D and D i ~ onto Ye gives

s D ! Ye = nA(ll) , (D i ~) ! Ye = L nA(il) .

i= 1

Since the representations A(il) with different i are inequivalent, (D i ~) ! 2 contains D only once.

5.6 The group C3v is decomposed as (2.37) with respect to the invariant subgroup C3 . Irreducible representations r 1, r 2, r 3 of the cyclic group C3 are readily obtained (Appendix B). g = 6 and h = 3 give k = g/h = 2, a prime number.

Choose first A = r 1, and show

A(u1 C3 ( 1) = A(C31) = 1, A(u1 C31Ud = A (C3 ) = 1

Then A(U1YeU1) is equivalent to A(Ye), which corresponds to case (b) of the text (2 = ~). Determine U = D(ud such that U2 = A(ui) = A(E) = 1. Then U = ± 1 will give the Al and A2 representations of C3v •

Choose next A = r 2, and show

A(U1C3 ud=w2 , A(UI C31Ul)=W.

This is case (a) (2 = Ye). r 2 i C3v gives the irreducible representation E of C3v •


i.7 (i) Single-valued representations: Begin by evaluating Pii of (5.44) for the ~ven factor system to get two ray classes E and ah' and hence r = 2. Now, ~2 + 22 = m = 8. The two 2-dimensional irreducible representations are con:tructed from the irreducible representations of the invariant subgroup D2. The ~roup D2 has only one ray class E, so that it has one 2-dimensional representaion (22 = 4). Its representation matrices A(S) are determined so as to satisfy the nultiplication table

A(C2z )A(C2x ) = - A(C2x )A(C2z ) = - A(C2y ) ,

A(C2X )A(C2y ) = - A(C2y )A(C2X ) = A(C2z ) ,

A(C2y )A(C2z ) = - A(C2z )A(C2y ) = A(C2J, A(C2x )2 = - A(C2y )2 = A(C2z )2 = i

)uch matrices are given by

n terms of the Pauli spin matrices (7.34). Next, use (5.56) to calculate

J(I)(S) = ~~I A(1SI) , SeD2 IXI,ISI

md find A(1)(S) is equivalent to A(S) [case (b)]. Determine the representation natrix for inversion 0 = .15(1) such that

J(C2x) 0 = -OA(C2x ) , A(C2z )0 = OJ(C2z ) ,

A(C2y)0 = - 0 A(C2y ) , 02 = i .

'{ou will have two such O's, namely, 0 = ±uz , each corresponding to an irreducible representation.

E C2• c2x c2y I CTh CTvx CTvy

Xl 1 U. Ux -iuy Uz 1 -iuy Ux

X2 1 Uz Ux -io-y -Ctz -1 io-y -fix

(ii) Double-valued representations: PH is nonvanishing for E and C2z' Notice the difference from (i). To find the two 2-dimensional irreducible representations, use again the invariant subgroup D2, which now has four I-dimensional representations given below. Induced representations X3 = r 1 i D2h and X4 = r 3 i D2h are irreducible [case (a)].


D2 E C2z C2x C2,

r1 1 r2 -i -1 r3 -i -i 1 r4 -i -1

E C2z C2x C2, I a h ayX ay,

X3 i it i& z Uz Ux iu;c -u, iu, X4 i -it -iuz Uz Ux -iax u, iu,

In Sects. 11.8 and 11.10, the above representations are obtaIned by means of Herring's method, which does not make use of ray representations. Although Herring's method is straightforward, it requires lengthy calculations as compared with the above method because ofthe doubled (and further doubled in the case of double-valued representations) number of group elements.

Chapter 6

6.1 Rt/I(r) = t/I(R -lr) = t/I(r R) requires (6.6). The proof can also be given explicitly using the operator expression R = exp( -irxlz ) (see Exercise 4.3). Evaluate RxR -1 and RPxR -1 using commutation relations like [Iz' x] = iy.

6.3 Use (6.32, 33) and (6.20) to find

pUI) j= "" c(a). dp " D<P)(R)* R,/,(a) I(m) i..J i..J p. i..J 1m 'l'p.

a p. 9 R

Use here the great orthogonality theorem (4.46).

6.S Use the representation matrices defined in (4.5, 6, 3).

6.10 (P!:!(m)f, g) = da L D!:!m(R) (Rf, g) 9 R

= da L D!:~, (R - 1 )* (I, R - 1 g) 9 R

= (I, P!:lm,)g) .


9.1 Let fPl' fP2' ••• , fP6 denote the six 2s orbitals of the carbon atoms (or six Is orbitals of the hydrogen atoms). The problem is to reduce the representation based on these six orbitals which transform into themselves according to (9.15) except that all the negative signs are dropped. Since the s orbitals themselves remain the same under symmetry operations, you have only to count the number of s orbitals that do not move.

X(R) 6 0 o o 2 o o 0 o 6 0 2

Reduction of this character gives A1g + B1u + E1u + E2g. The corresponding wavefunctions are obtained by means of the methods mentioned in the text:

t/J(a1g) = (fPl + fP2 + fP3 + fP4 + fPs + fP6)! J6 '

t/J(b1u) = (fPl - fP2 + fP3 - fP4 + fPs - fP6)!J6 '

t/J(elu 1) = ( - CfJ2 - fP3 + fPs + fP6)!2 ,

t/J(elu2) = (2fPl + fP2 - fP3 - 2fP4 - fPs + fP6)!fo. '

t/J(e2g 1) = ( - fP2 + fP3 - fPs + fP6)!2 ,

t/J(e2g2) = ( - 2fPl + fP2 + fP3 - 2fP4 + fPs + fP6)! fo. . Here t/J(elu 1), t/J(e1u2) transform like x, y and l/!(e2g 1), t/J(e2g2) like xy, and x2 - y2. a1g and e1u are bonding orbitals, while b1u and e2g are antibonding orbitals.

Consider the Px and Py orbitals in the following way: For each carbon atom, choose the y' -axis such that it goes through the center of the molecule and the x'-axis perpendicular to it. For the x'- and y'-axes defined in this way, the six Py' orbitals and the six Px' orbitals are closed within themselves under the symmetry operations. The six Py' orbitals, transforming like the s orbitals, give A1g + B1u + E1u + E2g, while the six Px' orbitals give the following character:

X(R) 6 0 o o -2 0 o 0 o 6 0 -2

9.2 Upon symmetry descent from Oh to C4, we have the compatibility relations,

Eg ! C4 = A + B, T 2g ! C4 = B + E .


As indicated in Table 9.6, we have the following symmetry-adapted functions:

A: r2 Y20 '" 3z2 - r2 ,

B: r2 Y2, ±2 '" (x + iy)2, (x - iy)2 ,

E: r2 Y2,±1 '" =+= (x ± iy)z

for the representations A, Band E of C4 from the five d functions. Since the representations A and E derive respectively from Eg and T 2g' we find that 3z2 - r2 and (x ± iy)z respectively belong to Eg and T 2g'

On the other hand, B can derive from Eg as well as from T 2g' so that the functions (x + iy)2 and (x - iy)2 cannot be assigned uniquely to Eg or T 2g by means of the above compatibility relations alone. The assignment calls for a somewhat higher symmetry than C4. Restrict the symmetry'Oh down to C4v, and so find

Eg! C4v = A1 + B1 , T 2g ! C4v = B2 + E .

The distinction becomes possible at this level. The characters of B 1 and B2 of C4v differ for mirror reflections (Iv and (ld' The symmetric linear combination of (x + iy)2 and (x - iy)2, being invariant under (Iv, belongs to B1 and hence to Eg, while the antisymmetric one belongs to T 2g'

9.3 For (I orbitals, just count the number of atoms left unmoved by the symmetry operations.

E 6C~

X(R) 6 2 2 o o 0 o 4 2 0

Reduction of this character gives A1g + Eg + T1u ' The basis function for the

totally symmetric representation A1g is (S1 + S2 + S3 + S4 + Ss + S6)/)6, while the three functions of T 1u that behave like x, y, z are (S4 - sd/)2, (S5 - S2)/)2, (S6 - S3)/)2. For p1t orbitals, examine also how the orbitals are transformed.

E 6C~

X(R) 12 0 -4 o o o o o o o

9.5 The wavefunctions for the t~t2 configuration, which has an inequivalent third electron, are constructed from (9.38). The allowed terms are

2A1 2A2 22E 32T1 4 2T2 4A2 4E 4T1 4T2 .


For the t~ configuration, use Table 15.4 to evaluate the character, and find

T~131=A2' T~211=E+Tl+T2'

0 E 6C4 3C~

51 3 -1 -1 52 3 -1 3 53 3 -1 -1

T~1'1 1 -1 1 T~211 8 0 0

10.1 Q(B2u) = U 1z + U 3z - U2z - U4z ,

Q(B2g) = u1y - u3y + U2x - U4x •

6C~ 8C3

1 0 3 0

3

-1 0 -1

For the Eu representation, the projection operator constructed from the diagonal elements of the representation matrices becomes

which may be factorized as

Operate this on u1x, and orthogonalize the result to the translational mode U 1x + U2x + U 3x + U4x to find

10.2 All modes are allowed.

A 1 : xx, yy, zz

10.4 The proof rests on the time-reversal degeneracy of odd-electron wavefunctions '1'. {'I', 0 'I'} will form a basis for the double-valued representation D. If a symmetry operation R transforms 'I' as

R'I' = a'l' + bOP,

then show

RO'l' = OR'I' = - b*'I' + a*O'l'


The representation matrix D(R) then becomes

~ [a D(R) = b - b*] * . a

Unitarity of this matrix requires 1 a 12 + 1 b 12 = 1. Now evaluate the character of the anti symmetric product {D x D} using

10.5 The center atom M gives T lu. Displacements of the surrounding six atoms give the following character X(R).

E

6 2 3

18 2

2 -1 -2

Find the normal coordinates

o o o

o o o

Q(T2g xy) = u1y + U2x - u4y - USx ,

Q(T2u z(X2 - y2» = U1z - U2z + U4z - Usz ,

o 4

o 4

Q(T lu z) = {U 1Z + U2z + U4z + USz - 2u3z - 2u6z ,

U1z + U2z + U3z + U4z + USz + U6z - 6U z ,

2 1 2

o

o

using the factorized projection operators given below. Here U z stands for the displacement of the center atom M.

Projection operators for the irreducible representations of the octahedral group Oh are as follows [for "u" representations, replace (E + 1) by (E - 1)]:

x (E + Ciz + C4z + Ci/) ,

x (E + Ciz + C4z + Ci/) ,


x (E + Ci% - C4% - Ci/) ,

T1g y: (E + 1)(E - C~[110])(E + ci% + C4% + Ci/) ,

T2g (: (E + 1)(E + C~[110])(E + Ci% - C4 % - Ci/) .

The above projection operators were constructed from the diagonal elements of the representation matrices using the basis functions given in Appendix B.

11.1 The lower-right block of the representation matrix for {C2y IT} should be, from (11.52), the small representation matrix for

{O"dyIO} -1 {C2y IT} {O"dfIO} = {C2x l T}

For {C4 IT}, use (11.55) to find

{O"dyIO}-l {C4 IT}{eIO} = {O"vyIT} ,

{eIO}-1{C4 IT}{O"dyIO} = {O"vxI T} .

An example of the basis of this four-dimensional representation is

{cos(ny/a), sin(ny/a), cos(nx/a), sin(nx/a)} .

You can also construct the representation matrices from the transformation of this basis.

11.2 For instance, {C2y IT} has the following effect on orbital functions f(x, y, z) and spin functions a, p.

( a a c ) {CZyIT} f(x, y, z) = f 2 - x, y - 2' 2 - z

a.-+p, {3-+-a.

so that

. ny ny astn--+ - pcos- ,

a a

ny . ny {3cos--+ - astn- .

a a

{eIO}


12.1 The representation M~ at k = (nla)(110) has a basis function like cos(nxla) cos(nYla) sin (2nzla). Displace the origin through (aI2, a12, aI2). The phase factor (12.16) takes on -1 for a = C4 , C;, IC4 and (Iv.

12.3 The rotation {CizIO} sends "'1' "'2' "'3' "'4 to

"'1-"'3' "'3-"'1' "'2-"'4' "'4-"'2' For the {C;Xy IT} operation, show using (11.9) that

{C;xy IT} f(x, y, z) = f(y - a14, x - a14, a/4 - z),

and hence

In addition,

{C;xy IT} f(x, y, z) = f(a/4 - y, a/4 - x, a/4 - z) ,

"'1 - "'1, "'3 - - "'3' "'2 - - "'4, "'4 - - "'2 . Comparing these transformation properties with the characters of Table 12.1, observe that

is a basis for the X3 representation. The same result can be obtained from the general theory of Sect. 12.12.

Characters of the Ll representation at kl' k2' k3' and k4 are given by [see (12.48)]

Xk.(Q) = X(Ld(S;':l QSp-d, Sy = {IC4 10}Y .

For instance, for Q = {l IT} and p = 2, we have

{IC4 10} -1 {IIT}{IC4 10} = {lIIC;l T} = {lIT} {Ill - t2 } ,

using the primitive vector t2 defined in (11.2), which yields

Use (1253) and find the character ofrepresentation at X contained in Ll x L1 :

Use further (12.69) to evaluate

4 XX(Ld(Q2) = L: b(kp + Qkp,kx )xk,(Q2) .

p=l


The table below shows the process of character evaluation. The columns kl '" k4 give the values ofthe characters Xk.(Q) for k p • Blank means that Q does not belong to the ~(kp). From this table and (12.68), we find[Ll x Ll ] = Xl' and {Ll x Ll } = X3.

Q E "§x/T kl k2 k3 k4 XX(LI x Ltl (Q) XX(Ltl(Q2)

{eIO} 1 1 1 1 4xl 0 {q",C!,IO} 0 0 {C!IO} 0 4 2{C4IT} 0 0 {C:z",IT} -1 -1 2 x 1 - -2 {C:Z'iyIT} 1 -1 2 x ( - 1) 2 {lIT} 1 -1 -1 -1 0 0 {a",a,IT} 0 0 {ahIT} 0 0 2{S410} 0 0 {ad",IO} 1 1 2 x 1 2 { a.u,IO} 1 1 2 x 1 2

12.4

r ls X4 r ls x X4

{eIO} 3 2 6 {C!", C!yIO} -1 0 0 {C!IO} - 1 -2 2 2{C4IT} 1 0 0 {C:Z"yIT} -1 -2 2 {C:Zj"yIT} -1 2 -2 {lIT} ":"3 0 0 {a",ayIT} 0 0 {ahIT} 0 0 2{S410} -1 0 0 2{adIO} 1 0 0

12.5 Only those vibrations that appear in the reduction ofRls x r 15 x Xl can give rise to indirect transitions. Show that

r lS XX1=X~+X;,

R15 X X~ = Ml + Ms ,

R15 X X; = Ml + M2 + M3 + M4 + M5 .

Now, observe from Table 12.4 that the normal modes at the M point are M~, M~, and 2M;.


12.6

{r} T: {sin 2Y sin2Z, sin2Z sin2X, sin2X sin2Y} {M} A: cos X cos Y sin2 Z, {M}Bl: sin X sin Y cos2 Z, where X = 'Ttx/a, Y = 'Tty/a, Z = 'Ttz/a .

Exercise to Table 13.8. The group of kat r, X, Z and A is identical to the space group D~~. In such cases, the general theory of Sect. 12.12 simplifies greatly: Characters of product representations may be evaluated as in the case of point groups without invoking complicated summations. Just one point to watch out for is that Q2 can contain primitive lattice translations, as seen in the table below.

Q Zl Zl X Zl Q2 Zl Al {Zl x zd {Al x Ad Al Al x Al

{eIO} 2 4 {eIO} 2 2 1 1 {C210} -2 4 {eIO} 2 2 1 1 {C2x IT} 0 0 {eltd 2 -2 -1 1 {C2.IT} 0 0 {elt2} 2 -2 -1 1 {flO} 0 0 {eIO} 2 2 -1 -1 {CThIO} 0 0 {eIO} 2 2 -1 -1 {CTvxIT} 0 0 {elt2+t3} -2 2 1 -1 {CTv.IT} 0 0 {e Itl + t3} -2 2 1 -1

15.2 For an 1= 1 electron, sp of (15.42) is

In the p3 configuration, the total spin angular momentum S takes on the values 1/2 and 3/2. The orbital part corresponding to the S = 1/2 spin state should belong to

[n/2 + S, n/2 - S]* = [21]* = [21]

of the product representation D(1) x D(1) X D(1). Evaluate its character using Table 15.4,

P1] = (s~ - s3)/3

= (e2iq> + eiq> + 1 + e-iq> + e- 2iq» + (eiq> + 1 + e-iq» ,

which consists of L = 2 and L = 1 orbital states. For S = 3/2, show

[n/2 + S, n/2 - S]* = [3]* = [1 3 ] ,

X[13] = (s~ - 3S1 S2 + 2s3 )/6 = 1 .

Motifs of the family crests (found at the beginning of each chapter)

1: Crane made from folded paper 2: Bound sheaves of the rice plant 3: Umbrellas 4: Tea berry 5: Hollyhock 6: Chinese flower 7: Arrow feathers 8: Feathers of the falcon 9: Cloves

10: Chinese balloon flower 11: Swallowtail butterfly 12: Small pouches 13: Scissors 14: Sword blades and leaves of the wood sorrel 15: Bell flower

References

Chapter 5

5.1 The following paper gives a slightly different treatment: W. G. Harter: J. Math. Phys. 10, 739 (1969)

5.2 T. Janssen: Crystallographic Groups (North-Holland, Amsterdam 1973) pp. 243-254

Chapter 7

7.1 L. D. Landau, E. M. Lifshitz: Quantum Mechanics, 3rd ed. (pergamon, Oxford 1976) pp.410-419

7.2 Tables of the Racah coefficients are available, for example, in: T. Ishidzu, H. Horie, M. Sato, Y. Tanabe, S. Yanagawa: Tables of Racah Coefficients (Pan Pacific, Tokyo 1960); M. Rotenberg, N. Metropolis, R. Bivins, J. K. Wooten, Jr.: The 3-j and 6-j Symbols (MIT, Cambridge, MA, 1959)

7.3 R. Kubo, T. Nagamiya (eds.): Solid State Physics (McGraw-Hill, New York 1969) pp. 446-448

7.4 Tables of the c.f.p. for p, d, and f electrons have been given by G. Racah: Phys. Rev. 63, 367 (1943); ibid. 76, 1352 (1949)

7.5 For details of the treatment, see, for example, J. C. Slater: Quantum Theory of Atomic Structure, Vol. 2 (McGraw-Hill, New York 1960) pp,. 95-157

Chapter 8

8.1 G. Herzberg: Molecular Spectra and Molecular Structure, Vol. 3: Electronic Spectra and Electronic Structure of Poly atomic Molecules (Van Nostrand, New York 1966)

Chapter 9

9.1 M. Kotani: J. Phys. Soc. Jpn. 19, 2150-2156 (1964) 9.2 J. S. Griffith: Irreducible Tensorial Methodfor Molecular Symmetry Groups (Prentice-Hall,

Englewood Cliffs, NJ 1962) pp. 4-31

Chapter 10

10.1 T. P. Wilson: J. Chern. Phys. 11, 369 (1943) 10.2 H. A. Jahn, E. Teller: Proc. R. Soc. London A 161, 220 (1937) 10.3 E. Ruch, A. Schonhofer: Theor. Chim. Acta 3, 291 (1965) 10.4 L. D. Landau, E. M. Lifshitz: Quantum Mechanics, 3rd ed. (Pergamon, Oxford 1976) p. 408 10.5 E. I. Blount: J. Math. Phys. 12, 1890 (1971)

Chapter 11

11.1 L. P. Bouckaert, R. Smoluchowski, E. Wigner: Phys. Rev. SO, 58 (1936) 11.2 C. Herring: J. Franklin Inst. 233,525 (1942). An error at the W point is corrected in [11.3] 11.3 R. J. Elliott: Phys. Rev. 96, 280 (1954) 11.4 R. H. Parmenter: Phys. Rev. 100,573 (1955) 11.5 R. C. Casella: Phys. Rev. 114, 1514 (1959) 11.6 J. J. Hopfield: J. Phys. Chern. Solids 10, 110 (1959)

392 References

11.7 E. I. Rashba: Soviet Phys.-Solid State 1,368 (1959) U.8 R. Knox, A. V. Gold: Symmetry in the Solid State (Benjamin, New York 1964). This book

contains reprints of the following original papers: [10.2], [11.1-4], [12.5]. 11.9 G. F. Koster: "Space Groups and Their Representations", in Solid State Physics, Vol. 5

(Academic, New York 1957) pp. 173-256 11.10 J. L. Warren: Rev. Mod. Phys. 40, 38 (1968) 11.11 C. J. Bradley, A. P. Cracknell: The Mathematical Theory of Symmetry in Solids (Oxford

University Press, Oxford 1972)

Chapter 12

12.1 J. L. Birman: Theory of Crystal Space Groups and Lattice Dynamics (Springer, Berlin, Heidelberg 1984)

12.2 A. A. Maradudin, E. W. Montroll, G. H. Weiss, I. P. Ipatova: Theory of Lattice Dynamics in the Harmonic Approximation, 2nd ed. (Academic, New York 1971) Chap. 3

12.3 J. L. Warren: Rev. Mod. Phys. 40, 38 (1968) 12.4 G. Nilsson, G. Nelin: Phys. Rev. B 3, 364 (1971) 12.5 M. Lax, J. J. Hopfield: Phys. Rev. 124, 115 (1961)

Chapter 13

13.1 R. E. Dietz, A. Misetich, H. J. Guggenheim: Phys. Rev. Lett. 16, 841 (1966) 13.2 Tables of the irreducible characters are due to J. O. Dimmock, R. G. Wheeler: Phys. Rev. 127,

391 (1962)

Chapter 14

14.1 L. D. Landau, E. M. Lifshitz: Statistical Physics, Part 1, 3rd ed. (Pergamon, Oxford 1980) pp.446-471

14.2 A detailed treatment without using group theory is given by T. Nagamiya: "Helical Spin Ordering" in Solid State Physics, Vol. 20, ed. by F. Seitz, D. Turnbull, H. Ehrenreich (Academic, New York 1967) pp. 305-411

14.3 J. M. Kosterlitz, D. J. Thouless: J. Phys. C 6, 1181 (1973) 14.4 W. F. Brinkman, R. J. Elliott: Proc. R. Soc. London 294A, 343 (1966)

Chapter 15

15.1 D. E. Rutherford: Substitutional Analysis (Edinburgh University Press, Edinburgh 1948) pp. 23-44

15.2 T. Yamanouchi: Proc. Phys.-Math. Soc. Jpn, 19,436 (1937) 15.3 H. Weyl: The Classical Groups (Princeton University Press, Princeton, 1939) pp. 115-136 15.4 G. Racah: Phys. Rev. 76, 1352 (1949)

Appendix

B.l L. P. Bouckaert, R. Smoluchowski, E. Wigner: Phys. Rev. 50, 58 (1936) B.2 Y. Onodera, M. Okazaki: J. Phys. Soc. Jpn.21, 2400 (1966)

Subject Index

Italic numbers indicate primary references.

Abelian groups 8 Accidental degeneracy 106 Active points 322 Addition of angular momenta - two 137 - three 151 Additional representations 178 Additive groups 10 Adiabatic approximation 229 Adiabatic potential 229 - 230 Adjoint representations 73 Alternating group 334 Angular momentum 52, 120-124, 137, 151 Antibonding orbitals 188 Antisymmetric product representations 68,

112, 289 Anti-unitary operators 295 Associative law

for group elements 7 - for vectors 31

Band theory 259 Basis - of a crystal structure 241 - of a group representation 46 - of a vector space 32 Basis functions 46 - generation of 112, 190-193 Benzene crystal, excitons in 281 - 283 Benzene molecule, molecular orbitals

in 189-195 Bethe notation - for double-valued representations 178 - for single-valued representations 175 Bloch functions 247, 260, 264, 269 Bloch theorem 247 Body-centered cubic lattice 245 Bonding orbitals 188 Branching rule 350 Bravais lattices 239 Breathing mode 222 Brillouin zone (BZ) 246

Center 88 c.f.p. 161

CO coefficients, see Clebsch-Oordan coeffi-cients

Character tables - construction of 71 - for point groups 363 - for ray representations 93 Characters 61

first orthogonality 62 - of ray representations 92 - second orthogonality 62 Class constants 25 Classes 22 - multiplication of 24 - in point groups 173 Clebsch-Oordan coefficients 67, 137, 212 Closed set 7 Commutative groups 8-10, 23, 73 Commutative law 8 Compact groups 57 Compatibility relations 65, 83, 201 - in a space group 264 Complementary subspace 37 Complex conjugate representations 74 Conjugate classes, see Classes Conjugate elements 21 Conjugate representations of the symmetric

group 340 Conjugate subgroups 25 Connectivity relations 264 Corepresentations 296 Cosets 20-21 - decomposition 21, 26 - representatives 21 Covering operations 2, 10 Crystal field 198 Crystal systems 241, 362 Crystallographic point groups 171, 360 CsCl structure 243, 262, 268 Cycles 333 Cyclic boundary conditions 235 Cyclic groups 8

Davydov splitting 281 Diamond structure 243, 262-264 Diatomic molecules 48, 185

394 Subject Index

Dimension - of a representation 44 - of a vector space 31 Direct-product groups 29, 176 - irreducible representations 69 Direct-product representations 66, 69 Direct sum - of representations 48 - of vector spaces 37 Direct transitions 276 Distributive law 10, 31 Double point groups 177 Double rotation group 129 Double space groups 256, 274 Double-valued representations - of point groups 94, 176 - of the rotation group 129 - of space groups 257, 274 Dynamical matrix 221

Eigenvalue problems 40 Electric field gradient 156 Electron-phonon interaction 274 Elements 7 Empty lattice 265 Energy bands 259 Equivalence transformations 47 Equivalent representations 47 Euler angles 117 Even permutations 334 Excitons - in antiferromagnets 308 - in molecular crystals 278 Extra representations 178

Face-centered cubic lattice 235, 245, 265 Factor groups 27 Factor systems 88, 90 Faithful representations 45 Field 10 Finite groups 8 Force constants 220 Four group 9 Frenkel exciton 278 Frobenius reciprocity theorem 83, 280 Frobenius-Schur criterion 74

Ge 260 Generating elements 9 Glides 238 Great orthogonality theorem 58 Group elements 7 Group table 8 Group of wavevector 85, 249 - irreducible representations 250 Groups 7

Half-odd representations 213 Heitler-London method 195 Hermitian conjugate matrix 40 Hermitian matrix 40 Herring criterion for time-reversal

degeneracy 76, 301 Herring's method 251, 257 Holes 163 Homomorphic mapping 19,27,44 Homomorphism 19 Homomorphism theorem 28 Hund's rule 162 Hybridized orbitals 195 Hydrogen molecule 185

Identity element 7 Identityoperation 2-3, 11, 169 Identity representation 45 Improper rotations 171 Index of coset decomposition 21, 27 Indirect transitions 277 Induced representations 82 Inequivalent representations 47 Infinite groups 9 Infrared absorption 227 Inner product of vectors 38 Integral representations 213 International symbols - for magnetic space groups 308 - for point groups 173, 362 - for space groups 242 Invariant subgroups 25, see also Factor

groups Invariant subspace 37, 46 - irreducible 48 Inverse elements 7 Inversion 104, 170 Irreducible characters 61 Irreducible decomposition, see Reduction Irreducible representations 48 - of a group having an invariant

subgroup 50, 84 Irreducible tensor operators 110, 142 Isomorphism 3, 18 Isospin 354

3j symbols 140 6j symbols 153 Jacobi polynomial 127 J ahn-Teller effect 231

Kernel 28 Kramers degeneracy 232, 294

Landau criterion 321 Lande g-factor 157

Lattice 239 Lattice vibrations 271 LCAO method 187 LCAO MO 188 Left cosets 21 Legendre polynomial 133 Length of a cycle 333 Lifshitz criterion 321, 329 Ligand field 198 Linear independence 31 Linear operators 34 Linear space, see Vector space Linear transformation groups 42 Linear transformations 34, 47 Little groups 85 - representations of, see Small representa

tions

Magnetic groups 308 Many-electron wave functions - in an atom 158, 352 - in a molecule 204 Mapping 19 Methane, hybridized orbitals 196 Metric vector space 38 Mirror reflections 1, 12 - symbols for 170 MO, see Molecular orbitals Modes, see Normal modes Molecular orbitals 184 Molecular vibrations 220 Mulliken notations - for double-valued representations 178 - for single-valued representations 175 Multiplets in molecules 204 Multiplication table 8

n-fold rotation axis 169, 173 NaCI structure 243, 268 Non-crossing rule 186 Nonsingular matrix 35 Nonsymmorphic groups 88, 239, 250 Nonunitary groups 295 Norm of vectors 38 Normal coordinates 221 Normal divisors, see Invariant subgroups Normal modes 221 Normal subgroups, see Invariant subgroups

0(3) 43, 130 Octahedral group 172 - character table 364 - Clebsch-Gordan coefficients 216-219 - projection operators 384- 385 Odd permutations 334 Onto-mapping 20

Subject Index 395

Operator equivalents 149 Optical transitions 276 Order

of a group 8 - of group elements 8 - of a star 248 Order-disorder transition of alloys 322- 324 Orthogonal groups 43 Orthogonality - of basis functions 108 - of characters 62 see also Great orthogonality theorem Orthonormal basis 38

ll-electron approximation 1-89 1l orbitals 186, 189 Paramagnetic groups 308 Partition 336 Partners 46 Pauli spin matrices 124 Permutation group, see Symmetric group Permutations 15-17,333 Perturbation, energy level splitting caused

by 107 Physically irreducible representations 317 Plane waves 265 Point groups 105, 169, 179-182 Primitive translation vectors 234 Product representations 66 Projection operators 112 Projective representations, see Ray represen-

tations Proper rotations 171 Proper subgroups 20 Pseudoreal representations 74

Quadrupole moments 150

Racah coefficients 153 Raman scattering 228 Ray classes 92 Ray representations 88, 90, 178 Real representations 74 Rearrangement theorem 17 Reciprocal lattice 244 Reduced matrix elements 144 Reducible representations 48 Reduction 48, 63 Regular matrix 35 Regulai' ray representation 91,

93 Regular representation 70 Representations 4, 44 - restriction to a subgroup 65 Right cosets 20

396 Subject Index

Rotation group 105, 107, 116 - full 130 - irreducible representations of 125, 357 Rotation-reflection group 130 Rotations - infinitesimal 119, 121 - in point groups 169 - in three dimensions 115-118 - in two dimensions 13, 32 Rotatory inversions 170, 173 Rotatory reflections 170 Rutile structure 237,251-253,256-258,

306

(1 orbitals 185, 189 SchOnflies symbols

for double groups 177 - for point groups 171, 362 - for space groups 242 Schur's lemma 58 - for ray representations 91 Screws 238 Second-order phase transitions 316 Selection rules 109 - for diagonal matrix elements 111, 232 - in a space group 283 Self-adjoint matrix, see Hermitian matrix Shubnikov groups 308 Signature 358 Similarity transformations 47 Simple characters, see Irreducible characters Simple cubic lattice 235, 245 Simply reducible groups 213 Single groups 177 Single-valued representations 176 Site groups 278 Slater determinants 159 Slater integrals 168 Small representations 85, 87, 250 SO(3) 43, 129 Solvable groups 87 sp3 hybridized orbitals 196 Space groups 237, 242 - irreducible representations 85, 253 Space inversion, see Inversion Special orthogonal groups 43 Special unitary groups 43 Spherical harmonics 133 Spin angular momentum 124 Spin configurations 324 Spin functions 176 - transformation by rotations 124, 179 - transformation by time reversal 293 Spin-orbit interaction

in atoms 149 - in crystals 273

Spin waves 312 Square, symmetry group of 14 Standard tableau 338 Star 85,248 SU(2) 42, 129, 355 Subduced representations 65, 83 Subgroups 20 Subspace of a vector space 36 Symmetric group 15, 105, 333 Symmetric product representations 68, 112,

289 Symmetry-adapted functions 112, 190 Symmetry descent, method of 192 Symmetry group 105 Symmetry operations 10 Symmorphic groups 239"

Thbleau, see Standard tableau Tetrahedral group 172 - character table 365 Tensor operators, see Irreducible tensor

operators Time reversal 291 Time-reversal degeneracy 299 - 306 Time-reversal operator 74, 163, 233,

292 TlBr 260 lransformation of functions 51 lranslation group 105, 235, 239 - irreducible representations 246 Translations 235 Transposed matrix 73 Transpositions 334 lriangle, symmetry group of 10-12 Twofold rotation axis 2

Umklappung 3 Unit cell 234 Unit element 7 Unitary groups 42, 342, 358 Unitary matrices 40 Unitary operators 39 Unitary representations 45, 57

Vector representations 227 Vector space 30 - metric 38 Vectors 30

components of 32 - transformation of 32

Wavevector 248 Wigner coefficients 137 Wigner criterion for time-reversal

degeneracy 299

Wigner-Eckart theorem - in point groups 213 - in the rotation group 144

Yamanouchi symbol (Y-symbol) 338

Subject Index 397

Young diagram 336 Young symmetrizer 341

Zero classes 92. 178 Zero element 9

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