Transcript Slide 1

Lecture 10
REPRESENTATIONS OF SYMMETRY POINT GROUPS
1) Basis functions, characters and representations
•
Each symmetry operation in a group can be represented by a matrix transforming a
particular object (basis function) like the symmetry operation in the group does.
Finding matrix representation of the operations E, C2, sv, and s’v available in the C2v
symmetry point group with vectors r , x, y or z as a basis function
z
E
(x,y,z)
(x,y,z)
y
rr
x
1 0 0   x 
0 1 0   y  =  y 
 

  
0 0 1   z 
 z 
x
matrix representing
symmetry operation
z
C2(z)
(x,y,z)
(-x,-y,z)
r
r
x
The operation E has the character
c(E) = 1
with any of the vectors x, y or z
chosen as a basis function
and the character 3 with r as a BF
y
coordinates
of original
coordinates
of product
  1 0 0  x 
 x
 0  1 0  y  =   y 

  
 
0 1  z 
 0
 z 
The operation C2(z) has c(C2) = 1
if z is chosen as a basis function,
-1 if the basis function is x or y,
and the character -1 with r as
a basis function
2) Representations of the C2v symmetry point group
sxz
z
r
r
y
x
matrix representing
symmetry operation
syz
z
(-x,y,z)
(x,y,z) r
r
•
•
of original
  1 0 0  x 
 0 1 0  y 

 
 0 0 1  z 
y
x
•
The operation sxz has the character
c (sxz) = -1
=
if y is chosen as a basis function
and 1 if the basis function is x or z
and the character 1 with r as a BF
coordinates coordinates
1 0 0   x 
0  1 0  y 

 
0 0 1  z 
(x,y,z)
(x,-y,z)
 x 
 y 
 
 z 
of product
The operation syz has c (syz) = -1
if x is chosen as a basis function
and 1 if the basis function is y or z
and the character 1 with r as a BF
 x 
 
=  y 
 z 
Virtually any function can be used as a basis function to build a representation of any
particular symmetry operation available in a given symmetry point group.
x, y and z in the examples above are the most commonly used basis functions.
Rotations about a particular axis, either x, or y or z, are other commonly used basis
functions. Their symbols are respectively Rx, Ry and Rz.
Rz
Rz
y
y
x
c(E)=1
x
c(C2)=1
-Rz
-Rz
y
y
x
c(sxz)=-1
x
c(syz)=-1
3) Character tables
The results can be summarized in the form of the following character table:
sxz
syz
1
1
1
G4
1
1
1
1
-1
-1
-1 -1
1 -1
-1 1
Rz
x
y
G5
3
-1
1
r
C2v
E
G1
1
G2
G3
C2
1
z
•
Sets of characters of all available symmetry operations, which correspond to one and the
same basis function are combined into representations Gn (rows).
•
In character tables symmetry operations are combined into classes (column headings).
For any two operations A and B inside one class the following is true: A=X-1BX, where X is
another operation available in the group. A and B are called conjugate.
E, i, C2 or sh is always in a class by itself. sv’s (sd’s) may be in several different classes.
In some symmetry point groups (Cnv, D etc.) operations Cnm and Cnn-m (Snm and Snn-m if
available) are in one class.
In other symmetry groups (Cn, Sn etc.) operations Cnm and Cnn-m (Snm and Snn-m) are not
identical and are in two separate classes.
•
•
•
•
Reducible representations (G5 above) can be decomposed into a combination of
irreducible ones (G1 –G4 above): G5 = G1 + G3 + G4 for the table above.
4) Some important relationships
•
The number of the irreducible representations in a group is equal to the
number of the classes constituting it (four in the case of C2v group). This is
simply the number of columns in a character table.
•
The order (dimension) of a symmetry group, h, is equal to the number of
symmetry operations X constituting the group (four in the case of C2v group).
To find it, add numbers of symmetry operations in all classes.
•
For any irreducible representation the sum of the squares of the characters
c(X) is equal to the group order h:
c2(X )  h

X
•
The sum of the squares of the orders (dimensions) of all irreducible
representations constituting a group is equal to h.
c 2 (E)  h

i
•
Irreducible representations of a group are orthogonal to each other:
 c ( X )c
i
X
j
(X )  0