Online ISSN: 1097461X Print ISSN: 00207608 Copyright © 2001 John Wiley & Sons, Inc. 

Resonance states of atomic anions 
Alexei V. Sergeev^{ *}, Sabre Kais 
Department of Chemistry, Purdue University, 1393 Brown Building, West Lafayette, Indiana 47907 
email: Alexei V. Sergeev (kais@power1.chem.purdue.edu) 
^{*}Correspondence to Alexei V. Sergeev, Department of Chemistry, Purdue University, 1393 Brown Building, West Lafayette, Indiana 47907
Funded by:
Office of Naval Research; Grant Number: N000149710192
NSF EarlyCareer Award
Keywords 
resonance states; negative ions; complex rotation method; screened Coulomb potential; ionization energy 
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 that 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 O^{2} and S^{2} 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 twoelectron atoms as the nuclear charge decreases. Results are used to check the less accurate oneelectron model. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 255261, 2001 
Digital Object Identifier (DOI) 
Article Text 
There has long been an interest in the existence of longlived doubly charged negative atomic ions [14]. The possibility of doubly charged negative ion resonances has been raised in the case of oxygen by experiments of Peart et al. [5] who observed resonancelike structures in electronimpact detachment cross section at energies of 19.5 and 26.5 eV. A HartreeFock calculation of the closedshell electronic configuration shows that the resonance energy of O^{2} is about 8 eV above O^{}, which later was confirmed and modified by configuration interaction and other methods [2]. Sommerfeld et al. [6] performed a largescale multireference configuration interaction calculation ion using the complex rotation technique to investigate resonance states of H^{2}. Their results predict the existence of a (2p^{3})^{4}S resonance state of H^{2} with a resonance position of about 1.4 eV above the (2p^{2})^{3}P state of H^{} and a lifetime of 3.8×10^{16} [7], which was confirmed later by complex coordinate rotation calculations of Bylicky and Nicolaides [8]. Recently, the multireference configuration interaction calculations using the complex absorbing potential were performed for O^{2}, S^{2}, B^{2}, and Al^{2} [9]. No experimental evidence of existence of dianions was found, probably due to insensitivity of scattering processes to dianionic resonances. But, in principle, these resonances could be observed in some scattering experiments [9].
In this work, we consider an Nelectron 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<N to negative ions and with Z=N to neutral atoms, where N is the number of electrons. A complex energy of a doubly charged negative ion (Z=N2) is found by extrapolating the ionization energy into the instability region of nuclear charges less than some critical nuclear charge Z_{c}. We use the reliable data for the ionization energy of a neutral atom (Z=N) and its isoelectronic negative ion (Z=N1), which were calculated or experimentally measured.
The simplest extrapolations by polynomial fits [10][11] or by analytic formulas with a few fitting parameters [12][13] give the real negative binding energy for closedshell anions such as O^{2} or S^{2}. Extrapolating formula of Herrick and Stillinger [10] includes a singular 3/2 power term and gives both positions and widths of the dianions. Although it correctly models a similar singularity of the energy calculated by a variational method, it fails for the exact energy, which has a less trivial singularity [14].
In our approach, we use the oneelectron model to approximate movement of a weakly bound electron that is going to escape when the charge approaches the critical charge. Free parameters of the model are fit in order to reproduce correctly the binding energies of a neutral atom and its isoelectronic anion. Since the existence of one loosely bound electron is a realistic approximation in the vicinity of the critical charge, we believe that the model realistically reproduces the nontrivial singularity at Z=Z_{c} [15]. Earlier, this model was used to determine the critical charges by finding zeroes of the binding energy [16]. It was found that the critical charges never drop below N3/2. The maximal surcharge defined as S_{e}=NZ_{c} was found for Rn isoelectronic series and is equal to 1.48. Here, we extrapolate the ionization energy deeper into the instability region, to Z=N2, corresponding to a dianion.
Although various model interaction potentials for a scattering state of Z+2 electrons can realistically describe a process of electronimpact detachment from a negative ion (see [17] and references therein), applicability of model potentials for a resonant state of Z+2 electrons remains to be studied. There is a common belief that due to a strong correlation effect among several equivalent electrons the underlying physics is much more complicated than a simple model could suggest, and fullscale calculations with multielectron trial wave functions are unavoidable. For H^{2}, there were studies of potential curves in the framework of hyperspherical coordinates [18] and studies of the interaction between the Nth electron and the (N1)electron target including exchange interaction [19].
In this work, we attempt for the first time to model an unstable multielectron dianion by a suitable oneelectron potential. The underlying assumption and the mapping of an atomic system with a weakly bound electron to a oneparticle system with a screened Coulomb potential is discussed in the next section. In Section 3, we find an approximate dependence of the energy on the nuclear charge by extrapolation from the known energies of a neutral atom and the corresponding negative ion. Energies and widths of doubly negative ions for isoelectronic series exhibiting a bound singly charged anion are evaluated systematically for N18. Accuracy of the model is estimated by comparison of the model results with exact numerical results at Z<Z_{c} for helium isoelectronic series. Section 4 is a summary and conclusion.
2. Description of OneElectron Model
A variational study of the helium isoelectronic series using the Chandrasekhar trial wave function e^{a1r1a2r2}e^{a1r2a2r1} [20] shows that a_{1}/a_{2}2 if Z2, where a_{2} is the smallest among the two variational parameters a_{1} and a_{2}, in a reasonable agreement that both electrons occupy equivalent orbitals in helium and twoelectron positive ions. The ratio of the exponents increases as the nuclear charge decreases and reaches 3.7 for a negative hydrogen ion (Z=1). It means that the orbital of one of electrons inflates in comparison with the orbital of another electron.
A similar variational study of the beryllium isoelectronic series using the function e^{a1r1a2r2a3r3a4r4} properly symmetrized over coordinates and spin functions shows that a_{3}=a_{4} for beryllium and fourelectron positive ions, but a_{3}/a_{4}=2.4 for the negative ion of lithium where a_{3} and a_{4} are the smallest exponents corresponding to outer electrons, and a_{3}a_{4} (a_{3}a_{4} if Z3.25). Variational study of the neon isoelectronic series by Herrick and Stillinger [10] shows a similar diffusivity of the outer orbital for F^{} (Z=9) and for charges slightly less than 9 (0.111<Z^{1}0.118).
We expect that for Nelectron atom in general, the outer electronic shell undergoes a profound change when the charge decreases and reaches values around N1, so that for a negative ion one of the electrons is held much farther from the nucleus than the others. We admit that some exceptions from this rule are possible. Consider, for example, the H^{2} resonance studied in [7][8]. It has (2p^{3})^{4}S electronic configuration corresponding to the excited state of the isoelectronic Li. In this system, three electrons occupy three different p orbitals. Since these orbitals are spatially separated along three different axes, three electrons are relatively weakly interacting between themselves. In fact, no electron is pushed out to a loosely bound orbital, so all electrons are equivalent. But at least for atoms in the ground state with an even number of electrons, the last two electrons occupy the same spatial orbital, and we expect that they become unequivalent when the electron repulsion becomes sufficiently strong in comparison with the attraction to the nucleus.
For a given atom with N electrons and a nuclear charge Z, let us consider a spherically symmetric potential (also known as Hellmann potential [21]) of the form
2.1 
To extrapolate the ionization energy, we suggest here a oneparticle model of the outer electron responsible for ionization process moving in an effective potential of the atomic core comprising of the nucleus and N1 electrons. In scaled coordinates rZr, this potential is approximated by our model potential, Eq. (2.1). Our approximation is asymptotically correct both at small and at large distances from the nucleus where the scaled atomic core potential tends to 1/r and to (ZN+1)/(Zr), respectively. The transition between the two different asymptotic regimes occurs at distances roughly equal to 1/, which is about the atomic core radius.
The second parameter of the model potential, Eq. (2.1), , is chosen so that the ionization energy in the potential (2.1) is equal to the scaled ionization energy Z^{2}E_{I}(Z) of the atom. For atoms with more than two electrons, we consider here an excited state in the potential (2.1) with the same spherical quantum numbers (n,l) as quantum numbers of the loosely bound electron on an external atomic shell (in this aspect our approach differs from the method of pseudopotentials [22] that deals with the ground state in a potential with an additional repulsive term necessary to satisfy orthogonality conditions). In this way, we map an arbitrary atom, which is characterized by a pair of numbers (N,Z) to the model oneparticle system (2.1), which is characterized by a pair of parameters (,). Results of fitting the parameter for elements with N10 in our previous study [16] give evidence that depends on 1/Z almost linearly.
In summary, our model (2.1) effectively eliminates the singularity in the energy function E_{I}(Z) to be extrapolated by replacing it with a weakly varying function (Z) that can be accurately extrapolated by a linear dependence on 1/Z without taking into account a complex singularity at Z=Z_{c}. In our previous study [16], we fitted the parameter to meet the known binding energy of the neutral atom and its isoelectronic negative ion and then found as a function of 1/Z by a linear extrapolation. After that, we solved the Schrödinger equation with the potential (2.1) and found some kind of extrapolation of the ionization energy of an atom to the range of Z<N1. Finally, by locating a zero of the ionization energy, we found critical charges for most of atoms with N86. In the present work, we use the same technique to find resonances of doubly charged negative ions by calculating the ionization energy at Z=N2.
For twoelectron isoelectronic series, the parameter is 1.066 and 0.881 for He and H^{}, respectively. Decreasing of for H^{} means increasing of the core radius that is induced by stronger repulsion between electrons. Results of extrapolating the ionization energy to the range Z<1 with assumed as a linear function of 1/Z are shown in Figure 1.
Figure 1. Ionization energy for the helium isoelectronic series as a function of the nuclear charge including an instability region Z<Z_{c}. Bold lines are accurate results found by the variational method with a large basis set. Thin lines are results of the simplified oneparticle model described in Section 2. Solid and dashed lines are real and imaginary parts, respectively. [Normal View 7K  Magnified View 11K] 
To verify our approximation, we performed more accurate variational calculations using the Hylleraas trial wave function of the form
3.1 
The above method is a more general version of the complex rotation or the complex stabilization method [23]. Instead of one nonlinear complex variational parameter, the rotation angle, we are using three nonlinear variational parameters a, b, and c. Junker [24] used a similar variant of the complex stabilization method with several nonlinear variational parameters. Like the method of Junker [24], our method gives correct resonance energies without taking into account an asymptotical form of the resonant wave function, or imposing specific constrains on the parameters a, b, and c.
We used the complex parameters a, b, and c calculated for the particular case of N=5 in order to extend calculations to higher N by optimizing only the linear coefficients C_{i,j,k}. Almost exact variational energy calculated at N=25 differs from the variational energy at N=5 in the amount of less than 10^{4}. It is compared in Figure 1 with results of oneelectron model calculations. The real part of the ionization energy is always negative at 0<Z<Z_{c}. It reaches its minimum at Z0.5, and the width reaches its maximum approximately at the same point. For small Z, convergence of the variational method becomes worse. It seems that both real and imaginary parts tend to zero when Z0.
As expected, the oneelectron model gives fairly accurate results in the vicinity of the critical charge where it models the threshold singularity. Here, the real part is approximated more accurately than the imaginary part because the singularity manifests mostly in the imaginary part.
Dubau and Ivanov [14] calculated the twoelectron atom resonance in the vicinity of the critical charge using 1/Z expansion and the complex rotation method. Their results agree with our calculation; see Table I. Since in the latter case the charge is close to the critical charge, we slightly corrected location of the singularity in our model by fitting at points Z=Z_{c}=0.911028225 and Z=2 instead of fitting it at points Z=1 and Z=2.
Table I. Resonances in the helium isoelectronic series for charges below the critical charge.  
 
For isoelectronic series of Li in its ground state (1s^{2}2s)^{2}S, our model cannot be used because He^{} does not exist in the (1s^{2}2s)^{2}S state. However, in the excited (2p^{3})^{4}S state the negative ion He^{} is bound nonrelativistically. We tried to use our model for this excited state by taking the ionization energies as a difference between the energies of (2p^{3})^{4}S states listed in [25] and the energies of (2p^{2})^{3}P states with the same Z listed in [26]. The parameter is 0.239 and 0.204 for Li and He^{}, respectively. Extrapolation of the ionization energy to the range Z<2 by the method described in Section 2 shows that E_{I}(Z) is zero at Z=1.92, and it has minimum at Z1.3, while the width has maximum at Z1.4. For H^{2}, our extrapolation gives E_{I}(1)=0.029+0.00002i (a.u.). Our values for the position of the resonance ReE_{I} and its halfwidth ImE_{I} are considerably lower than results of complex rotation calculations [8], E_{I}=0.053+0.031i. It is likely that our extrapolation becomes poorer as the charge drops below the point where the extrapolation has a minimum, as it can be seen on Figure 1 for heliumlike ions. A simple threepoint quadratic extrapolation gives E_{I}(1)3E_{I}(2)3E_{I}(3)+E_{I}(4)=0.0298, which is very close to the prediction of our model, while an extrapolation with use of Eldén's [12] threeparameter formula does not work because it has a nearby pole at Z1.3. It seems that any extrapolation over the parameter 1/Z from stable to unstable anions is unreliable for light atoms because 1/Z differs considerably for singly and doubly negative ions if N is small.
For fourelectron isoelectronic series, results of using our model for extrapolating the ionization energy are shown in Figure 2. We found that the curve hits the border of continuum spectrum E_{I}=0 at Z_{c}=2.864 [16] and again at Z_{c}=2.17. Very small imaginary part exists in the interval Z_{c}<Z<Z_{c}. The extrapolation gives positive ionization energy for He^{2}, E_{I}=0.064. Since a bound state of He^{2} does not exist, it means that we deal with a virtual state with an exponentially growing wave function (corresponding to a pole of the scattering function on the second sheet of the Riemann surface). It may be also possible that because of its limitations our method fails for the He^{2} ion. We found positive ionization energies for another doubly charged negative ions of noble gases, for example, Ne^{2} (see Table III).
Figure 2. Ionization energy for the beryllium isoelectronic series extrapolated from bound systems Be and Li^{} to a resonance state of He^{2}. A dashed line is an imaginary part of the energy times 100. The system He^{2} with a zero imaginary part is presumably in a virtual state. [Normal View 8K  Magnified View 12K] 
The doubly charged negative ion of atomic oxygen in the same electronic configuration as a closedshell 10electron configuration of a neutral atom of neon was extensively studied theoretically. Comparison of the ionization energy of O^{2} found by different methods is given in Table II. For critical discussion of some of these results, see [10]. Our result for the resonance position is in agreement with most of the above results. However, our prediction of the resonance width is significantly lower than that of Herrick. It may be attributed to the fact that Herrick's 3/2power term, which is responsible for the width, is far from the actual singular behavior or to the fact that our model systematically underestimates the width as it happens for twoelectron atoms. Recent multireference configuration interaction calculations of Sommerfeld [9] predict still larger width. However, this prediction may be only an upper bound to the true resonance width because of possible missing correlation effects that stabilize the metastable state [9]. The extrapolated ionization energy as a function of Z is shown in Figure 3. It has a relatively small nonlinear distortion and a weak singularity at Z_{c}=8.75.
Figure 3. Ionization energy for neon isoelectronic series extrapolated from bound systems Ne and F^{} to a resonance state of O^{2}. A dashed line is an imaginary part of the energy, which is halfwidth of the level. [Normal View 7K  Magnified View 10K] 
Table II. Energy of O^{2} resonance calculated by various methods.  
 
^{a} Huzinaga and HartDavis listed the total energy. We found ionization energy by subtracting the HartreeFock total energy of the corresponding singly charged negative ion that is listed in works of Clementi et al. [27][31]. ^{b} Estimated width is greater than 1 eV. 
We extended calculations of resonances of doubly charged negative ions to the first two rows of the periodic table. Here, we considered only the ions for which an isoelectronic singly charged negative ion is stable, for example, we omitted C^{2} because N^{} does not exist. Results are listed in Table III. For the closedshell ion S^{2}, our result for the resonance position ReE_{I} is somewhat larger than the previous estimates.
^{a} Tabulated energies are results of the present work unless marked otherwise. For energy of O^{2} ion, see Table II. ^{b} Complex absorbing potential method of Sommerfeld [9]. ^{c} HartreeFockRoothaan calculations of Huzinaga and HartDavis [28]. ^{d} HartreeFock calculations of Clementi and McLean [27]. ^{e} Polynomial fit of Robicheaux et al. [11]. 
Herrick and Stillinger [10] suggested that stability of negative ions increase with decreasing of =(N1)/Z, or the effective charge seen asymptotically by the outer electron. A greater stability would then be expected for the S^{2} ion (=1.0625) than for the O^{2} ion (1.125). Our results for halfwidth of S^{2} and O^{2}, 0.045 and 0.22 eV, respectively, as well as results of Sommerfeld [9] (0.45 and 0.87 eV) support this argument. However, the widths for B^{2}, Al^{2}, S^{2} calculated by Sommerfeld [9] are larger than our results by one order in magnitude.
We have investigated the existence of resonance states for doubly charged atomic negative ions by considering the behavior of the ground state with a decrease of the nuclear charge. When the charge drops below the critical charge, the bound state goes into a complex plane and becomes a resonance. We attempt to approximate reasonably accurate the function E_{I}(Z) in the vicinity of the critical charge Z_{c} where it exhibits a singularity. Approximating of isoelectronic ionization potentials by simple analytic formulas with several adjustable constants was widely used before appearance of powerful computers more than 40 years ago. However, these formulas seem inadequate for negative ionization potentials, which are complexvalued, with an imaginary part meaning a halfwidth. Herrick and Stillinger [10] introduced a singular term (ZZ_{c})^{3/2} to their extrapolating scheme to approximate the complex energy at Z<Z_{c}.
Our method is some kind of refinements of Herrick and Stillinger method of analytic continuation. We believe that our model reproduces the threshold singularity of the energy more realistically than the 3/2power singularity of Herrick and Stillinger because it is based on a physical model that is realistic at ZZ_{c}. Generally, the method of analytic continuation from bound to resonance states appears very accurate when the threshold behavior of the energy is incorporated into continuation scheme, as it was demonstrated earlier for twoparticle resonances in nuclear physics; see Chapter 5 of Ref. [32].
The developed method was checked for a twoelectron atom, for which the Hylleraas basis variational technique gives reliable results. Position and width of the resonance were compared with a prediction of the less accurate model approximation. Results were found in agreement.
The results of our approximate oneelectron model agree with earlier estimates of closedshell resonances of O^{2} and S^{2}. In addition, the model predicts similar resonances other dianions that are isoelectronic to some bound singly charged anion.
Our model invites further generalizations, for example, by considering a motion of two weakly bound electrons instead of only one.
Although no multiple negative ions exist in a bound state [16][33][34], some longlived resonances of doubly negative ions are observed experimentally [1]. Our results give evidence for some resonances that remain to be observed.
We would like to acknowledge the financial support of the Office of Naval Research (N000149710192). One of us (S.K.) acknowledges support of an NSF EarlyCareer Award.
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