to contain D
(α) belonging to a totally symmetric representation. Note that in group
theory we usually write D
(γ)
 D
(β) instead of D
(γ)
D
(β)
; see (18.191).
If ϕ
β
ð Þ
l belongs to a reducible representation, from (18.80) we have
D
β
ð Þ
¼
X
ω
q ω D
ω
ð Þ ,
where D
(ω) belongs to an irreducible representation ω. Then, we get
D
γ
ð Þ
 D
β
ð Þ
¼
X
ω
q ω D
γ
ð Þ
 D
ω
ð Þ
:
Thus, applying (18.199) and (18.200), we can obtain a direct sum of irreducible
representations. After that, we examine whether an irreducible representation α is
contained in D
(γ)
 D
(β) . Since the totally symmetric representation is
one-dimensional, s ¼ 1 in (19.51).
19.3 Calculation Procedures of Molecular Orbitals (MOs)
We describe a brief outline of the MO method based on LCAO (LCAOMO).
Suppose that a molecule consists of n atomic orbitals and that each MO ψ i (1 i n)
comprises a linear combination of those n atomic orbitals ϕ k (1 k n). That is, we
have
ψ i ¼
X n
k¼1
c ki ϕ k 1 i n
ð
Þ ,
ð19:52Þ
where c ki are complex coefficients. We assume that ϕ k are normalized. That is,
ϕ k jϕ k
h
i
ð
ϕ
Ã
k ϕ k dτ ¼ 1,
ð19:53Þ
where dτ implies that an integration should be taken over ℝ
3 . The notation hg| fi
means an inner product defined in Sect. 13.1. The inner product is usually defined by
a definite integral of g
à f whose integration range covers a part or all of ℝ
3 depending
on a constitution of a physical system.
First we try to solve Schrödinger equation given as an eigenvalue equation. The
said equation is described as
Hψ ¼ λψ or H À λψ
ð
Þ¼0:
ð19:54Þ
Replacing ψ with (19.52), we have
742
19 Applications of Group Theory to Physical Chemistry
(α) belonging to a totally symmetric representation. Note that in group
theory we usually write D
(γ)
 D
(β) instead of D
(γ)
D
(β)
; see (18.191).
If ϕ
β
ð Þ
l belongs to a reducible representation, from (18.80) we have
D
β
ð Þ
¼
X
ω
q ω D
ω
ð Þ ,
where D
(ω) belongs to an irreducible representation ω. Then, we get
D
γ
ð Þ
 D
β
ð Þ
¼
X
ω
q ω D
γ
ð Þ
 D
ω
ð Þ
:
Thus, applying (18.199) and (18.200), we can obtain a direct sum of irreducible
representations. After that, we examine whether an irreducible representation α is
contained in D
(γ)
 D
(β) . Since the totally symmetric representation is
one-dimensional, s ¼ 1 in (19.51).
19.3 Calculation Procedures of Molecular Orbitals (MOs)
We describe a brief outline of the MO method based on LCAO (LCAOMO).
Suppose that a molecule consists of n atomic orbitals and that each MO ψ i (1 i n)
comprises a linear combination of those n atomic orbitals ϕ k (1 k n). That is, we
have
ψ i ¼
X n
k¼1
c ki ϕ k 1 i n
ð
Þ ,
ð19:52Þ
where c ki are complex coefficients. We assume that ϕ k are normalized. That is,
ϕ k jϕ k
h
i
ð
ϕ
Ã
k ϕ k dτ ¼ 1,
ð19:53Þ
where dτ implies that an integration should be taken over ℝ
3 . The notation hg| fi
means an inner product defined in Sect. 13.1. The inner product is usually defined by
a definite integral of g
à f whose integration range covers a part or all of ℝ
3 depending
on a constitution of a physical system.
First we try to solve Schrödinger equation given as an eigenvalue equation. The
said equation is described as
Hψ ¼ λψ or H À λψ
ð
Þ¼0:
ð19:54Þ
Replacing ψ with (19.52), we have
742
19 Applications of Group Theory to Physical Chemistry