Ab initio calculations of energy levels in Be-like xenon: strong interference between electron-correlation and QED effects
A. V. Malyshev, D. A. Glazov, Y. S. Kozhedub, I. S. Anisimova, M. Y. Kaygorodov, V. M. Shabaev, I. I. Tupitsyn
aa r X i v : . [ phy s i c s . a t o m - ph ] S e p Ab initio calculations of energy levels in Be-like xenon:strong interference between electron-correlation and QED effects
A. V. Malyshev, D. A. Glazov, Y. S. Kozhedub, I. S. Anisimova, M. Y. Kaygorodov, V. M. Shabaev, and I. I. Tupitsyn Department of Physics, St. Petersburg State University, Universitetskaya 7/9, 199034 St. Petersburg, Russia
High-precision QED calculations of the ground and singly excited energy levels in Be-like xenonare performed. The close levels of the same symmetry which are strongly mixed by the electron–electron interaction are treated within the QED perturbation theory for quasidegenerate states.The contributions of all the Feynman diagrams up to the second order are taken into account. Themany-electron QED effects are rigorously evaluated in the framework of the extended Furry pictureto all orders in the nuclear-strength parameter αZ . The higher-order electron-correlation effects areconsidered within the Breit approximation. The nuclear recoil effect is accounted for as well. Themost accurate theoretical predictions for the binding and excitation energies are obtained. Theseresults deviate from the most precise experimental value by 3 σ but perfectly agree with a morerecent measurement. Be-like ions represent the simplest example of atomicsystems with more than one electron in a valence shell.Electromagnetic interaction of the L -shell electrons,combined with intershell correlation effects, leads to astrong mixing of the close energy levels of the same sym-metry. On the other hand, precise calculations of highlycharged Be-like ions must include QED effects, whichare of the same order of magnitude as in He- and Li-like ions (see, e.g., the recent review [1] and referencestherein). Therefore, highly charged Be-like ions pose aserious challenge to atomic structure calculations withinthe framework of bound-state QED. Despite numerousrelativistic calculations of the energy levels in these sys-tems [2–16], the QED effects have been included at bestwithin some one-electron approximations only. Differ-ent theoretical approaches show significant scatter of theresults when compared with the available experimentaldata [17–28]. The many-electron QED effects for theground state of Be-like ions were evaluated in our recentworks [29, 30], where the calculations were performedusing the QED perturbation theory for a single level.However, due to the proximity of all the n = 2 levels,including the ground state, the accuracy of this eval-uation remains unclear, unless the calculations basedon ab initio QED approach for quasidegenerate statesare performed. The present work is intended to solvethis long-standing and extremely difficult problem. Thedevelopment of the rigorous QED theory for the high-precision calculations of the ground and singly excitedstates in Be-like ions is the primary goal of this study.To the best of our knowledge, the QED calculations ofthe quasidegenerate states at such a level have neverbeen performed previously for the systems with morethan two electrons. Moreover, the three-dimensionalmodel subspace of quasidegenerate levels, which we em-ploy in the present paper, is also considered for the firsttime in the framework of ab initio approach. It is theapplication of such sophisticated methods that allowsus to accurately address an issue of strong interferencebetween electron–electron interaction and QED effects.The natural zeroth-order approximation for the sys- tematic QED description of highly charged ions is pro-vided by the Dirac equation[ α · p + βm + V ] ψ n = ε n ψ n . (1)By substituting the Coulomb potential of the nu-cleus V nucl as the binding potential V in Eq. (1) onecomes to the Furry picture of QED [31]. The electron–nucleus interaction is taken into account to all ordersin αZ in this way ( α is the fine-structure constant, Z is the nuclear charge number). In order to par-tially account for the electron–electron interaction ef-fects from the very beginning, the initial approxima-tion can be modified by adding some local screeningpotential, V = V nucl + V scr . This corresponds to theextended version of the Furry picture [32–43]. The re-maining part of the interelectronic interaction as well asthe interaction with the quantized electromagnetic fieldare to be considered within appropriate perturbationtheory (PT).Standard nondegenerate PT suits well for single (iso-lated) levels, such as the ground state 1 s of He-likeions [44], or when one studies, e.g., the fine-structuresplitting 2 p j –2 s in Li-like ions [40, 41]. However, con-sidering more complicated systems or excited states oneinevitably encounters close levels with the same symme-try which are mixed strongly by the electron–electroninteraction. Application of the extended Furry pictureallows one to lift the degeneracy in some cases. Never-theless, the proper treatment of these systems requiresemploying PT for quasidegenerate levels. Both types ofthe QED perturbation series can be conveniently con-structed in the framework of the two-time Green’s func-tion (TTGF) method [45]. For a set of s quasidegen-erate levels, the TTGF method implies the evaluationof the s × s matrix H which acts in the model sub-space Ω spanned by the unperturbed wave functions ofthe states under consideration. The energies can be ob-tained by diagonalizing the matrix H . PT for a singlelevel corresponds to s = 1.In the present work, we aim at the ab initio evalu-ation of the ground and singly excited energy levels in (a) (b) FIG. 1. Tho-photon exchange diagrams.
Be-like xenon. Despite the fact that our calculationsare fully relativistic, we employ the LS -coupling nota-tions. The excited states with the total angular mo-mentum equal to J = 0 and J = 2, namely 2 s p P and 2 s p P (here and in what follows, the K shell isomitted for brevity), are considered as the single levels.The corresponding unperturbed wave functions in the jj coupling read as (2 s p / ) and (2 s p / ) , respectively.The states 2 s p P and 2 s p P with J = 1 are stud-ied within the two approaches: (a) as the isolated lev-els, starting from the initial approximations (2 s p / ) and (2 s p / ) ; (b) as a pair of quasidegenerate levelswithin the two-dimensional subspace Ω. Finally, thebinding energy of the ground 2 s s S state is evalu-ated by means of three independent approaches: (a) asthe isolated (2 s s ) level; (b) as one of the two quaside-generate levels, (2 s s ) and (2 p / p / ) , within thetwo-dimensional subspace Ω; (c) as one of the threequasidegenerate levels, including the (2 p / p / ) con-figuration, for which the mixing coefficient can be evenlarger than for (2 p / p / ) [2, 46]. The approach (a)for the ground state reproduces the one which we haveused in Ref. [29]. In the Coulomb potential, the unper-turbed levels forming the quasidegenerate subspaces aresplit only by the nuclear-size and relativistic effects. Let us briefly formulate our approach. In order to de-rive all the relevant calculation formulas, we start withthe formalism in which the closed 1 s shell is regardedas belonging to the vacuum [45], and there are two L -shell “valence” electrons. The redefinition of the vac-uum is carried out by changing the sign before i G isreplaced as follows: G ( ω ) ≡ X n | n ih n | ω − ε n + iε n → X n | n ih n | ω − ε n + iη n , (2)where η n = ε n − ε F , and ε F is the Fermi energy. TheFermi energy is chosen to be higher than the energyof the closed-shell electrons but lower than the valence-electron energy. In this formalism, the one- and two-loop one-electron Feynman diagrams include the contri-butions describing the interaction between one valenceelectron and the closed shell. These contributions canbe separated by considering the difference with the stan-dard definition of the vacuum. For instance, the first-order self-energy and vacuum-polarization diagrams forthe valence state | v i lead to the one-photon exchangecontribution∆ E (1)int = X c [ I vcvc (0) − I cvvc ( ε v − ε c )] , (3)where I abcd ( ω ) = h ab | I ( ω ) | cd i , I ( ω ) = e α µ α ν D µν ( ω ), α µ = (1 , α ), and D µν ( ω ) is the photon propagator. Westress that the treatment of the one-electron diagramsdoes not depend on the type of PT. By employing thisformalism, we obtain the well-known expressions for theintershell-interaction corrections derived previously forLi-like systems, see, e.g., Refs. [41, 47–49].Now we turn to the discussion of the two-electron diagrams within the formalism with the redefined vacuum. Thesediagrams contain the three-electron contributions corresponding to the interaction of both the valence electrons withthe 1 s core. In case of He-like ions and two-dimensional subspace Ω, the formal expressions for the second-ordertwo-electron contributions were derived within the TTGF method in Refs. [50–52], see also Ref. [53]. In the presentwork, we have generalized these expressions to deal with an arbitrary number of quasidegenerate levels. The mostcomplicated and time-consuming is the derivation of the formulas for the two-photon exchange diagrams in Fig. 1.We restrict our consideration only to the contribution of the ladder diagram in Fig. 1(a). This contribution isnaturally divided into the reducible (“red”) and irreducible (“irr”) parts. The reducible part involves the termswith the intermediate states coinciding with the quasidegenerate levels under consideration, while the irreduciblepart corresponds to the remainder. The reducible part is not affected by the redefinition of the vacuum, and itscontribution to H , arising from the interaction between the valence electrons, reads as H red ik = − X P ( − PE (0) n = E (0)1 ...E (0) s X n n i π Z ∞−∞ dω " I P i P i n n ( ω − ε P i ) I n n k k ( ε k − ω ) (cid:0) ω − ε n − i (cid:1)(cid:0) ω + ε n − ¯ E (0) ik − i (cid:1) + { ↔ } , (4)where the indices i and k enumerate the quasidegenerate states, P is the permutation operator, E (0) n = ε n + ε n ,¯ E (0) ik = ( E (0) i + E (0) k ) /
2, and { ↔ } means the expression with the transposed indices 1 and 2. The contributionof the irreducible part of the ladder diagram within the employed formalism can be expressed as follows:˜ H irr ik = 12 X P ( − PE (0) n = E (0)1 ...E (0) s X n n i π Z ∞−∞ dω " I P i P i n n ( ω − ε P i ) I n n k k ( ε k − ω ) (cid:0) ω − ε n + iη n (cid:1)(cid:0) ¯ E (0) ik − ω − ε n + iη n (cid:1) + { ↔ } . (5)We readily extract the desired three-electron contribution from Eq. (5) δH ik = − X P ( − P X c X n E (0) ik − ε c − ε n " I P i P i nc ( ¯ E (0) ik − ε c − ε P i ) I nck k ( ε k + ε c − ¯ E (0) ik )+ I P i P i nc ( ε c − ε P i ) I nck k ( ε k − ε c ) + { ↔ } . (6)The total three-electron contribution can be obtained bystudying the crossed diagram in Fig. 1(b) and the two-electron self-energy and vacuum-polarization graphs.By considering the one- and two-electron diagramsin the formalism with the redefined vacuum, we takeinto account all the necessary contributions describingthe interaction between the L and K shells. In or-der to evaluate the total binding energies of Be-likeions, we have to add the QED contributions correspond-ing to the 1 s core. This issue is discussed in details,e.g., in Refs. [53, 54]. As a result, our numerical ap-proach rigorously takes into account all the contribu-tions of the first and second orders of QED PT. Theelectron-correlation contributions due to the exchangeby three or more photons are accounted for within theBreit approximation in the present work. The corre-sponding calculations are based on the Dirac–Coulomb–Breit (DCB) Hamiltonian and performed by means ofthe large-scale configuration-interaction (CI) method inthe basis of the Dirac–Sturm orbitals [55–57]. The pro-cedure how to merge the QED calculations with thehigher-order interelectronic-interaction contributions incase of quasidegenerate levels was suggested first inRef. [54] and described in more details in Ref. [53].Finally, we account for the nuclear recoil and nuclearpolarization effects which lie beyond the external-fieldapproximation, that is beyond the Furry picture.Let us now turn to the discussion of the numeri-cal results. In Table I, we present the binding ener-gies of the ground and singly excited states in Be-likexenon. The calculations are performed starting from theCoulomb potential as well as within the extended Furrypicture. In the latter case, the core-Hartree (CH) andlocal Dirac–Fock (LDF) screening potentials are incor-porated in Eq. (1). The description and applications ofthese potentials can be found, e.g., in Refs. [34, 43, 58].For the nuclear charge distribution, the Fermi modelis used. The root-mean-square radius and the nuclearmass of the isotope Xe are taken as in Ref. [59].As noted above, for the ground and J = 1 states weconstruct the alternative perturbation series for boththe single and the quasidegenerate levels. The type ofPT is shown in the first column, where the size of thesubspace Ω is indicated. The columns labeled with A,B, C, and D demonstrate how the energies change whenwe successively take into account different contributions.In the column A, we present the values obtained withinthe Breit approximation by means of the CI method. From Table I, one can see that for the specific potentialthese results do not depend on the size of Ω. We notethat the 1 × × × H which are constructed based onthe CI calculations in accordance with the prescriptionsfrom Ref. [53]. The energies for the specific subspace Ωin the column A vary slightly from the potential to po-tential. This variation is caused by the dependence ofthe positive-energy-states projectors in the DCB Hamil-tonian on the initial approximation in our approach, seethe discussion, e.g., in Refs. [53, 57].The results in the column B are obtained by addingthe first-order QED contributions, namely the self en-ergy, vacuum polarization, and frequency-dependentcorrection of the one-photon exchange contribution, aswell as the nuclear recoil and nuclear polarization con-tributions. We stress that for the quasidegenerate levelsthe inclusion of the terms is not quite additive becauseof the mixing. One can see that the results for dif-ferent potentials demonstrate significant scatter. Thescatter can be reduced by considering the second-orderQED corrections. This is done in the column C, wherewe add the contributions of the two-electron self-energyand vacuum-polarization diagrams, the nontrivial QEDpart of the two-photon exchange contribution (beyondthe Breit approximation), and the two-loop one-electroncorrections. The difference between the calculationswith the Coulomb, CH, and LDF potentials indeed de-creases going from B to C. The values in the column Cobtained for the screening potentials are shifted slightlywith respect to the Coulomb ones. This results froma rearrangement of the perturbation series within theextended Furry picture.From the column C, it is seen that the energy ofthe ground 2 s s S state considerably shifts as we passfrom 1 × ×
2. However, the value of this shift isalmost independent on the initial approximation. Theshift is explained by the accurate treatment of the mix-ing of the states within the model subspace. Whenwe extend Ω by including the (2 p / p / ) configu-ration, the ground-state energy acquires an additionalshift which is smaller by an order of magnitude. Theuncertainties of these calculations are determined, inparticular, by the uncalculated screened QED contri-butions of the second order in 1 /Z . Nowadays, thesecorrections are inaccessible by the rigorous QED meth-ods. We can estimate them approximately employing TABLE I. Binding energies (with the opposite sign) of theground and singly excited states in Be-like xenon (in eV).Comparison of the different approaches: A, B, C, and D.See the text for details. Ω V eff A B C D2 s s S × .
884 100 970 .
193 100 973 .
026 —CH 101 071 .
948 100 973 .
924 100 972 .
977 100 973 . .
928 100 973 .
451 100 972 .
981 100 973 . × .
884 100 970 .
443 100 973 .
244 100 973 . .
948 100 974 .
157 100 973 .
194 100 973 . .
928 100 973 .
682 100 973 .
198 100 973 . × .
884 100 970 .
487 100 973 .
278 100 973 . .
948 100 974 .
199 100 973 .
229 100 973 . .
928 100 973 .
724 100 973 .
233 100 973 . s p P × .
328 100 866 .
432 100 868 .
743 100 868 . .
373 100 869 .
738 100 868 .
699 100 868 . .
356 100 869 .
227 100 868 .
703 100 868 . s p P × .
533 100 843 .
637 100 845 .
987 100 845 . .
579 100 846 .
944 100 845 .
941 100 845 . .
562 100 846 .
433 100 845 .
946 100 845 . × .
533 100 843 .
635 100 845 .
975 100 845 . .
579 100 846 .
941 100 845 .
930 100 845 . .
562 100 846 .
430 100 845 .
934 100 845 . s p P × .
593 100 501 .
446 100 503 .
789 100 503 . .
656 100 504 .
784 100 503 .
744 100 503 . .
637 100 504 .
272 100 503 .
748 100 503 . s p P × .
196 100 438 .
050 100 440 .
465 100 440 . .
262 100 441 .
390 100 440 .
416 100 440 . .
242 100 440 .
878 100 440 .
420 100 440 . × .
197 100 438 .
053 100 440 .
477 100 440 . .
262 100 441 .
392 100 440 .
428 100 440 . .
242 100 440 .
880 100 440 .
432 100 440 . the model Lamb-shift (QEDMOD) operator which hasbeen suggested recently in Refs. [60, 61] and successfullyapplied to the QED calculations in various atomic sys-tems [57, 62–67]. In order to estimate the screened QEDeffects of the second order in 1 /Z , we have introducedthe QEDMOD operator into Eq. (1) and evaluated thetwo-photon exchange contribution in the Breit approx-imation using the related one-electron basis. The cor-rection of interest was obtained by subtracting the cor-responding contribution calculated without the QED-MOD operator. The binding energies with these cor-rections included are shown in the column D. One cansee that the 1 × s s S state tend to the eigenvalues of the matrices H . However, thediscrepancy of the results for two screening potentialsis large. Within nondegenerate PT, the higher-orderQED effects contribute significantly, and their contri-bution can not be neglected. On the other hand, in thecolumn D the difference between the 2 × × × J = 1 states,the situation, in principal, is the same. However, themixing is less pronounced than for the ground state.The values shown in the column D of Table I forthe LDF potential and the maximum size of thesubspace Ω are employed as the final results. Theuncertainties are obtained by summing quadraticallyseveral contributions. First, in addition to the numer-ical errors, we take into account the uncertainties ofthe nuclear-size effect and of the two-loop one-electroncorrections [59]. Second, we estimate the uncalculatedhigher-order QED contributions through several means.The QED corrections to the electron-correlation effectsof third and higher orders are estimated according tothe procedure from Ref. [53]. The screening of thetwo-loop contributions is estimated by multiplyingthe corresponding term for 1 s by the conservativefactor 2 /Z . We take into account the scatter of theresults obtained for the different potentials as well.Finally, having in mind that the calculations of thescreened QED effects of the second order in 1 /Z inthe column D are approximate, we take the corre-sponding correction with the 100% uncertainty. As aresult, for the ground-state binding energy we obtain E [2 s s S ] = − . − . × E [2 s p P ] = −
100 868 . E [2 s p P ] = −
100 845 . E [2 s p P ] = −
100 503 . E [2 s p P ] = −
100 440 . s p S +1 P J states pre-sented in Table II. The uncertainties are estimated sim-ilarly to the binding energies. In Table II, we compareour excitation energies with the results of the previ-ous relativistic calculations and recent measurements.In Ref. [57], these energies were evaluated by means of TABLE II. Excitation energies from the ground 2 s S to 2 s p S +1 P J states in Xe (in eV).2 s p P s p P s p P s p P Year ReferenceTheory104 . . . . . . . . et al. [57]104 .
475 127 .
282 469 .
449 532 .
877 2008 Cheng et al. [11]104 .
663 127 .
475 470 .
004 533 .
401 2005 Gu [9]127 .
168 469 .
25 532 .
62 2000 Safronova [7]127 .
301 532 .
854 1997 Chen & Cheng [6]104 .
482 127 .
267 469 .
386 532 .
759 1996 Safronova et al. [5]103 .
722 126 .
846 468 .
338 532 .
766 1979 Cheng et al. [3]Experiment127 . . . et al. [28]127 . et al. [26]127 . et al. [24]127 . et al. [21]126 . et al. [20]126 . et al. [19]126 . et al. [18] the CI+QEDMOD method, and the uncertainties wereestimated in a rather conservative way. However, thecomparison with the results of the present work showsthat the accuracy of this approach is at least one or-der of magnitude higher. As for the experiments, ourresults are in perfect agreement with the most recentmeasurements performed in Ref. [28], especially for the2 s p P state, for which the experimental uncertaintyis minimal. On the other hand, the most precise mea-surement for the 2 s p P state [24] deviates from ourresult by almost 4 times the experimental uncertaintyand by 5 times our theoretical uncertainty. The reasonof this discrepancy is unclear to us.To summarize, ab initio QED calculations of thebinding energies of the ground and singly excitedstates in Be-like xenon have been performed with themost advanced methods available to date. The cal-culations merge the rigorous QED treatment up tothe second-order contributions and the higher-order electron-correlation effects evaluated within the Breitapproximation. For the first time, the ground stateas well as the states with the total angular momen-tum equal to 1 are treated by means of perturbationtheory for quasidegenerate levels. As a result, we haveobtained the most precise theoretical predictions for theenergy levels and ∆ n = 0 intra- L -shell excitation ener-gies, which are in perfect agreement with the most re-cent measurements [28]. Meanwhile, some discrepancywith the previous experiment [24] is found. New mea-surements with Be-like xenon and other Be-like ions arein demand.This work was supported by the grant of the Presidentof the Russian Federation (Grant No. MK-1459.2020.2).A.V.M., M.Y.K., and V.M.S. acknowledge the supportfrom the Foundation for the Advancement of Theo-retical Physics and Mathematics “BASIS”. D.A.G. ac-knowledges the support by RFBR (Grant No. 19-02-00974). The work of I.I.T. was supported by RFBR(Grant No. 18-03-01220). [1] P. Indelicato, J. Phys. B: At. Mol. Opt. Phys. , 232001 (2019).[2] L. Armstrong, W. R. Fielder, and D. L. Lin, Phys. Rev.A , 1114 (1976).[3] K. T. Cheng, Y. K. Kim, and J. P. Desclaux, At. DataNucl. Data Tables , 111 (1979).[4] X.-W. Zhu and K. T. Chung, Phys. Rev. A , 3818(1994).[5] M. S. Safronova, W. R. Johnson, and U. I. Safronova,Phys. Rev. A , 4036 (1996).[6] M. H. Chen and K. T. Cheng, Phys. Rev. A , 166(1997). [7] U. I. Safronova, Mol. Phys. , 1213 (2000).[8] S. Majumder and B. P. Das, Phys. Rev. A , 042508(2000).[9] M. F. Gu, At. Data Nucl. Data Tables , 267 (2005).[10] H. C. Ho, W. R. Johnson, S. A. Blundell, andM. S. Safronova, Phys. Rev. A , 022510 (2006).[11] K. T. Cheng, M. H. Chen, and W. R. Johnson, Phys.Rev. A , 052504 (2008).[12] J. M. Sampaio, F. Parente, C. Naz´e, M. Gode-froid, P. Indelicato, and J. P. Marques, Phys. Scr. T156 , 014015 (2013).[13] V. A. Yerokhin, A. Surzhykov, and S. Fritzsche, Phys.
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