Three-electron systems with inner-shell vacancies

A computational method is described for obtaining inner-shell-vacancy states of three-electron atoms which combines a block-diagonalization procedure with generalized Feshbach projection operators applicable to systems with three or more electrons. Typically, the accuracy is about 1.5 parts per thousand (δE/E≊1.5×${10}^{\mathrm{-}3}$). The strength of the method is that it provides many energy levels for each Rydberg series. A quantum-defect analysis is then applied that identifies the members of each series and yields reliable quantum defects and series limits. The method is particularly important in symmetries for which multiple Rydberg series exist. The present work reports on ${}_{}{}^4{}_{}{}^{}$ ${\mathit{P}}^{\mathit{e}}$ states of three-electron systems with 3≤Z≤10, which are compared with other calculations. The energies of ${}_{}{}^2{}_{}{}^{}$ ${\mathit{S}}^{\mathit{e}}$ states of ${\mathrm{C}}^{3+}$ are also presented as an example requiring both the multielectron Feshbach projection operators and the quantum-defect analysis developed here. Four distinct Rydberg series are found for this case and their series limits obtained.