character table for T d , we find that ε e Á P belongs to the irreducible representation T 2
(see Table 19.12).
The ground-state electronic configuration is A 1 (totally symmetric). It is denoted
by
a
2
1 t
2
2 t
0 2
2 t
00 2
2 ,
where three T 2 states are distinguished by a prime and double prime. For possible
configuration of excited states, we have.
a
2
1 t 2 t
0 2
2 t
00 2
2 t
Ã
2 A 1 ! T 2 Â T 2
ð
Þ , a
2
1 t 2 t
0 2
2 t
00 2
2 a
Ã
1 A 1 ! T 2 Â A 1 ¼ T 2
ð
Þ ,
a 1 t
2
2 t
0 2
2 t
00 2
2 t
Ã
2 A 1 ! A 1 Â T 2 ¼ T 2
ð
Þ , a 1 t
2
2 t
0 2
2 t
00 2
2 a
Ã
1 A 1 ! A 1 Â A 1 ¼ A 1
ð
Þ :
ð19:236Þ
In (19.236), we have optical excitations of t 2 ! t
Ã
2 , t 2 ! a
Ã
1 , a 1 ! t
Ã
2 , and a 1 ! a
Ã
1 ,
respectively. Consequently, the excited states are denoted by T 2 Â T 2 , T 2 , T 2 , and A 1 ,
respectively. Since ε e Á P is Hermitian as seen in (19.51) of Sect. 19.2, we have
Θ
αÂβ
ð
Þ
jε e Á PjΘ
A 1
ð Þ
D
E
¼ Θ
A 1
ð Þ
jε e Á PjΘ
αÂβ
ð
Þ
D
E Ã :
ð19:237Þ
Therefore, according to the general theory of Sect. 19.2, we need to examine whether
T 2 Â D
(α)
 D
(β) contains A 1 to judge whether the optical transition is allowed. As
mentioned above, D
(α)
 D
(β) is chosen from among T 2 Â T 2 , T 2 , T 2 , and A 1 . The
results are given as below:
T 2 Â T 2 Â T 2 ¼ 3T 1 þ 4T 2 þ 2E þ A 2 þ A 1 ,
ð19:238Þ
T 2 Â T 2 ¼ T 1 þ T 2 þ E þ A 1 ,
ð19:239Þ
T 2 Â A 1 ¼ T 2 :
ð19:240Þ
Admittedly, (19.238) and (19.239) contain A 1 , and so the transition is allowed. As
for (19.240), however, the transition is forbidden because it does not contain A 1 . In
light of the character table of T d (Table 19.12), we find that the allowed transitions
(i.e., t 2 ! t
Ã
2 , t 2 ! a
Ã
1 , and a 1 ! t
Ã
2 ) equally take place in the direction polarized
along the x-, y-, z-axes. This is often the case with molecules having higher
symmetries such as methane. The transition a 1 ! a
Ã
1 , on the other hand, is
forbidden.
In the above argument including the energetic consideration, we could not tell
magnitude relationship of MO energies or photon energies associated with the
optical transition. Once again, this requires more accurate calculations and experiments. Yet, the discussion we have developed gives us strong guiding principles in
the investigation of molecular science.
19.5 MO Calculations of Methane
797
(see Table 19.12).
The ground-state electronic configuration is A 1 (totally symmetric). It is denoted
by
a
2
1 t
2
2 t
0 2
2 t
00 2
2 ,
where three T 2 states are distinguished by a prime and double prime. For possible
configuration of excited states, we have.
a
2
1 t 2 t
0 2
2 t
00 2
2 t
Ã
2 A 1 ! T 2 Â T 2
ð
Þ , a
2
1 t 2 t
0 2
2 t
00 2
2 a
Ã
1 A 1 ! T 2 Â A 1 ¼ T 2
ð
Þ ,
a 1 t
2
2 t
0 2
2 t
00 2
2 t
Ã
2 A 1 ! A 1 Â T 2 ¼ T 2
ð
Þ , a 1 t
2
2 t
0 2
2 t
00 2
2 a
Ã
1 A 1 ! A 1 Â A 1 ¼ A 1
ð
Þ :
ð19:236Þ
In (19.236), we have optical excitations of t 2 ! t
Ã
2 , t 2 ! a
Ã
1 , a 1 ! t
Ã
2 , and a 1 ! a
Ã
1 ,
respectively. Consequently, the excited states are denoted by T 2 Â T 2 , T 2 , T 2 , and A 1 ,
respectively. Since ε e Á P is Hermitian as seen in (19.51) of Sect. 19.2, we have
Θ
αÂβ
ð
Þ
jε e Á PjΘ
A 1
ð Þ
D
E
¼ Θ
A 1
ð Þ
jε e Á PjΘ
αÂβ
ð
Þ
D
E Ã :
ð19:237Þ
Therefore, according to the general theory of Sect. 19.2, we need to examine whether
T 2 Â D
(α)
 D
(β) contains A 1 to judge whether the optical transition is allowed. As
mentioned above, D
(α)
 D
(β) is chosen from among T 2 Â T 2 , T 2 , T 2 , and A 1 . The
results are given as below:
T 2 Â T 2 Â T 2 ¼ 3T 1 þ 4T 2 þ 2E þ A 2 þ A 1 ,
ð19:238Þ
T 2 Â T 2 ¼ T 1 þ T 2 þ E þ A 1 ,
ð19:239Þ
T 2 Â A 1 ¼ T 2 :
ð19:240Þ
Admittedly, (19.238) and (19.239) contain A 1 , and so the transition is allowed. As
for (19.240), however, the transition is forbidden because it does not contain A 1 . In
light of the character table of T d (Table 19.12), we find that the allowed transitions
(i.e., t 2 ! t
Ã
2 , t 2 ! a
Ã
1 , and a 1 ! t
Ã
2 ) equally take place in the direction polarized
along the x-, y-, z-axes. This is often the case with molecules having higher
symmetries such as methane. The transition a 1 ! a
Ã
1 , on the other hand, is
forbidden.
In the above argument including the energetic consideration, we could not tell
magnitude relationship of MO energies or photon energies associated with the
optical transition. Once again, this requires more accurate calculations and experiments. Yet, the discussion we have developed gives us strong guiding principles in
the investigation of molecular science.
19.5 MO Calculations of Methane
797