determine the number of times that individual representations take place. Calculating
χ
B 1u
ð Þ g
ð Þχ
B 3g
ð Þ g
ð Þ for each group element, we can readily get the result. Using a
character table, we have
Γ f ¼ B 2u :
ð19:107Þ
Thus, we find that the transition is A g ⟶ B 2u , where A g is called an initial state
electronic configuration and B 2u is called a final state electronic configuration. This
transition is characterized by an electric dipole transition moment operator P. Here
we have an important physical quantity of transition matrix element. This quantity T fi
is approximated by
T fi Θ f jε e Á PjΘ i
,
ð19:108Þ
where ε e is a unit polarization vector of the electric field; Θ i and Θ f are the initial state
and final sate electronic configuration, respectively. The description (19.108) is in
parallel with (4.5). Notice that in (4.5) we dealt with a single electron system such as
a particle confined in a square-well potential, a sole one-dimensional harmonic
oscillator, and an electron in a hydrogen atom.
In the present case, however, we are dealing with a two-electron system, ethylene.
Consequently, we cannot describe Θ f by a simple wave function, but must use a
more elaborated function. Nonetheless, when we discuss the optical transition of a
molecule, it is often the case that when we study optical absorption or optical
emission we first wish to know whether such a phenomenon truly takes place. In
such a case, qualitative prediction for this is of great importance. This can be done by
judging whether the integral (19.108) vanishes. If the integral does not vanish, the
relevant transition is said to be allowed. If, on the other hand, the integral vanishes,
the transition is called forbidden. In this context, a systematic approach based on
group theory is a powerful tool for this.
Let us consider optical absorption of ethylene. With the transition matrix element
for this, we have
T fi ¼ Θ f
B 2u
ð Þ
jε e Á PjΘ i
A g
ð Þ
D
E
:
ð19:109Þ
Again, note that a closed-shell electronic configuration belongs to a totally
symmetric representation [1]. Suppose that the position vector x belongs to an
irreducible representation η. A necessary condition for (19.109) not to vanish is
that D
η
ð Þ
 D
A g
ð Þ contains the irreducible representation D
B 2u
ð Þ . In the present case,
all the representations are one-dimensional, and so we can use χ
(ω) instead of D
(ω) ,
where ω shows an arbitrary irreducible representation of D 2h . This procedure is
straightforward as seen in Sect. 19.2.
However, if the character is real (it is the case with many symmetry groups and
with the point group D 2h as well), the situation will be easier. Suppose in general that
we are examining whether a following matrix element vanishes:
19.4 MO Calculations Based on π-Electron Approximation
755
χ
B 1u
ð Þ g
ð Þχ
B 3g
ð Þ g
ð Þ for each group element, we can readily get the result. Using a
character table, we have
Γ f ¼ B 2u :
ð19:107Þ
Thus, we find that the transition is A g ⟶ B 2u , where A g is called an initial state
electronic configuration and B 2u is called a final state electronic configuration. This
transition is characterized by an electric dipole transition moment operator P. Here
we have an important physical quantity of transition matrix element. This quantity T fi
is approximated by
T fi Θ f jε e Á PjΘ i
,
ð19:108Þ
where ε e is a unit polarization vector of the electric field; Θ i and Θ f are the initial state
and final sate electronic configuration, respectively. The description (19.108) is in
parallel with (4.5). Notice that in (4.5) we dealt with a single electron system such as
a particle confined in a square-well potential, a sole one-dimensional harmonic
oscillator, and an electron in a hydrogen atom.
In the present case, however, we are dealing with a two-electron system, ethylene.
Consequently, we cannot describe Θ f by a simple wave function, but must use a
more elaborated function. Nonetheless, when we discuss the optical transition of a
molecule, it is often the case that when we study optical absorption or optical
emission we first wish to know whether such a phenomenon truly takes place. In
such a case, qualitative prediction for this is of great importance. This can be done by
judging whether the integral (19.108) vanishes. If the integral does not vanish, the
relevant transition is said to be allowed. If, on the other hand, the integral vanishes,
the transition is called forbidden. In this context, a systematic approach based on
group theory is a powerful tool for this.
Let us consider optical absorption of ethylene. With the transition matrix element
for this, we have
T fi ¼ Θ f
B 2u
ð Þ
jε e Á PjΘ i
A g
ð Þ
D
E
:
ð19:109Þ
Again, note that a closed-shell electronic configuration belongs to a totally
symmetric representation [1]. Suppose that the position vector x belongs to an
irreducible representation η. A necessary condition for (19.109) not to vanish is
that D
η
ð Þ
 D
A g
ð Þ contains the irreducible representation D
B 2u
ð Þ . In the present case,
all the representations are one-dimensional, and so we can use χ
(ω) instead of D
(ω) ,
where ω shows an arbitrary irreducible representation of D 2h . This procedure is
straightforward as seen in Sect. 19.2.
However, if the character is real (it is the case with many symmetry groups and
with the point group D 2h as well), the situation will be easier. Suppose in general that
we are examining whether a following matrix element vanishes:
19.4 MO Calculations Based on π-Electron Approximation
755