Classification and supermultiplet structure of doubly excited states

Based upon the analysis of electron correlations in hyperspherical coordinates for arbitrary $L$, $S$, and $\pi$ states, a classification scheme for all doubly excited states of two-electron atoms is presented. A new set of internal correlation quantum numbers, $K$, $T$, and $A$, are introduced. Here ($K$,$T$) describe angular correlations and $A=\pm 1,0$ describes radial correlations. These quantum numbers are used to label the first-order wave functions which are approximated as $\Psi =F_{\mu }^n\mathrm{}\left(R\right)\mathrm{}{\Phi }_{\mu }(R;\Omega )$ in terms of hyperspherical coordinates. The channel index $\mu$ is $\mu \equiv |{\mathrm{}(K,T)\mathrm{}}_N^A{}_{}{}^{2S+1}{}_{}{}^{}L_{}^{\pi }{}_{}{}^{}〉$, where $N$ is the dissociation limit of the channel. Rules for constructing the correlation diagram for channel potential $U_{\mu }\mathrm{}\left(R\right)\mathrm{}$ and the labeling of each channel are discussed. By comparing the rotation-averaged surface charge densities, it is shown that channels which have identical ${\mathrm{}(K,T)\mathrm{}}_N^{+,-}$ have isomorphic correlation patterns, irrespective of the overall $L$, $S$, and $\pi$. Such isomorphism is shown to be the underlying origin of the supermultiplet structure of intrashell doubly excited states. In fact, it is shown that such supermultiplet structures actually extend to all states which have $A=+1\mathrm{} or -1\mathrm{}$. A new Grotrian diagram for energy levels grouped according to ${\mathrm{}(K,T)\mathrm{}}^{+}$ and ${\mathrm{}(K,T)\mathrm{}}^{-}$ displays rotor-like structure. Such diagrams can easily reveal missing or misclassified levels. It is also shown that all $A=0\mathrm{}$ states are similar to singly excited states where for a given ${\mathrm{}(K,T)\mathrm{}}_N^0$, the triplet state always has a slightly lower energy than the singlet state. Approximate selection rules for photoabsorption and for $e$-H and $e$-${\mathrm{He}}^{+}$ scatterings are discussed.