58
4 Representations
Table 4.1 Character table for
the group C 3v and reducible
characters of the hydrogen 1s
functions χ(1s) and
hydrogen bends χ((φ).The
χ(1s) row is equal to the sum
of the A 1 and E rows, and the
χ((φ) row is equal to the
sum of the A 2 and E rows
C 3v
ˆ
E
2 ˆ
C 3
3 ˆ
σ v
χ |χ
A 1
11
1
6
A 2
11
−16
E
2
−106
χ(1s)
30
11 2
χ((φ)
30
−11 2
concept comes in very useful. Indeed, since all elements belonging to the same class
are similarity transforms, their representation matrices are unitary transforms and,
hence, all have the same character. We can thus group elements together in classes.
In Table 4.1 we show the character table for C 3v as can be obtained by algebraic
techniques such as the one we used in the previous section. We recognize at once
the characters for the totally symmetric A 1 and the twofold-degenerate E irrep. In
addition, there is another one-dimensional irrep, A 2 , which is symmetric under the
threefold axis and antisymmetric under the reflection planes.
Let us denote an irrep as Γ i and the string of characters, arranged in a row over
the full group, in a Dirac form as |χ Γ i . The norm of this string 2 will then be denoted
as a bracket, i.e.,
χ
Γ i |χ
Γ i
=
R∈G
¯
χ
Γ i (R)χ
Γ i (R)
(4.28)
Since the matrices are unitary, we could also replace the complex-conjugate character by the character of the inverse element:
¯
χ
Γ i (R) = χ
Γ i
R
−1
(4.29)
The character strings obey the following character theorem:
Theorem 4 The norm of the character string is equal to the order of the group if
and only if the characters refer to an irreducible representation. The scalar product
of two character strings of different irreps is equal to zero.
This theorem can be expressed as follows:
χ
Γ i |χ
Γ j
=
R∈G
¯
χ
Γ i (R)χ
Γ j (R) = δ ij |G|
(4.30)
where Γ i and Γ j refer to irreducible representations, and |G| is the order of the
group. This theorem provides an elegant and simple solution for determining the ir2 Since elements in the same class have the same character, we can also simplify the expression to
a summation over all classes, provided that we then multiply each term by the number of elements
in the class under consideration.
4 Representations
Table 4.1 Character table for
the group C 3v and reducible
characters of the hydrogen 1s
functions χ(1s) and
hydrogen bends χ((φ).The
χ(1s) row is equal to the sum
of the A 1 and E rows, and the
χ((φ) row is equal to the
sum of the A 2 and E rows
C 3v
ˆ
E
2 ˆ
C 3
3 ˆ
σ v
χ |χ
A 1
11
1
6
A 2
11
−16
E
2
−106
χ(1s)
30
11 2
χ((φ)
30
−11 2
concept comes in very useful. Indeed, since all elements belonging to the same class
are similarity transforms, their representation matrices are unitary transforms and,
hence, all have the same character. We can thus group elements together in classes.
In Table 4.1 we show the character table for C 3v as can be obtained by algebraic
techniques such as the one we used in the previous section. We recognize at once
the characters for the totally symmetric A 1 and the twofold-degenerate E irrep. In
addition, there is another one-dimensional irrep, A 2 , which is symmetric under the
threefold axis and antisymmetric under the reflection planes.
Let us denote an irrep as Γ i and the string of characters, arranged in a row over
the full group, in a Dirac form as |χ Γ i . The norm of this string 2 will then be denoted
as a bracket, i.e.,
χ
Γ i |χ
Γ i
=
R∈G
¯
χ
Γ i (R)χ
Γ i (R)
(4.28)
Since the matrices are unitary, we could also replace the complex-conjugate character by the character of the inverse element:
¯
χ
Γ i (R) = χ
Γ i
R
−1
(4.29)
The character strings obey the following character theorem:
Theorem 4 The norm of the character string is equal to the order of the group if
and only if the characters refer to an irreducible representation. The scalar product
of two character strings of different irreps is equal to zero.
This theorem can be expressed as follows:
χ
Γ i |χ
Γ j
=
R∈G
¯
χ
Γ i (R)χ
Γ j (R) = δ ij |G|
(4.30)
where Γ i and Γ j refer to irreducible representations, and |G| is the order of the
group. This theorem provides an elegant and simple solution for determining the ir2 Since elements in the same class have the same character, we can also simplify the expression to
a summation over all classes, provided that we then multiply each term by the number of elements
in the class under consideration.