Chemistry Reference
In-Depth Information
or, with the use of Eq. (208),
−
2
i
2
W
i
2
W
1
2
m
+
1
,m
1
4
0
=
−
λ
+
λ
−
+
+
1
2
m
+
1
,m
λ
1
4
ω
0
+
i
4
W
λ
2
=
−
+
m
+
1
,m
(253)
that yields
1
2
m
+
1
,m
±
1
4
2
m
+
1
,m
−
1
2
i
W
ω
0
−
λ
±
=
m
+
1
,m
1
2
m
+
1
,m
±
i
1
2
1
4
2
m
+
1
,m
+
i
W
ω
0
+
=
−
m
+
1
,m
(254)
The solution of Eq. (247) has the form
a
(
+
)
−
a
(
−
)
−
S
e
−
λ
+
t
S
e
−
λ
−
t
c
S
(
t
)
=
+
−
a
(
+
)
a
(
−
)
e
−
λ
+
t
e
−
λ
−
t
c
m
(
t
)
=
+
(255)
m
m
The eigenvectors
a
(
±
)
−
follow from Eq. (247):
S
,a
(
±
)
m
i
2
W
i
2
a
(
±
)
−
a
(
±
)
−
a
(
±
)
m
−
λ
=−
S
−
(256)
±
S
that is,
W
a
(
±
)
−
=−
a
(
±
)
+
2
iλ
(257)
±
m
S
Thus Eq. (255) can rewritten as:
a
(
+
)
−
a
(
−
)
−
S
e
−
λ
+
t
S
e
−
λ
−
t
c
S
(
t
)
=
+
−
W/
+
2
iλ
W/
+
2
iλ
a
(
+
)
−
+
a
(
−
)
−
−
e
−
λ
+
t
e
−
λ
−
t
c
m
(
t
)
=−
−
(258)
S
S
/
/
From the initial conditions
c
S
(0)
=
1 and
c
m
(0)
=
0, one obtains
−
a
(
+
)
−
a
(
−
)
−
S
+
=
1
S
a
(
+
)
−
a
(
−
)
−
S
(
W/
+
2
iλ
)
+
S
(
W/
+
2
iλ
)
=
0
(259)
+
−
