Chemistry Reference
In-Depth Information
With the use of energy conservation, this can be rewritten as:
R
αβ,α
β
1
δ
α
β
ρ
α
β
d
dt
ρ
αβ
=−
iω
αβ
ρ
αβ
+
−
α
β
π
e
−
E
/
(
k
B
T
)
V
α,γ
V
γ
,β
+
Z
b
γ
δ
ε
β
−
E
ρ
γγ
−
e
(
ε
β
−
ε
γ
)
/
(
k
B
T
)
ρ
ββ
×
ε
γ
+
E
−
E
ρ
γγ
−
e
(
ε
α
−
ε
γ
)
/
(
k
B
T
)
ρ
αα
δ
ε
α
−
+
ε
γ
+
E
−
(71)
One can see that as the diagonal DM elements approach their equilibrium values
defined by Eq. (68), they cease to drive nondiagonal elements. Thus the equilibrium
solution of the full DME is Eq. (54).
F.
Semisecular Approximation
In a system with tunnel-split energy levels, such as molecular magnets, there are
very small transition frequencies
ω
αβ
and correspondingly very slow nondiagonal
matrix elements
ρ
αβ
whose dynamics is coupled to that of the diagonal elements.
While the secular approximation neglects the interaction between diagonal and
slow nondiagonal DM elements that can lead to nonphysical results for relaxation
rates, the full nonsecular formalism involves a big
N
2
N
2
matrix. In important
particular cases, such as thermal activation over a barrier or tunneling, the eigen-
values of the DM span a broad range from very fast to very slow, the latter being of
primary importance in relaxation. Because of this, one has to do numerical calcu-
lations with increased precision, which makes them very slow. This difficulty can
be overcome with the help of the semisecular approximation that considers cou-
pled equations for diagonal and slow nondiagonal DM elements plus decoupled
equations for the fast DM elements. The easiest way to implement it is to include
the diagonal and subdiagonal terms
ρ
αα
and
ρ
α,α
±
1
into the slow group, because
in most situations there are only two levels that come close to each other, making
ρ
α,α
+
1
or
ρ
α,α
−
1
slow. Implementation of the semisecular DME in the case of
time-independent
H
s
will be covered in Section III.A.3.
×
G.
Transformation to the Natural Basis
One can transform the DME, Eq. (63) to the natural basis using Eqs. (12) and (11).
The result has the form
αβ
ρ
mn
=
ψ
m
|
ρ
|
ψ
n
=
ψ
m
|
χ
α
ρ
αβ
χ
β
|
ψ
n
(72)
