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: : 1+ 2: : : :
) . ( . . : )9 ( 2:00 12:20 :
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: . . :
1- Group Theory for Atoms, Molecules and Solids, B. S. Wherrett, 1986, Prentic Hall International.
2- Finite Group and Quantum Mechanics D. B. Chesnut, 1974, John Wiley. 3- Elements of Group Theory for Physics, A. W. Joshi, 1973. 4- "Molecular Symmetry and Group Theory, A. Vincent, 2001, John Wiley. 5- Physical Chemistry P. W. Atkins, 5th edition, Oxford University, 1994. 6- Chemical Application of Group Theory, F. A. Cotton, third edition , 1998,
John Wiley. 7- Group Theory and Its Application to Physical Problems, M. Hamermesh,
Dover, 1989. 8- Atoms and Molecules, M. Weissbluth, Academic Press, 1978.
, " " "-1 .1994 -
- . . . " "-2 .) . (2003,
: /imnasser/PHYS/sa.edu.kfupm.users://http
: . .... , , : : : ( -1 ....) , : ( -2 . , : -3
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 1
http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/
-
: ) (
. . .
. ) ( : ) (
.) ( -1
.)( ) ( -2
. -3 . ) ( ).
. .
. .
. . ,
) : - nC
n
)
. 2 . ) (
n
90 x :) : , , ,P x y z )
z x
P (x,z,-y)
-y P(x,y,z)
z
yx
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 2
-
:
'
' 1 0 0' 0 0 1 '' 0 1 0
r rR
x xy yz z
=
r Rr=
| |R
1 : ( .1= = ).
) nC (
2( ), 180nC C n = =
C E , . nn =
3
2
3601 ( ) (120),3
2 22 ( ) (2
23 ( ) (2 ) ,
n n n n
nn
C C C
C C C C C C Cn n
C C n C En
4) ( ),n
= =
= = = =
= = =
( ) 1 2 2 ( )(2 ) ( )m nn nm n mC C C Cn nm
= = =
( )( )
134 4
156 6
1 ,
2 ,
C C
C C
=
=
) ( - ,v h ( , ,P x y z ( , :
( ,)z( :)
xyz
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 3
-
y x
P (x,y,-z)
-z
zz
r B r=
yx
P(x,y,z)
:
'
' 1 0 0' 0 1 0 '' 0 0 1
r rB
x xy yz z
=
1 | |B = | |R =h x
. 1= 1 y =
. . ( ) (
(.E ) ).
xy
2 = =
v
) .h n - )n nS C =n
( )n hS C=
( ) ( )
hh . ) ( n nC C ).(
2n
h =nC
2 2 2( )h hI S C C
. = = =
P(x,y,z)
z
yxP (-x,-y,-z)
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 4
-
:
'
' 1 0 0' 0 1 0 '' 0 0 1
r rB
x xy yz z
= =
r B r
B
1 | = . 3 - = |
2 2( ,h hI C IC
:
-1
) = =
C E2h 22 = E =
C 1 -2 .
.
2h
) (:
-12 .
'2 2
second rotation first rotation
(2 )U U C =
" C" -22 .
'
second reflection first reflection
(2 )v v C =
'
second reflection first reflection
( 2 )v v C =
' ' ' '(2 ) (2 )v v v v v vC C = =
.
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 5
-
.
) - )
E
nC2( )n
i
nS 2n
) ( :
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 6
-
:
G ( I ( ) G( .
: G
{ }, , , ,n nE C S=,
G a - b G . ,a b G) ) ( - ) ( )c a b ca b =, ,a b c G
E G .
E . - a a E=a Ga G
1a - G E 1 . 1a a a a = =a G
b
a ) ( b a,= . a b G
{ }2,G E S= ={
: :
I }, hG E =
h
: E x I E x
E E I E E E E I I
h
h h
: . :
( ) ( , . ).
. : ) (
. G . .
HG
Hhh g
z
.| A | r >
x
| r ' >
y
= + = = = + = + = +
cos ; sinx r y r . = =
' cos sin 0' sin cos 0 | ' ( ) |' 0 0 1
=
x xy y rz z
> = >R r
: : ) ( -1
' cos sin 0' sin cos 0 | ' ( ) |' 0 0 1
= > =
x xy y rz z
cos sin 0sin cos 00 0
>R r
-2 ,
:
1
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 18
-
0 00 00 0
i
i
ee
z
1
: z c
11 12 13
21 22 23
31 32 33
cos sin 0sin cos 00 0 1
z
a a ac a
a a a
= =
2 2 211 12 13 1a+ + =
11 21 12 22 13 23 0a a a a
a a
( )
a -1 a. -2
a a. + + + =
: z . y x -1
11 22 21 12det( ) 1zc a a a a = . -2 =
11 22 33trace( )zc a a = a -3 + +
r >2 2( , ) (=
. , ( |
( . ) 45 : . = o
f x y a x ya
' cos45 sin 45 0' sin 45 cos45 0' 0 0 1
=
x xy yz z
( ', ') 2 ' . ' ( , )f x y3, 2= =x y(3,2) 5= . .
' / 5 2, '= =x y 1/ 2(5 / 2) =g / 2, 1 5 .
2 2( , ) (= + )f x y x y ) .
2 2( '= +x y( 'g x
= g x y a x yf a
, ') 'y
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 19
-
c : C 2 2 ( )2
C = = =
2
1 0 00 10 0 1
c =
( )
0
z : xy 1 0 0
( ) 0 1 00 0 1
xy =
1 0 00 1 00 0 1
i =
xy
: i
zzz
c s z : xy
cos sin 0 1 0 0 cos sin( ) sin cos 0 0 1 0 sin cos 0
0 0 1 0 0 1 0 0z zs xy c
= = =
0
1
: E 1 0 00 1 00 0 1
E =
h
: : i2c2h E
i2c E E
E 2c2c 2ch E ii
E 2cih
h
ih
h
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 20
-
, ,E : 1 00 1
=
11 00 1
A = =
212
B =
, 1 33 1
=3C ,1 31
2 3 1
= =
1 3
3 1
,
.
312
D c
= =
23
1 312 3 1
F c
= = : -1
F D C B A E F D C B A E E C B F D E A A A C D E F B B B A E F D C C E F B A C D D D E A C B F F
-2
. ( ) ( ) ( ); , , ; ,E A B C D F
2v
2
1 0 0 1 0 0 1 0 0 1 0 00 1 0 , 0 1 0 , ( ) 0 1 0 , ( ) 0 1 00 0 1 0 0 1 0 0 1 0 0 1
E C xz yz
= = = =
( )
: :
yz 2c2v E 2c E E E 2c2c
E E
2c
( )xz
( )yz ( )xz( )xz ( )yz
2c( )yz ( )xz ( )xz( )xz ( )yz ( )yz
2Trace 3, Trace 1, Trace ( ) 1, Trace ( ) 1.E c yz xz = = = =
z( .
( ) ( ) ( )2; ; ( ), (E c xz y
: :
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 21
-
2
1 0 0 1 0 0 1 0 0( ) 0 1 0 0 1 0 0 1 0 (
0 0 1 0 0 1 0 0 1) xz C yz
= = =
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 22
-
. .
: : 1 cos sin
Rsin cos
=
. :
: , -1
1 2
cos sinR I 0 ,
sin cosi ie e
= = = =
1ie
. : . -2 =
11 2
2
0cos sin0
0sin cos
i
i
xeix x
xe
= +
1 2,
=
x x1 1x =2
1X 1i
.
x i=
1
2
1xx i
= = 112
X =
2 . 11
2X
i
=
2ie =
, 1 11
2X
i i
=
1 0R0
i
i
eX X
e
:
=
cos sin 0R sin cos
0 0
=
: :
01
.
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 23
-
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 24
-
: : 2 5 4
A1 2
=
. :
: , -32
1 2
5 4A I 0 7 6
1 21, 6
= = + =
= =
1 1
0
. : . -4 =
11 2
2
5 1 4 00
1 2 1 0x
x xx
= +
1 2,
=
x x1 1x =2 1x = 1
2
11
xx
= =
1X 11
. 112
X =
2 . 41117
X =
2
1 41 1
=
X
2 6 =
,
1 1 411 15
X
=
: 1 1 1 4 5 4 1 4 1 01A
1 1 1 2 1 1 0 65X X X
= = =
A
.X
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 25
-
GL(2) -1
1 2
3 4
' ;' ,
x a x a yy a x a= += +
y
:
1 2 1 2
3 4 3 4
x' x xR ,
y' y ya a a aa a a a
= =
1 0A
0 1
=
(2)SL
)
0
:
. 4
-2
. 2) GL
. 3
1 2
3 4
1a aa a
=
O(2) -3(2)2 2+ =
( )x GL y :
( )2 22 2 21 2 3 4' 'x y a x a y a x a y x y+ = + + + = +2 2 2 21 3 2 4 1 2 3 41; 1; 0;a a a a a a a a+ = + = + =
x' cos sin x; 0
y' sin cos y
2;
: : .
2
=
. z
.
O(3) -42 2 2 3
. x y z+ + =
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 26
-
, A BA B
11 12 11 12
22 22 22 22
,
= =
a a b bA B
a a b b
11 12
22 2211 12 11 12
22 22 21 22 11 12
21 22
0 00 00
0 0 00 0
a aa aa a b b A
a a b b
: : 1
A B B b b
b b
= = =
ace( ) Trace( ) Trace( ) :
Tr B = +A B A
11 12 11 12
22 22 22 22
,
= =
a a b bA B
a a b b
11 12 11 1211 12
21 22 21 2211 12 11 12
22 22 21 22 11 12 11 1222 22
21 22 21 22
11 11 11 12 12 11 12 12
11 21 11 22 12 21 12 22
22 11 22 1
= =
=
b b b ba a
b b b ba a b bA B
a a b b b b b ba a
b b b b
a b a b a b a ba b a b a b a ba b a b 2 22 11 22 12
22 21 22 22 22 21 22 22
a b a ba b a b a b a b
Trace( ) Trace( )Trace( )
: : 2
: =A B A
1 2 1 0,
1 0 0 1
= =
A B
1 2 0 01 2 1 0 1 0 0 01 0 0 1 0 0 1 0
0 0 0 1
= =
A B
Trace( ) 3race( ) Trace( ) 1 2 3
A BA B =+ = +
B
: : 3
T
=
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 27
-
11 12 11 12
22 22 22 22
1 0 1 0 1 0 2 01 20 1 0 1 0 1 0 2
1 0 0 01 0 1 01 0
0 1 0 00 1 0 1
= = =
a a b bA B
a a b b
Trace( ) 2Trace( )Trace( ) 1 2 2
A BA B =
= =
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 28
-
A : X= ,2B X= ,C X,,F X, 3456E X= = D X= =
: -1F D C B A E F D C B A E E E F D C B A A B B C C D D F F
) -2 ) ( ) ( ); , , ; ,E A B C D F
23 : 12b C
= = , ,
1c = = ,1 0
0 1
212
d = = 1 33 1
312
1 3
3 1f = = ,
1 3
3 1
3
1 312 3 1
a C
= =
1 00 1
E =
-1) -2 ) ( ) ( ); , ; , ,E a b c d f
2
1 2 3;
1 3 2p =
3
;
6p =
6
:
:
4p = 1 2 3
;3 1 2
1
1 2 3;
1 2 3p =
5p = 1 2 3
;3 2 1
3
1 22 1 3
p = , 1 2 3 ;2 3 1
p
1p 2p 3p 4p 5p
1p 1p 2p 3p 4p 5p 6p
2p2p
3p 3p
4p 4p
5p 5p
6p 6p
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 29
-
Prof. Dr. I. Nasser (imnasser@kfupm.edu.sa) 30
00yi
i
=
1 00 1
1 00 1
I =
z
, :
.
0: 11 0x
=
,
z = ,
I
I I I
I I
x y
x y z
x yizix
yy zi xi
yi xiz z
!
a : cAb d
=
ad 1 = 00 1
A A =
1
1 1 d cAb a
=
0bc
4
34c
24c4c4
:
E 34c
24c4c E E
E 34c
24c4c4c
4c34c
24c
24c E
24c4c
34c
34c E
2h :
h
i2c2h E i2c E E
E 2c2c 2ch E ii
E 2cih
h
ih
h
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