Correlations of excited electrons. The study of channels in hyperspherical coordinates

According to early experiments and calculations, doubly excited states of helium are grouped into channels (Rydberg series and continua) with very different excitation and decay probability. Macek has identified channels by using hyperspherical coordinates [$R={(r_1^2+r_2^2)}^{\frac{1}{2}},tan\alpha =\frac{r_2}{r_1},\Omega \equiv \{\alpha ,{\hat{r}}_1,{\hat{r}}_2\}$] and solving the Schrödinger equation initially at fixed $R$; each eigenvalue $U_{\mu }\mathrm{}\left(R\right)\mathrm{}$ and eigen-function ${\Phi }_{\mu }(R;\Omega )$ characterizes one channel and the approximate solution of the whole equation has the form $F_{\mu }\mathrm{}\left(R\right)\mathrm{}{\Phi }_{\mu }(R;\Omega )$. We study the electron correlations in each channel by obtaining explicit expressions for the ${\Phi }_{\mu }(R;\Omega )$, and extend the range of Macek's investigation to higher channels. Pairs of orbital quantum numbers ($l_1$, $l_2$) are found to be quasiconstants for each channel; channels with the same ($l_1$, $l_2$) and ($L$, $S$, $\pi$) differ mainly in the degree of excitation of the radial correlation. Channel functions ${\Phi }_{\mu }(R;\Omega )$ of ${}_{}{}^1{}_{}{}^{}S$, ${}_{}{}^1{}_{}{}^{}P+$, and ${}_{}{}^3{}_{}{}^{}P+$ channels have an antinode at or near $\alpha ={45}^{\circ }$, while those of ${}_{}{}^3{}_{}{}^{}S$, ${}_{}{}^1{}_{}{}^{}P-$, and ${}_{}{}^3{}_{}{}^{}P-$ channels have a node at or near $\alpha ={45}^{\circ }$. An antinode or a node appears to occur generally at or near $\alpha ={45}^{\circ }$. This location of nodes in the minus and ${}_{}{}^3{}_{}{}^{}S$ channels is shown to be related to the weakness of coupling both among these channels and to plus channels with the same ($L$, $S$, $\pi$) symmetry. Examples for ${\mathrm{H}}^{-}$ are discussed. The interactions between channels with different ($l_1$, $l_2$) are important where their potential curves are nearly degenerate. Examples of the effect of this degeneracy in the spectra of alkaline-earth atoms are indicated.