# Extended Photoionization Cross Section Calculations for C III
## 2 Computation Methods and Results
### Resonant Widths and Lifetime of Resonant Transitions  
Various experimental methods [44; 45; 46] have been developed for the measurement of radiative lifetimes in atoms. On the theoretical side, large scale calculations based on the configuration interaction (CIV3) [47], multiconfiguration Hartree-Fock (MCHF) [48], and multiconfiguration Dirac-Fock (MCDF) [49] methods, or the relativistic many-body theory [50] have shown that the agreement between the lengths and velocity gauges cannot be uniquely used for the accuracy assessment.  
The strong cancellations (up to three orders of magnitude) in the relativistic transition amplitudes between the \(2\mathrm{s}-2\mathrm{p}_{1/2}\) and \(2\mathrm{s}-2\mathrm{p}_{3/2}\) matrix elements in the \(LS\)-coupling limit ([51] and references therein) justify the difficulty of the transition rate calculations for the \(2\mathrm{s}^{2}(^{1}\mathrm{S})-1\mathrm{s}^{2}2\mathrm{s}2\mathrm{p}^{3} \mathrm{P}_{1}{}^{\circ}\). For CIII, due to the fact that the \(1\mathrm{s}^{2}2\mathrm{s}2\mathrm{p}^{1}\mathrm{P}_{1}{}^{\circ}\) state decays rapidly to the ground state C III \(2\mathrm{s}^{2}(^{1}\mathrm{S})\), in contrast to the long lifetime of the C III \(2\mathrm{s}2\mathrm{p}^{3}\mathrm{P}_{1}{}^{\circ}\), the \(1\mathrm{s}^{2}2\mathrm{s}2\mathrm{p}^{1}\mathrm{P}_{1}{}^{\circ}\) resonance lifetime is defined by its line strength. In R-matrix approaches, in the absence of measured values for physical quantities, one normally shifts the ab initio energies to the experimental or observed energy levels in the diagonalization of the Hamiltonian. This procedure provides a better positioning of resonance features in the corresponding cross sections. Table 4 presents a comparison of experimental and theoretical values of the weighted oscillator strengths in length gauge (gf\({}^{\prime}\)) and a lifetime calculation for the \(2\mathrm{s}^{2}\mathrm{s}^{2}\mathrm{s}^{2}-1\mathrm{s}^{2}2\mathrm{s}2\mathrm{p }^{1}\mathrm{P}_{1}{}^{\circ}\)- and \(2\mathrm{s}^{2}\mathrm{s}^{2}\mathrm{s}^{2}-1\mathrm{s}^{2}2\mathrm{s}3\mathrm{p }^{1}\mathrm{P}_{1}{}^{\circ}\)-allowed transitions.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{2s^{(^{1}\mathrm{S})}-1s^{2}2s2p^{1}P_{1}{}^{\circ}}\) & \(\mathbf{2s^{2}(^{1}\mathrm{S})-1s^{2}2s3p^{1}P_{1}{}^{\circ}}\) \\
**Source** & \(\mathbf{gf}^{\prime}\) & **Source** & \(\mathbf{gf}^{\prime}\) \\ \hline Experiment & \multicolumn{2}{c}{Experiment} \\ \hline [PERSON] et al. [46] & 0.754 & NIST & 0.232 \\ \hline [PERSON] et al. [52] & 0.760 & [PERSON] et al. [53] & 0.1979 \\ \hline [PERSON] and [PERSON] [54] & 0.753 & \\ \hline Theory & \multicolumn{2}{c}{Theory} \\ \hline Present & 0.767 & Present & 0.231 \\ \hline [PERSON] [47] & 0.796 & [PERSON] et. al. [50] & 0.267 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Electric-dipole weighted oscillator strength in length gauge, \(\mathbf{gf}^{\prime}\), for the \(2\mathrm{s}^{2}(^{1}\mathrm{S})\)-, \(1\mathrm{s}^{2}2\mathrm{s}2\mathrm{p}^{1}\mathrm{P}_{1}{}^{\circ}\)-, \(2\mathrm{s}^{2}(^{1}\mathrm{S})\)-, and \(1\mathrm{s}^{2}2\mathrm{s}3\mathrm{p}^{1}\mathrm{P}_{1}{}^{\circ}\)-allowed transitions and lifetime (ns). Comparison with experiments and other theoretical calculations.  
Figure 7: Photoionization (red dashed line) and photoabsorption (black dotted line) cross sections (in Mb) versus photon energy (Ry) representing the calculated spectra out of \(2\mathrm{s}5\mathrm{s}\) (\({}^{1}\mathrm{g}^{\circ}\)) of Be-like C III.