Biomedical Engineering Reference
In-Depth Information
by the long-range potential of the target. As all the molecules considered here have
permanent dipole moments, the long-range effect of the resulting dipole potentials
need to be treated.
The R-matrix, which relates the scattering wavefunctions to its derivative at a
given distance, is constructed on the boundary of the inner and outer region. The
R-matrix itself is energy-dependent but can be built from the results of energy-
independent inner-region calculations. This has the significant advantage that the
energy dependence is entirely obtained from rapid, outer-region calculations. This
is particularly useful for calculations aimed at characterizing resonances since such
calculations usually require the use of a dense grid of energies.
In this work the R-matrix method is used to obtain the eigenphase sums. To
obtain resonance positions and widths, these eigenphase sums were automatically
fitted to a Breit-Wigner form by the recursive resonance fitting [
47
]. The calculations
also give elastic electron collision cross sections and, for models which include elec-
tronically excited states in the close-coupling expansion, electron impact electronic
excitation cross sections.
The UK Molecular R-matrix codes [
26
] are designed to be very flexible and a
number of models have been tested in the course of the work considered here. The
simplest of these models is the so-called static exchange (SE) approximation in
which a full collision treatment is used for a target which is not allowed to relax
during the collision. Polarisation effects can be included using the static exchange
plus polarisation (SEP) approximation which allows for single excitations of the
target wavefunction. The only resonances that can be detected in an SE calculation
are shape resonances where the electron is temporarily trapped behind a centrifugal
barrier. The SEP model moves these resonances to lower energy and also can,
at least in principle, give Feshbach resonances. Feshbach resonances, which are
associated with simultaneous electronic excitation of the target and trapping of the
electron, are best given by calculations which explicitly include the parent target
state(s) for a given resonance, These are best considered by using several states in a
close-coupling (CC) expansion. Our best results presented below were all obtained
using CC models.
The use of CC methods for molecules with many valence electrons, such as
the ones considered here, leads to very large Hamiltonian matrices. That has led
to the development of special methods for treating the problems both in terms of
Hamiltonian construction [
42
] and diagonalisation [
16
,
44
]. In particular the use of
the so-called partitioned R-matrix [
44
] means that it is not necessary to explicitly
obtain all the eigenvalues and eigenvectors of the scattering Hamiltonian. The CC
results presented below are based on the use of all eigenvalues and vectors (typically
4000 to 6000) for “contracted” calculations but only the 5000 lowest solutions
for “uncontracted” calculations which lead to very large Hamiltonian matrices,
typically of dimension significantly bigger than 100 000.
The biomolecules considered here have also been the subject of R-matrix
calculations by Tonzani and Greene [
50
]. The R-matrix method developed by
these workers [
48
,
49
] has significant differences from the one used by us. Their
method involves using R-matrices to solve the electron-molecule scattering problem
defined entirely by potentials. Even if one ignores the issue that interactions
