Geoscience Reference
In-Depth Information
Here '
i
is the electron wave function of the XY
C
ion, and
v
.R/ is the
vibrational wavefunction. The total angular function of the system ˆ
JM
lN
.r R/ is
defined in the representation with total angular momentum
J
and its projection
M
,
angular momentum N of the rotational notion of the nuclei, and in the case LS-
coupling for configuration of XY
C
has the form (Golubkov and Ivanov
2001
)
lN
r R
Y
lm
.r/Y
N;Mn
R
.lNmM
n
j
JM/
D
X
m
ˆ
JM
( r,and R are spherical coordinates of the electron and nucleus, and .lNmM
n
j
JM/ are the vector summation coefficients,
J
D
l
C
N
). The
j
q
i
functions are
determined in the laboratory frame of reference, where the direction of the
z
-axis
coincides with the electron wave vector.
In the dissociative configuration the electrons are fast enough, so their motion
is quantized in the field of the fixed nuclei and is described in the adiabatic
approximation. For calculation of the configuration interaction matrix elements it is
necessary to expand the channel wave functions (Eq.
1.16
) into the adiabatic basis,
that is, to pass into the coordinate system associated with the molecule, in which the
absolute value of the projection of electronic angular momentumƒ on the molecular
axis is fixed. States of a diatomic molecule in this basis are classified according to
the values of
J
(the angular momentum of an electron l is not preserved in general).
The total wave function in the adiabatic approximation is a superposition of the
channel basis functions:
j
JMƒ
v
i D
X
l
a
Jƒ
l
j
JMlƒ
v
i
:
(1.17)
Here is the quantum number given by the largest coefficient of the expansion
(
1.17
) characterizing the effective electron angular momentum of the Rydberg con-
figuration with
l
mixing taken into account. The wave functions of the dissociative
ˇ configuration are determined quite similarly.
To calculate the elements t
lN
v
;l
0
N
0
v
0
and t
lN
v
;l
0
ˇƒ
in the system of Eqs.
1.11
and
1.12
and responsible for the rovibronic and configuration nonadiabatic transitions,
it is convenient to introduce the auxiliary
t
operator, i.e.:
P
Z
V
X
q
1
j
q
ih
q
j
E
E
q
"
t
d"
t
D
V
C
It describes the interaction of the electron with the ion core in an isolated Rydberg
e
C
XY
C
configuration. Electron parts of matrix elements of the
t
-operator in
the mixed basis (
1.17
), depending on
R
by definition, are diagonal with respect to
subscripts and ƒ, and are expressed via the diabatic quantum defects
N
ƒ
as
t
J
ƒ;
0
ƒ
0
.R/
D
tan
N
ƒ
.R/ı
0
ı
ƒƒ
0
:
(1.18)
