Variational-Bound Methods for Auto-Ionization States. II. Three-Electron Atoms

Singly auto-ionizing (SAI) states of three-electron atoms below the first core excitation threshold are defined as eigenfunctions of an operator $\mathrm{QHQ}$. It is shown that a particular type of multiconfiguration calculation provides upper-energy-bound estimates of such states, subject to a certain definition of $Q$ (and the assumption that a certain set of equations has a solution). For SAI states with $L\ensuremath{\ne}0$, $Q$ is the proper Feshbach operator which projects out all wave-function components which overlap the exact ground state of the two-electron core. For SAI states with $L=0$, the bound is only established by adopting a modified, approximate $Q$ which projects out only those wave-function components which overlap the angle-independent part of the core ground state. It is proposed that doubly auto-ionizing (DAI) states be defined as eigenfunctions of ${Q}^{\ensuremath{'}}H{Q}^{\ensuremath{'}}$, where ${Q}^{\ensuremath{'}}$ is the projection operator which eliminates all wave-function components which overlap the known $1s$ hydrogenic ground state of the one-electron system. The explicit form of ${P}^{\ensuremath{'}}$ and ${Q}^{\ensuremath{'}}$ is given, and a brief summary is given of various upper-energy-bound computational methods for DAI states; these closely parallel methods discussed previously for SAI states of two-electron atoms.