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, C 2,  v, and  ’ v available in the C 2v symmetry point group with vectors r, x, y or z as a basis function The operation E has the character  (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 C 2 (z) has  (C 2 ) = 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 = = coordinates of original coordinates of product matrix representing symmetry operation

2) Representations of the C 2v symmetry point group The operation  xz has the character  (  xz ) = -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  yz has  (  yz ) = -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 R x, R y and R z. = = coordinates of original matrix representing symmetry operation coordinates of 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 representations  n (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 -1 BX, where X is another operation available in the group. A and B are called conjugate. E, i, C 2 or  h is always in a class by itself.  v ’s (  d ’s) may be in several different classes. In some symmetry point groups (C nv, D etc.) operations C n m and C n n-m (S n m and S n n-m if available) are in one class. In other symmetry groups (C n, S n etc.) operations C n m and C n n-m (S n m and S n n-m ) are not identical and are in two separate classes. Reducible representations (  5 above) can be decomposed into a combination of irreducible ones (  1 –  4 above):  5 =  1 +  3 +  4 for the table above. C 2v EC2C2  xz  yz 11 22 33 44 55

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 C 2v 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 C 2v group). To find it, add numbers of symmetry operations in all classes. For any irreducible representation the sum of the squares of the characters  (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: