Geoscience Reference
In-Depth Information
In the continuum (
E
> 0) there are all the states of the intermediate Rydberg
complex XY
**
associated with the scattering channels, which are divided into the
open and closed (resonant states). Recall that in open channels E>E
v
C
E
N
the
function
cot
"
#
p
2.E
v
E
N
E/
)
i:
When the total energy of the system is
E
< 0, Eq.
1.28
describes the rovibronic
spectrum of the bound predissociative Rydberg states of the molecule XY
**
mixed
with a dissociative continuum. Depending on the values of the principal quantum
number
n
of the molecule, the energy spectrum can be roughly divided into three
main areas. They are determined by the type of the communication between the
electronic and nuclear motions and define a hierarchy of the characteristic times: the
electronic
T
e
n
3
, the vibrational
T
v
1/! and the rotational
T
N
1/
B
(here ! and
B
are the vibrational quantum frequency and the rotational constant of the XY
C
ion).
Under the condition T
e
T
v
T
N
for the low-lying electronically excited
states, the adiabatic Born-Oppenheimer approximation is true when the electron
motion is quantized in the fixed molecular ion, and the projection on the axis ƒ is a
good quantum number. When
n
increases, and
T
e
T
v
, the adiabatic coupling break
takes place with the vibrational motion first and then, if
T
N
T
e
, with rotating, or
at large values of Bn
3
1, the picture is reversed, i.e., significantly nonadiabatic,
when the electron becomes a slow subsystem and the ion core is fast. In this case
the states are classified by ranges of the data values of the vibrational
v
and the
rotational
N
quantum numbers.
Under the condition !n
3
1, the transition to the adiabatic terms of the
intermediate complex XY
**
is the subject to the semiclassical approximation by
reexpansion of the wave functions
j
l
v
N
i
over the adiabatic
j
v
ƒ
i
basis (
1.17
),
following by the summing over all possible quantum numbers
v
and
N
(Golubkov
and Ivanov
2001
). As a result, for each dissociative ˇƒ-term we can obtain from
Eq.
1.28
the following simple transcendental equation:
tan ..R//
t
ƒ;ƒ
.R/
E
U
ˇƒ
.R/
D
V
2
ƒ;ˇƒ
.R/
(1.29)
where
v
(
R
)isdefinedas
1
p
2ŒU
i
.R/E
;
.R/
D
U
i
.R/ and U
ˇƒ
.R/ are the potential curves of the molecular ion XY
C
,andthe
dissociative term t
ƒ;ƒ
.R/ is the matrix element (
1.18
). Equation
1.29
reproduces
completely the physical situation, which we discussed earlier (in the introduction to
this section). Indeed, the zero values of the first square brackets for different values
of and ƒ correspond to the set of the Rydberg series with the diabatic potential
curves:
