Ψ 2 ¼
j ξ 2 i
j ξ 2
j j j
¼
ϕ 1 þ εϕ 2 þ ε
Ã
ϕ 3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 1 À S
ð
Þ
p
:
ð19:139Þ
Also, with a MO corresponding to λ 3 (¼λ 2 ), we have
Ψ 3 ¼
j ξ 3 i
j ξ 3
j j j
¼
ϕ 1 þ ε
Ã
ϕ 2 þ εϕ 3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 1 À S
ð
Þ
p
:
ð19:140Þ
Equations (19.139) and (19.140) include complex numbers, and so it is inconvenient to computer analysis. In that case, we can convert it to real numbers. In Part III,
we examined properties of unitary transformations. Since the unitary transformation
keeps a norm of a vector unchanged, this is suited to our present purpose. This can be
done using a following unitary matrix U:
U ¼
1
ffiffi ffi
2
p
À
i ffiffi ffi
2
p
1
ffiffi ffi
2
p
i ffiffi ffi
2
p
0
B
B
@
1
C
C
A :
ð19:141Þ
Then we have
Ψ 2 Ψ 3
ð
ÞU ¼
1
ffiffi ffi
2
p Ψ 2 þ Ψ 3
ð
Þ
i ffiffi ffi
2
p ÀΨ 2 þ Ψ 3
ð
Þ
:
ð19:142Þ
Thus, defining f
Ψ 2 and f
Ψ 3 as
f
Ψ 2 ¼
1
ffiffi ffi
2
p Ψ 2 þ Ψ 3
ð
Þ and f
Ψ 3 ¼
i ffiffi ffi
2
p ÀΨ 2 þ Ψ 3
ð
Þ ,
ð19:143Þ
we get
f
Ψ 2 ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6 1 À S
ð
Þ
p
2ϕ 1 þ ε þ ε
Ã
ð
Þϕ 2 þ ε
Ã
þ ε
ð
Þϕ 3
½
Š
¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6 1 À S
ð
Þ
p
2ϕ 1 À ϕ 2 À ϕ 3
ð
Þ :
ð19:144Þ
Also, we have
f
Ψ 3 ¼
i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6 1 À S
ð
Þ
p
ε
Ã
À ε
ð
Þϕ 2 þ ε À ε
Ã
ð
Þϕ 3
½
Š ¼
i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6 1 À S
ð
Þ
p
Ài
ffiffi ffi
3
p ϕ 2 þ i
ffiffi ffi
3
p ϕ 3
h
i
¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 1 À S
ð
Þ
p
ϕ 2 À ϕ 3
ð
Þ :
ð19:145Þ
19.4 MO Calculations Based on π-Electron Approximation
763