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Lecture 10REPRESENTATIONS 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’vavailable in the C2v symmetry point group with vectors r , x, y or z as a basis function

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

The operation C2(z) hasc(C2) = 1

if zis chosen as a basis function,

-1 if the basis function is x or y,

and the character -1 with r as a basis function

=

matrix representing symmetry operation

coordinates of original

coordinatesof product

=

2) Representations of the C2v symmetry point group

=

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

The operation syz hasc (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

- 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.

matrix representing symmetry operation

coordinates of original

coordinatesof product

=

3) Character tables

The results can be summarized in the form of the following character table:

- Sets of characters of all available symmetry operations, which correspond to one and the same basis function are combined into representationsGn (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 oneclass.
- 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 (G5above) can be decomposed into a combination of irreducible ones (G1 –G4above): 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 charactersc(X) is equal to the group order h:
- The sum of the squares of the orders (dimensions) of all irreducible representations constituting a group is equal to h.
- Irreducible representations of a group are orthogonal to each other:

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