A projection operator is used to construct a linear combination of a set of basis functions that spans an irreducible representation of a point group .

Using eq55, we can write

where is the -th symmetry operation of ,

is a set of basis functions of the -th irreducible representation of , and is the -th row and -column matrix entry of .

Multiplying eq90 by and sum over ,

Substituting eq20a in the above equation,

where .

is a linear combination of the symmetry operators with coefficients that are entries of the matrix representations of . If and ,

We call the projection operator, which generates a basis of the irreducible representation from another basis of . The significance of this is that if we know one member of a set of basis functions of an irreducible representation, then we can project all the other members of the set.

###### Question

Use eq93 and the general function to show that the basis functions and belong to the degenerate irreducible representation of the point group .

###### Answer

For ,

Substituting the values of this table into the above equation and simplifying, we have . If we repeat the procedure for , we have . Both projections eliminate . It is obvious by inspecting the matrix entries of the matrices of that and also eliminate . Since is not projected out of , it does not belong to . This implies that transforms according to . In other words, ** any linear combination of a set of functions that transforms according to an irreducible representation of a point group is also a basis of the irreducible representation**.

We can also define another projection operator, , where and is the dimension of . This projection operator employs the characters of an irreducible representation instead of matrix entries of every matrix representation, i.e.

In contrast with (c.f. eq93), which projects or from for , projects out the same basis function from (easily shown using eq94). Since any linear combination of a set of basis functions of an irreducible representation of a point group is also a function that transforms according to the irreducible representation, we can express eq96 and eq97 as

where .

Finally, let’s examine the effect of a projection operator on a set of linearly independent basis functions of an -dimensional reducible representation of a point group . Since is a basis for , we have or equivalently,

where .

Let’s analyse the example for , where :

can undergo a similarity transformation to , which has the block-diagonal form of:

where each block is an irreducible representation.

It is obvious that is a direct sum of irreducible representations of . The direct sum can also be obtained using eq27a. Let’s denote the basis functions of the transformed reducible representation by

where refers to the dimension of the -th irreducible representation of .

Therefore, we have

Since a similarity transformation involves a change of basis, the old basis functions can be expressed as a linear combination of the new basis functions :

where , i.e. also transforms according to the -th irreducible representation of .

Applying the projection operator to eq102 and using eq98 and eq99,

With reference to the LHS of eq103, results in a linear combination of because is itself a linear combination of symmetry operators of (c.f. eq94). This linear combination of , according to the RHS of eq103, is equal to a function that transforms according to an irreducible representation of . Such a consequence is used to construct symmetry-adapted linear combination (SALC) of orbitals, which shall be discussed in the next article.