| Summary: | The convergent close-coupling (CCC) method was developed in order to resolve the long-standing discrepancy between two consistent experiments and all available theories for 2p excitation of atomic hydrogen [1] . The method was unable to resolve this discrepancy, but subsequent experiments [2,3] found much more in favor of theory than the previous experiments. There have been a number of reviews of the applications of the CCC theory with the most recent one being by Bray et al. [4] . The method has been extended to ionization [5], resulting in some controversy [6,7] that required further explanation [8,9]. Our own confidence in the ability of the CCC method to reproduce electron— hydrogen fully differential ionization cross sections was shaken by the less than satisfactory agreement with experiment [10] . However, this turned out to be primarily due to insufficient computational resources available at the time [11]. Consequently, we are now confident that the CCC method is able to solve the e—H, γ—He, and e—He (within the frozen-core model) collision systems at all energies with one or two outgoing electrons. We shall attempt to explain here the underlying foundations as clearly as possible. The example of the S-wave model will be used to demonstrate the method. A published program is available that shows the workings of the method discussed here [12] . We will finish by concentrating on the application of the method to fully differential ionization processes.
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