Effects of electron correlation on the time-delay in photoionization of atomic beryllium

We report on the effects of electron correlation on the relative time-delay in photoionization of atomic beryllium in the channels Be[($1s^22s^2$) $^1$S$^e$] $\rightarrow$ Be$^+$[($1s^22s$) $^2$S$^e$] + $e^-$ ($p$ electron), and Be[($1s^22s^2$) $^1$S$^e$] $\rightarrow$ Be$^+$[($1s^22p$) $^2$P$^o$] + $e^-$ ($s$ or $d$ electron). We use our recent three-dimensional implementation of the time-dependent restricted-active-space self-consistent-field method and study the changes in the value obtained for the time-delay when including more and more correlation. We find that the mean-field, time-dependent Hartree-Fock theory does not account accurately for the time-delay. A larger active orbital space is needed. We find that the relative time-delay between ionization into Be$^+$[($1s^22s$) $^2$S$^e$] and Be$^+$[($1s^22p$) $^2$P$^o$] is around 7-8 attoseconds.