## 1 Introduction  
During the last decade, the advent of new light sources, capable of delivering intense and/or ultrashort, coherent radiation in the soft- and hard-X-ray regime, either directly by free-electron lasers (FEL) or indirectly by high-harmonic generation techniques, has renewed interest in experimental and theoretical photoionization studies of multiply charged ions. The excitation of atoms/molecules involving inner-shell electrons with such light sources offers the possibility of investigating processes at their natural time-scale, such as Auger processes and its variations [1], double core-hole creation and ionization [2], resonant enhanced photoionization, etc. Beyond the viewpoint of accessing the dynamics of these processes, the availability of extremely high flux has made it feasible to experimentally create the required conditions and observe for the first time new processes, for example the two-photon single- and double-(direct) photoionization of helium, the complete stripping of neon [3] and multiphoton inner-shell ionization in noble gases [4] and solid targets [5]. Since a number of recent reviews elaborate on experimental and theoretical studies as well as on their numerous applications in this short wavelength regime [6; 7; 8; 9], we will only mention here that typical fluxes of FELs to date range between \(5\times 10^{11}\) Wcm\({}^{-2}\) to \(5\times 10^{18}\) Wcm\({}^{-2}\), photon energies range in (0.02-15) KeV, while average full width at half maximum (FWHM) durations vary from 10 fs to 85 fs. It is also worth noting that three new FELs are under development and set to start user operations soon (E-XFEL, Swiss FEL, PAL) with extra pulse parameter specifications compared to the current ones. Notably, the Swiss FEL will provide pulses as short as 2 fs while E-XFEL as long as 100 fs with a repetition rate between 50-100 times higher than any other existing FEL. From the above discussion, it appears that, nowadays, in these wavelength regimes, excitation/ionization processes involving more than one photon are routinely feasible. Accordingly, their quantitative description requires theoretical approaches capable of coping with the non-linear features of the interaction between X-ray radiation with atomic and molecular systems. Amongst the various theoretical approaches, one is based on a perturbation expansion of the interaction potential with respect to the electric field, known as lowest-order (non-vanishing) perturbation theory (LOPT) [10]. In the case of X-ray quasi-monochromatic radiation, the use of LOPT is well justified for the current available peak intensities. The main requirements are for (a) the ponderomotive potential, \(V_{p}=I_{0}/4\omega^{2}\), and (b) the bandwidth, \(\Delta\omega\), to be much smaller than the mean central photon frequency, \(\omega\), of the radiation. For our case, for example, if we choose a peak intensity at the higher end, \(I_{0}\simeq 3.51\times 10^{18}\) W/cm\({}^{2}\), and a mean photon energy at the lowest limit of soft X-ray spectrum, say \(\omega\sim 270\) eV, then \(V_{p}/\omega\simeq 0.025<<1\). Since for current FEL sources the bandwidth, \(\Delta\omega\), ranges in (\(10^{-3}\)-\(10^{-2}\))\(\omega\), LOPT is very well suited to provide reliable quantitative information about various quantities of experimental interest, for example, ion and fluorescence yields. The validity of LOPT to calculate few X-ray photon processes allows the development of an alternative theoretical framework in place of the direct, but much more demanding, integration of a multielectron time-dependent Schrodinger equation. In the present work, we have chosen to apply LOPT to three elements of high experimental interest, namely Li, Ne and Ar. As it is known, these three atomic elements have been the subject of numerous experiments and theoretical studies since the inception of quantum mechanics.