Chemistry Reference
In-Depth Information
Within the adiabatic approximation, it is necessary to determine the fixed-nuclei
scatteringamplitudeandtherovibrationalstatesofthemolecule,whichinturndepend
on the potential energy surface of the target molecule. The former can be obtained as
afunctionof
R
from the asymptotics of the
N
+
1 electron, fixed-nuclei Lippmann-
(
+
)
E
(
r
,
r
;
R
)
Schwingerequation,whichdetermines
,whilethelatterrequires,again
as a function of
R
, the solution of the
N
electron problem (9.77). In both cases, anti-
symmetrized wave functions have to be used because of the indistinguishability
of electrons. Thus, even when the nuclear motion is split off, the calculation of
cross sections for electron-molecule scattering remains a formidable many-body
problem [74].
,
,
n
i
9.2.1.4 Resonant Scattering
At electron energies of a few electron volts or less, the collision time is long compared
to the period of the internuclear motion and the adiabatic approximation fails. The
projectile electron is then so slow that it is captured by the molecule giving rise
to a bound state of the negatively charged molecular ion, which is the collision
compound for electron-molecule scattering.
∗
This state interacts with the electron-
molecule scattering continuum, acquires therefore a finite lifetime, and turns into
a quasi-bound (auto-detaching) state. Auto-detaching states play a central role in
(vibrational) excitation, attachment, recombination, and detachment collisions. From
a theoretical point of view, all these processes can hence be analyzed within a model
describing a discrete state (resonance) embedded in a continuum of scattering states.
A particularly elegant derivation of the resonance model for an electron colliding
with a diatomic molecule containing
N
electrons has been given by Domcke [83] who
uses many-particle Green functions to reduce the
N
1 electron scattering problem
to an effective single-electron problem. The reduction is achieved by two projections:
First, electronic states that are not accessible at the energy considered are eliminated
by introducing an optical potential for the incoming electron. Then, in a second step,
the fixed-nuclei
T
-matrix is split into a rapidly varying part due to the quasi-bound
resonant state and a slowly varying background term. The splitting of the
T
-matrix
can be shown to be equivalent to an effective single-electron Hamiltonian at fixed-
nuclei describing a resonance coupled to a continuum of states. Finally, the kinetic
energy of the nuclei is included and the electronic degrees of freedom are integrated
out to obtain an effective Lippmann-Schwinger equation for the nuclear dynamics
that, with appropriate boundary conditions, can then be used to calculate the collision
cross sections of interest.
It is essential for the formalism that the optical potential supports a resonance
and that the resonance can be extracted from the single-electron continuum such
that the scattering background contains no spurious resonances. The many electron
problem is then completely buried in an optical potential, which can be calculated
separately using, for instance, the many-body perturbation theory [84]. In principle,
the formalism can handle more than one resonance as well as electronically inelastic
+
∗
The electronic and vibrational properties of negative ions therefore play an important role in low-
temperature gas discharges with
k
B
T
e
∼
O
(
−
)
, even when the gas is not electronegative, that is,
when the negative ion is unstable on the plasma time scale and therefore irrelevant at the kinetic level.
1
10eV
