Hole projection, saddle points, and localization in the theory of autoionizing states

In view of a renewed interest in the theory of multiply excited resonances [M. Bylicki, Phys. Rev. A 39, 3316 (1989); 40, 1748 (1989); 41, 4093 (1990)] and the related comment by Chung [Phys. Rev. A 41, 4090 (1990)], I present a brief analysis of aspects of the theory for the identification and computation of resonances in many-electron systems. As examples, I use the ${\mathrm{He}}^{\mathrm{-}}$ 2s2${\mathit{p}}^2$ ${}_{}{}^4{}_{}{}^{}$P and ${}_{}{}^2{}_{}{}^{}$D triply excited resonances.