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.