(18.171). Even though those vectors belong to the same irreducible representation,
they are orthogonal if they are allocated to a different place. However, it is often the
case that the vectors belong to the same place of the same irreducible representation.
Then, it is always possible according to Theorem 13.2 to make them mutually
orthogonal by taking their linear combination. In such a way, we can construct an
orthonormal basis set throughout the representation space. In fact, the method is a
powerful tool for solving an energy eigenvalue problem and for determining associated eigenfunctions (or molecular orbitals).
18.8 Direct-Product Representation
In Sect. 16.5 we have studied basic properties of direct-product groups. Correspondingly, we examine in this section properties of direct-product representation. This
notion is very useful to investigate optical transitions in molecular systems and
selection rules relevant to those transitions.
Let D
(α) and D
(β) be two different irreducible representations whose dimensions
are d α and d β , respectively. Then, operating a group element g on the basis functions
we have
g ψ i
ð Þ ¼
X d α
k¼1
ψ k D
α
ð Þ
ki g
ð Þ 1 i d α
ð
Þ ,
ð18:185Þ
g ϕ j
À Á ¼
X d β
l¼1
ϕ l D
β
ð Þ
lj g
ð Þ 1 j d β
À
Á
,
ð18:186Þ
where ψ k (1 i d α ) and ϕ l (1 l d β ) are basis functions of D
(α) and D
(β) ,
respectively. We can construct d α d β new basis vectors using ψ k ϕ l . These functions
are transformed according to g such that
g ψ i ϕ j
À
Á ¼ g ψ i
ð Þg ϕ j
À Á ¼
X
k
ψ k D
α
ð Þ
ki g
ð Þ
h
i X
l
ϕ l D
β
ð Þ
lj g
ð Þ
h
i
¼
X
k
X
l
ψ k ϕ l D
α
ð Þ
ki g
ð ÞD
β
ð Þ
lj g
ð Þ:
ð18:187Þ
Here let us define the following matrix [D
(α Â β) (g)] kl, ij such that
D
αÂβ
ð
Þ g
ð Þ
h
i
kl,ij
D
α
ð Þ
ki g
ð ÞD
β
ð Þ
lj g
ð Þ:
ð18:188Þ
Then we have
g ψ i ϕ j
À
Á ¼
X
k
X
l
ψ k ϕ l D
αÂβ
ð
Þ g
ð Þ
h
i
kl,ij
:
ð18:189Þ
720
18 Representation Theory of Groups
they are orthogonal if they are allocated to a different place. However, it is often the
case that the vectors belong to the same place of the same irreducible representation.
Then, it is always possible according to Theorem 13.2 to make them mutually
orthogonal by taking their linear combination. In such a way, we can construct an
orthonormal basis set throughout the representation space. In fact, the method is a
powerful tool for solving an energy eigenvalue problem and for determining associated eigenfunctions (or molecular orbitals).
18.8 Direct-Product Representation
In Sect. 16.5 we have studied basic properties of direct-product groups. Correspondingly, we examine in this section properties of direct-product representation. This
notion is very useful to investigate optical transitions in molecular systems and
selection rules relevant to those transitions.
Let D
(α) and D
(β) be two different irreducible representations whose dimensions
are d α and d β , respectively. Then, operating a group element g on the basis functions
we have
g ψ i
ð Þ ¼
X d α
k¼1
ψ k D
α
ð Þ
ki g
ð Þ 1 i d α
ð
Þ ,
ð18:185Þ
g ϕ j
À Á ¼
X d β
l¼1
ϕ l D
β
ð Þ
lj g
ð Þ 1 j d β
À
Á
,
ð18:186Þ
where ψ k (1 i d α ) and ϕ l (1 l d β ) are basis functions of D
(α) and D
(β) ,
respectively. We can construct d α d β new basis vectors using ψ k ϕ l . These functions
are transformed according to g such that
g ψ i ϕ j
À
Á ¼ g ψ i
ð Þg ϕ j
À Á ¼
X
k
ψ k D
α
ð Þ
ki g
ð Þ
h
i X
l
ϕ l D
β
ð Þ
lj g
ð Þ
h
i
¼
X
k
X
l
ψ k ϕ l D
α
ð Þ
ki g
ð ÞD
β
ð Þ
lj g
ð Þ:
ð18:187Þ
Here let us define the following matrix [D
(α Â β) (g)] kl, ij such that
D
αÂβ
ð
Þ g
ð Þ
h
i
kl,ij
D
α
ð Þ
ki g
ð ÞD
β
ð Þ
lj g
ð Þ:
ð18:188Þ
Then we have
g ψ i ϕ j
À
Á ¼
X
k
X
l
ψ k ϕ l D
αÂβ
ð
Þ g
ð Þ
h
i
kl,ij
:
ð18:189Þ
720
18 Representation Theory of Groups