Angular correlation in He and He-like atomic ions: a manifestation of the genuine and conjugate Fermi holes

The ground and low-lying singly excited states of the two-dimensional He and He-like atomic ions have been studied by the full configuration interaction method focusing on the angular correlation between the two electrons involved. For small values of the nuclear charge ${Z}_{n}$, the two-electron angular-density distribution for the ground state strongly depends on the two-electron angle ${\ensuremath{\phi}}_{\ensuremath{-}}$ reaching a peak at ${\ensuremath{\phi}}_{\ensuremath{-}}$ = 0. This strong dependence on ${\ensuremath{\phi}}_{\ensuremath{-}}$ strongly decreases with increasing ${Z}_{n}$ along with the decreasing electron-electron interaction. In contrast, the probability-density distribution for the singlet-triplet pair of states of the $(1s)(2p)$ configuration becomes appreciable with increasing ${Z}_{n}$, reaching peaks at ${\ensuremath{\phi}}_{\ensuremath{-}}$ = 0 and at ${\ensuremath{\phi}}_{\ensuremath{-}}$ = $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ for the singlet and triplet states, respectively. This indicates a preference of the two electrons to be on the same side of the nucleus for the singlet $(1s)(2p){\phantom{\rule{0.28em}{0ex}}}^{1}P$ state and on opposite sides of the nucleus for the triplet $(1s)(2p){\phantom{\rule{0.28em}{0ex}}}^{3}P$ state. The origin of these angular dependences is rationalized on the basis of the genuine and conjugate Fermi hole concepts.