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: