Ionization of atoms under charged heavy particle impact is a fundamental phenomenon, both from the point of view of theory and practical applications. The process of single ionization of helium presents a difficult few-body Coulomb problem above the break-up threshold for three freely charged particles.

We recently developed an approach based on the expansion in terms of parabolic convoluted quasi-Sturmians (CQSs). These basis functions are obtained by applying the Green's function operators of two models (named 2C and IIC in PRA, 111, 052812 (2025)), which involve the interaction of two pairs of particles using the corresponding Jacobi coordinates. Our goal here is to extend the CQS approach by including all Coulomb interactions when constructing the basis set. Specifically, we propose an ansatz for the three-body Green's function in the form of a modification of the 2C model operator by equipping the latter with factors that are consistent with the 3C model. In this work, to ensure the effectiveness of this approach, we restrict ourselves to the zeroth-order approximation, which directly reduces to the 3C model.

Calculations are performed for proton-impact ionization of atomic helium at 75 keV for different regimes that have been explored experimentally and theoretically. The ejected electron energies are taken to be below, nearly equal to, and above the cusp energy, i.e., the energy corresponding to the proton–electron velocity matching regime. Comparison of the results of the present 3C calculations with the cross-sections obtained with the 2C (for electron and proton velocities, which are quite different) and IIC models (for velocity matching) allows us to reveal the role of various pairwise interactions in the possible ionization mechanisms.

The achieved agreement between the calculated cross-sections and the results obtained by other theoretical methods, both perturbative and ab initio (e.g., WP-CCC), confirms the effectiveness of the developed approach.