Coupled-channel calculations have been performed at 27.5, 30, and 50 eV for the reaction ${\mathrm{He}}^{+}+\mathrm{He}\to {\mathrm{He}}^{+}+\mathrm{He}\left(2\mathrm{}{}_{}{}^3{}_{}{}^{}S\right)+19.81\mathrm{}$ eV using presently available ab initio potentials for the two lowest ${}_{}{}^2{}_{}{}^{}\Sigma _g^{+}{}_{}{}^{}$ states. The inelastic-scattering differential cross sections have been calculated at these energies and are compared with recent experimental results. Several features are distinguished on the calculated cross sections. All possess low-frequency Stückelberg oscillations and a higher-frequency oscillation due to the nuclear symmetry of the reactants. At the two lowest energies of 27.5 and 30 eV, inelastic rainbow-type oscillatory structure is also observed. The inelastic rainbow structure was due to the maximum on the excited-state potential and the closeness of the collision energy to the thershold for this reaction. The elastic differential cross sections were calculated at 50 eV, and perturbations due to the crossing of the inelastic channel were found superimposed upon the electron and nuclear exchange oscillations.

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