Chemistry Reference
In-Depth Information
Computing the time derivative of
ρ
αβ
(
t
)
I
and using Eq. (52), one arrives at the
equation
e
i
ω
αβ
−
ω
α
β
t
R
αβ,α
β
ρ
α
β
(
t
)
I
d
dt
ρ
αβ
(
t
)
I
=
(56)
α
β
where the conservative term disappeared and the relaxation term has an explicit
time dependence. While the change of the density matrix due to the relaxation is
slow, the oscillation in the terms with
ω
αβ
/
ω
α
β
are generally fast. These fast
oscillating terms average out and make a negligible contribution into the dynamics
of the small system. In general, all transition frequencies are nondegenerate, so
that one can drop all terms with
α/
=
α
β
,
if
α/
β.
In the equations
for the diagonal terms
ρ
αα
(
t
)
I
one can keep only diagonal terms with
α
=
=
and
β/
=
=
β
.
This is the
secular approximation
that greatly simplifies the DME. In the secular
approximation, one has
δ
αβ
α
R
αα,α
α
ρ
α
α
(
t
)
I
+
1
δ
αβ
R
αβ,αβ
ρ
αβ
(
t
)
I
d
dt
ρ
αβ
(
t
)
I
=
−
δ
αβ
α
/
=
R
αα,α
α
ρ
α
α
(
t
)
I
+
R
αβ,αβ
ρ
αβ
(
t
)
I
(57)
=
α
V
α,α
,
one obtains
Simplifying Eq. (53) and using the Hermiticity
V
α
,α
=
Z
b
R
αα,α
α
α
=
α
=
E
)
V
α,α
2
π
2
e
−
E
/
(
k
B
T
)
δ
(
ε
α
−
ε
α
+
E
−
≡
αα
(58)
and
E
V
α,γ
e
−
E
/
(
k
B
T
)
δ
ε
α
−
π
2
R
αβ,αβ
=
−
ε
γ
+
E
−
Z
b
γ
e
−
E
/
(
k
B
T
)
δ
ε
β
−
E
V
β,γ
2
−
ε
γ
+
E
−
γ
E
)
V
α,α
V
β
,β
2
e
−
E
/
(
k
B
T
)
δ
(
E
−
+
(59)
Rearranging the terms one obtains
˜
αβ
R
αβ,αβ
=−
(60)
