Energy level structure and transition data of Er 2+
Gediminas Gaigalas, Pavel Rynkun, Laima Radžiūtė, Daiji Kato, Masaomi Tanaka, Per Jönsson
DDraft version February 19, 2020
Typeset using L A TEX twocolumn style in AASTeX61
ENERGY LEVEL STRUCTURE AND TRANSITION DATA OF E r Gediminas Gaigalas, Pavel Rynkun, Laima Radˇzi¯ut ˙e, Daiji Kato,
2, 3
Masaomi Tanaka, and P. J¨onsson Institute of Theoretical Physics and Astronomy, Vilnius University, Saul˙etekio Ave. 3, Lithuania National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, Japan Department of Advanced Energy Engineering, Kyushu University, Kasuga, Fukuoka 816-8580, Japan Astronomical Institute, Tohoku University, Sendai 980-8578, Japan Group for Materials Science and Applied Mathematics, Malm¨o University, SE-20506, Malm¨o, Sweden (Received February 19, 2020; Revised; Accepted)
Submitted to ApJSABSTRACTThe main aim of this paper is to present accurate energy levels of the ground [Xe]4 f and first excited [Xe]4 f d configurations of Er . The energy level structure of the Er ion was computed using the multiconfiguration Dirac-Hartree-Fock and relativistic configuration interaction (RCI) methods, as implemented in the GRASP2018 programpackage. The Breit interaction, self-energy and vacuum polarization corrections were included in the RCI computations.The zero-first-order approach was used in the computations. Energy levels with the identification in LS coupling for all(399) states belonging to the [Xe]4 f and [Xe]4 f d configurations are presented. Electric dipole (E1) transition databetween the levels of these two configurations are computed. The accuracy of the these data are evaluated by studyingthe behaviour of the transition rates as functions of the gauge parameter as well as by evaluating the cancellationfactors. The core electron correlations were studied using different strategies. Root-mean-square deviations obtainedin this study for states of the ground and excited configurations from the available experimental or semi-empiricaldata are 649 cm − , and 747 cm − , respectively. Keywords: atomic data, radiative transfer, opacity, chemically peculiar stars, neutron stars
Corresponding author: Gediminas [email protected] author: Pavel [email protected] a r X i v : . [ phy s i c s . a t o m - ph ] F e b Gaigalas et al. INTRODUCTIONErbium is a lanthanide element with Z = 68 and ithas 6 stable isotopes. The isotopes are generated by dif-ferent processes. Isotopes with A = 162 are producedby the p process (proton capture), with A = 167 , r process (rapid neutron capture), with A = 164by the p or the s process (slow neutron capture) andwith A = 166 ,
168 by the r or the s process (Jaschek& Jaschek 1995). Since Er can be generated by the r process, which can occur in the mergers of neutron star(NS), the atomic spectra of this element is of interest toa wide community of astrophysicists dealing with stellarnuclear synthesis. The contribution of this element tothe opacity of NS ejecta should be tested (e.g., Kasenet al. 2017; Tanaka et al. 2018, 2019), but even the en-ergy levels of first excited configuration have not beenfully presented.Ions of erbium have been observed in different typesof stars. In the chemically peculiar (CP) stars, highabundances of lanthanide elements compared with so-lar values are observed. In particular, Er III has beenidentified in the spectra of CP stars of the upper mainsequence (in the silicon star HD 192913 by Cowley &Crosswhite (1978); in the CP A star HR 465 by Cowley& Greenberg (1987)). Cowley & Mathys (1998) haveidentified lines in the range 5445-6587 ˚A in spectra ofthe extreme peculiar star HD 101065 (Przybylski’s star).In such stars the strongest spectral lines belong to thelanthanides rather than the iron group elements. In theabove spectral range lines of Er III at λ f s configuration.Bi´emont et al. (2001) have measured radiative life-times of seven excited states of the 4 f p configura-tion using time-resolved laser-induced fluorescence fol-lowing two-photon excitation. Theoretical computationwas done in frame of relativistic Hartree-Fock includingcore-polarization effects.The aim of this paper is to provide accurate calcula-tions of Er III, which can contribute to the stellar spec-troscopy and understanding of opacities in NS merg-ers. All levels of the ground [Xe]4 f and first ex- cited [Xe]4 f d configurations of Er are analysed inthis paper. Different core correlation effects and theirinclusion strategies are presented. The energy levelsof these configurations and the corresponding electricdipole (E1) transition parameters were computed usingthe GRASP2018 (Fischer et al. 2019) package. Com-putations are based on the multiconfiguration Dirac-Hartree-Fock (MCDHF) and relativistic configurationinteraction (RCI) methods. The zero-first-order methodwas tested for various cases. GENERAL THEORY2.1.
Computational procedure
The MCDHF method used in the present paper isbased on the Dirac-Coulomb (DC) Hamiltonian H DC = N (cid:88) i =1 (cid:0) c α i · p i + ( β i − c + V Ni (cid:1) + N (cid:88) i>j r ij , (1)where V N is the monopole part of the electron-nucleusCoulomb interaction, α and β are the 4 × c is the speed of light in atomic units. Theatomic state functions (ASFs) were obtained as linearcombinations of symmetry adapted configuration statefunctions (CSFs)Ψ( γP JM ) = N CSFs (cid:88) j =1 c j Φ( γ j P JM ) . (2)Here J and M are the angular quantum numbers and P is parity. γ j denotes other appropriate labeling of theconfiguration state function j , for example orbital oc-cupancy and coupling scheme. Normally the label γ ofthe atomic state function is the same as the label of thedominating CSF, see also section 2.3. For these calcu-lations the spin-angular approach (Gaigalas & Rudzikas1996; Gaigalas et al. 1997), which is based on the secondquantization in coupled tensorial form, on the angularmomentum theory in three spaces (orbital, spin, andquasispin) and on the reduced coefficients of fractionalparentage, was used. It allows us to study configurationswith open f -shells without any restrictions. The CSFsare built from products of one-electron Dirac orbitals.Based on a weighted energy average of several states, theso-called extended optimal level (EOL) scheme (Dyallet al. 1989), both the radial parts of the Dirac orbitalsand the expansion coefficients were optimized to self-consistency in the relativistic self-consistent field proce-dure (Fischer et al. 2016).2.2. Zero-first-order method
ASTEX Energy level structure and transition data of Er P ), which contains CSFs that ac-count for the major parts of the wave functions and isreferred to as a zero-order partitioning;ii) an orthogonal complementary part ( Q ), which con-tains CSFs that represent minor corrections and is re-ferred to as a first-order partitioning.Interaction between P and Q is assumed to be thelowest-order perturbation. The total energy functionalis partitioned into the zero-order part ( H (0) ) and theresidual part ( V ). The Dirac-Fock energy functional ischosen as the zero-order part; the residual part thenrepresents a correlation energy functional. The second-order Brillouin-Wigner perturbation theory then leadsto, ( E − H (0) QQ ) − V QP Ψ P = Ψ Q , (cid:104) H (0) P P + V P P + V P Q ( E − H (0) QQ ) − V QP (cid:105) Ψ P = E Ψ P . (3)The above equations define the first-order correlationoperator and the second-order effective Hamiltonian op-erator for the P -space, respectively. In the brackets ofthe second equation, the first and second terms composethe total energy functional in the P -space, and the thirdterm represents the second-order correction to the corre-lation energy functional in the P -space. The non-lineareffective Hamiltonian equation is written in a linearizedform, H (0) P P + V P P V P Q V QP H (0) QQ Ψ P Ψ Q = E Ψ P Ψ Q . (4)The requirement that the total energy functional ( E )is stationary with respect to variations in spin-orbitals( { φ } ) under the normalization and the orthogonalityconditions leads to a set of the Euler-Lagrange equa-tions, δE [ { φ } ] δφ a = µ a φ a + (cid:88) b (cid:54) = a µ ab φ b , (5)where { µ } are the Lagrange multipliers. The aboveequations are nothing but reduced MCDHF equa-tions. That is to say, an apparent connection betweenthe second-order Brillouin-Wigner perturbation energyfunctional and a set of reduced MCDHF equations isprovided.Block H (0) QQ is diagonal in the Hamiltonian matrix (eq.4). As a result, computation time and size required forthe construction of the Hamiltonian matrix are reduced. This method, named as zero-first-method (ZF), has thepotential for taking a very large configuration space intoaccount, which is almost unachievable by full MCDHFand RCI methods, and for allowing accurate calculationto be performed with relatively small computational re-sources provided the Q -space contributes perturbativelyto the P -space.2.3. Relativistic configuration interaction method
The RCI method taking into account Breit and quan-tum electrodynamic (QED) corrections (Grant 2007;Fischer et al. 2016), was used in the computations. Thetransverse photon interaction (Breit interaction) H Breit = − N (cid:88) i Computation of transition parameters The evaluation of radiative electric dipole (E1) transi-tion data (transition probabilities, oscillator strengths)between two states: γ (cid:48) P (cid:48) J (cid:48) M (cid:48) and γP JM , built on dif-ferent and independently optimized orbital sets is non-trivial. The transition data can be expressed in termsof the transition moment, which is defined as (cid:104) Ψ( γP J ) (cid:107) T (1) (cid:107) Ψ( γ (cid:48) P (cid:48) J (cid:48) ) (cid:105) = (cid:88) j,k c j c (cid:48) k (cid:104) Φ( γ j P J ) (cid:107) T (1) (cid:107) Φ( γ (cid:48) k P (cid:48) J (cid:48) ) (cid:105) , (7)where T (1) is the transition operator. The calculationof the transition moment breaks down to the task ofsumming up reduced matrix elements between differentCSFs. The reduced matrix elements can be evaluatedusing standard techniques assuming that both left andright hand CSFs are formed from the same orthonor-mal set of spin-orbitals. This constraint is severe, sincea high-quality and compact wave function requires or-bitals optimized for a specific electronic state, for anexample, see (Fritzsche & Grant 1994). To get around Gaigalas et al. the problems of having a single orthonormal set of spin-orbitals, the wave function representations of the twostates, i.e. γ (cid:48) P (cid:48) J (cid:48) M (cid:48) and γP JM were transformed insuch way that the orbital sets became biorthonormal(Olsen et al. 1995). Standard methods were then used toevaluate the matrix elements of the transformed CSFs.The reduced matrix elements are expressed via spin-angular coefficients d (1) ab and operator strengths as: (cid:104) Φ( γ j P J ) (cid:107) T (1) (cid:107) Φ( γ (cid:48) k P (cid:48) J (cid:48) ) (cid:105) = (cid:88) a,b d (1) ab (cid:104) n a l a j a (cid:107) T (1) (cid:107) n b l b j b (cid:105) . (8)Allowing for the fact that we are now using Brink-and-Satchler type reduced matrix elements, we have (cid:104) n a l a j a (cid:107) T (1) (cid:107) n b l b j b (cid:105) = (cid:18) (2 j b + 1) ωπc (cid:19) / ( − j a − / j a j b − M ab (9)where M ab , is the radiative transition integral defined byGrant (1974). The latter integral can be written M ab = M eab + GM lab , where G is the gauge parameter. When G = 0 we get the Coulomb (velocity) gauge, whereas for G = √ G axis) (Rudzikas2007; Gaigalas et al. 2010). This dependence may alsobe used for the evaluation of the accuracy of the results.The more accurate the wave functions, the closer theparabola is to a straight line.For electric dipole transitions the Babushkin andCoulomb gauges give the same value of the transi-tion moment for exact solutions of the Dirac-equation(Grant 1974). For approximate solutions the transitionmoments differ, and the quantity dT , defined as (Ekmanet al. 2014) dT = | A l − A v | max( A l , A v ) , (10)where A l and A v are transition rates in length and ve-locity form, can be used as a measure of the uncertaintyof the computed rate.In the present work also the cancellation factor (CF),which shows cancellation effects in the computation oftransition parameters was investigated. The cancella-tion factor is defined as (Cowan 1981; Zhang et al. 2013) CF = (cid:12)(cid:12)(cid:12)(cid:80) j (cid:80) k c j (cid:104) Φ( γ j P J ) (cid:107) T (1) (cid:107) Φ( γ (cid:48) k P (cid:48) J (cid:48) ) (cid:105) c (cid:48) k (cid:12)(cid:12)(cid:12)(cid:80) k (cid:80) j (cid:12)(cid:12) c j (cid:104) Φ( γ j P J ) (cid:107) T (1) (cid:107) Φ( γ (cid:48) k P (cid:48) J (cid:48) ) (cid:105) c (cid:48) k (cid:12)(cid:12) . (11)To calculate CFs some modifications to the GRASP2018(Fischer et al. 2019) package were done. A small value of the CF, for example less than 0.1 or 0.05 (values aregiven in (Cowan 1981)), indicates that the calculatedtransition parameter, such as transition rate or oscilla-tor strength, is affected by a strong cancellation effect.Transition parameters with small CF are often associ-ated with large uncertainties. COMPUTATIONAL STRATEGIESThe study of the Er ion, as well as of the other lan-tanides, is quite a complex task because of the open f shells. For systems with open f shells, the number ofCSFs increases very rapidly when including various elec-tron correlation effects. Computations for such systemsusing standard schemes are extremely demanding. Forthis reason new computational strategies were developedand tested for Er .To obtain good wave functions, various electron cor-relation effects were investigated. The ZF method wasapplied to reduce computational resources in differentsteps of the calculations and to facilitate the inclusionof more electron correlation effects. The final wave func-tions were used to compute electric dipole (E1) transi-tion data between the levels of the two configurations.The computational strategies will be discussed in moredetails in the sections below.3.1. Generation of initial wave functions and activespace construction The first step of the wave function generation was anMCDHF computation of the [Xe]4 f configuration. Inthe second step, orbitals from the first step were keptfrozen and used for the [Xe]4 f d configuration, forwhich only the 5 d orbitals (5 d + and 5 d − in relativisticnotation) were optimized. In the tables, such an initialcomputation in two steps will be referred to as a compu-tation for the multireference (MR) space of CSFs. Theorbitals belonging to the [Xe]4 f configuration werekept frozen to get correct order for the states of theground and excited configurations. A similar techniquefor the generation of the initial wave functions was al-ready applied for Nd ions (Gaigalas et al. 2019).In the following steps of the computation, activespaces (AS) of CSFs were generated by allowing single-double (SD) or single-restricted-double (SrD) sub-stitutions from only the valence shells or from va-lence and core shells of both configurations to the or-bital spaces (OS): OS = { s, p, d, f } , ..., OS = { s, p, d, f, g, h } . When a new OS is being com-puted, the previous orbitals are frozen. In Table 1 thenumber of CSFs used in the computations for the evenand odd states is given. The strategies mentioned inthis Table will be described below in greater detail. ASTEX Energy level structure and transition data of Er Table 1. Summary of Active Space Constructions forthe MCDHF and RCI Computations.No. of CSFsStrategy and AS Even Odd ZF SD 4f AS 25 618 407 606 PAS 115 146 2 414 665 P + QAS 326 187 7 986 088 P + QAS 649 673 16 859 203 P + Q SD 5d AS 25 618 538 902 PAS 115 146 2 868 718 P + QAS 326 187 8 958 563 P + QAS 649 673 18 527 744 P + Q SD 5p AS 369 343 11 769 255 SD 5s AS 193 028 4 745 781 SrD 5p 5d AS 337 325 10 720 590 SD 5s 5d AS 193 028 5 584 829 SrD 5s 5p 5d AS 414 383 13 402 965 SD 5s 5p 5d AS 476 274 19 482 860 Note —The number of CSFs for the even and odd pari-ties are given for each computational strategy and AS. Valence-valence electron correlations Two strategies for including valence-valence (VV)electron correlations were investigated. In the first, the SD 4f strategy, the orbitals of which were used in allother strategies ( SD 5d , SD 5p , SD 5s , SrD 5p 5d , SD 5s 5d , SrD 5s 5p 5d , SD 5s 5p 5d for these onlyRCI computations were performed), SD substitutionswere allowed only from the 4 f valence shell of bothconfigurations to the different orbital spaces. Later,separate computations were done for AS for the evenand odd parities and continued for the AS , built fromthe OS orbital space. In the second strategy, the SD5d strategy, SD substitutions were allowed from bothvalence (4 f and 5 d ) shells to the different orbital spaces.Results of these investigations are presented in Table 2and will be discussed in section 4.1.3.3. Core-valence and core-core electron correlations The contribution of core-valence (CV) and core-core(CC) electron correlation effects to the energy levels wasstudied in RCI calculations by allowing SD or SrD sub-stitutions from core (5 p, s ) shells. Results of these com-putations are presented in Table 3. The orbital spacesare the same as described in section 3.1. The columnlabeling is similar, for example, the notation SD 5p means that SD substitutions were done from the 4 f and5 p shells. In some computational schemes restrictionsfor the substitutions were applied. SrD substitutions inthe SrD 5p 5d strategy mean that SD substitutionswere done from the 4 f and 5 d shells, but from the 5 p shell only S substitutions were allowed. In the SrD 5s5p 5d strategy restrictions are applied to the 5 s and 5 p shells by allowing only S substitutions from these shells.A summary of the active spaces of the different strate-gies, including core-valence and core-core electron corre-lation, is displayed in Table 1. From the Table it is seenthat substitutions from core shells rapidly increase thenumber of CSFs. The contribution of these correlationseffects to energy levels will be presented in Section 4.2.3.4. Electron correlations using the zero-first-ordermethod The ZF method was applied to the SD 4f and SD5d strategies and tested at different steps of the compu-tations to reduce the computational load. These resultsare presented in Tables 4 and 5. Firstly, the ZF methodwas applied to the MCDHF calculation in the SD 4f strategy for AS . The results of these calculations, per-formed separately for the even and odd configurations,are marked as ZF MCDHF . For the AS , , active spacesthe AS space was used as the principal ( P ) part. Theprincipal part was selected based on the convergence ofthe energies, see section 4.1. The sizes of the P and P + Q spaces used in the calculations are given in Table1. Orbitals from the SD 4f ZF MCDHF strategy wereused in the RCI calculations for the SD 4f ZF MCDHFRCI , SD 5d ZF MCDHF , and SD 5d ZF MCDHFRCI strategies.The ZF approach was also used in the RCI calcula-tions. The results are displayed in Tables 4, 5 and re-ferred to as ZF RCI . The last columns of the Tablespresent the results of RCI computations using the ZFmethod based on orbitals from the ZF MCDHF calcula-tions. These results are referred to as ZF MCDHFRCI . ENERGY LEVELS RESULTSParts of the computed energy spectra from differentstrategies (described in section 3) are presented in Ta-bles 2 - 6. The labels of the energy levels are given in LS notation which are taken from NIST (Kramida et al.2019), or ordered by energy values for fixed J value Gaigalas et al. (POS). The notation 4 f N (2 S +1) L Nr n (cid:48) l (cid:48) (2 S (cid:48) +1) L (cid:48) isused for the level labels. Intermediate quantum num-bers define parents levels 4 f N (2 S +1) L Nr , where N is electron number in the 4 f shell, (2 S + 1) is mul-tiplicity, N r is a sequential index number represent-ing the group labels νW U for the term, and L is or-bital quantum number (see Gaigalas et al. (1998) inmore details). Energies in parentheses are from semi-empirical (SE) calculations by Wyart et al. (1997). Thetotal amount of energy levels presented in the NIST(Kramida et al. 2019) database and in the paper (Wyartet al. 1997) for the ground and first excited configu-ration is only 64. The accuracy of computed energyspectra was evaluated by comparing results with theNIST/(SE) data and calculating the relative difference∆ E/E = ( E NIST/ ( SE ) − E ) /E NIST/ ( SE ) .4.1. Convergence and valence-valence electroncorrelations Table 2 displays the results when just VV correlations( SD 4f and SD 5d strategies) are included. Using the SD 4f strategy we infer that the wave function relax-ation for AS , resulting from separate computations forthe even and odd parities, in comparison to the compu-tations where the even and odd parities are computedtogether, has small effect on the energy levels. It mod-erately increases the transition energy value by 0.15%(0.09% for levels of ground configuration and 0.15% forlevels of excited configuration). For this comparison all399 levels were included.The convergence of the obtained energies was eval-uated by the following equation ∆ E/E = ( E AS N − − E AS N ) /E AS N − . The relative difference (∆ E/E = (cid:80) | ∆ E i /E i | N ) between active space AS and AS usingthe SD 4f strategy (when all 399 levels are included) isabout 2.6%. By analyzing the results we observe thatenergies for some J values converge much faster thanfor others. This is seen from Figure 1, where the con-vergence for the lowest states of the 4 f d configurationwith J = 0 − 11 is presented. For example, the differencebetween AS and AS for J = 0 is about 5% while forlowest state with J = 6 it reaches 13%. After the stud-ies of energy levels with different J values, we observedthat the lower energy levels converge much slower thanthe higher energy levels for a fixed J value (see Figure2). From the Figure we see that even the third level con-verges much faster than the first one and the agreementbetween the energies for the last two active spaces is upto 0.3%. In conclusion, the active space has inconsider-able influence on the higher levels as compared to thelowest ones. The upper levels converge much faster. (cid:1) E/E ,% J M R - A S A S - A S A S - A S Figure 1. Convergence of the lowest states of the 4 f d configuration with J = 0 − 11 in the energy spectrum ( SD4f strategy). (cid:1) E/E ,% n u m b e r o f l e v e l w i t h J = 6 M R - A S A S - A S A S - A S Figure 2. Convergence of all energy values of the 4 f d configuration with J = 6 ( SD 4f strategy). The lowest levels according to Hund’s first rule havethe largest multiplicity. For a given set of eigenstates,the lowest state will have largest multiplicity. Almostall the lowest levels for each J in case of the 4 f d con-figuration have the largest multiplicity (except J = 9),and all these levels converge slower than the higher ones(as it can be seen from Figures 2 and 4). However, evenin the set of levels with the largest multiplicity, a largedifferences in convergence is observable (see Figures 1and 3). From these Figures it can also be seen that theCSFs from the AS (black squares) have the largest in-fluence. The first active space has a larger influence onenergy levels in the SD 5d strategy than in the SD 4f strategy. ASTEX Energy level structure and transition data of Er Table 2. Energy Levels from RCI Calculations Using the SD 4f and SD 5d Strategies. LS POS JP NIST/(SE) SD 4f SD 5d OrthogonalMR AS AS AS AS AS AS f 12 3 H f 12 3 F − − − − f 12 3 H f 12 3 H − − − − f 12 3 F − − − − f 12 3 F − − − − f 12 1 G f 11 (4 I 1) 5 d G − − f 11 (4 I 1) 5 d H − − f 11 (4 I 1) 5 d L − − f 11 (4 I 1) 5 d I − − f 11 (4 I 1) 5 d L − − f 11 (4 I 1) 5 d K − − f 11 (4 I 1) 5 d G − − f 11 (4 I 1) 5 d H − − f 11 (4 I 1) 5 d I − − f 11 (4 I 1) 5 d I − − f 11 (4 I 1) 5 d L − − f 11 (4 I 1) 5 d H − − f 11 (4 I 1) 5 d K − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − Note —The relative difference compared with NIST/(SE) data is given in percent. Gaigalas et al. (cid:1) E/E, % M R - A S A S - A S A S - A S A S - A S A S - A S A S - A S J Figure 3. Convergence of the lowest states of 4 f d config-uration with J = 0 , 11 in the energy spectrum. Results areobtained using the SD 5d strategy (open symbols mark theresults when the ZF MCDHF approach at AS , , is applied). M R - A S A S - A S A S - A S A S - A S A S - A S A S - A S (cid:1) E/E, % n u m b e r o f l e v e l w i t h J = 6 Figure 4. Convergence of all energy values of the 4 f d configuration with J = 6. Results are obtained using the SD 5d strategy (open symbols mark the results when the ZF MCDHF approach at AS , , is applied). The results of the SD 4f strategy substantially dis-agree with NIST/(SE) data (see Table 2) for states of the4 f d configuration, and after adding one more layer( AS ) to the computations, the disagreement increases.From the Table it is seen that after including substitu-tions from the 5 d shell ( SD 5d strategy) the resultsagree much better. The averaged uncertainty of ob-tained results from the SD 5d strategy at AS is around5.6% comparing with NIST or SE data. By studying the convergence of the results obtained using the SD5d strategy we see similar trends as those from the SD4f strategy. Firstly, energies for different J values con-verge differently. Secondly, lower energy levels convergemuch slower than the higher energy levels. But in caseof the SD 5d strategy the energies converge much fastercomparing with the SD 4f strategy (see Figures 3 and4). For example, the difference between AS and AS for J = 0 is about 2.3% and 9.5% for lowest state with J = 6.4.2. Studies of core-valence and core-core electroncorrelations The investigations of core-core and core-valence elec-tron correlations contributions to the transition energiesare presented in Table 3. From the Table it is seen thatby including substitutions just from the valence shell(4 f ) and core shells (5 p ) or (5 s ) ( SD 5p or SD 5s strat-egy) the results are in worse agreement with NIST/(SE).In case of the SD 5p strategy this disagreement is verylarge. The relative difference compared with NIST/(SE)data is reduced when substitutions from 4 f , 5 d and 5 p or 5 s shells are allowed. The averaged uncertainty of theobtained results from strategies SrD 5p 5d , SD 5s 5d , SrD 5s 5p 5d , SD 5s 5p 5d is similar, around 5-7%comparing with NIST or SE data. As was mentionedabove, inclusion of the substitutions from the core shells(5 p or 5 s ) increases the number of CSFs dramatically(see Table 1). So for further investigations substitutionsfrom the 5 p and 5 s shells were neglected.4.3. Optimal strategy for electron correlations The SD 5d strategy was chosen as the optimal strat-egy considering achieved accuracy of the results andthe computational resources needed for the calculations.The main goal of this work is to obtain accurate energylevels of the ground and first excited configurations ofEr . So we give priority to balanced electron correla-tion effects which improves the energy separations.4.4. Impact of the zero-first-order method The ZF method was applied at different stages of thecalculations to reduce computation resources, as it wasdescribed in Section 3.4. The impact of the ZF methodwas studied using the SD 4f and SD 5d strategies.In the investigations of the effect of ZF on the energylevels all 399 states were included. The zero-first-ordermethod (see SD 4f ZF MCDHF column in Table 4) hasup to 0.08% impact on the values of the energy levelsat AS if all levels are compared. From Table 5 we seethat the ZF method for MCDHF calculations (see SD5d ZF MCDHF column) affects on average the values of ASTEX Energy level structure and transition data of Er Table 3. Energy Levels from RCI Calculations Including CV and CC Electron Correlations. LS POS JP NIST/(SE) SD 5p SD 5s SrD 5p 5d SD 5s 5d SrD 5s 5p 5d SD 5s 5p 5d AS AS AS AS AS AS f 12 3 H f 12 3 F − − − − − − f 12 3 H f 12 3 H − − − − − − f 12 3 F − − − − − − f 12 3 F − − − − − − f 12 1 G f 11 (4 I 1) 5 d G − − − f 11 (4 I 1) 5 d H − − − f 11 (4 I 1) 5 d L − − − f 11 (4 I 1) 5 d I − − − f 11 (4 I 1) 5 d L − − − f 11 (4 I 1) 5 d K − − − f 11 (4 I 1) 5 d G − − − f 11 (4 I 1) 5 d H − − − f 11 (4 I 1) 5 d I − − − f 11 (4 I 1) 5 d I − − − f 11 (4 I 1) 5 d L − − − f 11 (4 I 1) 5 d H − − − f 11 (4 I 1) 5 d K − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − Note —The relative difference compared with NIST/(SE) data is given in percent. Gaigalas et al. the energy levels at AS by 0.29%, and in some cases upto 1.01% .The application of the ZF method for the RCI compu-tation only (see SD 4f ZF RCI column in Table 4), has alarger influence on the energy levels; it is up to 2.84% at AS and 1.61% in average for all states. Using the SD5d ZF RCI strategy (see Table 5) the contribution of ZFin RCI is up to 2.69% at AS and 1.45% in average forall states.When the ZF method was applied for the RCI compu-tations using orbitals from SD 4f ZF MCDHF the ener-gies changed in average about 0.5% ( SD 4f ZF MCDHFRCI )and up to 2.39% for some levels. Using the SD 5dZF MCDHFRCI strategy (see Table 5) the influence of theZF method is up to 3.95% at AS and 0.66% in averagefor all states.From the above study we infer that the impact of theZF order method on the energy levels is very small inself consistent field computations for both strategies. Inthe case of the SD 5d ZF MCDHF strategy, the effecton the energy levels at AS is only 0.29%.4.5. Final results Based on the analysis made in previous sections, the SD 5d strategy was chosen as the optimal strategy.Therefore this strategy with the orbitals taken from the ZF MCDHF strategy was used to continue computationsin AS basis. The final results of the present work aredisplayed in Table 6 together with NIST and SE data.In first column of the Table we give identifications ofenergy levels in LS or JJ (see definition in Gaigalas(2020), Eq. (10) and (16)) coupling from our computa-tions, in second column identifications of energy levelsare from Wyart et al. (1997). Labels in LS couplingagree with identification given in the NIST database.Labels in JJ coupling are given only for the part ofthe energy spectra that is used for the comparison withthe results of Wyart et al. The averaged uncertaintyof the computed energy levels is 5.24%, 2.68%, respec-tively for states of the ground and excited configura-tions (see Table 6 SD 5d ZF MCDHF strategy AS ).Root-mean-square (rms) deviations of these results forstates of the ground and excited configurations from theNIST/(SE) data are 649 cm − , and 1571 cm − , respec-tively. If the ZF method is used in both the MCDHFand RCI calculations ( SD 5d ZF MCDHFRCI strategy) theobtained data are in worse agreement (moderately about7%) with NIST or SE data.Figure 5 displays the differences between the NIST/(SE)energies and final results of the present study. As itcan be seen from Figure 5 and Table 6 (energy levelsmarked in gray color), there is a significant disagree- ment between states with the following identifications J =4 Pos=10, J =5 Pos 12, J =5 Pos=13, J =7 Pos=7,and J =2 Pos=1. Energy differences exceed 2000 cm − for these five energy levels. It is highly probable thatthe obtained differences result from incorrect orderingand incomplete identification of energy levels presentedby Wyart et al. (1997). Only for one level ( J =7 Pos=7)from the five above mentioned levels Wyart et al. (1997)give identification in JJ coupling, for the four othersonly configurations are given. The level is identified as F / d / ( J =7). We have transformed ASFs from LS to JJ coupling using the Coupling program devel-oped by Gaigalas (2020). The level J =7 Pos=7 has the4 f ( I / ) 5 d / (9 / , / 2) label in JJ coupling whichdisagree with Wyart et al. By looking at levels whichmatch the identification given by Wyart et al. we seethat there is a fit for J =7 Pos=8 with identification4 f ( F / ) 5 d / (9 / , / − . Bycomparing the labels of the levels for which Wyart et al.gives the full identification with our identification in JJ coupling, the labels from both studies agree except forthe levels (namely J =4 Pos=8, J =6 Pos 5, J =6 Pos=10, J =5 Pos=3, and J =3 Pos=1). Level J =4 Pos=8 inthe present work has the 4 f ( F / ) 5 d / (7 / , / J =6 Pos 5 – 4 f ( I / ) 5 d / (9 / , / J =6 Pos=10 – 4 f ( F / ) 5 d / (9 / , / J =5 Pos=3– 4 f ( I / ) 5 d / (15 / , / J =3 Pos=1 –4 f ( I / ) 5 d / (11 / , / J =3 Pos=1 is incorrect. That level was assigned as I / d / but such a label for J =3 is not consistentwith the selection rules. The deeper analysis of un-certainties estimation is complicated because completeidentification of energy levels was not given in the paperby Wyart et al. (1997).The full energy spectrum (energy levels for 399 states)with unique labels and with atomic state function com-position in LS coupling using the SD 5d ZF MCDHF strategy is presented in machine-readable format in Ta-ble 8. TRANSITION DATA RESULTSThe wave functions from the SD 5d and SD 5dZF MCDHF strategies, which were chosen as the optimal ASTEX Energy level structure and transition data of Er Table 4. Energy Levels from RCI Calculations Using the ZF Approach in Different Steps of the Calculations( SD 4f Strategy). LS POS JP NIST/(SE) SD 4f SD 4f ZF MCDHF SD 4f ZF RCI SD 4f ZF MCDHFRCIAS AS AS AS AS AS AS AS AS f 12 3 H f 12 3 F f 12 3 H f 12 3 H f 12 3 F f 12 3 F f 12 1 G f 11 (4 I 1) 5 d G f 11 (4 I 1) 5 d H f 11 (4 I 1) 5 d L f 11 (4 I 1) 5 d I f 11 (4 I 1) 5 d L f 11 (4 I 1) 5 d K f 11 (4 I 1) 5 d G f 11 (4 I 1) 5 d H f 11 (4 I 1) 5 d I f 11 (4 I 1) 5 d I f 11 (4 I 1) 5 d L f 11 (4 I 1) 5 d H f 11 (4 I 1) 5 d K Gaigalas et al. N I S T S E ( W y a r t e t a l . 1 9 9 7 ) E NIST/(SE) - E comp., cm-1 E N I S T / ( S E ) , c m - 1 Figure 5. A comparison of energy levels between the NISTor SE values Wyart et al. (1997) and results of the presentstudy. The dashed arrows indicate the improved agreementresulting from a re-identification of the levels in Wyart et al.(1997), see text for details. computational schemes, were used to compute E1 transi-tion data between states of the [Xe]4 f and [Xe]4 f d configurations. The accuracy of the transition data ob-tained in this work was evaluated by:1. calculating parameter dT , which shows the dis-agreement between the length and velocity formsof the computed transition rates;2. analyzing the convergence of the computed tran-sition rates in the length and velocity forms;3. analyzing the dependence of the transition rate onthe gauge parameter G ;4. analyzing the dependence of cancellation factor onthe gauge parameter G ;5. comparing computed transition data with otherexperimental or theoretical calculations.For these investigations a few strong transitions havebeen chosen as examples. The evaluation of transitiondata will be presented in the sections below.Computed transition data, such as wavelengths,weighted oscillator strengths, transition rates of E1along with the accuracy indicator dT , are given inmachine-readable format in Table 9.5.1. Disagreement between the length and velocity andtheir convergence In a variational approach the wave functions are op-timized on an energy expression. In general this givesa better representation of the outer part of the wave functions, thus favoring the length form. The velocityform contains a dependence on the transition energy inthe matrix element, which may affect the accuracy ofthe evaluation. Due to the above mentioned reasons,a much slower convergence of the velocity gauge is ex-pected (Ynnerman & Fischer 1995). However, a recentpaper by Papoulia et al. (2019), analyzing in detail theconvergence properties of transitions in light elements,suggests that transition probabilities in the Coulombgauge may give the more accurate values. Thus, it isimportant to systematically study the transition data tosee which gauge results in the most rapid convergence.The convergence of the transition rates in both gaugeswith the increasing active spaces is presented in Figures6 and 7. From these Figures it is seen that transitionprobabilities in the Babushkin gauge are more stable toelectron correlation effects than the probabilities in theCoulomb gauge. The dT for the analyzed transitionsbased on the final AS in the SD 5d ZF MCDHF strat-egy are 12% for 4 f 12 3 P – 4 f ( F ) 5 d P , 23%for 4 f 12 1 S – 4 f ( F ) 5 d P (Figure 6); 3% for4 f 12 3 P – 4 f ( F ) 5 d P and 5% for4 f 12 3 P –4 f ( F ) 5 d P (Figure 7).Analyzing the impact of the ZF method on the tran-sition rates, we see that ZF MCDHF AS reduces transi-tion rates compared to those from the SD 5d strategy.The transition rates in Coulomb gauge change even morethan those in the Babushkin gauge. Transition rates inBabushkin gauge decreases just by a few percent for theanalyzed transitions. The above analysis shows that theBabushkin gauge is the preferred one. A x 108, s-1 M R A S A S A S A S A B A B ( Z F ) A C A C ( Z F ) A B A B ( Z F ) A C A C ( Z F ) Figure 6. Convergence of E1 transition probabilities us-ing the SD 5d strategy (open symbols mark the resultswhen the ZF MCDHF approach is applied). The 4 f 12 3 P – 4 f ( F ) 5 d P transition is marked in black and the4 f 12 1 S – 4 f ( F ) 5 d P transition in red. ASTEX Energy level structure and transition data of Er Table 5. Energy Levels from RCI Calculations Using the ZF Approach in Different Steps of the Calculations( SD 5d Strategy). LS POS JP NIST/(SE) SD 5d SD 5d ZF MCDHF SD 5d ZF RCI SD 5d ZF MCDHFRCIAS AS AS AS AS AS AS AS f 12 3 H f 12 3 F f 12 3 H f 12 3 H f 12 3 F f 12 3 F f 12 1 G f 11 (4 I 1) 5 d G f 11 (4 I 1) 5 d H f 11 (4 I 1) 5 d L f 11 (4 I 1) 5 d I f 11 (4 I 1) 5 d L f 11 (4 I 1) 5 d K f 11 (4 I 1) 5 d G f 11 (4 I 1) 5 d H f 11 (4 I 1) 5 d I f 11 (4 I 1) 5 d I f 11 (4 I 1) 5 d L f 11 (4 I 1) 5 d H f 11 (4 I 1) 5 d K Gaigalas et al. M R A S A S A S A S A x 108, s-1 A B A B ( Z F ) A C A C ( Z F ) A B A B ( Z F ) A C A C ( Z F ) Figure 7. Convergence of E1 transition probabilities us-ing the SD 5d strategy (open symbols mark the resultswhen the ZF MCDHF approach is applied). The 4 f 12 3 P – 4 f ( F ) 5 d P transition is marked in black and the4 f 12 3 P – 4 f ( F ) 5 d P transition in red. Gauge dependence In Figures 8–11 the dependence of the transition prob-abilities for the different active space calculations on thegauge parameter G is displayed. In each of these Fig-ures the position of Coulomb and Babushkin gauges aremarked by dotted lines. For some of analyzed transi-tions the curves of gauge dependence intersect at somepoint. The cross points are marked by dotted lines andthe values are placed on the axis. The curves cross ataround G = 1 . f 12 1 S – 4 f ( F ) 5 d P (Figure 8) and 4 f 12 3 P – 4 f ( F ) 5 d P (Figure 10) transitions. For the4 f 12 3 P – 4 f ( F ) 5 d P transition (Figure 9) themost of curves (except the curve of gauge dependencewith AS ) intersect at around G = 3 . 4. In case of the4 f 12 3 P – 4 f ( F ) 5 d P transition (Figure 11) thecurves do not intersect at one point. From these Fig-ures we can see that by increasing the active space, thecurves of gauge dependence approach straight lines. At AS (final results) these curves are very close to straightlines. It means that the wave functions should be quiteaccurate. 5.3. Cancellation factor Figures 12 and 13 show the CF as a function of the in-creasing active space for the SD 5d strategy. From theFigures it is seen that CF in the Babushkin gauge for theanalyzed transitions in all active spaces are lager thanin the Coulomb gauge. In Figure 14 and 15 we presentthe dependence of CF on the gauge parameter G usingthe SD 5d ZF MCDHF strategy (at AS ). The CF is - 5 0 5 1 0 1 50 . 00 . 51 . 01 . 52 . 02 . 53 . 0 A x 109, s-1 G A S A S A S A S ( Z F ) A S ( Z F ) A S ( Z F ) C B Figure 8. The gauge dependence of the 4 f 12 1 S –4 f ( F ) 5 d P E1 transition probability for the differentactive space calculations using the SD 5d strategy (opensymbols mark the results when the ZF MCDHF approach isapplied). - 5 0 5 1 0 1 50 . 00 . 51 . 01 . 52 . 02 . 53 . 03 . 5 C B A x 108, s-1 G A S A S A S A S ( Z F ) A S ( Z F ) A S ( Z F ) Figure 9. The gauge dependence of the 4 f 12 3 P –4 f ( F ) 5 d P E1 transition probability for the differentactive space calculations using the SD 5d strategy (opensymbols mark the results when the ZF MCDHF approach isapplied). presented for the four analyzed transitions. The CFs inBabushkin gauge for these transitions are much largerthan 0.1 or 0.05, and in all cases they are the largestones. They are even larger than at the cross points,where gauge dependence curves from different activespaces intersect. The CFs in Coulomb gauge for thetransitions 4 f 12 3 P – 4 f ( F ) 5 d P and 4 f 12 3 P – 4 f ( F ) 5 d P (Figure 15) are smaller than 0.05, ASTEX Energy level structure and transition data of Er Table 6. Comparison of Energy Levels from the Present Calculations Based on the SD 5d Strategy and Usingthe ZF Approach with NIST/SE data. LS / JJ label in Wyart et al. (1997) POS JP NIST/(SE) SD 5d ZF MCDHF SD 5d ZF MCDHFRCIAS AS f 12 3 H f 12 3 F − − f 12 3 H f 12 3 H − f 12 3 F − − f 12 3 F − − f 12 1 G − f 11 (4 I 1) 5 d G − f 11 (4 I 1) 5 d H − f 11 (4 I 1) 5 d L − f 11 (4 I 1) 5 d I − f 11 (4 I 1) 5 d L − f 11 (4 I 1) 5 d K − f 11 (4 I 1) 5 d G − f 11 (4 I 1) 5 d H − f 11 (4 I 1) 5 d I − f 11 (4 I 1) 5 d I − − f 11 (4 I 1) 5 d L − f 11 (4 I 1) 5 d H − − f 11 (4 I 1) 5 d K − − f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − f 11 (4 I / 2) 5 d / / , / 2) 4 f 11 5 d − − f 11 (4 F / 2) 5 d / / , / 2) 4 F / d / − − f 11 (4 I / 2) 5 d / / , / 2) 4 f 11 5 d − − f 11 (4 F / 2) 5 d / / , / 2) 4 f 11 5 d − − f 11 (4 F / 2) 5 d / / , / 2) 4 F / d − − f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − − f 11 (4 F / 2) 5 d / / , / 2) 4 F / d / − − f 11 (4 S / 2) 5 d / / , / 2) 4 f 11 5 d − − f 11 (4 F / 2) 5 d / / , / 2) 4 f 11 5 d 10 4- (46937.23) 41524/ 11.53 43024/ 8.344 f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − − f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − − f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − − f 11 (4 I / 2) 5 d / / , / 2) 4 f 11 5 d − − f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − − f 11 (4 I / d / / , / 2) 4 f 11 5 d − − f 11 (4 F / d / / , / 2) 4 I / d / − − f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − − f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − − f 11 (4 I / 2) 5 d / / , / 2) 4 f 11 5 d − − f 11 (4 F / 2) 5 d / / , / 2) 4 f 11 5 d − − f 11 (4 F / 2) 5 d / / , / 2) 4 f 11 5 d − − f 11 (4 I / 2) 5 d / / , / 2) 4 f 11 5 d − − f 11 (4 F / 2) 5 d / / , / 2) 4 f 11 5 d − − f 11 (4 F / 2) 5 d / / , / 2) 4 f 11 5 d 10 5- (39265.81) 41260/ − − f 11 (2 H / 2) 5 d / / , / 2) 4 f 11 5 d 11 5- (40857.10) 41726/ − − f 11 (4 F / 2) 5 d / / , / 2) 4 f 11 5 d 12 5- (46552.18) 42297/ 9.14 43795/ 5.924 f 11 (2 H / 2) 5 d / / , / 2) 4 f 11 5 d 13 5- (48747.15) 44128/ 9.48 45642/ 6.374 f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − − f 11 (4 I / 2) 5 d / / , / 2) 4 F / d / f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − f 11 (4 I / 2) 5 d / / , / 2) 4 I / d / − − f 11 (4 F / 2) 5 d / / , / 2) 4 F / d / − − f 11 (4 I / 2) 5 d / / , / 2) 4 f 11 5 d − − f 11 (4 F / 2) 5 d / / , / 2) 4 f 11 5 d − − f 11 (4 F / 2) 5 d / / , / 2) 4 f 11 5 d − − f 11 (4 F / 2) 5 d / / , / 2) 4 f 11 5 d − − f 11 (4 I / 2) 5 d / / , / 2) 4 f 11 5 d Note —The relative difference compared with NIST/(SE) data is given in percent. Gaigalas et al. Table 7. The Proposed Energy Levels in Comparison with NIST/(SE) Data and Their Relative Difference (Given inPercent). JP NIST/(SE) iden. in Wyart et al. (1997) POS SD 5d ZF MCDHF ( AS ) iden. in present work4- (46937.23) 10 → 15 41524/11.53 → − → 16 42297/ 9.14 → − → 17 44128/ 9.48 → − F / d / → → − f ( F / ) 5 d / (9 / , / → → − Table 8. Energy Levels (in cm − ) and Atomic State Function Composition of the Ground [Xe]4 f and First Excited[Xe]4 f d Configurations for the Er ion. No. POS J P E label comp.1 1 6 + 0.00 4 f 12 3 H f 12 3 F f 12 1 G + 0.11 4 f 12 3 H f 12 3 H f 12 3 H f 12 3 F + 0.07 4 f 12 1 G f 12 3 F f 12 3 F f 12 1 D − f ( I ) 5 d G f ( I ) 5 d H + 0.02 4 f ( K ) 5 d H − f ( I ) 5 d H f ( I ) 5 d I + 0.02 4 f ( G ) 5 d H f 12 1 G f 12 3 H + 0.11 4 f 12 3 F 10 1 9 − f ( I ) 5 d L f ( I ) 5 d L + 0.17 4 f ( I ) 5 d K 11 1 10 − f ( I ) 5 d L f ( K ) 5 d M 12 1 8 − f ( I ) 5 d I f ( I ) 5 d K + 0.15 4 f ( I ) 5 d K 13 2 9 − f ( I ) 5 d K f ( I ) 5 d L 14 1 5 − f ( I ) 5 d G f ( I ) 5 d H + 0.05 4 f ( I ) 5 d G 15 2 6 − f ( I ) 5 d H f ( I ) 5 d G + 0.11 4 f ( I ) 5 d I 16 2 8 − f ( I ) 5 d K f ( I ) 5 d I + 0.09 4 f ( I ) 5 d L 17 2 7 − f ( I ) 5 d I f ( I ) 5 d I + 0.17 4 f ( I ) 5 d K 18 3 8 − f ( I ) 5 d L f ( I ) 5 d K + 0.20 4 f ( I ) 5 d L 19 1 4 − f ( I ) 5 d G f ( I ) 5 d H + 0.03 4 f ( H ) 5 d F 20 2 5 − f ( I ) 5 d H f ( I ) 5 d G + 0.09 4 f ( I ) 5 d I Note — Table 8 is published in its entirety in the machine-readable format. Part of the values are shown here for guidance regarding itsform and content. which means that in velocity form there is a strong can-cellation effect. For the 4 f 12 3 P – 4 f ( F ) 5 d P and 4 f 12 1 S – 4 f ( F ) 5 d P (Figure 14) transi-tions, the CF in the Coulomb gauge is around 0.05. Theanalysis shows that transition data in the Babushkin gauge are less affected by cancellation effects than tran-sition data in the velocity gauge.5.4. Comparison with other computations No experimental transition rates for the studied con-figurations of Er are available. The transition dataobtained using the SD 5d ZF MCDHF strategy (at ASTEX Energy level structure and transition data of Er - 5 0 5 1 0 1 50 . 00 . 20 . 40 . 60 . 81 . 01 . 2 A x 109, s-1 G A S A S A S A S ( Z F ) A S ( Z F ) A S ( Z F ) Figure 10. The gauge dependence of the 4 f 12 3 P –4 f ( F ) 5 d P E1 transition probability for the differentactive space calculations using the SD 5d strategy (opensymbols mark the results when the ZF MCDHF approach isapplied). - 5 0 5 1 0 1 50 . 00 . 30 . 60 . 91 . 21 . 5 A x 108, s-1 G A S A S A S A S ( Z F ) A S ( Z F ) A S ( Z F ) C B Figure 11. The gauge dependence of the 4 f 12 3 P –4 f ( F ) 5 d P E1 transition probability for the differentactive space calculations using the SD 5d strategy (opensymbols mark the results when the ZF MCDHF approach isapplied). AS ) are compared with rates presented by Wyart et al.(1997) and Bi´emont et al. (2001). They used experimen-tal transition wavelengths to compute transition data.Bi´emont et al. (2001) used the Cowan code and includedcore-polarization effects in the computations.Figure 16 presents a comparison of obtained transi-tion wavelengths with experimental data, which werepresented in the paper by Wyart et al. (1997). The M R A S A S A S A S cancellation factor A B A B ( Z F ) A C A C ( Z F ) A B A B ( Z F ) A C A C ( Z F ) Figure 12. Cancellation factor dependence on the activespace. The 4 f 12 3 P – 4 f ( F ) 5 d P transition is markedin black and the 4 f 12 1 S – 4 f ( F ) 5 d P transition inred. M R A S A S A S A S cancellation factor A B A B ( Z F ) A C A C ( Z F ) A B A B ( Z F ) A C A C ( Z F ) Figure 13. Cancellation factor dependence on the activespace. The 4 f 12 3 P – 4 f ( F ) 5 d P transition is markedin black and the 4 f 12 3 P – 4 f ( F ) 5 d P transition inred. agreement between the computed wavelengths and theexperimental ones is very good. Almost all comparedlines achieve 5% uncertainty. In Figure 17 the compari-son of transition rates (given in Babushkin gauge) of thepresent work with rates available from other computa-tions (Wyart et al. 1997; Bi´emont et al. 2001) is dis-played. It is seen that there is a good agreement withvalues from other authors for the stronger transitions.However, the transitions presented in the Figure are notthe strongest obtained in this work. The strongest tran-sition have rates of the order 10 s − . By applying re-placement in the energy levels discussed in Section 4.58 Gaigalas et al. - 5 0 5 1 0 1 50 . 0 00 . 0 50 . 1 00 . 1 50 . 2 00 . 2 50 . 3 00 . 3 50 . 4 00 . 4 50 . 5 00 . 5 5 cancellation faktor G C B Figure 14. Cancellation factor dependence on gauge usingthe SD 5d ZF MCDHF strategy (at AS ). The 4 f 12 3 P –4 f ( F ) 5 d P transition is marked and in black and the4 f 12 1 S – 4 f ( F ) 5 d P transition is marked in red. - 5 0 5 1 0 1 50 . 0 00 . 0 50 . 1 00 . 1 50 . 2 00 . 2 50 . 3 0 cancellation faktor G C B Figure 15. Cancellation factor dependence on gauge usingthe SD 5d ZF MCDHF strategy (at AS ). The 4 f 12 3 P –4 f ( F ) 5 d P transition is marked and in black and the4 f 12 3 P – 4 f ( F ) 5 d P transition is marked in red. we achieve better agreement for wavelength and tran-sition rate of marked transition (see open symbols inFigures 16 and 17). SUMMARY AND CONCLUSIONIn the present paper energy levels of the ground[Xe]4 f and first excited [Xe]4 f d configurations forEr ion were computed using the GRASP2018 package.Transition data for E1 transitions between computedstates are presented. The accuracy of the obtained re-sults is evaluated. (cid:1) comp. (A) (cid:1) e x p . ( A ) Figure 16. Comparison of transition wavelengths betweenour computed data (comp.) using the SD 5d ZF MCDHF strategy (at AS ) and experimental data presented in thepaper by Wyart et al. Wyart et al. (1997). The thick line cor-responds to perfect agreement, while thin solid and dashedlines correspond to 5% and 10% deviations. The dashed ar-rows indicate the improved agreement by applying replace-ment in the energy levels discussed in Section 4.5. (cid:1)(cid:2)(cid:13)(cid:27)(cid:14)(cid:23)(cid:25)(cid:1)(cid:15)(cid:25)(cid:1)(cid:14)(cid:18)(cid:6)(cid:1)(cid:8)(cid:11)(cid:11)(cid:10)(cid:3)(cid:1)(cid:2)(cid:12)(cid:16)(cid:28)(cid:19)(cid:21)(cid:20)(cid:25)(cid:1)(cid:15)(cid:25)(cid:1)(cid:14)(cid:18)(cid:6)(cid:1)(cid:9)(cid:7)(cid:7)(cid:8)(cid:3) (cid:1) A , s-1 (other computations) A B (cid:4)(cid:1)(cid:24) (cid:5)(cid:8) (cid:1)(cid:2)(cid:22)(cid:23)(cid:15)(cid:24)(cid:15)(cid:20)(cid:25)(cid:1)(cid:26)(cid:21)(cid:23)(cid:17)(cid:3) Figure 17. Comparison of transition rates of present work( A is given in Babushkin gauge) with rates presented inWyart et al. (1997) and Bi´emont et al. (2001). The datafrom Wyart et al. (1997) are marked by black squares andthe red circles correspond to the results by Bi´emont et al.Bi´emont et al. (2001). The thick line corresponds to perfectagreement, while the thin solid and dashed lines correspondto deviations by factors of 1.5 and 2.0, respectively. Thedashed arrows indicate the improved agreement by applyingreplacement in the energy levels discussed in Section 4.5. ASTEX Energy level structure and transition data of Er Table 9. Transition Energies ∆ E (in cm − ), Transition Wavelengths λ (in ˚A), Weighted Oscillator Strengths gf and TransitionRates A (in s − ) for E1 Transitions of the Er Ion. No.(u) Ju P u state(u) No.(l) Jl P l state(l) ∆ E λ A gf dT − f 11 (4 I 1) 5 d G f 12 3 H − 05 0.8557 6 − f 11 (4 I 1) 5 d G f 12 3 H − 06 0.9198 7 − f 11 (4 I 1) 5 d H f 12 3 H − 01 3.246E − 08 1.00014 5 − f 11 (4 I 1) 5 d G f 12 3 H − 03 0.71114 5 − f 11 (4 I 1) 5 d G f 12 3 F − 03 0.68514 5 − f 11 (4 I 1) 5 d G f 12 3 H − 04 0.77114 5 − f 11 (4 I 1) 5 d G f 12 3 H − 04 0.78014 5 − f 11 (4 I 1) 5 d G f 12 1 G − 05 0.96215 6 − f 11 (4 I 1) 5 d H f 12 3 H − 02 0.55115 6 − f 11 (4 I 1) 5 d H f 12 3 H − 04 0.70017 7 − f 11 (4 I 1) 5 d I f 12 3 H − 02 0.51719 4 − f 11 (4 I 1) 5 d G f 12 3 F − 05 0.08419 4 − f 11 (4 I 1) 5 d G f 12 3 H − 04 0.59519 4 − f 11 (4 I 1) 5 d G f 12 3 H − 04 0.79619 4 − f 11 (4 I 1) 5 d G f 12 3 F − 04 0.72819 4 − f 11 (4 I 1) 5 d G f 12 1 G − 05 0.68220 5 − f 11 (4 I 1) 5 d H f 12 3 H − 03 0.74620 5 − f 11 (4 I 1) 5 d H f 12 3 F − 02 0.61520 5 − f 11 (4 I 1) 5 d H f 12 3 H − 06 0.98320 5 − f 11 (4 I 1) 5 d H f 12 3 H − 03 0.701 Note —All transition data are in length form. dT is the relative difference of the transition rates in length and velocity form as given by equation 10. Table 9 is publishedin its entirety in the machine-readable format. Part of the values are shown here for guidance regarding its form and content. From the studies of the Er ion, and also from theprevious investigations of Nd ions, it was observed thatin such calculations to get the correct order of groundand excited configurations it is important to freeze thewave functions of ground configuration.The valence-valence, core-valence, and core-core elec-tron correlations were studied using different strategies.This analysis has led to the final results in which themain balance electron correlation effects (mainly fromVV substitutions) were included. This allows us to im-prove accuracy of the energy difference between differentconfigurations considering the computational resourcesneeded for the computations of such a complex system.The rms deviations of the final results (using SD 5dZF MCDHF strategy) from the NIST or SE data for states of the ground and excited configurations are 649cm − , and 747 cm − , respectively.Having analyzed convergence trends and dependenciesof the gauge parameter G , we propose, for the Er ion,to use transition rates in the Babushkin gauge.There is a lack of atomic data for the lanthanides. Thepresent study is a first step towards the goal to providethis data with an accuracy high enough for opacity mod-eling.This research was funded by a grant (No. S-LJB-18-1) from the Research Council of Lithuania. Com-putations presented in this paper were performed at theHigh Performance Computing Center “HPC Sauletekis”of the Faculty of Physics at Vilnius University. 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