Chemistry Reference
In-Depth Information
3. Direct Processes
To compute matrix elements
V
(1)
and so on, in Eq. (53) with respect to the
α,γ
phonon bath, one can label the state
|
φ
by the numbers of phonons
ν
k
λ
=
0
,
1
,
2
,...
in each phonon mode
k
λ
|
φ
=|
...,ν
k
λ
,...
⇒|
ν
k
λ
(162)
In the direct processes, the state
by creation or annihilation
of one phonon, according to Eq. (153). We will make use of the phonon matrix
elements
|
φ
differs from
|
φ
M
(
k
)
=
ν
k
λ
±
1
|
δ
φ
|
ν
k
λ
(163)
±
and their conjugates. From Eq. (153), one obtains
1
ν
k
λ
−
e
k
λ
]
e
i
k
·
r
√
ω
k
1
2
[
i
k
×
2
MN
M
(
k
)
=
a
k
λ
ν
k
λ
−
e
k
λ
]
e
i
k
·
r
√
ω
k
1
2
[
i
k
×
√
ν
k
λ
2
MN
=
ν
k
λ
1
e
k
λ
]
e
−
i
k
·
r
√
ω
k
1
2
2
MN
[
−
i
k
×
a
k
λ
M
−
(
k
)
=
ν
k
λ
−
e
k
λ
]
e
−
i
k
·
r
√
ω
k
1
2
2
MN
[
−
i
k
×
√
ν
k
λ
=
ν
k
λ
+
1
e
k
λ
]
e
−
i
k
·
r
√
ω
k
1
2
[
−
i
k
×
2
MN
a
k
λ
M
(
k
)
=
ν
k
λ
+
ν
k
λ
+
e
k
λ
]
e
−
i
k
·
r
√
ω
k
1
2
[
−
i
k
×
2
MN
=
1
ν
k
λ
1
e
k
λ
]
e
i
k
·
r
√
ω
k
1
2
[
i
k
×
2
MN
M
+
(
k
)
=
a
k
λ
ν
k
λ
+
e
k
λ
]
e
i
k
·
r
√
ω
k
ν
k
λ
+
1
2
[
i
k
×
2
MN
=
1
(164)
