Chemistry Reference
In-Depth Information
is the Green function for the nuclear motion on the potential energy surface of
the target,
V
0
(
R
)
=
R
|
V
0
|
R
, and μ is the reduced mass of the nuclei;
V
d
(
R
)
=
R
|
V
d
|
R
=
d
(
R
)
+
V
0
(
R
)
is the potential energy surface of the resonant state and
V
d
k
(
)
=
|
V
d
k
|
(
)
on the right-hand side of (9.79) depends
on the boundary conditions. It will be discussed in the following for particular
collision processes.
Equation 9.79 is an effective Lippmann-Schwinger equation for the nuclear
dynamics in the energy-dependent, nonlocal, and complex potential of the resonant
state. To make this more explicit,
G
(
+
)
0
R
R
R
. The inhomogeneity
J
R
(
R
,
R
,
E
)
is expressed in terms of a complete
set of target nuclear wave functions χ
ν
(
. Employing Dirac's identity and assuming
that the interaction between the resonance and the scattering continuum depends only
on
k
(isotropic interaction) leads to
R
)
i
2
(
F
(
+
)
(
R
,
R
;
E
R
,
R
;
E
R
,
R
;
E
)
=
(
)
−
)
(9.82)
with
P
dE
V
dE
(
4π
ν
)
χ
ν
(
)
χ
ν
(
R
)
V
dE
(
R
)
R
R
(
R
,
R
;
E
)
=
(9.83)
E
−
ω
ν
−
E
and
8π
ν
R
,
R
;
E
χ
ν
(
R
)
V
dE
−
ω
ν
(
R
)
(
)
=
V
dE
−
ω
ν
(
R
)
R
)
χ
ν
(
,
(9.84)
where symbol “P” denotes the principal value of the integral and ω
ν
stands for the
vibrational energies of the target molecule.
The inverse of
is the lifetime of the resonance. As expected, the
auto-detaching property of the resonance arises from its coupling to the scattering
continuum. In principle, the resonance has a finite lifetime even in the absence of this
coupling because the optical potential,
V
opt
, is complex. Its imaginary part therefore
induces a lifetime. This contribution, however, is much smaller than the one due to
V
dE
. Therefore, it is usually neglected.
The nonlocality of the potential complicates the numerical solution of Equa-
tion 9.79. In the early applications of the resonance model [59-64,72], the nonlocal
potential was thus replaced by a local one. The local approximation can be obtained
from (9.83) to (9.84) by identifying the energy available for the scattered electron
with an effective resonance energy:
E
(
R
,
R
;
E
)
−
ω
ν
≈
E
res
(
R
)
[83]. The completeness of the
(
)
=
L
(
)
(
−
R
)
vibrational target states can then be used to obtain
R
,
R
;
E
R
δ
R
and
(
R
,
R
;
E
)
=
L
(
)
(
−
R
)
R
δ
R
with
4π P
|
V
dE
res
(
R
)
(
R
)
|
2
2
,
L
(
R
)
=
8π
|
V
dE
res
(
R
)
(
R
)
|
L
(
R
)
=
,
(9.85)
E
res
(
R
)
−
E
which reduces the Lippmann-Schwinger equation to an ordinary differential
equation:
E
1
2μ
d
2
dR
2
−
i
2
L
(
(
+
)
dE
+
V
0
(
R
)
−
L
(
R
)
+
R
)
(
R
)
=
J
(
R
)
.
(9.86)
