Biomedical Engineering Reference
In-Depth Information
purposes, details of the derivation are not relevant, and we simply write down the
variational expression for the scattering amplitude:
D
E
D
E
2f.k
0
; k
j
/ D
j
.k
j
/ j V j
.C/
.k
0
/
./
.k
j
/ j V j
0
.k
0
/
C
D
E
./
.k
j
/ j A
.C/
j
.C/
.k
0
/
;
(5.2)
0;j
.k
0;j
/
where
represent the antisymmetrized products of target states
and plane
waves appearing in Eq.
5.1
,
V
is the electron-molecule scattering potential defined
in terms of the
N
-electron and
.N C1/
-electron Hamiltonians via
V D H
NC1
H
0
,
A
.C/
is
with
H
0
D H
N
C T
and
T
the kinetic energy operator of a free electron.
the operator
1
N C 1
P
.E H/ VG
.C
P
V;
A
.C/
D VP C
(5.3)
with
a projection operator onto open, i.e., energetically accessible, scattering
channels
P
G
.C/
P
the projected interaction-free Green's function subject to
outgoing-wave boundary conditions,
j
,and
P
E H
0
C
G
.C
P
D
lim
"!0
C
"
:
(5.4)
i
A key property that the SMC expression shares with the original Schwinger
principle is that, despite the appearance of a Hamiltonian term in Eq.
5.3
, all required
matrix elements are independent of the asymptotic form of
.˙/
. Therefore,
.˙/
; in particular,
they may be approximated in the same form as is used in conventional bound-state
quantum chemistry, that is, by linear expansions in terms of configuration state
functions that are, in turn, represented in terms of molecular orbitals constructed
from Gaussian-type atomic orbitals. This convenient property allows us to make
use of computational machinery developed for the bound-state problem to handle
much of the work. Writing those expansions explicitly as
one may use square-integrable wavefunctions to approximate
X
.C/
.k
0
/ D
x
m
.k
0
/
m
.r
1
;:::;r
NC1
/
m
X
./
.k
j
/ D
y
m
.k
j
/
m
.r
1
;:::;r
NC1
/
(5.5)
m
with
y
m
,we
proceed in the usual manner by inserting the expansions of Eq.
5.5
into Eq.
5.2
and
requiring that
m
a configuration state function and unknown coefficients
x
m
and
f
be stationary:
@x
m
D
@f
@f
@y
m
D 0;
(5.6)
