Collective and independent-particle motion in doubly excited two-electron atoms

Abstract

Configuration-interaction wave functions are calculated for all intrashell doubly excited states of the $N=2\mathrm{}$ shell for He and isoelectronic ions ${\mathrm{Li}}^{+}$, ${\mathrm{Be}}^{2+}$, and ${\mathrm{Ne}}^{8+}$ with the use of a basis of Sturmian functions. The density function $\rho (r_1,{\theta }_{12}|r_2=\alpha )$, which is the conditional probability that one electron will be found at distance $r_1$ from the nucleus with interelectronic angle ${\theta }_{12}$ given that the other electron is at distance $\alpha$ from the nucleus, is computed for each of the states and for various values of $\alpha$. Density plots for the ${}_{}{}^1{}_{}{}^{}S_{}^e{}_{}{}^{}$ states of He are compared with those from Hylleraas-Kinoshita wave functions to assess the quality of the Sturmian basis. Sequences of plots are then examined to investigate the possibility of collective motion in doubly excited states. For He the form of the densities $\rho (r_1,{\theta }_{12}|\alpha )$ is in qualitative accord with the molecular model proposed by Kellman and Herrick in that the collective rotational and bending vibrational states are readily identifiable. With increasing nuclear charge a transition occurs to less collective behavior, corresponding to superpositions of hydrogenic configurations associated solely with the $N=2\mathrm{}$ shell. Some states, e.g., ${}_{}{}^1{}_{}{}^{}D_{}^e{}_{}{}^{}$, correspond to single independent-particle configurations. Electron density distributions are also presented for the lowest $S$ and $P$ states of the He $n_1=2\mathrm{}$, $n_2=3\mathrm{}$ intershell manifold with a view to extension of the molecular interpretation to this case. Results are compared with those of calculations of the quantum states of model problems such as particles on concentric spheres.