Geoscience Reference
InDepth Information
h
'
"l
.r/
j
'
"
0
l
.r/
i
D
ı
"
"
0
:
After excluding the smoothed real part from (
1.13
):
g
l
r;r
0
I
E
E
i
Y
lm
r
0
r
0
i;lm
j
i
i
Y
lm
r
G
0
.E/
D
X
h
i
j
r
P
Z
X
ˇ
1
j
ˇ
ih
ˇ
j
E
E
ˇ
dE
ˇ
C
(
P
means the principal value of the integral), we can write the following equation
for the collision
T
operator:
8
<
9
=
j
q
ih
q
j
cot
h
2"
q
1
=
2
i
X
i
X
ˇ
T
D
t
C
t
j
ˇ
ih
ˇ
j
T
:
(1.15)
:
;
q
Here "
q
D
E
E
q
is the electron energy in the
q
channel of motion,
E
q
is the energy
of its vibrational and rotational excitation. For open channels
"
q
>0
, the function
cot Œ
2"
q
1=2
D
i.
Because of the linearity and the separable structure of the nucleus (which follows
from the unique properties of the Coulomb Green function), the integral equation
(
1.15
) reduces to a system of linear algebraic equations for the elements of
T

matrix, in which the dissociative channels are taken into account along with the
scattering channels, at the same time consistently ignoring the strong nonadiabatic
coupling of the electronic and nuclear motions, which forms a heterogeneous
continuum of intermediate Rydberg states interacting with the decaying dissociative
terms (Golubkov et al.
1996a
). These important properties follow from the formal
scattering theory, and automatically provide a unitary
S
matrix on an arbitrary basis,
accounting for the movement of channels, which ensures the controlled precision of
carried calculations.
1.3.3
Basic Wave Functions and Elements of Reaction
t Matrix
Basic wave functions
j
q
i
in the Rydberg configuration (taking into account the
vibrational and rotational motion of nuclei) are
lN
r R
:
j
q
i D j
JMlN
v
i D
'
"
v
l
.r/'
i
.
x
/
v
.R/ˆ
JM
(1.16)