Hindawi Publishing Corporation
Journal of Atomic, Molecular, and Optical Physics
Volume 2010, Article ID 604572, 3 pages
An Introduction to Continuum Distorted Wave Theory
D. S. F. Crothers
Department of Applied Mathematics and Theoretical Physics, Queen’s University Belfast, Belfast BT7 1NN, UK
Correspondence should be addressed to D. S. F. Crothers, email@example.com
Received 11 November 2009; Revised 25 February 2010; Accepted 1 June 2010
Academic Editor: Roberto Daniel Rivarola
Copyright © 2010 D. S. F. Crothers. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The author gives an introduction to Continuum Distorted Wave theory, in the form of a brief review.
Originally a continuum distorted wave (CDW) was a quan-
tum mechanical Coulomb wave associated with the motion
of an electron attached to an ion/atom relative to, and, while
simultaneously in, the continuum of another ion .
The context was charge transfer. Subsequently this was
generalized to the continuum of an electron , but not
before it was generalized to ionization in an ion-atom
collision . Based on these latter two papers, Crothers and
McCann  invented CDWEIS (eikonal initial state) which
guarantees unitarity of the propagating initial state. Other
later notations for the electron projectile case areC3and
BBK. CDW-CDW is appropriate for high impact energies
and small and medium impact parameters. CDW-EIS is
appropriate for large impact parameters and lower energies.
One of the principal advantages of CDW theory lies
in its Coulomb phases which guarantee the correct asymp-
totic/boundary behaviour [5–7].
Of course CDWEIS is intended to describe ionization in
intermediate and high-energy collisions. At lower energies
charge transfer is an important intermediate event.
The CDW theory is an on-shell theory and avoids artiﬁ-
cial logarithmic potentials and spurious nonlocal operators
. The latter accordingly used generalized nonorthogonal
coordinates (gnoc). If the electron has position coordinates
with respect to the target and projectile atomic (or
molecular) nuclei, respectively, then the net perturbation is
the non-orthogonal kinetic energy
Elastic-divergence-free CDW Neumann-Born series may
be derived  and generalization to more than one electron
is straight forward.
Rivarola and Fainstein  generalized the exact one-
electron formulation to the Roothan-Hartree-Fock treat-
ment of eﬀective one-electron problems. Jones and Madison
 applied the formulation of Crothers toactual
The independent-event model was introduced in 1987 by
Crothers and McCarroll . This should not be mistaken
for the more familiar independent-electron model. The
independent-event model is possible because an alternative
to the all-encompassing wave treatment is the impact
parameter treatment . It is a particular case of the time-
dependent Schroedinger (or Hartree-Fock) equation.
As described by Crothers , Crothers and Holt ,
and McDowell and Coleman , the transition between the
full-wave and impact-parameter treatments is facilitated by
a two-dimensional Fourier transform, the two dimensions
being orthogonal to the relative velocity. This leads to the
familiar eikonal representation only in the case of (total)
cross-sections with atomic spherical symmetry.
A further example of the power of the independent-event
model is in the description of Thomas double scattering
electron-capture which is well-described as a second-order
CDW event : CDW2, third-order being negligible and
given the necessary small impact parameters for large angle
scattering. In the case of capture and/or excitation, coupled
equations may be formulated variationally .
Generalized (G) CDWEIS was introduced by Crothers et
al.  in which a complete set of magnetically quantized
CDWs was introduced successfully. So, strong Coulomb
interaction, event-correlation, gauge and Galilean invari-
ance, detailed balance, intermediate continuum coupling,
stationary or nonstationary: electron capture, ionization,
transfer-ionization and other double events are all described.