Investigating properties of Cl^- and Au^- ions using relativistic many-body methods
aa r X i v : . [ phy s i c s . a t o m - ph ] J a n Investigating properties of Cl − and Au − ions using relativistic many-body methods B. K. Sahoo ∗ Atomic, Molecular and Optical Physics Division,Physical Research Laboratory, Navrangpura, Ahmedabad-380009, India (Dated: Received date; Accepted date)We investigate ground state properties of singly charged chlorine (Cl − ) and gold (Au − ) negativeions by employing four-component relativistic many-body methods. In our approach, we attachan electron to the respective outer orbitals of chlorine (Cl) and gold (Au) atoms to determine theDirac-Fock (DF) wave functions of the ground state configurations of Cl − and Au − , respectively.As a result, all the single-particle orbitals see the correlation effects due to the appended electronof the negative ion. After obtaining the DF wave functions, lower-order many-body perturbationmethods, random-phase approximation, and coupled-cluster (CC) theory in the singles and doublesapproximation are applied to obtain the ground state wave functions of both Cl − and Au − ions.Then, we adopt two different approaches to the CC theory – a perturbative approach due to thedipole operator to determine electric dipole polarizability and an electron detachment approach inthe Fock-space framework to estimate ionization potential. Our calculations are compared with theavailable experimental and other theoretical results. I. INTRODUCTION
A number of stable negative atomic ions have beenobserved in the laboratories [1, 2]. Their spectroscopicand scattering properties are of immense interest to boththe experimentalists and theoreticians [3, 4]. It is wellknown fact that the Sun looks yellow due to the black-body radiation from H − at the temperature T = 5780K [5]. Another prominent example is, radiation fromnight-sky is observed due to the reaction of O − with O +2 and N +2 ions [6]. Generally Penning traps are used tostore the negative ions in the laboratory [7, 8], but Paultraps combined with time-resolved detection techniquesare also useful to investigate photo-detachment processesof electrons [3, 4]. The Penning trap provides large mag-netic field to analyze its effect on the negative ions, whilethe Paul trap offers better signal-to-noise ratio to performhigh-precision measurements of spectroscopic properties.Though conducting experiments with negative ions areprecarious relative to positively charged ions, there arestill a number of negative ions undertaken in the labora-tories for investigations. Some of the prominent negativeatomic ions that are experimentally probed include H − ,Li − , B − , C − , Al − , Ca − , Cu − , Si − , Cl − , Au − and others(please see reviews in Refs. [3, 4, 9, 10]). Due to a lot ofdemand, negative atomic ion physics are being reviewedfrom time to time since 1970s. Massey was one of thefirst persons to update the information about the nega-tive ions in a monograph [11] followed by a review article[12]. The progresses made in the negative ion physicsduring 1980s were discussed by Bates [13], Esaulov [14],Schulz [15] among many others. The latest review articleby Andersen covers a wide range of topics relevant to thenegative ion physics [16].Owing to the complication in the experimental set up,only a few selective spectroscopic properties of the nega- ∗ [email protected] tive ions have been measured among which electron affin-ity (EA) or negative of the ionization potential (IP) of theoutermost electron is the most common [11, 12]. A largenumber of studies are focused on the photo-detachmentcross-sections using various techniques [3, 4, 9, 17, 18].The typical energy levels of negative ions are quite differ-ent than their isoelectronic neutral atoms. There is onlylittle knowledge revealed about the energetically excitedstates of negative atomic ions, but the general perceptionis that these states lie just above the ground state of theparent neutral atom. Recent studies reveal that someof the negative ions such as lanthanide sequence possessbound excited states [19]. Thus, these states are antici-pated to be extremely short-lived. Unlike the Coulombinteractions that are solely responsible for binding elec-trons in neutral atoms and positively charged ions, theexcess electron(s) in negative ions are known to be boundby short-range potentials [16]. As a result, negative ionsexhibit many exotic properties that are totally differentfrom neutral atoms and positively charged ions.Theoretical studies of spectroscopy properties ofatomic negative ions are very interesting to test the valid-ity of quantum many-body methods. The extrapolatedEA values from the IPs of neutral atoms and the pos-itive ions suggests that negative ions for the elementslike He, N, Ne, Mg, or Ar ions cannot exist, but thishas been disproved later [16]. Therefore, it is imper-ative to apply potential quantum many-body methodsfrom the first principle to study the properties of negativeions. A number of methods such as many-body perturba-tion theory (MBPT), multi-configuration Hartree-Fock(MCHF) method, random-phase approximation (RPA),R-matrix approach including coupled-cluster (CC) the-ory have been employed to investigate atomic propertiesand scattering cross-sections of negative ions [3, 4, 20].It is still challenging to match the theoretical values withthe experimental results using many-body calculationseven for the basic property like EA. Most of the previouscalculations are carried out in the non-relativistic theoryframework and some cases the relativistic effects are es-timated approximately [21]. There are also relativisticcalculations in the negative ions reported in Refs. [22–24]. The CC theory is considered to be the gold standardof electronic structure calculations in many-electron sys-tems [25, 26]. It captures electron correlation effects toa much better extent than other many-body methods atthe given level of approximation. Therefore, considera-tion of relativistic CC (RCC) methods are the naturalchoices to investigate both the relativistic and electroncorrelation effects in the determination of properties ofatomic systems in general and of negative ions in partic-ular.Accurate evaluation of electric dipole polarizabilities( α d ) of negative ions have been paid less attention. Theirapplications in crystals are tremendous as they help tofind out mobility of negative ions due to external elec-tric fields. The sizes of crystals can be estimated withthe knowledge of α d values of their negative ions [27–30]. α d values of ions are used as key parameters in the ex-planation of the Hofmeister series – the systematic trendof different ions with the same valency in their ability toprecipitate in macromolecules from aqueous solutions [31]and they can be useful to analyze the behaviors of nega-tive ions in external static fields, which are manifested inthe threshold photo-detachment studies [32]. Measure-ments of α d values of negative ions are extremely diffi-cult due to which only a very limited number of theoret-ical studies on these quantities are carried out thus far.Many of these calculations are available only for a fewelectron negative ions [33–35]. Theoretical studies on α d values of a number of heavier negative ions are reportedby Sadlej and coworkers [36, 37] by employing a varietyof methods including the CC methods. They have alsohighlighted unusually large contributions from the rela-tivistic effects to the determination of α d values. Butthe relativistic effects were estimated by taking the spin-averaged Douglas-Kroll atomic Hamiltonian with no-pair(DKnp) approximation.In this work, we intend to investigate the α d values andIPs of Cl − and Au − negative ions by considering the four-component Dirac-Coulomb (DC) Hamiltonian at differ-ent levels of approximations in the many-body methods.We have deliberately selected these two candidates to an-alyze the electron correlation trends in the above proper-ties. Cl − is isoelectronic to Ar noble gas atom, whereasAu − is isoelectronic to Hg atom. Our previous calcula-tions of α d values in Ar [38] and Hg [39, 40] atoms showa very contrast correlation contribution trends at differ-ent levels of approximations in the many-body methods.Since the outermost electron in a negative ion is veryweakly bound, the electron correlation effects can be-have completely different way in Cl − than Ar, and soas between Au − and Hg. This can be demonstrated byevaluating α d values of Cl − and Au − by applying meth-ods similar to that were employed earlier to determine α d values of Ar and Hg atoms, and making comparativeanalyses. There are precise measurements of EA of Cl − reported in Refs. [41, 42]. An interesting study togetherwith experimental and theoretical methods was carriedout to infer mass shift from the EAs of negative ions ofchlorine isotopes [43], following which another theoreticalwork was devoted to explain the discrepancy between theprevious experimental and theoretical data [44]. MBPTmethods in the finite-field (FF) approach were employedby Diercksen and Sadlej [36] to estimate the α d value ofCl − . Photo-detachment phenomena of Au − has been rig-orously studied by several experiments [3, 4, 18]. Thereare also many practical applications of Au − in materialscience and chemistry [45, 46]. Both the non-relativisticand approximated relativistic calculations of α d of Au − are reported in Refs. [37, 48]. A precise measurementof EA of Au − has been reported in Ref. [47], followingwhich a number of calculations have been carried out toexplain the experimental data. Earlier theoretical calcu-lations of energies in both the ions spanned over a widerange and disagree with each other [21]. However, recentsophisticated calculations considering higher-level exci-tations and higher-order relativistic corrections show ex-cellent agreement with the measurements [49, 50]. II. BASIC FORMALISM
The electronic configurations of Cl and Au are[2 p ]3 s p and [5 p ]4 f d s , respectively. This im-plies that the electronic configurations of Cl − and Au − are the [3 p ] and [5 p ]4 f d s closed-shell configu-rations, respectively. The ground state | Ψ (0)0 i and itsenergy E (0)0 due to atomic Hamiltonian ( H at ) withoutconsidering any external interaction can be obtained bysolving the equation H at | Ψ (0)0 i = E (0)0 | Ψ (0)0 i , (1)where we consider H at as sum of the Dirac Hamiltonian,nuclear potential, and Coulomb repulsion potential ( V C )seen by the electrons. Due to the two-body nature of V C = P i,j r ij (in atomic units (a.u.)), an exact solutionof the above equation is not feasible. Thus, we express H at = H + V res where H = P i h i contains Dirac Hamil-tonian, nuclear potential and an effective one-body mean-field potential U = P i u i constructed from V C and theresidual part is defined as V res = V C − U . We adopt theDirac-Fock (DF) method to define U . In this method,the approximated ground state wave function | Φ i andself-consistent Fock (SCF) energy (or DF energy E DF )are obtained by determining the wave functions for thesingle-particles as( h i + u i ) | φ (0) i i = ǫ (0) i | φ (0) i i , (2)where | φ (0) i i is the i th orbital wave function with energy ǫ (0) i . The Slater determinant of single-particle wave func-tions form | Φ i and E DF = P i ǫ (0) i + h Φ | V res | Φ i . Thesingle-particle mean-field potential is defined as u i | φ (0) i (1) i = N c X a (cid:20) h φ (0) a (2) | r | φ (0) a (2) i| φ (0) i (1) i−h φ (0) a (2) | r | φ (0) i (2) i| φ (0) a (1) i (cid:21) , (3)where N C represents for the number of electrons in therespective negative ions. It should be noted here that allthe orbitals see the correlation with the appended elec-tron of the respective negative ion in the above formal-ism. We have calculated nuclear potential for an electronat the distance r by assuming finite-size nuclear Fermicharge density distribution, given by [51] ρ ( r ) = ρ e ( r − c ) /a , (4)where ρ is the normalization constant, and the param-eter c and a = 4 t ln (3) are said to be half-charge-radius and skin thickness of the atomic nucleus, respec-tively. The radial components of the DF single-particlewave functions are expanded using Gaussian type orbitals(GTOs), defined for a given orbital angular momentum( l ) symmetry as [52] f l ( r ) = N l X k C k N k r l e − α β k − r , (5)where N l denotes number of GTOs, C k corresponds toexpansion coefficient, α and β are arbitrary parametersthat are chosen to optimize for the finite-size basis func-tions, and N k is the normalization factor of the respectiveGTO and defined in Ref. [53].The exact ground state wave function after includingthe electron correlation effects from the residual interac-tion V res can be obtained from the above mean-field wavefunction by operating the wave operator Ω (0)0 as [54] | Ψ (0)0 i = Ω (0)0 | Φ i . (6)In the presence of an external electric field ~ E , the wavefunction of the negative ion due to the total Hamiltonian H = H at + ~D · ~ E with dipole operator D = P i d i can beexpressed as | Ψ i = Ω | Φ i , (7)where Ω is the wave operator that is responsible foraccounting for electron correlation effects and effects dueto the electric field. For the weak electric field, we canexpand the wave function perturbatively as | Ψ i = | Ψ (0)0 i + | ~ E|| Ψ (1)0 i + · · · = h Ω (0)0 + | ~ E| Ω (1)0 + · · · i | Φ i (8)so that Ω = Ω (0)0 + | ~ E| Ω (1)0 + · · · . (9) Similarly, the modified energy can be expanded as E = E (0)0 + | ~ E| E (1)0 + 12 | ~ E| E (2)0 · · · . (10)In the above expressions, superscripts 0, 1, etc. denoteorder of ~ E in the expansion. The first-order energy shift( E (1)0 ) in atomic systems due to the presence of electric-field vanishes owing to spherical symmetry distributionof charges, but the second-order energy shift ( E (2)0 ) canbe given by E (2) = 12 α d | ~ E| . (11)This shift can be estimated with the knowledge of α d fora given value of ~ E . In molecular systems, α d is estimatedconveniently using the FF approach. To adopt the FF ap-proach for determining α d of atomic systems, it requiresto exploit the spherical symmetrical property. Thus, theprevious calculations of α d of Cl − and Au − are estimatedin the FF approach by breaking atomic spherical symme-try. To determine α d values of these ions by preservingspherical symmetry, we adopt the perturbative approachby expressing as [56, 57] α d = 2 h Ψ (0)0 | D | Ψ (1)0 ih Ψ (0)0 | Ψ (0)0 i = 2 h Φ | Ω (0) † D Ω (1)0 | Φ ih Φ | Ω (0) † Ω (0)0 | Φ i . (12)In the following section, we shall be discussing about howto define both the unperturbed and perturbed wave oper-ators in the DF, relativistic MBPT (RMBPT), relativis-tic RPA (RRPA) and RCC methods to fathom about thepropagation of electron correlation effects from lower- toall-order perturbative methods in the evaluation of α d ofthe undertaken negative ions.Now, we proceed to discuss the general procedure toobtain IP by removing the extra electron from the outermost orbital of the negative ions. For this purpose, wedefine the new working reference state as | Φ a i = a a | Φ i ,where a a denotes annihilation of an electron from theoutermost orbital | φ a i of | Φ i . Accordingly, the waveoperator due to H at is defined to obtain the exact stateas [65] | Ψ a i = Ω a | Φ a i . (13)By calculating energy difference between this state and | Ψ (0)0 i , one can get IP. Below, we discuss the RCC theoryin the Fock-space formalism to define Ω a . III. MANY-BODY METHODSA. α d evaluation In the k th order (R)MBPT method ((R)MBPT( k )),the wave operator can be expanded as [38, 54]Ω = k X m =1 Ω ( m ) = k X m =1 m X i =0 k − m X j =1 Ω ( i,j ) , (14)where i - orders of V res and j - orders of D are incorporatedin the expansion. Thus, the wave operators with zeroth-and first-order D in the RMBPT( n ) method are given byΩ (0) = k X m =0 Ω ( m, and Ω (1) = k − X m =0 Ω ( m, (15)with Ω (0 , = 1, Ω (1 , = 0 and Ω (0 , = P p,a h φ p | d | φ a i ǫ (0) p − ǫ (0) a for all the occupied orbitals denoted by the index a andunoccupied orbitals denoted by the index p . This followsthe expression to evaluate the lowest-order polarizabili-ties result in the DF method as α d = 2 h Φ | Ω (0 , † D Ω (0 , | Φ i = 2 h Φ | D Ω (0 , | Φ i . (16)The amplitudes of the finite-order unperturbed andperturbed wave operators are obtained using the Bloch’sequation [54, 55][Ω ( β, , H ] P = QV res Ω ( β − , P − β − X m =1 Ω ( β − m, P V res Ω ( m − ,l ) P (17)and using the modified Bloch’s equation [38][Ω ( β, , H ] P = QV res Ω ( β − , P + QD Ω ( β, P − β − X m =1 (cid:0) Ω ( β − m, P V res Ω ( m − , P − Ω ( β − m, P D Ω ( m, P (cid:1) , (18)respectively, with the definitions of model space P = | Φ ih Φ | and orthogonal space Q = 1 − P .This follows the expression for α d in the RMBPT(3)method as [38] α d = 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 (0 , | Φ i . It can be easily followed that thelowest-order term corresponds to the DF expression andterms containing up to one-order in V res and one D oper-ator will give rise expression for the RMBPT(2) method.Now, we move on to RRPA expression by expandingsingle-particle DF wave function and energy in the pres-ence of external electric field as [56] | φ i i = | φ (0) i i + | ~ E|| φ (1) i i + · · · (20)and ǫ i = ǫ (0) i + | ~ E| ǫ (1) i + · · · . (21)Since the single-particle dipole operator d is odd underparity, ǫ (1) i = 0. To obtain the first-order correctionto the single- particle wave function, the general single-particle equation is expanded by keeping up to linear in | ~ E| as (cid:16) h i + | ~ E| d i (cid:17) (cid:16) | φ (0) i (1) i + | ~ E|| φ (1) i (1) i (cid:17) + N c X b (cid:16) h φ (0) b (2)+ | ~ E| φ (1) b (2) | r | φ (0) b (2) + | ~ E| φ (1) b (2) i| φ (0) i (1)+ | ~ E| φ (1) i (1) i − h φ (0) b (2) + | ~ E| φ (1) b (2) | r | φ (0) i (2)+ | ~ E| φ (1) i (2) i| φ (0) b (1) + | ~ E| φ (1) b (1) i (cid:17) ≃ ǫ (0) i (cid:16) | φ (0) i (1) i + | ~ E|| φ (1) i (1) i (cid:17) . (22)Retaining only linear in | ~ E| terms from the above expres-sion, it yields (cid:16) h i + u i − ǫ (0) i (cid:17) | φ (1) i i = ( − d i − u (1) i ) | φ (0) i i , (23)where the modified DF potential u (1) i is given by u (1) i | φ (0) i (1) i = N c X b (cid:18) h φ (0) b (2) | r | φ (1) b (2) i| φ (0) i (1) i−h φ (0) b (2) | r | φ (0) i (2) i| φ (1) b (1) i + h φ (1) b (2) | r | φ (0) b (2) i×| φ (0) i (1) i − h φ (1) b (2) | r | φ (0) i (2) i| φ (0) b (1) i (cid:19) . (24)Using the completeness principle, we can write | φ (1) i i = X j = i C ji | φ j i , (25)where C ji s are the expansion coefficients. Thus, it can beexpressed as X j = i C ji (cid:16) h j + u j − ǫ (0) j (cid:17) | φ (0) j i = − (cid:16) d i + u (1) i (cid:17) | φ (0) i i . (26)This is solved self-consistently to obtain the C ji coeffi-cients, hence, | φ (1) i i to infinity order in Coulomb interac-tion and one order in the dipole operator by consideringcontributions only from the singly excited determinantsfrom | Φ i . In RRPA, the unperturbed wave operator is taken to be Ω (0 , = 1 and the first-order perturbed waveoperator is defined using the above expression byΩ (1) = Ω RPA = ∞ X k =0 X p,a Ω ( k, a → p = Ω (0 , a → p + ∞ X β =1 X pq,ab h h φ (0) p (1) φ (0) b (2) | r | φ (0) a (1) φ (0) q (2) i − h φ (0) p (1) φ (0) b (2) | r | φ (0) q (1) φ (0) a (2) i i Ω ( β − , b → q ǫ (0) p − ǫ (0) a + Ω ( β − , † b → q h h φ (0) p (1) φ (0) q (2) | r | φ (0) a (1) φ (0) b (2) i − h φ (0) p (1) φ (0) q (2) | r | φ (0) b (1) φ (0) a (2) i i ǫ (0) p − ǫ (0) a , (27)where a → p means replacement of an occupied orbital | φ a i from | Φ i by a virtual orbital | φ p i which alterna-tively refers to a singly excited state with respect to | Φ i .It can be understood from the above formulation thatthe RRPA method picks-up a certain class of single exci-tation configurations by capturing the core-polarizationcorrelation effects to all-orders. Again, contributions in-cluded in this perturbative approach is equivalent to theorbital relaxation effects that arise at the DF method inthe FF approach.Using the above wave operator, we evaluate α d inRRPA as α d = 2 h Φ | Ω (0 , † D Ω (1) | Φ i = 2 h Φ | D Ω RPA | Φ i . (28)In the RCC method, the wave operator including theexternal perturbation has the formΩ = e T , (29)where T is known as the excitation operator that is re-sponsible to take care of electron correlation effects fromthe reference state | Φ i due to V res and D operators. Byexpanding T in | ~ E| , and keeping zeroth and linear termsgives us [38, 39, 56, 58]Ω (0) = e T (0) and Ω (1) = e T (0) T (1) , (30)respectively. The amplitudes of the excitation operator T (0) and energy E (0) are determined by projecting theexcited determinants as [38, 58] h Φ τ | H at | Φ i = E (0)0 δ τ, , (31)where notation O = ( Oe T (0) ) c is used with subscript c means connected terms and | Φ τ i means excited Slaterdeterminants with respect to | Φ i . Similarly, the ampli-tudes of the excitation T (0) operator (note that energy E (1) = 0) are obtained by solving the equation h Φ τ | H at T (1) + D | Φ i = 0 . (32) In our calculations, we consider only the singles and dou-bles excited configurations in the RCC theory (RCCSDmethod) by denoting τ ≡ T (0) = T (0)1 + T (0)2 and T (1) = T (1)1 + T (1)2 . (33)In the RCC theory, the α d determining expression isgiven by [59, 60] α d = 2 h Φ | Ω (0) † D Ω (1) | Φ ih Φ | Ω (0) † Ω (0) | 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) ) c | Φ i , (34)where z}|{ D (0) = e T † (0) De T (0) is a non-truncating series.The above expression is derived from the property evalu-ation expression given by Refs. [61, 62]. We have adoptedan iterative procedure to take into accounting contribu-tions from this non-terminating series self-consistently asdescribed in our earlier works on α d calculations in theclosed-shell atoms [57, 58]. B. IP evaluation
In the Fock-space approach, the wave operator describ-ing removal of an electron from orbital | φ a i of | Φ i isdefined in the RCC theory by [63–67]Ω a = e T (0) (1 + R a ) , (35)where R a is another RCC operator introduced to takecare of the extra correlation effects that was includedthrough the detached electron. Then, the energy ( E a ) ofthe product state and amplitudes of the R a operator isobtained by solving h Φ η | H at R a + H at | Φ a i = h Φ η | [ δ η,a + R a ] | Φ a i E a , (36)where | Φ η i is designated as the excited configuration de-terminants from | Φ a i for the R a amplitude determinationelse it corresponds to | Φ a i to estimate E a . Hence, the IPof the electron removed from | φ a i is obtained by takingthe difference as ∆ E a = E (0)0 − E a . Here, we have alsoconsidered the RCCSD method approximation by consid-ering singles and doubles excited configurations for | Φ η i . C. Atomic Hamiltonian
The starting point of our calculation is the Dirac-Coulomb (DC) Hamiltonian [68] representing the leadingorder contributions to H em to calculate the zeroth-orderwave functions and energies which in atomic units (a.u.)is given by H DC = X i (cid:2) c α i · p i + ( β i − c + V n ( r i ) (cid:3) + X i,j>i r ij , (37)where α and β are the usual Dirac matrices, p is thesingle particle momentum operator, V n ( r ) denotes thenuclear potential, and P i,j r ij represents the Coulombpotential between the electrons located at the i th and j th positions. It should be noted that the above Hamil-tonian is scaled with respect to the rest mass energies ofelectrons. Contributions from the Breit interaction [69]to H em is determined by including the following potential V B = − X j>i [ α i · α j + ( α i · ˆr ij )( α j · ˆr ij )]2 r ij , (38)where ˆr ij is the unit vector along r ij .Contributions from the QED effects to H em are esti-mated by considering the lower-order vacuum polariza-tion (VP) interaction ( V V P ) and the self-energy (SE)interactions ( V SE ). We account for V V P through theUehling [70] and Wichmann-Kroll [71] potentials ( V V P = V Uehl + V W K ), given by V Uehl = − X i α e r i Z ∞ dx x ρ ( x ) Z ∞ dt p t − × (cid:18) t + 12 t (cid:19) h e − ct | r i − x | − e − ct ( r i + x ) i (39)and V W K = X i . Z πc (1 + (1 . cr i ) ) ρ ( r i ) , (40)respectively, where α e is the fine structure constant.The SE contribution V SE is estimated by including twoparts [72] V efSE = A l X i πZα e r i I ef ( r i ) − B l X i α e r i I ef ( r i ) (41) known as the effective electric form factor part and V mgSE = − X k iα e γ · ∇ k r k Z ∞ dx x ρ ( x ) Z ∞ dt t √ t − × h e − ct | r k − x | − e − ct ( r k + x ) − ct ( r k + x − | r k − x | ) i , (42)known as the effective magnetic form factor part. In theabove expressions, we use [73] A l = ( .
074 + 0 . Zα e for l = 0 , .
056 + 0 . Zα e + 0 . Z α e for l = 2 , (43)and B l = ( . − . y − . y + 0 . y for l = 0 ,
10 for l ≥ . (44)The integrals are given by I ef ( r ) = Z ∞ dx x ρ ( x )[( Z | r − x | + 1) e − Z | r − x | − ( Z ( r + x ) + 1) e − ct ( r + x ) ] (45)and I ef ( r ) = Z ∞ dx x ρ ( x ) Z ∞ dt √ t − (cid:26) (cid:18) − t (cid:19) × (cid:20) ln( t −
1) + 4 ln (cid:18) Zα e + 12 (cid:19)(cid:21) −
32 + 1 t (cid:9) × { α e t h e − ct | r − x | − e − ct ( r + x ) i + 2 r A e r A ct × [ E (2 ct ( | r − x | + r A )) − E (2 ct ( r + x + r A ))] (cid:27) (46)with the orbital quantum number l of the system, y =( Z − α e , r A = 0 . Z α e , and the exponential integral E ( r ) = R ∞ r dse − s /s . IV. RESULTS AND DISCUSSION
We would like to first discuss briefly about the pre-vious calculated values of α d in both the Cl − and Au − ions to understand the need of doing new theoretical re-sults by including correlations effects among all the elec-trons rigorously through relativistic many-body methods.For this purpose, we give the results for Cl − in Table Ifrom the only one calculation reported by Diercksen andSadlej [36] by incorporating electron correlation effectsusing the non-relativistic (NR) MBPT( k ) methods with k = 2 , s p d (basis I). Then, they had useda slightly larger basis (basis II) by appending a few more TABLE I. Calculated α d values (in a.u.) of Cl − by Dierck-sen and Sadlej [36] in the FF approach. Results from basis I(14 s p d ) and basis II (17 s p d ) are reported at differentlevels of approximations in MBPT method. Results from ba-sis I after including all core orbitals (All), and freezing coreorbitals from the K and L shells (Frozen) are also given.Method Basis I Basis IIAll Frozen K Frozen K+L AllHF 31.45 31.45 31.45 31.56MBPT D (2) 37.06 37.07 37.00 37.25MBPT D (3) 29.91 29.90 29.93 29.99MBPT SD (4) 36.71 36.71 36.62 36.87MBPT SD[1 / (4) 38.82 39.04 37.52 39.20MBPT SD (˜4) 35.47 35.44 35.22 35.63 high-lying s , p and d orbitals to basis I as 17 s p d toshow contributions from the high-lying orbitals. The dif-ferences between results from both the basis functionswere found to be insignificant. There are three majorlimitations of this calculation: (i) It uses NR theory, how-ever, later theoretical studies have exhibited quite largerelativistic effects in the determination of α d values of thenegative ions [37, 48, 78]. (ii) It considers only either dou-ble excitations (denoted by MBPT D ) or single and dou-ble excitations (denoted by MBPT SD ) excitations evenin the MBPT methods. (iii) It has completely ignoredcorrelation contributions from the higher-symmetry or-bitals such as f , g , etc.. Since α d involves E1 operator,whose matrix element is directly proportional to radialdistance, contributions from the higher angular momen-tum orbitals cannot be completely ignored. Nonetheless,a recommended value of α d of Cl − was reported as 37.5a.u. by Diercksen and Sadlej after analyzing correlationenergy trends and taking into account corrections fromthe Pade approximants though the MBPT SD [1 / SD (˜4) approxi-mation [36]. This recommended value was, however, notclose to any of the their results obtained using the first-principle calculations. Therefore, it is necessary to per-form more accurate calculation of α d of Cl − to ascertainits value by incorporating correlation effects among allelectrons more rigorously in the RCC theory.Now we turn to discuss about the earlier calculationsof α d for Au − . Compared to the Cl − ion, there are tworigorous calculations available for α d of Au − . Schwerdt-feger and Bowmaker [48] had carried out calculation ofthis quantity by using pseudoptentials in the NR theoryframework and using spin-orbit ( j )-averaged relativisticapproach. They had applied MBPT( k ) approximations,with k = 2 , TABLE II. Calculated α d values (in a.u.) of Au − bySchwerdtfeger and Bowmaker [48] using the MBPT, QCISDand QCISD(T) methods. Results obtained using the non-relativistic and j -averaged relativistic pseudopotentials aregiven separately. Number of active orbitals considered in thecalculations are denoted by N .Method N ResultNon-relativisticHF 660.07MBPT(2) 20 27.47MBPT(3) 20 199.02MBPT(4) 20 279.59QCISD 20 399.18QCISD(T) 20 257.12QCISD(T)/ f
20 362.87 j -averaged relativisticDF 204.95MBPT(2) 20 3.04MBPT(3) 20 62.29MBPT(4) 20 60.60QCISD 20 118.71QCISD(T) 20 96.03QCISD(T)/ f
20 121.88Here, “ f ” denotes for “without metal f functions”. sults from these methods, both in the NR and relativis-tic frameworks, are given in Table II. It can be seen thatthere are huge differences among the results from the NRand relativistic calculations at the same level of approx-imation in the many-body methods. These calculationshave also several limitations such as they use pseudopo-tentials instead of the HF potentials, relativistic effectsare approximated to j -averaged approach which cannotconsider the exact relativistic effects, and only N = 20number of active electrons were allowed to correlate outof total 80 electrons of Au − . Therefore, contributionsfrom the correlation effects from the remaining 60 elec-trons need to be investigated. Moreover, there are verylarge differences between results at various approxima-tions were seen (without showing any signature of con-vergence of result with the higher-order contributions).The difference in the results with and without consider-ing metal function f in the QICSD(T) method was alsofound to be quite large. Since it does not provide a rec-ommended value, one cannot be very confident to usethose results in any of the applications.Later, Kell¨o et al. have made a systematic analysisof α d of the negative ions of the coinage metal atomsincluding Au − [37]. These calculations were also car-ried out in the FF approach and they had investigatedrelativistic effects more judiciously by analyzing resultsfrom the the quasi-relativistic corrections from the mass-velocity and Darwin (MVD) corrections over NR resultsand DKnp Hamiltonian. They had used NpPolMe basisfunctions without and with fully uncontracted orbitalsand demonstrated roles of electron correlation effects by TABLE III. Calculated α d values (in a.u.) of Au − by Kell¨o etal. [37] using NR, MVD and DKnp Hamiltonians. Results us-ing the uncontracted basis with DKnp Hamiltonian are givenas DKnp ∗ . Calculations are carried out in the MBPT(2),CCSD and CCSD(T) approximations by considering threedifferent number of active orbitals N .Method N NR MVD DKnp DKnp ∗ SCF 630 101 193 195MBPT(2) 12 62 −
106 13 1518 − − − − applying MBPT(2) method, and CC method with sin-gles and doubles approximation (CCSD) and the CCSDmethod with partial triples approximation (CCSD(T))systematically. A qualitative agreement between thecalculations by Kell¨o et al. and that of Schwerdtfegerand Bowmaker was observed. Their finding reveals thatthe quasi-relativistic corrections from the MVD termsbring down the results by more than half to the NRresults in the HF method as well as in the CCSD andCCSD(T) methods. They also showed that the resultsfrom the DKnp Hamiltonian are very different from theirNR+MVD results. However, electron correlation effectsonly from a fewer electrons were included in their calcu-lations. They had considered active orbitals as N = 12,18 and 20 for the NR calculations, but they used only N = 12 and 18 for the relativistic calculations using theDKnp Hamiltonian. In fact, their basis functions werealso quite small, which considered only 3 s p d f diffusefunctions over the NpPolMe basis functions [37]. Resultsobtained by Kell¨o et al. at different levels of approxi-mations are given in Table III. This calculation also doesnot provide any recommended α d value of Au − and therewas no estimate of uncertainty.To improve the calculations of α d in Cl − and Au − , wehave considered the RCC theory in the perturbative ap-proach by using four-component relativistic Hamiltonian.We have used 40 GTOs for each angular momentum sym-metry up to l = 4 (i.e. g -symmetry) for the generation ofsingle particle orbitals. All the electrons are correlatedup to principal quantum number n = 20 virtual orbitalsto carry out calculations using the RMBPT, RRPA andRCCSD methods. The α d values are discussed and com-pared with the previous works first, then we present theIP results. In Table IV, we give the α d values of both theCl − and Au − negative ions that are obtained by usingDC Hamiltonian in the relativistic many-body methodsdescribed in the previous section and adding correctionsfrom the neglected effects. As can be seen, the trends TABLE IV. Our calculated α d values (in a.u.) of Cl − andAu − from different relativistic methods using the DC Hamil-tonian. Estimated corrections from higher-order effects anduncertainties are also given. The final recommended valuesare given after accounting for possible uncertainties and re-sults from the even-parity channel are shown with * mark.Method Cl − Au − N Result N ResultResults using DC HamiltonianDF 18 25.66 80 122.64RMBPT(2) 18 27.79 80 138.88RMBPT(3) 18 20.45 80 60.69RRPA 18 31.71 80 194.61RCCSD* 18 33.64 80 95.66RCCSD 18 35.68 80 94.30CorrectionsTriples 18 0.42 80 − . − .
01 80 3.32Final 35(1) 97(3) in the results from the DF to RCCSD methods usingthe DC Hamiltonian are quite different in both the ions.As described in Ref. [79], the even-parity channel mul-tipoles usually contribute predominantly to the electroncorrelation effects in the RCC calculations. To demon-strate their roles here, we have also presented results con-sidering only the even-parity multipoles in the RCCSDmethod (marked as RCCSD* to distinguish from the all-parity channel calculations). As can be seen there aresignificant differences in the results from both the chan-nels. It is worth mentioning that it is possible to evaluateresults from both the channels only when the sphericalcoordinate system is used to describe the atomic wavefunctions. In Cl − , the DF method gives a lower valueand RMBPT(2) increases it to a larger value. Then, theRMBPT(3) method brings it down and makes its valuelower than that of the DF value. After that the RRPAmakes it larger than the RMBPT(2) value and then, theRCCSD method gives the largest value. This trend is al-most similar to the calculation of α d in the isoelectronicatom Ar of Cl − , however, the α d value of Cl − is foundto be about three times larger than the value of Ar [38].Moreover, variation in the results from lower- to higher-order methods are not abrupt compared to the FF ap-proach discussed above. Since its previous calculation byDiercksen and Sadlej [36] was performed in the FF ap-proach through the molecular code, their mean-field re-sult using the Hartree-Fock (HF) method is equivalent toRPA approximation in the perturbative approach. Thisis why comparison between our RRPA value and the HFvalue of Ref. [36] shows a very good agreement. Further,Diercksen and Sadlej had employed NR method in con-trast to our relativistic calculation. So good agreementbetween our RRPA result with the above HF value ofFF approach indicates that the relativistic effects playless important roles in the determination of α d of Cl − .We have also given the calculated α d values of Au − from the considered relativistic many-body methods inTable IV. The trends in these results from the DF toRMBPT(3) methods look analogous to the calculationsin Cl − , but the RRPA result is found to be much higherthan the RCCSD value in this case. This trend has simi-larity with the calculation of α d of the isoelectronic atomHg of Au − , but the result of Au − is about three timeslarger than Hg [59, 60]. Our RRPA value is found to bein good agreement with the HF results of the earlier cal-culations using the j -averaged pseudopotential [48] andDK Hamiltonian [37]; better agreement with the laterone. We have also observed large differences in the re-sults from the RMBPT(2) and RMBPT(3) methods likethe previous studies. Though there are large differencesbetween our RCCSD results with the earlier discussedCCSD and QCISD results are observed, we find that ourRCCSD value agrees quite well with the CCSD(T) andQCISD(T) results. In a recent study on Cd atom [74], wehad observed that dipole polarizability value convergesfaster in the perturbative approach than FF approachwith respect to level of higher excitations. This mayhave been the reason for the good agreement between ourRCCSD result obtained in the perturbative approach andCCSD(T)/QCISD(T) results than the CCSD/QCISD re-sults of the FF approach.From the comparison between the α d results of boththe negative ions with their isoelectronic neutral atoms,it appears to us that these quantities change drasticallywhen there is imbalance between nuclear and electroniccharges. We have quoted the final α d values from ourcalculations by adding the estimated corrections fromthe Breit interaction, quantum electrodynamics (QED)effects and triple excitations. We have used RRPA toestimate the Breit and QED contributions, whereas thetriple excitation contributions are estimated by definingthe following excitation operators in the perturbative ap-proach [75, 76] T (0) ,pert = 1(3!) X abc,pqr ( H at T (0)2 ) pqrabc ǫ (0) a + ǫ (0) b + ǫ (0) c − ǫ (0) p − ǫ (0) q − ǫ (0) r (47) and T (1) ,pert = 1(3!) X abc,pqr ( H at T (1)2 ) pqrabc ǫ (0) a + ǫ (0) b + ǫ (0) c − ǫ (0) p − ǫ (0) q − ǫ (0) r , (48) where a, b, c and p, q, r subscripts denoting for the occu-pied and unoccupied orbitals, respectively, and subscripts0(1) correspond to (un)perturbed excitation operators.These operators are directly used in Eq. (34) as partof the T (0) and T (1) RCC operators to estimate the ap-proximated contributions due to the triple excitations.We have also estimated uncertainties to the final valuesdue to use of finite size basis functions. We obtain thefinal α d values of Cl − and Au − as 35(1) a.u. and 97(3)a.u. respectively. TABLE V. Contributions to α d values (in a.u.) of Cl − andAu − ions from different RCC terms. Terms that are notshown explicitly, their contributions are quoted together as‘Others’.RCC term Cl − Au − DT (1)1 T (0) † DT (1)1 − . T (0) † DT (1)1 − . − . T (0) † DT (1)2 T (0) † DT (1)2 − . − . We would also like to discuss the trends of electroncorrelation effects by comparing individual RCC termcontributions from the DC Hamiltonian to α d of boththe ions. In Table V, we give the contributions fromvarious RCC terms to α d of the Cl − and Au − negativeions. In both the cases, DT (1)1 contributes the highest asit contains the DF value and core-polarization effects toall-orders. These contributions are different than RRPAresults as here the core-polarization effects are also cou-pled with the pair-correlation correlations, and there arealso additional non-RPA contributions arising throughthe non-linear RCC terms [58]. We find that T (0) † DT (1)1 contributes negligibly small in Cl − , but it contributessignificantly to Au − . Contributions from T (0) † DT (1)1 arefound to be important in both the ions. The reasonfor this finding is that the lowest-order correlation ef-fects from the unperturbed and perturbed RCC operatorscome through T (0)2 and T (1)1 , respectively. The remainingterms are found to be less important in the determinationof α d values of both the ions.Now, we turn to discuss the IP values of Cl − and Au − ions. We give these values in Table VI from the DF,RMBPT(2), RCCSD* and RCCSD methods using theDC Hamiltonian. It can be noted that the extra elec-tron present in the 3 p / outer orbital in Cl − , whereasit is in the 6 s / orbital in Au − . Thus, the outer elec-tron in Cl − is more tightly bound than Au − . As can beseen in the above table, the correlation trends are differ-ent in both the cases because of the above said reason.The DF result in Cl − is higher than the RCCSD result,where the DF value is slightly higher than half of theRCCSD value in Au − . The RMBPT(2) method givesrelatively smaller correlation contributions to the deter-mination of IP in Cl − , whereas it gives comparativelylarger correlation contributions in Au − . We have alsoestimated corrections from the Breit and QED interac-tions using the RMBPT(2) method and quoted them inthe above table. In this case also we find that there arelarge differences between the results from the RCCSD*and RCCSD methods, and the results from the all-paritychannel are more reliable. To estimate the correctionsfrom the triple excitations, we construct a perturbative0 TABLE VI. IP values (in eV) of both Cl − and Au − nega-tive ions from various calculations by approximating many-body methods at different levels. Main results using the DCHamiltonian and corrections due to the higher-order effectsare given separately. Uncertainties are quoted along with thefinal results. The experimental results and previously calcu-lated values are also listed. Results from even-parity channelare shown with * mark. We have used conversion factor 1cm − = 0.00012397788 eV to mention all the results in thesame units.Method Cl − Au − ReferenceFrom the DC HamiltonianDHF 4.027 1.177 This workRMBPT(2) 3.070 2.297 This workRCCSD* 3.786 2.286 This workRCCSD 3.735 2.232 This workCorrectionsBreit 0.002 − . − .
113 0 . valence triple excitation as R pert a = 1(2!) X abc,pqr ( H at R a ) pqrabc ǫ (0) a + ǫ (0) b + ǫ (0) c − ǫ (0) p − ǫ (0) q − ǫ (0) r . (49) This is used only in the energy evaluating expression ofEq. (36) after obtaining amplitudes of the RCCSD oper-ators and the estimated contributions are given in TableVI. We have also estimated uncertainties from the finitesize basis functions to the RCCSD values using the DCHamiltonian. After taking into account all these contri-butions, we obtain the final IP values of Cl − and Au − as 3.63(5) eV and 2.33(5) eV respectively. We also com-pare our results with the available calculations and ex-perimental values. Two precise experimental values ofIP for Cl − have been reported in Refs. [41, 43] and ourresult agrees within the error bars of the experimentalvalues. A list of data for this quantities using variousmethods can be found in the National Institute of Sci-ence and Technology database [49]. We have quoted theresult from the CCSD(T) method with daug-cc-pVTZbasis from this list in the above table. We find a goodcomparison between both the calculations. Similarly, avery precise experimental value of IP for Au − is reported [80] and we find that our result matches well with theexperimental value. We also compare our results withthe other calculations that are reported using the DKnpHamiltonian in the CCSD(T) method [37] and using the j -averaged relativistic pseudo-potential in the QCISD(T)method [48] in the above table. There are also an-other two more precise calculations of energies reportedby Eliav et al. by considering four-component Dirac-Coulomb-Breit interaction Hamiltonian in the Fock-space RCCSD method with partial triples correction(RCCSD(T) method) [77] and by Pasteka et al byusing singles, doubles, triples, quadruples, and pen-tuples approximations in the relativistic equation-of-motion coupled-cluster method after including Breit andQED interactions with the Dirac-Coulomb Hamiltonian(DC-CCSDTQP+Breit+QED method) [50]. Our re-sults are in agreement with the values reported in Refs.[49, 50, 77]. V. SUMMARY
We have employed relativistic many-body methods inthe lower-order perturbation, random-phase approxima-tion, and coupled-cluster theory frameworks by consid-ering the four-component Dirac-Coulomb atomic Hamil-tonian to analyze the trends in the electron correlationeffects for the determination of dipole polarizabilities ofCl − and Au − ions. The relativistic coupled-cluster the-ory is approximated at the singles and doubles excitationslevel. We have evaluated these values in the perturba-tive approach by preserving atomic spherical symmetryin contrast to the previous studies. We have comparedour results with the previous calculations that were re-ported using the quasi-relativistic and scalar Douglas-Kroll spin-averaged (no-pair) Hamiltonians. We find rea-sonably good agreement among these results. We havealso given contributions from various terms of the rela-tivistic coupled-cluster theory and compared the trendsbetween both the considered negative ions. Moreover,we have analyzed ionization potentials of both the ions atdifferent levels of approximations in the relativistic many-body methods, and compared with the available preciseexperimental results and calculations. Our results matchwell with the previous calculations suggesting that ourmethods are also reliable to produce these values. Ourresults can be further improved by including higher-orderrelativistic effects and contributions from the full tripleexcitations through the relativistic coupled-cluster the-ory. ACKNOWLEDGEMENT
We acknowledge use of Vikram-100 HPC cluster ofPhysical Research Laboratory (PRL), Ahmedabad, In-dia to carry out computations for this work.1 [1] V. Dudinikov,
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