## 3 Results and Discussion
In other words, in the resonant case, the LOPT relation for the ionization yield,  
\[W_{\mathit{Sf}}=\sigma_{N}F^{N}, \tag{10}\]  
(\(F\) is the pulse's flux) becomes invalid. It is well established that, for resonant and near-resonant processes, while perturbation theory is still valid, an alternative formulation is required for the calculation of the expected ionization yields. This formulation, in addition to the ionization of the system directly from the ground state, takes into account the stepwise formation and the subsequent ionization of the intermediate states. Without going into the details, we shall only mention that in such cases the formulation should be developed in terms of a density matrix representation in combination with a proper representation of the spatiotemporal profile of the laser field. At this point, it is worth noting the pulse properties for which the LOPT cross sections presented here are valid. Equation (10) is only just marginally applicable for a pulse with its detuning from a resonance of the same order as its bandwidth, \(\Delta E_{P}\)\(\sim\)\(\gamma_{L}\), where \(\gamma_{L}\) is the bandwidth. If we take as a rough rule that the bandwidth of the pulse is, say, a 1/100-th of its average photon energy (\(\gamma_{L}\)\(\sim\)\(\omega\)/100), that would mean that the cross section values within the range \(61.47\pm 0.61\) eV (first peak in the two-photon cross section for Li\({}^{+}\)) cannot be safely used in combination with Equation (10). Similar considerations should be assumed for the higher peaks. For completeness, for Ne\({}^{8+}\) and Ar\({}^{16+}\), if based again on the appearance of the first peaks in the two-photon cross section (see the first row for the \(1s2p\) state in Table 3), the corresponding intervals are scaled upwards to \(\pm 9.2\) eV (\(\sim\)920/100) and \(\pm 31\) eV (\(\sim\)3126/100), respectively. A case that such a discrepancy between the LOPT two-photon ionization cross section [16] and the experimental value [29] is attributed to the bandwidth of the X-ray pulse can be found in Ref. [17]. More specifically, for an X-ray photon energy of 1110 eV (in between (b) and (c) peaks in Figure 2), the reported experimental value was \(7\times 10^{-54}\) cm\({}^{4}\)\(\cdot\)s [29] while the theoretical cross section based on a Hartree-Fock-Slater (HFS) model was found to be equal to \(4.0\times 10^{-57}\) cm\({}^{4}\)s. Our calculated value for this photon energy is about \(4.6\times 10^{-57}\) cm\({}^{4}\)s for the \({}^{1}\)\(D\) wave while the \({}^{1}\)\(S\) value makes a negligible contribution to the total cross section. For Ne\({}^{8+}\) and Ar\({}^{16+}\), the 'twin' peaks (see \(A_{0}\) peaks in Figure 2), exclusive to the \({}^{1}\)\(S\) symmetry, are due to the strong coupling between the \(2s2s\) and \(2p2p\) configurations in the expansion Equation (4). Since in the \({}^{1}\)\(D\) symmetry the \(2s2s\) configuration is missing, we observe only one peak (\(A_{2}\) in Figure 2), in between the \(A_{0}\) ones. To confirm this, we have performed some further tests where, for example, we excluded the \(\Phi_{2s2s}^{15}\), \(\Phi_{2p2p}^{15}\) zero-order states (separately each time) from the CI wavefunction, Equation (4). By doing this, we obtain a cross section with only one \(A_{0}\) peak at the same position where the \(A_{2}\) peak appears. This suggests that the observed (two) \(A_{0}\) peaks are the result of strong-mixing of the \(2s2s\) and \(2p2p\) configurations, mainly due to their proximity in energy. Similar considerations hold for the peaks \(B_{0},B_{2}\) at higher photon energies. The \(B_{0}\) twin peaks are due to the mixing of the \(\Phi_{2s3s}\),\(\Phi_{2p3p}\) configurations in the \({}^{1}\)\(S\) symmetry, while the \(B_{2}\) peak is due to the \(3p^{2}\) state exclusively.