## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**State** & **Li\({}^{2+}\)** & **Ne\({}^{9+}\)** & **Ar\({}^{17+}\)** \\ \hline \(1s\) & \(-122.451\) & \(-1360.57\) & \(-4408.2\) \\ \(2s,2p\) & \(-30.613\) & \(-340.14\) & \(-1102.1\) \\ \(3p,3d\) & \(-13.606\) & \(-151.18\) & \(-489.8\) \\ \(4d,4f\) & \(-7.653\) & \(-85.04\) & \(-275.5\) \\ \(5f\) & \(-4.898\) & \(-54.42\) & \(-176.3\) \\ \hline \hline \(\delta E(1s)\) & \(2.5\times 10^{-5}\) & \(1.2\times 10^{-3}\) & \(4.1\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Energies of the few lowest states of the one-electron Li\({}^{2+}\), Ne\({}^{9+}\) and Ar\({}^{17+}\) ions. Energies are given in units of eV. Energies are relative to the single-ionization threshold. \(\delta E(1s)\equiv|E(1s)-E_{g}^{Z+}|/|E_{g}^{Z+}|\) is the relative discrepancy (between our calculated ground state values with the energies (\(E_{g}^{Z+}\)) listed in the NIST database [28]). The relative discrepancy for excited states is generally less than \(\delta E(1s)\). given in SI units, cm\({}^{2N}\)s\({}^{N-1}\), where \(N\) is the order of the process (here equivalent with the number of the photons absorbed). The final angular momenta, following two-photon absorption are the \({}^{1}\)S,\({}^{1}\)\(D\) continua, while in the case of three-photon absorption are the \({}^{1}\)P,\({}^{1}\)\(F\) continua. The cross sections have been evaluated using both the length and the velocity forms of the dipole operator. They generally have excellent agreement throughout all the spectral regions considered, especially for the non-resonant ones. Relative agreement between the length and the velocity forms is important since it provides strong evidence that the dipole matrix elements, contributing in the multiphoton transition amplitude, have been converged. In Figures 1-3, we show the photon energy dependence of the calculated two photon partial-ionization cross sections, \(\sigma_{S}^{(2)}(\omega),\sigma_{D}^{(2)}(\omega)\), of Li\({}^{+}\), Ne\({}^{8+}\) and Ar\({}^{16+}\), respectively, from the ground state, 1\(s^{2}\)(\({}^{1}\)S), to final states of symmetry \({}^{1}\)S, \({}^{1}\)\(D\). Summing the latter, we obtain the corresponding total two-photon ionization cross sections, \(\sigma_{2}(\omega)\). For clarity, we have plotted only the length-form results. Generally, for all three ions, the dominant two-photon ionization channel is the \({}^{1}\)\(D\) symmetry. The cross sections exhibit strong peak structures, which appear in both the \({}^{1}\)S and \({}^{1}\)\(D\) final symmetries, due to one-photon resonance with the intermediate states 1\(snp\)\({}^{1}\)P, \(n=2,3,...\). Apart from these intermediate-resonance peaks, there are further peaks due to strong configuration mixing of the type \(npn^{\prime}p\), \(n,n^{\prime}=2,3...\), associated with the \({}^{1}\)S, \({}^{1}\)\(D\) continua. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**State** & **Li\({}^{+}\)** & **Ne\({}^{8+}\)** & **Ar\({}^{16+}\)** \\ \hline \(1s2s\) & 60.42 & 914.0 & 3114.1 \\ \(1s2p\) & 61.67 & 920.4 & 3126.7 \\ \hline \(1s3s\) & 68.73 & 1070.1 & 3666.1 \\ \(1s3d\) & 69.02 & 1071.5 & 3668.7 \\ \(1s3p\) & 69.09 & 1072.0 & 3669.7 \\ \hline \(1s4s\) & 71.54 & 1124.5 & 3858.7 \\ \(1s4d\) & 71.66 & 1125.1 & 3859.9 \\ \(1s4p\) & 71.67 & 1125.3 & 3860.2 \\ \(1s4f\) & 71.67 & 1125.1 & 3859.8 \\ \hline \(1s5s\) & 72.83 & 1149.6 & 3947.7 \\ \(1s5d\) & 72.89 & 1148.9 & 3948.3 \\ \(1s5p\) & 72.90 & 1150.0 & 3948.5 \\ \(1s5f\) & 72.89 & 1149.9 & 3948.3 \\ \hline \(E(1s^{2})\) & \(-197.518\) & \(-2554.53\) & \(-8513.84\) \\ \(\delta E(1s^{2})\) & \(2.9\times 10^{-3}\) & \(1.36\times 10^{-3}\) & \(3.87\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Energy differences of the few lowest states of Li\({}^{+}\), Ne\({}^{8+}\) and Ar\({}^{16+}\) ions with respect to their respective ground state (1\(s^{2}\)\({}^{1}\)S\({}_{0}\)) energy value.