Ab initio electronic factors of the A and B hyperfine structure constants for the 5 s 2 5p6s 1,3 P o 1 states in Sn I
Asimina Papoulia, Sacha Schiffmann, Jacek Bieroń, Gediminas Gaigalas, Michel Godefroid, Zoltán Harman, Per Jönsson, Natalia S. Oreshkina, Pekka Pyykkö, Ilya I. Tupitsyn
AAb initio electronic factors of the A and B hyperfine structureconstants for the s p s , P o1 states in Sn I Asimina Papoulia § ,
1, 2, ∗ Sacha Schiffmann § ,
2, 3, † Jacek Bieroń, Gediminas Gaigalas, Michel Godefroid, Zoltán Harman, Per Jönsson, Natalia S. Oreshkina, ‡ Pekka Pyykkö, and Ilya I. Tupitsyn Department of Materials Science and Applied Mathematics,Malmö University, SE-20506 Malmö, Sweden Division of Mathematical Physics, Department of Physics,Lund University, SE-22100 Lund, Sweden Spectroscopy, Quantum Chemistry and Atmospheric Remote Sensing (SQUARES),CP160/09, Université libre de Bruxelles (ULB), 1050 Brussels, Belgium Instytut Fizyki Teoretycznej, Uniwersytet Jagielloński,ul. prof. Stanisława Łojasiewicza 11, Kraków, Poland Institute of Theoretical Physics and Astronomy,Vilnius University, Saul˙etekio av. 3, LT-10222, Vilnius, Lithuania Max Planck Institute for Nuclear Physics,Saupfercheckweg 1, 69117 Heidelberg, Germany Department of Chemistry, University of Helsinki,PO Box 55 (A. I. Virtasen aukio 1), FIN-00014 Helsinki, Finland Department of Physics, St. Petersburg State University, 198504 St. Petersburg, Russia (Dated: July 23, 2020) § Contributed equally to this work. a r X i v : . [ phy s i c s . a t o m - ph ] J u l bstract Large-scale ab initio calculations of the electric field gradient, which constitutes the electroniccontribution to the electric quadrupole hyperfine constant B , were performed for the s p s , P o1 excited states of tin, using three independent computational strategies of the variational multicon-figuration Dirac-Hartree-Fock method and a fourth approach based on the configuration interactionDirac-Fock-Sturm theory. For the s p s P o1 state, the final value of B/Q = 703(50)
MHz/b differsby . from the one recently used by Yordanov et al. [Communications Physics , 107 (2020)] toextract the nuclear quadrupole moments, Q , for tin isotopes in the range (117 − Sn from collinearlaser spectroscopy measurements. Efforts were made to provide a realistic theoretical uncertaintyfor the final
B/Q value of the s p s P o1 state based on statistical principles and on correlationwith the magnetic dipole hyperfine constant A . PACS numbers: 31.15.A-, 31.30.Gs, 32.10.Fn, 21.10.Ky ∗ [email protected] † saschiff@ulb.ac.be ‡ [email protected] . INTRODUCTION Nuclear quadrupole moments, Q , are important characteristics of nuclei that provide ameasure of the deviation of the nuclear charge distribution from a spherical shape. They canbe determined from nuclear, atomic, molecular or solid-state spectroscopies, such as high-resolution laser spectroscopy [1], muonic or pionic x-ray spectroscopy [2], Nuclear MagneticResonance (NMR) [3, 4], Nuclear Quadrupole Resonance (NQR) [5, 6], Mössbauer measure-ments [7, 8] or Perturbed Angular Correlation (PAC) of nuclei passing thin foils [9, 10].Most of these techniques involve the knowledge of the electric field gradient (EFG) due tothe charges of the constituting environment of the considered nuclei. Three compilations ofavailable Q -values are provided by Raghavan [11], Stone [12], and Pyykkö [13].In this work, we focus on tin, with an atomic number Z = 50 . The proton shells atthis magic number are closed, but the incomplete neutron shells can still induce a Q withquadratic dependence on the neutron number N , which will not become magic until theknown Sn isotope, with N = 82 and nuclear spin I = 0 . The nuclear trends of Q andthe charge radii among ten isotopes in the range − Sn, which is below the doubly magicisotope, have just been published by Yordanov et al. [1]. Continuous, linear and quadratictrends were found as functions of the mass number for cadmium and tin isotopes, respec-tively. The Q (Sn)-values published in Ref. [1] were based on measured atomic hyperfinestructures for odd- N isotopes and were obtained by combining the calculated EFG in the[Pd] s p s P o1 state of the neutral atom with the measured electric quadrupole (E2) hyper-fine coupling constant, B , for each isotope. (Note that [Pd] is used, for brevity, to indicatethe palladium-like core and will be omitted in the following.) The relative accuracy of thecalculated EFG value in the s p s P o1 state is of the order of 7%. The accuracy of themeasured B -values varies, depending on the isotope, between 1.5% for Sn and 33% for
Sn, as reflected in Table 1 of Ref. [1]. As a result, the accuracy of the evaluated “atomic” Q (Sn)-values ranges between 7% and 34%.No “pionic” nor “muonic” Q (Sn)-values are available. There exists a quoted value of − I = 3/2, 24-keV Mössbauer state of Sn by Barone et al. [14], usingdata from 34 solids. A closely similar | Q | value of 128(7) mb was reported by Svane et al. [15].Moreover, the same 24-keV, I = 3 / state of Sn was observed by Dimmling et al. [16]using PAC, together with other tin isotopes embedded in elemental Cd, Sn, or Sb. Taking3heir B -measurements at face value and renormalizing to the − et al. [17], we obtain Q ( Sn , I = 11 / ,
90 keV) = − Q (11 / Q (3 /
2) = − · . − mb . (1)In fact, such a value was already used in the review by Stone [12]. It can be compared withthe Q ( Sn) = − et al. [1], where a direct atomic measurementwas combined with a calculation of the EFG. Thus, some coupling between the PAC andMössbauer Q -values and the present “atomic” ones seems possible. We, finally, note thatRef. [12] gives some Q -values for the even- N isotopes within the range − Sn , and also Sn , in various nuclear states. Yet, none of the observed hyperfine interaction constants B that were used for extracting these Q (Sn)-values have very high accuracy.The aim of the present paper is to report the details of the atomic calculations of the EFGvalues for the s p s , P o1 excited states in neutral tin. The EFG constitutes the electronicpart of the E2 hyperfine constant B . This electronic contribution is, in Sec. II C, definedas B/Q ≡ B el . A well-grounded estimate of the theoretical uncertainties is provided byevaluating the computed values of both E2 and magnetic dipole (M1) hyperfine structures.Following the multiconfiguration Dirac-Hartree-Fock (MCDHF) method [18, 19] describedin Sec. II A, three different computational schemes for optimizing the Dirac one-electron ra-dial functions were employed. The first optimization strategy, denoted S-MR-MCDHF, isbased on configuration state function (CSF) expansions that were built from single (S)electron substitutions from a set of multi-reference (MR) configurations, while the secondapproach, denoted SrD-SR-MCDHF, is based on CSF expansions produced by S and re-stricted double (rD) substitutions from a single-reference (SR) configuration. Finally, thethird optimization scheme, named SrD-MR-MCDHF, is similar to the second approach, re-garding the electron substitutions that build the CSF space, but it is instead applied to aselected MR. All three approaches include higher-order electron substitutions in the sub-sequent relativistic configuration interaction steps. The sets of calculations based on theabove-mentioned optimization schemes are, respectively, discussed in Secs. III, IV, and V.A fourth independent set of calculations was performed using the configuration interactionDirac-Fock-Sturmian (CI-DFS) method [20–23] described in Sec. II B, and the respectivevalues of the computed electronic hyperfine factors are reported in Sec. VI.The results from all four different approaches are combined to provide theoretical uncer-4ainties in Sec. VII. Note that the latest of the four independent calculations that are pre-sented here was carried out after the work by Yordanov et al. [1] had been sent to the printerand represent new, previously unpublished, results. That being so, in the current paper, thefinal B/Q ≡ B el value for the s p s P o state is slightly shifted from 706(50) MHz/b [1]to 703(50) MHz/b . II. THEORYA. MCDHF-RCI multiconfiguration methods
The principles of the MCDHF method are fully discussed in, e.g., the book by Grant [18]and the review article by Froese Fischer et al. [19]. With this section, we provide the readerwith a short introduction of the main concepts.In the relativistic framework, the MCDHF method describes an atomic state function(ASF), Ψ( γ Π J M ) , as an expansion over a set of jj -coupled relativistic CSFs, Φ µ ( γ µ Π J M ) ,characterized by the parity Π , the total electronic angular momentum J , and the projectionquantum numbers M , i.e., Ψ( γ Π J M ) = N CSF (cid:88) µ =1 c µ Φ µ ( γ µ Π J M ) , where N CSF (cid:88) µ =1 c µ = 1 . (2)In the expression above, γ µ represents the configuration, the angular momentum couplingtree, and other quantum numbers that are necessary to uniquely describe each CSF. TheCSFs are built on one-electron Dirac spinors that have the general form: ψ nκm ( r, θ, ϕ ) = 1 r P nκ ( r ) Ω κm ( θ, ϕ ) i Q nκ ( r ) Ω − κm ( θ, ϕ ) , (3)where Ω ± κm ( θ, ϕ ) are the two-component spin-angular functions and { P nκ ( r ) , Q nκ ( r ) } are,respectively, the radial functions of the so-called large and small components with principalquantum number n and relativistic angular momentum quantum number κ defined as κ = − ( l + 1) for j = l + 1 / κ negative )+ l for j = l − / κ positive ) , (4) Although the electronic contribution
B/Q is proportional to the computed EFG value (see also Sec. II C),in Ref. [1], the quantities EFG and
B/Q are used interchangeably. l is the orbital angular momentum quantum number. The quantum number m de-scribing the Dirac spinors (3) is the projection of the total angular momentum j .To derive the MCDHF equations, the variational principle is applied on the functional: F ( { c } , { P } , { Q } ; γ Π J ) ≡ (cid:104) Ψ | H DC | Ψ (cid:105) + (cid:88) ab δ κ a κ b λ ab C ab , (5)where the indices a and b , respectively, represent the orbitals n a κ a and n b κ b , which are usedto construct the ASF of Eq. (2). The energy functional is estimated from the expectationvalue of the N e -electron Dirac-Coulomb Hamiltonian H DC = N e (cid:88) i =1 h D ( i ) + N e (cid:88) j>i =1 r ij = N e (cid:88) i =1 (cid:2) c α i · p i + c ( β i −
1) + V nuc ( r i ) (cid:3) + N e (cid:88) j>i =1 r ij , (6)where V nuc ( r i ) is the potential from an extended nuclear charge distribution, α and β arethe × Dirac matrices, c is the speed of light in atomic units, and p ≡ − i ∇ is the electronmomentum operator. Lagrange multipliers λ ab are introduced for constraining the variations ( δP nκ , δQ nκ ) to satisfy the orthonormality of the one-electron function, i.e., C ab ≡ (cid:90) ∞ [ P a ( r ) P b ( r ) + Q a ( r ) Q b ( r )] dr − δ n a n b = 0 , (7)and to also ensure the orthonormality of the CSFs. The resulting coupled integro-differentialequations have the form w a V ( a ; r ) − c (cid:0) ddr − κ a r (cid:1) c (cid:0) ddr + κ a r (cid:1) V ( a ; r ) − c P a ( r ) Q a ( r ) = (cid:88) b (cid:15) ab δ κ a κ b P b ( r ) Q b ( r ) , (8)where w a is the generalized occupation number of the orbital a and V ( a ; r ) = V nuc ( r ) + Y ( a ; r )+ ¯ X ( a ; r ) is the average and central field MCDHF potential built from the nuclear, di-rect, and exchange contributions that arise from both diagonal and off-diagonal (cid:104) Φ µ | H DC | Φ ν (cid:105) matrix elements. In each κ -space, Lagrange-related energy parameters (cid:15) ab ≡ (cid:15) n a n b are intro-duced to impose the orthonormality constraints (7) in the variational process.The resulting coupled radial equations are solved iteratively in the self-consistent field(SCF) procedure. Once the radial functions have been determined, a relativistic config-uration interaction (RCI) calculation is performed over the set of configuration states todetermine the expansion coefficients c µ for building the next-iteration potentials. The SCFand RCI coupled processes are repeated until convergence of the total wave function (2) andcorresponding energy, (cid:104) Ψ | H DC | Ψ (cid:105) , is reached.6s mentioned in the introduction, these MCDHF calculations provide the one-electronorbital basis that can be used to investigate the effect of higher-order electron substitutionsthrough RCI calculations that use larger CSF spaces. At this subsequent RCI step, thetransverse photon interaction, which reduces to the Breit interaction at the low-frequencylimit, and the leading quantum electrodynamic (QED) corrections are added to the Dirac-Coulomb Hamiltonian (see Refs. [25, 26] for more details).The MCDHF method is implemented in the Grasp2K [27] and
Grasp2018 [28] com-puter packages. The description of the numerical methods, virtual orbital sets, electronsubstitutions, and other details of the computations can be found in Refs. [19, 27, 29–32].The wave function representation in jj -coupling is transformed to an approximate repre-sentation in LSJ -coupling, using the methods and program developed by Gaigalas andco-workers [33, 34].
B. CI-DFS method
The detailed description of the CI-DFS method can be found in Refs. [20–24]. We resumehereafter the underlying theoretical background.
1. Dirac-Fock-Sturm orbitals.
Dirac-Fock-Sturm orbitals of a general type ϕ j can be obtained as the solutions of thefollowing eigenvalue problem: ( h D − ε ) ϕ j = λ j W ( r ) ϕ j , (9)where h D is the one-electron Hamiltonian from Eq. (6), W ( r ) is the so-called weight functionof the positive sign, ε is a reference energy, and λ j is the eigenvalue. Analytic CoulombSturmian functions, introduced in Ref. [35], can be obtained for the non-relativistic H-likeHamiltonian h D , W ( r ) = 1 /r and ε is equal to the energy of the s state. This method ofSturmian-function generation is equivalent to the charge-quantization formalism, i.e., to thereplacement Z → Z ∗ j = Z + λ j . The Coulomb Sturmian basis has been shown to be avaluable tool in non-relativistic atomic physics (see, e.g., Ref. [36]).In the relativistic case, this method fails, since the Dirac equation has no solution for Z > . Therefore, early attempts to define the relativistic Sturmians were based on re-7riting the Dirac equation for a H-like ion to the second-order equation. Later, more suitableconstructions based on the Dirac equation were proposed by Drake and Goldman [37], byGrant [38] (L-spinors), and by Szmytkowski [39] (see also references therein).A much more flexible Sturmian basis can be constructed by solving the Sturm equation (9)numerically for a general form of the weight function W ( r ) . In Refs. [20, 21], it was proposedto use the Hartree-Fock or Dirac-Fock (DF) operator, in non-relativistic and relativisticcases, respectively, and the following weight function: W ( r ) = (cid:20) − exp( − ( βr ) )( βr ) (cid:21) , (10)where the parameter β is chosen to speed up the convergence of Sturmian series. In contrastto the function /r , the weight function (10) is regular at the origin. One can also seethat, for λ j = 0 , the Sturmian function coincides with the reference DF orbital. Since theweight function W ( r ) → at r → ∞ , all Sturmian functions ϕ j have the same exponentialasymptotic behavior at r → ∞ : ϕ j ( r ) → A j e −√− ε r . (11)Therefore, all Sturmian functions have approximately the same radial size, which is deter-mined by the reference energy ε . It is well known that the Sturmian operator is Hermitianand, in contrast to the Fock operator, it does not contain continuous spectra. Thus, the setof the Sturmian eigenfunctions forms a discrete and complete orthonormalized basis set ofone-electron wave functions with the weight W ( r ) .
2. One-electron basis
The numerical solution of the Dirac-Hartree-Fock equation in the non-relativistic configu-ration average ( LS -average) approximation [40, 41] yields one-electron radial wave functionsfor the occupied (spectroscopic) and low-lying excited orbitals. The remaining virtual or-bitals with positive energies and all orbitals with negative energies have been obtained bysolving the generalized Dirac-Fock-Sturm equation (9) with the weight function (10) and DFoperator as the one-electron Hamiltonian. The basis constructed in such a way automaticallysatisfies the dual kinetic balance condition [42].The next step is the construction of an orthonormalized set of one-electron wave functionsby the solution of DF equations in the DFS orbital basis. One-electron wave functions8btained before by the DF method stay intact, whereas the virtual Sturmian orbitals aremodified to be eigenfunctions of the DF operator and, therefore, they can be used for theconstruction of determinants in the configuration interaction (CI) method.
3. The Restricted Active Space concept
In Refs. [43, 44], the Restricted Active Space (RAS) concept for electronic structurecalculations has been proposed. All non-relativistic atomic shells are divided into threesubspaces, namely, RAS1, RAS2, and RAS3. The subspace RAS1 contains the low-lyingoccupied orbitals (except for the inactive ones), the subspace RAS2 is built from activeoccupied and low-lying valence orbitals, and all other obitals form RAS3.The non-relativistic configurations are constructed by allowing electron substitutions fromone or a few orbitals. From the subspace RAS1, N h substitutions are possible to RAS2 orRAS3, usually at most two. Therefore, the configurations with one or two holes in theRAS1 shells are included. For RAS2, the substitutions are allowed within RAS2 and toRAS3, with the total number of substitutions N ex = 2 , , . . . . The number of substitutionsto subspace RAS3 is limited to N el electrons, which is usually ≤ . Subsequently, fromthe list of non-relativistic configurations a list of relativistic ones is built, including all rel-ativistic configurations that correspond to a given non-relativistic one. For each relativisticconfiguration, the CSF set is constructed as linear combinations of Slater determinants,thus forming a many-electron basis for CI calculations. The substitutions from RAS1 andto RAS3, which are not expected to contribute much compared to substitutions involvingthe subspace RAS2, can also be taken into account by perturbation theory (PT) [24]. C. Hyperfine structure
The hyperfine contribution to the Hamiltonian is represented by a multipole expansion H hfs = (cid:88) k ≥ T ( k ) · M ( k ) , (12)where T ( k ) and M ( k ) are spherical tensor operators of rank k in the electronic and nuclearspaces, respectively. The k = 1 and k = 2 terms represent the M1 and the E2 interactions. Inthe fully relativistic approach, the electronic contributions are obtained from the expectation9alues of the irreducible spherical tensor operators [40, 45] T (1) = − i α N e (cid:88) j =1 (cid:16) α j · l j C (1) ( j ) (cid:17) r j , (13)and T (2) = − N e (cid:88) j =1 C (2) ( j ) 1 r j , (14)where l is the electronic orbital angular momentum and C (1) , C (2) are the renormalizedspherical harmonics of rank and rank , respectively. The EFG, also denoted q [46], isobtained from the reduced matrix element of the operator (14) using the electronic wavefunction of the electronic state in question (see Refs. [30, 45] for details). It corresponds tothe electronic part of the E2 hyperfine interaction constant B . The latter is expressed inunits of MHz and can be calculated using the following equation B/ MHz = 234 . q/a − )( Q/ b ) , (15)where the EFG, or q , and the nuclear quadrupole moment, Q , are expressed in a − and barns,respectively. We introduce the isotope-independent hyperfine constants of an electronic stateof total angular momentum J as follows A el ≡ (cid:112) J ( J + 1)(2 J + 1) (cid:104) Ψ || T (1) || Ψ (cid:105) = A (cid:18) Iµ I (cid:19) [ MHz/ µ N ] , (16) B el ≡ (cid:115) J (2 J − J + 1)(2 J + 1)(2 J + 3) (cid:104) Ψ || T (2) || Ψ (cid:105) = B/Q [ MHz/b ] . (17)In the equations above, we adopted the definition of the reduced matrix element, whichis compatible with the Wigner-Eckart theorem of Edmonds [47], as used in most of theatomic physics textbooks [48]. The electronic quantities (16) and (17) are closely related tothe hyperfine constants A and B , which can then easily be computed for a given isotopecharacterized by the ( µ I , I , Q ) set of nuclear parameters. III. S-MR-MCDHF CALCULATIONS
In this first set of calculations, the initial approximation of the atomic states was acquiredby using
Grasp2018 [28] to perform an MCDHF calculation on expansions that are built10rom a set of reference configurations. For this calculation, we used an equally weighted setof the s p s P o1 and s p s P o1 configuration states, together with the lowest s p dJ Π = 1 − state, which is close to the s p s J Π = 1 − levels. Following [49], mixings with the s p configurations were also taken into account, since these were found to strongly influencethe odd levels of Sn I [49]. These configurations – s p s , s p d , s p , represented bynine CSFs – were treated in the “extended optimal level” (EOL) scheme [50]. As will bediscussed below, the aforementioned set of reference configurations was further extended.The spectroscopic (occupied) orbitals that took part in the initial MCDHF calculationwere kept fixed in all subsequent MCDHF (and RCI) calculations. As a next step, virtualorbitals were generated in MCDHF calculations based on CSF expansions that were pro-duced by allowing single (S) electron substitutions from all spectroscopic orbitals to thesubspace of virtual orbitals. Therefore, we label the method presented in this section asS-MR-MCDHF, where S stands for the single electron substitutions and MR indicates thatmore than one reference configuration were accounted for.Due to the one-body nature of the hyperfine operators (13) and (14), the S substitutionsare known to play an important role in the calculations of hyperfine structures, which is,also, in agreement with the perturbative analysis conducted by, e.g., Verdebout et al. [51].CSFs generated by S electron substitutions interact, i.e., the corresponding matrix elementsof the hyperfine operators are non-zero, with at least one of the CSFs that are built from theMR configurations. Based on similar arguments, the triple (T) substitutions are quite ascrucial. The T substitutions may be decomposed into single and double (SD) substitutionsfollowed by S substitutions. This implies that the CSFs built from configurations that differby T electron substitutions from the configurations in the MR interact through the one-bodyhyperfine operators with the energetically important CSFs that are generated by double (D)substitutions. By using more than one reference configuration, the current computationalstrategy takes into account important D and T electron substitutions from the targeted s p s configuration.The sequence of added layers of virtual orbitals were added is given in column 2 of Table I.In column 3 of the same table, where the label MR3 underlines the number of referenceconfigurations, the resulting numbers of CSFs, N CSF , are displayed for every additionalvirtual orbital layer i . At every step, the previously generated virtual orbitals were keptfixed and, in the subsequent step, only the newly added virtual orbitals were variationally11ptimized. Hence, the p and f orbitals that make up the first layer of virtual orbitals wereobtained by keeping the core orbitals fixed, the s , p , d , and f orbitals that correspondto the second virtual orbital layer were generated in the next step by also keeping the p and f orbitals fixed, and so forth. Overall, eight virtual orbital layers were built, with thelast virtual orbital layer i = 8 corresponding to the s p d f g h i set of orbitals. Table I: The sequence of the layers of virtual orbitals that were optimized in the S-MR3-MCDHF andS-MR4-MCDHF calculations. The former optimization scheme is based on S electron substitutionsfrom the MR3 set of reference configurations, i.e., { s p s, s p d, s p }, whereas the latterscheme also includes the s p s configuration in the so-called MR4 multi-reference. When all fourconfigurations are included in the MR, the s orbital is part of the spectroscopic orbitals and it is,thus, placed in parentheses in row 3, which displays the i = 2 virtual orbital layer. In columns 3and 4, the numbers of generated CSFs, N CSF , are, respectively, given for each of the two differentoptimization strategies. N CSF i Layers of virtual orbitals MR3 MR4none (MR) 9 111 p , f s ,) p , d , f s , p , d , f , g s , p , d , f , g s , p , d , f , g , h
12 563 14 1206 s , p , d , f , g , h
15 367 17 2887 s , p , d , f , g , h , i
18 242 20 5298 s , p , d , f , g , h , i
21 117 23 770
Following the MCDHF calculations, valence-valence (VV) correlations were ultimatelyincluded in the RCI calculations by also allowing D substitutions of electrons from a smallerset of valence subshells, i.e., s , p , d , and s , to the space of virtual orbitals. D sub-stitutions from lower-lying subshells were not included to keep the number of CSFs at amanageable level. The resulting values of the isotope-independent hyperfine constants A el and B el are shown in Fig. 1 with the label S-MR3-MCDHF+RCI.12 R 1 2 3 4 5 6 7 8 A e l [ P ] ( M H z / µ Ν ) S-MR3-MCDHF+RCIS-MR4-MCDHF+RCI
MR 1 2 3 4 5 6 7 8 -350-300-250-200-150-100 B e l [ P ] ( M H z / b ) S-MR3-MCDHF+RCIS-MR4-MCDHF+RCI
MR 1 2 3 4 5 6 7 8
Virtual layers A e l [ P ] ( M H z / µ Ν ) S-MR3-MCDHF+RCIS-MR4-MCDHF+RCI
MR 1 2 3 4 5 6 7 8
Virtual layers B e l [ P ] ( M H z / b ) S-MR3-MCDHF+RCIS-MR4-MCDHF+RCI
Figure 1: The convergence patterns of the electronic hyperfine factors A el [ P o1 ] , A el [ P o1 ] (left panels)and B el [ P o1 ] , B el [ P o1 ] (right panels) as functions of the virtual orbital layers. The radial orbitalbasis was obtained by applying two different optimization strategies with respect to the selectedMR configurations. The dashed lines connect the values resulting from the S-MR3-MCDHF op-timization, where three reference configurations are included in the MR, and the solid lines linkthe resulting values from the S-MR4-MCDHF scheme, where the MR was extended to include fourreference configurations. Both sets of values are the results from the RCI calculations that followedthe orbital optimization step. For further details, see text in Sec. III. In a succeeding set of calculations, the number of reference configurations was extended toinclude the s p s J Π = 1 − configuration in the initial multi-reference optimization of thespectroscopic orbitals. The sequence of the virtual orbitals that were progressively optimizedis similar to the previous calculations (see Table I). Since, at this point, the s orbital ispart of the spectroscopic orbitals, the second virtual orbital layer consists of the p , d , and f orbitals alone. For every additional virtual orbital layer, the resulting numbers of CSFs, N CSF , are displayed in column 4 of Table I, under the name MR4. Likewise the S-MR3-MCDHF calculations, these MCDHF calculations were completed with RCI calculationsthat included SD electron substitutions from the valence subshells, i.e., s , p , d , s , and13 s . The respective results of the electronic hyperfine factors are illustrated in Fig. 1 withthe label S-MR4-MCDHF+RCI.As seen in Fig. 1, the computed electronic hyperfine factors for the two targeted s p s , P o1 states are effectively converged. For the largest CSF expansions, no notice-able change is observed between the results from the S-MR3-MCDHF and S-MR4-MCDHFoptimization strategies. That being so, the final results are taken from the largest RCIcalculation using the s p d f g h i orbital basis set that was optimized within theS-MR4-MCDHF scheme (corresponding to 103 403 CSFs). For this first set of calculations,the final values of the isotope-independent hyperfine constants are: A el (cid:2) P o1 (cid:3) = 2 180 MHz /µ N ; B el (cid:2) P o1 (cid:3) = −
166 MHz / b ; A el (cid:2) P o1 (cid:3) = 174 MHz /µ N ; B el (cid:2) P o1 (cid:3) = 622 MHz / b . (18)It is noteworthy that the above-displayed final A el [ P o1 ] and B el [ P o1 ] values are significantlylarger than the results of the initial MR3(MR4) calculations that were based only on the9(11) reference CSFs. The relative discrepancies amount to ( ) and ( ), re-spectively. The final A el [ P o1 ] and B el [ P o1 ] values are, however, surprisingly close to theMR3(MR4) results. IV. SrD-SR-MCDHF CALCULATIONSA. Convergence of the electronic hyperfine structure factors
Experimental data (and state compositions) indicate that the hyperfine constant A islarge for the P o1 state and small for the P o1 state. Oppositely, the EFG is small, in absolutevalue, for the P o1 state and large for the P o1 state. When spectroscopic data are used forthe extraction of nuclear parameters, only these large values are of interest. Therefore, thesingle-reference (SR) calculations of this section focus on the “large” results, presenting onlythe electronic hyperfine factors A el [ P o1 ] and B el [ P o1 ] ∝ EFG[ P o1 ] in the following Fig. 2 andTable II.All calculations were performed together for the targeted s p s P o1 and s p s P o1 excited states. The spectroscopic and virtual orbitals were optimized based on an EOLenergy functional that was built over the CSFs of the two lowest J Π = 1 − levels. Thegeneration of the wave functions essentially followed the scheme described in our previous14apers [29, 52, 53]. The virtual orbitals were generated in a series of SCF calculations,where the virtual orbital set was systematically increased by one layer. Each layer includedone additional virtual orbital of the s , p , d , f , and g angular symmetries. In the very last,eighth layer, only the s , p , d , and f symmetries were represented. Therefore, the largestmulticonfiguration expansion was built on the s p d f g set of orbitals.The occupied orbital shells were systematically opened for substitutions, starting from the p and s valence orbitals and, eventually, opening all occupied orbital shells, down to the s core orbital. All virtual orbitals were generated in calculations with single and restricteddouble substitutions (SrD), which means (1) unrestricted single substitutions and (2) doublesubstitutions restricted by the limitation that at most one electron might be substituted fromoccupied, closed shells with n < . The other (or both) electron(s) must be substituted fromthe s , p , and s subshells. The largest expansion contains substitutions from all occupiedorbital shells to eight layers of virtual orbitals of the s , p , d , f , and g symmetries, exceptfor the very last layer that was limited to l max = 3 as mentioned above.The orbitals generated in this first phase were frozen and used in subsequent RCI calcu-lations that constituted the next two phases of the calculations. The second phase allowedsingle and (unrestricted) double substitutions (SD), while the third phase allowed single,double, and triple substitutions (SDT). For the latter two phases, only six layers of virtualorbitals were retained, since both A el [ P o1 ] and B el [ P o1 ] values saturated at the 7 th and 8 th layers of the first phase, as also illustrated in Fig. 2 (see magenta circles). Accordingly, the A el [ P o1 ] and B el [ P o1 ] values resulting from the 6 th layer of phase 1 are taken as a referenceand shown in the second row of Table II. In the same table, the first row displays the A el [ P o1 ]and B el [ P o1 ] values from the DHF computation, restricted to two CSFs. The DHF valuesdiffer substantially from the converged values of phase 1, i.e., by and , for A el [ P o1 ]and B el [ P o1 ], respectively. The large deviations from the simplest DHF calculations validatethe significance of electron correlation effects in the computation of the electronic hyperfinestructure factors.Similarly to any other expectation value, the isotope-independent hyperfine constants A el [ P o1 ] and B el [ P o1 ] are functions of four variables, called “dimensions” in Ref. [29]. These“dimensions” refer to the number of virtual orbital layers, the virtual space angular momen-tum, the opened occupied shells, and the number of substitutions, i.e., S, D, T, quadruple(Q), or higher. In the second, or else SD, phase, the multiconfiguration expansions were15 SrD layers 0-8 SD/SDT phases A e l [ P o 1 ] ( M H z / µ Ν ) Expt. SrDSDSDT
SrD layers 0-8 SD/SDT phases B e l [ P o 1 ] ( M H z / b ) SrDSDSDT
Figure 2: (Color online) The electronic hyperfine factors A el [ P o1 ] (in MHz/ µ N ) and B el [ P o1 ] (inMHz/b) obtained in three approximations: the SrD (magenta circles), the SD (green squares), andthe SDT (blue triangles) computational approaches. The symbols represent the results at eachstep of the calculations, while the lines are only for the guidance of the eyes. On the x-axes, thenumber indicates the DHF computation, the numbers 1-8 represent the consecutive layers ofvirtual orbitals developed in the SrD phase of the calculations, and the numbers 10-26 match thelabels of the multiconfiguration expansions presented in the first column of Table II, correspondingto the calculations performed in the SD and SDT phases. The red straight horizontal line on thetop of the left graph represents the experimental value A exptel [ P o1 ] = 2 398 MHz/ µ N from Ref. [1].For further details, see text in Sec. IV. systematically increased in the first three of the “dimensions” above. The resulting valuesof the electronic hyperfine factors A el [ P o1 ] and B el [ P o1 ] from the SD phase are presented inTable II as functions of the increasing multiconfiguration expansions. The computed valuesare also displayed graphically in Fig. 2, in which the points on the x axis represent the labelsdisplayed in the first column of Table II. For each labelled calculation, the second column ofTable II gives the principal quantum number n of the deepest orbital shell that was openedfor substitutions. For instance, n ≥ involves substitutions from the s, p, d, s, p , and s orbitals. The third column of the same table provides the active set of orbitals to whichthe electron substitutions were allowed.Saturation of the SD phase is observed for the n ≥ → s p d f g calculation (seelabel 24 in Table II). The saturation is demonstrated by comparing the results obtainedfrom the label-24 calculation with the results obtained from the two larger expansions:the label-25 calculation, which was extended by one additional virtual orbital layer, and16 able II: The computed electronic hyperfine factors A el [ P o1 ] (in MHz/ µ N ) and B el [ P o1 ] (in MHz/b)for various multiconfiguration expansions. The considered CSFs were generated based on SD(columns 4 and 6) and SDT (columns 5 and 7) substitutions from the opened shells displayedin column 2 to the active set of orbitals given in column 3. The first row contains the resulting A el [ P o1 ] and B el [ P o1 ] values from the DHF computation, where only the CSFs of the two targetedstates were considered, and the second row displays the converged results from the SrD compu-tational phase after the sixth layer of virtual orbitals was added. The labels given in column 1correspond to the labels used on the horizontal axes of Fig. 2. A el [ P o1 ] (MHz/ µ N ) B el [ P o1 ] (MHz/b)Label open shells active orbital set SD SDT SD SDT0 phase 1: DHF computation 1 869 6076 phase 1: SrD virtual layer 6 2 353 75710 n ≥ s p n ≥ s p d f n ≥ s p d f n ≥ s p d f g n ≥ s p n ≥ s p d n ≥ s p d f n ≥ s p d f g n ≥ s p d f g n ≥ s p d f g n ≥ s p d f g n ≥ s p d f g n ≥ s p d f g n ≥ s p d f g n ≥ s p d f g n ≥ s p d f g n ≥ s p d f g n = 2 shells were added. All thesethree different expansions yield similar values and, therefore, the expansion with label 24( n ≥ → s p d f g ) was carried over to the SDT phase of the calculations.In the third phase, involving SDT electron substitutions, we tried to follow a pattern forgenerating the multiconfiguration expansions that was similar to the SD phase. However,the number of the generated CSFs based on T substitutions was growing very rapidly andthe limits of the computational resources available to us were reached before the computedproperties were fully saturated. The results from the SDT phase are presented alongside theSD results in Table II and graphically in Fig. 2. It should be mentioned that the SDT calcu-lations included the CSFs produced during both the SrD phase ( n ≥ → s p d f g )and the SD phase ( n ≥ → s p d f g ). B. Correlation between A el [ P o1 ] and B el [ P o1 ] The electronic M1 hyperfine factor A el appears to be quite sensitive to D and T sub-stitutions. The dependence of the computed A el values on the different classes of elec-tron substitutions, i.e., S, D, and T substitutions, is illustrated in the left graph of Fig. 2.The overall dependence follows the trends that had been observed in many earlier calcula-tions [29, 53–58], where the effect of the D substitutions was to decrease the absolute valuesof the computed electronic hyperfine factors and the effect of the T substitutions was todecrease the effect of the D substitutions. In fact, these dependencies are visible in Fig. 2for both A el [ P o1 ] and B el [ P o1 ].The comparison between the two graphs of Fig. 2 further illustrates the correlation be-tween the A el [ P o1 ] and B el [ P o1 ], which is evident not only in the SrD phase of the calculations(magenta circles), but also in the case of the SD approximation (green squares). This obser-vation will be exploited in Sec. IV C as a tool for determining a more reliable value for theelectronic E2 hyperfine factor B el [ P o1 ]. As seen in Fig. 2, the correlation between A el [ P o1 ]and B el [ P o1 ] is less pronounced in the case of SDT approximation (blue triangles). It appearsthat B el [ P o1 ] is insensitive to T substitutions. However, the comparison between the resultsobtained in the SD and SDT approximations shows that the T substitutions are still quiteeffective in decreasing the effects of the D substitutions.18 . Estimating B el [ P o1 ] It is often the case that an estimate of an error bar of a calculated expectation value of aparticular atomic property X is based on a calculated value of another atomic property Y .This situation often occurs when the accuracy of X cannot be easily obtained, while theaccuracy of Y can be obtained from comparison with experiment or by other means. Ex-amples of such indirect estimations include: error bars of transition rates [59, 60] or isotopeshifts [61, 62] inferred from the accuracy of the corresponding transition energies; error barsin the calculations of amplitudes involved in parity- and time-reversal symmetry violationsinferred from hyperfine calculations [63, 64]; error bars of E2 hyperfine constants inferredfrom calculations of M1 hyperfine constants [29, 53].In the case of the hyperfine interaction constants, when a series of multiconfiguration ex-pansions are employed, both A el and B el factors exhibit similar and synchronous dependenceon the size of the expansion. This is illustrated in Fig. 2 and in numerous previous calcula-tions of hyperfine structures [31, 65–68]. Synchronous oscillations that are observed in thecomputed A el and B el values may be qualitatively explained by comparing two consecutivemulticonfiguration expansions. Within the MCDHF methodology employed in the presentpaper, two neighboring multiconfiguration expansions, N CSF ( i ) and N CSF ( i +1) , differ by thepresence of an additional i + 1 layer of virtual orbitals in the N CSF ( i + 1) expansion. Theorbitals represented in the N CSF ( i ) expansion are frozen, and only the orbitals included inthe additional i +1 layer are optimized in the SCF procedure described in Sec. II. The addi-tional layer of virtual orbitals introduces contributions to the expectation values of the M1and E2 hyperfine operators, and these contributions, in turn, depend on the radial formsof the additional virtual orbitals. A “layer” is composed of a set of virtual orbitals withprincipal quantum numbers increased by one with respect to the previous layer. Each layerincludes orbitals with different angular quantum numbers (typically one orbital per angularsymmetry is represented in each consecutive layer). The evaluation of hyperfine structuresinvolves radial integrals, in which a radial factor r − induces a strong dependence on theinner part of the electronic orbitals [40, 69, 70], which results in the well known statementthat the hyperfine interaction happens within the innermost one-half of the first oscillationof one-electron orbitals [71, 72]. This common r − dependence of the radial hyperfine inte-grals for both M1 and E2 interactions, although not immediately obvious from Eqs. (13) and1914), may be explained from the different structures of the relevant one-electron matrix ele-ments [73]. The additional layer of virtual orbitals induces a ∆ A shift of the A constant and,respectively, a ∆ B shift of the B constant (for the purpose of this discussion we used the M1hyperfine constant A and the E2 hyperfine constant B , but the arguments apply similarlyto the isotope-independent hyperfine constants A el and B el ). From the above considerationsone should expect that these shifts are approximately proportional, i.e., ∆ A/A ≈ ∆ B/B. (19)This may be illustrated in Fig. 8 of Ref. [51], where the curves a dip (top right) and b quad (bottom right) apparently oscillate synchronously as functions of the multiconfigurationexpansion, and in numerous previous calculations of hyperfine structures [31, 65–68].The above equation may be transformed into a relation in which the calculated values A calc and B calc are related to the experimental values A expt and B expt , so that | A calc − A expt | /A expt ≈ | B calc − B expt | /B expt . (20)The equation above can, then, be used to correct the calculated value of the E2 hyperfinefactor B el (or the calculated EFG ∝ B el ) by a semiempirical shift arising from the (known)error in the calculated value of the M1 hyperfine factor A el .As already discussed in Sec. IV A, the resulting A el [ P o1 ] and B el [ P o1 ] values from the SDapproximation were fully converged, whereas the multiconfiguration calculations involvingSDT substitutions were not entirely saturated due to the exceptionally large CSF expansions.It should be pointed out that the largest completed SDT calculation, with 4 406 086 CSFs,took 37 days of wall time on the computer cluster at our disposal (6x96 CPU @ 2.4GHz with6x256 GB RAM). The next in line, with 17 817 617 CSFs, eventually exceeded the capacityof that cluster. However, as mentioned in Sec. IV B and illustrated in Fig. 2, B el [ P o1 ] is ratherinsensitive to T substitutions. On this ground, the B el [ P o1 ] = 760 MHz/b value obtainedfrom the largest completed SDT calculation is still taken into consideration in the analysisbelow for determining the final B el [ P o1 ] value of the SrD-SR-MCDHF and RCI calculationsdescribed in this section. In the following, the latter result is labeled B el (SDT).When applying the semiempirical shift of Eq. (20) to the final results obtained fromthe SrD, SD, and SDT calculations described in Sec. IV, three more B el [ P o1 ] values areobtained, which are labeled B el ( SrD ) shifted , B el ( SD ) shifted , and B el ( SDT ) shifted , respectively.20o estimate these three values, the experimental A expt el [ P o1 ] = 2 398 MHz/ µ N value was takenfrom Ref. [1]. We ultimately arrive at the following four figures: B el ( SDT ) = 760 MHz/b, B el ( SrD ) shifted = 759 MHz/b, B el ( SD ) shifted = 800 MHz/b and B el ( SDT ) shifted = 793 MHz/b.By taking their average, we obtain: B el [ P o1 ] = 778 MHz/b . (21) V. SrD-MR-MCDHF CALCULATIONSA. Selecting the multi-reference set
In this third approach, we go beyond the single-reference approximation described inSec. IV by defining a set of MR configurations. These configurations must be selected sothat they produce a number of CSFs that account for the major electron correlation effects,or else static correlation [19]. Further, by allowing S and D substitutions of electrons fromthe MR configurations, important T and Q substitutions from the targeted s p s con-figuration are taken into account. Starting from single-reference SD-MCDHF calculations,where the two targeted s p s , P o1 states were optimized according to the EOL scheme, the LS -composition of the resulting ASFs was analyzed. After allowing, for instance, SD sub-stitutions from the valence orbitals ( n ≥ ) to a first layer of virtual orbitals, i.e., s, p, d ,and f , the LS -composition of each of the targeted states can be seen in Table III. From thefirst analysis based on Table III, we, thus, defined an MR that is composed of the followingthree non-relativistic configurations: s p s , p s , and s p d s . The above configura-tions correspond to 17 relativistic CSFs for the J Π = 1 − symmetry. Enlarging further theMR set would lead to very large CSF expansions that are beyond the reach of the compu-tational resources that are available to us. For that reason, the MR was restricted to thethree leading configurations. B. Orbital optimization strategies
The active set of virtual orbitals was systematically increased by one layer, with eachlayer i including orbitals with quantum numbers nl equal to ( i + 6) s, ( i + 5) p, ( i + 5) d, and ( i +3) f . Therefore, the last virtual orbital layer i = 9 corresponds to the s p d f set of21 able III: The LS -composition of the two targeted s p s , P o1 states after performing initialSD-MCDHF calculations (see also text in Sec. V A). The percentages of the four most dominant LS -components are solely displayed, with the first percentage corresponding to the assigned con-figuration and term.Pos. Conf. LSJ LS -composition1 s p s P o1 s p s P o + 0.016 p s P o + 0.010 s p d s P o s p s P o1 s p s P o + 0.017 p s P o + 0.009 s p d s P o orbitals. The orbital basis was obtained by performing two different sets of MCDHF compu-tations, which are, respectively, denoted VV-SD-MR-MCDHF and CV-SrD-MR-MCDHF.The former orbital optimization strategy included SD substitutions from the s, p, d, and s valence orbitals of the MR configurations, capturing valence-valenve (VV) correlationeffects. On the other side, the latter optimization strategy allowed SrD substitutions fromthe d, s, p, d, and s orbitals in the MR, with the limitation that there was at most onesubstitution from the d subshell. All other inner subshells were kept closed. Since the d orbital is considered part of the core, the generated CSF expansions based on the above-mentioned SrD substitutions further accounted for core-valence (CV) correlation effects.By using the orbitals optimized in the VV-SD-MR-MCDHF scheme and, for each virtualorbital layer, performing an RCI calculation based on the CSF expansion that was generatedin the CV-SrD-MR-MCDHF computations, the values of the electronic factor A el [ P o1 ] werecompared for the two different orbital optimization strategies. The upper left panel of Fig. 3illustrates the convergence of A el [ P o1 ] as a function of the number of virtual orbital layers forboth sets of computations; the results from the VV-SD-MR-MCDHF optimization strategyare represented by RCI-(VV+CV). Although the two approaches exhibit similar trends, thediscrepancies between the resulting A el [ P o1 ] values strongly suggest to use the CV-SrD-MR-MCDHF orbital set to perform additional RCI computations (see Sec. V F). After addingnine layers of virtual orbitals, the absolute discrepancy is ∼ MHz/ µ N . As will be seen,this value is much larger than the variations induced by the effects of the “ f -limit” andthe reduction of the CSF space, which are, respectively, investigated in the following twosubsections. 22 A e l [ P o 1 ] ( M H z / µ Ν ) RCI-(VV+CV)CV-SrD-MR-MCDHF -144-140-136-132 B e l [ P o 1 ] ( M H z / b ) CV-SrD-MR-MCDHF
Virtual layers A e l [ P o 1 ] ( M H z / µ Ν ) CV-SrD-MR-MCDHF
Virtual layers B e l [ P o 1 ] ( M H z / b ) CV-SrD-MR-MCDHF
Figure 3: (Color online) The convergence of the electronic hyperfine factors A el [ P o1 ] , A el [ P o1 ] (left panels) and B el [ P o1 ] , B el [ P o1 ] (right panels) with the increasing number of virtual orbitallayers, optimized using the CV-SrD-MR-MCDHF scheme. In the upper left panel, the com-puted A el [ P o1 ] values using the CV-SrD-MR-MCDHF scheme (black circles) are compared withthe A el [ P o1 ] values resulting from the VV-SD-MR-MCDHF orbital optimization strategy and ad-ditional RCI-(VV+CV) computations, where CV correlation was included (orange triangles). Fordetails, see text in Secs. V B and V E. C. The “ f -limit” The angular quantum number l of the orbital basis was limited to l max = 3 so thatonly orbitals up to the f symmetry were included in the computations presented in thissection. The effects of orbitals with higher angular symmetry on the computation of theelectronic hyperfine factors are, generally, known to be small [74]. In the present work, thiswas verified by comparing two sets of VV-SD-MR-MCDHF computations that, respectively,utilized l max = 3 and l max = 5 (the latter includes orbitals of the g and h symmetries). TheVV-SD-MR-MCDHF computational scheme results in smaller number of CSFs (comparedto the CV-SrD-MR-MCDHF scheme) and it, therefore, serves better the purpose of this23ubsection, which is to quantify the error (cid:15) f induced by the choice of the “ f -limit”.These VV-SD-MR-MCDHF computations were carried out for the first five virtual or-bital layers, and the resulting A h el [ P o1 ] and A f el [ P o1 ] values, respectively, corresponding tothe l max = 5 and l max = 3 active spaces, are presented in Table IV. Looking at Ta-ble IV, after adding five layers of virtual orbitals, the difference between the two setsof values is ∆ A f el [ P o1 ] = A h el [ P o1 ] − A f el [ P o1 ] (cid:39) MHz/ µ N . A similar analysis yields ∆ A f el [ P o1 ] (cid:39) MHz/ µ N , ∆ B f el [ P o1 ] (cid:39) MHz/b, and ∆ B f el [ P o1 ] (cid:39) MHz/b. The differences ∆ A f el and ∆ B f el are expected to remain almost unvarying as the number of virtual orbitallayers increases further, and when presenting the final results of the computed electronichyperfine factors in Sec. V F, the above-mentioned values are assumed to be the corrections (cid:15) f due to the “ f -limit”. Table IV: The influence of orbitals with g and h angular symmetries on the computation of theelectronic hyperfine factor A el [ P o1 ] . By, respectively, using l max = 5 and l max = 3 , the A h el [ P o1 ] and A f el [ P o1 ] values were computed in MHz/ µ N for five virtual orbital layers that were optimized in theVV-SD-MR-MCDHF scheme, and the results are, respectively, shown in columns 2 and 3. In thelast column, the differences ∆ A f el [ P o1 ] = A h el [ P o1 ] − A f el [ P o1 ] are also presented.Virtual layer A h el [ P o1 ] A f el [ P o1 ] ∆ A f el [ P o1 ] † † Note that the roundings were performedafter evaluating the difference.
D. Reduction of CSF space
When we open the d subshell to include CV correlation in the CV-SrD-MR-MCDHFoptimization strategy, the number of CSFs grows very rapidly with the increasing active setof virtual orbitals. It is, then, advisable to restrict the atomic state expansions to CSFsthat interact, i.e., have non-zero Hamiltonian matrix elements, with the ones generated by24 able V: The effect of reducing the configuration space, to CSFs that interact with the ones thatare part of the MR, on the computation of the electronic hyperfine factor A el [ P o1 ] . Using the fulland reduced CSF spaces, two different sets of computations were performed, in which nine virtualorbital layers were optimized in the VV-SD-MR-MCDHF scheme. The numbers of CSFs beforeand after the reduction are, respectively, given in columns 2 and 3. The corresponding A full el [ P o1 ] and A redel [ P o1 ] values are given in MHz/ µ N in columns 4 and 5, and their discrepancies ∆ A redel [ P o1 ] are displayed in the last column. In the second portion of the table, results from additional RCI-(VV+CV) computations up to the fifth virtual orbital layer are included.Virtual layer N CSF N redCSF A full el [ P o1 ] A redel [ P o1 ] ∆ A redel [ P o1 ] † † † † RCI-(VV+CV)2 71 747 57 086 2 007 2 000 8 † † Note that the roundings were performed after evaluating the difference. the MR configurations. This was done by utilizing the tool rcsfinteract , which is includedin both
Grasp2K [27] and
Grasp2018 [28] computer packages. Such reduction of CSFsnormally does not bring any major losses in accuracy [75–77].The effect of limiting the number of CSFs to the “interacting” ones is evaluated by compar-ing the results from two sets of computations, where for each one the full and reduced CSFspaces were used. Similarly to the investigation of the effect of the “ f -limit”, the electronichyperfine factors were computed in the VV-SD-MR-MCDHF scheme, which is computa-25ionally faster. The resulting A full el [ P o1 ] and A red el [ P o1 ] values, respectively, corresponding tothe full and reduced CSF spaces, together with the associated numbers of generated CSFs,are displayed in Table V. Looking at Table V, we observe that, after building nine layersof virtual orbitals, the discrepancy ∆ A redel [ P o1 ] = A full el [ P o1 ] − A redel [ P o1 ] converges to about MHz/ µ N . A similar analysis yields ∆ A redel [ P o1 ] (cid:39) − MHz/ µ N , ∆ B redel [ P o1 ] (cid:39) − MHz/b,and ∆ B redel [ P o1 ] (cid:39) MHz/b.To further evaluate the possible influence that the reduction of the CSF space mighthave on CV correlation, additional RCI-(VV+CV) computations were performed includingfive virtual orbital layers. The resulting A full el [ P o1 ] and A redel [ P o1 ] values from the latter RCIcomputations, together with the numbers of the considered CSFs, are displayed in the secondportion of Table V. It is seen that ∆ A redel [ P o1 ] is kept almost unchanged and, thus, unaffectedby CV correlation, and the same applies to ∆ A redel [ P o1 ] . That being so, when inferring thefinal values of the computed electronic hyperfine factors A el in Sec. V F, it is assumed thatthe corrections due to the reduction of the CSF space, (cid:15) red , are, respectively, equivalent to ∆ A redel [ P o1 ] = 7 MHz/ µ N and ∆ A redel [ P o1 ] = − MHz/ µ N , just as estimated above. On theother hand, a noticeable increase in the variations of the electronic factors B el was observedand, for those, it was concluded that the corrections (cid:15) red equal ∆ B redel [ P o1 ] = − MHz/b and ∆ B redel [ P o1 ] = 1 MHz/b, respectively.
E. Convergence of the CV-SrD-MR-MCDHF computations
The importance of optimizing the orbital basis using the CV-SrD-MR-MCDHF computa-tional strategy, which also takes into account the polarization of the d core orbital throughCV substitutions, was already highlighted in Sec. V B (see also Fig. 3). The purpose ofthis subsection is to examine the convergence of the properties of interest within the CV-SrD-MR-MCDHF approach. In Table VI, the computed excitation energies and electronichyperfine factors A el and B el of the , P o1 states are given as functions of the increasing activeset of virtual orbitals. As a reference point, the values associated with the initial MR calcu-lation are also given in the first row of Table VI. One notices that these values are close tothe resulting A el [ P o1 ] and B el [ P o1 ] values from the DHF calculations presented in Table II.Table VI shows that nine virtual orbital layers are required to ensure the convergence ofall computed properties. We note that, in the final atomic state expansion for the J = 1 − Table VI: The convergence of energies and electronic hyperfine factors A el and B el for the , P o1 statesafter extending the MR orbital basis to include nine layers of virtual orbitals that were optimizedusing the CV-SrD-MR-MCDHF scheme. The computed excitation energies of the P o1 and P o1 states are, respectively, presented in columns 2 and 3, whereas the evaluated energy separationsare displayed in column 4. For comparison, the observed energies are shown at the bottom part ofthe table. In each of the columns 5 and 6, the values of the electronic factors A el [ P o1 ] and A el [ P o1 ] are given, and columns 7 and 8, respectively, contain the values of the electronic factors B el [ P o1 ] and B el [ P o1 ] . The last column exhibits the numbers of generated CSFs for every additional virtualorbital layer. Energies (cm − ) A el (MHz/ µ N ) B el (MHz/b)Virtual layer P o1 1 P o1 1 P o1 − P o1 3 P o1 1 P o1 3 P o1 1 P o1 N CSFs none (MR) 33 301 38 002 4 701 1 869 515 −
154 613 171 34 327 38 990 4 663 2 044 192 −
134 586 16 5932 34 789 39 323 4 534 2 055 206 −
134 586 57 0863 34 644 39 154 4 510 2 092 179 −
135 587 122 6104 34 587 39 072 4 485 2 109 191 −
143 591 212 9465 34 544 39 020 4 476 2 120 190 −
140 596 328 0946 34 529 39 004 4 475 2 117 188 −
140 590 468 0547 34 521 38 993 4 472 2 121 188 −
139 593 632 8268 34 519 38 987 4 468 2 119 187 −
139 590 822 4109 34 517 38 984 4 467 2 120 186 −
139 592 1 036 806Expt. [78, 79] 34 914 39 257 4 343
At this point, the excitation energies of the P o1 and P o1 states are well converged andthe predicted energy separation between the two states agrees with the observed value towithin (see column 4 in Table VI). The quality of the obtained energies provides aninitial assessment of the computed electronic hyperfine factors, which, as seen in Table VI,are also effectively converged. The convergence patterns of the A el [ P o1 ] , A el [ P o1 ] , B el [ P o1 ] ,27nd B el [ P o1 ] values with respect to the increasing number of virtual orbital layers are alsoillustrated in Fig. 3. One should bear in mind that the y-axes in the two right-hand sidegraphs of Fig. 3 span a considerably smaller range of values compared to the graphs on theleft side. That being so, once the sixth virtual layer has been added, the convergences ofthe A el and B el factors should be considered equally smooth. F. Final CV-SrD-MR-MCDHF + RCI computations
After ensuring the convergence of the computed properties within the CV-SrD-MR-MCDHF orbital optimization scheme, a final RCI computation was carried out, in which theatomic state expansions were augmented to include CSFs accounting for additional electroncorrelation effects. Due to limited computational power, the CSF space must be generatedso that the most dominant correlation effects are, first and foremost, efficiently captured.To assess the relative importance of the various correlation effects on the computation of theelectronic hyperfine factors, preliminary RCI computations, which used the s p d f setof orbitals optimized within the CV-SrD-MR-MCDHF scheme, were performed, and theirresults are discussed below.VV and some CV electron correlation effects were already captured during the optimiza-tion of the orbital basis. To also account for core (C) and core-core (CC) correlation effects,inner subshells were progressively opened to allow substitutions of electrons to the s p d f active set of orbitals. At first, additional D substitutions were allowed from the d subshell.Then, S substitutions from inner core subshells were gradually considered, starting from the p subshell and eventually opening all occupied subshells down to the innermost s coreorbital. Further CV correlation effects were considered by allowing rD substitutions fromthe n ≥ orbitals with the limitation that there is maximum one substitution, initially, onlyfrom the p orbital, and then, also from the s orbital. For every additional type of elec-tron substitutions described above, RCI computations using the respective CSF expansionswere performed. For each of these RCI computations, the resulting values of the electronichyperfine factors A el and B el are shown in Table VII.Looking at Table VII, the D substitutions from the d orbital are clearly the most im-portant to consider for the final RCI computation. The S substitutions from np orbitalsalso have a rather substantial effect, although it becomes less significant as n decreases. All28 able VII: The effect of different types of electron substitutions on the computation of the electronichyperfine factors A el and B el for the , P o1 states. The reference values displayed in the first linewere computed after optimizing four virtual orbital layers using the CV-SrD-MR-MCDHF scheme(maximum one hole allowed in the d orbital). Contributions from additional substitutions wereevaluated in subsequent RCI computations, in which the configuration space was enlarged by,first, including D substitutions from the d orbital and, then, progressively adding S substitutionsfrom the p orbital down to the s orbital. In the last portion of the table, contributions fromCV correlation effects were evaluated in RCI computations, where the configuration space wasgenerated by allowing SrD substitutions from n ≥ orbitals with the restriction that there was atmost one substitution, initially, only from the p orbital, and then, from the s orbital as well. A el (MHz/ µ N ) B el (MHz/b)Subst. Orbital P o1 1 P o1 3 P o1 1 P o1 SrD d −
143 591SD d −
149 605 + S p −
166 671 + S s −
166 671 + S d −
167 672 + S p −
170 684 + S s −
170 684 + S p −
170 687 + S s −
170 687 + S s −
170 687SD d −
149 605 + SrD p −
164 674 + SrD s −
164 675
S substitutions from ns orbitals are less important, but computationally cheap, and theywere, thus, considered in the CSF expansion of the final RCI computation. Looking at thelast portion of Table VII, one realizes that the CV correlation effects from both p and s subshells are important. However, our computational resources allowed us to include rD29ubstitutions only from the p orbital, and not from the s orbital, when constructing thefinal expansions of the atomic states.The final RCI computation was carried out by applying the above-mentioned rules ofelectron substitutions to the largest active set, corresponding to the s p d f set oforbitals. The number of CSFs in the final expansions was . In Table VIII, theconcluding values of the electronic hyperfine factors A el and B el are presented together withthe corrections (cid:15) red and (cid:15) f , respectively, induced by the reduction of the configuration space(see Sec. V D) and by the omission of l ≥ orbitals in the active set (see Sec. V C). Table VIII: The resulting values of the electronic hyperfine factors A el and B el for the P o1 and P o1 states from the final CV-SrD-MR-MCDHF+RCI computations. The corrections (cid:15) red and (cid:15) f arefurther added to account for reducing the active space to configurations that interact with the MRand for limiting the angular quantum number l of the orbitals basis to l max = 3 . A el (MHz/ µ N ) B el (MHz/b) P o1 1 P o1 3 P o1 1 P o1 Final RCI 2 169 409 −
173 716 + (cid:15) red −
174 717 + (cid:15) f −
172 717
G. Sensitivity to orbital bases and CSF expansions
As seen in the previous sections, different calculations based on the same general method,the MCDHF method, and performed with the same program, the
Grasp2018 package, leadto different results. The A el [ P o1 ], B el [ P o1 ], and B el [ P o1 ] values are in agreement withina relative error of approximately , while the A el [ P o1 ] values differ significantly. Thedifferences in the S-MR-MCDHF, SrD-SR-MCDHF, and SrD-MR-MCDHF approaches liein the choice of their respective orbital bases and CSF expansions, each with its benefitsand drawbacks. In this subsection, we investigate the sensitivity of the SrD-SR-MCDHFand SrD-MR-MCDHF approaches by arbitrarily interchanging their orbital bases and CSFexpansions.Each of the above-mentioned methods provides the final results by ultimately performingRCI computations. As explained before, the largest RCI expansion in the CV-SrD-MR-30 able IX: The electronic hyperfine factors A el and B el for the P o1 and P o1 states, computed forsix different combinations of orbital basis sets and CSF spaces. The SrD-SR-MCDHF and SrD-MR-MCDHF computational approaches are compared by expanding the total wave function overthe largest CSF expansion of the one method and using the orbital basis of the other method.Adjustments were made in the CSF expansions due to the specific properties of the orbital basesobtained in the two different approaches. For details, see text in Sec. V G. A el (MHz/ µ N ) B el (MHz/b)Orb. basis CSF expansions P o1 1 P o1 3 P o1 1 P o1 B el [ P o1 ] /B el [ P o1 ]SrD-MR SrD-MR 2 179 404 −
172 717 − −
190 760 − spdf limit) 2 303 289 −
188 739 − spdf limit) 2 297 289 −
190 722 − layers) 2 161 396 −
172 709 − layers) 2 168 342 −
194 718 − MCDHF method accounts for electron substitutions to the s p d f set of orbitalsincluding S substitutions from all occupied orbitals, D substitutions from the d , s , and p subshells, and rD substitutions from the p subshell. The SrD-SR-MCDHF approachallows all D and T substitutions from orbital shells with n = 4 , , and to three layers of s and p virtual orbitals and to one layer of d, f , and g virtual orbitals. This was mergedwith the CSFs generated by allowing SrD substitutions from all core subshells to six layersof s, p, d, f, and g virtual orbitals. These results are presented in Table IX under the labelsSrD-MR/SrD-MR and SrD-SR/SrD-SR, for the SrD-MR-MCDHF and SrD-SR-MCDHFapproaches, respectively (where the notation X/Y defines the orbital basis from X and CSFexpansion from Y). Two additional sets of computations were performed: one combiningthe SrD-SR-MCDHF orbital basis and the SrD-MR-MCDHF CSF space (see SrD-SR/SrD-MR in Table IX) and one combining the SrD-MR-MCDHF orbital basis and the SrD-SR-MCDHF CSF space (see SrD-MR/SR-SrD in Table IX). Minor changes in the CSF spaceswere required, e.g., the SR active space was restricted to the s, p, d , and f symmetries as theMR orbital basis is limited to l max = 3 and the SrD-MR-MCDHF CSF space was limited to31nly six virtual layers.Table IX shows that the results are consistent, although far to be in perfect agreement.The effect of replacing the orbital set for a given electron correlation model (i.e., for agiven CSF expansion) is (surprisingly) small, which is reassuring. The largest discrepanciesbetween the two models (SrD-SR-MCDHF and SrD-MR-MCDHF) are observed for A el [ P o1 ](40%) and B el [ P o1 ] (10%). The extreme sensitivity of A el [ P o1 ] is expected, as shown inAppendix A utilizing the framework of non-relativistic theory. A similar analysis for the B el [ P o1 ] /B el [ P o1 ] ratio reveals a steadier value, which can be derived from a simple P o − P o mixing. This analysis is consistent with the B el [ P o1 ] /B el [ P o1 ] ratios presented in Table IXand the computed B el [ P o1 ] /B el [ P o1 ] ratio from the final B el values given by Eq. (18) inSec. III. The values of all these ratios range from − . to − . , which is in excellentagreement with the experimental value: − . [1]. VI. CI-DFS CALCULATIONS
In this last set of calculations, which is based on the CI-DFS theory, we used for allSturmian functions the same reference energy, namely, that of the s state. All orbitals upto d form the occupied shells’ space RAS1, orbitals with n = 4 , and the s, p orbitalsbelong to the active space RAS2, and orbitals from d and beyond form the open shells’space RAS3. To assess the uncertainty of the calculations presented in this section, weperformed a series of calculations using an increasing orbital basis set. SD substitutionsfrom the occupied shells’ space and from the open shells’ space, i.e., N h = N ex = N el = 2 (see Sec. II B 3 for their definition), were included in the calculations, leading to a largenumber of configurations and huge matrices for the numerical diagonalization. By freezingthe s , s , and p states, and by using PT to build low-lying closed shells and highly-excitedstates, we were able to extend the one-electron basis to the s p d f set of orbitals. Forthe three smallest orbital basis sets, T substitutions into RAS2, i.e., N ex = 3 , were alsoincluded, however, their influence turned out to be smaller than the uncertainty level weaim at.For each virtual orbital layer that was used in the CI-DFS calculations, the correspondingorbital basis set, numbers of configurations and computed energy separations ∆ E betweenthe targeted P o1 and P o1 states are listed in Table X. One can see that the resulting energy32 able X: The numbers of configurations and the resulting energy separations between the targeted P o1 and P o1 states for each virtual orbital layer used in the CI-DFS calculations. Two approaches,the direct (full basis) and the one based on perturbation theory (PT), were implemented. N NonRel stands for the numbers of non-relativistic configurations, and when followed by “(PT)”, the numbersof configurations built using PT are also displayed in parentheses, N Total indicates the total numbersof relativistic configurations, and the energy separations ∆ E = E [ P o1 ] − E [ P o1 ] are given in cm − .The numbers of virtual orbital layers that are given in the first column correspond to the labelsused on the horizontal axes of Fig. 4, and the respective orbital basis sets are displayed in column 2.See also text in Secs. II B and VI.Virtual Orbital N NonRel N Total ∆ E (cm − ) N NonRel (PT) N Total ∆ E (cm − )layer basis set Full basis Perturbation theory1 s p d
73 425 4 537 41 (16) 247 4 5382 s p d f
335 3 267 4 868 188 (74) 1 897 4 8683 s p d f
749 6 915 4 930 421 (166) 4 019 4 9304 s p d f s p d f s p d f s p d f s p d f s p d f s p d f – – – 4 460 (3 121) 50 749 5 508 difference ∆ E = E [ P o1 ] − E [ P o1 ] from the direct (full basis) calculations is well converged,in contrast to the PT calculations, where the ∆ E value is not saturated. By constructingthe reduced one-particle density matrix, one can calculate the isotope-independent M1 andE2 hyperfine splitting constants, respectively, given by Eqs. (16) and (17). For both non-PT (solid circles) and PT (empty circles) bases, the convergence patterns of the isotope-independent hyperfine constants A el [ P o1 ] , A el [ P o1 ] , B el [ P o1 ] , and B el [ P o1 ] are shown in Fig. 4.It is seen in the figure that the electronic E2 hyperfine factors B el are more sensitive tovariations of the orbital basis set, in comparison to the electronic M1 hyperfine factors33
1 2 3 4 5 6 7 8 9 1840 1920 2000 2080 A e l [ P ] ( M H z / µ N ) CI-DFS, SDCI-DFS, SD-PT
1 2 3 4 5 6 7 8 9 1840 1920 2000 2080 1 2 3 4 5 6 7 8 9 -210-200-190-180 B e l [ P ] ( M H z / b ) CI-DFS, SDCI-DFS, SD-PT
1 2 3 4 5 6 7 8 9 -210-200-190-180 1 2 3 4 5 6 7 8 9 200 260 320 380 A e l [ P ] ( M H z / µ N ) Virtual layers
CI-DFS, SDCI-DFS, SD-PT
1 2 3 4 5 6 7 8 9 200 260 320 380 1 2 3 4 5 6 7 8 9 610 640 670 700 B e l [ P ] ( M H z / b ) Virtual layers
CI-DFS, SDCI-DFS, SD-PT
1 2 3 4 5 6 7 8 9 610 640 670 700
Figure 4: The convergence of the electronic hyperfine factors A el [ P o1 ] , A el [ P o1 ] (left panels) and B el [ P o1 ] , B el [ P o1 ] (right panels) with the increasing number of virtual orbital layers, optimized byemploying the CI-DFS method. The solid circles represent the results from the direct(full basis)calculations, while the empty circles illustrate the values obtained using perturbation theory (PT).The numbers of virtual orbital layers on the x-axes are equivalent to the numbers given in column 1of Table X. For more details, see text in Secs. II B and VI. A el , and for that reason, their theoretical uncertainties are larger. In addition, we observethat the results from the non-PT and PT calculations progressively diverge as the numberof virtual PT orbitals increases. That being so, and taking also into account the weakerstability of the PT energy separation value, the results from the perturbative treatment canonly be used for estimating the theoretical error bars (see Sec. VII), and not for extendingthe basis further. The final results of the CI-DFS calculations are based on the largestnon-PT orbital basis set (corresponding to the s p d f set of orbitals) and are shownbelow: A el (cid:2) P o1 (cid:3) = 2 082 MHz /µ N B el (cid:2) P o1 (cid:3) = −
202 MHz / b ; A el (cid:2) P o1 (cid:3) = 294 MHz /µ N B el (cid:2) P o1 (cid:3) = 693 MHz / b . (22)34 II. FINAL VALUE AND EVALUATION OF ACCURACY
In Secs. III–VI, four different computational approaches for evaluating the electronic hy-perfine factors A el and B el of the , P o1 excited states in neutral tin were presented. In thissection, we solely focus on the B el [ P o1 ] value that can be used to extract the quadrupole mo-ments, Q , of tin isotopes for which spectroscopic data are available. The four independentsets of calculations yielded four individual values of B el [ P o1 ] . In summary, we obtained B el [ P o1 ] = 622 MHz/b from the S-MR-MCDHF calculations (see Eq. (18) in Sec. III), B el [ P o1 ] = 778 MHz/b from the SrD-SR-MCDHF calculations (see Eq. (21) in Sec. IV), B el [ P o1 ] = 717 MHz/b from the SrD-MR-MCDHF calculations (see Table IX in Sec. V), and B el [ P o1 ] = 693 MHz/b from the CI-DFS calculations (see Eq. (22) in Sec. VI). We ultimatelyarrive at B el [ P o1 ] = 703 MHz/b by taking the average of these values.As a crude estimate of the uncertainty of the concluding B el [ P o1 ] value, we can considerthe half-range of the aforementioned individual results, i.e., 78 MHz/b. Yet, if one wantsto be in a position to discuss atomic, or nuclear, properties and their underlying physics,a rigorous assessment of the uncertainties of the computed values is required. In recentyears, atomic physicists have been putting great efforts into providing reliable uncertaintieson their theoretical results [53, 80–83]. In line with these efforts, the following subsectionstake into account a number of considerations to determine the accuracy of the final B el [ P o1 ] value. Some of these considerations are only applicable to one (or more) particular set(s)of calculations (see Sec. VII A), while others are analogously applied to all four separateresults (see Sec. VII B). Statistical principles are implemented (see Sec. VII C), and formeroutcomes from computations of electronic hyperfine factors, regarding atomic states withelectronic structure similar to the structure of the s p s , P o1 states in Sn I, are also usedas an estimate of the accuracy of the B el [ P o1 ] value deduced in this work (see Sec. VII D). A. Model-specific uncertainties
In each of the four independent approaches that were discussed in the previous sections,the wave functions (radial orbitals and configuration mixing coefficients) that describe theatomic states were obtained based on various approximations with respect to the orbital basisand the list of configuration states. In Sec. V G, the sensitivity of the SrD-SR-MCDHF+RCI35nd SrD-MR-MCDHF+RCI results to the orbital basis was investigated by combining theradial orbital basis obtained in one of these two computational approaches with the CSFexpansions used in the RCI computations of the other approach. As seen in Table IX, thesecombinations gave rise to B el [ P o1 ] values that range from 709 MHz to 760 MHz/b. The half-range of these values yields an uncertainty of 26 Hz/b. Further, in the CI-DFS calculations,the electronic hyperfine factors were computed using both non-PT and PT orbital bases.The comparison between the non-PT and PT results for different orbital basis sets suggestsan uncertainty of 70 MHz/b in the deduced value of B el [ P o1 ] . Lastly, in the instance ofthe SrD-SR-MCHDF calculations, the outcome for the B el [ P o1 ] value is the average of fourseparate values. One can, thus, assume an error bar corresponding to the half-range of thesevalues, i.e., 20 MHz/b. B. Difference between the theoretical and experimental A el values The deviation of the calculated M1 hyperfine constant from the experimental value,A theor - A expt , is often assumed to be a measure of the overall accuracy of the hyperfinestructure calculations [29, 53, 84]. In Sec. IV C, the experimental A expt [ P o1 ] value was usedto accordingly shift the resulting B el [ P o1 ] values from all three approximations, i.e., SrD,SD, and SDT, that were used in the SrD-SR-MCDHF+RCI calculations. Considering onlythe B el [ P o1 ] result from the most extensive SDT calculation and evaluating the difference B el ( SDT ) − B el ( SDT ) shifted yields an error estimate of 32 MHz/b. When applying this shiftto the final results of the remaining calculations, we acquire three more error bars; 54 MHz/bfrom the S-MR-MCDHF calculations, 64 MHz/b from the SrD-MR-MCDHF calculations,and 91 MHz/b from the CI-DFS calculations. C. Statistical standard deviation
The individual results provided by the four independent sets of calculations could beregarded as a statistical sample and, in that case, the average value µ and the standarddeviation σ can be evaluated. For the { , , , } set of B el [ P o1 ] -values, it is µ ± σ = 703 ± MHz/b, which positions B el [ P o1 ] between 591 MHz/b and 815 MHz/bwithin the 2 σ condition ( ). In this manner, we obtain another uncertainty estimate36quivalent to MHz.
D. Zinc analogy
In a recent paper [53], the quadrupole moment Q ( Zn) was evaluated based on in-dependent multiconfiguration calculations of the EFG ∝ B el for the s p P o1 and s p P o2 states in Zn I. The final accuracy of the calculated EFGs was estimated using the scatterof the individual results of these 11 calculations, resulting in a relative error of about .The valence structure of the s p P o1 , states in neutral zinc is quite similar to the structureof the s p s , P o1 tin states, which are of interest in this work; in both cases, there aretwo electrons outside the closed shells, and the orbitals of these valence electrons have sim-ilar angular symmetries. Thereby, one expects that, for atomic calculations using similarcomputational approaches, the relative error bars of the computed hyperfine factors will becomparable. Adopting the relative error bar, an uncertainty of MHz/b is inferred forthe B el [ P o1 ] value deduced in the present paper. E. Final accuracy
The considerations above lead to diverse error bars, which, according to the order of theirappearance in the text, are (in units of MHz/b): , , , , , , , , , and . The largest of these uncertainties, i.e., MHz/b, places B el [ P o1 ] between 521 MHz/band 885 MHz/b within the σ condition, which is a rather conservative choice. On the otherhand, the smallest of all these error estimates, i.e., 20 MHz/b, positions B el [ P o1 ] between663 MHz/b and 743 MHz/b within the σ condition. This interval does not overlap withall individual B el [ P o1 ] -values resulting from the four independent sets of calculations andtherefore, such an error bar is not appropriate. Assuming that some of the obtained errorbars possibly overestimate the uncertainty in our concluding B el [ P o1 ] value, and that a fewothers might underestimate it, we believe that the rounded value of MHz/b is a reasonablechoice. The final result of the present paper, then, becomes B el [ P o1 ] = 703 ± MHz/b , (23)localizing B el [ P o1 ] between 603 MHz/b and 803 MHz/b with confidence. Comparingthe value above with the B el [ P o1 ] value resulting from the simplest DHF calculation yields37 difference of ∼ , which is significantly higher than the ( / (cid:39) ) uncertaintyclaimed in this work. VIII. QUADRUPOLE MOMENTS
The computed B el [ P o1 ] ∝ EFG value can be used to deduce the nuclear quadrupolemoments Q ( A Sn ) = B/B el of the tin isotopes for which the E2 hyperfine constant B wasmeasured. The B el value resulting from the present multiconfiguration calculations wasrecently employed by Yordanov et al. [1] to extract the nuclear quadrupole moments ofodd- A tin isotopes. As mentioned in the introduction, the final value of the EFG for the s p s P o state has been slightly shifted from 706(50) MHz/b that was reported and usedin Ref. [1] to 703(50) MHz/b in the present paper. The Q -values listed in the last columnof Table 1 in Ref. [1] should, therefore, be increased by a factor of 706/703.For a few tin isotopes, more than one experimental values of E2 hyperfine constants areavailable, allowing us to compare the extracted quadrupole moments. The E2 hyperfineconstant B [ P o1 ] for the I = 5 / state of Sn was measured independently in Refs. [1, 89],and their results are, respectively, B [ P o1 ] = 212 . . MHz and B [ P o1 ] = 154(5) MHz. InRef. [89], the computed B el [ P o1 ] = 596 MHz/b value is also available, despite the fact thatthey used the data related to the P o1 state, i.e., B [ P o1 ] = − . . MHz and B el [ P o1 ] = − MHz/b, to extract the quadrupole moment Q ( Sn ) = 310(100) mb. By combiningthe experimental B [ P o1 ] result of Ref. [1] with the presently computed B el [ P o1 ] = 703(50) ,we obtain Q ( Sn ) = 219(7)(16) , which significantly differs from the above-mentioned valueof Q ( Sn ) = 310(100) mb. These two quadrupole moments merely overlap with each otherdue to the large uncertainty of 100 mb in the latter value. Further, the B el [ P o1 ] valuegiven in Ref. [89] barely overlaps with the present B el [ P o1 ] value, which strengthens theneed for re-computing the electronic hyperfine factors by using state-of-the-art programsand performing large-scale calculations.Finally, taking the Sn isotope as an example, we propose Q ( Sn) = − . b , (24)where (12) represents the theoretical uncertainty of 7% of the B el [ P o1 ] deduced in this workand (4) represents the experimental uncertainty of the measured B [ P o1 ] in Ref. [1]. We38otice that the theoretical uncertainty suggested above is about three times larger than theexperimental uncertainty. In the previous section, a number of considerations was takeninto account to provide a well-grounded estimate of the theoretical uncertainties in our final B el [ P o1 ] value. Nonetheless, one should always remain cautious towards error estimatesof electronic hyperfine factors deduced from atomic calculations and of the correspondingerror bars in the evaluated nuclear quadrupole moments Q . The element bismuth is a goodexample of the difficulties in estimating such error bars. Table I in Ref. [66] lists the proposedvalues of the nuclear quadrupole moment Q for the Bi isotope. The error bars of severalof those values not only do not overlap, but they do not even touch each other (to make allof the error bars overlap, the relative uncertainties would have to exceed 50%). We should,however, also note here that the valence structure of the tin atom is less complicated andless demanding computationally than the valence structure of the bismuth atom, and we areconfident enough that the deduced error bars in this paper are trustworthy.
IX. CONCLUSIONS
We presented the details of the theoretical calculations of the isotope-independent elec-tric quadrupole hyperfine constant B el ∝ EFG, which was recently used to extract nuclearquadrupole moments, Q , of tin isotopes [1]. Four independent computational approacheswere employed to finally provide the value of B el = 703(50) MHz/b for the s p s P o1 ex-cited state of Sn I. Three of these approaches were based on the variational MCDHF methodas implemented in the Grasp packages, whilst the fourth approach relied on the CI-DFStheory. The convergence of B el [ P o1 ] was monitored as the CSF expansions were enlargedby allowing single, double, and, depending on the correlation model, also triple, electronsubstitutions from the reference configuration(s). Efforts were made to provide a realistictheoretical uncertainty for the final B el [ P o1 ] value by accounting for statistical principles, thecorrelation with the isotope-independent magnetic dipole hyperfine constant A el , and previ-ous calculations of electronic hyperfine factors on systems with electronic structure similarto that of Sn I.The deduced relative accuracy of the present atomic ab initio calculations of the EFGsis of the order of , leading to even larger uncertainties in the extracted Q (Sn)-values dueto the uncertainty in the measured B . This level of accuracy is certainly inferior to the de-39uced Q (Sn)-values from the solid-state density functional calculations performed by Barone et al. [14], which are about an order of magnitude more accurate. In general, the accuracyof the atomic ab initio calculations of EFGs strongly depends on the valence structure of theatom, or ion, in question. We should note that, in the extreme case of lithium-like systems,the relative uncertainties of the atomic calculations of hyperfine structures can be limitedto . − . [31, 65, 85–88]. Even though the tin atom is far more demanding compu-tationally than the lithium-like systems, an atomic calculation of hyperfine structures withlower uncertainty would be possible for singly-ionized tin, with one electron outside closedshells, and it would be even more accurate, for triply-ionized tin, which has one electronoutside the n < core. Such calculations, as the latter, would likely challenge the accuracyof the solid-state methods. We hereby encourage experimentalists to consider one, or both,of the above-mentioned ions.Interestingly, we observed that all computed A el [ P o1 ] values are always smaller than theexperimental A expel [ P o1 ] = 2 398 MHz/ µ N value, independently of the computational methodor the correlation model. This could be explained by the lack of variational freedom intrinsicto the layer-by-layer optimization strategy, which hinders the contraction of spectroscopicorbitals when CV correlation is accounted for. In the specific case of tin, the spectroscopic d soft shell, i.e., lying between the core and the valence orbitals, is expected to be highly sensi-tive to CV correlation that might not be effectively captured. Natural orbitals were recentlyused, as an efficient tool to overcome the limitation of the layer-by-layer optimization scheme,to estimate hyperfine structure constants in Na I. Thanks to the radial re-organization of theorbitals, the spectroscopic orbitals are ultimately contracted, which affects both the mag-netic dipole and electric quadrupole electronic hyperfine factors [84]. Further investigationson the usefulness of the natural orbitals in the calculations of hyperfine structures are inprogress. Acknowledgments
SS is a FRIA grantee of the Fonds de la Recherche Scientifique − FNRS. MG acknowledgessupport from the FWO & FNRS Excellence of Science Programme (EOS-O022818F), PJacknowledges support from the Swedish Research Council (VR) under contract 2015-04842,and IIT acknowledges support by the RFBR Grant No. 18-03-01220.40 ppendix A: Sensitivity of the hyperfine constants and stability of the EFGs ratio
The extreme sensitivity of A el [ P o1 ] to correlation models is not really surprising if oneperforms calculations using the quasi-relativistic Hartree-Fock+Breit-Pauli [70] method inthe single configuration approximation. In the Breit-Pauli (BP) scheme, the low value ofthe ratio A el [ P o1 ] / A el [ P o1 ] can indeed be understood. The A el value of the pure P o1 i.e.,without considering any relativistic LS -term mixing, arises from the addition of the threecontributions [90], orbital, spin-dipole and contact term, which interfere positively. On theother hand, the A el value of the pure P o1 is only made of a (larger) orbital contribution, thetotal spin value ( S = 0 ) forbidding the two other contributions. For J = 1 , the two singletand triplet symmetries are mixed with relative phases that result from the orthogonalityconstraints. The eigenvector dominated by the triplet character has the same signs ofboth components, which makes the A el value even larger than the one of the pure triplet(increase of 40%). For the state dominated by the singlet, strong cancellation occurs dueto the triplet contamination, reducing the A el value by 61%. Strong cancellation in theestimation of a property usually involves high uncertainty.The “sharing rule” [91, 92] that is used to quantify configuration mixing from the measuredisotope shifts can be applied to the term-mixing analysis of the electric field gradients. In thesingle-configuration approximation, the ratio EFG[ P o1 ] / EFG [ P o1 ] is exactly − / − . ,resulting from angular momentum algebra, when using the same orbital sets for describingboth levels. Assuming a simple P o − P o mixing for J = 1 , we have Ψ(“ P o1 ”) = a | P o1 (cid:105) + b | P o1 (cid:105) , (A1) Ψ(“ P o1 ”) = b | P o1 (cid:105) − a | P o1 (cid:105) , where | , P o1 (cid:105) are the two lowest J Π = 1 − states resulting from pure LS -terms and Ψ(“ , P o1 ”) are the corresponding mixed states. Using the analytical ratio EFG[ P o1 ] / EFG [ P o1 ] = − ,one can estimate EFG [“ P o1 ”] = EFG [ P o1 ]( a − b ) , (A2)EFG [“ P o1 ”] = EFG [ P o1 ]( b − a ) , from which we deduce R = EFG [“ P o1 ”] / EFG [“ P o1 ”] = a − b b − a . (A3)41dopting for this ratio a reasonable guess that is guided by the experimental result fromRef. [1] and that offers numerical simplicity, R = − / , one gets the following analyticaleigenvector compositions Ψ(“ P o1 ”) = √ | P o1 (cid:105) + √ | P o1 (cid:105) , (A4) Ψ(“ P o1 ”) = √ | P o1 (cid:105) − √ | P o1 (cid:105) . In other terms, the ratio EFG [“ P o1 ”] / EFG [“ P o1 ”] only reflects the singlet-triplet mixingin this simple model. We should not be surprised by its relative stability for more elaboratemodels. Extracting the P o character ( a ) from the lowest ( “ P o1 ” ) BP eigenvector obtainedin a simple MR model mixing the { s p s, s p d s, p s } configurations, we get afterrenormalization a = 0 . from which we determine EFG [“ P o1 ”] / EFG [“ P o1 ”] = − . according to EFG [“ P o1 ”] / EFG [“ P o1 ”] = a − b b − a = − a − a − . (A5)For the other BP eigenvector ( “ P o1 ” ), we have a = 0 . from which one confirms theratio EFG [“ P o1 ”] / EFG [“ P o1 ”] = − . . The latter value is not too far from the above R = − / ratio value that would be obtained from the hypothetical ( a = 7 / b = 2 / singlet-triplet mixing, taking into account that (i) one trusts the nonrelativistic ratioEFG[ P o1 ] / EFG [ P o1 ] = − / of the single-configuration approximation, (ii) the BP eigenvec-tor has to be renormalized, and (iii) one assumes no contamination by other LS symmetries( D o1 , P o1 , D o1 , F o1 , D o1 , . . . , X o1 ). [1] D. T. Yordanov, L. V. Rodriguez, D. L. Balabanski, J. Bieroń, M. L. Bissell, K. Blaum,B. Cheal, G. Gaigalas, R. F. Garcia Ruiz, G. Georgiev, W. Gins, M. R. Godefroid, C. Gorges,Z. Harman, H. Heylen, P. Jönsson, A. Kanellakopoulos, S. Kaufmann, C. H. Keitel, V. La-gaki, S. Lechner, B. Maass, S. Malbrunot-Ettenauer, W. Nazarewicz, R. Neugart, G. Neyens,W. Nörtershäuser, N. S. Oreshkina, A. Papoulia, P. Pyykkö, P.-G. Reinhard, S. Sailer,R. Sánchez, S. Schiffmann, S. Schmidt, L. Wehner, C. Wraith, L. Xie, Z.-Y. Xu, and X.-F. Yang, Commun. Phys. , 107 (2020).[2] A. Antognini, N. Berger, T. E. Cocolios, R. Dressler, R. Eichler, A. Eggenberger, P. Indelicato,K. Jungmann, C. H. Keitel, K. Kirch, A. Knecht, N. Michel, J. Nuber, N. S. Oreshkina, A. Ouf, . Papa, R. Pohl, M. Pospelov, E. Rapisarda, N. Ritjoho, S. Roccia, N. Severijns, A. Skawran,S. M. Vogiatzi, F. Wauters, and L. Willmann, Phys. Rev. C , 054313 (2020).[3] M. Cohen and F. Reif, in Solid State Physics Series, Volume 5, edited by F. Seitz and D. Turn-bull (Academic Press, 1957) pp. 321–438.[4] P. Schwerdtfeger, M. Pernpointner, and W. Nazarewicz, inCalculation of NMR and EPR Parameters. Theory and Applications, edited by M. Kaupp,M. Bühl, and V. G. Malkin (Wiley-VCH, Weinheim, 2004) pp. 279–291.[5] E. A. C Lucken, Nuclear Quadrupole Coupling Constants (Academic Press, London and NewYork, 1969).[6] T. Das and E. Hahn, Solid State Physics: Supplement 1: Nuclear Quadrupole Resonance Spectroscopy(Academic Press, Inc., NY, 1958).[7] Mössbauer Spectroscopy, edited by D. Dickson and F. Berry (Cambridge University Press,2005).[8] Y. Chen and D. Yang, Mössbauer Effect in Lattice Dynamics: Experimental Techniques and Applications(Wiley, 2007).[9] H. Haas and D. A. Shirley, J. Chem. Phys. , 3339 (1973).[10] H. Haas, S. P. A. Sauer, L. Hemmingsen, V. Kellö, and P. W. Zhao, EPL , 62001 (2017).[11] P. Raghavan, At. Data Nucl. Data Tables , 189 (1989).[12] N. J. Stone, At. Data Nucl. Data Tables , 1 (2016).[13] P. Pyykkö, Mol. Phys. , 1328 (2018).[14] G. Barone, R. Mastalerz, M. Reiher, and R. Lindh, J. Phys. Chem. A , 1666 (2008).[15] A. Svane, N. E. Christensen, C. O. Rodriguez, and M. Methfessel, Phys. Rev. B , 12572(1997).[16] F. Dimmling, D. Riegel, K.-G. Rensfelt, and C. J. Herrlander, Phys. Lett. B , 293 (1975).[17] G. N. Beloserski, D. Gumprecht, and P. Steiner, Phys. Lett. B , 349 (1972).[18] I. P. Grant, Relativistic Quantum Theory of Atoms and Molecules: Theory and Computation(Springer, New York, 2007).[19] C. Froese Fischer, M. Godefroid, T. Brage, P. Jönsson, and G. Gaigalas,J. Phys. B: At. Mol. Opt. Phys. , 182004 (2016).[20] I. I. Tupitsyn and A. V. Loginov, Opt. Spectrosc. , 319 (2003).[21] I. I. Tupitsyn, V. M. Shabaev, J. R. Crespo López-Urrutia, I. Draganić, R. Soria Orts, and . Ullrich, Phys. Rev. A , 022511 (2003).[22] I. I. Tupitsyn, A. V. Volotka, D. A. Glazov, V. M. Shabaev, G. Plunien, J. R. Crespo López-Urrutia, A. Lapierre, and J. Ullrich, Phys. Rev. A , 062503 (2005).[23] R. Soria Orts, Z. Harman, J. R. Crespo López-Urrutia, A. N. Artemyev, H. Bruhns, A. J.González Martínez, U. D. Jentschura, C. H. Keitel, A. Lapierre, V. Mironov, V. M. Shabaev,H. Tawara, I. I. Tupitsyn, J. Ullrich, and A. V. Volotka, Phys. Rev. Lett. , 103002 (2006).[24] I. Tupitsyn, Dirac-Fock-Sturm method in relativistic calculations for atoms and two-atomic molecules,Ph.D. thesis, Saint-Petersburg State University, Saint-Petersburg (2008), in Russian.[25] I. P. Grant and N. C. Pyper, J. Phys. B: At. Mol. Phys. , 761 (1976).[26] C. Y. Zhang, K. Wang, M. Godefroid, P. Jönsson, R. Si, and C. Y. Chen, Phys. Rev. A ,032509 (2020).[27] P. Jönsson, G. Gaigalas, J. Bieroń, C. Froese Fischer, and I. P. Grant, Comput. Phys. Com-mun. , 2197 (2013).[28] C. Froese Fischer, G. Gaigalas, P. Jönsson, and J. Bieroń, Comput. Phys. Commun. , 184(2019).[29] J. Bieroń, C. Froese Fischer, P. Indelicato, P. Jönsson, and P. Pyykkö, Phys. Rev. A ,052502 (2009).[30] J. Bieroń, C. Froese Fischer, S. Fritzsche, G. Gaigalas, I. P. Grant, P. Indelicato, P. Jönsson,and P. Pyykkö, Phys. Scr. , 054011 (2015).[31] J. Bieroń, P. Jönsson, and C. F. Fischer, Phys. Rev. A , 3547 (1999).[32] P. Jönsson and J. Bieroń, J. Phys. B: At. Mol. Opt. Phys. , 074023 (2010).[33] G. Gaigalas, T. Žalandauskas, and Z. Rudzikas, At. Data Nucl. Data Tables , 99 (2003).[34] G. Gaigalas, C. Froese Fischer, P. Rynkun, and P. Jönsson, Atoms , 1 (2017).[35] M. Rotenberg, Adv. Atom. and Molec. Phys. , 233 (1970).[36] J. Avery and J. Avery, Generalized Sturmians and atomic spectra (World Scientific PublishingCo. Pte. Ltd., 2006).[37] G. W. F. Drake and S. P. Goldman, Adv. Atom. and Molec. Phys. , 393 (1988).[38] I. P. Grant, The Effects of Relativity in Atoms, Molecules, and the Solid State (New York:Plenum, 1991) pp. 17–43, ed S Wilson, I P Grant and B L Gyorffy.[39] R. Szmytkowski, J. Phys. B: At. Mol. Opt. Phys. , 825 (1997).[40] I. Lindgren and A. Rosén, Case Stud. At. Phys. , 93 (1974).
41] I. I. Tupitsyn, N. A. Zubova, V. M. Shabaev, G. Plunien, and T. Stöhlker, Phys. Rev. A ,022517 (2018).[42] V. M. Shabaev, I. I. Tupitsyn, V. A. Yerokhin, G. Plunien, and G. Soff, Phys. Rev. Lett. ,130405 (2004).[43] J. Olsen, B. O. Roos, P. Jørgenssen, and H. J. A. Jensen, J. Chem. Phys. , 2185 (1988).[44] P. Å. Malmqvist, B. O. Roos, and B. Schimmelpfennig, Chem. Phys. Lett. , 230 (2002).[45] P. Jönsson, F. Parpia, and C. Froese Fischer, Comput. Phys. Commun. , 301 (1996).[46] P. Pyykkö and M. Seth, Theor. Chem. Acc. , 92 (1997).[47] A. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, NewJersey, 1957).[48] M. Godefroid and G. Gaigalas, “On the different versions of the Wigner-Eckart reduced matrixelements” , work in progress, July 2020.[49] J. Dembczyński and H. Rebel, Physica B+C , 341 (1984).[50] I. P. Grant, B. J. McKenzie, P. H. Norrington, D. F. Mayers, and N. C. Pyper, Com-put. Phys. Commun. , 207 (1980).[51] S. Verdebout, P. Rynkun, P. Jönsson, G. Gaigalas, C. Froese Fischer, and M. Godefroid,J. Phys. B: At. Mol. Opt. Phys. , 085003 (2013).[52] J. Bieroń, P. Jönsson, and C. Froese Fischer, Phys. Rev. A , 3547 (1999).[53] J. Bieroń, L. Filippin, G. Gaigalas, M. Godefroid, P. Jönsson, and P. Pyykkö, Phys. Rev. A , 062505 (2018).[54] B. Engels, Theor. Chim. Acta , 429 (1993).[55] B. Engels, L. A. Eriksson, and S. Lunell, Adv. At. Mol. Phys. , 297 (1996).[56] P. Jönsson, A. Ynnerman, C. Froese Fischer, M. R. Godefroid, and J. Olsen, Phys. Rev. A , 4021 (1996).[57] M. R. Godefroid, G. Van Meulebeke, P. Jönsson, and C. Froese Fischer, Z. Phys. , 193(1997).[58] J. Bieroń, C. Froese Fischer, P. Jönsson, and P. Pyykkö, J. Phys. B: At. Mol. Opt. Phys. ,115002 (2008).[59] J. Ekman, P. Jönsson, , S. Gustafsson, H. Hartman, G. Gaigalas, M. Godefroid, and C. FroeseFischer, A&A , A24 (2014).[60] J. Ekman, M. R. Godefroid, and H. Hartman, Atoms , 215 (2014).
61] M. L. Bissell, T. Carette, K. T. Flanagan, P. Vingerhoets, J. Billowes, K. Blaum, B. Cheal,S. Fritzsche, M. Godefroid, M. Kowalska, J. Krämer, R. Neugart, G. Neyens, W. Nörtershäuser,and D. T. Yordanov, Phys. Rev. C , 064318 (2016).[62] L. Filippin, J. Bieroń, G. Gaigalas, M. Godefroid, and P. Jönsson, Phys. Rev. A , 042502(2017).[63] J. S. M. Ginges and V. V. Flambaum, Phys. Rep. , 63 (2004).[64] L. Radži¯ut˙e, G. Gaigalas, P. Jönsson, and J. Bieroń, Phys. Rev. A , 012528 (2014).[65] J. Bieroń, P. Jönsson, and C. F. Fischer, Phys. Rev. A , 2181 (1996).[66] J. Bieroń and P. Pyykkö, Phys. Rev. Lett. , 133003 (2001).[67] J. Bieroń, P. Pyykkö, and P. Jönsson, Phys. Rev. A , 012502 (2005).[68] N. Frömmgen, D. L. Balabanski, M. L. Bissell, J. Bieroń, K. Blaum, B. Cheal, K. Flanagan,S. Fritzsche, C. Geppert, M. Hammen, M. Kowalska, K. Kreim, A. Krieger, R. Neugart,G. Neyens, M. M. Rajabali, W. Nörtershäuser, J. Papuga, and D. T. Yordanov, Eur. Phys. J. D , 164 (2015).[69] H. Kopfermann, Nuclear moments (Academic Press Inc, New York, 1958).[70] C. Froese Fischer, T. Brage, and P. Jönsson, Computational Atomic Structure. An MCHF Approach(Institute of Physics Publishing, Bristol and Philadelphia, 1997).[71] P. Pyykkö, J. Magnetic Reson. , 15 (1972).[72] P. Pyykkö, E. Pajanne, and M. Inokuti, Int. J. Quantum Chem. , 785 (1973).[73] W. R. Johnson, Atomic Structure Theory: Lectures on Atomic Physics (Springer, Berlin,2007).[74] D. Sundholm and J. Olsen, Phys. Rev. A , 2672 (1993).[75] T. Carette, C. Drag, O. Scharf, C. Blondel, C. Delsart, C. Froese Fischer, and M. Godefroid,Phys. Rev. A , 042522 (2010).[76] J. Li, P. Jönsson, M. Godefroid, C. Dong, and G. Gaigalas, Phys. Rev. A , 052523 (2012).[77] G. Gaigalas, P. Rynkun, L. Radži¯ut˙e, D. Kato, M. Tanaka, and P. Jönsson, Astrophys. J.,Suppl. Ser. , 13 (2020).[78] A. Kramida, Yu. Ralchenko, J. Reader, and NIST ASD Team, NIST Atomic Spectra Database(ver. 5.7.1), [Online]. Available: https://physics.nist.gov/asd [2020, July 17]. NationalInstitute of Standards and Technology, Gaithersburg, MD. (2019).[79] W. Brill, The Arc Spectrum of Tin, Ph.D. thesis, Purdue University, Lafayette, IN (1964).
80] M. S. Safronova and W. R. Johnson, Adv. At. Mol. Opt. Phys. , 191 (2008).[81] M. S. Safronova and U. I. Safronova, Phys. Rev. A , 052508 (2011).[82] H.-K. Chung, B. J. Braams, K. Bartschat, A. G. Császár, G. W. F. Drake, T. Kirchner,V. Kokoouline, and J. Tennyson, J. Phys. D , 363002 (2016).[83] K. Wang, C. X. Song, P. Jönsson, G. D. Zanna, S. Schiffmann, M. Godefroid, G. Gaigalas,X. H. Zhao, R. Si, C. Y. Chen, and J. Yan, Astrophys. J., Suppl. Ser. , 30 (2018).[84] S. Schiffmann, M. Godefroid, J. Ekman, P. Jönsson, and C. F. Fischer, Phys. Rev. A ,062510 (2020).[85] M. Tong, P. Jönsson, and C. Froese Fischer, Phys. Scr. , 446 (1993).[86] D. K. M. Zong-Chao Yan and G. W. F. Drake, Phys. Rev. A , 1322 (1996).[87] V. A. Yerokhin, Phys. Rev. A , 020501(R) (2008).[88] V. A. Yerokhin, Phys. Rev. A , 012513 (2008).[89] J. Eberz, U. Dinger, G. Huber, H. Lochmann, R. Menges, G. Ulm, R. Kirchner, O. Klepper,T. Kühl, and D. Marx, Z. Phys. A , 121 (1987).[90] P. Jönsson, C.-G. Wahlström, and C. F. Fischer, Comput. Phys. Commun. , 399 (1993).[91] J. Bauche and R.-J. Champeau, Adv. At. Mol. Phys. , 39 (1976).[92] Y. Ishida, H. Iimura, S. Ichikawa, and T. Horiguchi, J. Phys. B: At. Mol. Opt. Phys. , 2569(1997)., 2569(1997).