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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.

Ionization of atoms by electrons is one of the most common atomic scattering processes. We review the progress in modeling the ionization process from early quantum models in the 1920s to the present day. Despite the fact that ionization is so fundamental, the path to precision calculations even for the simplest systems has been slow and tortuous. In the past decade major progress has been achieved with the aid of new methods and rapid increase in computing power and we illustrate some of the important developments by focusing on total ionization cross-sections.

We formulate scattering theory in the framework of a surface-integral approach utilizing analytically known asymptotic forms of the two-body and three-body scattering wavefunctions. This formulation is valid for both short-range and long-range Coulombic interactions. New general definitions for the potential scattering amplitude are presented. For the Coulombic potentials, the generalized amplitude gives the physical on-shell amplitude without recourse to a renormalization procedure. New post and prior forms for the Coulomb three-body breakup amplitude are derived. This resolves the problem of the inability of the conventional scattering theory to define the post form of the breakup amplitude for charged particles. The new definitions can be written as surface-integrals convenient for practical calculations. The surface-integral representations are extended to amplitudes of direct and rearrangement scattering processes taking place in an arbitrary three-body system. General definitions for the wave operators are given that unify the currently used channel-dependent definitions.

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.

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.

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).

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.

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.

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.

We consider positron scattering on the alkali-metal atoms of Li, Na, and K at very low energies, where only the elastic scattering and positronium formation in the ground state are the two open channels. Utilizing the recently developed two-center convergent close-coupling method [Lugovskoy, Kadyrov, Bray, and Stelbovics, Phys. Rev. A 82, 062708 (2010)] we investigate the behavior of the cross sections as the impact energy goes to zero and demonstrate their convergence. The study sets quantitative benchmarks for any rigorous theoretical treatment of the collision problems.

Positron scattering from atomic hydrogen is studied at energies ranging from 4 to 100 Ryd using the close-coupling approach and cross sections from both the - H and Ps - p entrance channels are reported in a basis comprising 1s, 2s and 2p states of the hydrogen and positronium atoms. Comparative studies with widely used approximation methods such as the first Born, unitarized Born and second Born approximations are presented. With the exception of elastic scattering for the Ps(1s) - p entrance channel, it was found that on-shell approximations such as the first Born and the unitarized Born approximations are reasonably accurate for collisions not involving particle transfer provided the energy is sufficiently high. Rearrangement collisions on the other hand are dominated by off-the-energy-shell scattering and the accurate approximation at high energies is the second Born approximation. Further, the effects of positronium channels on the direct reactions are investigated and we conclude that the positronium formation channels can be omitted for energies greater than 20 Ryd. Elastic scattering in the Ps(1s) - p entrance channel is seen to be unique, the high-energy behaviour of the phase shifts is different from any other quantum system.

The recent technological advances in electron-atom scattering experiments enable scientists to examine experimentally the excitation of not only the lowest, but also higher-lying states. The important question then arises as to whether these higher states contain significant new information. It has been suggested that the scattering amplitudes should scale with principal quantum number and thus contain no new information. The purpose of this work is to examine the N-dependence of the electron-hydrogen scattering. Two different approaches, the convergent close-coupling and second-order distorted-wave methods are used to calculate scattering amplitudes for excitation of the N=2, 3 and 4 levels of the atomic hydrogen. The integrated and differential cross sections, polarization fractions and angular-correlation parameters are discussed for energies from 20 to 1000 eV.

The accurate solution of the Schrodinger equation (SE) for electron-impact collisions leading to discrete elastic and inelastic scattering progressed rapidly with the increase in computing power from the 1970s. A review of the principal methods, including second Born, distorted wave, R-matrix, intermediate-energy R-matrix, pseudo-state close coupling and optical model is given in [1]. However, electron impact collisions leading to ionization on even the simplest atom, hydrogen, were by comparison poorly described; significant progress dates only from the early 1990s when Bray and Stelbovics [2] developed a technique called convergent close coupling (CCC). In this approach they used an in-principle complete set of functions to approximate the hydrogenic target states, both bound and continuous, and used the coupled channels formalism to expand the scattering wave function in these discretized states, reducing the solution of the SE to a set of coupled linear equations in a single co-ordinate. The method was tested in a non-trivial model [3] and shown to provide convergent cross sections not only for discrete elastic and inelastic processes but also for the total ionization cross section. Shortly thereafter the method was applied to the full collision problem from atomic hydrogen and one of the major achievements of the method was that it yielded essentially complete agreement with the (then) recent experiment for total ionization cross section [4]. In the following years, the method was applied to other atoms with considerable success; the range of applications of CCC are covered in the review of Bray et al. [5].

Calculations of positron-hydrogen scattering using the close-coupling approach are reported for incident positron energies below the ionization threshold. The channel space includes nine physical hydrogen and positronium states and, in addition, eleven hydrogen and positronium pseudostates. The accuracy of our model has been validated by computations of elastic phase shifts below the positronium threshold, and elastic and positronium-formation cross sections in the Ore gap. Calculations are performed for sufficient partial waves to obtain converged elastic, positronium, and total cross sections.

The simplified model of electron-hydrogen atom scattering which has all orbital angular momenta set to zero is solved to convergence at a range of energies from 1 to 400 eV using a general coupled-channel approach. The results are in complete agreement with the solution of this problem by Poet using a method that is applicable to this model only. The corresponding singlet and triplet T-matrix elements for the transitions from the ground state to each one of 1s, 2s, and 3s states are presented in tabular form. The 1s, 2s, 3s, ionization, and total cross sections are also presented for each energy.

Close coupling calculations of positron-hydrogen scattering using a six state model (H(1s, 2s, 2p), Ps(1s, 2s, 2p)) are completed for incident energies from zero to 4.0 Rydberg. The calculations are very comprehensive and all cross sections and phaseshifts for the e+-H(1s) entrance channels have been computed. The cross sections for positronium formation in the Ps(1s), Ps(2s) and Ps(2p) levels underestimate the latest experimental values. Cross sections are also computed for the Ps(1s)-p entrance channel, and these to our knowledge represent the first detailed calculation of the Ps(1s)-p scattering. Particular attention is paid to the cross sections for electron transfer since these are relevant to proposed experiments to form antihydrogen in sufficient quantities to perform spectroscopic experiments. Resonances are identified at incident positron energies both below and above the ionization threshold.

Positron scattering from atomic hydrogen is analysed above the ionization threshold and resonances for coupled-channels calculations comprising bases of up to three physical states of hydrogen H(1s,2s,2p) and three physical states of positronium Ps(1s,2s,2p) are determined. The calculations have been performed over the entire energy range from the ionization threshold up to a maximum incident positron energy of 4 Ryd. Additional resonances appear as the models increase in complexity. For the six-state model the authors report the identification of one new resonance for the J=0 partial wave, four for J=1, one for J=2 and one for J=3. No resonances are found in the higher partial waves. The positions and widths of the resonances are given.

The calculation of electron–atom excitation and ionization cross section is considered in both the non‐relativistic and relativistic scattering theory. We consider electron collisions with H, He, Cs, and Hg. Differential cross sections for elastic scattering and ionization are presented.

We report the first measurements and associated theory for angular correlations of sequential cascading photons for an atomic system in which a scattering plane is defined. The cascade radiation from the [ital n]=3 levels of atomic hydrogen excited by 290 eV electrons is observed. The first determinations of a rank-4 multipole and of the magnetic sublevels of the 3[ital d] state have been made. The measured angular correlations and the deduced multipole moments have order-of-magnitude agreement with Glauber, Born, and six-state close-coupling model predictions.

We present the total ionization cross section and spin asymmetry at projectile energies ranging from threshold to 500 eV for electron-impact excitation of atomic hydrogen. They are calculated using the convergent close-coupling formalism of Bray and Stelbovics [Phys. Rev. A 46, 6995 (1992)]. Both observables are found to be in complete quantitative agreement with measurements over almost the entire energy range. This is the only electron-atom scattering theory that is able to achieve this result to date.

The convergent close-coupling method has been applied to positron scattering on a helium atom in the 2 {sup 3}S metastable state. For this system the positronium (Ps) formation channel is open even at zero scattering energy making the inclusion of the Ps channels especially important. Spin algebra is presented for the general case of arbitrary spins. A proof is given of the often-used assumption about the relationship between the amplitudes for ortho-positronium and para-positronium formation. The cross sections for scattering from 2 {sup 3}S are shown to be significantly larger than those obtained for the ground state.

We present main features of a surface-integral approach to the Coulomb few-body scattering problem. This approach is valid for both short-range and Coulombic longe-range interactions. We give new general definitions for the potential scattering amplitude. For the Coulombic potentials the generalized amplitude gives the physical on-shell amplitude without recourse to a renormalization procedure. New post and prior forms for the amplitudes of breakup, direct and rearrangement scattering in a Coulomb three-body system are also presented. The Green’s functions and formal solutions of the Schrödinger equation in integral form are not used. Therefore, for the purpose of defining the scattering amplitudes the knowledge of a complicated analytic structure of the Green’s function in the complex-energy plane is not required.

The convergence of the close-coupling formalism is studied by expanding the target states in an orthogonal ${\mathit{L}}^{2}$ Laguerre basis. The theory is without approximation, and convergence is established by simply increasing the basis size. We present convergent elastic, 2s, and 2p differential cross sections, spin asymmetries, and angular-correlation parameters for the 2p excitation at 35, 54.4, and 100 eV. Integrated and total cross sections as well as T-matrix elements for the first five partial waves are also given.

Electron impact excitation of atomic hydrogen is reviewed with the aim of determining the maximum amount of information about the inelastic scattering from the ground state to the n=2,3 levels. In particular the information obtainable from (e,γ) (e,γ,γ) and (γ,γ) coincidence measurements is discussed.

Proton and antiproton collisions with atomic hydrogen are investigated using the atomic-orbital close-coupling method. After a suitable transformation, the method leads to coupled three-dimensional momentum-space integral equations for the half-off-shell scattering amplitudes. These equations are further reduced to two-dimensional ones using the symmetry properties of the transition amplitudes. The resulting equations are solved directly without partial-wave expansion. The total and differential cross sections for electron transfer in proton collisions with the ground state of atomic hydrogen are calculated. The contribution of off-shell effects is fully taken into account and shown to be significant. We find that the off-shell effects can both increase or decrease the cross sections substantially depending on energy. At high energies as much as a half of the total electron transfer cross section comes from the off-shell effects. This contradicts classical and iterative estimates which predict the off-shell effects to be small. The significance of the off-shell effects in elastic scattering of protons falls with increasing energy to the estimated levels. The calculated cross section results agree well with experiment over a wide energy range. The total and differential cross sections for elastic scattering of antiprotons are also calculated and compared with the corresponding results for protons. Above 200 keV there is practically no difference between the cross sections for scattering of protons or antiprotons. The off-shell effects for elastic scattering of antiprotons are generally smaller than those for scattering of protons.

We formulate scattering theory in the framework of a surface-integral approach utilizing analytically known asymptotic forms of the three-body wave functions. This formulation is valid for both short-range and Coulombic potentials. The post and prior forms of the breakup amplitude are derived without any reference to renormalization procedures.

The three-body approach to the single-scattering optical potential proposed by Tandy, Redish, and Bolle is tested in the context of ..pi..-d scattering where the various approximations are compared with the exact solution of the Faddeev equations. For ..pi..-d scattering we find that the integral equations derived for the Kerman-McManus-Thaler form of the single-scattering optical potential are markedly superior to those of the Watson form. Our analysis includes a study of the convergence properties of the various multiple-scattering series encountered.

The application of an exterior complex scaling method to an atomic scattering problem with distinct rearrangement channels is reported. Calculations are performed for positron-hydrogen collisions in an s -wave model employing an electron-positron potential of V12 =- [8+ (r1 - r2) 2] -1/2, using the time-independent propagating exterior complex scaling method. This potential has the correct long-range Coulomb tail of the full problem and the results demonstrate that exterior-complex-scaling-based methods can accurately calculate scattering, ionization,and positronium formation cross sections in this three-body rearrangement collision.

The scattering theory implied by the close-coupling equations is studied using a Lippmann-Schwinger formalism. The new results derived can be summarized as follows: An alternative form of the equations that ensures there are no spurious solutions in the scattering region can be constructed, and moreover there is an infinite number of such forms. The Neumann- (perturbation-) series expansion diverges in general for most energies for both the old and new forms. The Born limit nevertheless holds and can be recovered by appropriate rearrangement of the Neumann series. The original integral formulation may give convergent scattering amplitudes despite the lack of uniqueness of the solutions. The conditions under which this happens are examined.

Antihydrogen formation in positronium scattering on antiprotons is investigated using the two-centre convergent close-coupling method. A several orders of magnitude enhancement in the formation of antihydrogen is found when positronium is in an excited 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 existing formulations of electron-impact ionization of a hydrogenic target suffer from a number of formal problems including an ambiguous and phase-divergent definition of the ionization amplitude. An alternative formulation of the theory is given. An integral representation for the ionization amplitude which is free of ambiguity and divergence problems is derived and is shown to have four alternative, but equivalent, forms well suited for practical calculations. The extension to amplitudes of all possible scattering processes taking place in an arbitrary three-body system follows. A well-defined conventional post form of the breakup amplitude valid for arbitrary potentials including the long-range Coulomb interaction is given. Practical approaches are based on partial-wave expansions, so the formulation is also recast in terms of partial waves and partial-wave expansions of the asymptotic wave functions are presented. In particular, expansions of the asymptotic forms of the total scattering wave function, developed from both the initial and the final state, for electron-impact ionization of hydrogen are given. Finally, the utility of the present formulation is demonstrated on some well-known model problems.

The differential ionization cross sections for charged hydrogenic targets with low Z at low to moderate energies were calculated using propagating exterior complex scaling (PECS) method. The PECS method with iterative coupling proved to be highly efficient, providing an estimated 100-fold reduction in total computation time compared with the PECS method using direct coupling. Scattering wave functions were calculated for the electron impact of hydrogenic targets with nuclear charge Z≤4. The results show that the angular distributions of the differential cross sections change systematically with increasing nuclear charge for energies above the peak total ionization cross section.

We present an efficient generalization of the exterior complex scaling (ECS) method to extract discrete inelastic and ionization amplitudes for electron-impact scattering of atomic hydrogen. This fully quantal method is demonstrated over a range of energies for the collinear and Temkin-Poet models and near-threshold ionization is examined in detail for singlet and triplet scattering. Our numerical calculations for total ionization cross sections near threshold strongly support the classical threshold law of Wannier [Phys. Rev. 90, 817 (1953)] (σ∝E1.128±0.004) for the L=0 singlet collinear model and the semiclassical threshold law of Peterkop [J. Phys. B 16, L587 (1983)] (σ∝E3.37±0.02) for the L=0 triplet collinear model, and are consistent with the semiclassical threshold law of Macek and Ihra [Phys. Rev. A 55, 2024 (1997)] (σ∝exp[(-6.87±0.01)E-1/6]) for the singlet Temkin-Poet model.

A fully quantal calculation of the {sup 1}S three-body wave function is performed for the Wannier model of electron-hydrogen scattering in the near-ionization threshold region using an exterior complex scaling method. The region of configuration space of the wave function that provides the dominant contribution to the total ionization cross section is demonstrated to be r{sub 1}{approx_equal}r{sub 2} in accord with the argument of Rau [Phys. Rev. A 4, 207 (1971)], but only in the so-called Coulomb zone. The work confirms to a high precision (E{sup 1.128{+-}}{sup 0.004}) the Wannier threshold law for the total ionization cross section, which is strictly valid only at the threshold. A threshold law for the {sup 3}S total ionization cross section is determined to be E{sup 3.37{+-}}{sup 0.02}, in agreement with the semiclassical calculations of Peterkop [J. Phys. B 16, L587 (1983)].

According to quantum collision theory, scattering amplitudes are complex numbers, which are completely defined by their magnitude and phase. Although the phase information is generally not determined entirely in collision experiments, the phases are well defined and can be used to check computational models. We use four state-of-the-art approaches to calculate the magnitude and phase of the electron-hydrogen ionization amplitude in the Temkin-Poet S-wave model. We demonstrate that the correct phase can be extracted for each method by using the appropriate final-state continuum functions.

We outline progress in a broad range of research projects that are currently being undertaken in our group. These include the formulation of ionisation amplitudes in Coulomb few‐body problems, a new convergent close‐coupling implementation for electron‐helium collisions, elastic electron‐magnesium scattering at very low energies, and the calculation of electron‐cesium spin asymmetries and differential cross sections.

The total and total ionization cross sections for positron scattering on atomic hydrogen are calculated by applying the convergent-close-coupling method to the model where positronium-formation channels are omitted. This model accurately describes the physics of the scattering whenever the positronium formation cross section is negligible, in particular, above 100 eV for this system. The total ionization cross section results in this energy region are in excellent agreement with the recent measurements of Jones et al. [J. Phys. B 26, L483 (1993)], and so lie below the earlier measurements of Spicher [Phys. Rev. Lett. 64, 1019 (1990)], and the recent calculations of Acacia et al. [Phys. Rev. Lett. (to be published)]. The total cross section is in very good agreement with the recent measurements of Zhou et al. (unpublished) down to 30 eV.

The electron-helium scattering system is studied within the S-wave framework with a particular emphasis on one-electron ionization processes. Total ionization and the singly differential cross section for the equal-energy-sharing point are given for three initial states. In addition, discrete excitation cross sections are also given. Whereas generally there is no a priori relationship between cross sections for excitation of corresponding singlet and triplet states, we find that for the case of ionization with equal-energy outgoing electrons the triplet cross section is approximately three times bigger than the corresponding singlet one, irrespective of the initial state or energy.

We briefly review recent progress in the field of electron-impact ionization of light atoms concentrating on those theories which attempt to fully solve the underlying scattering problem. Comparison between competing theories and experiment shows up some unexpected discrepancies.

The convergent close-coupling approach developed by the authors is applied to positron scattering from atomic hydrogen below the first excitation threshold. In this approach the multichannel-expansion one-electron states are obtained by diagonalizing the target Hamiltonian in a large Laguerre basis. It is demonstrated that this expansion of the scattering wave function is sufficient to reproduce the very accurate low-energy variational results, provided target states with l\ensuremath{\le}15 are included in the expansions.

An overview is given of recent progress in the calculation of positron scattering on atoms and molecules using the convergent close-coupling method. Particular emphasis is given to those cases where positronium formation is one of the reaction channels, as well as the importance of demonstrating convergence with increasing orbital angular momentum of the bases used. Targets considered are atomic hydrogen, lithium, and molecular hydrogen. The last two decades have seen extraordinary progress in the field of electron, positron and photon scattering on atoms and ions. The problems of electron and photon scattering on atoms are very closely related. In the typical case of single photon absorption, the interaction proceeds by the resulting photo-electron scattering on the residual ion. For example, photon-helium scattering is essentially electron scattering on the singly charged helium ion. Positron-atom scattering is a little more interesting due to the possibility of positronium (Ps) formation. This is a rearrangement collision that considerably increases the complexity of the problem. Though it has been often claimed that positron-atom scattering is simpler than the corresponding electron-scattering problem due to the absence of exchange, in practice the introduction of the Ps-formation channel creates considerably more significant challenges. Historically, computational approaches to the problems have been subdivided into the low-, intermediate- and high-energy regimes. In addition, excitation and ionisation processes have also received different treatments. However, our interest in developing the convergent close-coupling (CCC) method has been to unify the approach to all such problems to be valid for the three projectiles across all energies and for the major excitation and ionisation processes. In developing the CCC method for excitation we took note of the techniques used specifically in their regimes of validity. At low energies the R-matrix close-coupling approach (1) has yielded outstanding results. At the higher energies the perturbative approach (2) has been particularly successful. The CCC method (3) combines the two techniques because it formulates close- coupling as coupled Lippmann-Schwinger equations in momentum space, which may be readily expanded in a perturbative series. Furthermore, the coupled equations may be solved in a distorted-wave formalism (4). However, unlike distorted-wave approximations the CCC results are independent of the choice of the distorting potential. In this sense the usage of such a potential is solely for numerical ease of solution. Following the pioneering implementation of the close-coupling method to ionisation processes (5), we developed an even simpler CCC approach (6). Rather than reconstructing the total wavefunction of the electron-atom system, we associated ionisation amplitudes with excitation of the positive-energy pseudostates. In other words, we extracted the required

An adaptation of the convergent close-coupling method (CCC) to calculation of differential ionization cross sections is analyzed in the context of the Temkin-Poet model. The asymptotic scattering wave functions and the unitarity relation are given for the model. It is concluded the use of "distinguishable" electrons as proposed in the CCC model is not essential in the close-coupling formulation of ionization., Comment: 4 pages, 1 table, 1 figure, 16 references

The convergent-close-coupling (CCC) method is illustrated for the Temkin-Poet model of electron-hydrogen scattering. This model treats only states of zero orbital angular momentum, but has played an important role in the development and testing of methods of solution for general electron-atom scattering systems. This chapter describes the features of the model and discusses the application of the CCC approach to its numerical solution. All scattering processes, elastic, inelastic excitation, and ionization, are found from a single unified calculation. The computer program associated with this chapter is the much reduced version of the general CCC code, but contains the essential techniques for solving large sets of coupled equations.

We concentrate in this chapter on one of the latest electron scattering methods, the covergent close-coupling (CCC) method, developed by the authors. It takes the close-coupling (CC) formalism to completeness, and the present indications are that it satisfies the requirements of a general, reliable, and practical scattering theory. The target states are obtained by diagonalizing the target Hamiltonian in a large truncated orthogonal Laguerre basis. This ensures that the obtained target states are all square integrable. The orthogonality of the basis ensures that {open_quotes}completeness{close_quotes} of the expansion is approached as the basis size is increased. We say we have convergence whenever further increase in basis size does not have a significant effect on the results of interest. The CCC method has proved to be one of the most successful in describing electron scattering from light targets. We take this opportunity to combine here the ideas that appear in a number of publications and provide further detail not given elsewhere. This chapter is structured in the following way. First, we will give an outline of the various electron scattering methods currently in use, then discuss their strengths and weaknesses, and contrast these with the CCC method. This will be followed bymore » a section devoted to the detailed description of the CCC method. Subsequently, various comparisons of experimentation, the CCC method, and other available theories will be presented for a number of targets. We will not attempt to give a comprehensive survey, but rather concentrate on issues of greatest interest to us, namely where treatment of the target continuum is of great importance or where there are unresolved discrepancies with experiment.« less