Compound-Atom States for Two-Electron Systems

The energies of the lowest He-like compound-atom or resonance states lying below the $n=2\mathrm{}$ hydrogenic level are evaluated variationally for He and ${\mathrm{H}}^{-}$. In order to do this, we employ the definition of such states given by Feshbach as eigenvalues of the Hamiltonian with the open channel projected out, together with the modification required by the Pauli principle. Our trial wave function uses a Legendre expansion in the relative angle and a sum of exponentials in the radial coordinates, with appropriate angular factors to obtain the desired symmetry, parity, and orbital-angular-momentum eigenvalues associated with the ${}_{}{}^{1,3}{}_{}{}^{}S_{}^e{}_{}{}^{}$ and ${}_{}{}^{1,3}{}_{}{}^{}P_{}^o{}_{}{}^{}$ states. Our results are approximations to the actual physical resonances in that the shift and finite width caused by coupling to the neighboring continuum are not included. It appears, however, that the actual shifts are small, so that the positions of these compound-atom states are believed to give a close indication of where the physical resonances may be expected to occur. The results for He and ${\mathrm{H}}^{-}$ are compared in detail with the calculations of resonances in $e$-H and $e$-${\mathrm{He}}^{+}$ elastic scattering and with the observation of these states by ultraviolet photon absorption in He (where they are called autoionizing levels) and by inelastic scattering of electrons from He.