Chemistry Reference
In-Depth Information
natural basis that is simpler than the DME and can be written in the form
d
dt
c
i
2
=−
c
m
−
S
iW
c
m
−
d
dt
c
m
1
2
m
+
1
,m
i
2
=
−
c
(247)
−
S
where the level
m
S
is undamped. The decay rate
between the adjacent
m
-states in the generic MM model is given by Eq. (A9) of
[13] that can be rewritten as:
−
is damped and the level
)
2
ω
m
+
1
,m
3
1)
2
l
2
m
+
1
,m
24
π
(2
m
+
(
D/
m
+
1
,m
=
(248)
t
Here
l
m
+
1
,m
is defined below Eq. (182) and
t
is defined by Eq. (175). In the case
where
m
=
S
−
1 and
m
+
1
=
S,
Eq. (248) simplifies to the elegant form
ω
S
−
1
,S
t
S
2
12
π
S,S
−
1
=
(249)
where
1)
D.
The DME can be obtained from Eq. (247) by setting
ω
S
−
1
,S
=
(2
S
−
S
c
−
S
,
c
m
c
m
,
S
c
m
ρ
=
c
m
m
=
ρ
S,m
=
c
(250)
−
S,
−
S
−
−
−
and calculating time derivatives. It has the form
ρ
ρ
m
,
−
S
d
dt
ρ
i
2
=
S,m
−
−
S,
−
S
−
ρ
ρ
m
,
−
S
d
dt
ρ
m
,m
i
2
=−
m
+
1
,m
ρ
m
,m
−
S,m
−
−
ρ
ρ
ρ
m
,m
(251)
d
dt
ρ
i
W
1
2
m
+
1
,m
i
2
=
−
−
S,m
+
−
S,m
−
S,
−
S
−
−
that coincides with the results of [11]. (In the latter, the precession goes in the
wrong direction, however). It should be stressed once more that this tunneling
DME is non-secular.
Of course, Eq. (247) is easier to solve than Eq. (251). We search for the solution
of Eq. (247) in the form
e
−
λt
.
Eigenvalues
λ
of Eq. (247) satisfy the equation
=
i
2
W
i
2
−
λ
+
0
(252)
i
2
i
2
W
1
−
λ
−
+
2
m
+
1
,m
