Question

Show using the C4v point group character table the orthogonality of any two irreducible representations.

Using the same point group demonstrate the property of the character table that two irreducible representations can be used to deduce the characters for another (ex combinations of the irreducible representations for px and py can be used to determine the characters for the irreducible representation of dxy)

Answer 1

The derivation of C4v character table, the symeetry operation for this point group are:

there are five classes of symmetry operation derived using multiplication table

i.e five irreducible representations.

A totally symmetric represenatation denoted by a set of 1x1 matrices we shall have all symmetry operations

the sum of the squares of the dimensions of the group must be equal to the order of the group because

1²+1²+1²+1²+2² = 8 order of the group i.e one of the irreducible representation will be two dimensional. The characters of all operation in the dame class are the same in each of the irreducible representations and the sum of the squares of each row must be = 8 and the point product of any two rows must be = 0 i.e. orthogonal

this can be written as

We can now write down 4 equation that uniquely determine the remaining 4 unknown characters a, b, c, d

= 2+2a+b+2c+2d = 0

from this we find a = 0, b = -2, c = 0, d = 0

using Mulliken symbols, we can complete the character table by the naming the reducible representations