南京大学结构化学双语课件CH4LEC2

b c a
c a b
2. Conjugacy Class
In Group G, element B is said to be conjugated to element
A, if there exists another element R ( R is not the identity
The Subgroup (子群)
A group (K) fห้องสมุดไป่ตู้rmed by a subset of the elements of Group G is called a subgroup of G. h(G)/h(K) = Integer
{I, C31, C32} is a subgroup of {I, C31, C32, a, b, c} {I, a} is a subgroup of {I, C31, C32, a, b, c} {I, b} is a subgroup of {I, C31, C32, a, b, c} {I, c} is a subgroup of {I, C31, C32, a, b, c}
I
R-1 I R = R-1 R = I
C31, C32
a C31 a = C32
a , b, c
C31 a C32 = b , a b a = c
3. Character of Group Elements
Definition: The character of group element R is the trace of its representation matrix.
element, such that R-1 A R = B or A = R-1 B R
Elements in a group which are conjugate to each other form a conjugacy class.
Example
In C3V group, the elements are grouped into three conjugacy classes,
Section 4.3 The Group and its Representation
1. Definition 2. Conjugacy class (共轭类) 3. Character of group elements (表示) 4. Irreducible representation (不可约表示) 5. Character Table (特征标表)
2. Cyclic Group (循环群)
Definition: A group is a cyclic group if every element can be expressed as a power (乘幂) of a single element.
{I, } {I, Cn1, Cn2 , …, Cnn-1} ()1 = ()2 = I (Cn1)i = Cni
Proof:
A) k ( B ab i k i
i 1k 1
n
n
Identical
B) k ( A ba i k i
i 1k 1
n
n
Property II:
Conjugated elements have equal value of character.
Proof:
(3) The associative law: A(BC)=(AB)C.
(4) The inverse element: For every element A G, there
exists an element A-1 G, so that AA-1=I.
Example I
A set composed of the whole integer numbers from - to +, with the defined operation of “add”, forms a group. The closure: I + J = k Integer The identity element: 0 0 + I = I + 0 =I The associate element: I+J+K=(I+J)+K=I+(J+K) The inverse element: the inverse element of I is –I
1. Definition
A set of elements G{ A, B, C, …}, satisfy, (1) The closure: for any AG and B G, there exists AB G. (2) The identity element: I G, for any A G, IA=AI. Then I is the identity element of set G.
a b c
a b c
b c a
c a b
The Order (阶) of a Group
The number of elements in the group.
Labeled h
Finite group: if h is finite,
Infinite group: if h is infinite.
Section 4.3 The Group and its Representation
1. Definition 2. Conjugacy class (共轭类) 3. Character of group elements (表示) 4. Irreducible representation (不可约表示) 5. Character Table (特征标表) Section 4.4 Classification of Point Group
(Cn1)n = Cnn =I
The Cyclic Group is Abelian
Cni Cnj = (Cn1)i+j = (Cn1)j+i = Cnj Cni
The Inverse elements in the cyclic group
()-1 =
(Cni)-1 = Cnn-i
Cs and Cn Group, two Cyclic Groups
a 11 a 21 R a n1 a 12 a22 an2 an 1 a2n ann
n
Notation :
(R)
(R) ii a
i 1
Property I: Commutative
(A B) = (B A)
S6 = {I, S61 , C3(S62 ), i (S63), C32(S64 ), S65}
Case 2: Sn Group, n is an odd number Order h = 2n
Sn = {Sn1 , Sn2 , Sn3 ,…, Snn = I}
For even powers
S C h n (n ) C n
2 k 2 k 2 k
For odd powers
S n
21 k
nh ( C)
21 k
n C h
21 k
S n n. n ICn ,, C n , h. n h C. , ,, .C , C C . . n h
Sn = {Sn1 , Sn2 , Sn3 ,…, Snn = I}, for even n
Sn = {Sn1 , Sn2 ,…, Snn = h, …, Sn2n = I }, for odd n
Generator:
Sn = C nσ h = σ h C n
Case 1: Sn Group, n is an even number
24 n 1 3 n 1

Subgroup Cn/2
When n=4k+2, there exists an inversion center
1 i ( S 42kk 2 )
S2 = {I, i (S21)} S4 = {I, S41 , C2(S42 ), S43}
Notation: Group Ci
Example
C3 , A subgroup of C3V
AB I C31 C32
I I C31 C32
C31 C31 C32 I
C32 C32 I C31
a a c b
I C32 C31
b b a c
C31 I C32
c c b a
C32 C31 I
a b c
a b c
Example II and III are finite groups, Example I is an infinite group.
The Abelian Group (阿贝尔群)
The group multiplication is commutative. AB=BA
Example I and II are Abelian Groups, Example III is Non-Abelian
n
n
n
Section 4.4 Classification of Point Group 0. Generation Elements 1. C1, Cs 2. Abelian Groups (Cn, Sn) 3. Cnv, Cnh, Cv 4. Dn, Dnh, Dnd, Dh 5. Td, Oh, and Ih
The Closure
The multiplication table of C3V
A
AB I C31 2 B C3
I I C31 C32
C31 C31 C32 I
C32 C32 I C31
a a c b
I C32 C31
b b a c
C31 I C32
c c b a
C32 C31 I
I + ( -I )=0
Example II
i 2 l
N complex numbers f (l ) e
, l = 0, 1, …, N-1 forms a GROUP if the operation is defined as “a multiplication” The closure: f(I) * f(J) = f(I+J) The identity element: f(0)=1, f(0) * f(I) = f(I) * 0 =f(I) The associate element: f(I)*f(J)*f(K)=(f(I)*f(J))*f(K) =f(I)*(f(J)*f(K))=f(I+J+K) The inverse element: the inverse element of f(I )is f(N-I) f(I) * f((N-I )=f(N)=f(0)
Cs Group {I, }
Generator:
Cn Group {I, Cn1, Cn2 , …, Cnn-1}
Generator: Cn
Sn Group, Another Cyclic Group
Sn Group, the group generated by the
improper-rotation axis.
N
Example III
All of the symmetry operations in NH3 molecule span a GROUP (Point group C3V) {I, C31, C32, a, b, c}
Identity element: I Inverse element: I-1=I, [C31]-1 = C32 , [a]-1 = a , [b]-1 = b , [c]-1 = c
(R B ) ik bjaji R a k
1 1 i k j 1 1 1 n n n
1 (ajia ) kj ik b j k i 1 1 1
n n n
b jkj kk (B b ) k
j k 1 1 k 1