Tensorial form and matrix elements of the relativistic nuclear recoil operator
E. Gaidamauskas, C. Nazé, P. Rynkun, G. Gaigalas, P. Jönsson, M. Godefroid
aa r X i v : . [ phy s i c s . a t o m - ph ] J un Tensorial form and matrix elements of therelativistic nuclear recoil operator
E. Gaidamauskas , C. Naz´e , P. Rynkun , G. Gaigalas , ,P. J¨onsson and M. Godefroid Vilnius University Research Institute of Theoretical Physics and Astronomy,A. Goˇstauto 12, LT-01108 Vilnius, Lithuania Chimie Quantique et Photophysique, CP160/09, Universit´e Libre de Bruxelles,Av. F.D. Roosevelt 50, B-1050 Brussels, Belgium Vilnius Pedagogical University,Student¸u 39, LT-08106 Vilnius, Lithuania School of Technology, Malm¨o University, 205-06 Malm¨o, SwedenE-mail: [email protected], [email protected]
Abstract.
Within the lowest-order relativistic approximation ( ∼ v /c ) and to firstorder in m e /M , the tensorial form of the relativistic corrections of the nuclear recoilHamiltonian is derived, opening interesting perspectives for calculating isotope shifts inthe multiconfiguration Dirac-Hartree-Fock framework. Their calculation is illustratedfor selected Li-, B- and C-like ions. The present work underlines the fact that therelativistic corrections to the nuclear recoil are definitively necessary for getting reliableisotope shift values.PACS numbers: 31.15.ac , 31.15.aj, 31.15.am, 31.15.V-, 31.30.Gs, 31.30.J-, 31.30.jc,32.10.Fn Submitted to:
J. Phys. B: At. Mol. Opt. Phys.
Keywords : Isotope shifts, mass shift, relativistic nuclear recoilSaturday 13 th October, 2018@ 16:45 ensorial form and matrix elements of the relativistic nuclear recoil operator
1. Introduction
Nuclear and relativistic effects in atomic spectra are treated in the pioneer works ofStone [1, 2] and Veseth [3]. The theory of the mass shift has then been reformulatedby Palmer [4]. Calculations of nuclear motional effects in many-electron atoms havebeen performed by Parpia and co-workers [5, 6] in the relativistic scheme, using fullyrelativistic wave functions, but adopting the non-relativistic form of the recoil operator.Relativistic nuclear recoil corrections to the energy levels of multicharged ions have beenestimated by Shabaev and Artemyev [7] who derived the relativistic corrections of therecoil Hamiltonian. In a study of isotope shifts of forbidden transitions in Be- and B-likeargon ions, Tupitsyn et al [8] showed that a proper evaluation of the mass isotope shiftrequires the use of this relativistic recoil operator. The latter has also been shown to becrucial by Porsev et al [9] for calculating isotope shifts of transitions between the finestructure energy levels of the ground multiplets of Fe I and Fe II.As far as computational atomic structure is concerned, the extension of the availablerelativistic codes such as grasp2k [10] or mcdf -gme [11, 12] is needed for estimatingthese mass corrections properly for any many-electron system. Programs to calculatepure angular momentum coefficients for any scalar one- and two- particle operator areavailable [13] but do require the knowledge of the tensorial structure of the operatorsto be integrated between the many-electron atomic wave functions [14]. The tensorialform of the nuclear recoil Hamiltonian is derived in the present work, opening interestingperspectives for calculating isotope shifts in the multiconfiguration Dirac-Hartree-Fock(MCDHF) framework.
2. The relativistic mass shift operator
In the MCDHF method, the atomic state function (ASF) Ψ( γP J M J ), of a stationarystate of an atom, is expressed as a linear combination of symmetry-adapted configurationstate functions (CSFs) Φ( γ p P J M J ), i.e.Ψ( γP J M J ) = X p c p Φ( γ p P J M J ) , (1)where J is the total electronic angular momentum of the state, γ represents the electronicconfiguration and intermediate quantum numbers, and P stands for the parity. Themixing coefficients c p and the one-electron radial wave functions spanning the CSFs areoptimized by solving the MCDHF equations iteratively until self-consistency. The latterare derived by applying the variational principle to the energy functional based on theDirac-Coulomb Hamiltonian [14] H DC = N X i =1 (cid:16) c α i · p i + ( β i − c + V ( r i ) (cid:17) + N X i 4) Dirac matrices and c is the speed of light ( c = 1 /α in atomic units, where α isthe fine-structure constant). ensorial form and matrix elements of the relativistic nuclear recoil operator M is caused bythe recoil motion of the atomic nucleus. The corresponding recoil Hamiltonian H MS = 12 M N X i,j (cid:18) p i · p j − αZr i (cid:18) α i + ( α i · r i ) r i r i (cid:19) · p j (cid:19) , (3)has been derived within the lowest-order relativistic approximation and to first order in m/M by Shabaev and collaborators [7, 8]. Rewriting it as the sum of the normal massshift (NMS) and specific mass shift (SMS) contributions and using the tensorial form r = r C , (3) becomes H MS = H NMS + H SMS , (4)with H NMS = 12 M N X i =1 (cid:18) p i − αZr i α i · p i − αZr i (cid:0) α i · C i (cid:1) C i · p i (cid:19) , (5) H SMS = 1 M N X i 1) is defined by : π ( l a , l c , 1) = ( l a + 1 + l c even,0 otherwise. (48) 3. Applications We wrote a new program, hereafter referred as rms2 , for estimating the expectationvalues of the relativistic nuclear recoil operators using MCDHF wave functionscalculated with the grasp2K package [10]. This code is based on the previous program sms92 [6] in which • for the NMS, the one-electron radial integrals (expression (39) of the originalpaper [6]) are replaced by the corresponding relativistic expression (26), • for the SMS, the first contribution X ( abcd ) (expression (40) of the originalpaper [6]) is corrected by adding the relativistic contributions (41) and (44).It is important to notice that the program sms 92 calculates the uncorrectedNMS as the expectation value h P i T i i , where T i is the Dirac kinetic energy operator T i = c α i · p i + ( β i − c associated to electron i , while the program rms2 uses more accurately h H i = h P i p i / M i , which is consistent with section 2.1. Anequivalent version has been written for the code mcdf -gme.In the present work, we evaluate the NMS and SMS parameters (9) and (27) forsome low-lying levels of neutral lithium, boron-like argon and two medium- Z carbon-likeions (Ca XV and Sc XVI) to investigate the importance of the relativistic corrections.The nuclear charge distribution is described by a Fermi model. Nuclear masses ( M N )are calculated by taking away the mass of the electrons and the binding energy fromthe atomic mass ( M A ), using the formula: M N ( A, Z ) = M A ( A, Z ) − Zm e + B el ( Z ) (49)where the total binding energy of the electrons (expressed in eV) is estimatedusing [18, 19] B el ( Z ) = 14 . Z . + 1 . · − Z . . (50)The atomic and nuclear masses relevant to the present work are reported in Tables 1. ensorial form and matrix elements of the relativistic nuclear recoil operator Table 1. Atomic masses ( M A ) [20] and nuclear masses ( M N ) (in u) calculated from(49) and (50) for lithium and argon isotopes.Isotope M A M N Li 6.015122795(16) 6.01386737 Li 7.01600455(8) 7.01474907 Ar 35.967545106(29) 35.9576862 Ar 39.9623831225(29) 39.9525242 When discussing a transition mass isotope shift, one needs to consider the variationof the mass parameter from one level to another. The line k frequency isotope shift, δν A ,A k = ( δE A ,A j − δE A ,A i ) /h , between the isotopes A and A , of nuclear masses M and M respectively, is usually written as the sum of the normal mass shift (NMS),specific mass shift (SMS) and field shift (FS) contributions : δν A ,A k = δν A ,A k, NMS + δν A ,A k, SMS | {z } δν A ,A k, MS + δν A ,A k, FS , (51)with δν A ,A k, MS = (cid:18) M − M M M (cid:19) ∆ K MS h = (cid:18) M − M M M (cid:19) ∆ e K MS , (52)where ∆ K MS is the difference of the K MS parameters of the levels involved intransition k . As far as conversion factors are concerned, we use § ∆ e K MS [GHz u] =3609 . K MS [ m e E h ]. Note that thanks to the separability enhanced in (4), (52) canbe applied to both the mass contributions NMS and SMS, separately. Below we present some relevant calculations of the expression (26) for a heavy one-electron ion (Se XXXIV, Z = 34). This choice is motivated by the interestingcomparison with the unpublished work of Kozlov [22]. The normal mass shift valuescalculated with the operators H and ( H + H ), using the rms2 program, arereported in Table 2. In the second and third column respectively, comparison is madewith the numerical results of Kozlov together with our analytical values. The latter arebased on analytical hydrogenic wave functions k [23]. The agreement is very satisfactory. § This conversion factor is calculated as ( m e /u )2 R ∞ c × . − = 3609 . k The values reported in Tables 2 and 3 are based on α − = 137 . grasp2K .For 1 s / , the analytical result for K NMS becomes 656.358886872 if adopting the α − = 137 . ensorial form and matrix elements of the relativistic nuclear recoil operator Table 2. Contributions to the normal mass shift K NMS parameters (in m e E h ) forhydrogen-like selenium ( Z = 34). rms K s / . . . p / . . . p / . . . K + K s / − . − . − . p / − . − . − . p / − . − . − . The SMS parameters for Li-like iron ( Z = 26) and selenium ( Z = 34) are calculatedin the single configuration approximation using three-electron wave functions built onunscreened Dirac solutions. The results are reported in Table 3 and compared withindependent estimations using an adapted version of mcdf -gme [11, 24] and with theanalytical results. The three sets are consistent with each other but sensitively differentfrom Kozlov’s values [22] reported in the last column of the table. Note that thecomparison is somewhat unfair to Kozlov since the grid parameters used for the discreterepresentation of orbital wave functions have been adapted in both programs ( rms mcdf -gme) to achieve a better accuracy. ensorial form and matrix elements of the relativistic nuclear recoil operator Table 3. Contributions to the specific mass shift K SMS (in m e E h ) parameters forLi-like iron ( Z = 26) and selenium ( Z = 34) using unscreened Dirac one-electron wavefunctions. rms mcdf -gme Analytic Kozlov [22]Li-like iron K s p / P o / − . − . − . − . s p / P o / − . − . − . − . K + K s p / P o / . . . . s p / P o / . . . . K s p / P o / − . − . − . − . s p / P o / − . − . − . − . K + K s p / P o / . . . s p / P o / . . . The MCDHF active space method consists in writing the total wavefunction as aconfiguration state function expansion built on a set of active one-electron orbitals. Toinvestigate the convergence of the property, the orbital set is systematically expandedup to n = 10, but imposing the angular restriction l max = 6 ( i orbitals). The sequenceof CSFs Active Spaces (AS) is resumed as followsAS = 1 s s, AS = AS + { p } , AS = AS + { s, p, d } , AS = AS + { s, p, d, f } , AS = AS + { s, p, d, f, g } , AS = AS + { s, p, d, f, g, h } , AS = AS + { s, p, d, f, g, h, i } , AS = AS + { s, p, d, f, g, h, i } , AS = AS + { s, p, d, f, g, h, i } , AS = AS + { s, p, d, f, g, h, i } , ensorial form and matrix elements of the relativistic nuclear recoil operator nl )-notation implies the relativistic shell structure j = l ± / 2. Theconfiguration space is increased progressively, by adding at each step a new layer ofvariational orbitals, keeping the previous ones frozen from the ( n − 1) calculation.The MCDHF expansions are based on single and double (SD) excitations from theconfiguration reference. Triple excitations are investigated through SDT-configurationinteraction (CI) calculations. Table 4. Uncorrected ( K ) and corrected ( K NMS ) normal mass shift parameters(in m e E h ) for Li I. AS n SD SDTK K NMS K K NMS s s S / n =5 7.479955285 7.473188966 7.480387512 7.473620714 n =6 7.480757179 7.473989593 7.481294538 7.474526401 n =7 7.480843823 7.474076167 7.481413156 7.474644913 n =8 7.482617678 7.475849092 7.483709525 7.476940080 n =9 7.482764972 7.475996085 7.483865298 7.477095534 n =10 7.482767804 7.475998981 7.483872626 7.4771029331 s p P o / n =5 7.411878601 7.405201316 7.412125687 7.405448151 n =6 7.412307495 7.405629843 7.412599631 7.405921698 n =7 7.412593434 7.405916172 7.413034981 7.406357367 n =8 7.414193990 7.407516244 7.415203718 7.408525555 n =9 7.414351543 7.407673608 7.415377292 7.408698936 n =10 7.414365017 7.407687009 7.415402512 7.4087240811 s p P o / n =5 7.411871260 7.405208436 7.412118064 7.405455006 n =6 7.412300503 7.405637317 7.412592555 7.405929108 n =7 7.412584979 7.405922413 7.413026010 7.406363146 n =8 7.414185599 7.407522728 7.415193271 7.408530126 n =9 7.414343399 7.407680348 7.415366987 7.408703663 n =10 7.414356793 7.407693666 7.415392260 7.408728857 Tables 4 and 5 present the evolution of the NMS and the SMS parameter,respectively. In each table both the uncorrected ( K ) and corrected ( K MS ) valuesare reported. Comparing the SD and SDT calculations, we observe that the influenceof the triple excitations reaches more than 1 % for the SMS while it is one order ofmagnitude smaller (0.1%) for the NMS.In Table 6 the individual contributions to the mass shift ∆ e K MS (= ∆ K MS /h )parameters are reported for the 2 p / P o / − s S / ( D line) and 2 p / P o / − s S / ( D line) transitions in lithium. Values are calculated with the SD and SDT n = 10 active space final results of Tables 4 and 5. Although many robust theoreticalstudies on the resonance line transition isotope shifts are available (see Table 7 anddiscussion below), the comparison with other theoretical works presented in Table 6 is ensorial form and matrix elements of the relativistic nuclear recoil operator Table 5. Uncorrected ( K ) and corrected ( K SMS ) specific mass shift parameters(in m e E h ) for Li I. AS n SD SDTK K SMS K K SMS s s S / n =5 0.3010343291 0.3008225633 0.3013767853 0.3011648528 n =6 0.3010361585 0.3008243666 0.3014579841 0.3012459847 n =7 0.3019544951 0.3017423943 0.3024396569 0.3022273237 n =8 0.3018617523 0.3016497153 0.3024115843 0.3021992791 n =9 0.3017987398 0.3015867742 0.3023512554 0.3021390203 n =10 0.3018561821 0.3016442119 0.3024141615 0.30220192001 s p P o / n =5 0.2489342617 0.2487564343 0.2490604867 0.2488826064 n =6 0.2482881614 0.2481107089 0.2484282378 0.2482507247 n =7 0.2482993836 0.2481222326 0.2484107486 0.2482335819 n =8 0.2476015693 0.2474248242 0.2474557225 0.2472791143 n =9 0.2475207029 0.2473440589 0.2473719731 0.2471954806 n =10 0.2476566450 0.2474799659 0.2475131224 0.24733658651 s p P o / n =5 0.2489331884 0.2487377216 0.2490592224 0.2488636586 n =6 0.2482892038 0.2480941432 0.2484289962 0.2482338261 n =7 0.2483022860 0.2481070388 0.2484139538 0.2482185782 n =8 0.2476039924 0.2474089620 0.2474585594 0.2472634414 n =9 0.2475233735 0.2473284034 0.2473750253 0.2471799634 n =10 0.2476606363 0.2474656360 0.2475176569 0.2473225644 limited to the recent large-scale configuration-interaction Dirac-Fock-Sturm calculationsof Kozhedub et al [25] since these authors precisely focus on the estimation of therelativistic nuclear recoil corrections. Kozhedub et al ’s values are very consistent withour results: They report ∆( e K + e K ) = 0 . 33 and 0.38 GHz u for the D and D transitions respectively. However, the uncorrected NMS contribution and therefore, thetotal NMS values, sensitively differ from each other by around 1.6 GHz u. This latterdiscrepancy is not understood yet and clearly deserves further investigations.The uncorrected contribution of the SMS is also compared with the non-relativisticresult of Godefroid et al [26] using the multiconfiguration Hartree-Fock method. Moreinteresting is the comparison with the recent SMS values of Kozhedub et al [25]investigating the relativistic recoil corrections and using the same NMS and SMSpartition according to (5) and (6). As for the NMS, the relativistic corrections arein very nice agreement (they report ∆( e K + e K ) = 0 . 12 and 0.06 GHz u for the D and D lines ) but the uncorrected forms do differ substantially with our estimation(they report for instance for the D line, ∆ e K = − . 78, against our value of − . 164 GHz u). ensorial form and matrix elements of the relativistic nuclear recoil operator ab initio calculations and the observed transition IS allows toextract the change in the mean square charge radius of the nuclear charge distributionsfor all isotopes, as illustrated by the very recent and complete work of N¨ortersh¨auser etal [27]. Another good reason is that once the FS “eliminated”, a clean separation of theNMS and SMS contributions could be criticized, as pointed out by Palmer [4]. However,remembering that for lithium, the FS is roughly 10 times smaller than the MS, it isworthwhile to neglect it for trying the mass separation exercise. There is indeed oneexperimental work by Radziemski et al [28] discussing the NMS and SMS separationin this line but as we will observe later (see Table 7), the corresponding experimentaltransition IS values are not aligned with most of the other observed values. In theirwork, these authors separate the two mass shift contributions from the experimentaltransition IS in , Li, neglecting the field shift contribution and approximating the Bohrmass shift by the experimental observed level energy, as suggested by M˚artensson andSalomonson [29],∆ BMS = − m e M E BM ≃ − m e M E exp . (53)From the same expression, we build the transition Bohr mass shift for the , Li isotopepair δE BMS ≃ (cid:18) m e M ( Li ) − m e M ( Li ) (cid:19) ∆ E exp (54)from the obserbed transition energy. Combining (52) and (54), one finds∆ e K NMS ≃ m e ∆ E exp h = m e ν exp (55)from which we estimate the “observed” NMS values reported in Table 6, usingthe most recent absolute frequency measurements of Das and Natarajan [30]. Thecorresponding “observed” ∆ e K SMS values are calculated by substracting the so-estimatedNMS contribution from the experimental IS line shifts ( − . − . D and D , respectively). Note that we did not take theliberty of reporting the frequency uncertainties estimated by Das and Natarajan on theseparate contributions, the separability of NMS and SMS being by itself questionable.Cleaner and in principle less problematic should be the comparison of the totalmass shifts, as reported in Table 7. On the theoretical side, we refer to the studyof Korol and Kozlov [31] treating electron correlation with configuration interaction(CI) and many-body perturbation theory (MBPT) methods with Dirac-Fock orbitals,to the calculations of Kozhedub et al [25] using large-scale configuration-interactionDirac-Fock-Sturm method and to the Yan et al [32] calculations estimating the mass ensorial form and matrix elements of the relativistic nuclear recoil operator Table 6. Individual contributions to the mass shift ∆ e K MS (GHz u) parameters forthe 2 p P o / − s S / and 2 p P o / − s S / transitions in lithium P o / − S / P o / − S / NMS ∆ e K SD − . − . − . − . e K + e K ) SD 0 . 328 0 . . 333 0 . e K NMS SDT − . − . a − . − . b − . − . e K SD − . − . − . − . e K + e K ) SD 0 . 127 0 . . 129 0 . e K SMS SDT − . − . a − . − . c − . − . d − . − . a CI Dirac-Fock-Sturm calculation of Kozhedub et al [25]. b NMS values deduced from the transition frequencies [30] using (55) (see text). c Non-relativistic MCHF calculations [26]. d SMS values obtained by subtracting the “observed” NMS (see footnote b above) fromthe IS measured by Das and Natarajan [30]. corrections from highly correlated non-relativistic wave functions expressed in Hylleraascoordinates ¶ . From all these elaborate results, we only kept the mass contributions,systematically excluding the contributions from the nuclear size corrections. We alreadynoticed the differences between Kozhedub et al ’s results and ours appearing in theseparate NMS and SMS contributions. As commented above, these differences do notarise from the relativistic corrections ( K + K ), but rather from the “uncorrected” K values, and should be further investigated. Our results seem to be of higher quality than ¶ The values of Yan et al reported in the Kozhedub’s paper [25] suggest that the atomic mass hasbeen used in order to evaluate the mass shift parameter. In table 7, the Yan et al ’s values have beenreevaluated using the nuclear mass. ensorial form and matrix elements of the relativistic nuclear recoil operator et al ’sresults are concerned, we should keep in mind i) that our orbital active set is truncatedto l max = 6, ii) that the layer approach adopted in the SD-MCDHF optimization couldbe a limiting factor and iii) that the convergence of the ∆ e K MS parameter as a functionof the size of the active set is slow and not yet achieved at n = 10, as illustrated by thecomparison of the two n = 9 and n = 10 sets of results reported in the Table 7.On the experimental side, we display in the same Table 7, the experimental isotopeshift values somewhat abusively converted in ∆ e K MS parameters, ie. neglecting the FScontribution and inverting (52), ∆ e K MS = δν k ( M M ) / ( M − M ). As already mentioned,this conversion is unfair to physicists who do some huge efforts to extract the nuclearcharge radii from the FS [27], but has the merit of illustrating where the present modestcontribution lies in the distribution of experimental values. From this not exhaustivechronological list ([33, 28, 34, 35, 36, 37, 30]), it is clear that Radziemski et al ’s resultslie a bit outside the experimental distribution. Table 7. Mass shift ∆ e K MS (GHz u) for the 2 p / P o / − s S / and 2 p / P o / − s S / transitions in lithium, compared with experimental IS. P o / − S / P o / − S / Ref. n = 9 − . − . n = 10 − . − . − − − . − . et al [25] − . − . et al [32]Experiment a − . − . et al [33] − . − . et al [28] − . − . et al [34] − . et al [35] − . et al [36] − . − . et al [37] − . − . a inverting (52), ie. using ∆ e K MS = δν ( M M ) / ( M − M ) (see text) Large-scale calculations are performed for 1 s s p P o / , / of B-like argon ( Z = 18).The radial orbital basis is obtained from SD-MCDHF calculations, including singleand double excitations from all shells of the { s s p, s p } complex to increasingorbital active sets, up to the { s p d f g h i } . Subsequently to this layer-by-layerSD-MCDHF orbital optimization, RCI calculations are performed including the Breit ensorial form and matrix elements of the relativistic nuclear recoil operator { s s p ,1 s p , 1 s s p d , 1 s p d } multireference set to the full orbital set. The expansionfor the two J values includes more than 200 000 relativistic CSFs. This computationalstrategy has been developed by Rynkun et al [38] for the evaluation of transition ratesin boron-like ions, from N III to Zn XXVI.Table 8 illustrates the convergence of the NMS and SMS contributions with theincreasing of the active set. In Table 9, the isotope shifts of the forbidden transitions Table 8. NMS and SMS parameters (in m e E h ) values for the states 1 s s p P o / and 1 s s p P o / states of B-like Ar. AS n K (cid:0) K + K (cid:1) K (cid:0) K + K (cid:1) s s p P o / n = 3 417 . − . − . . n = 4 418 . − . − . . n = 5 418 . − . − . . n = 6 418 . − . − . . n = 7 418 . − . − . . n = 8 418 . − . − . . n = 9 418 . − . − . . n = 10 418 . − . − . . n = 10 expand . − . − . . s s p P o / n = 3 417 . − . − . . n = 4 417 . − . − . . n = 5 417 . − . − . . n = 6 417 . − . − . . n = 7 417 . − . − . . n = 8 417 . − . − . . n = 9 417 . − . − . . n = 10 417 . − . − . . n = 10 expand . − . − . . s s p P o / − P o / in , Ar are presented and compared with the results ofTupitsyn et al [8]. In their work, the CI Dirac-Fock method was used to solve theDirac-Coulomb-Breit equation and to calculate the energies and the isotope shifts. TheCSFs expansions were generated including “all single and double excitations and somepart of triple excitations”. The nuclear charge distribution is described by a Fermi model ensorial form and matrix elements of the relativistic nuclear recoil operator h H + H i . The nice agreement withTupitsyn et al ’s results is also a good sign of reliability for the tensorial form derivationof the nuclear recoil Hamiltonian of section 2. Table 9. Individual contributions to the wavenumber mass shift δσ (cm − ) for theforbidden transition 1 s s p P o / − P o / in boron-like , Ar. δσ h H i h H + H i h H i h H + H i TotalThis work 0 . − . − . . et al [8] 0 . − . − . . As another illustration of the importance of the relativistic corrections to the recoiloperator, the values of the SMS, NMS and total level mass shift parameters arereported in Table 10 for the levels arising from the ground configuration 1 s s p in Ca XV and Sc XVI. As far as the calculations are concerned, the orbitals areobtained by SD-MCDHF calculations, considering single and double excitations fromall shells of the { s s p , s p } Layzer complex to the { s p d f g h } active set.These MCDHF calculations are followed by relativistic configuration interaction (RCI)calculations, including the Breit interaction and the QED corrections, using the enlargedmultireference { s s p , s p , s s p d, s s d } set. The size of the expansionsis around 350 000 relativistic CSFs. This computational method has been used byJ¨onsson et al [39] to calculate transition rates, hyperfine structures and Land´e g factorsfor all carbon-like ions between F IV and Ni XXIII.On the absolute scale of level shift parameters, one observes that the relativisticcorrections ( K + K ) to the NMS have the same order of magnitude than theuncorrected SMS contribution. Transition isotope shifts are more interesting propertiessince they are the real observables if the resolution is good enough. These are monitoredby the differential effects on the level IS. It is interesting to infer from Table 10 thepossible mass isotope shifts on the intraconfiguration (M1/E2) transition frequencies.Considering for example the Ca XV P → P transition, the uncorrected total mass ensorial form and matrix elements of the relativistic nuclear recoil operator K + K ) relativisticcorrections. For the P → D transition, a similar increase of the mass shift ispredicted but of “only” 20%. Some reduction could occur: this is the case of P → S (13%). For the P → D transition, the relativistic recoil corrections reach 48%. Table 10. Specific mass shift K SMS , normal mass shift K NMS , total mass shifts K MS parameters (all in m e E h ) for 2 s p levels of Ca XV and Sc XVI from multireferenceRCI calculations. K = K + K + K .SMS NMS TotalLevel J K K SMS K K NMS K K MS K + K Ca XV2 s p P − − − − − − − − − s p D − − − s p S − − − s p P − − − − − − − − − s p D − − − s p S − − − 4. Conclusion and outlook The irreducible tensorial form of the nuclear recoil Hamiltonian is derived in thepresent work, opening interesting perspectives for calculating isotope shifts in themulticonfiguration Dirac-Hartree-Fock framework. We implemented the formalism inthe relativistic package grasp2K by writing a dedicated code ( rms2 ) for estimatingthe expectation values of the relativistic nuclear recoil operators. The comparisonwith other works is satisfactory and the results are promising, although not achievingthe accuracy of the state-of-the-art methodology available for a few-electron systems.Electron correlation remains the major problem that might be solved in our schemawith the use of “localized pair-correlation functions interaction method”, as proposedby Verdebout et al [40]. The present work enhances the fact that the relativisticcorrections to the nuclear recoil are definitively necessary for getting reliable isotopeshift calculations. The new computational tool, that we developed on the basis of theirreducible tensorial operator techniques, will hopefully provide valuable mass isotopeshift data for large systems for which there are no reliable theoretical or experimental ensorial form and matrix elements of the relativistic nuclear recoil operator Acknowledgements C´edric Naz´e is grateful to the “Fonds pour la formation `a la Recherche dans l’Industrieet dans l’Agriculture” of Belgium for a Ph.D. grant (Boursier F.R.S.-FNRS). MichelGodefroid and C´edric Naz´e thank the Communaut´e fran¸caise of Belgium (Actionde Recherche Concert´ee) for financial support. MRG also acknowledges the BelgianNational Fund for Scientific Research (FRFC/IISN Convention). 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