g i ψ ν
ð Þ ¼
X d
μ¼1
ψ μ D μν g i
ð Þ:
ð18:22Þ
We may well ask then how we can construct such basis vectors. In this section we
address this question. A central concept about this is a projection operator. We have
already studied the definition and basic properties of the projection operators. In this
section, we deal with them bearing in mind that we apply the group theory to
molecular science, especially to quantum chemical calculations.
In Sect. 18.6, we examined the permissible number of irreducible representations.
We have reached a conclusion that the number (n r ) of inequivalent irreducible
representations is equal to that (n c ) of conjugacy classes of the group. According
to this conclusion, we modify the above equation (18.22) such that
g ψ
α
ð Þ
i
¼
X d α
j¼1
ψ
α
ð Þ
j D
α
ð Þ
ji g
ð Þ,
ð18:145Þ
where α and d α denote the α-th irreducible representation and its dimension,
respectively; the subscript i is omitted from g i for simplicity. This naturally leads
to the next question of how the basis vectors ψ
α
ð Þ
j
are related to ψ
β
ð Þ
j
that are basis
vectors as well, but belong to a different irreducible representation β.
Suppose that we have an arbitrarily chosen function f. Then, f is assumed to
contain various components of different irreducible representations. Thus, let us
assume that f is decomposed into the component such that
f ¼
X
α
X
m
c
α
ð Þ
m ψ
α
ð Þ
m ,
ð18:146Þ
where c
α
ð Þ
m is a coefficient of the expansion and ψ
α
ð Þ
m is the m-th component of α-th
irreducible representation.
Now, let us define the following operator:
P
α
ð Þ
l m
ð Þ
d α
n
X
g
D
α
ð Þ
lm g
ð Þ
à g,
ð18:147Þ
where Σ g means that the summation should be taken over all n group elements g; d α
denotes a dimension of the representation D
(α) . Operating P
α
ð Þ
l m
ð Þ on f, we have
18.7 Projection Operators
711
ð Þ ¼
X d
μ¼1
ψ μ D μν g i
ð Þ:
ð18:22Þ
We may well ask then how we can construct such basis vectors. In this section we
address this question. A central concept about this is a projection operator. We have
already studied the definition and basic properties of the projection operators. In this
section, we deal with them bearing in mind that we apply the group theory to
molecular science, especially to quantum chemical calculations.
In Sect. 18.6, we examined the permissible number of irreducible representations.
We have reached a conclusion that the number (n r ) of inequivalent irreducible
representations is equal to that (n c ) of conjugacy classes of the group. According
to this conclusion, we modify the above equation (18.22) such that
g ψ
α
ð Þ
i
¼
X d α
j¼1
ψ
α
ð Þ
j D
α
ð Þ
ji g
ð Þ,
ð18:145Þ
where α and d α denote the α-th irreducible representation and its dimension,
respectively; the subscript i is omitted from g i for simplicity. This naturally leads
to the next question of how the basis vectors ψ
α
ð Þ
j
are related to ψ
β
ð Þ
j
that are basis
vectors as well, but belong to a different irreducible representation β.
Suppose that we have an arbitrarily chosen function f. Then, f is assumed to
contain various components of different irreducible representations. Thus, let us
assume that f is decomposed into the component such that
f ¼
X
α
X
m
c
α
ð Þ
m ψ
α
ð Þ
m ,
ð18:146Þ
where c
α
ð Þ
m is a coefficient of the expansion and ψ
α
ð Þ
m is the m-th component of α-th
irreducible representation.
Now, let us define the following operator:
P
α
ð Þ
l m
ð Þ
d α
n
X
g
D
α
ð Þ
lm g
ð Þ
à g,
ð18:147Þ
where Σ g means that the summation should be taken over all n group elements g; d α
denotes a dimension of the representation D
(α) . Operating P
α
ð Þ
l m
ð Þ on f, we have
18.7 Projection Operators
711