Chemistry Reference
In-Depth Information
In Eq. (53), one has, for example,
e
−
E
/
(
k
B
T
)
δ
ε
α
−
E
V
(1)
π
V
(1)
γ
,α
ε
γ
+
E
−
α,γ
Z
b
e
−
E
ν
k
λ
,...
/
(
k
B
T
)
δ
ε
α
−
ω
k
π
=
ε
γ
+
Z
b
ν
k
λ
,...
k
λ
(
k
)
M
−
(
k
)
(1)
γα
·
(1)
αγ
M
−
×
·
e
−
E
ν
k
λ
,...
/
(
k
B
T
)
δ
ε
α
−
ω
k
π
+
ε
γ
−
Z
b
ν
k
λ
,...
k
λ
(
k
)
(
k
)
(1)
γα
·
(1)
αγ
M
+
×
·
M
(165)
+
After averaging over phonon populations,
1
Z
b
1
e
ω
k
/
(
k
B
T
)
e
−
E
ν
k
λ
,...
/
(
k
B
T
)
ν
k
λ
≡
ν
k
λ
=
n
k
λ
=
n
k
=
(166)
−
1
ν
k
λ
,...
this becomes
8
MN
k
λ
π
×
×
[
k
e
k
λ
]
√
ω
k
[
k
e
k
λ
]
√
ω
k
(1)
(1)
αγ
·
γα
·
δ
ε
α
−
ω
k
n
k
×
ε
γ
+
[
k
e
k
λ
]
√
ω
k
×
[
k
e
k
λ
]
√
ω
k
×
(1)
γα
·
(1)
+
αγ
·
δ
ε
α
−
ω
k
(
n
k
+
1)
×
ε
γ
−
(167)
Further simplifications can be achieved using
k
e
k
t
1
=±
×
k
e
k
t
2
(168)
and, for the summation over the two transverse polarizations,
(
k
·
a
)(
k
·
b
)
(
e
k
t
·
a
)(
e
k
t
·
b
)
=
(
a
·
b
)
−
(169)
k
2
t
=
t
1
,t
2
with
a
=
b
=
e
x
,
and then with
e
y.
Averaging over the directions of the vector
k
:
k
2
3
(
k
·
a
)(
k
·
b
)
=
(
a
·
b
)
(170)
