## 3 Results and Discussion
We use boldface for the states’ notation to emphasize that they are listed according the dominant configuration in the CI expansion of Equation (4). In the last row, \(\delta E(1s^{2})\equiv|E(1s^{2})-E_{g}^{Z+}|/|E_{g}^{Z+}|\) is the relative discrepancy (between our calculated ground state values and the energies (\(E_{g}^{Z+}\)) listed in the NIST database [28]). \(E(1s^{2})\) is given relative to the double ionization threshold. Figure 1: Two-photon partial ionization cross sections of Li\({}^{+}\) from its ground state with linearly polarized light. The characteristic peaks in the cross section are associated with the intermediate bound states of the symmetry \({}^{1}\!P\). Figure 2: Two-photon partial ionization cross sections of Ne\({}^{8+}\) from its ground state with linearly polarized light. The small letters \((a,b,c)\) indicate features associated with the intermediate bound states of symmetry \({}^{1}\!P\) while the capital letters \((A_{i},B_{i},i=0,2)\) are associated with final state correlations. The configuration-mixing peaks are absent for two-photon ionization of Li\({}^{+}\) ionization since these would occur at higher photon energy, past about 75 eV, as the lowest post-ionization energy levels are about 150 eV and half that value (two photons) is about 75 eV. When we add the \({}^{1}\)S and \({}^{1}\)\(D\) cross sections to obtain the total cross section, we find a relatively good agreement with that of Ref. [15], which models two-photon ionization of Li\({}^{+}\) employing a less elaborate approach, namely, single-channel quantum defect theory; when our shift is accounted for, our peaks occur close to that work (62.2 eV \(1s2p(^{1}P)\), 69.7 eV \(1s3p(^{1}P)\), 72.3 eV \(1s4p(^{1}P)\) and 73.5 eV \(1s5p(^{1}P)\)), and our (partial-wave sum) cross section baseline between 50-55 eV is within the same order of magnitude as in Ref. [15] (both between \(10^{-53}\)-\(10^{-52}\) cm\({}^{4}\cdot\)s); however, our shape is slightly different here, being slightly convex (downward) in this region. For Ne\({}^{8+}\) and Ar\({}^{16+}\), our values are also in good agreement with the Green-function calculations in Ref. [18]. In addition, for Ne\({}^{8+}\), we have also compared our values with the second-order perturbation theory calculation of [PERSON] and [PERSON] [16]; their \(1s2p(^{1}P)\) and \(1s3p(^{1}P)\) one-photon resonance peaks (these are their only peaks) are comparable to ours occurring at around 920 eV and 1070 eV, respectively. Their cross section base-line (non-resonant part) circa 600-800 eV is also close to ours, i.e., within the same order of magnitude (both between \(10^{-56}\)-\(10^{-55}\) cm\({}^{4}\cdot\)s). In relation to the intermediate resonance peaks in Table 3, we show the energy differences of the few lowest states of Li\({}^{+}\), Ne\({}^{8+}\) and Ar\({}^{16+}\) ions from their respective ground state (\(1s^{2}\)\(1S_{0}\)) energy, \(E_{g}=E_{1s^{2}}\), namely, \(\Delta E_{P}\equiv E_{P}-E_{g}\), (i.e., corresponding to peaks (a-d) in Figure 2). Their importance is derived from the fact that these energy differences appear in the denominator of the two-photon cross section expression, Equation (7), i.e., \(E_{g}+\omega-E_{p}^{\prime}=\omega-\Delta E_{P}\). It is then immediately evident that the photon energy detuning from these energy differences generates a series of characteristic features in the cross section. A word of caution is necessary at this point: within the current formulation, the height of these peaks becomes infinite in the exact on-resonance case, \(\Delta E_{P}=\omega\). The first point to note is that we have ignored the inherent spontaneous decay width of the intermediate bound states. This would have served only to remove the unphysical singularities that occur at the resonances  
Figure 3: Two-photon partial ionization cross sections of Ar\({}^{16+}\) from its ground state with linearly polarized light. positions. However, most importantly, the LOPT cross sections fail to provide the correct ionization yields.