A method is presented to remedy the defects of the projection-operator technique for calculating electron resonances in scattering from many-electron targets. Specifically it is shown that if the projection operator (i.e., idempotent) $Q$ is replaced by a quasi-projection operator $\hat{Q}$ such that $\lim \hat{Q}\Psi =0\mathrm{}$ as any ${\mathcal{r}}_i\to \infty$, then the spectrum of $\hat{Q}H\hat{Q}$ is discrete, and can be made to be in essentially a unique correspondence with resonance energies. Relaxation of the idempotency requirement allows us to define two forms of $\hat{Q}$ operator. The simpler of the two forms is tested on $e$-H and $e$-${\mathrm{He}}^{+}$ systems; the two lowest resonant energies differ by less than 0.01 eV from rigorous $\mathrm{QHQ}$ results. For many-electron targets it is further argued that replacement of the exact target eigenfunction (${\varphi }_0$) by reasonable approximations (${\tilde{\varphi }}_0$) in constructing $\hat{Q}$ will affect neither the discreteness of the spectrum $\hat{Q}H\hat{Q}$ nor the proximity of its eigenvalues to the resonant energies. Calculations of ${\mathrm{He}}^{-}$ using two different (open and closed shell) ${\varphi }_0\text{'}\mathrm{s}$ and an angle-independent total wave function as well as a configuration-interaction wave function containing up to 40 configurations are carried out. The difference between open- and closed-shell ground-states results is about 0.02 eV, and the latter yields $E_{\mathrm{res}}\left({}_{}{}^2{}_{}{}^{}S\right)=19.363\mathrm{}$ eV plus a width $\Gamma =0.014\mathrm{}$ eV. No other resonances are found below the first excited ($2{}_{}{}^3{}_{}{}^{}S$) He threshold.

• Received 10 December 1971

DOI:https://doi.org/10.1103/PhysRevA.5.1663