ϕ
β
ð Þ
s jψ
α
ð Þ
l
D
E
¼ δ αβ δ ls ϕ
β
ð Þ
s jψ
α
ð Þ
l
D
E
:
ð18:171Þ
At the first glance this equation seems trivial, but the equation gives us a
powerful tool to save us a lot of troublesome calculations. In fact, if we
encounter a series of inner product calculations (i.e., definite integrals), we
ignore many of them. It is because matrix elements of Hamiltonian do not
vanish only if α ¼ β and l ¼ s. That is, we only have to estimate ϕ
α
ð Þ
s jψ
α
ð Þ
s
.
Otherwise the inner products vanish. The functional forms of ϕ
α
ð Þ
s and ψ
α
ð Þ
s are
determined depending upon individual practical problems. We will show tangible examples later (Sect. 19.4).
(ii) Next, let us consider matrix elements of the optical transition. In this case, we
are thinking of transition probability between quantum states. Assuming the
dipole approximation, we use ε e Á P
(γ) for O
(γ) in ϕ
β
ð Þ
l jO
γ
ð Þ
ψ
α
ð Þ
s
D
E
of (19.48). The
quantity P
(γ) represents an electric dipole moment associated with the position
vector. Normally, we can readily find it in a character table, in which we
examine which irreducible representation γ the position vector components
x, y, and z indicated in a rightmost column correspond to. Table 18.4 is an
example. If we take a unit polarization vector ε e in parallel to the position vector
component that the character table designates, ε e Á P
(γ) is nonvanishing. Suppose
that ψ
α
ð Þ
s
and ϕ
β
ð Þ
l
are an initial state and final state, respectively. At the first
glance, we would wish to use (18.200) and count how many times a representation β occurs for a direct-product representation D
(γ Â α) . It is often the case,
however, where β is not an irreducible representation.
Even in such a case, if either α or β belongs to a totally symmetric representation,
the handling will be easier. This situation corresponds to that an initial electronic
configuration or a final electronic configuration forms a closed shell an electronic
state of which is totally symmetric [1]. The former occurs when we consider the
optical absorption that takes place, e.g., in a molecule of a ground electronic state.
The latter corresponds to an optical emission that ends up with a ground state. Let us
consider the former case. Since O
γ
ð Þ
j is Hermitian, we rewrite (19.48) as
ϕ
β
ð Þ
l jO
γ
ð Þ
j ψ
α
ð Þ
s
D
E
¼ O
γ
ð Þ
j ϕ
β
ð Þ
l jψ
α
ð Þ
s
D
E
¼ ψ
α
ð Þ
s jO
γ
ð Þ
j ϕ
β
ð Þ
l
D
E Ã
,
ð19:51Þ
where we assume that ψ
α
ð Þ
s
is a ground state having a closed shell electronic
configuration. Therefore, ψ
α
ð Þ
s
belongs to a totally symmetric irreducible representation. For (19.48) not to vanish, therefore, we may alternatively state that it is
necessary for
D
γÂβ
ð
Þ
¼ D
γ
ð Þ
D
β
ð Þ
19.2 Method of Molecular Orbitals (MOs)
741
β
ð Þ
s jψ
α
ð Þ
l
D
E
¼ δ αβ δ ls ϕ
β
ð Þ
s jψ
α
ð Þ
l
D
E
:
ð18:171Þ
At the first glance this equation seems trivial, but the equation gives us a
powerful tool to save us a lot of troublesome calculations. In fact, if we
encounter a series of inner product calculations (i.e., definite integrals), we
ignore many of them. It is because matrix elements of Hamiltonian do not
vanish only if α ¼ β and l ¼ s. That is, we only have to estimate ϕ
α
ð Þ
s jψ
α
ð Þ
s
.
Otherwise the inner products vanish. The functional forms of ϕ
α
ð Þ
s and ψ
α
ð Þ
s are
determined depending upon individual practical problems. We will show tangible examples later (Sect. 19.4).
(ii) Next, let us consider matrix elements of the optical transition. In this case, we
are thinking of transition probability between quantum states. Assuming the
dipole approximation, we use ε e Á P
(γ) for O
(γ) in ϕ
β
ð Þ
l jO
γ
ð Þ
ψ
α
ð Þ
s
D
E
of (19.48). The
quantity P
(γ) represents an electric dipole moment associated with the position
vector. Normally, we can readily find it in a character table, in which we
examine which irreducible representation γ the position vector components
x, y, and z indicated in a rightmost column correspond to. Table 18.4 is an
example. If we take a unit polarization vector ε e in parallel to the position vector
component that the character table designates, ε e Á P
(γ) is nonvanishing. Suppose
that ψ
α
ð Þ
s
and ϕ
β
ð Þ
l
are an initial state and final state, respectively. At the first
glance, we would wish to use (18.200) and count how many times a representation β occurs for a direct-product representation D
(γ Â α) . It is often the case,
however, where β is not an irreducible representation.
Even in such a case, if either α or β belongs to a totally symmetric representation,
the handling will be easier. This situation corresponds to that an initial electronic
configuration or a final electronic configuration forms a closed shell an electronic
state of which is totally symmetric [1]. The former occurs when we consider the
optical absorption that takes place, e.g., in a molecule of a ground electronic state.
The latter corresponds to an optical emission that ends up with a ground state. Let us
consider the former case. Since O
γ
ð Þ
j is Hermitian, we rewrite (19.48) as
ϕ
β
ð Þ
l jO
γ
ð Þ
j ψ
α
ð Þ
s
D
E
¼ O
γ
ð Þ
j ϕ
β
ð Þ
l jψ
α
ð Þ
s
D
E
¼ ψ
α
ð Þ
s jO
γ
ð Þ
j ϕ
β
ð Þ
l
D
E Ã
,
ð19:51Þ
where we assume that ψ
α
ð Þ
s
is a ground state having a closed shell electronic
configuration. Therefore, ψ
α
ð Þ
s
belongs to a totally symmetric irreducible representation. For (19.48) not to vanish, therefore, we may alternatively state that it is
necessary for
D
γÂβ
ð
Þ
¼ D
γ
ð Þ
D
β
ð Þ
19.2 Method of Molecular Orbitals (MOs)
741