## 1 Introduction
Excitation and ionization of the valence shells of the neutral species of these elements have been extensively studied and experimental reports and theoretical calculations about their photoionization cross sections are available in literature. In contrast to this plethora of data, the availability of analogous studies of their ionized species are scarce, especially when few-photon ionization is involved. To provide one example illustrating the need for such data, we mention the case of neon's multiple ionization in the pioneering experiment of [PERSON] et al. [3] at LCLS with a FEL pulse at a photon energy circa 1110 eV (pulse peak intensity and FWHM duration were estimated to be \(10^{17}\) W/cm\({}^{2}\) and 100 fs, respectively). In this experiment, the ionization of Ne\({}^{8+}\), following the sequential one-photon stripping of the lower charged neon ions, proceeds mainly through a two-photon absorption (the ionization potential of Ne\({}^{8+}\) is circa 1362 eV). It is needless to say that the same two-photon ionization channel would have been the dominant ionization channel for any photon in the energy range between 681 eV \(\leq\omega\leq 1362\) eV; thus, the need for two-photon ionization cross sections for a range of photon energies. Of course, similar considerations can be carried over to any atomic system and of any degree of charge in the presence of X-ray radiation. In the simplest case of the one-electron charged ions, multiphoton cross sections can be straightforwardly calculated by scaling the corresponding hydrogen cross sections according to the relation \(\sigma_{N}(Z^{2}\omega)=\sigma_{N}^{(H)}/Z^{4N-2}\)[11], where \(\sigma_{N}^{(H)}\) is the \(N\)-photon cross section of hydrogen and \(Z\) is the atomic number of the element. Throughout the years, a number of reports have appeared in literature with multiphoton cross sections of hydrogen involving quite a high number of photons as, for example, the calculations by [PERSON] [12] and by [PERSON] and [PERSON] (\(N=20\)) [13]. Next in line are charged ions with only their K-shell electrons remaining, making them helium-like (two-electron) systems. For helium, systematic studies exist where two, three- and four-photon LOPT ionization cross sections are calculated [14]; it appears, despite the significance of Li, Ne and Ar as experimental and theoretical targets, that no similar detailed study has been extended to these systems. To the best of our knowledge, we are aware of the two- and three-photon total ionization cross sections that have been calculated on Li\({}^{+}\) with the use of single-channel quantum-defect theory (SQDT) [15], the work of [PERSON] and [PERSON] in neon ions [16], the more recent work by the group of [PERSON] [17] (Ne\({}^{8+}\)) and the two-photon total cross sections on Ne\({}^{8+}\) and Ar\({}^{16+}\), obtained through a Greens-function method calculation [18]. More specifically, in the present study, we apply an ab initio configuration interaction (CI) approach and make use of LOPT to calculate the two- and three-photon partial photoionization cross sections for Li\({}^{+}\), Ne\({}^{8+}\) and Ar\({}^{16+}\). In addition to this, we have calculated details of their electronic structure. The structure of the text is as follows: in Section 2, we present the theoretical method in sufficient detail for a self-contained formulation of the present study; in Section 3, we present and discuss our results about the calculated energies and the LOPT two- and three-photon ionization partial cross sections. Finally, we conclude with a summary of our findings and a brief discussion of possible further investigations within the present context. In the presentation of the theoretical formulas, we use the atomic-Gaussian unit system (\(\hbar=m_{e}=e=1/4\pi\epsilon_{0}=1\)). In the figures, the cross sections and the energies are presented in more traditional units, namely, eV for the energies, and cm\({}^{4}\)s and cm\({}^{6}\)s\({}^{2}\) for the two- and three-photon cross sections, respectively.