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

Abstract

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 $L\ne 0$, $Q$ 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 $L=0\mathrm{}$, 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 $Q^{\prime }H{Q}^{\prime }$, where $Q^{\prime }$ 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 $P^{\prime }$ and $Q^{\prime }$ 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.