## 3 Results and Discussion
We mention here that these doubly excited (highly correlated) states are also known as autoionizing states as they are associated with a temporal trap of the two excited electrons in the core's region, eventually leading to the ejection of one of them and the residual (higher-charged ion) to its ground state. In the present (static) context of the CI calculation, these doubly excited states are degenerate with the \(1s^{2}\)\({}^{1}\)\(S\) or \(1 sd\)\({}^{1}\)\(D\) continua, which eventually cause their radiationless (auto)-ionization [30]. In Figures 4-6, we present our calculated three photon partial-ionization cross sections of Li\({}^{+}\), Ne\({}^{8+}\) and Ar\({}^{16+}\), respectively, from the ground state \(1s^{2}(^{1}\)\(S\)). The final angular momentum of the ions, following three-photon absorption, are the \({}^{1}P_{\mathrm{}}\)\({}^{1}\)\(F\) continua, all being of singlet symmetry. Similarly, as in the two-photon case, the total three-photon ionization cross section is obtained by the addition of the \({}^{1}\)\(P\) and \({}^{1}\)\(F\) partial-wave cross sections. Again, the final state is dominated by configurations with the residual ion in its ground state and the ejected electron with angular momentum \(l=1\) for the \({}^{1}\)\(P\) and \(l=3\) for the \({}^{1}\)\(F\) symmetry. The three-photon cross sections exhibit strong peak structures, which appear in both \({}^{1}\!P\) and \({}^{1}\!F\) final symmetries, due to two-photon resonances with the intermediate states \(1s3d,1s4d,...\) (see denominators (\(E_{g}+2\omega-E_{L},L=S,D\)) in Equations (8) and (9)). In the \({}^{1}\!P\) symmetry, there are additional peaks due to two-photon resonance states of the type \(1 sns^{1}S,n=2,3,...\). Because the \(1s3s\) and \(1s3d\) states have slightly different energy positions, the intermediate resonance peaks for the \({}^{1}\!P\) and \({}^{1}\!F\) (for example circa 34.4 eV for Li\({}^{+}\)) do not generally coincide. Note that the \({}^{1}\!P\) final states are reached by the coherent superposition of two ionization absorption channels: \(S\to P\to S\to P\) and \(S\to P\to D\to P\). In contrast, the \({}^{1}\!F\) states are reached only via one ionization channel, namely: \(S\to P\to D\to F\). Figure 4: Three-photon ionization partial cross sections of Li\({}^{+}\) from its ground state with linearly polarized light. Figure 5: Three-photon ionization partial cross sections of Ne\({}^{8+}\) from its ground state with linearly polarized light. At this point, it might be worth comparing the cross sections among the three investigated ions. It is clearly visible from the figures that generally the ionization cross section decreases from Li\({}^{+}\) towards Ar\({}^{16+}\). This observation is rather consistent with the (exact) scaling, \(1/Z^{4N-2}\), of the \(N\)-photon cross section for hydrogenic systems [11]. A second point worth mentioning is that the cross sections ending on the higher symmetry, \({}^{1}\!D\) for two-photon and \({}^{1}\!F\) for three-photon ionization, are proportional to the cross sections for circularly polarized light. To be more specific, due to the dipole selection rules, ionization by circularly polarized light will proceed through intermediate states where \(M_{L}\) will change either by \(+1\) or by \(-1\) monotonously. For example, let us assume circularly polarized light which causes a change of the magnetic quantum number by \(\Delta M_{L}=+1\). This means that if we start from the ground state, where \(L=0\) and \(M_{L}=0\), then the first ionization step will involve only states with \(M_{L}=+1\). Similarly, the next ionization step will involve states that differ by \(+1\) from the previous step, meaning that only states with \(M_{L}=+2\) will be accessed in this step. In this case, these states will necessarily have \(L=2\). Accordingly, if further ionization occurs (three-photon ionization), for the same reason, only states with \(M_{L}=+3\) will be reached, thus ending necessarily with an \(L=3\) total orbital angular momentum.