Higher-order perturbative relativistic calculations for few-electron atoms and ions

Nuclear Instruments and Methods in Physics Research B 408 (2017) 46–49

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Higher-order perturbative relativistic calculations for few-electron atoms and ions D.A. Glazov a,b,⇑, A.V. Malyshev a,b, A.V. Volotka a,c, V.M. Shabaev a, I.I. Tupitsyn a,d, G. Plunien e a

Department of Physics, St. Petersburg State University, Oulianovskaya 1, Petrodvorets, 198504 St. Petersburg, Russia State Scientific Centre ‘‘Institute for Theoretical and Experimental Physics” of National Research Centre ‘‘Kurchatov Institute”, B. Cheremushkinskaya st. 25, 117218 Moscow, Russia c Helmholtz-Institut Jena, Fröbelstieg 3, D-07743 Jena, Germany d Center for Advanced Studies, Peter the Great St. Petersburg Polytechnic University, Polytekhnicheskaja 29, 195251 St. Petersburg, Russia e Institut für Theoretische Physik, Technische Universität Dresden, Mommsenstraße 13, D-01062 Dresden, Germany b

a r t i c l e

i n f o

Article history: Received 15 December 2016 Received in revised form 27 April 2017 Accepted 28 April 2017 Available online 30 May 2017 Keywords: Highly charged ions Electronic structure Perturbation theory

a b s t r a c t An effective computational method is developed for electronic-structure calculations in few-electron atoms and ions on the basis of the Dirac-Coulomb-Breit Hamiltonian. The recursive formulation of the perturbation theory provides an efficient access to the higher-order contributions of the interelectronic interaction. Application of the presented approach to the binding energies of lithiumlike and boronlike systems is demonstrated. The results obtained are in agreement with the large-scale configuration interaction Dirac-Fock-Sturm method and other all-order calculations. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Highly charged ions provide a unique scenario for probing QED effects in the strongest electromagnetic fields and give an access to accurate determination of the fundamental physical constants and the nuclear parameters [1,2]. The most stringent test of boundstate QED for highly charged ions is achieved with the Lamb shift in lithiumlike uranium [3,4]. Investigations of the bound-electron g factor in hydrogenlike and lithiumlike ions provided the most accurate value of the electron mass [5] and the stringent tests of the various bound-state QED effects in the presence of magnetic field [6,7], including the relativistic nuclear recoil effect [8]. The g factor and the hyperfine splitting in lithiumlike and boronlike ions play the leading role for proposed tests of bound-state QED [9,10] and determination of the fine structure constant [11,12]. Evaluation of the interelectronic-interaction effects is required to provide accurate theoretical predictions for many-electron ions. Rigorous treatment of these effects within the framework of bound-state QED yields the correct results to all orders in the parameter aZ (a is the fine structure constant, Z is the nuclear charge). Corresponding calculations through the second order of perturbation theory (one- and two-photon-exchange diagrams)

⇑ Corresponding author at: Department of Physics, St. Petersburg State University, Oulianovskaya 1, Petrodvorets, 198504 St. Petersburg, Russia. E-mail address: [email protected] (D.A. Glazov). http://dx.doi.org/10.1016/j.nimb.2017.04.089 0168-583X/Ó 2017 Elsevier B.V. All rights reserved.

have been accomplished for the binding energies in heliumlike [13], lithiumlike [14–17], berryliumlike [18,19] and boronlike [20–22] ions. The contributions of the third and higher orders of perturbation theory are available to date only within the Breit approximation, which corresponds to the leading orders in aZ. There are various highly elaborate methods to evaluate these contributions. The starting point of any of these methods is the DiracCoulomb-Breit equation. The many-body perturbation theory (MBPT) (see, e.g., [23]) assumes evaluation of the perturbationtheory diagrams and is generally bound by the 2nd or 3rd order. Most of the other methods treat the interelectronic interaction to all orders. We mention here the multiconfiguration Dirac-Fock (MCDF) method (see, e.g., [24]), the coupled-cluster method (see, e.g., [25]), and many variations of those, which are successfully applied to relativistic calculations of the neutral atoms and ions with few valence electrons. However, the most accurate to date results for few-electron ions have been obtained within the large-scale configuration interaction Dirac-Fock-Sturm method (CI-DFS) [26,27]. It has been used to evaluate the third and higher orders for the binding energies in lithiumlike [17], berryliumlike [18,19] and boronlike [20–22] ions. It was also applied to the corresponding calculations for the g factor [7,28,29], the hyperfine splitting [30], the transition probabilities [31] and many others. In the present paper we develop the novel method of calculation of the interelectronic-interaction contributions to the binding energies in the Breit approximation. It is based on the perturbation

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D.A. Glazov et al. / Nuclear Instruments and Methods in Physics Research B 408 (2017) 46–49

theory and is not bound to the lowest orders. In contrast to MBPT, it allows for evaluation of arbitrary orders on equal footing, without consideration of the individual diagrams. In contrast to allorder methods, it provides the perturbation-theory terms separately, which is advantageous in many cases. It is based on the recursive formulation of the perturbation theory with the finite basis set of many-electron wave functions constructed as the Slater determinants. The one-electron wave functions are found within the DKB method [32]. This method can be generalized to the calculations of the g factor, hyperfine splitting and other atomic properties.

ð1Þ

where Kþ is the positive-energy-states projection operator, constructed as the product of the one-electron projectors. In order to formulate the perturbation theory (PT) we split the DiracCoulomb-Breit Hamiltonian H into parts H0 and H1 . The zerothapproximation part H0 is the sum of the one-electron Dirac Hamiltonians,

X hðjÞ;

ð2Þ

j

h ¼ a  p þ bm þ V nuc ðrÞ þ V scr ðrÞ;

ð3Þ

where the local screening potential V scr ðrÞ is introduced in order to improve the convergence of the perturbation series. The solutions to the zeroth-order equation,

Kþ H0 Kþ jAð0Þ i ¼ EAð0Þ jAð0Þ i;

ð4Þ

are the Slater determinants of the one-electron eigenfunctions of the Dirac Hamiltonian (2). In this work, we consider only the nondegenerate states described by one Slater determinant, e.g. closed shells or one electron beyond closed shells. The perturbation part H1 represents the interelectronic interaction in the Breit approximation with the screening potential subtracted,

X X V Breit ðj; kÞ  V scr ðr j Þ;

ð5Þ ð6Þ

The perturbation theory in H1 leads to the following expansions for the energy EA and the wave function jAi, 1 X ðkÞ EA ;

ð7Þ

k¼0

jAi ¼

1 1 X X X jAðkÞ i ¼ jNð0Þ ihNð0Þ jAðkÞ i: k¼0

ð8Þ

k¼0 N

The Slater determinants jN ð0Þ i form the orthogonal set of the ðkÞ

solutions to the zeroth-order Eq. (4). The energy corrections EA ð0Þ

ðkÞ

and the coefficients hN jA i can be found via the recursive system of equations, ðkÞ

EA

k1 E X D ð0Þ X ðjÞ ¼ A jH1 jM ð0Þ hM ð0Þ jAðk1Þ i  EA hAð0Þ jAðkjÞ i; M



k1 X

ð0Þ EN

E Nð0Þ jH1 jMð0Þ hM ð0Þ jAðk1Þ i

M

#

ðjÞ EA hNð0Þ jAðkjÞ i

;

ð10Þ

j¼1

hAjAðkÞ i ¼ 

k1 X 1X hAðjÞ jM ð0Þ ihM ð0Þ jAðkjÞ i; 2 j¼1 M

ð11Þ

with the obvious initial values,

  hNð0Þ jAð0Þ i

¼ 0;

ð12Þ

ð0Þ

j¼1

ð9Þ

ð13Þ

The Eqs. (9) and (10) follow immediately from the standard equations of the perturbation theory [33]. Eq. (11) is the consequence of the normalisation condition hAjAi ¼ 1. One can see that the Eqs. (9)–(11) lead to the standard expressions for the perturbation theory terms, which assume ðk  1Þ-fold summation for the energy and k-fold summation for the wave function in the kth order. In contrast to these expressions, the recursive equations comprise one summation for each state jNi, so it is twofold effectively. This fact provides the indisputable computational efficiency for the wave-function corrections of the 3rd and higher orders, and for the energy corrections of the 4th and higher orders. Within the standard methods, e.g. MBPT, the matrix elements with the many-electron wave functions are expanded in terms of the one-electron functions. On the one hand, it cuts down the major part of the summations. On the other hand, it forces to consider the corresponding set of diagrams, which multiply exponentially with the PT order. Instead, we work with the presented equations as is, using the well-known formulae for the matrix

Table 1 Interelectronic-interaction contributions to the ionization energy (in a.u.) of the 2s; 2p1=2 and 2p3=2 states in neutral lithium with the Coulomb potential. The results of the MBPT calculations of the 0th, 1st, 2nd and 3rd orders from Ref. [16] (a) and of the all-order variational calculations from Ref. [39] (b) are given for comparison. PT order

2s

2p1=2

2p3=2

0

1.12517 1.12517a

1.12517 1.12517a

1.12503 1.12503a

1

1.19370 1.19370a

1.40602 1.40602a

1.40571 1.40571a

2

0.25063 0.25067a

0.37128 0.37129a

0.37098 0.37105a

3

0.00840 0.00838a

0.02610 0.02608a

0.02625 0.02617a

0–3

0.19050 0.19051a

0.11653 0.11651a

0.11655 0.11654a

4 5 6 7 8 9 10 11 12 13 14 15

0.004363 0.001812 0.000741 0.000328 0.000187 0.000100 0.000029 0.000007 0.000013 0.000014 0.000007 0.000001

0.010045 0.003373 0.000803 0.000043 0.000199 0.000129 0.000101 0.000122 0.000089 0.000008 0.000044 0.000037

0.110036 0.003369 0.000800 0.000045 0.000200 0.000130 0.000101 0.000122 0.000089 0.000008 0.000044 0.000037

4–1

0.00763

0.01368

0.01363

0–1

0.19813

0.13021

0.13018

all-order

0.19815972b

0.13024269b

0.13024117b

j

  1 a1  a2 1 V Breit ¼ a   ða1  $1 Þða2  $2 Þr 12 : r12 2 r 12

EA ¼

ð0Þ EA

hA jA i ¼ 1:

Kþ ðH0 þ H1 ÞKþ jAi ¼ EA jAi;

j
" XD

1



ð0Þ

We assume that the few-electron ion under consideration is described by the Dirac-Coulomb-Breit equation written in the following form,

H1 ¼

N–A

¼

N–A

2. Methods and Results

H0 ¼

  hNð0Þ jAðkÞ i

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D.A. Glazov et al. / Nuclear Instruments and Methods in Physics Research B 408 (2017) 46–49

Table 2 Interelectronic-interaction contributions of the 3rd and higher orders to the ionization energies (in eV) of the 2p1=2 and 2p3=2 states in boronlike argon with the LDF and PZ potentials. The results of the CI-DFS calculations are given for comparison. 2p1=2

2p3=2

PT order

LDF

PZ

LDF

PZ

3 4 5 6 7 8 9 10 11 12 13 14 15

0.3560 0.5623 0.1644 0.0581 0.0608 0.0106 0.0100 0.0062 0.0001 0.0016 0.0005 0.0003 0.0002

0.4209 0.5219 0.1284 0.0583 0.0474 0.0043 0.0086 0.0038 0.0008 0.0011 0.0001 0.0002 0.0001

0.4706 0.5970 0.1575 0.0710 0.0669 0.0109 0.0114 0.0072 0.0001 0.0018 0.0006 0.0002 0.0002

0.5294 0.5535 0.1222 0.0689 0.0520 0.0044 0.0097 0.0044 0.0007 0.0013 0.0002 0.0002 0.0001

3–1

0.0296

0.0471

0.0516

0.1249

CI-DFS

0.0295

0.0470

0.0515

0.1246

elements of the one-electron and two-electron operators with the Slater determinants. The DKB method [32] provides the finite basis set of the one-electron wave functions. The recursive equations, essentially equivalent to Eqs. (9)–(11), have been employed for variational calculations, e.g. by Drake and Dalgarno [34]. Somewhat similar approach with the basis of B-splines was developed by Plante and co-authors [35], however it doesn’t correspond to the PT expansion. The presented approach has been successfully applied to the calculations of the ground-state energy of heliumlike ions, the energies of the ground and first excited states in lithiumlike and boronlike ions. For heliumlike ions the 1=Z-expansion has been developed in a number of papers, which corresponds to the perturbation expansion without the screening potential. The PT terms obtained within our method for the ground-state energy of heliumlike ions with the Coulomb potential are in full agreement with the nonrelativistic [36] and relativistic [37] coefficients of the 1=Zexpansion. In order to obtain the nonrelativistic limit, we extrapolate the results to Z ¼ 0. For lithiumlike ions the nonrelativistic 1=Z-expansions for 2s and 2p states were presented in Ref. [38], and we find our results with the Coulomb potential in agreement as well. In Table 1 the ionization energies of the 2s; 2p1=2 and 2p3=2 states in neutral lithium are presented for the Coulomb potential. The values of Ref. [16] for the contributions of up to the 3rd order evaluated within the MBPT approach are given for comparison. We also present the high-accuracy variational results for the total energies from Ref. [39] used as a reference by Yerokhin and co-authors. One can see that the convergence of the PT expansion is good enough, and that our results are in full agreement with those of Refs. [16,39]. The ionization energies of the 2p1=2 and 2p3=2 states in boronlike argon are investigated in our recent study [22]. The contributions of the interelectronic interaction are calculated within the rigorous QED approach through the 2nd order, while the 3rd and higher orders are considered in the Breit approximation. This is accomplished within the two independent methods: the large-scale configuration-interaction Dirac-Fock-Sturm method (CI-DFS) and the recursive perturbation theory discussed in the present paper. In Table 2 we present the results of our calculations within both methods, and one can see that the results are in good agreement. The number of the PT orders presented is not limited by the method we use, it is chosen according to the total accuracy that we aim at.

In calculations with the CI-DFS method the contribution of the 3rd and higher orders is obtained by subtraction of the lowest orders from the total energy. For this purpose, the factor k is introduced in the interaction Hamiltonian, and the leading orders are found as the derivatives with respect to k. It can lead to the loss of accuracy, even if the total energy is calculated precisely. In contrast, the presented perturbative method provides the contributions of the individual orders to be used immediately. This advantage is particularly useful in the cases when the higherorder remainder is small in comparison to the leading-order part, e.g. for the g factor of highly charged ions. In view of this, we plan to generalize this approach to calculations of the matrix elements of various operators. 3. Conclusions The recursive formulation of the perturbation theory combined with the finite basis set of the many-electron wave functions constructed as Slater determinants comprises an efficient tool for the electronic-structure calculations for few-electron atoms and ions. Applications to the ionization energies of neutral lithium and boronlike argon are presented. The results obtained are found in full agreement with the other methods. One of the advantages of this approach is the direct access to the higher-order part, which is useful when the lowest orders are evaluated within the framework of bound-state QED. This method can be applied also to the calculations of the matrix elements of various operators, e.g. for the hyperfine and Zeeman splittings. Acknowledgments The work reported in this paper was supported by RFBR (Grant No. 16-02-00334), by SPbSU (Grant Nos. 11.42.1059.2016, 11.38.237.2015, and 11.38.269.2014), and by the FAIR – Russia Research Center. The numerical computations were performed at the St. Petersburg State University Computing Center. References [1] V.M. Shabaev, Phys. Usp. 51 (2008) 1175. [2] A.V. Volotka, D.A. Glazov, G. Plunien, V.M. Shabaev, Ann. Phys. (Berlin) 525 (2013) 636. [3] P. Beiersdorfer, H. Chen, D.B. Thorn, E. Träbert, Phys. Rev. Lett. 95 (2005) 233003.

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