Determination of the dipole polarizability of the alkali-metal negative ions
aa r X i v : . [ phy s i c s . a t o m - ph ] J a n Determination of the dipole polarizability of the alkali-metal negative ions
B. K. Sahoo ∗ Atomic, Molecular and Optical Physics Division,Physical Research Laboratory, Navrangpura, Ahmedabad-380009, India (Dated: Received date; Accepted date)We present electric dipole polarizabilities ( α d ) of the alkali-metal negative ions, from H − to Fr − , byemploying four-component relativistic many-body methods. Differences in the results are shown byconsidering Dirac-Coulomb (DC) Hamiltonian, DC Hamiltonian with the Breit interaction, and DCHamiltonian with the lower-order quantum electrodynamics interactions. At first, these interactionsare included self-consistently in the Dirac-Hartree-Fock (DHF) method, and then electron correlationeffects are incorporated over the DHF wave functions in the second-order many-body perturbationtheory, random phase approximation and coupled-cluster (CC) theory. Roles of electron correlationeffects and relativistic corrections are analyzed using the above many-body methods with size of theions. We finally quote precise values of α d of the above negative ions by estimating uncertainties tothe CC results, and compare them with other calculations wherever available. There are several experimental techniques available toproduce negative alkali ions in the laboratory. The tech-niques to produce these ions have been engineered timeto time over the several decades [1, 2]. Electron affinities(EAs) of these systems have been measured very pre-cisely [3–7]. Photoabsorption spectra of these ions havealso been extensively investigated both theoretically andexperimentally, in order to understand their structures[8–11]. Starting from seventies, a series of studies onthe photodetachment of negative lithium (Li − ), sodium(Na − ) and potassium (K − ) ions have been conductedby several groups [5, 12–15]. Theoretical results fromthese lighter ions were in excellent agreement with thecorresponding experimental values. By solving a set ofcoupled equations, Norcross had predicted the existenceof bound excited states in negative alkali ions [16]. Thiswas later conformed by Greene [17], by applying a com-bined approach of jj -coupling in the R -matrix methodand generalized quantum-defect theory while attempt-ing to describe photodetachment spectra of negative ru-bidium (Rb − ), cesium (Cs − ) and francium (Fr − ) ions.Later, another study [18] disproved the existence of suchstates in these ions by reanalyzing the calculations us-ing the Dirac R -matrix method. Instead, the authors ofthe work suggested that the lowest excited state of theabove alkali negative ions is a multiplet of P oJ -shape res-onance. This work clearly demonstrated the importanceof relativistic effects for accurate calculations of atomicproperties in these ions. Similarly, several studies on neg-ative hydrogen (H − ) ion have been carried out [19, 20],and its applications [21, 22] and production sources arewell-known to the physicists [23, 24].Apart from EAs and photodetachment cross-sections,there is scarcity in the atomic data of the negative alkaliions. Alkali atoms have a closed-shell and a valence or-bital in their electronic configurations. Owing to this, itis relatively simpler to calculate atomic wave functionsof these atoms. However, it is challenging to determine ∗ [email protected] atomic wave functions of the alkaline earth-metal atomsdue to strong electron correlations among the valenceelectrons in such systems. The negative alkali ions areisoelectronic to the alkaline earth-metal atoms.Electric dipole polarizability ( α d ) is a very useful prop-erty of any atomic system. This quantity has been mea-sured very precisely [25] in the alkali atoms as well asin the singly charged positive alkaline earth-metal ions,and with reasonable accuracy in the alkaline earth-metalatoms. However, there has not been a single measure-ment of α d carried out thus far in any of the negativealkali ions due to difficulties in setting up their exper-iments. There are no full relativistic calculation of α d available in these ions, and only a few non-relativisticcalculations have been reported in the lighter H − [26–28], Li − [13, 29, 30], Na − [13] and K − [13] ions. Exceptthe high-precision calculations in H − , the reported valuesof other ions are not very reliable.Calculations of α d in the alkali atoms are in very goodagreement with the experiments [31, 32]. The reason forthis is that one can easily use their experimental ener-gies and electric dipole (E1) matrix elements inferringfrom the lifetime measurements of their atomic statesin the evaluation of α d values using the sum-over-statesapproach. It is possible to adopt the sum-over-states ap-proach in these atoms because they possess a large num-ber of bound states in contrast to the negative alkali ions. Ab initio procedures demonstrate that the electron cor-relation and the relativistic effects are pronounced, andthey need to be accounted for, in order to determine α d values of alkaline earth-metal atoms [33–35], which areisoelectronic systems to the negative alkali ions. A fewcalculations of the α d values of some heavier negative ionsof the coinage metal atoms have been reported by Sadlejand coworkers [36, 37], by employing a variety of methodsincluding the coupled-cluster (CC) theory. In anotherstudy by Schwerdtfeger and Bowmaker [38], these quan-tities were also evaluated by using total angular momen-tum j -averaged relativistic pseudo-potentials in the con-figuration interaction (CI) method. These works high-light about the unusually large electron correlation andrelativistic effects in the determination of α d values inthe negative ions compared to their counter isoelectronicneutral atoms. But the relativistic effects were estimatedonly approximately in these calculations. Recently, wehad evaluated α d values of Cl − and Au − by applying anumber of relativistic many-body methods at differentlevels of approximation to demonstrate the roles of elec-tron correlations for their accurate determination. Here,we intend to determine α d values of all the negative alkaliions very accurately.It is not possible to adopt a finite-field (FF) approachto determine α d of atomic states by preserving spher-ical symmetry property and treating parity as a goodquantum number. Thus, the spherical symmetry of thesystems is exploited in Refs. [36–38] in order to adoptthe FF procedure for the determination of α d of nega-tive ions. Also, the FF approach introduces large uncer-tainty to the calculation of α d , which stems from both nu-merical differentiation as well as neglecting higher-orderperturbation corrections in the evaluation of the second-order perturbed energy due to external electric-field, in abrute-force manner. To overcome these problems in thedetermination of α d while retaining spherical symmetrybehavior of atomic orbitals, we follow a perturbative ap-proach in which the total Hamiltonian of the system isdefined in the presence of a weak external electric field ~ E as H = H at + ~D · ~ E with the atomic Hamiltonian H at and electric dipole operator D in a similar framework asDalgarno and Lewis [39]. In such case, the wave functionand energy of an atomic state can be expressed as | Ψ i = | Ψ (0)0 i + | ~ E|| Ψ (1)0 i + · · · (1)and E = E (0)0 + | ~ E| E (1)0 + 12 | ~ E| E (2)0 · · · , (2)respectively, where superscripts (0), (1), etc. denote or-der of ~ E in the expansion. Since D is an odd-parity opera-tor, E (1)0 = 0 and the second-order energy is traditionallygiven by E (2)0 ≡ α d . It follows that α d can be evaluatedin the perturbative approach as [34] α d = 2 h Ψ (0)0 | D | Ψ (1)0 ih Ψ (0)0 | Ψ (0)0 i . (3)Thus, it is imperative to determine both the unperturbedwave function | Ψ (0)0 i of H at and the first-order perturbedwave function | Ψ (1)0 i due to D very reliably for an ac-curate evaluation of α d . Instead of using the sum-over-states approach to determine | Ψ (1)0 i , we would like tosolve it as the solution to the first-order inhomogeneousperturbed equation given by( H at − E (0)0 ) | Ψ (1)0 i = − D | Ψ (0)0 i . (4)Though the solution of this equation appears to be sim-ilar to the procedure adopted by Dalgarno and Lewis[39], but it can be kept in mind that we only obtain the first-order wave function | Ψ (1)0 i for Eq. (3) instead ofdetermining E (2)0 directly.The many-electron atomic wave function can be ob-tained by | Ψ i = Ω | Φ i , (5)where | Φ i is a mean-field wave function, which is ob-tained here by the Dirac-Hartree-Fock (DHF) method,and Ω is known as the wave operator that is responsiblefor accounting for electron correlation effects due to theinteractions that are neglected in the determination of | Φ i . Likewise, for the wave function, we can expand Ω in the presence of weak electric field ~ E asΩ = Ω (0)0 + | ~ E| Ω (1)0 + · · · . (6)Using this, we can write the unperturbed and the first-order perturbed wave function as | Ψ (0)0 i = Ω (0)0 | Φ i and | Ψ (1)0 i = Ω (1)0 | Φ i . (7)In the n th -order perturbation theory, we express [35]Ω (0) = n X k =0 Ω ( k, and Ω (1) = n − X k =0 Ω ( k, (8)with Ω (0 , = 1, Ω (1 , = 0 and Ω (0 , = P p,a h Φ pa | D | Φ i ǫ (0) a − ǫ (0) p for all the occupied orbitals denoted by the index a and unoccupied orbitals denoted by the index p .In the second-order relativistic perturbation theory(RMBPT(2) method) that accounts for the lowest-orderelectron correlation effects [35] in the many-body theory,it corresponds to n = 1 in the above summations.We present results from two all-order many-bodymethods: relativistic random-phase approximation(RRPA) and relativistic CC (RCC) theory. The RCCtheory incorporates electron correlation effects more rig-orously, while RRPA has traditionally been employed tocapture these effects due to the core-polarization only,which can be done to all-orders in a computationallymuch less expensive way. The correlation effects aris-ing through RRPA also represent the orbital relaxationeffects that arise naturally in the mixed-parity orbitals ofthe DHF method in the FF procedure. In our RRPA im-plementation [40, 41], they are contained in Ω (0)0 = 1 andΩ (1) = P ∞ k =0 P p,a Ω ( k, a → p . Here, a → p means replace-ment of an occupied orbital a from | Φ i by a virtualorbital p , which alternatively refers to a singly excitedstate with respect to | Φ i . The RCC theory implicitlyincludes correlation effects arising through RRPA alongwith other correlation effects such as pair-correlation ef-fects to all-orders and is known as the gold standardmethod of many-body theory for its capabilities of pro-ducing accurate results in multi-electron systems. In thistheory, the wave operators are given by [40, 42]Ω (0) = e T (0) and Ω (1) = e T (0) T (1) , (9)respectively. We consider only singles and doubles ex-citations in the RCC calculations (RCCSD method) byexpressing T (0) = T (0)1 + T (0)2 and T (1) = T (1)1 + T (1)2 , (10)where subscripts (1) and (2) denote the level of excita-tion. In this method, α d determined as α d = 2 h Φ | Ω (0) † D Ω (1) | Φ ih Φ | Ω (0) † Ω (0) | Φ i = 2 h Φ | ( z}|{ D (0) T (1) ) c | Φ i , (11)where z}|{ D (0) = e T † (0) De T (0) is a non-truncating series.We have adopted an iterative procedure to take into ac-count contributions from this non-terminating series self-consistently, as described in our earlier works on α d cal-culations in the closed-shell atoms [42, 43].For the evaluation of | Ψ i , we consider first the Dirac-Coulomb (DC) Hamiltonian, given by H DC = X i (cid:2) c α i · p i + ( β i − c + V n ( r i ) (cid:3) + X i,j>i r ij , (12)where c is the speed of light, α and β are the usual Diracmatrices, p i is the single particle momentum operator, V n ( r i ) denotes the nuclear potential, and r ij representsthe Coulomb potential between two electrons located atthe i th and j th positions. We estimate the Breit interac-tion by using the Dirac-Coulomb-Breit (DCB) Hamilto-nian ( H DCB = H DC + V B ) by defining the potential V B = − X j>i [ α i · α j + ( α i · ˆr ij )( α j · ˆr ij )]2 r ij , (13)where ˆr ij is the unit vector along r ij . Similarly, contribu-tions from the quantum electrodynamics (QED) effectsare estimated using the Dirac-Coulomb-QED (DCQ)Hamiltonian ( H DCQ = H DC + V Q ) by considering V Q = V V P + V SE with the vacuum polarization interaction po-tential V V P and the self-energy interaction potential V SE .We use the model potentials for V V P and V SE as definedin Refs. [44, 45].We use Gaussian type orbitals (GTOs), as defined inRef. [46], to obtain the single particle orbitals. Wehave considered orbitals up to h -angular momentum sym-metry (orbital angular momentum l = 5) to carry outall the calculations. We have used 40 GTOs for eachsymmetry to obtain the DHF wave function. However,we have frozen high-lying orbitals beyond energy 3000atomic units (a.u.) to account for electron correlationeffects through the employed many-body methods. Wehave verified contributions from these neglected orbitalsusing RRPA and they are found to be extremely small.These contributions are included in the uncertainty esti-mation later.In Table I, we present α d values of all the negative al-kali ions in a.u., ea , from the DHF, RMBPT(2), RRPAand RCCSD methods using the DC, DCB and DCQHamiltonians. It can be seen from this table that the TABLE I. Calculated α d values (in ea ) of the negative alkali-metal ions from the DHF, RMBPT(2), RRPA and RCCSDmethods. Results from the DC, DCB and DCQ Hamiltoniansare listed separately to highlight the roles of Breit and QEDinteractions in the determination of α d of the above ions.Method DC DCB DCQH − ionDHF 44.41 44.61 44.41RMBPT(2) 66.35 66.35 66.35RRPA 91.13 91.13 91.13RCCSD 206.14 206.16 206.15Li − ionDHF 500.90 500.94 500.91RMBPT(2) 764.68 764.74 764.70RRPA 1176.68 1176.76 1176.70RCCSD 794.06 794.08 794.07Na − ionDHF 605.91 606.00 606.03RMBPT(2) 923.52 923.66 923.72RRPA 1447.69 1447.92 1448.01RCCSD 952.54 952.63 952.67K − ionDHF 1053.39 1053.58 1053.89RMBPT(2) 1586.91 1587.18 1587.68RRPA 2565.41 2565.88 2566.66RCCSD 1353.69 1353.84 1354.19Rb − ionDHF 1214.16 1214.43 1215.40RMBPT(2) 1816.88 1817.22 1818.75RRPA 2968.55 2969.21 2971.66RCCSD 1506.57 1506.75 1507.88Cs − ionDHF 1534.15 1534.48 1537.06RMBPT(2) 2271.42 2271.78 2275.81RRPA 3770.88 3771.71 3778.25RCCSD 1800.42 1800.54 1803.59Fr − ionDHF 1357.23 1357.78 1357.33RMBPT(2) 1990.66 1991.21 1990.77RRPA 3308.49 3309.76 3308.69RCCSD 1619.04 1619.16 1619.03 DHF method gives lower values in all the ions, and theelectron correlation contributions enhance their magni-tudes. Except in the H − ion, the RRPA yields the largestvalue for each ion. The results from the RMBPT(2)method are also larger than the values obtained using theRCCSD method, except in the lighter H − and Li − ions.It can be recalled that the results from the RMBPT(2)method are actually the lowest-order core-polarizationterms and results from RRPA are the contributions fromthe all-order core-polarization effects including the DHF −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 10 20 30 40 50 60 70 80 90 R a t i o ( X − DH F ) / DH F Atomic number (Z)DFRMBPT(2)RRPARCCSD
FIG. 1. (X-DHF)/DHF values, where X represents contribu-tions from the DHF, RMBPT(2), RRPA and RCCSD meth-ods, against the atomic number ( Z ) of the negative alkali ions.Values from the DHF method act as reference (zero line on x-axis) for comparison of correlation contributions incorporatedat different levels of many-body methods. value. Thus, the huge differences between the resultsfrom the RMBPT(2) and RRPA methods imply that thecore-polarization effects arising through the higher-orderperturbation theory are quite strong in the negative alkaliions. Since RRPA values are the mean-field contributions(i.e. DHF values) in the FF procedure, it would requireimmense efforts to attain convergence in the values byevaluating the property in the FF framework than in theperturbation approach through a many-body method asadopted here. Nonetheless, the RCCSD method implic-itly incorporates the RRPA contributions in addition tocorrelation effects due to non-RRPA effects such as thosefrom the pair-correlations. Large differences between theRRPA and RCCSD results indicate that the non-RRPAcontributions are also substantially large, but with op-posite sign than that of the RRPA contributions. As aresult, the final results in the RCCSD method come outto be smaller than the RMBPT(2) values in the heavierions.The electron correlation effects in the simplest twoelectron H − ion, which is analogous to He atom, showsa unique trend than the rest of the ions. In this sys-tem, the inclusion of electron correlation effects increasesthe α d values gradually through the RMBPT(2), RRPAand RCCSD methods. Recently, we have evaluated thisproperty for the negative chlorine (Cl − ) and gold (Au − )ions by employing the above many-body methods [41].We find similar trends in the electron correlation effectsin H − and Cl − , but the correlation contributions at dif-ferent levels of approximations in the many-body theoryare found to be quite large in H − compared to Cl − . TheDHF value of α d was larger than the RCCSD value inAu − , in contrast to the trend seen in the negative alkaliions. In Fig. 1, we plot the fractional differences of theresults from the DC Hamiltonian from different meth-ods against the atomic number ( Z ) of the negative alkaliions. This shows that scaling of correlation contributions TABLE II. Contributions from the DC Hamiltonian throughdifferent RCC terms in the determination of α d (in ea ) of thenegative alkali ions. The differences between the sum of thecontributions from the mentioned terms and the final valuesfrom the RCCSD method given using the DC Hamiltonian inTable I correspond to the contributions from the remainingRCC terms that are not shown explicitly here. Ion DT (1)1 T (0) † DT (1)1 T (0) † DT (1)1 T (0) † DT (1)2 T (0) † DT (1)2 H − − . − .
99 32.25Li − − . − .
09 25.45 127.48Na − − . − .
33 34.24 133.65K − − . − .
67 69.89 197.74Rb − − . − .
50 82.39 209.90Cs − − . − .
24 105.36 250.74Fr − − . − .