4.2 Character Theorems
59
Fig. 4.2 Symmetry
coordinates for the in-plane
bending of the hydrogen
atoms in ammonia. The
length of the arrows is
proportional to the SALC
coefficients
rep content of a function space. One first determines the characters of the representation matrices, χ(R). They will be equal to the sums of the traces of the individual
irreducible symmetry blocks. This can be expressed as follows:
χ(R) =
i
c i χ
Γ i (R)
(4.31)
Here, the important quantities are the c i coefficients. These are integers that tell
how many times a given irrep Γ i is contained in the function space. They can easily
be calculated by using the character theorems. All one has to do is to evaluate the
scalar product of a given irreducible character, say Γ k , and the reducible character,
and then divide by the group order.
1
|G|
χ
Γ k |χ
=
1
|G|
i
c i
χ
Γ k |χ
Γ i
=
i
c i δ ki = c k
(4.32)
It is also clear that the norm of the reducible character is a multiple of the group
order, since
χ|χ=
ij
c i c j
χ
Γ i |χ
Γ j
=
i
c
2
i |G|
(4.33)
This procedure may seem quite complicated, but it is in fact very simple. Let us
demonstrate this for the previous example, the set of the three 1s-orbitals on the
hydrogens in NH 3 . We first determine the reducible character of this set (see χ(1s)
in Table 4.1). We do not have to do this for all six elements of C 3v but only for one
representative of each class since conjugate elements have the same characters. For
the unit element, the representation matrix is of course the 3 × 3 unit matrix, and its
trace is equal to three, the dimension of the set. For the other elements, we do not
need to know the full matrix representation; indeed, we need only the elements on
the diagonal. Now a diagonal entry in a representation matrix can differ from zero
only if a component function is turned into itself, or at least into a fraction of itself.
59
Fig. 4.2 Symmetry
coordinates for the in-plane
bending of the hydrogen
atoms in ammonia. The
length of the arrows is
proportional to the SALC
coefficients
rep content of a function space. One first determines the characters of the representation matrices, χ(R). They will be equal to the sums of the traces of the individual
irreducible symmetry blocks. This can be expressed as follows:
χ(R) =
i
c i χ
Γ i (R)
(4.31)
Here, the important quantities are the c i coefficients. These are integers that tell
how many times a given irrep Γ i is contained in the function space. They can easily
be calculated by using the character theorems. All one has to do is to evaluate the
scalar product of a given irreducible character, say Γ k , and the reducible character,
and then divide by the group order.
1
|G|
χ
Γ k |χ
=
1
|G|
i
c i
χ
Γ k |χ
Γ i
=
i
c i δ ki = c k
(4.32)
It is also clear that the norm of the reducible character is a multiple of the group
order, since
χ|χ=
ij
c i c j
χ
Γ i |χ
Γ j
=
i
c
2
i |G|
(4.33)
This procedure may seem quite complicated, but it is in fact very simple. Let us
demonstrate this for the previous example, the set of the three 1s-orbitals on the
hydrogens in NH 3 . We first determine the reducible character of this set (see χ(1s)
in Table 4.1). We do not have to do this for all six elements of C 3v but only for one
representative of each class since conjugate elements have the same characters. For
the unit element, the representation matrix is of course the 3 × 3 unit matrix, and its
trace is equal to three, the dimension of the set. For the other elements, we do not
need to know the full matrix representation; indeed, we need only the elements on
the diagonal. Now a diagonal entry in a representation matrix can differ from zero
only if a component function is turned into itself, or at least into a fraction of itself.