Analytical channel functions for two-electron atoms in hyperspherical coordinates

Abstract

By expressing the two-electron wave functions in hyperspherical coordinates ($R$,$\Omega$) in an adiabatic approximation $F_{\mu }\mathrm{}\left(R\right)\mathrm{}{\Phi }_{\mu }(R;\Omega )$, I describe a simple procedure for obtaining the channel functions ${\Phi }_{\mu }(R;\Omega )$ analytically. These analytical functions, ${\Phi }_{\mu }(R;\Omega )$ in the finite-$R$ region, are obtained by generalizing the known hydrogenic solutions in the asymptotic ($R\to \infty$) limit for each channel $\mu$ and are required to satisfy proper boundary conditions in the hyperangles $\Omega$ rigorously. It is shown that these analytical functions compare well with the channel functions obtained previously from numerical calculations and, in a straightforward manner, describe the + and - channels of doubly excited states. The implication of this result as a method of generalizing hyperspherical coordinates to many-electron problems is discussed.