The distorted-wave Born approximation (DWBA) has been one of the most successful theoretical approaches for treating electron collisions with complicated atoms, and recently the DWBA has been successfully extended to treat electron-impact ionization of molecules. The purpose of this paper is to give an overview of that development and to provide a summary of the recent experimental and theoretical works examining low to intermediate energy electron-impact single ionization of molecules.

One of the most fundamental, unsolved problems in physics is the few-body problem. The few-body problem arises from the fact that the Schrödinger equation is not analytically solvable for more than two mutually interacting particles. As a result, for three or more particles, theory must resort to significant modeling efforts using approximations, the validity of which is determined by comparison with experiment.

Atomic collisions present a valuable test of these theoretical treatments of the few-body problem for two main reasons. First, the underlying fundamental interaction between the particles in atomic systems, the electromagnetic force, is well understood. In particular, for any two particles, it is known exactly. Consequently, for collisions involving more than two particles, discrepancies between experiment and theory must be attributed to the few-body aspects of the theoretical model. Second, the recent advancements in experimental techniques, such as cold-target recoil-ion momentum spectroscopy (COLTRIMS) [

Advances on the theoretical side now allow for an essentially exact numerical calculation of one of the simplest three-body problems, namely, electron-impact ionization of hydrogen [

One of the most successful theoretical approaches for electron-impact ionization of more complicated targets is the distorted-wave Born approximation (DWBA). The DWBA treats single ionization of a complex target as a 3-body problem with the effect of the spectator electrons being represented by a spherically symmetric potential which is used in the Schrödinger equation to calculate the continuum wave functions for the continuum particles. Madison et al. [

In the standard DWBA for ionization, the final state wave function is represented as a product of two wave functions which contain no mutual electron-electron repulsion (normally called postcollision interaction or PCI). In 1989, Brauner et al. [

The BBK final state wave function for ionization of hydrogen is a product of three Coulomb functions—a Coulomb wave for each of the two continuum electrons in the field of a proton and the Coulomb interaction between the two electrons. This 3-Coulomb (3C) wave function was first proposed by Redmond [

The

If the final state Coulomb interaction is included in the standard DWBA approximation using numerical bound and continuum wave functions, a full 6D integral is required for the _{1}F_{1} hypergeometric function and it is the hypergeometric function that causes the problem. Botero and Macek [

All of the above work focused on approximations for the final state wave function. The BBK approximation uses a plane wave for the incoming electron which is assumed to be OK at least for high energies since it is asymptotically correct. However, for heavy ion impact, Crothers and McCann [

Subsequently, we successfully applied the same EIS approach to recent 3D FDCS measurements for heavy particle ionization [

All of the CDW-EIS calculations performed in the 1990s for less differential cross-sections used a straight-line semiclassical trajectory for the projectile while we use a full quantum-mechanical treatment. These straight-line semiclassical trajectory models have been very effective in predicting experimental data for less differential cross-sections. However, these approximations have been shown to be not as reliable for cross-sections differential in the projectile scattering angle [

We use distorted-waves for the continuum states of the active electron. The previous studies have shown that distorted-waves can improve CDW-EIS calculations in less differential cross-sections [

The exact

Consequently, we labeled our calculation 3DW-EIS to distinguish from the previous CDW-EIS calculations.

For electron-impact ionization of heavier atoms or molecules, it is necessary to use numerical bound state wave functions and numerical distorted-waves for the continuum electrons. If one additionally wants to include the final state electron-electron Coulomb interaction in the final state, one is forced to perform a full numerical 6D integral for the

All the above calculations are based upon first-order perturbation theory. However, one has to be very careful with terminology since any physics contained in the initial or final state wave function is contained to all orders of perturbation theory while the interactions in the perturbation are contained to only first-order. As a result, a first-order calculation will normally contain some physics to all orders. For example, a distorted-wave is an elastic scattering wave function from the target. Consequently, a normal DWBA calculation will contain elastic scattering of the incoming electron from the neutral target to all orders of perturbation theory as well as elastic scattering of the two final continuum electrons. The inclusion of the full Coulomb PCI in the final state wave function means that the final-state electron-electron repulsion is included to all orders of perturbation theory.

Although there were some second-order plane-wave Born calculations (PWB2) using closure reported in the 1980s for ionization of hydrogen [

Electron-impact single ionization of molecules has been extensively studied for 2-3 decades using high-energy incident electrons. This work has been nicely summarized by Weigold and McCarthy [

Although there were some experimental papers reported for low to intermediate energy ionization of molecules in the 90s, atoms received more attention probably due to limited theoretical support for the molecular measurements. The challenge for molecules is to use a multicenter wave function that is normally not spherically symmetric as compared to atoms which have one scattering center and spherically symmetric wave functions. Another challenge arises from the fact that the experiments to date do not align the molecules before the collision. The traditional (e,2e) measurements therefore represent an average over all molecular orientations which must be taken into account in the theoretical approach. In spite of these difficulties, there has been an increased interest in low energy molecular ionization in the last ten years [

On the theoretical side, the plane-wave impulse approximation (PWIA) developed by McCarthy and coworkers [_{2}H_{6} by Deutch and Becker [

The distorted-wave impulse approximation (DWIA) is similar to the PWIA except that the plane waves are replaced with distorted-waves. Gao et al. presented DWIA results for electron-impact ionization of N_{2} [

In the first Born approximation (FBA), the ejected electron is treated as a Coulomb wave whereas the incident and scattered electrons are treated as plane waves. In 2001, Champion et al. [_{2}O. Monzani et al. [_{2} in which all incoming and outgoing continuum electrons are represented as distorted-waves calculated in a single-center static-exchange potential. Weck et al. [_{2} absolute experimental data of Chérid et al. [

We generalized the 3DW approach we had developed for atoms to molecules which we called the molecular 3-body distorted-wave approach (M3DW) [

The purpose of this paper is to give an overview of the M3DW method for ionization of molecules and to show the current status of agreement between experiment and theory for low to intermediate energy electron-impact ionization of molecules.

The theoretical basis for the 3-body distorted-wave approximation was described by Prideaux and Madison [

One of the most successful approximations for calculating atomic ionization by electron impact has been the first-order distorted-wave Born approximation (DWBA). In the standard DWBA, the initial-state Hamiltonian is chosen to be

The full Hamiltonian is given by

As mentioned above, any physics contained in the wavefunctions of the

The DWBA has been highly successful for calculating the FDCS for ionization by higher-energy electrons. However, as the energy of the electron decreased, the DWBA starts to fail. One source of this failure is an inadequate treatment of the final-state interaction between the projectile electron and ejected electron. As mentioned in the introduction, BBK demonstrated that better agreement with experiment for electron-hydrogen scattering could be achieved for lower incident electron energies by including the final-state projectile-electron interaction in the approximation for the final-state wave function. In the BBK approach, the exact final state for electron-hydrogen scattering is approximated as

The physics contained in M3DW approximation is the following. The final-state Coulomb interaction between the projectile and a screened nuclear charge, the Coulomb interaction between the ejected-electron and a screened nuclear charge, and the Coulomb interaction between the projectile and ejected-electron are contained to all orders of perturbation theory. For the initial state, the Coulomb interaction between the projectile and a screened nuclear charge for a neutral target is contained to all orders of perturbation theory. Similar to the DW, the only interaction contained only to first-order in the 3DW is the initial-state nonspherical projectile-active-electron interaction as will be demonstrated below.

In the DW approach, ionization of more complex targets is treated as a three-body problem. In the 3-body approach, the initial-state interaction is approximated as

From (

As mentioned in the introduction, evaluating the DW

The spherically symmetric distorting potentials for molecules are calculated similar to the atomic case. The starting point is the molecular charge density for the neutral molecule which is obtained from the Dyson orbitals

In addition to the static distorting potential, it is standard practice to add additional terms designed to approximate known important physical effects. Two such effects are exchange distortion (

For the correlation-polarization potential

As mentioned in the introduction, including the full final state Coulomb interaction

For the lower energies, it has become clear that using the full Coulomb interaction of (

Botero and Macek [

Finally, (

There have been numerous high and low energy experimental papers published over the last 20–30 years examining electron-impact ionization of molecules with limited theoretical support. In the last few years, the level of theoretical support has increased and here we will give an overview of the current status of the agreement between experiment and theory. In particular, we will present the recent experimental and theoretical results for H_{2}, N_{2}, H_{2}O, and formic acid (FA) or (HCOOH). We would like to emphasize that we are not trying to provide an overview of all the low to intermediate energy work that has been done—only the work performed in the last few years.

Two sets of experimental data have been recently reported for electron-impact ionization of H_{2}. Azzedine Lahmam-Bennani’s group in Paris, France has performed some measurements for relatively high energies and Andrew Murray’s group at Manchester, England has reported low energy measurements. Figure

FDCS for H_{2} as a function of the ejected electron angle. The scattered electron energy was 500 eV and the scattering electron angle was

The M3DW results were calculated using the orientation-averaged molecular orbital (OAMO) approximation and, as mentioned in the theory section, this approximation is expected to be valid for highly symmetric molecular states and scattering angles for which the momentum transferred to the residual ion is small (less than unity). The momentum transferred to the ion is zero in the binary peak direction and small for angles in the binary peak. For the kinematics of the Paris data, the momentum transferred to the ion is quite large for angles in the recoil peak so the OAMO approximation is expected to be valid for the binary peak but not the recoil peak. Figure

Figure

FDCS for electron impact ionization of H_{2} for equal final state energies

Three theoretical results are presented in Figure

One of the interesting aspects of the experimental data lies in the fact that the largest cross-section occurs for

Low to intermediate energy experimental (e,2e) fully differential cross-section measurements for N_{2} have recently been made in three different laboratories—those of Azzedine Lahmam-Bennani (Paris, France), Andrew Murray (Manchester, England), and Birgit Lohmann (Adelaide, Australia). The higher energy experiments measured in Paris were able to resolve two features [

Figure _{2}, the experimental recoil peak is relatively much stronger (as big as the binary peak for the

FDCS for electron-impact ionization of N_{2} for the

The experimental measurements performed at Manchester and Adelaide were for lower incident-electron energies. In both cases, the _{2} to our knowledge except using the Stia et al. [_{2} results, the agreement between experiment and the M3DW theory is not very good for the recoil peak. Interestingly, the Stia et al. [

FDCS for electron-impact ionization of 3_{g} orbital of N_{2}. The incident electron energy for (a) was 150 eV, the ejected electron energy was 10 eV, and the projectile scattering angle was 15

For 75 eV incident electrons, the M3DW predicts a significant peak at 180

Fully differential cross-sections for electron-impaction ionization of H_{2}O have been recently measured at both Adelaide and Manchester. Milne-Brownlie et al. [_{2} measurements presented in Figure

FDCS for electron-impact the ionization of the 3a_{1} state of H_{2}O for equal final state energies

Figure _{2} results of Figure _{2}, the best agreement between experiment and theory was in the perpendicular plane and here we see the best agreement between experiment and theory is in the scattering plane and the worst agreement is in the perpendicular plane. For H_{2}, the largest cross-section was found for

The most puzzling aspect of Figure _{2}, N_{2}, and CO_{2}. Here we see that the theoretical results are consistent with the Al-Hagan et al. [

Finally, we looked at the biomolecule formic acid (FA) (HCOOH). Very recently, Colyer et al. [

There have been several high energy EMS studies performed for FA with the most recent being a study of seven of the highest occupied orbitals including the

Theoretical and experimental FDCS for electron-impact ionization of the

The EMS theoretical results presented in the figure from the Nixon et al. [

Figure

Experimental FDCSs for the sum of the

There has been considerable interest in whether a Young’s type double-slit interference effect could be seen for scattering from diatomic molecules. Stia et al. [_{2} could be expressed as the atomic cross-section multiplied by an “interference” factor. The interference factor depends on the molecular separation and the momentum transferred to the residual ion. As a result, the shape of the molecular FDCS was predicted to be different from the shape of the atomic FDCS as modified by the interference factor.

Milne-Brownlie et al. [_{2} FDCS. For the cases they examined, the interference factor predicted that, relative to the binary peak, the molecular recoil peak should be smaller than the atomic recoil peak and this was verified by the experimental data. Consequently, this was interpreted as an observation of double-slit interference effects. A similar study was performed by Staicu-Casagrande et al. [_{2}. Although the Stia et al. [_{2}, it was assumed that the same idea would be applicable for any diatomic molecule. For the three cases considered, the interference factor predicted a reduction in the molecular recoil peak relative to the atomic case and very good agreement between experiment and theory was found using DWBA atomic cross-sections giving further evidence for two-slit effects.

As mentioned in the introduction, there are now numerical techniques which can yield accurate FDCS for single ionization of atomic hydrogen, helium, and more recently molecular hydrogen. In terms of calculating reliable cross-sections, these methods should be clearly superior and preferable to any perturbation technique. Perturbation techniques such as the DWBA are expected to be valid for higher energies and we are amazed (and obviously pleased) at the accuracy of these techniques for the low energy results presented here. On the other hand, these perturbation techniques have some important advantages over numerical techniques. Even though the 6D integrals required for the M3DW are numerically intensive, they are still faster than the numerical methods particularly near threshold. Secondly, the M3DW can be applied to atoms or molecules of any size with equal ease. It is likely to be a very long time before numerical methods will be developed which can treat ionization of FA! Of course, the important question concerns the accuracy of the M3DW predictions and that is still an unanswered question. Finally, and most importantly, one of the great strengths of a perturbation calculation lies in the fact that one can easily investigate physical effects causing observed phenomena.

For example, Al-Hagan et al. [_{2} as shown in Figure _{2} and He) have peaks around 90_{2} has a deep minimum at 180_{2} and it is seen that the 90

Experimental and theoretical FDCS for electron-impact ionization of H_{2} and He targets in the perpendicular plane. The outgoing electron energies were

Experimental and theoretical FDCS for electron-impact ionization of H_{2} and He in the perpendicular plane. The outgoing electron energies were

The next question concerns why there is a maximum at 180_{2}. One of the important (e,2e) mechanisms is backscattering from the nucleus so we decided to investigate the effect of the nuclear charge distributions on the FDCS. Since the distorting potential is a sum of the electronic part plus the nuclear part, one thing we can easily do is to leave the electronic part alone and change the nuclear part. Recall that the nuclear part consists of the charge of the nucleus distributed uniformly on a thin spherical shell centered on the center of mass. For H_{2}, this means that we have a charge of +2 distributed on a thin spherical shell of radius 0.7 a_{0}. To investigate the effect of the nuclear charge distribution, we performed calculations with different sized radii decreasing to a point charge and the results are shown in Figure

Experimental and theoretical FDCS for electron-impact ionization of H_{2} in the perpendicular plane. The outgoing electron energies were _{0} as shown in the figure.

So, it is clear that the nuclei are responsible for the difference between the two targets but what is the physics behind this? If one performs a simple classical impact parameter calculation, it is seen that, for the kinematics of the experiment, an impact parameter of about 0.5 a_{0} is required to scatter into the perpendicular plane. If the classical collision between the projectile electron and target electron took place inside a spherical shell of charge, there would be no attractive force on the electrons. On the other hand, if there were a point nuclei close by, there would be a strong attractive force. Consequently, these observations suggest the following collision mechanisms. For He, elastic scattering places the projectile electron near the nuclei where it has a binary collision with the target electron which produces the 90_{2}, we have the same binary collision but now the two electrons are inside a spherical shell so there is no attractive force which can cause backscattering so all we get are the 90_{2}. However, we would also expect a 180_{2}O as mentioned above and the theory predicts the maximum but it was not found in the experiment.

Although significant theoretical progress for calculating FDCS for electron-impact ionization of molecules has been made in the last few years, there is still much to be done. While the experimental techniques are significantly ahead of the theoretical developments, this is an exciting time since experiments are able to produce excellent data with great detail which provides very stringent tests for theoretical models. The work that has been done so far has provided some valuable insights into the mechanisms of molecular ionization as well as provided some unanswered questions. For example, is the simple model of Al-Hagan et al. [

For the simplest molecule H_{2} and high incident energies, both the M3DW and FBA-TCC results are in qualitative agreement with the binary peak and poor agreement for the recoil peak. For low incident energies, both the M3DW and TDCC provide very good agreement with the shape of the data in the perpendicular plane. The TDCC gives good shape agreement and relative normalization for out-of-plane angles greater than 45

The larger diatomic molecule that has received the most attention recently is N_{2}. For intermediate energies (75 eV and 150 eV), there is reasonable good agreement between experiment and theory for the binary peak and very bad agreement for the recoil peak. For higher incident energies, one would expect the theoretical approaches to be better but this is not the case. For higher energies, the M3DW and FBA-TCC binary peaks are in better agreement with each other than with experiment. The experimental location for the

For the case of low energy ionization of the _{2}O, results completely opposite from H_{2} were found. For H_{2}, the best agreement between experiment and theory was in the perpendicular plane. For H_{2}O, the best agreement between experiment and theory is in the scattering plane and the worst agreement is in the perpendicular plane. For H_{2}, the largest cross-section was found for _{2}O the largest cross-section is in the scattering plane. Since the M3DW has been moderately successful for N_{2}, the big question is whether or not it will also work for even larger molecules and more theoretical and experimental works are required to answer this question.

Finally for the even larger molecule of FA, the fact that the M3DW produced reasonably good agreement with the high energy EMS measurements is very encouraging and indicates the validity of the OAMO at least for the

There has been considerable interest in the possibility of two-center double-slit interference effects for scattering from diatomic molecules. Stia et al. [_{2} and one experiment for N_{2} have found evidence for interference using this method. However, this represents indirect evidence for interference at best and it would be much more satisfying to have a more direct method for observing a double slit interference effect.

Finally, the M3DW results presented so far all rely on the OAMO approximation which is potentially valid for a limited number of states and a limited range of scattering angles. Although the approximation has proved to be surprisingly successful for several cases, it is clearly highly desirable to develop a M3DW calculation that does not use this approximation.

This work was supported by the American NSF under Grant no. 0757749. Ola Al-Hagan would like to acknowledge the support of the Saudi Ministry of Higher Education and King Abdullah Bin Abdul-Aziz Scholarship.