Dipole polarizabilities of the transition and post-transition metallic systems
aa r X i v : . [ phy s i c s . a t o m - ph ] M a y Dipole polarizabilities of the transition and post-transition metallic systems
Yashpal Singh ∗ and B. K. Sahoo † Theoretical Physics Division, Physical Research Laboratory, Navrangpura, Ahmedabad - 380009, India
We investigate the role of the electron correlation effects in the calculations of the electric dipolepolarizabilities ( α ) of elements belonging to three different groups of periodic table. To understandthe propagation of the electron correlation effects at different levels of approximations, we employthe relativistic many-body methods developed, based on the first principles, at mean-field Dirac-Fock (DF), third order many-body perturbation theory (MBPT(3)), random-phase approximation(RPA) and the singly and doubly approximated coupled-cluster methods at the linearized (LCCSD)and non-linearized (CCSD) levels. We observe variance in the trends of the contributions fromthe correlation effects in a particular group of elements through a employed many-body method;however they resemble similar tendency among the isoelectronic systems. Our CCSD results arewithin sub-one percent agreement with the experimental values which are further ameliorated byincluding the contributions from the important triple excitations (CCSD p T method).
PACS numbers: 31.15.ap, 31.15.bw, 31.15.ve, 31.15.xp
I. INTRODUCTION
Static electric dipole polarizability ( α ) of an atomicsystem is the measure of distortion of the electron cloudwhen the system is subjected to a stray electric field.Some of the notable applications with the accurateknowledge of α are in the studies of new generation fre-quency standards, atomic interactions in optical lattices,quantum information along with many others in the areasof atomic and molecular physics [1–8]. Various sophis-ticated experimental techniques have been exercised tomeasure α in different atomic systems having their ownmerits and disadvantages [9–15]. Despite of the techno-logical advancements, it is still remained to attain high-precision measurements of α in the ground states of manyatomic systems. In fact, there are also some systemswhere no experimental results are yet available. Never-theless, accurate evaluation of α can serve as a good testof the potential of any developed many-body method andto peruse the underlying interplay of the electron corre-lation effects in their determination.A seminal work on the calculations of polarizabilities ofthe many-electron systems in the ab initio framework wasfirst introduced by Dalgarno and his collaborators aboutmore than five decades ago [16, 17]. Since then variants ofadvanced many-body methods have been developed andapplied successfully in the same philosophical stratagemto evaluate this atomic property meticulously. Examplesof few well-known many-body methods that are oftenemployed in the studies of α are the random phase ap-proximation (RPA), coupled-cluster method in the linearresponse theory (CCLRT), configuration-interaction (CI)method etc. [18–25]; however many of these methodsare developed in the non-relativistic mechanics. Lim andcoworkers have demonstrated, by employing the coupled- ∗ [email protected] † [email protected] cluster (CC) methods developed using the Cartesian co-ordinates for the molecular calculations, that the rela-tivistic contributions to determine α values are signif-icant, especially in the heavier atomic systems [26, 27].In their CC method, the relativistic effects are accountedby using a two-component Douglas-Kroll Hamiltonian.To encompass both the correlation and the relativisticeffects in the α determination of the closed-shell atomicsystems, we have developed a CC method consideringthe Dirac-Coulomb (DC) Hamiltonian described by thefour-component atomic wave functions in the sphericalcoordinate system [28, 29]. Ground state α of a numberof closed-shell systems have been successfully evaluatedusing such methodology in the last couple of years [30–32]. Moreover, we have set-up methods in the third ordermany-body perturbation theory [MBPT(3)] and RPA inthe relativistic formalism with the intention of includingthe correlation effects through the first principle calcu-lations as has been employed in [16, 17]. The focus ofthe present work is to apprehend the role of the electroncorrelation effects using the above many-body methodsthat are restricted at different levels of approximationsand to demonstrate furtherance in the preciseness of theresults by carrying out large scale computations involvedin some of these employed methods. We apply thesemethods to determine polarizabilities of B + , C , Al + ,Si +2 , Zn, Ga + , Ge +2 , Cd, In + and Sn +2 , those belong tothe transition and post-transition metallic groups of theperiodic table. We also explicitly investigate the con-tributions arising through the non-linear mathematicalexpressions constituting the higher order excitation pro-cesses by setting-up intermediate maneuver to curtail thecomputational time at the expense of large memory re-quirements for the goal of promoting accuracies in theresults compared to the available measurements.The rest of the paper is organized as follows: In thenext section we give briefly the theory of the atomicdipole polarizability. In section III we describe many-body methods at different levels of approximations. Be-fore concluding the present work, we give our results insection IV and compare them with the other availablecalculations and measurements. Unless stated otherwiseatomic unites are considered throughout this paper. II. THEORY OF DIPOLE POLARIZABILITY
The change in energy of the ground state in an atomicsystem due to the application of an external electric field ~ E is given by ∆ E = − α | ~ E| , (1)where α is known as the dipole polarizability of the state.In the mathematical expression, we can write α = − h Ψ (0)0 | D | Ψ (1)0 ih Ψ (0)0 | Ψ (0)0 i , (2)with | Ψ (0)0 i and | Ψ (1)0 i are the unperturbed and the first-order perturbed ground state wave functions due to theinteraction Hamiltonian ~D.~ E due to the dipole operator D . The arduous part of calculating α using the aboveexpression lies in the evaluation of | Ψ (1)0 i which entailsmixing of different parity states. On the other hand, it issometimes facile to use a sum-over-states approach givenby α = − h Ψ (0)0 | Ψ (0)0 i X I |h Ψ (0)0 | D | Ψ (0) I i| E (0)0 − E (0) I , (3)where I represents all possible intermediate states | Ψ (0) I i and E (0) K s are the energies of the respective K states de-noted by the indices in the subscripts. The above ap-proach is convenient to use if the electric dipole (E1)matrix elements between the ground state and a suffi-cient number of intermediate states are known or canbe calculated to the reasonable accuracies. However,it is extremely difficult to determine these matrix ele-ments accurately with confidence as it requires carefulhandling of a large number of configuration state func-tions (CSFs). Moreover, contributions coming from thecore, doubly excited states, continuum etc. cannot beaccounted correctly through a sum-over-states approachand estimating these contributions approximately maybe an extortionate practice at times when the systemsare almost quasi-degenerate in nature.The other famous approach for determining α is usingthe finite ~ E perturbation method in which the second or-der differentiation of the total energy ( E ) of the groundstate need to be estimated in the presence of the electricfield (finite field method); i.e. α = − ∂ E ( | ~ E| ) ∂ | ~ E| ∂ | ~ E| ! | ~ E| =0 , (4) which requires numerical calculations for a smaller arbi-trary value of ~ E . This is a typical procedure of calculating α using the molecular methods based on the Cartesian co-ordinate systems where the atomic states do not possessdefinite parity. In contrast, it is a convoluted procedureof determining α of the atomic systems in the relativisticformalism if we wish to describe the method exclusivelyin the spherical coordinates.Our methodology to determine α lies in the techniqueof calculating | Ψ (1)0 i and to supplant the ideology of ob-taining it as the solution of the following inhomogeneousequation ( H − E (0)0 ) | Ψ (1)0 i = − D | Ψ (0)0 i (5)through the matrix mechanism in the four-componentrelativistic theory described by the spherical polar coor-dinate system. Approximating the total wave function ofthe ground state to | Ψ i ≃ | Ψ (0)0 i + λ | Ψ (1)0 i , we have α = h Ψ | D | Ψ ih Ψ | Ψ i , (6)where λ is an arbitrary parameter to identify the orderof perturbation in D . III. FEW MANY-BODY METHODS FOR | Ψ (1)0 i We consider the DC atomic Hamiltonian in our calcula-tions that is scaled with respect to the rest mass energiesof the electrons and is given by H DC = X i c α i · p i + ( β i − c + V nuc ( r i ) + X j>i r ij (7)where c is the velocity of light, α and β are the Dirac ma-trices in their fundamental representations, r ij representthe inter-electronic distances and V nuc ( r ) is the nuclearpotential calculated by considering the finite-size nuclearFermi charge distribution as given by ρ nuc ( r ) = ρ e ( r − c ) /a (8)with the parameter c and a = 4 tln (3) are said to bethe half-charge-radius and skin thickness of the nucleus,respectively.We determine the approximated wave function ( | Φ i )for the ground state using the mean-field method bydefining the Dirac-Fock (DF) Hamiltonian as H DF = X i [ c α i · p i + ( β i − c + V nuc ( r i ) + U DF ( r i )]= X i [ h ( r i ) + U DF ( r i )] , (9)with an average DF potential U DF ( r ) and disregardingcontributions from the residual interaction V es = N X j>i ~r ij − X i U DF ( ~r i ) . (10)The DF potential and the single particle wave function | φ i i of | Φ i are obtained by solving the following equa-tions h φ i | U DF | φ j i = occ X b [ h φ i φ b | r | φ b φ j i − h φ i φ b | r | φ j φ b i ](11)and ( h + U DF ) | φ i i = ǫ i | φ i i (12)simultaneously in a self-consistent procedure, where b issummed over all the occupied orbitals ( occ ).To get the wave function | Ψ (0)0 i , we follow the Blochequation formalism [33] in which we express | Ψ (0)0 i = Ω (0) | Φ i = n X k Ω ( k, | Φ i , (13)where Ω (0) is known as wave operator containing n (say)orders of Coulomb interactions and k represents order of V es in the wave operator in a perturbative expansion ofΩ (0) . In the presence of another external interaction, likethe operator D , the exact state can be written as | Ψ i = Ω | Φ i = n X β m X δ Ω ( β,δ ) | Φ i , (14)where the perturbation expansion is again described by n (say) orders of V es and m (say) orders of D . For ourrequirement of obtaining the first order wave functiondue to D , we have | Ψ (1)0 i = n X β Ω ( β, | Φ i . (15)To obtain the solutions for the wave operators, we usethe following generalized Bloch equations[Ω ( β, , H DF ] P = QV es Ω ( β − , P − β − X m =1 Ω ( β − m, P V es Ω ( m − ,l ) P (16)and[Ω ( β, , H DF ] P = QV es Ω ( β − , P + QD Ω ( β, P − β − X m =1 (cid:0) Ω ( β − m, P V es Ω ( m − , P − Ω ( β − m, P D Ω ( m, P (cid:1) , (17) with the definitions of P = | Φ ih Φ | and Q = 1 − P .It implies that Ω (0 , = 1, Ω (1 , = P p,a h Φ pa | V es | Φ i E pa − E = 0and Ω (0 , = P p,a h Φ pa | D | Φ i E pa − E = P p,a h p | D | a i ǫ p − ǫ a with a and p representing the occupied and unoccupied orbitals, re-spectively.Below we discuss various many-body methods in theDF, MBPT(3), RPA and CC approaches employed in thepresent work based on the above discussed Bloch’s equa-tion formalism to calculate α s in the considered atomicsystems for their case studies. Among these methods, wehave already described the DF, MBPT(3) and CC meth-ods elaborately and given the final α evaluating diagramsfor the closed-shell atomic systems in our earlier work[30]. For the sake of completeness we would like to out-line these methods here, but describe the RPA methodextensively. A. The DF method
The lowest order polarizabilities results in the DFmethod are evaluated by using the expression α = 2 h Φ | Ω (0 , † D Ω (0 , | Φ i = 2 h Φ | D Ω (0 , | Φ i . (18) B. The MBPT(3) method
In this approximation, we have considered two ordersof Coulomb ( β = 2) for which the expression for α isgiven by α = 2 P β =0 h Φ | Ω (2 − β, † D Ω ( β, | Φ i P β =0 h Φ | Ω (2 − β, † Ω ( β, | Φ i = 2 N h Φ | [Ω (0 , + Ω (1 , + Ω (2 , ] † D × [Ω (0 , + Ω (1 , + Ω (2 , ] | Φ i = 2 N h Φ | D Ω (0 , + D Ω (1 , + D Ω (2 , + Ω (1 , † D Ω (0 , +Ω (1 , † D Ω (1 , + Ω (2 , † D Ω (0 , | Φ i , (19)with the normalization constant N = h Φ | (1 , +Ω (2 , + Ω (1 , † + Ω (2 , † + Ω (1 , † Ω (0 , | Φ i . This clearlymeans that its lowest order corresponds to the MBPT(1)method which is nothing but the DF contribution. Termscontaining up to one order of Coulomb and one D oper-ator is referred to the MBPT(2) method. C. The RPA method
To arrive at the final working equation for the RPAmethod, we start by perturbing the DF orbitals and sin- + · · · + · · · + · · · DD DD D DD DD DDDD D D D
FIG. 1: Dominant direct core-polarization diagrams con-tributing to the α calculations in the RPA method. gle particle energies due to the perturbation D . i.e. | φ i i → | φ i i + λ | φ i i (20)and ǫ i → ǫ i + λǫ i , (21)where | φ i i and ǫ i are the first order corrections to theparticle wave function and energy, respectively. Owing tothe fact that D is an odd parity operator, ǫ i = 0. Nowin the presence of the perturbative source, the modifiedDF equation for the single particle wave function yieldsthe form( h + λD )( | φ i i + λ | φ i i ) + occ X b ( h φ b + λφ b | r | φ b + λφ b i| φ i + λφ i i − h φ b + λφ b | r | φ i + λφ i i| φ b + λφ b i ) = ǫ i ( | φ i i + λ | φ i i ) . (22)Collecting the terms only those are linear in λ , we get( h + U DF − ǫ i ) | φ i i = ( − D − U DF ) | φ i i , (23)where we use the notation U DF for U DF | φ i i = occ X b ( h φ b | r | φ b i| φ i i − h φ b | r | φ i i| φ b i + h φ b | r | φ b i| φ i i − h φ b | r | φ i i| φ b i ) . (24)Following basic principles we can write the single par-ticle perturbed wave function in terms of unperturbedwave functions as | φ i i = X j = i C ji | φ j i , (25)where C ji s are the expansion coefficients. In the RPAapproach, we write X j = i C ii ( h + U DF − ǫ j ) | φ j i = ( − D − U DF ) | φ i i , (26) and solve this equation self-consistently to obtain the C ji coefficients with infinity orders of contributions from theCoulomb interaction considering their initial solutions asthe above perturbed DF method.In the Bloch’s wave operator representation, we canexpressΩ (1)RPA = ∞ X k X p,a Ω ( k, a → p = ∞ X β =1 X pq,ab { [ h pb | r | aq i − h pb | r | qa i ]Ω ( β − , b → q ǫ p − ǫ a + Ω ( β − , † b → q [ h pq | r | ab i − h pq | r | ba i ] ǫ p − ǫ a } , (27)where a → p means replacement of an occupied orbital a from | Φ i by a virtual orbital p which alternatively refersto a singly excited state with respect to | Φ i . It can beshown from the above formulation that the RPA methodpicks-up a certain class of singly excited configurationscongregating the core-polarization correlation effects toall orders.Using the above RPA wave operator, we evaluate α by α = 2 h Φ | Ω (0 , † D Ω (1)RPA | Φ i = 2 h Φ | D Ω (1)RPA | Φ i . (28)Impediment of this method is that it encapsulates contri-butions to | Ψ (1)0 i from the correlation effects due to theCoulomb interaction to all orders, but only from the core-polarization effects through the singly excited configura-tions. However, it approximates the bra state | Ψ (0)0 i ofEq. (2) to the mean-field wave function | Φ i . Diagram-matic representation of the core-polarization correlationsembraced through RPA are given (without the exchangeinteractions) in Fig. 1. D. The CC method
In the CC method, we express the unperturbed atomicwave function as | Ψ (0)0 i = Ω (0)RCC | Φ i = ∞ X k Ω ( k, | Φ i = e T (0) | Φ i (29)and the first order perturbed wave function as [30] | Ψ (1)0 i = Ω (1)RCC | Φ i = ∞ X k Ω ( k, | Φ i = e T (0) T (1) | Φ i , (30) P − PH − HP − H ( i ) ( ii ) ( iii )( iv ) ( v )( viii ) ( ix ) ( x )( xi ) ( xii )( xv ) ( xvi )( vi ) ( vii )( xiii ) ( xiv ) FIG. 2: Effective one-body intermediate diagrams con-structed from H DCN in order to evaluate the T (1) amplitudes.Here broken lines represent the Coulomb interaction and thesolid line without arrows symbolize the T (0) operators. where T (0) and T (1) are the excitation operators fromthe reference state | Φ i that take care of contributionsfrom the Coulomb interactions and Coulomb interactionsalong with from the perturbed D operator, respectively.The amplitudes of the excitation T (0) and T (1) opera-tors are determined using the equations h Φ τ | H DCN | Φ i = 0 (31)and h Φ τ | H DCN T (1) | Φ i = −h Φ τ | D | Φ i , (32)where H DCN is the normal ordered DC Hamiltonian, O = ( Oe T (0) ) con with con means only the connectedterms and | Φ τ i corresponds to the excited configurationswith τ referring to level of excitations from | Φ i . In ourcalculations, we only consider the singly and doubly ex-cited configurations ( τ = 1 ,
2) by defining T (0) = T (0)1 + T (0)2 and T (1) = T (1)1 + T (1)2 , (33)which is known as the CCSD method in the literature.When we consider the approximation O = O + OT , werefer it as the LCCSD method.To carry out calculations in an optimum computationalrequirements, we construct the intermediate diagramsfor the effective operators by dividing the non-linear CCterms. The intermediate diagrams for the computationof the T (0) amplitudes are described at length in our pre-vious work [30]. Here, we discuss only about the inter-mediate diagrams used for the evaluation of the T (1) am-plitudes. For this purpose, we express H DCN into the ef-fective one-body, two-body and three-body diagrams. It ( a PP − PPHP − PP ( a
2) ( a b
1) ( a b
14) ( b
15) ( b
12) ( b b
6) ( b b b b
9) ( b
3) ( b
4) ( b b b
5) ( b b
16) ( b HH − PP ( c
4) ( c c
1) ( c
2) ( c d
1) ( d
2) ( d HP − PH ( d
5) ( d
6) ( d
7) ( d e
1) ( e
3) ( e e HH − HHHH − PH ( f
1) ( f ( g
1) ( g
17) ( g g
6) ( g g g g
14) ( g
3) ( g g g HP − HH ( g
9) ( g
10) ( g PP − PH ( g
2) ( g
11) ( g h
1) ( h FIG. 3: Effective two-body intermediate diagrams con-structed from H DCN in order to evaluate the T (1) amplitudes.Here broken lines represent the Coulomb interaction and thesolid line without arrows symbolize the T (0) operators. is worth while to note that there is a technical differencebetween the construction of the intermediate diagramsfrom H DCN for the T (0) and T (1) amplitude solving equa-tions. In Eq. (32), H DCN contains all the non-linear termswhile for solving Eq. (31) it is required to express as ( i ) ( ii )( iii ) ( iv ) T (0)2 T (0)2 T (0)2 T (0)2 P P − P H HH − P HP H − P H P H − P H
FIG. 4: Effective three-body intermediate diagrams con-structed from H DCN T (0)2 to solve for the T (1) amplitudes. T (0)2 T (0)2 ( i ) ( ii ) P P − HP HH − HP FIG. 5: Diagrams representing T (0) ,pert operator. H DCN = H DCN ′ ⊗ T τ . Thus the intermediate diagrams inthis case are comprised terms from H DCN ′ which requiresspecial scrutiny of the diagrams to avoid repetition in thesingles and doubles amplitude calculations. The effectiveintermediate diagrams used for the T (1) amplitude deter-mining equations are shown in Figs. 2, 3 and 4. Theseeffective diagrams are finally connected with the respec-tive T (1) operators to obtain the amplitudes of the singlesand doubles excitations and the corresponding diagramsare presented in Figs. 6 and 7. Contributions from theterms of D are evaluated directly for the T (1) amplitudecalculations and the corresponding diagrams are shownin Figs. 8 and 9.In order to estimate the dominant contributions fromthe triple excited configurations, we define an excitationoperator perturbatively in the CC framework as following T (0) ,pert = 13! pqr X abc ( H DCN T (0)2 ) pqrabc ǫ a + ǫ b + ǫ c − ǫ p − ǫ q − ǫ r (34)which diagrammatically shown in Fig. 5 and contract itwith the D operator to calculate the amplitudes of the T (1)2 perturbed CC operator in a self-consistent procedureconsidering it in Eq. (32) as part of D . We refer thisapproach as the CCSD p T method in this work.Using the above formulation, the expression for the T (1)1 T (1)2 T (1)2 T (1)2 T (1)2 T (1)2 T (1)2 T (1)1 T (1)1 T (1)1 ( a ) ( b ) ( c ) ( d ) ( e )( f ) ( g ) ( h ) ( i ) ( j ) P − P H − H P − H P − H HP − PHHH − PP PP − PH PP − PH HH − PH HH − PH FIG. 6: Final contributing diagrams for the T (1)1 amplitudecalculations which are constructed by the contraction of ef-fective one and two body intermediate diagrams with T (1) H − HP − P PP − PP HP − PPHH − PP HH − PP HP − PH HP − PHHH − HH HP − HH T (1)2 T (1)2 T (1)2 T (1)2 T (1)2 T (1)2 T (1)2 T (1)2 T (1)2 T (1)1 T (1)1 ( a ) ( b ) ( c ) ( d )( e ) ( f ) ( g ) ( h )( i ) ( j ) FIG. 7: Final contributing diagrams for the T (1)2 amplitudecalculations which are constructed by the contraction of ef-fective one and two body intermediate diagrams with T (1) polarizability is given by [30] α = 2 h Φ | Ω (0) † RCC D Ω (1)RCC | Φ ih Φ | Ω (0) † RCC Ω (0)RCC | Φ i = 2 h Φ | e T (0) † De T (0) T (1) | Φ ih Φ | e T (0) † e T (0) | Φ i = 2 h Φ | ( z}|{ D (0) T (1) ) con | Φ i , (35)where z}|{ D (0) = e T † (0) De T (0) is a non-truncating series. Inthe LCCSD method, we only consider the terms z}|{ D (0) = O + OT (0) + T † (0) O + T † (0) OT (0) . Computational steps toaccount all the significant contributions from z}|{ D (0) N have D D D DDD T (0)1 T (0)1 T (0)1 T (0)1 T (0)2 T (0)2 FIG. 8: Single excitation diagrams from D that contribute tothe calculations of the T (1) amplitudes. DD D D T (0)1 T (0)1 T (0)2 T (0)2 T (0)2 T (0)2 FIG. 9: Double excitation diagrams from D that contributeto the calculations of the T (1) amplitudes. been described in detail in our previous work [30]. IV. RESULTS AND DISCUSSION
Our final results using the CCSD p T method along withthe available experimental values for Al + , Si , Zn andCd and from the other calculations are given in Table I.To ascertain lucidity in the accuracies of the results fromour calculations, we also provide the estimated uncer-tainties associated with our results by estimating contri-butions from various neglected sources and give them inthe parentheses alongside the CCSD p T results in abovetable. The value that is referred to as the experimentalresults for Al + is not directly obtained from the mea-surement, rather it is estimated by summing over theexperimental values of the oscillator strengths and hasrelatively large uncertainty compared to some of the re-ported calculations [39]. There are two high-precisionresults reported as the experimental values for the Si ion [43, 44], however the value reported in [44] is ob-tained from the direct analysis of the energy intervalsmeasurement using the resonant Stark ionization spec-troscopy (RESIS) technique while the other value [43]is reported by reanalyzing the data of Ref. [44]; whichis about 0.03% larger than the former value. The onlyavailable experimental result of the ground state α of Znis measured using an interferometric technique by Goebel D Ω (2 , ( i ) D Ω (2 , ( iv ) Ω (1 , † ( v ) D Ω (1 , ( vi ) D Ω (2 , ( vii ) D Ω (0 , D Ω (2 , ( viii ) D Ω (2 , † Ω (2 , ( ii ) D Ω (2 , ( iii ) FIG. 10: Few significantly contributing non-RPA typeMBPT(3) diagrams. et al. [46]. Similarly there is also one measurement of α available for Cd using a technique of dispersive Fourier-transform spectroscopy, but the reported uncertainty inthis experimental value is comparatively large [51]. Nev-ertheless when we compare our CCSD p T results with allthese experimental values, they match very well withintheir reported error bars except for Cd. In fact, our cal-culations are more precise in all the systems apart for theSi ion. There are no experimental results available forthe other considered ions to compare them against ourcalculations.There are also a number of calculations of α avail-able by many groups using varieties of many-body ap-proaches among which some of them are based on ei-ther the lower order methods or considering the non-relativistic mechanics. An old calculation of α in B + was reported by Epstein et al [34] based on the cou-pled perturbed Hartree-Fock (CHF) method while Cheng et al had employed a configuration interaction methodconsidering a semi-empirical core-polarization potential(CICP method) to evaluate it more precisely [35]. LaterSafronova et al used a combined CI and LCCSD meth-ods (CI+all order method) to determine α of B + ion[36]. However, the CCSD p T result seems to be largerthan all other calculations. Our analysis suggests thatthe differences in these results are mainly due to inclu-sion of the pair-correlation effects to all orders in our CCmethod. In C ion, we find only one theoretical resultreported by Epstein et al using the same CHF method.Our result for C +2 is also slightly larger than the value TABLE I: Results for the dipole polarizabilities from ourCCSD p T method along with the available measurements andothers calculations. Uncertainties in the results are given inthe parentheses and the references are cited in the squarebrackets.System Present OthersCCSD p T Theory ExperimentB + + a b + + a Estimated from the measured oscillator strengths. b Obtained by reanalyzing data of Ref. [44]. reported by the above calculation. Till date Al + is themost precise ion clock in the world [5] for which a cou-ple of high-precision calculations have been reported onthe determination of α of this ion by attempting to pushdown the uncertainty in the black-body radiation (BBR)shift of the respective ion-clock transition [36, 41, 42].Among them calculations carried out by Mihaly et al is based on the relativistic CC method considering upto quadrupole excitations and finite field approach [41].However, calculations carried out in this work is basedon the Cartesian coordinate system and minimizing theenergies in the numerical differentiation approach in con-trast to the present CCSD p T method, where the matrixelements of D are evaluated in the spherical coordinatesystem. Calculations reported by Yu et al is using thesame approach of Ref. [41], but by considering a differ-ent set of single particle orbitals [42]. Safronova et al have employed the CI+all order approach to calculate α of Al + . There are also other theoretical results havebeen reported based on varieties of many-body methodssuch as CCSD, CICP, CI etc. both in the non-relativisticand relativistic mechanics [37, 38, 40]. We find an excel-lent agreement among all the theoretical results. Someof these methods have also been employed to calculate α of Si [38, 40] which are in perfect agreement with theexperimental results. However, our CCSD p T value seemsto be little larger then the experimental result but fallswithin the estimated uncertainty. We found only onemore calculation of α in Ga + using the CICP method TABLE II: Dipole polarizabilities of the considered atomicsystems are presented using different many-body methods.System DF MBPT(3) RPA LCCSD CCSDB + + + + [50] to compare with our result. Although values fromboth the calculations are very close but they do not agreewithin their reported uncertainties. Calculations in Cdare reported by many groups including the latest one us-ing the Douglas-Kroll-Hess (DKH) Hamiltonian by Roos et al [47]. Calculations carried out by Ye et al [45] isbased on the relativistic formalism in the CICP method.All the theoretical results are consistent and show goodagreement with each other suggesting that the experi-mental result could have been overestimated. Thereforeit is important to have another measurement of the po-larizability of Cd to resolve this ambiguity. Again therehas also been an effort made for the precise determina-tion of α in In + to estimate the BBR shift accuratelyfor its proposed atomic clock transition [36]. Our resultagrees nicely with this calculation. As discussed earlier,calculations carried out in [36] are based on the CI+allorder method. We could not find any other calculationsof α of the ground states of the Ge and Sn ions tomake comparative analyses with our results.To assimilate the underlying roles of the electron corre-lation behavior in the evaluation of α of the ground statesof the considered systems, we systematically present thecalculated values of the dipole polarizabilities in TableII from the DF, MBPT(3), RPA, LCCSD and CCSDmethods. So, the differences between the CCSD resultsand the values quoted from the CCSD p T method in Ta-ble I are the contributions from the partial triple excita-tions. Obviously, these differences are small in magnitudeimplying that the contributions from the unaccountedhigher order excitations are very small. The lowest orderDF results are smaller in magnitudes in the lighter sys-tems but their trends revert in the Cd isoelectronic sys-tems with respect to their corresponding CCSD results.Also, the MBPT(3) results do not follow a steady trend.In the B + , C , Al + and Si ions, the correlation ef-fects enhance the α values in the MBPT(3) method fromtheir DF results while the MBPT(3) results are smallerthan the DF values in the other systems. As has beenstated earlier RPA is a non-perturbative method embrac-ing the core-polarization effects to all orders, but we findthat the results are over estimated in this method com-pared to the CCSD results; more precisely from the ex-perimental values given in Table I. We understand thesedifferences as the contributions from the pair-correlationeffects that are absent in the RPA method, but they areaccounted intrinsically to all orders as the integral partof the CCSD method. The role of the pair-correlationeffects in the determination of α are verified by exam-ining contributions from the individual MBPT(3) dia-grams. The dominant contributing non-RPA diagramsappearing in the MBPT(3) method that take care of thepair-correlation effects are shown in Fig. 10. In fact, con-tributions from these non-RPA diagrams are found to belarger than the differences between the RPA and CCSDresults reported in Table II. This finding advocates thatthere are large cancellations among the lower order andhigher order pair-correlation contributions in the CCSDmethod bestowing modest size of contributions to α , butthey appear to be very significant in the heavier systemsto attribute accuracies in the results. To demonstrate theroles of the non-linear terms to procure high precision α values in the considered ions, we have also given theresults from the LCCSD method in the above table. Al-though LCCSD is an all order perturbative method, butit clearly omits higher order core-polarization and pair-correlation effects that crop-up through the non-linearterms involving T (0) T (0) or higher powers of T (0) . Con-sequently, this method also over estimates the results likethe RPA method. The LCCSD results in B + and C are larger than the RPA values, but the LCCSD valuesare smaller than the RPA results in the other cases. Thisclearly demonstrates intermittent trends of the correla-tion effects in the determination of α of the systems be-longing to a particular group of elements in the periodictable to another through a given many-body method aswell as when they are studied using the methods withdifferent levels of approximations. To manifest contribu-tions from the correlations effects through various many-body methods quantitatively, we portray the results ob-tained for α of the considered systems using these meth-ods in a histogram as shown in Fig. 11. This clearlybespeaks about the lopsided trend in the estimation of α of the considered systems. Again, we also plot the α values of the singly and doubly charged ions separatelyin Figs. 12 and 13 in order to make a comparative anal-ysis in the propagation of correlation effects through theemployed methods in these elements that belong to twodifferent groups of the periodic table. This figure showsthat the contributions from the correlation effects in thesingly charged and doubly charged ions do not exactlyfollow similar trends.Finally, we would like to discuss about the trends inthe correlation effects coming through various CCSD p Tterms. We give contributions explicitly from the indi-vidual CC terms of linear form and the rest as “Others”in Table III. Clearly, this table shows that the first term DT (1) gives the dominant contributions as it subsumes allthe leading order core-polarization and pair-correlationeffects along with the DF result. The next dominant -30-20-10 0 10 20 30 40 50 60 B + C +2 Al + Si +2 Zn Ga + Ge +2 Cd In + Sn +2 ( α - α D F ) / α D F ( % ) MBPT(3)RPALCCSDCCSD p T FIG. 11: (color online) Histogram showing ( α − α D ) /α D (in%) with different many-body methods against the consideredatomic systems.
10 15 20 25 30 B + Al + Ga + In + D i po l e P o l a r i za b ilit y ( α ) DFMBPT(3)RPALCCSDCCSD p T FIG. 12: (color online) Trends in the calculations of dipolepolarizabilities ( α ) from the employed many-body methodsin the considered singly charged ions. contributing term is T (0) † DT (1)1 which incorporates somecontributions from the correlation effects emanated atthe MBPT(2) level and possess opposite signs from the DT (1) contributions causing cancellations among them.It is also worthy to mention that contributions comingfrom the T (0) † DT (1)2 term corresponds to the higher or-der perturbation and also accounts contributions fromthe doubly excited intermediate states. As seen from thetable, these contributions are non-negligible suggestingthat they should also be estimated accurately for accom-plishing high precision results and the sum-over-statesapproach may not be able to augment these contribu-tions suitably in the considered systems. Contributionsfrom the other non-linear CC terms at the final propertyevaluation level seem to be slender, although the differ-ences between the LCCSD and CCSD results emphasistheir importance for accurate calculations of the atomicwave functions in the considered systems.0 C +2 Si +2 Ge +2 Sn +2 D i po l e P o l a r i za b ilit y ( α ) DFMBPT(3)RPALCCSDCCSD p T FIG. 13: (color online) Trends in the calculations of dipolepolarizabilities ( α ) from the employed many-body methodsin the considered doubly charged ions.TABLE III: Contributions to the α of the ground state ofconsidered atomic systems from various CCSD p T terms.System DT (1)1 T (0) † DT (1)1 T (0) † DT (1)1 T (0) † DT (1)2 Others+c.c +c.c +c.c +c.cB + − − − − + − − − − − − + − − − − − − + − − − − V. CONCLUSION
We have employed a variety of many body methodsto incorporate the correlation effects at different levels of approximations to unravel the role of the correlationeffects and follow-up their trends to achieve very accu-rate calculations of the dipole polarizabilities of threegroups of elements in the periodic table. We find thepatterns in which the correlation effects behave with re-spect to the mean-field level of calculations are divergentin the individual isoelectronic systems through a particu-lar employed many-body method. Also, our calculationsreveal that inclusion of both the core-polarization andpair-correlation effects to all orders are equally impor-tant for securing high precision dipole polarizabilities inthe considered systems and the core-polarization effectsplay the pivotal role among them. Contributions fromthe doubly excited states are found to be non-negligibleimplying that a sum-over-states approach may not bepertinent to carry out these studies. Our results ob-tained using the singles, doubles and important triplesapproximation in the coupled-cluster method agree verywell with the available experimental values in some of thesystems except for cadmium. In fact none of the reportedtheoretical results for cadmium agree with the measure-ment, however there seem to be reasonable agreementamong all theoretical results. This urges for further ex-perimental investigation of the cadmium result. In fewsystems, there are no experimental results available yetand the reported precise values in the present work can beserved as exemplars for their prospective measurements.
VI. ACKNOWLEDGMENT
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