RCC calculation of electric dipole polarizability and correlation energy of Cn, Nh^+ and Og: Correlation effects from lighter to superheavy elements
aa r X i v : . [ phy s i c s . a t o m - ph ] F e b RCC calculation of electric dipole polarizability and correlation energy of Cn, Nh + and Og:Correlation effects from lighter to superheavy elements Ravi Kumar, S. Chattopadhyay, D. Angom, and B. K. Mani ∗ Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India Department of Physics, Kansas State University, Manhattan, Kansas 66506, USA Department of Physics, Manipur University, Canchipur 795003, Manipur, India Physical Research Laboratory, Ahmedabad - 380009, Gujarat, India
We employ a fully relativistic coupled-cluster theory to calculate the ground-state electric dipole polarizabil-ity and electron correlation energy of superheavy elements Cn, Nh + and Og. To assess the trend of electroncorrelation as function of Z , we also calculate the correlation energies for three lighter homologs–Zn, Cd andHg; Ga + , In + and Tl + ; Kr, Xe and Rn–for each superheavy elements. The relativistic effects and quantum elec-trodynamical corrections are included using the Dirac-Coulomb-Breit Hamiltonian with the corrections fromthe Uehling potential and the self-energy. The effects of triple excitations are considered perturbatively in thetheory. Furthermore, large bases are used to test the convergence of results. Our recommended values of polar-izability are in good agreement with previous theoretical results for all SHEs. From our calculations we find thatthe dominant contribution to polarizability is from the valence electrons in all superheavy elements. Except forCn and Og, we observe a decreasing contribution from lighter to superheavy elements from the Breit interaction.For the corrections from the vacuum polarization and self-energy, we observe a trend of increasing contributionswith Z . From energy calculations, we find that the second-order many-body perturbation theory overestimatesthe electron correlation energy for all the elements considered in this work. I. INTRODUCTION
The study of superheavy elements (SHEs) is a multidisci-plinary research area which provides a roadmap to investigateand understand several properties related to physics and chem-istry [1–6]. There is however a lack of experimental data onatomic properties of SHEs due to various challenges, suchas low production rate, short half-lives of elements and thelack of the state of the art one-atom-at-a-time experimentalfacility, associated with atomic experiments [1, 7, 8]. More-over, the properties of SHEs can not be predicted based onlighter homologs, as they often differ due to relativistic ef-fects in SHEs [5] . In such cases, the theoretical investigationsof physical and chemical properties provide an important in-sight to the properties of SHEs. Moreover, the benchmark dataon these properties from accurate theoretical predications isimportant for future experiments. Calculating accurate prop-erties of SHEs is, however, a difficult task. The reason for thiscould be attributed to the competing nature of the relativisticand correlation effects in these systems. For a reliable pre-diction of the properties of SHEs both of these effects shouldbe incorporated at the highest level of accuracy. In addition,large basis sets should be used to obtain the converged prop-erties results.The electric dipole polarizability, α , of an atom or ion isa key parameter which used to probe several fundamental aswell as technologically relevant properties in atoms and ions[9–14]. The α for SHEs Cn and Og has been calculated in pre-vious works, Refs. [15–18] and [16, 17, 19, 20], respectively.Though most of these results are using the CCSD(T), there isa large variation in the reported values for both Cn and Og.For example, the value of α reported in CCSD(T) calculation ∗ [email protected] [20] is ≈
25% larger than the similar calculation [19]. Thereason for this can, perhaps, be attributed to the complex na-ture of the electron correlation and relativistic effects in thesesystems. The other point to be mentioned here is that, thebasis used in these calculations are not large. Moreover, theinclusion of the contributions from the Breit interaction andQED corrections is crucial to obtain the accurate and reliablevalues of α for SHEs.In this work, we employ a fully relativistic coupled cluster(RCC) theory based method to calculate the electric dipole po-larizability and the electron correlation energy of superheavyelements Cn, Nh + and Og. RCC is one of the most pow-erful many-body theories for atomic structure calculations asit accounts for the electron correlation to all-orders of resid-ual Coulomb interaction. We have used this to calculate themany-electron wavefunction and the electron correlation en-ergy. The effect of external electric field, in the case of α ,is accounted using the perturbed relativistic coupled-cluster(PRCC) theory [21–25]. One of the key merits of PRCC inthe properties calculation is that it does not employ the sum-over-state [26, 27] approach to incorporate the effects of aperturbation. The summation over all the possible intermedi-ate states is subsumed in the perturbed cluster operators. Theleading order relativistic effects are accounted using the fourcomponent Dirac-Coulomb-Breit no-virtual-pair Hamiltonian[28]. And, the effects of Breit, QED and triples excitationsin coupled-cluster are computed using the implementations inour previous works [21–24].To assess the trend of various electron correlation effectsfrom lighter to SHEs, we also calculate the correlation energyand the contributions from the Breit and QED corrections to α for three lighter homologs in each SHEs: Zn, Cd and Hg ingroup-12; Ga + , In + and Tl + in group-13; and Kr, Xe, and Rnin group-18. In this work, however, we do not report the val-ues of α for these lighter homologs as these have been alreadyreported in our previous works [21, 24, 25]. Here, our mainfocus is to get deeper insight of various correlation effects asa function Z in each of these SHEs. More precisely, we aimedto: accurately calculate the value of α and correlation energyof SHEs Cn, Nh + and Og using RCC and test the convergenceof results with very large basis; study the electron correlationin α for SHEs and assess the trend from lighter to SHEs; andexamine in detail the contributions from the Breit and QEDcorrections to α for SHEs elements and get a deeper insightto the trend of contributions from lighter homologs to SHEs.The remaining part of the paper is organized into five sec-tions. In Sec. II we provide a brief description of the methodused in the polarizability calculation. In Sec. III we providethe calculational details such as the single-electron basis andcomputational challenges associated with polarizability calcu-lation of SHEs. In Sec. IV we analyze and discuss the resultsfrom our calculations. The theoretical uncertainty in our cal-culation is discussed in Sec. V. Unless stated otherwise, allthe results and equations presented in this paper are in atomicunits ( ¯ h = m e = e = / πε = II. METHOD OF CALCULATION
The ground state wavefunction of an N-electron atom or ionin relativistic coupled-cluster (RCC) theory is | Ψ i = e T ( ) | Φ i , (1) where | Φ i is the Dirac-Fock reference wavefunction and T ( ) is the closed-shell coupled-cluster (CC) excitation operator.It is an eigenfunction of the Dirac-Coulomb-Breit no-virtual-pair Hamiltonian H DCB = N ∑ i = (cid:2) c α i · p i + ( β i − ) c − V N ( r i ) (cid:3) + ∑ i < j (cid:20) r i j + g B ( r i j ) (cid:21) , (2)where α and β are the Dirac matrices. The negative-energycontinuum states of the Hamiltonian are projected out by us-ing the kinetically balanced finite GTO basis sets [29, 30], andselecting only the positive energy states from the finite size ba-sis set [31, 32]. The last two terms, 1 / r i j and g B ( r i j ) , are theCoulomb and Breit interactions, respectively.The operators T ( ) in Eq. (1) are the solutions of the cou-pled nonlinear equations h Φ pa | H N + h H N , T ( ) i + hh H N , T ( ) i , T ( ) i + hhh H N , T ( ) i , T ( ) i , T ( ) i | Φ i = , (3a) h Φ pqab | H N + hh H N , T ( ) i , T ( ) i + hhh H N , T ( ) i , T ( ) i , T ( ) i + hhhh H N , T ( ) i , T ( ) i , T ( ) i , T ( ) i | Φ i = . (3b)Here, the states | Φ pa i and | Φ pqab i are the singly- and doubly-excited determinants obtained by replacing one and two elec-trons from the core orbitals in | Φ i with virtual orbitals, re-spectively. And, H N = H DCB − h Φ | H DCB | Φ i is the normal-ordered Hamiltonian.In the presence of an external electric field, E ext , the groundstate wavefunction | Ψ i is modified due to interaction be-tween induced electric dipole moment D of the atom and E ext .We call the modified wavefunction as the perturbed wavefunc-tion, which in PRCC is defined as | e Ψ i = e T ( ) h + λ T ( ) · E ext i | Φ i , (4)where T ( ) is the perturbed CC operator, and λ is a pertur-bation parameter. The wavefunction | e Ψ i is an eigenstate ofthe modified Hamiltonian H Tot = H DCB − λ D · E ext . The per-turbed CC operators T ( ) in Eq. (4) are the solutions of thelinearized PRCC equations [21, 22, 24, 25, 33, 34] h Φ pa | H N + h H N , T ( ) i | Φ i = h Φ pa | h D , T ( ) i | Φ i , (5a) h Φ pqab | H N + h H N , T ( ) i | Φ i = h Φ pqab | h D , T ( ) i | Φ i . (5b) The single and double excitations in the couple-cluster the-ory capture most of the correlation effects and hence, the op-erators T ( ) and T ( ) are approximated as T ( ) = T ( ) + T ( ) and T ( ) = T ( ) + T ( ) , respectively. This is referred to asthe coupled-cluster single and double (CCSD) approximation[35]. In the present work we, however, also incorporate thetriple excitations perturbatively [24].The perturbed wavefunction from Eq. (4) is used to calcu-late the ground state polarizability. In PRCC α = − h e Ψ | D | e Ψ ih e Ψ | e Ψ i . (6)Using the expression of | e Ψ i from Eq. (4) we can write α = − h Φ | T ( ) † ¯ D + ¯ DT ( ) | Φ ih Ψ | Ψ i , (7)where ¯ D = e T ( ) † D e T ( ) , and in the denominator h Ψ | Ψ i isthe normalization factor. Considering the computational com-plexity, we truncate ¯ D and the normalization factor to secondorder in the cluster operators T ( ) . From our previous study[36], using an iteration scheme we found that the terms withthird and higher orders contribute very less to the properties. III. CALCULATIONAL DETAILA. Single-electron basis
In the RCC and PRCC calculations, it is crucial to use an or-bital basis set which provides a good description of the single-electron wave functions and energies. In the present work, weuse Gaussian-type orbitals (GTOs) [29] as the basis. We op-timize the orbitals as well as the self-consistent-field energiesof GTOs to match the GRASP2K [37] results. In the TableI, we provide the values of the exponents α and β [29] ofthe occupied orbital symmetries for Cn, Nh + and Og. Forfurther improvement, we incorporate the effects of Breit inter-action, vacuum polarization and the self-energy corrections.This is crucial to obtain the value of the dipole polarizabil-ity of SHEs accurately, where the relativistic effects are largerdue to higher Z . The effects of finite charge distribution ofthe nucleus are incorporated using a two-parameter finite sizeFermi density distribution.In the Appendix, we compare the orbital energies of Cn (Ta-ble IX), Nh + (Table X) and Og (XI) with GRASP2K [37] andB-spline [38] data. As seen from the tables, the GTOs orbitalenergies are in excellent agreement with the numerical val-ues from GRASP2K. The largest differences are 3 . × − ,4 . × − and 9 . × − hartree in the case of 4 f / , 1 s / and 2 p / orbitals of Cn, Nh + and Og, respectively. Thecorrections from the vacuum polarization, ∆ε Ue , and the self-energy, ∆ε SE , to the orbital energies are provided in the TableXII. Our results match well with the previous calculation [39]for Cn and Og. B. Computational challenges with SHEs
The calculation of α for SHEs is a computationally chal-lenging task. This is due to the large number of core elec-trons and the need for larger basis size to obtain convergedproperties results. The latter pose three main hurdles in thecalculations. First, the number of cluster amplitudes is verylarge, and solving the cluster equations require long computetimes. To give an example, as shown in Fig. 1, in the caseof Cn using a converge basis of 200 orbitals leads to morethan 31 millions cluster amplitudes. This is about 2.3 timeslarger than the lighter atom Rn. Second, convergence of α with basis size is slow. This is in stark contrast to the con-vergence trends of α for lighter atoms and ions reported inour previous works [23–25]. For the lighter atoms and ionsconvergence is achieved with a basis of 160 or less orbitals.However, for SHEs convergence of α requires ≈
200 orbitals.And third, storing the two-electron integrals for efficient com-putation require of large memory. For instance, the numberof two-electron integrals in the case of Cn is morethan 427 millions. This is about 1.3 times larger than the case of Rn. Moreover, in general parallelization, solving the clusterequations require storing the same set of integrals are storedacross all nodes. This leads to replication of data across com-pute nodes and puts severe restrictions on basis size in thePRCC calculations. To mitigate this problem, we have im-plemented a memory-parallel-storage algorithm [40] whichavoids the storage replication of the integrals across differ-ent nodes. This allows efficient memory usage and use largeorbital basis in the PRCC computations. The inclusion of per-turbative triples to the computation of α enhances the com-putational complexity further. This is due to the evaluationof numerous additional polarizability diagrams arising fromthe perturbative triples. To illustrate the compute time, thecomputation of α for Cn using a basis of 200 orbitals withouttriples takes 120 hrs with 144 threads. Whereas, with partialtriples included, it requires 280 hrs with 200 threads. Thus theruntime more than doubled. IV. RESULTS AND DISCUSSIONA. Convergence of α and correlation energy Since the GTO basis are mathematically incomplete [32],it is essential to check the convergence of polarizability andcorrelation energy with basis size. The convergence trends of α and electron correlation energy are shown in the Fig. 2. Inthese calculations we have used the Dirac-Coulomb Hamil-tonian as it is computationally less expensive than using theDCB Hamiltonian. To determine the converged basis set westart with a moderate basis size and add orbitals in each sym-metry systematically. This is continued till the change in α and correlation energy is − in their respective units. Forexample, as discernible from the Table VIII in Appendix, thechange in α for Cn is 4 . × − a.u. when the basis set isaugmented from 191 to 200. So, to optimize the computetime, we consider the basis set with 191 orbitals as the op-timal one for α . Once the optimal basis set is chosen, forfurther computations we incorporate the Breit interaction andQED corrections. As discernible from the Figs. 2(a) and (b),one key observation is, the correlation energies converge withmuch larger bases than α . For example, for Cn, the con-verged second-order energy is obtained with the basis size of439 (31 s p d f g h i j k l ) orbitals. A simi-lar trend is also observed for other two SHEs and all lighterhomologs considered in this work. B. Correlation energy
The electron correlation energy in RCC is expressed as ∆ E = h Φ | ¯ H N | Φ i , (8)where ¯ H N , = e − T ( ) H N e T ( ) , is the similarity transformedHamiltonian. In Table II, we list ∆ E for SHEs and three lighterelements in each group. Since the correlation energies con-verge with very large basis sizes, it is therefore not practical TABLE I. The α and β parameters of the even tempered GTO basis used in our calculations.Atom s p d f α β α β α β α β Cn 0.00545 1.870 0.00475 1.952 0.00105 1.970 0.00380 1.965Og 0.00410 1.910 0.00396 1.963 0.00305 1.925 0.00271 1.830Nh + CC a m p lit ud e s ( ) (a) - p a r ti c l e S I ( ) (b) R A M ( G B ) (c) FIG. 1. (a) Number of cluster amplitudes, (b) number of 4-particle Slater integrals and (c) memory required to store 4-particle Slater integrals,as a function of Z for group-12 elements. to use such a large basis in the RCC calculations due to sev-eral computational limitations. Some of the limitations are asmentioned in the previous section. To mitigate this, and toaccount for correlation energy from the virtuals not includedin the RCC calculations, we resort to the second-order MBPTmethod. The RCC results for ∆ E listed in Table II are calcu-lated using the expression ∆ E RCC ≈ ∆ E nconvRCC + (cid:16) ∆ E conv , − ∆ E nconv , (cid:17) , (9)where ∆ E nconvRCC is the correlation energy computed using RCCwith orbitals up to j -symmetry. And, ∆ E nconv , and ∆ E conv , are the second-order energies calculated using RCC basis anda converged basis which includes orbitals up to l -symmetry,respectively.For all the elements listed in Table II, we observe an im-portant trend in the correlation energy. The RCC energy issmaller in magnitude than the second-order energy. A sim-ilar trend is also observed in the previous work [46]. Fromour calculations we find that the reason for this is the cancel-lations from the higher order corrections embedded in RCC.As discernible from the Fig. 3(a) for group-12 elements as anexample, contributions from the third and fifth order correc-tions are positive. And, as a result, correlation energy oscil-lates initially before converging to the RCC value. This canbe observed from the Fig. 3(b) where we have shown the cu-mulative contribution. Comparing our results with previouscalculations, to the best of our knowledge, there are no resultsfor SHEs. For lighter elements, our RCC energy agrees wellwith the previous RCC results [45, 46]. The small differencein the energies, however, could be attributed to the correctionsfrom the Breit and QED included in the present work and thedifference in the basis employed. For second-order energies, there are four results from the previous calculations [41–44].And, our results match very well with them for all the ele-ments.Examining the contributions from different symmetries ofvirtual orbitals, we find that all three SHEs show a similartrend. This is not surprising as all three are closed-shell sys-tems. As discernible from the histograms in Fig. 3, contribu-tion to correlation energy increases initially as a function oforbital symmetry and then decreases. The first two dominantchanges, ≈
35% and 26% of the total correlation energy, arefrom the g and f orbitals. The next two are from the d and h -symmetries, they contribute ≈
12% each to all the SHEs.The contribution from the virtuals with l -symmetry is about0.8%. This non negligible contribution from l -symmetry or-bitals indicates that the inclusion of the orbitals from highersymmetries are important to obtained the accurate energiesfor SHEs. C. Polarizability
The values of α with different methods subsumed in thePRCC theory are listed in the Table III. The Dirac-Fock (DF)contribution is computed using Eq. (7) with T ( ) and ¯ D re-placed by the bare dipole operator D , and is expected tohave the dominant contribution. The PRCC values are theconverged values from Table VIII, calculated using the DCHamiltonian with basis up to h symmetry. The values listedas Estimated include the estimated contribution from the or-bitals of i , j and k symmetries. For this, we use a basis set ofmoderate size from the Table VIII and then, augment it withorbitals from i , j and k symmetries to calculate percentagecontribution, this is added to the PRCC value. To the best of -3.2-2.4-1.6-0.80.0
120 220 320 420 520 D E ( ) Basis size (a)
CnNh + Og -2.4-1.8-1.2-0.60.0
150 200 250 300 350 D E ( RCC ) Basis size (b)
CnNh + Og -6.0-4.5-3.0-1.50.0
90 120 150 180 210 Da Basis size (c)
CnNh + Og FIG. 2. Convergence of second-order correlation energy (panel (a)), the RCC energy (panel (b)) and α (panel (c)) as function of the basissize. -0.40.00.40.81.2 D E (3) D E (4) D E (5) D E (6) C o rr . e n e r gy ( a . u . ) (a) ZnCdHgCn -8.0-6.0-4.0-2.00.0 D E (2) D E (3) D E (4) D E (5) D E (6) D E ( ¥ ) C o rr . e n e r gy ( a . u . ) (b) ZnCdHgCn s p d f g h i j k l | D E | (c) Cn FIG. 3. (a) Third, fourth, fifth and sixth-order correlation energies, (b) cumulative correlation energy and (c) contribution to correlation energyfrom orbitals of different symmetries. ∆ E ( ∞ ) in panel (b) represents the infinite-order correlation energy and is equivalent to RCC energy. our knowledge, there are no experimental data on α for SHEsconsidered in the present work. However, to understand thetrend of electron correlation effects, we compare our resultswith the previous theoretical results. One important and cru-cial difference between the previous studies and the presentwork is the absence of QED corrections in the previous works.Though the Breit interaction is included in the previous work[15] for Cn, the contribution is not given explicitly. Thesecorrections are, however, important to obtain the accurate andreliable values of α for SHEs. From our calculations, we findthat the combine Breit+QED contributions are ≈ + and Og. Considering theimportant prospects associated with accurate data on α forSHEs, these are significant contributions and can not be ne-glected.For Cn three of the previous studies, similar to the presentwork, are using CCSD(T). There are, however, important dif-ferences in terms of the basis used in these calculations. And,this could account for the difference in the values of α re-ported in these works. In Ref. [15], a relativistic basis with11 s p d f orbitals optimized using pseudopotential Hartree-Fock energy is used and reports the smallest value 25.82. Theother CCSD(T) result 27.40 from the Ref. [18] is using a rela- tively larger basis of 26 s p d f g h . In terms of method-ology and basis, Ref. [18] is the closest to the present work.Our recommended value 27 . ( ) is close this. The otherCCSD(T) result of 28.68 is from the Ref. [16], which is ob-tained using an uncontracted Cartesian basis. The is the high-est theoretical value reported in the literature. The other value28 ± α after the DF. For the triples contribution to α , there isno clear trend in the previous RCC results. In Ref. [15] and[18] the contribution from triples reported to as − .
08 and − .
07% of the CCSD value, respectively, and decrease thevalue of α . However, a positive contribution of ≈ ≈ α further.For Nh + , there are no previous theoretical results. Thepresent work reports the first theoretical result of α , usingPRCC theory. As we observed from the Table III, thoughit has the same electronic structure as Cn, the value of α issmaller. This is attributed to the relativistic contraction of7 s / orbital due to increased nuclear potential. Like the case TABLE II. Electron correlation and total energies in atomic units forgroup-12, group-13 and group-18 elements. Listed RCC energiesalso include the contributions from the Breit and QED corrections.Basis ∆ E DC E total ∆ E Others
MBPT RCC MBPT RCCGroup-12Zn 336 206 − . − . − . − . a , − . d − . f Cd 461 223 − . − . − . − . b , − . d − . f Hg 413 227 − . − . − . − . b , − . d − . f Cn 439 289 − . − . − . +
411 227 − . − . − . +
447 235 − . − . − . +
409 220 − . − . − . +
453 304 − . . − . − . − . − . − . c , − . d − . e − . f Xe 419 255 − . − . − . − . c , − . d − . e − . f Rn 372 245 − . − . − . − . c , − . d − . f Og 492 313 − . − . − . a Ref.[41][MP2], b Ref.[42][MP2], c Ref.[43][MP2], d Ref.[44][MP2], e Ref.[45][RCC], f Ref.[46][RCC], of Cn, the inclusion of partial triples increases the value of α further.For Og, there are three previous results based on calcula-tions using CCSD(T). Though the same methods are used,there is a large difference in the values of α reported in theseworks. For instance, the CCSD(T) value 57.98 reported inRef. [20] is ≈
25% larger than the result in Ref. [19]. The rea-son for this could be attributed to the different types of basisused. In Ref. [19] the computations used the Faegri basis with26 s p d f g h orbitals, however, in Ref. [20] an uncon-tracted relativistic quadrupole-zeta basis is used. The other TABLE III. Final value of α (a. u.) from PRCC calculation com-pared with other theoretical data in the literature.Element Present work Other cal.Method α Cn DF 35 .
234 25 . a , 28 . b ,PRCC 26 .
944 27 . d , 28 ± c PRCC(T) 27 . . . . . ( ) Nh + DF 23 . . . . . . . ( ) Og DF 56 .
197 52 . b , 46 . e ,PRCC 55 .
941 57 . f , 57 ± c PRCC(T) 56 . . . . . ( ) a Ref.[15][CCSD(T)], b Ref.[16][CCSD(T)], c Ref.[17][RRPA], d Ref.[18][DC-CCSD(T)], e Ref.[19][R, DC-CCSD(T)], f Ref.[20][R, Dirac + Gaunt, CCSD(T)],TABLE IV. Contributions to α (in a.u.) from different terms in PRCCtheory.Terms + h.c. Cn Nh + Og T ( ) †1 D . . . T ( ) † D T ( ) − . − . − . T ( ) † D T ( ) . . . T ( ) † D T ( ) − . − . − . T ( ) † D T ( ) − . − . . . . . . . . CCSD(T) result 52.43 from Ref. [16] lies between the othertwo results. Our recommended value 56 . ( ) is closer tothe RRPA value 57 ( ) from Ref. [17] and CCSD(T) value,57.98, from Ref. [20]. As mentioned in the case of Cn, thisis due to core-polarization effect accounted to all orders inboth CCSD and RRPA. The obtained contribution from par-tial triples 0.47% is consistent with the contribution 0.66%reported in Ref. [19]. TABLE V. Five leading contribution to { T ( ) † D + H . c . } (in a.u.) for α from core orbitals. This includes the DF and core-polarizationcontributions.Cn Nh + Og17 . ( s / ) . ( s / ) . ( p / ) . ( d / ) . ( d / ) . ( p / ) . ( d / ) . ( d / ) . ( d / ) . ( p / ) . ( p / ) . ( d / ) . ( f / ) . ( f / ) . ( s / ) TABLE VI. Five leading contributions to NLO term { T ( ) † D T ( ) + H . c . } (in a.u.) for α from core-core orbitalpairs. This includes the pair-correlation contributions.Cn Nh + − . ( s / , d / ) − . ( s / , d / ) − . ( s / , s / ) − . ( s / , s / ) − . ( d / , d / ) − . ( s / , d / ) − . ( d / , s / ) − . ( d / , d / ) − . ( s / , d / ) − . ( d / , s / ) Og − . ( p / , p / ) − . ( p / , p / ) − . ( p / , d / ) − . ( p / , d / ) − . ( p / , p / ) V. ELECTRON CORRELATION, BREIT AND QEDCORRECTIONS
In this section we analyze and present the trends of elec-tron correlation effects from the residual Coulomb interaction,Breit interaction and QED corrections to α as function of Z . A. Residual Coulomb interaction
To assess the correlation effects from residual Coulomb in-teraction we define relative-DF-contribution (RDFC) asRDFC = α PRCC − α DF α DF , and plot this for each of the groups in Fig. 4 for all the four el-ements. As observed from the figure, we obtain similar trendsfor the group-12 and group-13 elements. For these groups,the RDFC is positive initially and then changes to negative.The reason for this is the drastic change in the nature of thecore polarization contribution as function of Z, due to differ-ent screening of nuclear potential. The core polarization con-tribution is positive for first two elements and negative for thelast two. This negative contribution reduces the PRCC valueto lower than the DF value. A similar trend is also reported in the previous works [15, 16, 18] where the DF value for Cnis higher than the CCSD value. For the group-18 elements,the RDFC shows a slightly different trend. Except for Kr, itis negative for all the remaining elements. In addition, themagnitude decreases from Xe to Og. This could be attributedto the negative and decreasing core polarization contributionsfrom Xe to Og. Our higher DF value, 56.20, for Og is con-sistent with the previous results in Refs. [16, 19] in which thereported DF values of 54.46 and 50.01, respectively, are largerthan the CCSD values. The difference in the DF values couldbe due to the different basis used in these calculations, whichalso led to the different α values.To gain further insights on the electron correlations effectssubsumed in the PRCC theory, we examine the contributionsfrom different terms. And, these are listed in the Table IV. Asseen from the table, for all the SHEs, the LO contribution isfrom the term { T ( ) †1 D + h . c . } . This is to be expected, as itsubsumes the contributions from DF and RPA. The contribu-tions are larger than PRCC by ≈ + and Og, respectively. The contribution from the NLOterm { T ( ) † D T ( ) } is small and opposite in phase to the LOterm. It accounts for ≈ -11%, -10% and -7% of the PRCCvalue for Cn, Nh + and Og, respectively. The next to NLO(NNLO) term is T ( ) † D T ( ) and contributes ≈ T ( ) CC operators.To examine in more detail, we assess the contributionsfrom the core-polarization and pair-correlation effects. Forthe core-polarization, we identify five dominant contributionsto the LO term and these are listed in the Table V. Since theCn and Nh + have the same ground state electronic configura-tion, both show similar correlation trends. For both, the mostdominant contribution is from the valence orbital 7 s / andthis is due to its larger radial extent. As shown in the Fig. 5,the contribution from 7 s / is ≈
50% and 53% of the LO valuefor Cn and Nh + , respectively. For Cn, we find that more than60% of the 7 s / contribution arise from T ( ) involving the8 p / , 10 p / and 9 p / orbitals. Whereas for Nh + , the 7 p / and 7 p / together contribute more than 87% of the total con-tribution. The next two important contributions are from thecore orbitals 6 d / and 6 d / . The contribution from the for-mer is almost double of the latter. In particular, for the Cnand Nh + the contributions from 6 d / is 35% and 30%, re-spectively. Whereas, the contribution from 6 d / is ≈ d / could be attributed to the strong dipolar mixing with 10 p / and 9 p / for Cn, and 7 p / and 11 f / for Nh + (see the Fig.6).For Og, compare to Cn and Nh + , we observe a differenttrend of core-polarization effect. More than 95% of the con-tribution from the LO term arises from valence orbital 7 p / .The other valence and core orbitals contribute less than 5%and 7 p / contributes only ≈
3% of the LO term. The reasonfor this could be the larger radial extent of the 7 p / orbitalas 7 p / orbital contracts due to relativistic effects. The fivedominant contributions arise from the dipolar mixing of 7 p / -28-14 01428 Zn Cd Hg Cn R D F C ( % ) Group-12 -30-20-10 010 Ga + In + Tl + Nh + R D F C ( % ) Group-13 -6-3 0 3 6
Kr Xe Rn Og R D F C ( % ) Group-18
FIG. 4. In percentage, the relative-DF-contribution for group-12, group-13 and group-18 elements. s / / / / f / % c on t r i bu ti on Cn s / / / / f / % c on t r i bu ti on Nh + / / / / s / % c on t r i bu ti on Og FIG. 5. Five largest percentage contribution from core orbitals to LO term { T ( ) †1 D + h . c . } . with 10 d / , 9 d / , 10 s / , 9 s / and 11 s / orbitals. Theseorbitals together contribute ≈
73% of the total contribution(see the Fig. 6).To assess the contribution from pair-correlation effects weconsider the NLO term and identify the dominant contribu-tions to it. These are listed in the Table VI in terms of thepairs of core orbitals and these correspond to the T ( ) withdominant contributions. This is an appropriate approach as themost dominant term involving doubly excited cluster operatoris the NLO term. For better illustration the percentage contri-bution to those listed in Table VI are plotted in the Fig. 7. Forboth Cn and Nh + , the first two dominant contributions arefrom the ( s / , d / ) and ( s / , s / ) core-orbital pairs.In percentage, these are ≈ -20% and -16% for Cn, whereas ≈ -24% and -14% for Nh + . Though the next three contri-butions are from the same core-orbital pairs, ( d / , d / ) , ( d / , s / ) and ( s / , d / ) , in both the elements, thereare differences in terms of the order in which they contribute.Like in the core-polarization effect, we observe a differenttrend for Og. About 70% of the total contribution is fromonly the ( p / , p / ) orbital pair. TABLE VII. Contributions to α from Breit interaction, vacuum po-larization and the self-energy corrections in atomic units.Elements Z Breit int. Self-ene. Vacuum-pol.Group-12Zn 30 − . . − . − . . − . . . − . . . − . + − . . − . + − . . − . + − . . − . +
113 0 . . − . . . . . . . . . . . . . B. Breit and QED corrections
To analyze the trend of correlation effects arising from theBreit interaction, vacuum polarization and the self-energy cor- s / - / s / - / s / - / / - / / - / % c on t r i bu ti on Cn s / - / s / - / / - / / - / / - f / % c on t r i bu ti on Nh + / - / / - / / - s / / - s / / - s / % c on t r i bu ti on Og FIG. 6. In percentage, five dominant dipolar mixing of cores with virtuals. s / - / s / - s / / - / / - s / s / - / % c on t r i bu ti on Cn s / - / s / - s / s / - / / - / / - s / % c on t r i bu ti on Nh + / - / / - / / - / / - / / - / % c on t r i bu ti on Og FIG. 7. Five largest percentage contribution from core-core orbital pairs to NLO term. rections as function of Z , we separate the contributions fromthese interactions. And, these are listed in the Table VII. Inaddition, for comparison and to show the trends in the group,the percentage contributions from the corresponding groups inthe periodic table of the SHEs are shown in Figs. 8, 9 and 10,respectively. For the Breit interaction, as we see from the Fig.8, except for the Cn and Og, we observe a trend of decreasingcontributions with increasing Z within the groups. One com-mon trend we observe in the contributions to SHEs is that, allare in the same phase as PRCC, and hence increase the valueof α . For lighter elements, however, we get a mixed phase forcontributions.For the corrections from the vacuum polarization and selfenergy, from the Figs. 9 and 10 we see that, the contributionfrom both the vacuum polarization and self energy increaseswith Z for all the three groups. This is as expected. For thevacuum polarization, the effect is larger due to higher nuclearcharge Z . And, for the self energy, the correction depends onthe energy of the orbital, which again depends on the nuclearcharge. In terms of the phase of the contributions from vac-uum polarization, these are in opposite to PRCC value for allthe elements of group-12 and group-13, and hence lowers thevalue of α . For group-18, however, we observe the contribu-tions of the same phase as PRCC. In terms of magnitude, the contributions are ≈ + and Og, respectively. For the self energy,one prominent feature of the contributions we observe is thatit is positive for all elements in all the three groups, and there-fore increase the value of α . The contributions in the case ofCn, Nh + and Og are ≈ C. Theoretical uncertainty
In this section we discuss the theoretical uncertainty asso-ciated with our results for α . For this, we have identified fourdifferent sources which can contribute. The first source of un-certainty is the truncation of the basis set in our calculations.The recommended values of α in Table III are obtained fromthe sum of the converged value with basis up to h -symmetry(see the convergence Table VIII) and the estimated contribu-tion from i , j and k -symmetries. The combine contributionfrom i , j and k -symmetries are ≈ + and Og. Though the contributionsfrom the virtuals beyond k -symmetry is expected to be muchlower, we select the highest contribution of 0.5% from the caseof Cn and attribute this as an upper bound to this source of un-0 % c on t r i bu ti on Group-12 + In + Tl + Nh + % c on t r i bu ti on Group-13 % c on t r i bu ti on Group-18
FIG. 8. Percentage contribution from Breit interaction to group-12, group-13 and group-18 elements. % c on t r i bu ti on Group-12 + In + Tl + Nh + % c on t r i bu ti on Group-13 % c on t r i bu ti on Group-18
FIG. 9. Percentage contribution from vacuum polarization to group-12, group-13 and group-18 elements. certainty. The second source is the truncation of the dressedoperator ¯ D in Eq. (7) to second order in T ( ) . In our previ-ous work [36], we concluded that the contribution from theremaining higher order terms is less than 0.1%. So, from thissource of uncertainty we consider 0.1% as an upper bound.The third source is the partial inclusion of triple excitations inPRCC theory. The partial triples contribute ≈ + and Og, respectively.Since the perturbative triples account for the dominant contri-bution, we choose the highest contribution of 1 .
9% from Cnand take it as an upper bound. And, the last source of theo-retical uncertainty is associated with the frequency-dependentBreit interaction which is not included in the present work.To estimate an upper bound to from source we use our pre-vious work [23] where using GRASP2K calculations we esti-mated an upper bound of 0 .
13% for Ra. Using this with theBreit contributions, we derive ≈ + and Og, respectively. Amongthese, we select the highest contribution of 0.62% from thecase of Cn and attribute this as an upper bound. There couldbe other sources of theoretical uncertainty, such as the higherorder coupled perturbation of vacuum polarization and self-energy terms, quadruply and higher excited cluster operators,etc. These, however, have much lower contributions and theircombined uncertainty could be below 0.1%. Finally, combin-ing the upper bounds of all four sources of uncertainties, weestimate a theoretical uncertainty of 3.2% in the recommended values of α . VI. CONCLUSION
We have employed a fully relativistic coupled-cluster the-ory to compute the ground state electric dipole polarizabilityand electron correlation energy of SHEs Cn, Nh + and Og.In addition, to understand the trend of electron correlation asfunction of Z , we have calculated the correlation energies ofthree lighter homologs for each SHEs. To improve the ac-curacy of our results, contributions from the Breit interac-tion, QED corrections and partial triple excitations are alsoincluded. Moreover, in all calculations, very large bases up to l -symmetry are used to check the convergence of the results.Our recommended values of α for SHEs lie between theprevious results, more closer to the values from CCSD(T) [18,20] and RPA [17] calculations. From our calculations we findthat the dominant contribution to α comes from the valenceelectrons, viz, 7 s / for both Cn and Nh + , and 7 p / for Og.While 7 s / contributes more than 50% of the total value forCn and Nh + , the contribution from 7 p / orbital to Og is morethan 95%. This could be attributed to the larger radial extentof these orbitals.From the analysis of electron correlation effects, we findthat, for all three groups, the core polarization contribution de-creases as function of Z for the lighter homologs. For SHEs,1 % c on t r i bu ti on Group-12 + In + Tl + Nh + % c on t r i bu ti on Group-13 % c on t r i bu ti on Group-18
FIG. 10. Percentage contribution from self-energy correction to group-12, group-13 and group-18 elements. however, we observed an increased contribution, of the orderof the first element considered in each group. For the cor-rections from the Breit interaction, except for Cn and Og, atrend of decreasing contributions as function of Z is observed.On contrary, for Uehling potential and the self-energy correc-tions, we observed a trend of increasing contributions fromlighter homologs to SHEs. The largest contributions fromthe Uehling potential are ≈ . .
3% and 0 .
3% of PRCCvalue for Cn, Nh + and Og, respectively. And, the same fromthe self-energy corrections are ≈ . .
2% and 0 . α areobserved to be ≈ . .
4% and 0 .
6% for Cn, Nh + and Og,respectively. Considering the importance of accurate proper-ties results of SHEs, these are significant contributions, andcan not be neglected.We report the first result on the electron correlation energyfor SHEs Cn, Nh + and Og using RCC. From our detailedanalysis on correlation energy, we find that the second-orderMBPT calculations overestimate the electron correlation en-ergy for all superheavy elements and lighter homologs con-sidered in this work. ACKNOWLEDGMENTS
We would like to thank Chandan Kumar Vishwakarma foruseful discussions. One of the authors, BKM, acknowledgesthe funding support from the SERB (ECR/2016/001454). Theresults presented in the paper are based on the computationsusing the High Performance Computing cluster, Padum, at theIndian Institute of Technology Delhi, New Delhi.
Appendix A: Convergence table for α In Table VIII we provide the convergence trend of α asfunction of basis size. As it is evident from the table, the valueof α converges to 10 − a.u. for all three SHEs. TABLE VIII. Convergence trend of α calculated using the Dirac-Coulomb Hamiltonian as a function of basis size.Basis Orbitals α Cn90 14s, 11p, 10d, 8f, 6g, 3h 32 . . . . . . . . +
101 15s, 13p, 12d, 8f, 5g, 5h 19 . . . . . . . . . . . . . . . Appendix B: Single-electron energies
The single-electron energies of GTOs for SHEs Cn, Nh + and Og are listed in the Tables IX, X and XI, respectively, andcompared with the numerical data from GRASP2k [37] and2the energies from the B-spline [] basis. In Table XII, we listthe contributions from the Breit interaction, Uehling potentialand the self-energy corrections to the single-electron energies. TABLE IX. Orbital energies for core orbitals (in Hartree) from GTOis compared with the GRASP2K and B-spline energies for Cn. Here[x] represents multiplication by 10 x .Orbital GRASP2K B-spline GTO1 s / s / s / s / s / s / s / p / p / p / p / p / p / p / p / p / p / d / d / d / d / d / d / d / d / f / f / f / f / , 1237–1312 (2013), pMID: 23402305,https://doi.org/10.1021/cr3002438.[2] Matthias Schädel, “Chemistry of the superheavy elements,”Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences , 20140191 (2015),https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2014.0191.[3] V. Pershina, “Electronic structure and properties of superheavyelements,” Nuclear Physics A , 578 – 613 (2015), specialIssue on Superheavy Elements.[4] Peter Schwerdtfeger, Lukáš F. Pašteka, Andrew Pun-nett, and Patrick O. Bowman, “Relativistic and quan-tum electrodynamic effects in superheavy elements,”Nuclear Physics A , 551 – 577 (2015), special Issue onSuperheavy Elements. [5] Ephraim Eliav, Stephan Fritzsche, and Uzi Kaldor,“Electronic structure theory of the superheavy elements,”Nuclear Physics A , 518 – 550 (2015), special Issue on Su-perheavy Elements.[6] S. A. Giuliani, Z. Matheson, W. Nazarewicz, E. Olsen, P.-G.Reinhard, J. Sadhukhan, B. Schuetrumpf, N. Schunck, andP. Schwerdtfeger, “Colloquium: Superheavy elements: Oganes-son and beyond,” Rev. Mod. Phys. , 011001 (2019).[7] D. Shaughnessy and M. Schadel, The Chemistry of the Super-heavy Elements , 2nd ed. (Springer, Heidelberg, 2014).[8] V. Pershina and D.C. Hoffman,
Transactinide Elements and Fu-ture Elements (Springer, Dordrecht, 2008).[9] I.B. Khriplovich,
Parity Nonconservation in Atomic Phenom-ena (Gordon and Breach Science Publishers, Philadelphia,1991). TABLE X. Orbital energies for core orbitals (in Hartree) from GTO iscompared with the GRASP2K and B-spline energies for Nh + . Here[x] represents multiplication by 10 x .Orbital GRASP2K B-spline GTO1 s / s / s / s / s / s / s / p / p / p / p / p / p / p / p / p / p / d / d / d / d / d / d / d / d / f / f / f / f / ,”Phys. Rev. Lett. , 101601 (2009).[11] Th. Udem, R. Holzwarth, and T. W. Hansch, “Optical fre-quency metrology,” Nature , 233 (2002).[12] M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, AnneL’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,”Phys. Rev. A , 2117–2132 (1994).[13] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman,and E. A. Cornell, “Observation of Bose-Einstein condensationin a dilute atomic vapor,” Science , 198–201 (1995).[14] S. G. Karshenboim and E Peik, Astrophysics, Clocks and Fun-damental Constants, Lecture Notes in Physics (Springer, NewYork, 2010).[15] Michael Seth, Peter Schwerdtfeger, and Michael Dolg,“The chemistry of the superheavy elements. i. pseu-dopotentials for 111 and 112 and relativistic coupledcluster calculations for (112)h+, (112)f2, and (112)f4,”The Journal of Chemical Physics , 3623–3632 (1997),https://doi.org/10.1063/1.473437.[16] Clinton S. Nash, “Atomic and molecularproperties of elements 112, 114, and 118,” TABLE XI. Orbital energies for core orbitals (in Hartree) from GTOis compared with the GRASP2K and B-spline energies for Og. Here[x] represents multiplication by 10 x .Orbital GRASP2K B-spline GTO1 s / s / s / s / s / s / s / p / p / p / p / p / p / p / p / p / p / p / p / d / d / d / d / d / d / d / d / f / f / f / f / , 3493–3500 (2005),pMID: 16833687, https://doi.org/10.1021/jp050736o.[17] V. A. Dzuba, “Ionization potentials and polarizabilitiesof superheavy elements from Db to Cn (Z=105–112),”Phys. Rev. A , 032519 (2016).[18] V. Pershina, A. Borschevsky, E. Eliav, and U. Kaldor, “Pre-diction of the adsorption behavior of elements 112 and 114on inert surfaces from ab initio dirac-coulomb atomic calcu-lations,” The Journal of Chemical Physics , 024707 (2008),https://doi.org/10.1063/1.2814242.[19] V. Pershina, A. Borschevsky, E. Eliav, and U. Kaldor, “Ad-sorption of inert gases including element 118 on noble metaland inert surfaces from ab initio dirac–coulomb atomic calcu-lations,” The Journal of Chemical Physics , 144106 (2008),https://doi.org/10.1063/1.2988318.[20] Paul Jerabek, Bastian Schuetrumpf, Peter Schwerdtfeger, andWitold Nazarewicz, “Electron and nucleon localization func-tions of oganesson: Approaching the thomas-fermi limit,”Phys. Rev. Lett. , 053001 (2018).[21] S. Chattopadhyay, B. K. Mani, and D. Angom, “Perturbedcoupled-cluster theory to calculate dipole polarizabilities ofclosed-shell systems: Application to ar, kr, xe, and rn,” Phys. Rev. A , 062508 (2012).[22] S. Chattopadhyay, B. K. Mani, and D. Angom, “Elec-tric dipole polarizabilities of doubly ionized alkaline-earth-metal ions from perturbed relativistic coupled-cluster theory,”Phys. Rev. A , 062504 (2013).[23] S. Chattopadhyay, B. K. Mani, and D. Angom, “Electric dipolepolarizability of alkaline-earth-metal atoms from perturbed rel-ativistic coupled-cluster theory with triples,” Phys. Rev. A ,022506 (2014).[24] S. Chattopadhyay, B. K. Mani, and D. Angom, “Triple excita-tions in perturbed relativistic coupled-cluster theory and elec-tric dipole polarizability of groupiib elements,” Phys. Rev. A , 052504 (2015).[25] Ravi Kumar, S. Chattopadhyay, B. K. Mani, and D. An-gom, “Electric dipole polarizability of group-13 ions using per-turbed relativistic coupled-cluster theory: Importance of non-linear terms,” Phys. Rev. A , 012503 (2020).[26] M. S. Safronova, W. R. Johnson, and A. Derevianko,“Relativistic many-body calculations of energy lev-els, hyperfine constants, electric-dipole matrix ele-ments, and static polarizabilities for alkali-metal atoms,”Phys. Rev. A , 4476–4487 (1999).[27] A. Derevianko, W. R. Johnson, M. S. Safronova, and J. F. Babb,“High-precision calculations of dispersion coefficients, staticdipole polarizabilities, and atom-wall interaction constants foralkali-metal atoms,” Phys. Rev. Lett. , 3589–3592 (1999).[28] J. Sucher, “Foundations of the relativistic theory of many-electron atoms,” Phys. Rev. A , 348–362 (1980).[29] A. K. Mohanty, F. A. Parpia, and E. Clementi, “Kineticallybalanced geometric gaussian basis set calculations for relativis-tic many-electron atoms,” in Modern Techniques in Computa-tional Chemistry: MOTECC-91 , edited by E. Clementi (ES-COM, 1991).[30] Richard E. Stanton and Stephen Havriliak, “Kinetic balance:A partial solution to the problem of variational safety in diraccalculations,” J. Chem. Phys. , 1910–1918 (1984).[31] I. P. Grant, Relativistic Quantum Theory of Atoms andMolecules: Theory and Computation (Springer, New York,2010).[32] Ian Grant, “Relativistic atomic structure,” in
Springer Handbook of Atomic, Molecular, and Optical Physics ,edited by Gordon Drake (Springer, New York, 2006) pp. 325–357.[33] S. Chattopadhyay, B. K. Mani, and D. Angom, “Electric dipolepolarizability from perturbed relativistic coupled-cluster the-ory: Application to neon,” Phys. Rev. A , 022522 (2012).[34] S. Chattopadhyay, B. K. Mani, and D. Angom, “Electric dipolepolarizabilities of alkali-metal ions from perturbed relativisticcoupled-cluster theory,” Phys. Rev. A , 042520 (2013).[35] George D. Purvis and Rodney J. Bartlett, “A full coupled-cluster singles and doubles model: The inclusion of discon- nected triples,” J. Chem. Phys. , 1910–1918 (1982).[36] B. K. Mani and D. Angom, “Atomic properties calcu-lated by relativistic coupled-cluster theory without trunca-tion: Hyperfine constants of mg + , ca + , sr + , and ba + ,”Phys. Rev. A , 042514 (2010).[37] P. Jönsson, G. Gaigalas, J. Biero´n, C. Froese Fischer, andI. P. Grant, “New version: Grasp2k relativistic atomic structurepackage,” Comp. Phys. Comm. , 2197 – 2203 (2013).[38] Oleg Zatsarinny and Charlotte Froese Fischer, “DBSR-HF: A B-spline Dirac–Hartree–Fock program,”Computer Physics Communications , 287 – 303 (2016).[39] Karol Kozioł and Gustavo A. Aucar, “Qed ef-fects on individual atomic orbital energies,”The Journal of Chemical Physics , 134101 (2018),https://doi.org/10.1063/1.5026193.[40] B.K. Mani, S. Chattopadhyay, and D. Angom, “Rcc-pac: A parallel relativistic coupled-cluster program forclosed-shell and one-valence atoms and ions in fortran,”Computer Physics Communications , 136 – 154 (2017).[41] J R Flores and P Redondo, “Computation ofsecond-order correlation energies using a finiteelement method for atoms with d electrons,”Journal of Physics B: Atomic, Molecular and Optical Physics , 2251–2261 (1993).[42] Jesús R. Flores and P. Redondo, “High-precision atomiccomputations from finite element techniques: Second-order correlation energies for be, ca, sr, cd, ba, yb, andhg,” Journal of Computational Chemistry , 782–790 (1994),https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.540150710.[43] Jesús R. Flores, “High precision atomic compu-tations from finite element techniques: Second-order correlation energies of rare gas atoms,”The Journal of Chemical Physics , 5642–5647 (1993),https://doi.org/10.1063/1.464908.[44] Yasuyuki Ishikawa and Konrad Koc, “Relativistic many-bodyperturbation theory based on the no-pair Dirac-Coulomb-Breit Hamiltonian: Relativistic correlation energies for thenoble-gas sequence through Rn (Z=86), the group-iib atomsthrough Hg, and the ions of Ne isoelectronic sequence,”Phys. Rev. A , 4733–4742 (1994).[45] B. K. Mani, K. V. P. Latha, and D. Angom, “Rela-tivistic coupled-cluster calculations of Ne, Ar, Kr,and
Xe: Correlation energies and dipole polarizabilities,”Phys. Rev. A , 062505 (2009).[46] Shane P. McCarthy and Ajit J. Thakkar, “Accurate all-electroncorrelation energies for the closed-shell atoms from Ar to Rnand their relationship to the corresponding MP2 correlation en-ergies,” The Journal of Chemical Physics , 044102 (2011),https://doi.org/10.1063/1.3547262. TABLE XII. The orbital energies for core orbitals from vacuum polarization and self energy correction for Cn, Nh + and Og.Cn Nh + OgOrbital ∆ε Ue ∆ε SE ∆ε Ue ∆ε SE ∆ε Ue ∆ε SE Ours Ref. [39] Ours Ref. [39] Ours Ours Ours Ref. [39] Ours Ref. [39]1 s / -11.1193 -11.4416 30.6243 30.5752 -11.8402 31.9902 -16.2622 -16.7082 39.9045 39.78252 s / -2.1505 - 2.2283 5.9196 5.7672 -2.3160 6.2339 -3.3693 -3.4810 8.1051 7.77433 s / -0.5193 - 0.5410 1.4424 1.5061 -0.5597 1.5198 -0.8134 -0.8487 1.9781 2.16094 s / -0.1494 - 0.1560 0.4165 0.4375 -0.1614 0.4400 -0.2377 -0.2482 0.5795 0.63695 s / -0.0436 0.1214 -0.0474 0.1291 -0.0718 0.17836 s / -0.0108 0.0297 -0.0119 0.0322 -0.0197 0.05007 s / -0.0015 0.0037 -0.0019 0.0047 -0.0040 0.00982 p / -0.5505 -0.6632 1.6011 1.6611 -0.6457 1.7455 -1.0858 -1.2661 2.7124 2.78533 p / -0.1486 -0.1801 0.4445 0.5091 -0.1662 0.4835 -0.2926 -0.3430 0.7399 0.88124 p / -0.0422 -0.0523 0.1278 0.1551 -0.0473 0.1394 -0.0845 -0.1010 0.2156 0.27125 p / -0.0115 0.0359 -0.0130 0.0395 -0.0242 0.06746 p / -0.0024 0.0080 -0.0028 0.0090 -0.0059 0.01777 p / -0.0010 0.00272 p / p / p / p / p / p / d / d / d / d / d / d / d / d / f / f / f / f /2