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QED shifts in multivalent heavy ions

The quantum electrodynamics (QED) corrections are directly incorporated into the most accurate treatment of the correlation corrections for ions with complex electronic structure of interest to metrology and tests of fundamental physics. We compared the performance of four different QED potentials for various systems to access the accuracy of QED calculations and to make prediction of highly charged ion properties urgently needed for planning future experiments. We find that all four potentials give consistent and reliable results for ions of interest. For the strongly bound electrons the nonlocal potentials are more accurate than the local potential.

The quantum electrodynamics (QED) corrections are directly incorporated into the most accurate treatment of the correlation corrections for ions with complex electronic structure of interest to metrology and tests of fundamental physics. We compared the performance of four different QED potentials for various systems to access the accuracy of QED calculations and to make prediction of highly charged ion properties urgently needed for planning future experiments. We find that all four potentials give consistent and reliable results for ions of interest. For the strongly bound electrons the nonlocal potentials are more accurate than the local potential.
PACS numbers: 31.30.J-, 12.20.Ds Optical transitions in heavy many-electron highly charged ions (HCI) have been recently proposed for the development of ultra-precision atomic clocks and tests of fundamental physics [1][2][3][4][5]. From the experimental standpoint, locating these ultra-narrow optical transitions is particularly difficult. For most of these ions, with the degrees of ionization ranging from 8 + to 18 + , no experimental data exist and identification of their complicated atomic spectra is a very difficult task [6] unless accurate theoretical predictions are available. Therefore, it is crucial to develop methodologies for reliable prediction of their properties for rapid experimental progress toward the new applications.
In 2015, sympathetic cooling of Ar 13+ with laser cooled Be + ions have been demonstrated [7], elevating HCIs to the realm of applications previously limited to singly-charged ions currently used for atomic clocks [8], quantum information [9], and other applications requiring laser cooling and trapping. Accurate prediction of wavelength of optical transitions suitable for clock development is a difficult task due to very large cancelations of the energies of upper and lower states. In these ions, high-order correlation, Breit, and radiative quantum electrodynamic (QED) corrections are all important, with cancelation of these contributions making accurate computations even more difficult [4]. As a result, it has become urgent to accurately take into account QED corrections in calculations of the electronic structure of such manyelectron ions.
In this work, we combined the most accurate treatment of the correlation corrections for multivalent atoms [42] with four different QED potentials, which allows us for the first time to accurately calculate and systematically study QED corrections in heavy ions with complex electronic structure. To check the accuracy of all these potentials we also calculated self-energy (SE) corrections to the one-electron energies of the valence states of the neutral alkali metals and to the transition energies in Cu-like ions and compared our results with the ab initio calculations.
We selected three representative HCIs with different electronic configurations as the test cases for the QED contributions to the DCB Hamiltonian. All of these ions were included in the studies of the applications of HCIs to the development of clocks and tests of the variation of the fundamental constants [4,5,43,44]. Ba 8+ was selected owing to the availability of the experimental values for comparison, Eu 14+ was chosen as the test case with the f 3 configuration, and Cf 15+ has the largest sensitivity to the alpha-variation in a system which satisfies all the requirement for the development of accurate optical atomic clocks [5].
We use a high-precision relativistic hybrid approach that combines configuration interaction and a linearized variant of the single-double coupled-cluster method, generally referred to as CI+all-order approach [42]. This method allows to include dominant correlation correction to all orders of perturbation theory. Breit corrections were included into the cal-culations. To separate the QED corrections, the CI+all-order computations were carried out with and without the QED corrections and difference was taken to be the QED contribution.
The main goals of this study were to answer the following questions for the type of ions that are of interest to the applications mentioned above, i.e. many-electron ions with a few valence electrons: (i) How important is QED correction for accurate prediction of the energy levels of these ions for future experimental exploration? (ii) How much the QED correction depends on the version of the model potential being used? (iii) Is it important to include the QED correction when constructing the basis set orbitals? (iv) Does QED contribution in such many-electron system depend on the accuracy of the inclusion of the correlation corrections, i.e. will the QED corrections calculated in the CI+MBPT and CI+all-order approximations differ?
We present one-electron QED potential as a following sum where V SE is the self-energy operator, V Uehl and V WK are the Uehling and Wichmann-Kroll parts of the vacuum polarization respectively. Both V Uehl and V WK are local potentials, so their treatment is rather straightforward and is the same in all four versions of the calculations, which differ by the treatment of the SE potential. The Uehling potential can be evaluated by a direct numerical integration of the well-known formula [46], or, more easily, by using the approximate formulas from Ref. [47]. A direct numerical evaluation of the Wichmann-Kroll potential V WK is rather complicated. For the purpose of the present work, it is sufficient to use the approximate formulas for the point-like nucleus from Ref. [48]. Method M1. Model self-energy potential. Following [39,49] we approximate the one-electron SE operator as the sum of local V SE loc and nonlocal V nl potentials where nonlocal potential is given in a separable form Here φ i are so-called projector functions. The choice of these functions depends on the method of construction of the nonlocal potential V nl and is described in details in [39]. The constants B ik are chosen so that the matrix elements of the model operator V SE ik calculated with hydrogen like wave functions ψ i have to be equal to matrix elements Q ik of the symmetrized exact one-loop energy-dependent SE operator Σ(ε) [50]: Introducing two matrices ∆Q ik = Q ik − ψ i |V SE loc |ψ k and D ik = φ i |ψ k , we find that The local part of the SE potential in [39] was taken in the simple form where the constant A κ is chosen to reproduce the SE shift for the lowest energy level at the given κ in the corresponding Hlike ion, and λ C = /(mc). The computation code based on this method is presented in Ref. [49]. Method M2. Self-energy nonlocal potential. In this approach we use the same equations (2), (3), (5) to construct the SE potential, but use radiative potential developed in [33,51] as the local part. In [33], the self-energy part of the total radiative potential is divided into three terms: where the potentials Φ mag , Φ lf and Φ hf are referred to as the magnetic form factor, the low-and high-frequency parts of the electric form factor, respectively, according to [33] . The expressions for these potentials are given by Eqs. (7,9,10) in [33]. Then, we obtain for the total SE potential The electric form factor contains some fitting parameters to reproduce the SE corrections for 5s and 5p states of heavy Hlike ions. However the local radiative potential Φ rad gives the SE contribution for the 1s state with only 10% accuracy [33] (see method M3 below). The SE potential (8) which contains the nonlocal part in addition to the local radiative potential reproduces the low lying SE corrections of the H-like ions exactly. Method M3. Local radiation potential. Here, we neglect the nonlocal term in (8) and use local radiative potential V SE = V SE loc = Φ rad from Eq. (7) as a full SE one-electron potential [33]. This radiative potential was widely used in many-electron calculations, for example, see [34,38,41,52] and references therein. Note that this local potential was optimized for weakly bound valence states of heavy neutral atoms and may be less accurate for strongly bound ionic, or core states.
Method M4. Nonlocal self-energy potential. This approach developed in [36] is similar to method M2, but is simpler: it uses only diagonal matrix elements Q ii of the exact one-loop SE operator Σ(ε) and different projector functions: where V SE loc = Φ rad and φ i = V SE loc ψ i . The expectation value of this potential, calculated with the wave functions ψ i of H-like ions is equal to the self-energy corrections Q ii : kj were obtained in [36]: where ∆Q ii = Q ii − D ii and D ij = ψ i |Φ rad |ψ j .
In Tables I and II we compare the SE values obtained using methods M1, M2, M3, and M4 described above with the ab initio calculations of Refs. [24] and [28] respectively, to which we refer as "exact".
Calculations of the SE shifts in Refs. [24,28] were performed with the local potential V eff (r): where V nuc (r) is nuclear potential and ρ(r) is total electron charge density. The choice x α = 0 corresponds to the Dirac-Hartree potential, x α = 2/3 is the Kohn-Sham potential, and x α = 1 is the DFS potential. Our data were obtained by averaging the SE operator V SE with the wave function of the valence state determined from the Dirac equation with the potential V eff (r).
In Table I, the SE shifts for the valence s-state of the neutral alkali atoms are given in terms of function F (αZ), defined by  Table I illustrates that the SE shifts obtained using M1, M2, and M4 methods are in very good agreement with exact results. We find some discrepancies between the data calculated using the local radiative potential (method M3) and exact values, especially for low Z.
In Table II we present the SE corrections calculated for the 4s − 4p and 4p − 4d transition energies of Cu-like ions. The results obtained within methods M1, M2, and M4 are in very good agreement with the exact ones. There is slight deviation of the data obtained in method M3 for high Z. Note that method M3 was recently modified in Ref. [41], where more complicated and accurate finite size correction to the radiative potential and additional fitting for d states were introduced.
Comparison of the QED corrections to the energies of Ba 8+ , Eu 14+ , and Cf 15+ obtained using all four QED potentials is given in Table III. The results in column labeled CI-M1 are obtained by including the QED potential only in the CI Hamiltonian, using the first variant of the QED potential. In this version of the calculations, the finite basis set is constructed with no QED corrections. Respectively, the QED corrections for the 4f and 5f orbitals are zero owing to no overlap with the nucleus. In all other calculations QED po-  tential is added in both CI Hamiltonian and in the construction of the basis set, which effectively changes the nf orbitals via the modification of the self-consistent potential. The results of such calculation are listed in column labelled CI-M1. Comparison of these values with full QED calculations (column M1) shows that while the differences between these approaches are minor for Ba 8+ , they are very significant for heavier ions with higher degree of ionization. When the QED contribution to the ground state is subtracted, the differences between CI-M1 and M1 approaches are still significant, 5%, 14%, and 25% for Ba 8+ , Eu 14+ , and Cf 15+ , respectively. We also carried out the same calculations using the less accurate method that combines CI and MBPT [45] to evaluate if the accurate treatment of the electronic correlation is important for the QED calculation. In the CI+MBPT method, core-valence correlation are treated in the second order of MBPT. CI+MBPT results are listed in column labelled M1 ′ The differences between the QED contributions calculated in the CI+MBPT and CI-all-order methods are small for Ba 8+ and Eu 14+ , but significant for J = 7 2 5f 6p 2 and 5f 2 6p levels of Cf 15+ . These J = 7 2 levels are strongly mixed and all-order corrections change weights of 6p and 5f electrons in the many-electron wavefunction, which affects the QED contributions. The differences between the calculations carried out with different QED potentials are generally small, with the biggest difference for the QED M3 potential. The differences increase for Cf 15+ , where QED corrections are the largest.
The QED corrections to the energies of Ba 8+ , Eu 14+ , Cf 15+ calculated using the CI+all-order method with the first version of the QED potential are given in Table IV to show the relative size of the QED corrections to the energy levels. All values are given relative to the corresponding ground state. Final values that include QED corrections are given in columns "Total". Non-QED part of the calculation is the same as in [4,5,43,44]. Our final results for Ba 8+ are in excellent agreement with experiment [53]. The QED corrections are very significant for low-lying 4f 3 levels of Eu 14+ , so we have also included the CI+all-order values without QED for clarity.
In the CI+MBPT and CI+all order calculations for the systems with three or more valence electrons there is an additional contribution to the valence energy from an effective three-electron (3e) interaction between valence electrons [54]. This contribution may be enhanced for the systems with an open f shell [55]. Respective 3e corrections appear for Eu 14+ and Cf 15+ ions and are listed in Table IV. These corrections are comparable to QED corrections for Eu 14+ .
In summary, we find that accurate treatment of the QED effects is essential for reliable prediction of the transition energies in HCIs with optical transitions of interest to the clock development and tests of fundamental physics. The QED corrections in these ions are large enough to significantly affect the predictions of the transition wavelengths. Our results show that the QED corrections obtained by all four QED potentials are very similar, with the difference being smaller than the estimated uncertainty in the treatment of the correlation correction. We find that it is imperative to include the QED correction both in the construction of the basis set orbitals and into the CI Hamiltonian, in particular for the configurations involving 5f electrons, as in the example of Cf ions. In the case of strong configuration mixing, QED corrections calculated in the CI+MBPT (M1 ′ ) and CI+all-order (M1) approximations may differ by as much as 100 cm −1 . We demonstrate that QED effects can be reliably accounted for by incorporating the modern QED potentials into the CI+all order method. Finally, high precision calculations of the systems with more than two valence electrons should include contribution of the effective three-electron interactions between valence electrons together with QED effects.
This work is partly supported by the Russian Foundation for Basic Research Grants No. 14-02-00241, 15  IV: Transition energies (cm −1 ) for Ba 8+ , Eu 14+ , and Cf 15+ calculated using the CI+all-order method and M1 version of QED potential. Experimental results for Ba 8+ are from Ref. [53]. Columns QED, 3e, and Total present QED corrections, contribution of the effective threeelectron interactions [55], and final theoretical values, respectively.