e
H 11 ¼
ð
ξ
Ã
1 Hξ 1 dτ ¼ ξ 1 jHξ 1
h
i¼ 6 α þ 2β þ 2β
0
þ β
00
ð
Þ :
ð19:153Þ
In (19.153), we used the same α and β defined in (19.94) as in the case of
cyclopropenyl radical. That is, α is a Coulomb integral and β is a resonance integral
between two adjacent 2p z orbital of carbon. Meanwhile, β
0 is a resonance integral
between orbitals of “meta” positions such as ϕ 1 and ϕ 3 . A quantity β
00 is a resonance
integral between orbitals of “para” positions such as ϕ 1 and ϕ 4 . It is unfamiliar to
include such kind resonance integrals of β
0 and β
00 at a simple π-electron approximation level. To ignore such resonance integrals is because of a practical purpose to
simplify the calculations. However, we have no reason to exclude them. Or rather,
the use of appropriate SALCs makes it feasible to include β
0 and β
00
.
In a similar manner, we get
e
H 22 ¼
ð
ξ
Ã
2 Hξ 2 dτ ¼ ξ 2 jHξ 2
h
i¼ 6 α À 2β þ 2β
0
À β
00
ð
Þ ,
e
H 33 ¼ e
H 44 ¼ 6 α þ β À β
0
À β
00
ð
Þ ,
e
H 55 ¼ e
H 66 ¼ 6 α À β À β
0
þ β
00
ð
Þ :
ð19:154Þ
Meanwhile, we have
e S 11 ¼ ξ 1 jξ 1
h
i¼ 6 1 þ 2S þ 2S
0
þ S
00
ð
Þ ,
e S 22 ¼ ξ 2 jξ 2
h
i¼ 6 1 À 2S þ 2S
0
À S
00
ð
Þ ,
e S 33 ¼ e S 44 ¼ 6 1 þ S À S
0
À S
00
ð
Þ ,
e S 55 ¼ e S 66 ¼ 6 1 À S À S
0
þ S
00
ð
Þ ,
ð19:155Þ
where S, S
0 , and S
00 are overlap integrals between the ortho, meta, and para positions,
respectively. Substituting (19.153) through (19.155) for (19.152), the energy eigenvalues are readily obtained as
Table 19.9 Character table of C 6
C 6
E
C 6
C 3
C 2
C
2
3
C
5
6
ε ¼ exp (iπ/3)
A
1
1
1
1
1
1
z; x
2 + y
2
, z
2
B
1
À1
1
À1
1
À1
E 1
1
1
&
ε
ε
Ã
Àε
Ã
Àε
À1
À1
Àε
Àε
Ã
ε
Ã
ε
'
(x, y); (yz, zx)
E 2
1
1
&
Àε
Ã
Àε
Àε
Àε
Ã
1
1
Àε
Ã
Àε
Àε
Àε
Ã
'
(x
2 À y
2
, xy)
19.4 MO Calculations Based on π-Electron Approximation
767
H 11 ¼
ð
ξ
Ã
1 Hξ 1 dτ ¼ ξ 1 jHξ 1
h
i¼ 6 α þ 2β þ 2β
0
þ β
00
ð
Þ :
ð19:153Þ
In (19.153), we used the same α and β defined in (19.94) as in the case of
cyclopropenyl radical. That is, α is a Coulomb integral and β is a resonance integral
between two adjacent 2p z orbital of carbon. Meanwhile, β
0 is a resonance integral
between orbitals of “meta” positions such as ϕ 1 and ϕ 3 . A quantity β
00 is a resonance
integral between orbitals of “para” positions such as ϕ 1 and ϕ 4 . It is unfamiliar to
include such kind resonance integrals of β
0 and β
00 at a simple π-electron approximation level. To ignore such resonance integrals is because of a practical purpose to
simplify the calculations. However, we have no reason to exclude them. Or rather,
the use of appropriate SALCs makes it feasible to include β
0 and β
00
.
In a similar manner, we get
e
H 22 ¼
ð
ξ
Ã
2 Hξ 2 dτ ¼ ξ 2 jHξ 2
h
i¼ 6 α À 2β þ 2β
0
À β
00
ð
Þ ,
e
H 33 ¼ e
H 44 ¼ 6 α þ β À β
0
À β
00
ð
Þ ,
e
H 55 ¼ e
H 66 ¼ 6 α À β À β
0
þ β
00
ð
Þ :
ð19:154Þ
Meanwhile, we have
e S 11 ¼ ξ 1 jξ 1
h
i¼ 6 1 þ 2S þ 2S
0
þ S
00
ð
Þ ,
e S 22 ¼ ξ 2 jξ 2
h
i¼ 6 1 À 2S þ 2S
0
À S
00
ð
Þ ,
e S 33 ¼ e S 44 ¼ 6 1 þ S À S
0
À S
00
ð
Þ ,
e S 55 ¼ e S 66 ¼ 6 1 À S À S
0
þ S
00
ð
Þ ,
ð19:155Þ
where S, S
0 , and S
00 are overlap integrals between the ortho, meta, and para positions,
respectively. Substituting (19.153) through (19.155) for (19.152), the energy eigenvalues are readily obtained as
Table 19.9 Character table of C 6
C 6
E
C 6
C 3
C 2
C
2
3
C
5
6
ε ¼ exp (iπ/3)
A
1
1
1
1
1
1
z; x
2 + y
2
, z
2
B
1
À1
1
À1
1
À1
E 1
1
1
&
ε
ε
Ã
Àε
Ã
Àε
À1
À1
Àε
Àε
Ã
ε
Ã
ε
'
(x, y); (yz, zx)
E 2
1
1
&
Àε
Ã
Àε
Àε
Àε
Ã
1
1
Àε
Ã
Àε
Àε
Àε
Ã
'
(x
2 À y
2
, xy)
19.4 MO Calculations Based on π-Electron Approximation
767