Geoscience Reference
In-Depth Information
The unperturbed Hamiltonian
H
0
for Rydberg channels is chosen in such a way
that all interactions in the dissociative X
C
Y configurations are exactly taken into
account, but in the scattering e
C
XY
C
channel only the Coulomb part V
c
1
r
D
is contained. Thus, in the total Hamiltonian (Eq.
1.10
), the operator
V
D
V
nc
C
V
CI
includes a non-Coulomb interaction of the electron with the ionic core
V
nc
and
interaction
V
CI
responsible for the corresponding nonadiabatic transitions between
e
C
XY
C
and X
*
C
Y configurations.
A formal solution of the AI problem (or of the inverse process of the dissociative
recombination) is reduced to define the collision
T
operator (related to the
S
matrix by the relationship
S
D
1
2i
T
), which satisfies the system of the
rearranged integral Lippman-Schwinger equations (Golubkov and Ivanov
2001
):
T
D
t
C
t
.
G
G
0
/
T
;
(1.11)
t
D
V
C
VG
0
t
;
(1.12)
where
G
D
.E
H
0
/
1
is the Green's operator with the interaction
V
turned
off, and
G
0
is weakly dependent on the total energy
E
operator. We denote the
basic wave functions of the unperturbed Hamiltonian
H
0
by
j
q
i
for the e
C
XY
C
configuration and
j
ˇ
i
for the dissociative X
C
Y
*
configuration. The corresponding
matrix elements
h
q
j
T
j
ˇ
i
are the amplitudes of the transitions ˛
,
ˇ transitions
that are symmetrical under permutation of the indices.
The Green's
G
operator in Eq.
1.11
is represented by the contribution of
noninteracting e
C
XY
C
and X
*
C
Y configurations and has the following form:
Z
G
.E/
D
X
i
X
ˇ
1
j
ˇ
ih
ˇ
j
E
E
ˇ
C
i
dE
ˇ
:
j
i
i
G
.c/
.E
E
i
/
h
i
j C
(1.13)
Here
E
i
and
j
i
i
are the excitation energies and corresponding wave functions of
the XY
C
ion, and
G
(c)
is the Green function describing the electron motion in a
Coloumb field. In the representation of spherical harmonics, it has the form
l
.r;r
0
I
"/Y
lm
r
0
;
Y
lm
r
r
G
.c/
G
.c/
.
r
;
r
0
I
"/
D
X
lm
r
0
where Y
lm
r
is the spherical function. It is important for further consideration that
the radial function G
.c
l
.r;r
0
I
"/ can be separated into terms with weak and strong
energy dependences (Demkov and Komarov
1966
):
G
.c
l
r;r
0
I
"
D
cot ."/
j
'
"l
.r/
i
˝
'
"l
r
0
ˇ
ˇ
C
g
l
r;r
0
I
"
:
(1.14)
The first term in Eq.
1.14
reproduces the position of the Coulomb levels ."/
D
.
2"/
1
=
2
at "<0and is expressed through the regular at-zero Coulomb wave
functions '
"l
.r/ normalized in accordance with
