%\documentstyle[preprint,eqsecnum,aps]{revtex} \documentstyle[twocolumn,aps]{revtex} %\documentstyle[preprint,aps]{revtex} \begin{document} \draft \title{Resonance States of Atomic Anions} \author{Alexei V. Sergeev and Sabre Kais} \address{Department of Chemistry, Purdue University, 1393 Brown Building, West Lafayette, IN 47907} \date{\today} \maketitle \begin{abstract} We study destabilization of an atom in its ground state with decrease of its nuclear charge. By analytic continuation from bound to resonance states, we obtain complex energies of unstable atomic anions with nuclear charge which is less than the minimum "critical" charge necessary to bind $N$ electrons. We use an extrapolating scheme with a simple model potential for the electron which is loosely bound outside the atomic core. Results for ${\rm O}^{--}$ and ${\rm S}^{--}$ are in good agreement with earlier estimates. Alternatively, we use the Hylleraas-basis variational technique with three complex nonlinear parameters to find accurately the energy of two-electron atoms as the nuclear charge decreases. Results are used to check the less accurate one-electron model. {\bf Key words}: resonance states, negative ions, complex rotation method, screened Coulomb potential, ionization energy \end{abstract} \pacs{31.15.-p, 32.10.Hq, 02.60.Gf} %% http://www.th.physik.uni-frankfurt.de/~cbest/pacs.numbers.00, http://www.physica.org/pacs9830.html %% 31.15.-p Calculations and mathematical techniques in atomic and molecular physics (excluding electron correlation calculations) (for computational techniques, see 02.70) %% 31.15.Pf Variational techniques %% 32.10.Hq Ionization potentials, electron affinities %% 02.30.Mv Approximations and expansions %% 02.60.-x Numerical approximation and analysis %% 02.60.Gf Algorithms for functional approximation %% 03.65.-w Quantum theory; quantum mechanics %% 03.65.Db Functional analytical methods %% 03.65.Ge Solutions of wave equations: bound states \narrowtext \section{Introduction} There has long been an interest in the existence of long lived doubly charged negative atomic ions\cite{buc,bates,kalcher,scheller}. The possibility of doubly charged negative ion resonances has been raised in the case of oxygen by experiments of Peart et al. \cite{peart} who observed resonance like structures in electron-impact detachment cross section at energies of 19.5 and 26.5 eV. A Hartree Fock calculation of the closed-shell electronic configuration shows that the resonance energy of ${\rm O}^{--}$ is about 8 eV above ${\rm O}^-$, which later was confirmed and modified by configuration interaction and other methods\cite{bates}. Sommerfeld et al.\cite{som} performed a large scale multireference configuration interaction calculation using the complex rotation technique to investigate resonance states of ${\rm H}^{--}$. Their results predict the existence of a $(2p^3)^4S$ resonance state of ${\rm H}^{--}$ with a resonance position of about $1.4$ eV above the $(2p^2)^3P$ state of ${\rm H}^{-}$ and a lifetime of $3.8\cdot 10^{-16}$~sec\cite{sommerfeld}, which was confirmed later by complex coordinate rotation calculations of Bylicky and Nicolaides \cite{byl}. Recently, the multireference configuration interaction calculations using the complex absorbing potential were performed for ${\rm O}^{--}$, ${\rm S}^{--}$, ${\rm B}^{--}$, and ${\rm Al}^{--}$ \cite{som1}. No experimental evidence of existence of atomic di-anions was found, probably due to insensitivity of scattering processes to di-anionic resonances. But in principle, these resonances could be observed in some scattering experiments \cite{som1}. In this paper, we consider an $N$-electron atom in its ground electronic configuration state with the nuclear charge $Z$ as a variable. Integer values of $Z$ with $Z>N$ correspond to positive ions, with $Z