Updated square-well potential model for $A$@C$_{60}$ photoionization*

The empirical square-well potential model for photoionization of atoms $A$ associated with endo-fullerenes $A$@C$_{60}$ [1] has played an important role in providing the initial understanding of their spectra. The model approximates the C$_ {60}$ cage by a square-well potential of a certain inner radius $r_{o}$, width $\Delta$, and depth $U_{o}$. Their values were determined by approximately matching the thus theoretically calculated and experimentally measured photoionization cross section of \textit{empty} C$_{60}$ [2]. The recent, first of its kind, experiment [3] on $A$@C$_{60}$ (namely, Xe@C$_{60}^{+} $) photoionization has revealed \textit{quantitative} discrepancies between predictions of the model and experimental data. To us, this implies that $r_{o}$, $\Delta$, and $U_{o}$ for \textit{empty} C$_{60}$ might not be the same as for an \textit{endo-fullerene} $A$@C$_{60}$. We thus find, and report on, updated $r_{o}$, $\Delta$, and $U_{o}$ which bring on a better quantitative agreement between the model and experiment. \\[4pt] [1] V. K. Dolmatov, Adv. Quant. Chem. \textbf{58}, 13 (2009).\\[0pt] [2] Y. B. Xu \textit{et.\ al.}, PRL \textbf{76}, 3538 (1996).\\[0pt] [3] A. L. D. Kilcoyne \textit{et.\ al.}, PRL \textbf{105}, 213001 (2010).