Ab initio MCDHF calculations of the In and Tl electron affinities and their isotope shifts
Ran Si, Sacha Schiffmann, Kai Wang, Chong Yang Chen, Michel Godefroid
AAb initio MCDHF calculations of the In and Tl electron affinitiesand their isotope shifts
Ran Si,
1, 2
Sacha Schiffmann,
2, 3
Kai Wang, Chong Yang Chen, ∗ and Michel Godefroid † Shanghai EBIT Lab, Key Laboratory of Nuclear Physics and Ion-beam Application,Institute of Modern Physics, Department of Nuclear Science and Technology,Fudan University, Shanghai 200433, Peoples Republic of China Spectroscopy, Quantum Chemistry and Atmospheric Remote Sensing (SQUARES),CP160/09, Universit´e libre de Bruxelles, 1050 Brussels, Belgium Division of Mathematical Physics, Department of Physics,Lund University, Box 118, SE-22100 Lund, Sweden Hebei Key Lab of Optic-electronic Information and Materials,The College of Physics Science and Technology,Hebei University, Baoding 071002, Peoples Republic of China a r X i v : . [ phy s i c s . a t o m - ph ] F e b bstract We report multiconfiguration Dirac-Hartree-Fock and relativistic configuration interaction cal-culations on the Thallium (Tl) electron affinity, as well as on the excited energy levels arisingfrom the ground configuration of Tl − . The results are compared with the available experimentalvalues and further validated by extending the study to its homologous, lighter element, Indium(In), belonging to Group 13 (III.A) of the periodic table. The calculated electron affinities of Inand Tl, 383.4 and 322.8 meV, agree with the latest measurements by within 1%. Three boundstates P , , are confirmed in the 5 s p configuration of In − while only the ground state P isbound in the 6 s p configuration of Tl − . The isotope shifts on the In and Tl electron affinitiesare also estimated. The E2/M1 intraconfiguration radiative transition rates within 5 s p P , , of In − are used to calculate the radiative lifetimes of the metastable P , levels. I. INTRODUCTION
Negative ions play a major role in a number of areas of physics and chemistry involvingionized gases and plasma. Since there is no long-range Coulomb interaction between the out-ermost electron and the atomic core, their properties critically depend on electron-electroncorrelation and polarization and negative ions will only have a few bound states. In mostcases, the latter have the same parity or even belong to the same electronic configuration.Only in a few cases, namely, Os − [1–3], Ce − [4, 5], La − [6–8], and Th − [9], negative ionshave excited bound states of opposite parity to that of the ground state making them goodcandidates for laser cooling. The most promising ones are so far La − and Th − .Several experimental techniques are possible to measure atomic electron affinities (EAs)and excited energy levels of negative ions with high precision. Nevertheless some atomicelectron affinities and anion fine-structure splittings are still known with limited accuracy.All elements of Group 13 (B, Al, Ga, In, and Tl) form stable negative ions with electronaffinities of less than 0.5 eV. The latter are therefore challenging to determine with accuracy,especially for the heavier elements. Recently, tunable laser photodetachment threshold spec-troscopy (LPTS) was used to measure the electron affinity of the 6 s p P ground state of ∗ [email protected] † [email protected] Tl − to be 320.053(19) meV [10], which differs significantly from the value of 377(13) meVobtained by a fixed-frequency laser photodetachment electron spectroscopy (LPES) mea-surement [11]. Both experiments indicate that the excited levels are either unbound or tooweakly bound to be detected, although the three fine-structure levels P , , were detectedas bound states for the lighter elements of the same Group 13 (B − , Al − , Ga − and In − ) [12–17]. Number of theoretical studies on the EA of Tl have been reported using a variety ofcomputational methods [18–25], but their results show poor agreement. For example, usingdifferent theoretical methods, Arnau et al. [18] and Felfli et al. [24] predicted the EA of Tlto be 270 meV and 2415 meV, respectively.In the present study, we resolve the disagreement between experimental and theoreticalEA values of Tl, and explore the existence of bound excited states of Tl − . Since the groundconfigurations of In (5 s p ) and In − (5 s p ) are analogous to those of Tl (6 s p ) and Tl − (6 s p ), we use the In/In − system as a benchmark to support our Tl/Tl − analysis. Wealso estimate the balance between the nuclear mass and volume contributions to the isotopeshift (IS) on electron affinities of different isotopes of In and Tl. The radiative lifetimes ofthe excited In − s p P , fine structure levels based on the intraconfiguration radiativedecay rates are reported. II. THEORYA. Multiconfiguration Dirac-Hartree-Fock approach
The MCDHF method [26], as implemented in the
Grasp computer package [27, 28], isemployed to obtain wave functions that are referred to as atomic state functions (ASFs),i.e., approximate eigenfunctions of the Dirac-Coulomb Hamiltonian given by H DC = N (cid:88) i =1 ( c α i · p i + ( β i − c + V i ) + N (cid:88) i P J ) = M (cid:88) i =1 c i Φ ( γ i P J ) , (2)where γ i represents all the coupling tree quantum numbers needed to uniquely define theCSF, besides the parity P and the total angular momentum J . The CSFs are four-componentspin-angular coupled, antisymmetric products of Dirac orbitals of the form φ ( r ) = 1 r P nκ ( r ) χ κm ( θ, φ ) iQ nκ ( r ) χ − κm ( θ, φ ) . (3)The radial parts of the one-electron orbitals and the expansion coefficients c i of the CSFsare obtained by the relativistic self-consistent field (RSCF) procedure. In the present paper,the CSF expansions are obtained using the restricted active set (RAS) method, by allowingsingle and double (SD) substitutions from a selected set of reference configurations to agiven orbital active set. The latter is systematically expanded by the addition of successiveorbital layers to monitor the convergence of the calculated energies or any other relevantobservable.Each RSCF calculation is followed by a relativistic configuration interaction (RCI) cal-culation, where the Dirac orbitals are kept fixed and only the expansion coefficients of theCSFs are determined for selected eigenvalues and eigenvectors of the complete interactionmatrix. In this procedure, the Breit interaction and leading quantum electrodynamic (QED)effects (vacuum polarization and self-energy) are included.In addition to energy levels, lifetimes τ and transition parameters, such as transition rates A and line strengths S , are also computed. The transition parameters between two states γ (cid:48) P (cid:48) J (cid:48) and γP J are expressed in terms of reduced matrix elements of the relevant transitionoperators [29, 30]: (cid:104) Ψ( γP J ) || T || Ψ( γ (cid:48) P (cid:48) J (cid:48) ) (cid:105) = (cid:88) k,l c k c (cid:48) l (cid:104) Φ( γ k P J ) || T || Φ( γ (cid:48) l P (cid:48) J (cid:48) ) (cid:105) . (4)Biorthogonal orbital transformations and CSF-expansion counter-transformations are used[31] when radial non-orthogonalities arise from the independent optimization of the twoASFs involved in (4). 4 . Isotope shift We define the isotope shift (IS) on the EA between two isotopes of mass A and A (cid:48) asIS(EA) AA = EA( A ) − EA( A (cid:48) ) , (5)in agreement with the frequency isotope shift definition adopted in the description of theRIS3 [32] and RIS4 [33] codes. In the A > A (cid:48) case, a positive isotope shift on the EA impliesa larger electron affinity for the heavier isotope. Such an IS is qualified as a “normal”IS, referring to the normal mass shift encountered for the one-electron atomic hydrogencharacterized by a blueshift of the spectral lines of deuterium H compared with hydrogen H. The IS can be decomposed into two contributions: the mass shift (MS) and the field shift(FS). They arise, respectively, from the recoil effect due to the finite mass of the nucleus, andfrom the difference in nuclear charge distributions between the two isotopes. The revisedversion of the RIS3 code [32], RIS4, based on a reformulation of the field shift [33], is usedfor the computations of IS parameters in the present paper. 1. Mass shift The isotope mass shift of an atomic level i is obtained by evaluating the expectationvalues of the operator, H MS = 12 M N (cid:88) j,k (cid:18) p j · p k − αZr j (cid:18) α j + ( α j · r j ) r j r j (cid:19) · p k (cid:19) , (6)where M is the nuclear mass of the isotope [34].Separating the operator into one-body and two-body terms, H MS can be rewritten as thesum of normal mass shift (NMS) and specific mass shift (SMS) contributions, H MS = H NMS + H SMS . (7)The (mass-independent) normal mass shift K NMS and specific mass shift K SMS parametersfor a level i are defined by the following expressions K i , NMS M ≡ (cid:104) Ψ i ( γP J ) | H NMS | Ψ i ( γP J ) (cid:105) , (8) K i , SMS M ≡ (cid:104) Ψ i ( γP J ) | H SMS | Ψ i ( γP J ) (cid:105) . (9)5he mass shift parameters can be decomposed into three parts, K i , NMS = K i , NMS + K i , NMS + K i , NMS , (10) K i , SMS = K i , SMS + K i , SMS + K i , SMS . (11)where the K terms refer to the “uncorrected” relativistic contributions (first term of (6))and the sum K + K (2nd and 3rd terms of (6)) to the lowest-order relativistic correctionsin the Breit approximation [35, 36]. The mass shift contribution to the IS on the EAIS(EA) AA MS = ∆ K MS (cid:18) M − M (cid:48) (cid:19) , (12)is therefore directly proportional to the difference of the MS electronic parameters∆ K MS = K g . s ., MS − K − g . s ., MS , (13)where the minus exponent refers to quantities related to the negative ion and g.s. standsfor ground state. 2. Field shift In the first order perturbation approximation, the field shift for a given level i can beexpressed as δE (1) A,A (cid:48) i, FS = − (cid:90) R (cid:104) V A ( r ) − V A (cid:48) ( r ) (cid:105) ρ ei ( r ) d r , (14)where V A ( r ) and V A (cid:48) ( r ) are the potentials arising from the nuclear charge distributions ofthe two isotopes and ρ ei ( r ) is the electron density. By approximating the electron density atthe origin with a spherically symmetric even polynomial function ρ ei ( r ) ≈ b i, + b i, r + b i, r + b i, r , (15)Eq. (14) can be expressed as δE (1) A,A (cid:48) i, FS ≈ (cid:88) ≤ n ≤ ,even F i,n δ (cid:104) r n +2 (cid:105) A,A (cid:48) (16)where F i,n are level electronic factors and δ (cid:104) r n (cid:105) A,A (cid:48) = (cid:104) r n (cid:105) A − (cid:104) r n (cid:105) A (cid:48) . Assuming a constantelectron density within the nuclear volume, we get from the first term of Eq. (16) δE (1) A,A (cid:48) i, FS ≈ F i, δ (cid:104) r (cid:105) A,A (cid:48) . (17)6s suggested in Ref [33], we can include the effect of a varying electronic density (ved)to evaluate the FS, by introducing the appropriate corrected level electronic factors, F (0)ved i, and F (1)ved i, , without considering higher order nuclear moments. Eq. (16) is then replaced by δE (1) A,A (cid:48) i, FS ≈ (cid:16) F (0)ved i, + F (1)ved i, δ (cid:104) r (cid:105) A,A (cid:48) (cid:17) δ (cid:104) r (cid:105) A,A (cid:48) . (18)The FS contribution to the IS on the EA can therefore be estimated fromIS(EA) AA FS = (cid:0) F g . s ., − F − g . s ., (cid:1) δ (cid:104) r (cid:105) A,A (cid:48) (19)with (cid:0) F g . s ., − F − g . s ., (cid:1) = ∆ F (20)in the constant electron density approximation, or (cid:0) F g . s ., − F − g . s ., (cid:1) = (cid:16) ∆ F (0)ved0 + ∆ F (1)ved0 δ (cid:104) r (cid:105) A,A (cid:48) (cid:17) (21)using the varying electronic density model. III. RESULTS AND DISCUSSIONSA. Electron affinities To evaluate the ground state energies of the Tl neutral atom ([Xe]4 f d s p P o / )and Tl − anion ([Xe]4 f d s p P ), we start from single-reference (SR) calculations,where CSFs lists are generated by allowing single and double (SD) excitations from the 5 d ,6 s and 6 p electrons to orbitals with n ≤ l ≤ 5. The CSFs that contribute by more than0.1% in weight ( w i = | c i | ) to the ground states wave functions of Tl and Tl − are reportedin Table I. These configurationsTl (odd) : 5 d s p, d f s p, d p , Tl − (even) : 5 d s p , d f s p , d p , d s p d, d f s p p (22)form the multireference spaces used in the following.In the MR calculations, SD excitations are allowed from all MR configurations to anincreasing active set (AS) of orbitals. The calculations are performed layer by layer, intro-ducing at each step at most one new correlation orbital per angular κ -symmetry. Excitations7rom the SR configuration are ultimately allowed to an active set of orbitals with n ≤ 13 and l ≤ 5, noted 13 h . Excitations from the remaining MR configurations are, however, limitedto smaller active sets, i.e., 6 s p d f for the neutral atom and 8 s p d f for the anion, tokeep the number of CSFs manageable. The slightly larger active set for the anion is re-quired to balance correlation effects between the atom and its anion. Two sets of calculatedEAs with increasing ASs are listed in Table II. One is EA(∆ n =0)= E n (Tl) − E n (Tl − ), where E n labels the energy obtained with the AS of maximum principal quantum number n , theother is EA(∆=1)= E n (Tl) − E n +1 (Tl − ), that can be justified by the fact that more orbitalsare needed to describe electron correlation for the negative ion than for the neutral atom.The former is increasing with active sets and provides a lower bound to the calculated EA,whereas the latter is decreasing and hence provides an upper bound. We used a non-linearexponential decay function to extrapolate the last four EA(∆ n =0) and EA(∆ n =1) values,and adopted the intersection as our final theoretical EA value [37]. The theoretical uncer-tainty of the ab initio EA inevitably depends on the correlation models used for tailoringthe ASF expansions. Based on our passed experience on complex systems [9, 37] and onthe comparison with observation [9, 38], we estimated it to be less than 2%. This con-servative estimation covers the 0.8% corresponding to half of the interval between the twoEA(∆ n = 0 , 1) values and the smaller uncertainty (0.65%) associated with the extrapolationprocedure. With this uncertainty estimation, our final thallium EA-value is 322.8(6.5) meV.A comparison between experimental and theoretical EA values is presented in Table III.One can see that our final thallium EA-value of 322.8(6.5) meV agrees with the very recentLPTS experimental value of 320.053(19) meV [10], but definitely lies outside the error barsof the previous LPES measurement 377(13) meV [11]. As mentioned in the introduction, thescattering of theoretical results is surprisingly large. Amongst the most recent works, ourMCDHF-RCI value is in good agreement with the results of Finneyet et al. [25] using therelativistic coupled-cluster version of the Feller-Peterson-Dixon composite method (RCC-FPD). Our theoretical estimation is definitely smaller - by almost one order of magnitude -than the complex angular momentum (CAM) electron elastic total cross-sections result ofFelfli et al. [24]. The last authors suggested that all the EA values of thallium reported inthe literature before their work should be considered as the binding energy of an excitedstate of the anion. All RCI calculations performed in the present work for the odd parityexclude this possibility, the lowest state, 5 d s p S , being estimated to lie around 5.2 eV8 ABLE I. Wave function compositions for the ground states of Tl and Tl − . See text for thedefinition of the weights w i .Tl Tl − Configuration Term w i (%) Configuration Term w i (%)5 d s p P o d s p P d ( P )5 f ( S )6 s p P o d s p S d ( F )5 f ( S )6 s p P o d ( P )5 f ( S )6 s p P d p P o d ( F )5 f ( S )6 s p P d ( D )5 f ( S )6 s p P o d ( D )5 f ( S )6 s p P d ( S )5 f ( S )6 s p P o d p P d ( G )5 f ( S )6 s p P o d ( S )5 f ( S )6 s p P d ( G )5 f ( S )6 s p P d s p ( D )6 d P d f ( P )6 s ( P o )6 p ( D )7 p P d f ( P )6 s ( P o )6 p ( D )7 p P d f ( P )6 s ( P o )6 p ( P )7 p P above the ground level of Tl − .Since In and In − ( Z = 49) are homologous elements to Tl and Tl − ( Z = 81), we per-formed similar calculations for the electron affinity of In to further validate our calculated EAof thallium. SD excitations from 4 d s p and 4 d s p are allowed up to orbitals with n ≤ l ≤ 5; SD excitations from 4 d f s p and 4 d p are allowed up to the 5 s p d f AS for the In neutral atom; SD excitations from 4 d f s p , 4 d p , 4 d s p d and4 d f s p p are included up to the 7 s p d f AS for the In − anion. The calculatedEA(∆ n =0) and EA(∆ n =1) values are also listed in Table II. Using the same extrapolationmethod as for Tl − , we obtain a final value of 383.(7.7) meV for which we adopted the 2% un-certainty estimation, as discussed above, that largely covers half of the interval between thetwo EA(∆ n = 0 , 1) values (1.1%). This theoretical result is also in excellent agreement withthe LPTS experimental value of 383.92(6) meV [17]. Similarly as for thallium, our indiumEA value lies outside the confidence interval of the LPES measurement of 404(9) meV [12].9 ABLE II. Theoretical electron affinities of In and Tl (present work). All values in meV. In TlAS EA(∆ n =0) EA(∆ n =1) AS EA(∆ n =0) EA(∆ n =1)9h 356.2 428.6 10h 303.3 342.110h 368.5 398.6 11h 311.9 327.911h 374.4 390.0 12h 315.9 324.712h 378.1 386.5 13h 318.5 323.7Final 383.4(7.7) 322.8(6.5) B. Isotope shifts on the Indium and Thallium electron affinities Both In and Tl have many isotopes. While the stable isotope In is only 4.3% ofnaturally occuring indium, in lower abundance than the long-lived radioactive isotopes, Tlhas two stable isotopes, Tl (30% natural abundance) and Tl (70%). In this section, wereport the isotope shifts on the electron affinity of In and Tl by using the wave functionsobtained for estimating the electron affinity (see previous section). The differences of theisotope shift mass and field electronic parameters that make the IS on the EA (see Eqs. (13and (19)) are listed in Table IV.From this table, we can see that for In, the sum of the relativistic corrections to the un-corrected one-electron normal mass shift operator, ∆( K + K ), reinforces the ∆ K value by around 50%, while for the SMS, ∆( K + K ) counterbalances ∆ K by 85%,leading to a large dominance of ∆ K NMS over ∆ K SMS .Relativistic corrections to the recoil operator play an even more important role in Tl, forwhich ∆( K + K ) is 2.6 times larger than the uncorrected ∆ K value and stronglystrengthen it. For the SMS contribution, the ∆( K + K ) is 3 . K but of opposite sign. Oppositely to In, the total ∆ K SMS value is large, 66% of the∆ K NMS , and the constructive addition of both contributions makes the total ∆ K MS value1 . 66 times larger than the NMS contribution.The differences of the electronic FS parameters are reported in the same Table IV forboth In/In − and Tl/Tl − systems. Positive ∆ F values reveal a gain in electron density at10 ABLE III. Comparison of the present calculated electron affinities of In and Tl with the ex-perimental results and other theoretical values. LPTS: laser photoelectron threshold spectrocopy,LPES: laser photodetachment electron spectroscopy, MCDHF: Multiconfiguration Dirac-Hartree-Fock calculations (present work), CIPSI: multireference single and double configuration-interactionmethod, RCC: relativistic coupled cluster, HFR-DFT: pseudo-relativistic HartreeCFock and den-sity functional theory, IHFSCC: intermediate-Hamiltonian Fock-space coupled cluster method,CAM: complex angular momentum method, RCC-FPD: relativistic coupled-cluster version of theFeller-Peterson-Dixon composite method. (All values in meV.) Method EA(In) EA(Tl)ExperimentWalter et al. [17] LPTS 383.92(6)Walter et al. [10] LPTS 320.053(19)Williams et al. [12] LPES 404(9)Carpenter et al. [11] LPES 377(13)TheoryPresent work MCDHF 383.4(7.7) 322.8(6.5)Wijesundera [19] MCDHF 393 291Li et al. [23] MCDHF 397.83 290.20Arnau et al. [18] CIPSI 380 270Eliav et al. [20] RCC 419 400Chen and Ong [21] HFR-DFT 429 388Figgen et al. [22] IHFSCC 403 347Felfli et al. [24] CAM 380 2415Finneyet al. [25] RCC-FPD 386 320 the nucleus when detaching the outer electron from the anion. One should observe thatthis gain factor is 10 times larger for thallium than for indium. The ratio of the Tl andIn nuclear charges Z (Tl) /Z (In) ≈ . 65, arising from the explicit linear Z -dependence ofthe FS factor [32, 33], can only explain a little portion of this difference. The much largerremaining part of this factor ten is simply due to the Tl-In difference of the electron density11ithin the nuclear volume. The FS contribution to the IS on the EA has been estimatedfrom Eqs.(19) and (21) to include the effect of a varying electronic density, using the rootmean square (rms) nuclear radii of from Ref. [39].12 ABLE IV. Isotope shift parameters on the electron affinities of In and Tl. ∆ K values in GHz*u.∆ F and ∆ F ved in GHz/fm , ∆ F (1) ved in GHz/fm . In/In − Tl/Tl − ∆ K -134 -74∆( K + K ) -65 -194∆ K NMS -199 -268∆ K 41 68∆( K + K ) -34 -245∆ K SMS K MS = ∆ K NMS + ∆ K SMS -192 -445∆ F F (0) ved F (1) ved − − The mass, field and total isotope shifts on electron affinities are reported in Table V anddisplayed in Fig. 1 for a large range of isotopes relative to the In and Tl stable isotopes.We observe that for EA(In), the FS is already more important than the MS, while forEA(Tl), the FS largely dominates the MS that becomes almost negligible. This is expectedfor heavy elements, as the mass factor 1 /M − /M (cid:48) = ( M (cid:48) − M ) /M M (cid:48) decreases rapidly withthe nuclear mass. For example, the MS contribution to the electron affinities between the twostable isotopes of Tl, Tl and Tl, is IS(EA) , = 0 . , = 0 . Tl) is 0.7014 GHz higherthan EA( Tl), i.e. IS(EA) , = 0 . ABLE V. Mass (MS), Field (FS) and total (MS+FS) isotope shifts on EAs relative to the mostabundant isotopes, i.e, EA( A In) - EA( In) and EA( A Tl) - EA( Th). All shifts in GHz.In TlA MS FS MS+FS A MS FS MS+FS104 -0.147 -0.596 -0.744 188 -0.197 -5.397 -5.594105 -0.130 -0.505 -0.635 190 -0.172 -4.645 -4.816106 -0.113 -0.459 -0.572 191 -0.159 -4.297 -4.456107 -0.096 -0.374 -0.469 192 -0.147 -4.137 -4.285108 -0.079 -0.318 -0.397 193 -0.135 -3.76 -3.895109 -0.063 -0.236 -0.298 194 -0.123 -3.644 -3.767110 -0.047 -0.195 -0.241 195 -0.112 -3.165 -3.276111 -0.031 -0.112 -0.143 196 -0.1 -3.15 -3.250112 -0.015 -0.075 -0.090 197 -0.088 -2.707 -2.795 113 0.0 0.0 0.0 198 -0.077 -2.649 -2.725114 0.015 0.034 0.048 199 -0.066 -2.044 -2.110115 0.03 0.106 0.136 200 -0.054 -1.957 -2.011116 0.044 0.147 0.191 201 -0.043 -1.359 -1.402117 0.058 0.206 0.264 202 -0.032 -1.198 -1.231118 0.072 0.237 0.310 203 -0.021 -0.680 -0.701119 0.086 0.290 0.376 204 -0.011 -0.402 -0.413120 0.100 0.317 0.416 205 0.0 0.0 0.0 121 0.113 0.362 0.475 207 0.021 0.688 0.709122 0.126 0.384 0.510 208 0.031 1.370 1.402123 0.139 0.428 0.567124 0.152 0.451 0.602125 0.164 0.484 0.648126 0.176 0.507 0.684127 0.188 0.530 0.719 IG. 1. Mass (MS), Field (FS) and total (MS+FS) isotope shifts (IS) on EAs relative to In and Th, i.e., EA( A In) - EA( In) and EA( A Tl) - EA( Th)). C. ns np levels of In − ( n = 5) and Tl − ( n = 6) The RCI excitation energies of the four levels 5 s p P , , D and S of In − basedon the wave functions described in section III B, are reported in Table VI. The In − energylevels relative to the ground state of In (5 s p P o / ) are also displayed in Fig. 2(a).The excited energy levels excited levels, 5 s p P and P of In − have been ob-served as being stable using the techniques of laser-photodetachment electron spectroscopy(LPES) [12] and laser-photodetachment threshold spectroscopy (LPTS) [17]. Our theoret-ical work confirms the existence of three bound states in In − , all belonging to the P finestructure. The theory-observation agreement with observation is quite satisfactory. Thepresently calculated P energy level agrees within 0.5 meV with the two experimental valueswhile the P energy level is predicted to be 9 meV higher than the result of the LPTSmeasurement [12]. 15imilar RCI calculations were performed for the energy levels 6 s p P , , D and S of Tl − . The corresponding excitation energies are displayed relatively to the Tl groundstate (6 s p P o / ) in Fig. 2(b). We can see that due to the large fine-structure splitting of6 s p P , , , the P level is the only bound state in Tl − , which agrees with the interpre-tation of the recent threshold spectroscopy measurements [10].The D and S levels are both unbound in In − and Tl − (see Fig. 2). For both systems,the D level of the anion and the P o / level of the corresponding neutral atom are almostdegenerate.Recent progress has been done in the measurements of radiative lifetimes of metastablelevels of negative ions using cold storage techniques [41, 42]. It is therefore worthwhile toreport the theoretical lifetimes of the In − s p P , levels that have not been measuredyet so far. Our predicted lifetimes, based on our theoretical M1 and E2 radiative decayrates are reported in Table VI. Looking at the selection rules [30], the P level can onlydecay to the ground state P through a magnetic dipole (M1) process. The P level candecay to P via an electric quadrupole (E2) radiative transition, and to P through bothM1/E2 de-excitations but the E2 transition probabilities are found to be much smaller thanthe M1 amplitudes by at least 2-3 orders of magnitude. This means that the theoreticallifetimes mostly depend on the M1 rates that usually quickly converge with the correlationmodels [43]. The quality of the transition energies is however an important ingredient dueto the λ scaling factor appearing in the M1 spontaneous emission A rate. For this reason,we also report the adjusted lifetimes to the experimental excitation energies [44], measuredin the LPTS [17] experiment. The resulting lifetimes of a few hundreds of seconds, couldbe measured in a cryogenic ion storage ring that was demonstrated to be efficient to storenegative ion beams in the hour time domain [41, 42].16 ABLE VI. Excitation energies (in meV) and radiative lifetimes ( τ , in s) of the In − s p P , lev-els. MCDHF: present work, LPTS: laser photoelectron threshold spectroscopy measurements [17],LPES: laser photodetachment electron spectroscopy [12], MCDHF (adj.): adjusted lifetimes usingthe LPTS experimental transition energies [17]. Excitation energies (meV) τ (s)MCDHF LPTS LPES MCDHF MCDHF (adj.) P P FIG. 2. Energy diagram of 5 s p levels of In − (a) and 6 s p levels of Tl − (b) relative to theground states of In and Tl, respectively. The fine-structure splitting of P o / , / for the neutralatoms are from NIST ASD [45]. V. CONCLUSION In summary, we calculated the EAs of In and Tl to be 383.4 and 322.8 meV, respectively.These results agree with the latest experimental measurements [10, 17] within 1%. Thesignificant disagreement between the present theoretical EAs and the (too large) LPESvalues [11, 12] for both In and Tl systems allows us to discard the latter values againstthe LPTS [10, 17] results. As far as the suggestion made by Felfli et al. [24] is concerned,interpreting the previous thallium electron affinities as the binding energy of the first excitedstates of Tl − , the present MCDHF-RCI results firmly confirm that the experimental value of320.053(19) meV [10] should be definitely assigned to the real electron affinity. The presentcalculations indeed definitely exclude the possibility of a more bound state than 6 s p P .Excitation ground configuration energies of In − and Tl − and their radiative lifetimes arealso estimated. We confirm that In − has three bound states P , , , while Tl − only has onebound state P .The isotope shift on the EAs of along In and Tl isotopes are estimated using the currentlyavailable rms nuclear radii. Although the MS contributes significantly to the indium EAs,it is already smaller than the FS contribution. For the EA(Tl), the FS largely dominatethe MS that becomes almost negligible. The isotope shift between the two stable Tl andTl isotopes is estimated to be IS(EA) − IS(EA) = +0 . − fine structure 6 s p P , levels are ratherlong but could be measured using a cryogenic ion storage ring. We hope that the presentwork will stimulate such experiments in that line. ACKNOWLEDGEMENT We acknowledge support from the Belgian FWO and FNRS Excellence of Science Pro-gramme (EOS-O022818F). SS is a FRIA grantee of the F.R.S.-FNRS. CYC acknowledgessupport from the National Natural Science Foundation of China (Grant No. 12074081 and11974080). 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