Radial and angular correlations of doubly excited electrons

The nature of electron correlations between two excited electrons is examined in terms of surface plots of charge-density distributions on suitably chosen planes. Initially the angular correlation is neglected such that charge distributions for singly and doubly excited states are displayed on the conventional ($r_1,r_2$) plane. It is shown that all singly excited states exhibit circular nodal lines on the ($r_1,r_2$) plane along $R={(r_1^2+r_2^2)}^{\frac{1}{2}}=\mathrm{const}$ and no radial nodal structure in the $\alpha =arctan\left(\frac{r_1}{r_2}\right)$ coordinate, while doubly excited states are characterized by additional radial nodal structure. The interplay between angular and radial correlations is examined in terms of plots of surface charge density at each $R$ on the ($\alpha ,{\theta }_{12}$) plane for each channel. It is shown that channels which have attractive potential curves are characterized by large charge concentration for large ${\theta }_{12}$, while other channels which have less attractive or repulsive potentials are characterized by large charge concentration in the small-${\theta }_{12}$ region.