Finite field calculations of static polarizabilities and hyperpolarizabilities of In+ and Sr Yan-mei Yu1 ∗, Bing-bing Suo2†, Hui-hui Feng1, Heng Fan1, Wu-Ming Liu1 1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190,China and 2Institute of Modern Physics, Northwest University, Xi’an, Shanxi 710069, China (Dated: September30, 2015) The finite field calculations are performed for two heavy frequency-standard candidates In+ and Sr. The progressive hierarchy of electron correlations is implemented by the relativistic coupled- 5 clusterandconfigurationinteractionmethodscombinedwithbasissetofincreasingsize. Thedipole 1 polarizabilities, dipole hyperpolarizabilities, quadrupole moments, and quadrupole polarizabilities 0 2 arerecommendedforthegroundstate5s2 1S0 andlow-lyingstates5s5p3P0o,1,2 ofIn+andSr. Com- parative study of the fully and scalar relativistic electron correlation calculations reveals the effect p of the spin-orbit interaction on the dipole polarizabilities of In+ and Sr. Finally, the blackbody- e radiation shifts dueto the dipole polarizability, dipole hyperpolarizability, and quadrupolepolariz- S ability are evaluated for theclock transition 5s2 1S0 - 5s5p 3P0o of In+ and Sr. 9 2 PACSnumbers: 31.15.ap,31.15.aj,32.10.Dk ] h I. INTRODUCTION givenhighaccuracydataforthedipolepolarizabilitiesof p 5s2 1S and 5s5p 3Po of In+ and Sr using the configura- - 0 0 m tion interaction (CI) +all-order method [10, 12]. Mitroy Polarizabilitiesdescribe the ability ofatoms,ions, and et al. have calculated the 5s2 1S and 5s5p 3Po states o molecules to be polarized in an external electric field, 0 1 of Sr by using CI with a semiempirical core polarization t which is very useful in many areas of atomic and molec- a potential [18, 19]. Porsev et al. have calculated the Sr cs. uhlaavrepshhyowsicnsa[1t–re4m].eInndoreucsepnrtoygeraesrss,[5o–p8t]i,cawlhliacthtihceascsloticmks- 5msa2n1yS-b0oadnydp5esr5tpur3bPa0to,i1onsttahteeosruysi[n2g0].thVeeCryIrmeceetnhtoldy wthiethy si ulated a great deal of interest in performing precision havecalculatedthe 5s5p3Po stateby applying the more y calculations of atomic polarizabilities and hyperpolar- accurate CI + all-order met1hod [21]. From these works, h izabilities. For instance, the difference of the static one can see that the most currently available dipole po- p dipole polarizabilities between two associated states of larizability data for In+ and Sr are concentrated on the [ an atomic clock transition determines the blackbody- 3Po component, whereas the data for the 3Po and 3Po 2 radiation (BBR) shift that is crucial in evaluating the com0ponents remain scarce. Besides, the quadr1upole mo2- v error of an atomic clock, which has been calculated for ment of Sr in the 5s5p 3Po state and its quadrupole po- 9 Al+, Ca+, Sr, Yb, In+, Sr+, Hg+, Mg, Ca and so on larizabilities for the 5s2 1S and 5s5p 3Po states have 3 [1,3,9–12]. Thequadrupolepolarizabilityisanotherim- been calculated in earlier works by Mitroy, et al. us- 4 portant quantity that is related to the multipolar BBR 4 ing the CICP method [18, 23] and Porsev et al. using shift, which has been evaluated for Sr, Ca+, Sr+, and so 0 CI+MBPTandCI+all-ordermethods[21,22]. However, on [13, 14]. As a higher-order response to the applied . the recommended data for the quadrupole moments and 1 electric field, the hyperpolarizability also contributes to 0 quadrupole polarizabilities of In+ remain not available. the energy shift of an optical frequency standard, being 5 small but not necessarily negligible, which has been in- Among various theoretical approaches in calculating 1 : vestigated by Ovsiannikov,et al [15, 16]. polarizabilities, the finite-field method can provide reli- v able dataif the field-dependent energiesareevaluatedto i Amongthecurrentatomicopticalclockcandidates,an X important category is based on the ns2 1S - nsnp 3Po a high precision. This method is often implemented in 0 0 r optical transition [1–3], where the upper state is one of computational codes for atomic and molecular property a the metastable triplet states 3Po with J=0, 1, and 2. calculations [24–26]. In this method the employed ex- J Two suchexamples are the In+ andSr opticalclocks us- ternal field breaks the degeneracy of multiple states and ingthe5s2 1S -5s5p3Po transition[8,10]. Thepolariz- thus the J and MJ-resolved polarizabilities of the nsnp abilitiesofsuc0hsystemsh0avebeentargetedwithincreas- 3PJo metastable states can be obtained directly without ing theoretical efforts to perform precision calculations. usingbasicvectoralgebra,whereJ andMJ arethetotal For example, Sahoo et al. have calculated the ground- angular momentum and its magnetic quantum number state dipole polarizability of Sr by using relativistic cou- [27]. In particular, the influence of the spin-orbit cou- pled cluster (CC) method [17]. Safronova et al. have pling interaction on the J-resolved polarizability can be revealed through comparative studies between full and scalar relativistic calculations. Successful application of the finite-field method to atoms and ions of optical fre- ∗E-mail: [email protected] quencystandardhasbeendemonstratedforAl+ [28,29]. †E-mail: [email protected] However,the finite-fieldmethod hasnotbeen widely ap- 2 plied to calculate the polarizabilitiesof heavyatoms and polarizabilities can be established by using the LS cou- ions, for which relativistic effects and electron correla- pling approximation [27] tions are expected to be significant. It remains challeng- ing in treating these effects in this finite-field approach. α¯J = α¯L, In this work, the dipole polarizabilities, dipole hy- αJ(3Po) = −αJ(3Po)/2=−αL/2. (4) a 1 a 2 a perpolarizabilities,quadrupolemoments,andquadrupole polarizabilities for the groundstate 5s2 1S andthe low- The actual relationship between the J- and L-resolved 0 lying excited states 5s5p 3Po of In+ and Sr are calcu- polarizabilities can deviate from Eq. (4) when the spin- 0,1,2 latedbyapplyingthefinite-fieldmethod. Theconvergent orbit interaction is strong and the LS coupling scheme hierarchiesofelectroncorrelationandtheconvergentba- fails to describe the polarizabilities. sis sets are adopted in the relativistic CC and CI cal- In a pure quadrupole electric field, the corresponding culations in order to obtain properties of high accuracy. energy shift is The dipole polarizabilities of the ground-state 5s2 1S 0 ∆E (F )=−θF /2−α F2 /8−··· , (5) and the excited state 5s5p 3Po, as well as the scalar and d zz zz 2 zz 0 tensorpolarizabilitiesoftheexcitedstates5s5p3Po ,are 1,2 where Fzz is the electric field gradientin the z direction, obtained for In+ and Sr. Comparisons with previously andθandα arethequadrupolemomentandquadrupole 2 reporteddatafor5s2 1S0 and5s5p3P0o showgoodagree- polarizability, respectively. The quadrupole moment θ ment. The effect of the spin-orbit coupling interaction is calculated under the condition |M | = J or |M | = J L on the studied property is analyzed through comparing L. For the 5s5p 3P state of In+ and Sr, the L-resolved the J and L-resolved values of the 5s5p 3P0o,1,2 states, quadrupole moment is given for the state of 3Po with whereListhetotalorbitalangularmomentum. Besides, M =1,andtheJ-resolvedquadrupolemomentisgiven L werecommendthevaluesofthedipolehyperpolarizabili- for the state of 3Po with M =2. 2 J ties,thequadrupolemoments,andthequadrupolepolar- izabilitiesofIn+5s21S and5s5p3Po statesandSr5s2 0 0,1,2 1S and5s5p3Po states,forwhichthepreviousavailable III. METHOD OF CALCULATION 0 0 data are rather scarce. Finally, the BBR shifts for the dipolepolarizability,hyperpolarizability,andquadrupole The electric field-dependent energy is calculated at polarizability are evaluated for the clock transition 5s2 different levels of theories. Dirac-Hartree-Fock (DHF) 1S0 - 5s5p 3P0o of In+ and Sr. calculations are performed that generate the reference states as well as optimized atomic orbitals. These cal- culations are implemented by the SCF module in the II. THEORY DIRAC package [31]. Both the Dirac-CoulombHamilto- nian and the Dyall Hamiltonian [33] are employed. The Dirac-CoulombHamiltoniangivesacompletedescription The energy shift of an atom or ion in a homogeneous oftherelativisticeffects. WecarryouttherelativisticCC electric field can be expressed as [30] andCI calculations that are built on the Dirac-Coulomb Hamiltonian and yield fully relativistic results. The rel- ∆E (F )=−αF2/2−γF4/24−··· , (1) d z z z ativistic CC calculations are implemented by applying the MRCC suite [32], and the relativistic CI calculations where F is the electric field strength along the z direc- z are implemented by applying the the KRCI module in tion,andαandγ are,respectively,thedipolepolarizabil- the DIRAC package. The Dyall Hamiltonian contains ity and dipole hyperpolarizability. The J-resolvedscalar only the spin-free terms with the spin-orbit interaction and tensor polarizabilities are given by neglected. We carry out the scalar relativistic CI calcu- lationsthatarebuiltontheDyallHamiltonianandyield 1 Q¯J = ΣQ(J,M ) (2) the scalar relativistic results. The scalar relativistic CI J 2J +1MJ calculations are implemented by applying the LUCITA module in the DIRAC package. Due to escalating computational time for adopting QJa =Q(J,|MJ|=J)−Q¯J, (3) higher level correlation methods, the generalized active spaces (GAS) technique [34] is used to restrict the num- where Q denotes either α or γ, and Q(J,M ) denotes ber of correlated electrons and atomic orbitals, which J either α orγ foreachM componentwith M being the makesthecomputationtractable. IntheGAS,theDirac- J J projectionoftheangularmomentumJ inthez direction. Fock orbitals are divided into the inner-core, outer-core, The L-resolvedscalarand tensor polarizabilities are also valence, and virtual shells. No excitation is allowed in defined by Eqs. (2) and (3) with J and M replaced by the inner-coreshells,but the excitationin the outer-core J L and M respectively. If the spin-orbit coupling inter- and valence shells can be defined to any order. In or- L actionis notstrongsothatitcanbe treatedasapertur- der to obtain convergent results, we choose the outer- bation, the relationship between the J- and L-resolved core that includes increasing number of core electrons. 3 In particular, the 4d, 4s4p4d, and 3d4s4p4d shells that than 20 a.u. for In+ and 10 a.u. for Sr are neglected in consist of the ‘(core10)’, ‘(core18)’, and ‘core(28)’ outer- the correlation calculations. We also carry out the cor- cores, respectively, are considered for In+. Similarly the respondingcalculationsofthe cut-offvalue ofthe virtual 4s4p, 3d4s4p, and 3s3p3d4s4p shells that consist of the orbitalsof100a.u. forIn+ andSr. Thetruncationofvir- ‘(core8)’, ‘(core18)’, and ‘(core26)’ outer-cores, respec- tualorbitalsofthecut-offvalue100a.u. resultsinchange tively, are considered for Sr. For both In+ and Sr, the of αJ less than 0.5%in comparisonwith the cases of the valenceshellsarecomprisedof5s5p. Thevirtualorbitals cut-offvalue 20a.u. forIn+ and10a.u. forSr. The con- with the orbital energy larger than a given cut-off value tribution of the virtual orbitals with the orbital energy are neglected in the correlation calculations. We have largerthan20a.u. forIn+ and10a.u. forSristherefore carriedoutthe calculationswiththedifferentcut-offval- omitted in the composite scheme. ues in order to check the convergence of the obtained The uncertainty in P is mainly caused by three Final results with the truncation of the virtual orbitals. possible error sources. The first error is due to the finite In the following, a CC calculation is referred to as basis set used for calculating P and P . The con- SD SDT ‘(core n)SD’ and ‘(core n)SDT’ that include single and vergence of P with respect to the basis set is very quick double (SD) excitations and single, double, and triple whenthe basis set is largerthanX=3ζ,as shownby the (SDT) excitations fromthe outer-coreandvalence shells Al+ results [29]. Thus, we assume empirically that the into the virtual orbitals, where n means the number of error for P and for P is equal to half of the dif- SD SDT outer-core electrons that are involved in electronic cor- ference of the P values calculated by using X=4ζ and relation calculations. A CI calculation is referred to 3ζ basis sets. This error considers the possible correc- as ‘(core n)SD(2in4)SDT’ and ‘(core n)SD(2in4)SDTQ’, tion with respect to the value computed in the infinite where the outer-core shells are restricted to single and basis. The second error comes from the estimation of double (SD) excitations, and the valence shells, where ∆P and∆P . Previousexperience [29]has shownthat T Q two electrons are distributed in the four 5s5p orbitals, the∆P and∆P contributionscomputedevenwiththe T Q denotedby ‘(2in4)’,addexcitedelectronsfromthe outer smaller 2ζ basis set arenever in errorby more than 50% core,arerestrictedtoSDTandsingle,double,triple,and with respect to the basis-set limit, and hence the error quadruple (SDTQ) excitations into the virtual orbitals. bar of ∆P and of ∆P is taken to be half of itself, i.e., T Q The Dyall’s uncontracted correlation-consistent 0.5×∆P and0.5×∆P . Thethirderrorisduetothees- T Q double-, triple-, and quadruple-ζ basis sets are used, timationof∆P . As wewillshowinthe following,the core which are called Xζ with X=2, 3, and 4, respectively studied properties start to converge even with medium- [35, 36]. Each shell is augmented by two additional sizeoutercoressuchascore(18),whichindicatesthatthe diffused functions. The exponential coefficients of the error of ∆P is not larger than P −P for core (core28) (core18) augmented functions are determined according to In+ and P −P for Sr. Hence, the overall (core26) (core18) uncertaintyinP canbe estimatedtobe Root-Mean- Final ζ ζ = N ζ , (6) Square(RMS) of such three errors. N+1 N (cid:20)ζN−1(cid:21) Throughout this paper, atomic units (a.u.) are used, unless otherwise stated. The atomic units of α, α , γ, where ζ and ζ are the two most diffused exponents 2 N N−1 F , andF are,respectively,1.648778×10−41C2m2J−1, for the respective atomic shells in the original basis set. z zz 4.617048 × 10−62C2m4J−1, 6.235378 × 10−65C4m4J−3, Arbitrary4-8finitefieldstrengthsarechosenintherange 5.142250×109 V/cm, and 5.142250×1011 V/cm2. ofF =(0,4.5×10−3)andofF =(0,4.5×10−5)inatomic z zz units. Thefittingischeckedtoremovethedependenceof the properties studied on sampling. In our calculations, IV. RESULTS AND DISCUSSION the criterion for energy convergence is set to be 10−10 Hartree. A. Dipole polarizability and hyperpolarizability Weusethecompositescheme[28]togivethefinalvalue P of a studied property. In the CC calculation, Final Table I summarizes the results of αJ for the 5s2 1S 0 PFinal =PSD+∆PT +∆Pcore, (7) and5s5p3P0o,1,2 statesofIn+ calculatedbyusingtherel- ativistic CC method. Firstly, the obtained values of αJ where P is the value calculated for property P using SD inthe(core10)SDcalculationwiththeX=2ζ,3ζ,and4ζ the (core10)SD method with the X=4ζ basis set, ∆P T basis sets show a good convergence. The error in P and∆P arethecorrectionsduetothetripleexcitation SD core is only about 0.01∼0.06, which implies that the correc- and more outer-core electrons. In the CI calculation, tion with respect to the infinite basis set is very small. PFinal =PSDT +∆PQ+∆Pcore, (8) Then, the ∆Pcore correctionis estimatedwith the X=3ζ basis set. Upon inclusion of the 4s4p4d core electrons where P is the value calculated for property P using intothe (core18)SDcalculation,αJ decreasesincompar- SDT the (core10)SD(2in4)SDT method with the X=4ζ basis isonwiththe(core10)SDcase. Theeffectofaddingmore set,and∆P isthecorrectionduetothequadrupleexci- coreelectrons,3d4s4p4d,isillustratedbytherathersmall Q tation. Thevirtualorbitalswiththeorbitalenergylarger differencebetweenthe(core28)SDand(core18)SDcalcu- 4 TABLEI:Dipolepolarizabilities αJ ofIn+ forthestatesof5s2 1S0 and5s5p 3P0o,1,2 calculated bytherelativistic CCmethod, where (core10), (core18), and (core28) correspond respectively to the 4d, 4s4p4d, and 3d4s4p4d core shells included in the electron correlation calculations. Levelofexcitationa 1S0 3P0o |3MP1Jo|=1 |MJ|=0 |MJ|=1 |MJ|3=P22o α¯J αJa Basis:2ζ(23s,17p,13d,4f) (core10)SD 24.83 27.91 29.12 32.34 31.47 28.88 30.61 –1.73 (core10)SDT 24.54 27.50 28.75 32.26 31.13 28.42 30.27 –1.85 ∆PT –0.29 –0.41 –0.37 –0.08 –0.35 –0.46 –0.34 –0.12 Errorin∆PT ±0.15 ±0.21 ±0.19 ±0.04 ±0.18 ±0.23 ±0.17 ±0.06 Basis:3ζ(30s,23p,17d,5f,3g) (core10)SD 24.90 27.01 27.98 30.75 30.09 28.10 29.42 –1.33 (core18)SD 24.71 26.81 27.83 30.68 30.09 27.84 29.31 –1.47 (core28)SD 24.67 26.77 27.79 30.63 29.93 29.79 29.21 –1.42 ∆Pcore –0.24 –0.25 –0.19 –0.12 –0.16 –0.30 –0.21 –0.09 Errorin∆Pcore ±0.04 ±0.04 ±0.04 ±0.05 ±0.16 ±0.05 ±0.09 ±0.05 Basis:4ζ(35s,29p,20d,7f,5g,3h) (core10)SD,PSD 24.86 26.91 27.87 30.64 29.98 28.01 29.32 –1.31 ErrorinPSD ±0.03 ±0.05 ±0.06 ±0.06 ±0.05 ±0.04 ±0.05 ±0.01 PFinal=PSD+∆Pcore+∆PT Finaldata,PFinal 24.33 26.25 27.31 30.44 29.48 27.25 28.78 –1.53 Uncertainty(%) 0.62 0.82 0.73 0.28 0.82 0.87 0.69 4.99 Ref.[10] 24.01 26.02 a TherelativisticCIcalculationisperformedusingthe3ζ basissetat(core10)SD(2in4)SD<2level,whichyields α=25.06and27.93for1S0 and3P0o,α¯J=28.59andαJa=0.34for3P1o,andα¯J=30.30andαJa=–1.38for3P2o. TABLE II: Dipole hyperpolarizabilities γJ of In+ for the states of 5s2 1S0 and 5s5p 3P0o,1,2 calculated by the relativistic CC method, where (core10), (core18), and (core28) correspond respectively to the 4d, 4s4p4d, and 3d4s4p4d core shells included in theelectron correlation calculations. Levelofexcitationa 1S0 3P0o |3MP1Jo|=1 |MJ|=0 |MJ|=1 |MJ|3=P22o γ¯J γaJ Basis:2ζ(23s,17p,13d,4f) (core10)SD 3695 14752 21119 27116 20729 7294 16632 –9338 (core10)SDT 3640 15134 21611 27546 20989 7773 17014 –9241 ∆PT –55 381 492 431 260 479 382 97 Errorin∆PT ±28 ±191 ±246 ±216 ±130 ±240 ±191 ±49 Basis:3ζ(30s,23p,17d,5f,3g) (core10)SD 3143 13435 19265 26644 20951 6820 16437 –9617 (core18)SD 3072 13467 19368 26472 20755 6968 16384 –9645 (core28)SD 3065 13464 19327 26480 20703 7056 16400 –9344 ∆Pcore –78 28 62 –164 –248 236 –38 273 Errorin∆Pcore ±7 ±3 ±42 ±9 ±52 ±88 ±16 ±72 Basis:4ζ(35s,29p,20d,7f,5g,3h) (core10)SD,PSD 3122 13057 19125 26265 20571 6765 16187 –9422 ErrorinPSD ±11 ±189 ±70 ±190 ±190 ±28 ±125 ±97 PFinal=PSD+∆Pcore+∆PT Finaldata,PFinal 2989 13467 19679 26532 20583 7479 16531 –9052 Uncertainty(%) 1.01 1.99 1.32 1.08 1.15 3.43 1.38 1.44 a TherelativisticCIcalculationisperformedusingthe3ζ basissetat(core10)SD(2in4)SD<2level,whichyields α=2715and16164for1S0 and3P0o,α¯J=17247andαJa=4263for3P1o,andα¯J=18614andαJa=–8518for3P2o. lations, indicating that αJ has entered into convergence minimum number of core electrons that needs to be cor- region. Thus, ∆P is estimated as the difference be- related for In+ is 18; in other words, the 4s4p4d core core tweenthe(core28)SDand(core10)SDcalculations,which electrons need to be correlated in order to obtain accu- is about 0.09∼0.30. By comparison of the (core10)SD, rate αJ. Next, ∆P is estimated by the difference be- T (core18)SD,and(core28)SDcalculations,weseethatthe tween the (core10)SD and (core10)SDT values with the 5 TABLE III: Dipole polarizabilities αL and hyperpolarizabilities γL for the 5s2 1S and 5s5p 3Po states of In+ obtained bythe scalar relativistic CI method, where (core10), (core18), and (core28) correspond respectively to the 4d, 4s4p4d, and 3d4s4p4d core shells included in theelectron correlation calculations. α γ Levelofexcitation 3Po 3Po 1S0 |ML|=0 |ML|=1 α¯L αLa 1S0 |ML|=0 |ML|=1 γ¯L γaL Basis:2ζ(23s,17p,13d,4f) (core10)SD(2in4)SDT 24.42 32.24 27.98 29.40 –1.42 2832 35528 7099 16576 –9476 (core18)SD(2in4)SDT 24.28 32.23 27.84 29.31 –1.46 2521 35712 6972 16552 –9580 (core28)SD(2in4)SDT 24.26 32.19 27.82 29.28 –1.46 2567 35679 6967 16538 –9571 (core10)SDT(2in4)SDTQ 24.41 32.17 27.82 29.27 –1.45 3650 35577 7381 16780 –9398 ∆PQ –0.015 –0.07 –0.16 –0.13 –0.03 789 49 282 204 78 ErrorinPQ ±0.008 ±0.035 ±0.08 ±0.065 0.015 ±395 ±25 ±141 ±102 ±39 ∆Pcore –0.16 –0.05 –0.16 –0.12 –0.04 –294 151 –132 –38 –94 ErrorinPcore ±0.02 ±0.05 ±0.0.02 ±0.03 ±0.01 ±46 ±33 ±5 ±14 ±9 Basis:3ζ(30s,23p,17d,5f,3g) (core10)SD(2in4)SDT 24.38 30.43 27.17 28.26 –1.08 2182 34283 6742 15922 –9180 Basis:4ζ(35s,29p,20d,7f,5g,3h) (core10)SD(2in4)SDT,PSDT 24.34 30.35 27.12 28.20 –1.08 2270 34257 6761 15926 –9165 ErrorinPSDT ±0.02 ±0.04 ±0.03 ±0.03 ±0.005 ±44 ±13 ±9 ±2 ±8 PFinal=PSDT+∆Pcore+∆PQ Finaldata,PFinal 24.16 30.22 26.80 27.94 –1.14 2765 34457 6910 16092 –9182 Uncertainty(%) 0.13 0.23 0.33 0.28 1.56 14.46 0.13 2.05 0.64 0.44 X =2ζ basis set, which is around −0.08∼−0.46. Both of (core10)SD(2in4)SD<2, where the cutoff for the vir- the ∆P and ∆P correctionsare large in comparison tual orbitals is 2 a.u. in energy, and the basis set is 3ζ, T core with the error in P , and thus an omission of such a as indicated at the footnotes of Tables I and II. SD correction would lead to an underestimation of αJ. The The results of αL and γL for the 5s2 1S and 5s5p referencedataforthe5s2 1S0 and5s5p3P0o statesofIn+ 3Po states of In+, calculated by the scalar relativis- arealsogiveninTableI,ascalculatedbySafronovaetal. tic CI method, are contained in Table III. P is SDT by using the CI+all-order method [10]. Our results are determined using the X=4ζ basis set, and ∆P and Q consistentwiththeirdatawithadiscrepancyaround1%. ∆P are determined using the X=2ζ basis set as core Finally,werecommendthatα¯J=27.31for5s5p3P1o with the differences between the (core10)SD(2in4)SDT |MJ|=1 and α¯J=28.78 and αJa =−1.53 for 5s5p 3P2o. and (core10)SDT(2in4)SDTQ, and between the OurcalculatedresultsforγJ arelistedinTableII.The (core10)SD(2in4)SDT and (core28)SD(2in4)SDT, γJ values show a similar convergence trend with respect respectively. The triple excitation is considered in to the size of basis set. The values of ∆P , and ∆P the scalar relativistic CI calculations, which reduces T core are also larger than the error in P for γJ, similar to the uncertainties in the final values of αL and γL, SD the case of αJ. As shown in Tables I and II, the largest as compared with the relativistic CC calculations. In source of uncertainty in the final data comes from the TableIII,the∆PQ and∆Pcore correctionsarealsolarger error in ∆PT for both αJ and γJ. As mentioned above, than the error in PSDT. This trend is in accordance ∆P hasimportantcontributiontothefinaldata. Inthis with that in the relativistic CC calculations. Because T work, due to very high computational demand in using ∆PQ is computed only with a much smaller basis set, larger basis sets, ∆PT is calculated only with X = 2ζ. an estimation of error ∆PQ can not be exact (like ∆PT This basissetismuchsmaller,whichmaycauseanover- in Tables I and II), which leads to an anomalously large estimationoftheuncertaintyof∆PT,likethecasesofαJa uncertainty in the final data for the 5s2 1S0 state. Up for 5s5p 3Po and γJ for 5s5p 3Po with M = 2. Using to now, we can find that, for In+, increasing basis set 2 2 J the same composite scheme of convergence, we arrive at up to X = 4ζ has decreased the error in PSD with the recommended values of γJ=2989, 13467, and 19679 respect to the infinite basis set. However, the ∆PT and for the 5s2 1S0, 5s5p 3P0o, and 5s5p 3P1o, |MJ|=1 states, ∆Pcore corrections are larger than the error in PSD. In respectively,andγ¯J=16531andγaJ =−9052forthe5s5p this situation, both ∆PT and ∆Pcore become crucial for 3Po state. The uncertainties of the final values of γ are an accurate evaluation of the dipole polarizability and 2 slightlylargerthanthoseforαJ,whichiscomprehensible hyperpolarizability of In+. becausethehyperpolarizabilityisahigher-orderresponse Table IV summarizes the results of αJ for the states thatismoresensitivetoasmallenergyvariationandthus of5s2 1S and5s5p3Po ofSr,calculatedbyusing the 0 0,1,2 variouscontributionsbringsubstantialcorrectionstoγJ. relativistic CC method. The relativistic CI calculations, TherelativisticCIcalculationisimplementedatthelevel implemented at the (core8)SD(2in4)SD<2 levelwith the 6 TABLEIV:Dipolepolarizabilities αJ forthestatesof5s2 1S0 and5s5p3P0o,1,2 ofSrcalculated bytherelativisticCCmethod, where (core8), (core18), and (core26) correspond respectively to the 4s4p, 3d4s4p, and 3s3p3d4s4p core shells included in the electron correlation calculations. Levelofexcitationa 1S0 3P0o |3MP1Jo|=1 |MJ|=0 |MJ|=1 |M3PJ2o|=2 α¯J αJa Basis:2ζ(23s,17p,12d,3f) (core8)SD 205.1 349.3 377.1 418.7 397.1 331.7 375.3 –43.5 (core8)SDT 200.1 348.8 379.5 422.2 499.8 332.0 377.2 –45.2 ∆PT –5.0 –0.5 2.4 3.5 2.8 0.2 1.9 –1.7 Errorin∆PT ±2.5 ±0.3 ±1.2 ±1.8 ±1.4 ±0.1 ±1.0 ±0.9 Basis:3ζ(31s,22p,15d,7f,3g) (core8)SD 204.8 423.8 460.8 515.1 490.5 415.7 465.5 –49.8 (core18)SD 204.6 415.3 450.3 503.8 478.8 406.3 454.8 –48.5 (core26)SD 204.7 418.8 454.4 509.0 486.9 410.1 460.6 –50.5 ∆Pcore –0.2 –5.0 –6.4 –6.2 –7.6 –5.6 –6.5 0.9 Errorin∆Pcore ±0.01 ±3.5 ±4.0 ±5.2 ±4.1 ±3.9 ±4.2 ±0.3 Basis:4ζ(35s,27p,17d,9f,7g,3h) (core8)SD,PSD 204.8 446.4 484.9 546.6 520.0 441.9 494.1 –52.2 ErrorinPSD 0.0 ±11.3 ±12.1 ±15.7 ±14.8 ±13.1 ±14.3 ±1.2 PFinal=PSD+∆Pcore+∆PT Finaldata,PFinal 199.7 444.1 480.9 543.9 519.2 436.5 491.1 –54.5 Uncertainty(%) 1.2 2.7 2.7 3.1 3.0 3.1 3.1 2.8 RCC[17] 199.7 CI+MBPT[20] 197.2 457.0 498.8 CI+all-order[12] 194.4 441.9 CICP[18] 204.5 497.0(27.7)b CI+all-order[21] 459.2(26.0)b Expt. [37] (24.5)b a TherelativisticCIcalculationisperformedusingthe3ζ basissetat(core8)SD(2in4)SD<2level,whichyields α=179.5and394.5for5s2 1S0 and5s5p3P0o,α¯J=409.7andαJa=23for5s5p3P1o,andα¯J=455.2andαJa=–51.8 for5s5p3Po. 2 b Heregivenarethescalarandtensor(inparentheses)polarizabilitiesfor5s5p3Po. 1 TABLE V: Dipole hyperpolarizabilities γJ for the states of TABLE VI: Dipole polarizabilities αL for the states of 5s2 5s2 1S0 and 5s5p 3P0o of Sr obtained by the relativistic CC 1S and 5s5p 3Po of Sr obtained by the scalar relativistic CI method,where(core8),(core18), and(core26) correspond re- method,where(core8), (core18),and(core26) correspondre- spectively to the 4s4p, 3d4s4p, and 3s3p3d4s4p core shells spectively to the 4s4p, 3d4s4p, and 3s3p3d4s4p core shells included in theelectron correlation calculations. included in theelectron correlation calculations. LBeavseisl:2oζf(e2x3csi,t1a7tpio,1n2d,3f) 1S0 3P0o Levelofexcitation 1S0 |ML|=0|ML|=3P1oα¯L αLa (core8)SD 484500 7050552 Basis:2ζ(23s,17p,12d,3f) (core8)SDT 510144 6905904 (core8)SD(2in4)SDT 199.5 443.7 312.6 356.3 –43.7 ∆PT 25644 –144648 (core18)SD(2in4)SDT 198.2 427.5 302.3 344.0 –41.8 Errorin∆PT ±12822 ±72324 (core26)SD(2in4)SDT 197.8 427.6 302.3 344.1 –41.7 Basis:3ζ(31s,22p,15d,7f,3g) (core8)SDT(2in4)SDTQ 198.7 457.2 325.2 369.3 –44.0 (core8)SD 680965 3568344 ∆PQ –0.7 13.5 12.6 12.9 –0.3 (core18)SD 672720 3452688 Errorin∆PQ ±0.4 ±6.8 ±6.3 ±6.5 ±0.2 (core26)SD 674592 3289040 ∆Pcore –1.6 –16.2 –10.3 –12.3 2.0 ∆Pcore –6373 –279304 Errorin∆Pcore ±0.35 ±0.01 ±0.05 ±0.04 ±0.04 Errorin∆Pcore ±1872 ±163648 Basis:3ζ(31s,22p,15d,7f,3g) Basis:4ζ(35s,27p,17d,9f,7g,3h) (core8)SD(2in4)SDT 198.4 528.4 381.9 430.7 –48.8 (core8)SD,PSD 672686 3652171 Basis:4ζ(35s,27p,17d,9f,7g,3h) ErrorinPSD ±8279 ±83827 (core8)SD(2in4)SDT,PSD 197.1 547.7 396.7 447.0 –50.3 PFinal=PSD+∆Pcore+∆PT ErrorinPSD ±0.7 ±9.6 ±7.4 ±8.1 ±0.7 Finaldata,PFinal 691957 3228219 PFinal=PSDT+∆Pcore+∆PQ Uncertainty(%) 2.22 6.12 Finaldata,PFinal 194.7 544.9 398.9 447.6 –48.7 Uncertainty(%) 1.3 1.8 2.5 2.1 3.5 7 calculation of Sahoo et al. using the relativistic couple TABLE VII: Quadrupole moments θ and polarizabilities α¯2, cluster (RCC) method [17], the calculation of Porsev et where the L-resolved values are obtained by the scalar rela- al. using the CI method with many-body perturbation tivistic CI method andtheJ-resolved values areobtained by therelativistic CC method for 5s5p 3Po and therelativistic theory (CI+MBPT) [20], the calculation of Safronova et 0,2 CI method for 5s5p 3Po. al. using the CI + all-order method [12], the calculation 1 of Mitroy et al. using the CI method with semiempiri- θL θJ cal core polarization potential (CICP) [18], and the cal- In+ 1S culation of Porsev et al. using CI + all-order method 3Po 4.36 4.64(3Po) 2 [21]. Noteworthy to mention are the CI+all-order re- Sr 1S sults of Safronova et al. that give α=194.4 and 441.9 3Po 15.56 15.76(3P2o) for 5s2 1S0 and 5s5p 3P0o, respectively. For such two 15.6[22],15.52[23] states, we obtain αJ=199.7 and 444.1 in the relativistic α¯L α¯J CC calculation, which are slightly larger than the values 2 2 of Safronova et al. The possible reason for such discrep- In+ 1S 127 129 anciesmaybe due to underestimationofthe magnitudes 3Po 145 1425(3P0o) of ∆PT and ∆Pcore for which our basis sets are much 1678(3P1o) smaller. There are fewer reported data available for the –859.3(3P2o) 5s5p 3Po state. Porsev et al. have obtained αJ=498.8 1 Sr 1S 4688 4608 for the 5s5p 3Po state with M = 1 state by using the 1 J 4640[18] 4545[21] CI+MBPT method, which is close to our calculated re- 3Po 6756 9.75×104 (3P0o) sult αJ=480.9. The scalar and tensor polarizabilities of 6949[18] 1.17×105 (3P1o) 5s5p3P1o havealsobeengivenasα¯J=497.0andαJa=27.7 1.05×105 [21] byMitroyetal. [18]andα¯J=459.2andαJ=26.0byPor- a –7.39×104 (3P2o) sev et al. [21]. The experimental value of αJ for 5s5p a 3Po is 24.5. By using the relativistic CI method we ob- 1 tain α¯J=409.7 and αJ=23.0 for Sr 5s5p 3Po, which dif- a 1 3ζ basis set, are presented at the footnote area for com- fers from Porsev’s values by 11%. Such a relativistic CI parison. Asseenfromthe(core8)SDdatawiththeX=2, calculation is implemented with much smaller basis set 3, and 4ζ basis sets in Table IV, the changes of αJ for and has about 10% error in αJ in comparison with our Sr 5s2 1S state are very small , while the αJ values for relativistic CC calculation. Considering this error, our 0 Sr 5s5p 3Po states tend to increase obviously for the resultsaremuchcloserto the valuesofPorsevetal.[21]. 0,1,2 larger basis sets, implying the basis set effect is rather Thereisnoreporteddataavailableforthedipolepolariz- large for these states. The error in PSD for the 5s5p abilityoftheSr5s5p3P2ostate. Inthepresentrelativistic 3P0o,1,2 states of Sr is about 12.1∼15.7, except for αJa of CC calculations we obtain α¯J=491.1 and αJa=–54.5 for 5s5p 3P2o, which is larger than ∆PT and ∆Pcore. This m5se5npd3γPJ2o=.6F9o1r95th7eanhdyp3e2r2p8o2la1r9izfoarbi5list2ie1sSofaSnrdw5es5rpec3oPmo-, indicates that the primary dominant factor that affects 0 0 the final αJ of Sr comes from the size of the basis set. respectively. This trend differs from the results mentioned earlier for The results of αL for the states of 5s2 1S and 5s5p In+. Note that the error in ∆PSD is very small and is 3Po of Sr are givenin Table VI. In the scalar relativistic far less than ∆PT and ∆Pcore for the case of In+. The CI calculations, αL for the 5s2 1S groundstate seems to possiblereasonforsuchadifferenceisbecauseofthedif- be insensitive to the basis set, ∆P , and ∆P ; on the Q core ferentelectrondistributionofIn+ andSr. Asapositively otherhand,αL forthe5s5p3Postateissensitivetothese charged ion, in general In+ has compact electron distri- factors. αL of the 5s5p 3Po state increases with the size bution around nucleus. However, as a neutral atom, Sr of basis set and it reaches convergence at X=4ζ. ∆P T would have more diffused electron distribution than In+ and ∆P are substantial in magnitude but they are in core in an external electric field, indicating that larger basis opposite sign, resulting in a cancellationwith eachother sets are required for computing Sr. For Sr, the uncer- in the final value of αL. Notice that this phenomenon is taintyinαJ isaround1.2∼3.1,mostofwhichisfromthe alsofound inTable IV for the relativistic CC calculation error in PSD. This reflects that the accurate calculation but the cancelation is not as strong as that in Table VI of α of Sr replies greatly on the basis set. The values of for the scalar relativistic CI calculation. Combining the γJ for the 5s2 1S0 and 5s5p 3P0o states of Sr, calculated relativistic CC and scalar relativistic CI results, we can by using the relativistic CC method, are summarized in conclude that the basis set has very important effect on Table V. As a higher-orderproperty,the finaldata ofγJ the final values of αJ and αL. A further expansion of for the 5s2 1S0 and 5s5p 3P0o states of Sr show a large the basis set will no double improve the accuracy of α. dependence on ∆PT and ∆Pcore, and also show larger As shown in Tables IV and VI, for both the relativistic uncertainties than that for αJ. CCandthescalarrelativisticCIcalculations,∆P has core ThepreviouslyreporteddipolepolarizabilitiesofSrare reached convergence upon inclusion of 3d4s4p into the also listed in Table IV for comparison. They include the electron correlations. This indicates that the minimum 8 number of the core electrons is 18 for the case of Sr, out with the X=3ζ basis set. From Table VII, we can i.e.,thecorrelationcontributionfrom3d4s4pneedstobe foundthat,thevaluesofθLandθJ forthe5s5p3Postate, includedinall-electroncalculationinordertoachievean as well as the values of αL and αJ for the 5s2 1S state, 2 2 accurate dipole polarizability. areingoodaccord,whichindicatesthatthese properties The relativistic effect in the four-component relativis- areinsensitivetothespin-orbitcoupling. Incontrast,the ticformalismcanbeunderstoodasacombinationofspin- values of αL and αJ for the 5s5p 3Po state show clear 2 2 orbit coupling effect and contraction or decontraction of discrepancy,and αJ changes greatly for different J com- 2 the radial electron density, i.e., the scalar relativistic ef- ponents. This means that the spin-orbit coupling plays fect. Among the 5s5p 3Po states, the 3Po component an important role on the quadrupole polarizabilities of 0 has a spherically symmetric electronic density. In this the 5s5p3Po states of In+ and Sr. case, the effect of the pure spin-orbit interaction on the The quadrupole moments of Sr in the 5s5p 3Po state polarizability can be reflected by comparing the results were calculated in earlier worksby using the CI+MBPT obtained by fully and scalar relativistic calculations for method[22]andtheCICPmethod[23]. Ourresultsagree the3Po component,aswehavedoneforAl+,intermsof with their data within 1%. The quadrupole polarizabili- 0 ties for the 5s2 1S and 5s5p 3Po states of Sr were calcu- Q¯L(3Po)−QJ(3Po) lated using the CICP method [18] and the CI+all-order 0 . (9) QJ(3Po) method with the random phase approximation (RPA) 0 applied[21]. Forthe groundstate5s2 1S,ourresultsare Such a fractional difference is about 8% in α and 22% in agreement with their data within 1%. For the 5s5p in γ for the 5s5p 3Po states of In+, showing an evident 3Po states, our L-resolved values agree with the CICP spin-orbit effect in α and γ. The fractional difference data at the level of 2%, and our J-resolved result gives [α¯L−αJ(3Po)]/αJ(3Po) amounts to 2% for Sr, which is θJ=1.17×105fortheSr5s5p3Postate,inconsistentwith 0 0 1 smaller than that of In+. This implies that the effect of the CI+all-order+RPAvalue 1.05×105 [21] within 11%. the spin-orbit coupling on α of Sr is weaker than In+. The relativisticeffect canbe elucidatedby the fractional difference as follows, C. Black-body Radiation Shift Q¯J(3Po)−QJ(3Po) 1 0 (10) Finally, our results for the dipole polarizabilities, the QJ(3Po) second dipole polarizabilities, and the quadrupole polar- 0 izabilitiesareusedtoestimatetheBBRshiftintheclock and transition frequency of In+ and Sr. The BBR shift can Q¯J(3Po)−QJ(3Po) be written in the form [13, 14] 2 0 , (11) QJ(3Po) 1 1 1 0 δE =− ∆αhE2 i− ∆γhE2 i2− ∆α hE2 i, BBR 2 E1 24 E1 2 2 E2 where Q stands for either α or γ. Such two fractional (12) differences are about 8.9% and 9.6% for α and 28% and where hE2 i and hE2 i are the averaged electric fields 23% for γ in the case of In+. Such variations of α¯J for induced bEy1 the elecEt2ric dipole E1 and the electric different J components for Sr, as reflected by Eqs. (10) quadrupole E2 and they are respectively and (11), are about 3.9% using the values in Refs. [21] and [12], and 10.5% using our values. The above values 4π3α3 k T hE2 i= fs( B )4 (13) of the fractional differences of 5s5p 3P1o and 3P2o with E1 15 Eh respect to 5s5p 3Po are substantial that reveal the im- 0 portant role of the relativistic effect on α and γ for the and states of 5s5p 3Po of both In+ and Sr. 8π5α5 k T hE2 i= fs( B )6. (14) E2 189 E h B. Quadrupole moment and polarizability In the above, α is the fine structure constant, fs k T/E ≈10−9 forthetemperatureT=300K,k isthe B h B Table VII presents θ and α for the 5s2 1S and Boltzmann constant, E is the Hartree energy, and ∆α, 2 h 5s5p 3Po states of In+ and Sr. The L-resolved ∆α and ∆γ, expressedin atomic units, are respectively 2 θL and αL are obtained by the scalar relativis- the differences of the dipole polarizability, quadrupole 2 tic calculations (core10)SD(2in4)SDTQ for In+ and polarizabilityanddipolehyperpolarizabilitybetween5s2 (core8)SD(2in4)SDTQ for Sr, and the J-resolved θJ 1S and 5s5p 3Po of In+ and Sr. In Eq. (12) we have 0 0 and αJ are obtained by the relativistic CC calculations neglected the dynamic fractional correction to the total 2 (core10)SD for In+ and (core8)SD for Sr. The data shift [13] and assume that the contributionof the hyper- for the 5s5p 3Po state are obtained by the relativis- polarizability to the BBR shift can be approximated by 1 tic CI calculations (core10)SD(2in4)SD<2 for In+ and the AC-stark shift hE2 i2 for a given electric field. Us- E1 (core8)SD(2in4)SD<2forSr. Allcalculationsarecarried ing the data of α, γ, and α obtained by the relativistic 2 9 CC calculations in Tables I, II, and VII for the states of and thus a more accurate evaluation of the contribution 5s2 1S and 5s5p 3Po of In+ , the BBR shifts due to α, ofthetripleexcitationisneededforhigheraccuracy. For 0 0 α , and γ are determined to be 0.017, 8.33×10−10, and example, the relativistic CCSDT calculationwith the 4ζ 2 1.93×10−17 Hz, respectively,for the In+ clock transition basis set may give more accurate results, although such frequency; and using the data of α, γ, and α obtained a study is prohibited at this moment given the present 2 by the relativistic CC calculations in Tables IV, V, and computingresources. ForthecaseofSr,the convergence VIIforthestatesof5s21S and5s5p3Po ofSr,theBBR of α is highly sensitive to the quality of the basis set 0 0 shifts due to α, γ, and α are determined to be 2.09 and because a neutral atom has more diffused electron den- 2 5.82×10−8, and 1.69×10−15 Hz, respectively, for the Sr sity in an external field than a positive charged ion. In clock transition frequency. It is clear that the contri- this case, the predominant factor will be the expansion butions from γ and α are far less important than that and optimization of the basis sets beyond X=4ζ. The 2 fromα in the BBRshifts andcan thus be safely omitted higher-order corrections of the quadruple excitation of according to the current precision of the quoted 10−18 the electron correlation and the Breit and QED correc- uncertainty of the time frequency standard. tionsshouldberoughlytwo-andone-orderofmagnitude smallerthan∆P inthe relativisticCC calculationsand T accordingly the resulted uncertainties should be at the V. CONCLUSION level of 0.01% and of 0.1%, which are very small. Our fully relativistic calculations have shown that the In summary, we have calculated α, γ, θ, and α for 5s5p 3Po states of In+ and Sr have obviously differ- 2 0,1,2 thegroundstate5s2 1S andthelow-lyingexcitedstates entvaluesofpolarizabilities,whichreflectstheimportant 0 5s5p3Po ofIn+andSrbyusingthefinitefieldmethod. contributions of the relativistic and spin-orbit coupling 0,1,2 A satisfactory accuracy is achieved through convergence interactions. Somegeneraltrendsaboutthesoleeffectof studies for the basis sets and sufficient inclusion of the the spin-orbit coupling are worth noticing through com- electron correlations. This method can also be applied parativestudiesusingthefully andscalarrelativisticap- with similar accuracy to the calculations of the polariz- proaches. The fractional difference between the L- and abilities of atomic cores, see Supplemental Material at J-resolved α of 5s5p 3Po is about 8% for In+, but only 0 [URL will be inserted by publisher] for α, γ, and α of 2% for Sr, implying that the effect of the spin-orbit in- 2 In3+ and Sr2+. Thus, it will be useful to employ the fi- teraction on α of In+ is stronger than Sr. nitefieldmethodtoperformafastevaluationofrequired properties, especially when high-precision experimental studies and sophistical sum-over-state calculations are VI. ACKNOWLEDGEMENTS not available or all available results are not in complete agreement. Itisnoteworthytomentionthattheerrorsof thefinitefieldcalculationsneedtobeexaminedcarefully TheauthorswouldliketothankProf. Zong-ChaoYan andminimizedforsuchapplications,whichrequiresade- for careful reading and revising the manuscript and to tailed knowledge about the rate of convergence of basis Dr. Jun Jiang and Dr. Chengbin Li for valuable com- set and electron correlations for a property of interest. ments. Thisworkissupportedby2012CB821305,NSFC Inourfinitefieldcalculations,wehaveinvestigatedthe 61275129,NFSC 21203147,and CAS KJZD-EW-W02. influences of the basis set and the levelof electroncorre- lations on the computed properties. For the case of In+, theconvergenceforαcanbeeasilyreachedwhentheba- VII. AUTHOR CONTRIBUTION STATEMENT sis set is increased up to X=4ζ and the core electrons includedintothecorrelationcalculationareincreasedup to3d4s4p4d. Thedominantcorrectiontoαisfrom∆P , All authors contributed equally to the paper. 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