Biomedical Engineering Reference
In-Depth Information
The initial assumption that the target molecule is adequately described by
its ground-state Slater determinant (static exchange approximation) leads to the
following simplified form for the scattering equations at each chosen (positive)
energy
1
2
5
/
F.
r
/
Z
2
V.
r
;
r
0
/F .
r
0
/d
r
0
C
D
.E
(4.2)
that can be solved (once defined the correct potential for our system) using the
Schwinger variational method as described in our earlier publications (e.g. see ref.
[
18
]), and subject to the standard boundary conditions of the scattering problem [
19
]
on the continuum solutions.
4.2.2
Correlation-polarization potential with model exchange
forces
The static exchange approximation does not include the response of the target to the
impinging electron, i.e. the correlation and polarization effects acting at short and at
large electron-target distances respectively. That part of the overall interaction can
be modelled by writing
V
corr
.
r
/ for
r
r
match
V
pol
.
r
/ for
r
>
r
match
V
cp
.
r
/
D
(4.3)
both in the past and in the present work we have used the Lee-Yang-Parr form for
V
corr
.
r
/ [
20
] and further employed the V
pol
.
r
/ as a function of the polarizability
tensor, to describe successfully via the V
cp
model of eq. (
4.3
) this additional part of
the overall interaction potential [
21
]ofeq.(
4.2
), as discussed below.
In order to perform scattering calculations at many energies and for fairly
large molecular targets, we replaced an exact nonlocal bound-continuum exchange
interaction of eq. (
4.2
) with a local energy-dependent exchange model potential. We
have chosen the Free-Electron-Gas-Exchange (FEGE) potential proposed long ago
[
21
,
22
]:
k
F
.
r
/
1
ln
ˇ
ˇ
2
4
2
1
1
C
V
FEGE
D
2
C
(4.4)
1
where
k
F
is the Fermi momentum and is a parameter related to the molecule's
ionization potential [
22
]. We have already found before that eq. (
4.4
) yielded
a reasonably realistic description of such forces in a great variety of molecular
systems [
23
-
25
] and therefore decided to employ the above modelling also in the
present study.
