Hartree-Fock Calculations of Autoionization States

Abstract

It is argued that Hartree-Fock (HF), and certain HF-like, calculations provide upper bounds to energies of certain atomic autoionization states, defined as eigenfunctions of Feshbach's $\mathrm{QHQ}$. A detailed discussion and supporting calculations are given for the $2{s}^{2}^{1}S$ state of helium. The $Q$-space constraint can be satisfied by assignment of a suitable value to an orbital-expansion parameter. A slightly higher, and hence upper-bound, value of energy is obtained by instead maximizing energy with respect to this parameter. It is argued that a HF calculation can be reduced to such a parameter-maximization process, on the assumption that the solution is unique. Several autoionization states in two-electron atoms are discussed, and it is noted that the HF energy bounds are useful for some such states but are useless for others because of configuration mixing. The general approach is not inherently limited to two-electron atoms, and a similar argument is given for the lowest $^{2}S$ autoionization state of a three-electron atom.