Biomedical Engineering Reference
In-Depth Information
of electron-induced DNA damage, but we will not often have occasion to refer to
that work; for some recent perspectives, see [
51
-
53
]. Our focus is much narrower:
electron collision studies, primarily elastic, of constituents of RNA and DNA or
close analogues of those constituents. Though we mostly review our own results,
we attempt to situate them alongside complementary work by other researchers,
experimental and computational, and to provide sufficient references to guide
the reader to the relevant literature. We apologize in advance for the inevitable
oversights and errors.
5.2
The Schwinger Multichannel (SMC) Method
As mentioned in Sect.
5.1
, the many-electron scattering problem, though fully as
difficult as the bound-state problem of quantum chemistry, cannot be treated by
the energy-minimization procedure applicable to the latter. Instead, one typically
applies some other optimization procedure that produces accurate results efficiently.
In the time-independent picture, scattering is described by a wavefunction that
behaves asymptotically as
"
i
k
0
r
NC1
/
.C/
.r
1
;:::;r
NC1
/ !
A
0
.r
1
;:::;r
N
/
exp
.
3
X
exp
.
i
k
j
r
NC1
/
r
f.k
0
; k
j
/
j
.r
1
;:::;r
N
/
5
;
C
(5.1)
j
k
0
describes the direction and energy of incidence and
where the wave vector
k
j
describes the direction and energy of the outgoing scattered electron.
likewise
The
N
-particle wavefunctions
0
and
j
are the initial and final states of the target
molecule, and
is an antisymmetrizer between the projectile electron and those of
the target. Energy conservation, though not explicitly indicated, is assumed: that is,
E
0
C k
0
=2 D E
j
C k
j
=2
A
,where
E
0
is the energy of
0
,
E
j
the energy of
j
,and
k
0;j
=2
„Dc D e D 1
are the electron kinetic energies (Hartree atomic units,
,
j
are henceforward asssumed). The summation over
includes all open electronic-
.C/
excitation channels (we neglect ionization). The
denotes
that only outgoing spherical waves are included in the second term on the right,
which describes the scattered flux. The information about every possible scattering
process
superscript on
k
0
! k
j
is contained in the coefficient
f.k
0
; k
j
/
, known as the scattering
amplitude, that modulates those outgoing waves. The central role of the scattering
amplitude naturally suggests developing variational principles in which it is the
stationary quantity.
The SMC method was developed by Takatsuka and McKoy [
54
,
55
]asan
adaptation of the Schwinger variational procedure [
56
] to the specific situation of
low-energy scattering involving multiple indistinguishable particles. For the present
