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The past decade has seen extraordinary theoretical progress in the field of fully kinematically resolved atomic electron-impact ionisation, and its closely related field of double photoionisation. These processes were challenging to calculate due to formal and computational difficulties associated with break-up problems involving the long-range Coulomb potential. Presently, however, these processes can be calculated accurately for simple targets such as atomic hydrogen and helium, irrespective of the kinematics considered or the geometry of detectors. We report on the computational progress, and how it has resulted in a deeper understanding of the formalism of Coulomb few-body problems.

We present application examples of a graphical method for the efficient construction of potential matrix elements in quantum physics or quantum chemistry. The simplicity and power of this method are illustrated through several examples. In particular, a complete set of potential matrix elements for electron-lithium scattering are derived for the first time using this method, which removes the frozen core approximation adopted by previous studies. This method can be readily adapted to study other many-body quantum systems.

The two-center convergent close-coupling method is used to calculate antihydrogen $(\overline{\mathrm{H}})$ formation via positronium (Ps) scattering on antiprotons $(\overline{p})$ at near threshold energies. For excited Ps of energy $\ensuremath{\varepsilon}$, the $1/\ensuremath{\varepsilon}$ behavior of the $\overline{\mathrm{H}}$ formation cross sections is valid strictly only at the respective threshold, as is the $1/\sqrt{\ensuremath{\varepsilon}}$ behavior for Ps in the ground state. Simple equations are given for the $\overline{\mathrm{H}}(n\ensuremath{\le}4)$ formation cross sections from $\text{Ps}(n\ensuremath{\le}3)$ from zero to around 0.1 eV above threshold. Some of the implications of using $\overline{p}$-Ps collisions to form antihydrogen in beams, and held in traps, are discussed.

Utilizing the two-center convergent close-coupling method, we find a several order of magnitude enhancement in the formation of antihydrogen via antiproton scattering with positronium in an excited state over the ground state. The effect is greatest at the lowest energies considered, which encompass those achievable in experiment. This suggests a practical approach to creating neutral antimatter for testing its interaction with gravity and for spectroscopic measurements.

The excitation of atomic hydrogen from the ground state to each of the 3s, 3p and 3d states has been studied for incident electron energies from 16.5 to 54 eV. Measurements of total cross sections and differential excitation cross sections (DCSs) for scattered electron angles from 5° to 150° have been made using a variety of coincidence methods. Within the summed experimental uncertainties of about 10% at best from statistical sources and 25% from absolute calibration sources the values calculated using the propagating exterior complex scaling and convergent close-coupling methods show good agreement with observations. DCS calculations for all s, p, d and f final states with n ≤ 4 are also presented over this energy range and reveal a systematic trend with respect to increasing n, final-state angular momentum and incident energy.

In response to the recent measurements of the 54.4 eV e-H(2p) reduced Stokes parameters by Gradziel and O'Neill (2004 J. Phys. B: At. Mol. Opt. Phys. 37 1893) we perform corresponding calculations using two recently developed techniques. These are the propagating exterior complex scaling direct approach to the solution of the underlying Schrodinger equation, and the box-based convergent close-coupling method. We find the results to be almost identical to the Laguerre-based convergent close-coupling method of Bray and Stelbovics (1992 Phys. Rev. A 49 1066). Hence we are unable to explain the substantial discrepancy with experiment for the P∼ 3 parameter in the vicinity of 30°.

A standard technique for solving three-dimensional momentum-space integral equations in scattering theory is their transformation into one-dimensional equations in terms of partial waves. However, for some scattering systems where a large number of partial waves contribute this technique is not efficient. In this work we explore the alternative approach of solving these equations directly without partial-wave expansion. For illustrative purposes we adopt the coupled-channel approach and consider a well-studied static-exchange model of electron-hydrogen scattering.

The Laguerre-based convergent close coupling method is used to calculate the electron-impact excitation cross sections of the auto-ionizing states 2s2s1S, 2s3s1S and 2s3s3S, various ionization, as well as other cross sections for scattering from the 11S, 2 1S and 23S initial states of helium. The calculations are done within the S-wave model. Unlike the case of bound-state excitation, convergence studies need to take into account occasional mixing between the core manifolds which can severely affect the structure of the auto-ionizing states and those corresponding to ionization with excitation. A method to take this into account is given and it is shown that convergent results may be obtained, Comparison with the 'Z = 2' time-dependent calculations of Pindzola et al (1999 Phys. Rev. A 59 4390-8) is satisfactory.

We use three different implementations of the non-relativistic close-coupling method to investigate the convergence of theoretical predictions for electron scattering from magnesium at impact energies below 1 eV. It is found that the convergence of the close-coupling expansion is extremely slow, particularly for the L = 1 partial wave that is dominated by a shape-resonance common to many similar quasi-two-electron systems. An accurate prediction of the scattering length (we obtain −2.4 ± 0.1 a0) is also a nontrivial problem. Although the shape resonance in the L = 1 channel can be predicted in agreement with experiment already in small close-coupling expansions with three or nine coupled states (see Bartschat and Sadeghpour 2003 J. Phys. B: At. Mol. Opt. Phys. 36 L9 for details), this fortuitous agreement is due to simultaneously inaccurate descriptions of both the N-electron target and the (N + 1)-electron collision problems. We demonstrate how a more systematic though slow convergence pattern of the close-coupling expansion can be achieved by concentrating on a highly accurate target description before attempting to solve the collision problem.

The single-centred convergent close-coupling method has been applied to positron-helium scattering at energies from the first ionization threshold to 1 keV for calculating one-electron excitation and ionization processes. Total scattering, positronium formation plus single ionization (unresolved), and target excitation cross sections have been calculated. It is found that the often-used frozen-core approximation for the target structure leads to a small but systematic overestimation of the cross sections. Generally, the agreement between the present and most recent calculations, as well as experiment, is satisfactory.

A newly-derived iterative coupling procedure for the propagating exterior complex scaling (PECS) method is used to efficiently calculate the electron-impact wavefunctions for atomic hydrogen. An overview of this method is given along with methods for extracting scattering cross sections. Differential scattering cross sections at 30 eV are presented for the electron-impact excitation to the n = 1, 2, 3 and 4 final states, for both PECS and convergent close coupling (CCC), which are in excellent agreement with each other and with experiment. PECS results are presented at 27.2 eV and 30 eV for symmetric and asymmetric energy-sharing triple differential cross sections, which are in excellent agreement with CCC and exterior complex scaling calculations, and with experimental data. At these intermediate energies, the efficiency of the PECS method with iterative coupling has allowed highly accurate partial-wave solutions of the full Schrodinger equation, for L ≤ 50 and a large number of coupled angular momentum states, to be obtained with minimal computing resources.

The convergent close coupling method has been applied to positron?helium scattering for energies below the positronium formation threshold (17.8?eV). Convergence is studied as a function of the Laguerre basis size. Rapid convergence to the accurate phase shifts given by Van Reeth and Humberston (1999 J.?Phys.?B:?At.?Mol.?Opt.?Phys.?32 3651?67) is demonstrated. The resultant cross sections are in excellent agreement with the experimental data of Stein et al (1978 Phys.?Rev.?A?17?1600?8) and Mizogawa et al (1985 Phys.?Rev.?A 31 2171?9).

Born total ionization cross section was calculated by an algebraic computing program using Maple. The atomic orbitals were approximated by Hartree-Fock Slater functions, and the ejected electron was modeled by a hydrogenic Coulomb wave. The numerical integrations were performed using FORTRAN by automatically converting the analytic matrix element for each orbital into a FORTRAN subroutine.

We apply a new implementation of the convergent close-coupling (CCC) method to electron–helium scattering. The target states are obtained from one-electron He+ box-based eigenstates rather than the usual Laguerre-based orbitals. The utility of the new method is demonstrated for 50 eV electron-impact ionization of helium with three different energy sharings between the two outgoing electrons. Excellent agreement is found between previous and new CCC predictions, and also with experimental data.

The convergent close-coupling (CCC) method is applied to the calculation of electron–helium ionization doubly differential cross sections (DDCSs) at low to intermediate incident energies. We re-examine the CCC calculations and measurements of Roder et al (1997 J. Phys. B: At. Mol. Opt. Phys. 30 1309–22) by making allowance for the step-function behaviour of the underlying CCC-calculated amplitudes. As done previously, the experimental DDCS were normalized at energies below 100 eV using the 100 eV CCC calculation to determine analyser properties at several secondary energies. In addition, substantially larger calculations are presented, to check the convergence. The agreement between the experiment and the calculations as a whole is much improved on the situation reported earlier.

The substantial progress that has occurred during the 1990s in the field of electron-atom-collision theory is discussed. We show how a solution of a small-model three-body problem, using the convergent close-coupling method, has led to numerous applications involving real atomic collision systems. Consequently many fundamental electron-atom collision processes are considered as 'solved', and accurate collision data of interest to science and industry have become available. However, we suggest that the present has only just seen the birth of modern atomic collision theory. There are many more important collision problems to be tackled, with guidance coming from experiment being as important as ever.

Recently Zatsarinny and Bartschat [Phys. Rev. Lett. 107, 023203 (2011)] have given an ansatz for extracting ionization amplitudes from close-coupling calculations of electron-impact ionization of atoms. They applied it with extraordinary success to a fully differential cross section of electron-helium single ionization leaving the residual ion in an $n=2$ state. By considering electron-impact ionization of atomic hydrogen we explain the origin of the ansatz and show that it forms an effective interpolation scheme for determining the amplitudes, so long as the pseudostate energy distribution is sufficiently dense.

Benchmark results are presented for electrons colliding with hydrogen atoms in the S-wave (Temkin-Poet) model collision problem, which neglects angular momentum. Complete results (elastic, inelastic, and ionization), accurate to 1%, are obtained by numerically integrating Schrodinger's equation subject to correct asymptotic boundary conditions. This marks the first time direct matching to asymptotic boundary conditions has been shown to yield convergent ionization amplitudes for a Coulomb three-body problem. Results are presented for impact energies of 54.4 and 40.8 eV, where comparison with other theories is available.

Scattering amplitudes in the first Born approximation for electron-hydrogenic ion scattering are readily solved analytically for target's undergoing near ground state transitions. However, analytic solutions involving transitions to higher energy states are not so easily performed. We present a computer algebra solution and a Maple V Release 4 software package for analytically solving the first Born approximation to the electron-hydrogenic atom scattering problem. Given initial and final states for the e−target system, symbolic solutions for scattering amplitudes, differential cross sections and total cross sections can be calculated using this Maple package.

Dipole series in H− and in the positron-hydrogen system are studied in a coupled-channel calculation, in which the short-range part of the interaction between the electron or positron and the excited hydrogen core is modeled by a local potential, adjusted to reproduce low-lying states of the respective series. Consideration of the fine-structure splitting and radiative corrections enables us to describe the termination of the series towards the nondegenerate threshold. For the 1Se series below the N=2 threshold we give quantitative predictions for the number of resonances and their energies.

The two-center approach to positron-impact ionization of atomic hydrogen is shown to follow from the exact post form of the breakup amplitude [Kadyrov, Bray, Mukhamedzhanov, and Stelbovics, Phys. Rev. Lett. 101, 230405 (2008)]. In such approaches distinct ionization amplitudes arise from each center for the same ionization process. The fully differential cross section for the positron-impact breakup of atomic hydrogen is calculated including direct ionization of the target and electron capture into the positronium continuum. We show that the coherent combination of the amplitudes leads to oscillations in the differential cross sections, whereas the incoherent combination does not. The latter has also the advantage of being consistent with the unitary close-coupling formalism.

We apply a convergent close-coupling study of 54.4 eV electron-impact excitation of the n = 3 states of atomic hydrogen to circular polarization arising from the 3D state. There have been recent measurements followed by several theory calculations, some of which provide better agreement with the data than others. The most recent DWBA models suggest fair agreement with the data. In order to resolve the disagreements we have carried out a substantial set of convergent close-coupling calculations in order to estimate the effects of approximations present in the earlier theory models. Our conclusion is that the substantial discrepancy with experiment, first identified by Kumar et al, remains.

We present a general analysis of the close-coupling equations for e-He scattering. We show why the standard close-coupling expansion gives rise to nonunique solutions and, by characterizing the manifold of nonunique solutions, are able to construct equations with a unique solution. Our analysis concentrates on two models of target states; a frozen-core model in which one of the electrons is restricted to the 1s He+ orbital, as has been used with great success recently, and also a model which uses full configuration-interaction target states. We develop modified close-coupling equations by projecting out spurious solutions. A subtle difference between frozen-core and configuration-interaction models with respect to the manner in which antisymmetrization is incorporated is noted. A calculation to illustrate the theory is presented.

Msezane and colleagues have developed a `universal formula' to interpolate between the optical oscillator strength (OOS) and generalized oscillator strengths (GOS) as a function of the square of the momentum transfer K. They have inferred regions of applicability of their formula, then made deductions about theoretical and experimental differential cross sections and estimated integral cross sections for optically allowed transitions in atoms and molecules. This comment raises some questions about their work.