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Relativistic coupled-cluster calculations of Ne-20, Ar-40, Kr-84, and Xe-129: Correlation energies and dipole polarizabilities

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Relativistic coupled-cluster calculations of

20

Ne,

40

Ar,

84

Kr, and

129

Xe:

Correlation energies and dipole polarizabilities

B. K. Mani,1K. V. P. Latha,2and D. Angom1

1

Physical Research Laboratory, Navarangpura, 380009 Gujarat, India 2

Department of Electrophysics, National Chiao-Tung University, 1001 University Road, Hsinchu, Taiwan 300, Republic of China

共Received 13 August 2009; published 4 December 2009兲

We have carried out a detailed and systematic study of the correlation energies of inert gas atoms Ne, Ar, Kr, and Xe using relativistic many-body perturbation theory and relativistic coupled-cluster theory. In the relativ-istic coupled-cluster calculations, we implement perturbative triples and include these in the correlation energy calculations. We then calculate the dipole polarizability of the ground states using perturbed coupled-cluster theory.

DOI:10.1103/PhysRevA.80.062505 PACS number共s兲: 31.15.bw, 31.15.ve, 31.15.ap, 31.15.am

I. INTRODUCTION

High precision atomic experiments are at the core of sev-eral investigations into fundamental physics and high end technology developments. Selected examples are search for electric dipole moment共EDM兲 关1兴 and observation of parity

nonconservation 关2兴. These endeavors, in general, require

precision atomic theory calculations to analyze the results and understand systematics. The challenging part of preci-sion atomic structure and properties calculations is obtaining accurate wave functions. In the case of high Z atoms, the need to incorporate relativity adds to the difficulty. A system-atic study of the correlation energy is one of the possible methods to test the accuracy of the atomic wave function. In this paper, we report the results of correlation energy calcu-lations of inert gas atoms Ne, Ar, Kr, and Xe. For this we employ many-body perturbation theory 共MBPT兲 and calcu-late the second-order correlation energy. A comparative study reveals the changing nature of electron correlations in the group. Our interest in particular is Xe, which is a candidate for EDM experiments关3兴 and theoretical calculations 关4兴.

For completeness, in the presentation of the paper, we give an overview of MBPT. It is a powerful theory and forms the basis of other more sophisticated and elaborate many-body methods. However, one drawback of MBPT is the com-plexity of expressions at higher orders. This renders the theory inappropriate to incorporate strong correlation effects in heavy atoms. Yet, at lower orders its simplicity makes it an ideal choice to test and optimize basis sets. We use this insight to generate basis sets for coupled-cluster calculations. The coupled-cluster theory, first developed in nuclear many body physics 关5,6兴, is considered the most accurate

many body theory. In recent times, it has been used with great success in nuclear 关7兴, atomic 关8,9兴, molecular 关10兴,

and condensed matter 关11兴 calculations. It is equivalent to

incorporating electron correlation effects to all orders in per-turbation. The theory has been used in performing high pre-cision calculations to study the atomic structure and proper-ties. These include atomic electric dipole moments 关8,12兴,

parity nonconservation 关13兴, hyperfine structure constants

关9,14兴, and electromagnetic transition properties 关15,16兴. In

the present work we use the relativistic coupled-cluster singles and doubles共CCSD兲 approximation to calculate

cor-relation energy and dipole polarizability of inert gas atoms Ne, Ar, Kr, and Xe. In the dipole polarizability calculations, the dipole interaction Hamiltonian is introduced as a pertur-bation. A modified theory, recently developed关17兴,

incorpo-rates the perturbation within the coupled-cluster theory. This theory has the advantage of subsuming correlation effects more accurately. The results provide a stringent test on the quality of the wave functions as the dipole polarizability of inert gas atoms are known to high accuracy关18兴. Based on

the CCSD method, we also estimate the third-order correla-tion energy. Further more, perturbative triples are incorpo-rated in the coupled-cluster calculations.

In the paper we give a brief description of MBPT in Sec.

II and discuss the method to calculate electron correlation energy to the second and third order in residual Coulomb interaction. The coupled-cluster theory is described in Sec.

III, where we also discuss linearized coupled-cluster theory and correlation energy calculation using coupled-cluster theory. Then the inclusion of approximate triples to the cor-relation energy is explained and illustrated. Section IVis a condensed description of the perturbed coupled-cluster theory and provide details of how to incorporate the effects of an additional perturbation to the residual Coulomb inter-action in atomic systems. Results are presented and dis-cussed in Sec.V. In the paper, all the calculations and math-ematical expressions are in atomic units 共e=ប=me= 1兲.

II. CORRELATION ENERGY FROM MBPT

In this section, to illustrate the stages of our calculations and compare with coupled-cluster theory, we provide a brief description of many-body perturbation theory. Detailed and complete exposition of the method, in the context of atomic many-body theory, can be found in Ref.关19兴.

The Dirac-Coulomb Hamiltonian HDC is an appropriate choice to incorporate relativistic effects in atoms. This is particularly true for heavy atoms, where the relativistic ef-fects are large for the inner core electrons due to the high nuclear charge. For an N electron atom

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HDC=

i=1 N 关ci· pi+共␤− 1兲c2− VN共ri兲兴 +

i⬍j 1 rij , 共1兲

where␣iand ␤are the Dirac matrices. For the nuclear

po-tential VN共r兲, we consider the finite size Fermi density

distri-bution

␳nuc共r兲 =

␳0

1 + e共r−c兲/a, 共2兲

here, a = t4 ln 3. The parameter c is the half-charge radius, that is ␳nuc共c兲=0/2 and t is the skin thickness. The eigen states of HDC are 兩⌿

i典, the correlated many-particle states

with eigenvalues Ei. The eigenvalue equation is HDC兩⌿

i典 = Ei兩⌿i典. 共3兲

It is however impossible to solve this equation exactly due to the relative coordinates in the electron-electron Coulomb in-teraction. MBPT is one approach which, starting from a mean-field approximation, incorporates the electron correla-tion effects systematically.

The starting point of perturbative scheme in MBPT is to split the Hamiltonian as

HDC= H0+ V, 共4兲

where H0=兺i关ci· pi+共␤i− 1兲c2− VN共ri兲+u共ri兲兴, is the

unper-turbed or zeroth order Hamiltonian. It is the exactly solvable part of the total Hamiltonian and correspond to independent particle model. The average field of the other electrons is the Dirac-Fock central potential u共ri兲. The remaining part of the

electron-electron Coulomb interaction V =i⬍j N 1

rij−兺iu共ri兲, is

the residual Coulomb interaction. The purpose of any atomic many-body theory is to account for this part as accurately as possible. The Hamiltonian H0 satisfies the eigenvalue

equa-tion

H0兩⌽i典 = Ei 0兩⌽

i典, 共5兲

where兩⌽i典 is a many-particle state and Ei 0

is the eigenvalue. The eigenstates are generally Slater determinants, antisym-metrized direct product of single particle states and Ei0is the

sum of the single particle energies. The difference between the exact and mean field energy,⌬Ei= Ei− Ei

0

, is the correla-tion energy of the ith state. At the single particle level, the

relativistic spin orbitals are of the form

n␬m共r兲 = 1 r

Pn共r兲␬m共r/r兲 IQn共r兲␹−␬m共r/r兲

, 共6兲

where Pn共r兲 and Qn共r兲 are the large and small component

radial wave functions,␬ is the relativistic total angular mo-mentum quantum number and ␹␬m共r/r兲 are the spin or spherical harmonics. One representation of the radial compo-nents is to define these as linear combination of Gaussian-like functions and are referred to as Gaussian type orbitals 共GTOs兲. Then, the large and small components 关20,21兴 are

Pn共r兲 =

p C␬pL g␬pL 共r兲, Qn共r兲 =

p C␬pS g␬pS 共r兲. 共7兲

The index p varies over the number of the basis functions. For large component we choose

g␬pL共r兲 = C␬iLrne−␣pr2, 共8兲

here n is an integer. Similarly, the small component are derived from the large components using kinetic balance condition. The exponents in the above expression follow the general relation

p=␣0␤p−1. 共9兲

The parameters ␣0 and␤are optimized for an atom to

pro-vide good description of the atomic properties. In our case the optimization is to reproduce the numerical result of the total and orbital energies. Besides GTO, B splines is another class of basis functions widely used in relativistic atomic many-body calculations关22兴. A description of B splines with

details of implementation and examples are given in Ref. 关23兴. The other important types of basis used in atomic

cal-culations are finite discrete spectrum 关24兴, Slater type

orbit-als关25兴 and r multiplied virtuals 关26兴.

The next step in perturbative calculations is to divide the entire Hilbert space of H0 into two manifolds: model and

complementary spaces P and Q, respectively. The model space has, in single reference calculation, the eigenstate兩⌽i

of H0which is a good approximation of the exact eigenstate

兩⌿i典 to be calculated. The other eigenstates constitute the

complementary space. The corresponding projection opera-tors are

P =兩⌽i典具⌽i兩, Q =

兩⌽j典苸P

兩⌽j典具⌽j兩, 共10兲

and P + Q = 1. In the present paper, we restrict to calculating the ground state 兩⌿0典 of the closed-shell inert gas atoms.

From here on, for a consistent description, the model space consist of兩⌽0典.

The most crucial part of perturbation theory is to define a wave operator⍀ which operates on 兩⌽0典 and transform it to

兩⌿0典 as

兩⌿0典 = ⍀兩⌽0典. 共11兲

Then, with the intermediate normalization approximation 具⌿0兩⌽0典=1, the wave operator is evaluated in orders of the

perturbation as ⍀=兺i=0⬁ ⍀共i兲 with ⍀共0兲= 1. It is possible to evaluate⍀共i兲 iteratively or recursively from the Bloch equa-tion

关⍀,H0兴P = QV⍀P −PV⍀P, 共12兲

where␹=兺i=1⬁ ⍀共i兲is the correlation operator. For simplifica-tion, in the normal form the perturbation is separated as关19兴 V = V0+ V1+ V2. These are zero-, one- and two-body

opera-tors. From these definitions, the first-order wave operator can be separated as ⍀共1兲= 1 共1兲+ 2 共1兲. 共13兲

Here, ⍀1共1兲 and ⍀2共1兲 are one- and two-body components of the first-order wave operator. We obtain singly 共doubly兲

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ex-cited states兩⌽ap典共兩⌽abpq典兲 when ⍀1共1兲共⍀2共1兲兲 operates on the ref-erence state 兩⌽0典. The complexity of the expressions

in-creases with order of perturbation and is hard to manage. One powerful tool in many-body perturbation theory is the diagrammatic evaluation of the perturbation expansion. The diagrammatic representation of the⍀1共1兲and⍀2共1兲are shown in Fig.1. Then, the tedious algebraic evaluations are reduced to a sequence of diagrams and equivalent algebraic expres-sions are derived with simple rules. Even with this approach, it is computationally not practical to go beyond fourth order.

Second- and third-order correlation energy

The ground-state correlation energy⌬E0, in MBPT, is the

sum total of the energy corrections from all orders in pertur-bation. At the nth order, the energy correction E

corr 共n兲

=具⌽0兩V⍀共n−1兲兩⌽0典 and ⌬E0=兺nEcorr共n兲. Then the second-order

correlation energy is

Ecorr共2兲 =具⌽0兩共V1+ V2兲共⍀1共1兲+⍀2共1兲兲兩⌽0典. 共14兲

When Dirac-Fock orbitals are used, the diagonal matrix ele-ments of V1are the orbital energies and off diagonal matrix

elements are zero. For this reason, it does not contribute to the second-order energy. Then, the second-order correlation energy is

Ecorr共2兲 =具⌽0兩V2⍀2共1兲兩⌽0典. 共15兲

The singles are nonzero starting from the second order when Dirac-Fock orbitals are used. And, the triples and qua-druples also begin to contribute from this order. The triples consist of connected diagrams, whereas all the quadruples are disconnected. The third-order correlation energy is

Ecorr共3兲 =具⌽0兩共V1+ V2兲共⍀1共2兲+⍀2共2兲+⍀3共2兲+⍀4共2兲兲兩⌽0典.

共16兲 The triple and quadruple excitations do not contribute as V at the most can contract with double excitations. For the same reason mentioned earlier, in second order, V1 also does not contribute. Then the third-order correlation energy is simpli-fied to

Ecorr共3兲 =具⌽0兩共V2⍀2共2兲兲兩⌽0典. 共17兲

This is similar in form to the second-order correlation en-ergy. In general, the nthorder correlation energy has nonzero

contribution from the term V22共n−1兲 only. It must be men-tioned that the connected triples begin to contribute from the fourth-order energy. This is utilized in perturbative inclusion of triples, in later sections of the paper, while discussing coupled-cluster calculations.

III. COUPLED-CLUSTER THEORY

The coupled-cluster theory is a nonperturbative many-body theory and considered as one of the best. A recent re-view关27兴 provides an excellent overview of recent

develop-ments and different variations. In the context of

diagrammatic analysis of MBPT, coupled-cluster theory is equivalent to a selective evaluation of the connected dia-grams to all orders. In coupled-cluster theory, for a closed-shell atom, the exact ground state is

兩⌿0典 = eT

共0兲

兩⌽0典, 共18兲

where T共0兲is the cluster operator. The superscript is a tag to identify cluster operators arising from different perturba-tions. For the case of N electron atoms, the cluster operator is

T共0兲=

i=1 N

Ti共0兲. 共19兲

In closed-shell atoms, the single and doubles provide a good approximation of the exact ground state. Then, the cluster operator T共0兲= T1共0兲+ T2共0兲 and is referred to as the coupled-cluster single and doubles 共Fig.2兲. The cluster operators in

the second quantized notations are

T1共0兲=

a,p ta p apaa, 共20兲 T2共0兲= 1 2!a,b,p,q

tab pq apaqabaa. 共21兲 Here, ta p and tab pq

are the single and double cluster amplitudes, respectively, and ab共pq兲 denote core共virtual兲 orbitals. Sub-tracting具⌽0兩H兩⌽0典 from both sides of Eq. 共3兲 and using the

normal form of an operator, ON= O −具⌽0兩O兩⌽0典, we get

HN兩⌿0典 = ⌬E兩⌿0典, 共22兲

where ⌬E=E−具⌽0兩H兩⌽0典, as defined earlier, is the correla-tion energy. Operating with e−T共0兲 and projecting the above

equation on excited states we get the cluster amplitude equa-tions 具⌽a p兩H¯ N兩⌽0典 = 0, 共23兲 具⌽ab pq兩H¯ N兩⌽0典 = 0, 共24兲 where H¯N= e−T 共0兲 HNeT 共0兲

is the similarity transformed or dressed Hamiltonian. Following Wick’s theorem and struc-ture of HN, in general

Ω1 Ω2

FIG. 1. The diagrammatic representations of the one- and two-body wave operator. Lines with downward共upward兲 arrows repre-sent core共virtual兲 single particle states.

T1(0) T2(0)

FIG. 2. Diagrammatic representation of unperturbed single and double cluster operators.

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共25兲 Here

denote contraction between two operators A and B. The single and double cluster amplitudes are solutions of Eqs. 共23兲 and 共24兲, respectively. These are set of coupled

nonlin-ear equations and iterative methods are the ideal choice to solve these equations.

A. Linearized coupled-cluster

The nonlinearity in the cluster amplitude equation arises from the two and higher contractions in the dressed Hamil-tonian. An approximation often used as a starting point of coupled-cluster calculations is to retain only the first two terms in H¯N, then

共26兲 In the CCSD approximation T共0兲= T1共0兲+ T2共0兲, the cluster equa-tions are

共27兲 These are the linearized coupled-cluster equations of single and double cluster amplitudes. This can be combined as the matrix equation

H11 H12 H21 H22

冊冉

t1 t2

= −

H10 H20

, 共28兲 where H11=具⌽a p兩H N兩⌽b s典, H 12=具⌽a p兩H N兩⌽bc

st典 and so on. The

equations are set of coupled linear equations and solved us-ing standard or specialized linear algebra solvers. In the lit-erature several authors refer to linearized coupled-cluster as all-order method. A description of the all-order method and applications are given in Ref. 关28兴. In a recent work, the

authors report the combination of all-order method and con-figuration interaction关29兴.

B. Correlation energy and approximate triples

From Eq. 共22兲 the ground-state correlation energy, in

coupled-cluster theory

⌬E = 具⌽0兩H¯N兩⌽0典. 共29兲

The diagrams arising from the above expression are shown in the Fig. 3. The dominant contributions are from the dia-grams 共a兲 and 共b兲, which is natural as the doubles cluster amplitudes are larger in value than the singles. Diagram 共e兲 does not contribute to the correlation energy when Dirac-Fock orbitals are used.

To go beyond the CCSD approximation, we incorporate selected correlation energy diagrams arising from approxi-mate triples. The approxiapproxi-mate triples are perturbative con-traction of V2 with the T共0兲 cluster amplitudes 关30,31兴.

Ex-ample diagrams of the approximate triples and correlation energies are shown in Fig. 4. There are two categories of triples, first is V2contracted with T共0兲through a hole line, and

second contraction through a particle line关Fig.4共a兲兴. To cal-culate the correlation energy from the triples, these are con-tracted perturbatively with V2and reduced to a double

exci-tation diagram. Then the correlation energy is obtained after another contraction with V2. These two contractions generate

several diagrams. The triples correlation energy diagrams are separated into three categories based on the number of inter-nal lines. These are: two particle and two hole interinter-nal lines 共2p-2h兲, three particle one hole internal lines 共3p-1h兲, and one particle three hole internal lines共1p-3h兲. In the present calculations eight diagrams from the first category and two each from other remaining two categories are considered.

IV. PERTURBED COUPLED-CLUSTER

The atomic properties of interest are, in general, associ-ated with additional interactions. The interaction are either internal such as hyperfine interaction or external such as static electric field. These are treated as perturbations which modify the wave function and energy of the atom. This sec-tion briefly describes a method to incorporate an addisec-tional perturbation within the frame work of relativistic cluster. The scheme is referred to as perturbed coupled-cluster theory. It has been tried and tested in precision atomic properties and structure calculations. In the presence of a perturbation H1, the eigen value equation is

共HDC+␭H

1兲兩⌿˜0典 = E˜兩⌿˜0典. 共30兲

Here 兩⌿˜0典 is the perturbed wave function, E˜ is the corre-sponding eigenvalue and␭ is the perturbation parameter. The perturbed wave function is the sum of the unperturbed wave function and a correction兩⌿¯01典 arising from H1. That is

兩⌿˜

0典 = 兩⌿0典 + ␭兩⌿¯0

1典. 共31兲

(a) (b) (c) (d) (e)

FIG. 3. Coupled-cluster correlation energy diagrams. The dia-gram共e兲 is equal to zero when Dirac-Fock orbitals are used.

(a) (b)

FIG. 4. Diagrams of approximate triples calculated perturba-tively: 共a兲 approximate triples cluster operator and 共b兲 correlation energy arising from approximate triples.

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Following the earlier description of coupled-cluster wave function, the perturbed wave function is

兩⌿˜ 0典 = eT

共0兲+␭T共1兲

兩⌽0典. 共32兲

The cluster operators T共0兲, as defined earlier, incorporate the effects of residual Coulomb interaction. For clarity these are referred as unperturbed cluster operator. The T共1兲cluster op-erators arise from H1 and are referred to as the perturbed cluster operators共Fig.5兲. It acts on the reference state 兩⌽0典

to generate the correction. Consider the perturbation expan-sion to first order in␭, we get

兩⌿˜ 0典 = eT

共0兲

共1 + ␭T共1兲兲兩⌽

0典. 共33兲

To derive the cluster equations use this in Eq. 共30兲, then

operate with e−T共0兲and project on excited states. We get the equations for singles and doubles perturbed cluster ampli-tudes

共34兲 共35兲 The dressed Hamiltonian H¯Nis same as in Eq.共25兲. Like in

linearized coupled-cluster, these form a set of linear alge-braic equations.

A. Approximate triples

Like in T共0兲, a perturbed triple cluster Fig. 6共a兲 is a per-turbative contraction between V2and T2共1兲. As in the case of

unperturbed approximate triples discussed earlier, there are two types of diagrams in the present case as well. One arises from particle line contraction and the other from hole line contraction between V2 and T共1兲diagrams. In this work we implement approximate triples while calculating properties.

In particular, to calculate dipole polarizability and an ex-ample diagram is shown in Fig. 6共b兲. The algebraic expres-sion of this diagram is

a,b,c,p,q,r,s 具ab兩T2共0兲†兩pq典具c兩d兩s典具qs兩V2兩rc典具pr兩T2共1兲兩ab典a+⑀b+⑀c−⑀p−⑀q−⑀s , 共36兲 here, d is the dipole operator. In total there are twentyfour properties diagrams arising from the perturbative triples and we include all of these diagrams in the calculations.

B. Dipole polarizability

When an atom is placed in an external electric fieldE, the charge distribution of electron cloud is distorted and an elec-tric dipole moment Dindis induced. The dipole polarizability

of the atom␣is then the ratio of the induced dipole moment to the applied electric field, that is

Dind=␣E. 共37兲

By definition, the dipole polarizability of the ground state is

␣= − 2

I

兩具⌿0兩D兩⌿I典兩2 E0− EI

, 共38兲

where兩⌿I典 are intermediate atomic states. These are opposite

in parity to the ground state兩⌿0典. The expression of␣can be

rewritten as

␣= − 2具⌿0兩D兩⌿¯0

1典. 共39兲

Here兩⌿¯01典=兺I共兩⌿I典具⌿I兩D兩⌿0典兲/共E0− EI兲, which follows from

the first-order time independent perturbation theory. The per-turbation Hamiltonian is H1= −D ·E and external field E is the

perturbation parameter. One short coming of calculating ␣ from Eq.共38兲 is, for practical reasons, the summation over I

is limited to the most dominant intermediate states. However, the summation is avoided altogether when the perturbed coupled-cluster wave functions are used in the calculations. From Eq.共32兲 the perturbed wave function

兩⌿¯ 0 1典 = eT共0兲

T共1兲兩⌽0典. 共40兲

In a more compact form, the dipole polarizability in terms of the perturbed coupled-cluster wave function is

␣=具⌿˜0兩D兩⌿˜0典. 共41兲

After simplification, using the perturbed wave function in Eq. 共32兲, we get

␣=具⌿¯01兩D兩⌿0典 + 具⌿0兩D兩⌿¯01典. 共42兲

The correction兩⌿¯01典, as described earlier, is opposite in parity to 兩⌽0典. Hence the matrix elements 具⌿0兩D兩⌿0典 and

具⌿¯ 0 1兩D兩⌿¯

0

1典 are zero. As D is Hermitian, the two terms on the

left-hand side are identical and the above expression is same as Eq.共39兲. Considering the leading terms

T1(1) T2(1)

FIG. 5. Diagrams of single and double perturbed cluster operators. a p b r q c s (a) a p b q r c s (b)

FIG. 6. Diagrams of approximate triples calculated perturba-tively:共a兲 representation of approximate perturbed triples. 共b兲 Con-tribution of approximate perturbed triples to the dipole polarizability.

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␣=具⌽0兩T共1兲†共0兲+ D¯共0兲T共1兲兩⌽0典. 共43兲

Here, the operator D¯共0兲= eT共0兲†DeT共0兲 is the unitary

trans-formed electric dipole operator. It is explicitly evident that the dipole polarizability, in terms of perturbed cluster opera-tor, does not have a sum over states. In this scheme, contri-butions from all intermediate states within the chosen con-figuration space are included. For precision calculations, this is a very important advantage.

V. RESULTS

In order to get reliable results, in atomic structure and properties calculations, one prerequisite is good quality or-bital basis set. In all calculations described in the paper, we employ GTOs as orbital functions. In particular, we use even tempered basis in which the parameters␣0and␤, in Eq.共9兲,

are different for each symmetries. We use the basis param-eters of Tatewaki and Watanabe 关32兴 as starting values and

optimized further to obtain EDC共0兲 共ground state Dirac-Fock energy兲 and⑀a 共single particle energies of core orbitals兲 in

agreement with the numerical results. The values of␣and␤ chosen for the occupied symmetries are given in TableI. For the symmetries which occur only as virtual orbitals we chose the optimal values of 0.0070 and 2.6950 for␣and␤, respec-tively. The numerical results are obtained from the GRASP92

关33兴 code. In order to obtain converged Ecorr共2兲, we consider orbital basis set consisting of all the core orbitals and virtual orbitals up to 10 000– 11 000 in single particle energies.

The working equations of coupled-cluster theory are coupled nonlinear equations. Solving these equations is a computational challenge. The number of unknowns, cluster amplitudes, are in the order of millions. In addition, imple-menting fast and efficient algorithms demand huge memory to tabulate and store two-electron integrals. This is essential as the two-electron integrals are needed repeatedly and are compute intensive. For the larger basis sets in the present work, the number of two-electron integrals exceed 2⫻108.

In order to utilize memory efficiently, we have developed a scheme which parallelize the tabulation and storage of two-electron integrals.

The unperturbed and perturbed cluster amplitude equa-tions are solved iteratively using Jacobi method. We chose the method as it is relatively simple to parallelize. However, one drawback of the method is slow convergence. To obtain faster convergence, we employ direct inversion in iterated subspace共DIIS兲 关34兴 convergence acceleration.

A. Second-order correlation energy

The SCF energy EDC共0兲, second-order correlation energy

Ecorr共2兲 and the total energy E from our calculations are listed in TableII. For comparison the results of previous calculations are also listed. It is evident that our second-order correlation energy and total energy, sum of the SCF and second-order correlation energy, are in agreement with the results of Ish-ikawa et al.关35兴 for all the atoms studied. The results in the

table other than 关35兴 are from nonrelativistic calculations.

For all the atoms, our SCF energy EDC0 are lower and there are no discernible trends, as a function of nuclear charge, in the difference. Interestingly, except for Xe, our second-order correlation energies are higher. This compensates the lower

EDC共0兲 and subsequently, the total energies E of the two calcu-lations are in excellent agreement.

The TableIIIlists the cumulative contributions from vari-ous symmetries to Ecorr共2兲. Among the previous works, Lindgren and collaborators关36兴 provide cumulative Ecorr共2兲 for Ne up to i symmetry. Their converged result, with orbitals up to i symmetry, is −0.3836. However, in our calculation, we get converged results of −0.3830 after including j and k sym-metry orbitals. The contribution from k symsym-metry to Ecorr共2兲 for Ar, Kr, and Xe are −0.0017, −0.0083, and −0.0176 respec-tively. These are larger than that of Ne, which is −0.0007. However, these correspond to 0.24%, 0.45% and 0.59% for Ar, Kr and Xe, respectively, these compare very well with that of Ne 0.20%. For Ar there is a variation in the previous values of Ecorr共2兲, these range from the lowest value of Clem-enti关37兴 −0.790 to that of Ishikawa 关35兴 −0.6981. Our value

of −0.6938 is closer to that of Ishikawa.

There is a pattern in the change of the correlation energy with symmetry wise augmentation, without changing ␣ and

␤, of the virtual orbital set. There is an initial increase, reaches a maximum and then decreases. The maximum change occurs with the addition of p, d, d, and f symmetry for Ne, Ar, Kr, and Xe, respectively. The pattern is evident in Fig. 7, which plots the change in Ecorr共2兲 with symmetry wise augmentation of the virtual space. The pattern arise from the distribution of the contributions from each of the core orbit-als. Extrapolating the results to l→⬁, Ecorr共2兲 are −0.3838, −0.6966, −1.8549 and −2.9969 for Ne, Ar, Kr, and Xe, re-spectively. Depending on the core orbital combination ab, there are two types of correlation effects. These are inter- and intracore shell correlations corresponding to a = b and a⫽b,

respectively. Among the various combinations, the

2p3/22p3/2, 3p3/23p3/2, 3d5/23d5/2, and 4d5/24d5/2 core orbital

pairings have leading contributions in Ne, Ar, Kr, and Xe, respectively. Here for Ne and Ar the leading pairs correspond TABLE I. Values of the parameters,␣ and ␤, used in the calculations, and p is the number of basis functions.

Symmetry Ne Ar Kr Xe ␣ ␤ p ␣ ␤ p ␣ ␤ p ␣ ␤ p s 0.0925 1.4500 38 0.0985 1.8900 38 0.0002 2.0220 30 0.0001 2.0220 32 p 0.1951 2.7103 35 0.0072 2.9650 35 0.0072 2.3650 28 0.0072 2.3650 28 d 0.0070 2.7000 25 0.0070 2.7000 28 0.0070 2.5500 25 0.0070 2.5500 25

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to the valence shell but it is the last d shell for Kr and Xe. This correlates with the pattern observed in the symmetry wise augmentation.

B. Third-order correlation energy

We calculate the third-order correlation energy Ecorr共3兲 from the linearized CCSD equations. This is possible when the first-order MBPT wave operator⍀共1兲is chosen as the initial guess and iterate the linearized coupled-cluster equations once. The two-body wave operator so obtained is ⍀2共2兲 and from Eq. 共17兲 Ecorr共3兲=具⌽0兩V2⍀2共2兲兩⌽0典. This approach,

how-ever, is not applicable beyond third order. The reason is, starting from the fourth order correlation energy the triples contribute to ⌬E0 and triples are not part of the linearized

CCSD equations. The results of Ecorr共3兲, obtained from our cal-culations, are listed in Table IV. For comparison, results

from previous works are also listed. For Ne, Jankowski and Malinowski 关42兴 reported a value of 0.0024. Their

calcula-tions were done with a limited basis set and hence, could leave out less significant contributions. The results of Lindgren and collaborators 关36兴 0.0035 is perhaps more

ac-curate and reliable on account of larger basis set. In our calculations, we include virtual orbitals up to i symmetry, then extrapolate up to k symmetry based on Ecorr共2兲 results. We obtain 0.0019, which is in better agreement with the result of Jankowski and Malinowski关42兴. As expected, Ecorr共3兲 increases with Z and to our knowledge, our results of Ar, Kr, and Xe are the first reported calculations in literature. Interestingly,

Ecorr共3兲 is positive for Ne, Kr, and Xe but it is negative for Ar.

C. Coupled-cluster correlation energy

The MBPT correlation energies Ecorr共i兲 converges with rela-tively large basis set. For example, the Ecorr共2兲 of Ne converge TABLE II. The SCF EDC共0兲, the second-order correlation Ecorr共2兲 and the total energies E of Ne, Ar, Kr and Xe.

All the values listed are in atomic units共hartrees兲.

Atom Z Atomic mass

This work Other work

EDC共0兲 Ecorr共2兲 E EDC共0兲 Ecorr共2兲 E Ne 10 20.18 −128.6932 −0.3830 −129.0762 −128.6919 −0.3834a −129.0753 −0.3836b −0.3822c −0.3697d −0.3804e Ar 18 39.95 −528.6882 −0.6938 −529.3820 −528.6838 −0.6981a −529.3819 −0.6822e −0.685 f −0.790 g Kr 36 83.80 −2788.8659 −1.8426 −2790.7085 −2788.8615 −1.8468a −2790.7083 Xe 54 131.29 −7446.8887 −2.9767 −7449.8654 −7446.8880 −2.9587a −7449.8467 a Reference关35兴. b Reference关36兴. c Reference关38兴. d Reference关39兴. e Reference关40兴. f Reference关41兴. g Reference关37兴.

TABLE III. Cumulative second-order correlation energy when orbitals up to a particular symmetry are included in the virtual space. All the values are in atomic units.

Symmetry Ne Ar Kr Xe s −0.0194 −0.0210 −0.0236 −0.0247 p −0.1920 −0.2043 −0.2479 −0.2687 d −0.3216 −0.5401 −0.9512 −1.0419 f −0.3589 −0.6330 −1.5213 −2.2972 g −0.3732 −0.6695 −1.7077 −2.6879 h −0.3786 −0.6830 −1.7843 −2.8520 i −0.3811 −0.6891 −1.8179 −2.9238 j −0.3823 −0.6921 −1.8343 −2.9591 k −0.3830 −0.6938 −1.8426 −2.9767 Orbital symmetries 0.0 0.5 1.0 1.5 Xe Xe Kr Kr Ar Ar Ne Ne s p d f g h i j k

FIG. 7. 共Color online兲 The second-order energy 共in Hartrees兲 when orbitals up to a particular symmetry are included in the virtual space.

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when virtual orbitals up to k symmetry are included in the calculations. This correspond to a total of 224 virtual orbit-als. Similar or larger number of virtual orbitals are required to obtain converged Ecorr共2兲 of Ar, Kr, and Xe as well. However, it is not practical to have such large basis sets in relativistic coupled-cluster calculations. The nv4nc3, where nv and ncare

the number of the virtual and core orbitals, respectively, scal-ing of arithmetic operations in CCSD render computations with large nvbeyond the scope of detailed studies. Hence, in the CCSD calculations, the size of the virtual orbital set is reduced to a manageable level and restrict up to the h sym-metry. To choose the optimal set, after considering the most dominant ones, the virtual orbitals are augmented in layers. Where one layer consists of one virtual orbital each from all the symmetries considered.

The CCSD correlation energies for two basis sets are listed in Table V. The first is with a basis set considered optimal and manageable size for CCSD calculations after a series of calculations. Then the next is with an additional layer of virtual orbitals. The change in the linearized CCSD

⌬E0 with the additional layer of virtual orbitals are 0.2%,

1.7%, 6.0% and 5.0% for Ne, Ar, Kr, and Xe, respectively. Changes of similar order are observed in the corresponding ⌬E0 of the nonlinear CCSD calculations. It must be

men-tioned that, though the difference in⌬E0 is small, the

com-putational cost of nonlinear CCSD is much higher than the linearized CCSD calculations. The percentage changes indi-cate the basis size of Kr and Xe are not large enough. The orbital basis of Xe, with the additional layer, consists of 17 core and 129 virtual orbitals. This translates to ⬃5.0⫻106

cluster amplitudes, which follows from the nv2nc2 scaling of

the number of cluster amplitudes. At this stage, the compu-tational efforts and costs far out weight the gain in accuracy. To account for the correlation energy from the other virtual orbitals, not included in the CCSD calculations, we resort to the second-order correlation energy. For this we calculate

Ecorr共2兲 with the basis set chosen in CCSD calculations and subtract from the converged Ecorr共2兲. The estimated⌬E in Table

Vis the sum of this difference and CCSD⌬E. This includes the correlation effects from i, j, and k symmetries as well. For Ne, the estimated experimental value of correlation en-ergy lies between the range 0.385 and 0.390 关39,43兴. Our

coupled-cluster result, estimated value, is in excellent agree-ment.

The contributions to the correlation energy arising from the approximate triples are listed in TableVI. As discussed in Sec. III B, the correlation energy diagrams corresponding to the approximate triples are grouped into three classes. Out of these we have selected a few: eight from 2p-2h and two each from 3p-1h and 1p-3h. In Table VI,⌬E0arising from these are listed. It is evident from the table, the contribution from 1p-3h and 3p-1h are negative and adds to the magnitude of ⌬E0. Whereas, the contribution from 2p-2h is positive and

reduces the magnitude of⌬E0.

D. Dipole polarizability

One constraint while using perturbed coupled-cluster theory to calculate dipole polarizability is the form of D¯ . It is a unitary transformation of the dipole operator and expands to a nonterminating series. For the present calculations we consider the leading terms in T共1兲†D¯ . That is, we use the

approximation

T共1兲†D¯ ⬇ T1共1兲†关D + DT1共0兲+ DT共0兲2 兴 + T2共1兲†关DT2共0兲+ DT1共0兲兴. 共44兲 The ground state dipole polarizabilities of Ne, Ar, Kr, and Xe calculated with this approximation are given in Table VII. TABLE IV. Third-order correlation energy in atomic units.

Atom

E3

This work Other work

Ne 0.0019 0.0035a 0.0024b Ar −0.0127 Kr 0.0789 Xe 0.1526 aReference36兴. b Reference关42兴.

TABLE V. Correlation energy from coupled-cluster. All the val-ues are in atomic units.

Atom Active Orbitals

⌬E 共CCSD兲 Linear Nonlinear Ne 17s10p10d9f9g8h −0.3783 −0.3760 18s11p11d10f10g9h −0.3805 −0.3782 Estimated −0.3905 −0.3882 Ar 17s11p11d9f9g9h −0.6884 −0.6829 18s12p12d10f10g10h −0.7001 −0.6945 Estimated −0.7258 −0.7202 Kr 22s13p11d9f9g9h −1.5700 −1.5688 23s14p12d10f10g10h −1.6730 −1.6716 Estimated −1.8480 −1.8466 Xe 23s14p12d10f10g10h −2.5500 −2.5509 24s15p13d11f11g11h −2.6874 −2.6881 Estimated −2.9973 −2.9979

TABLE VI. Correlation energy arising from the approximate triples in the coupled-cluster theory. All the values are in atomic units.

Atom Basis size

⌬E 2p-2h 1p-3h 3p-1h Ne 18s11p11d10f10g9h 0.00672 −0.00145 −0.00164 Ar 18s12p12d10f10g10h 0.00805 −0.00066 −0.00192 Kr 22s13p11d9f9g9h 0.01546 −0.00171 −0.00305 Xe 19s15p10d9f5g2h 0.02011 −0.00148 −0.00260

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Among the various terms, the first term T共1兲†D subsumes

contributions arising from Dirac-Fock and random phase ap-proximation. We can thus expect this term to have the most dominant contribution. This is evident in Table VII, which shows that the contribution from T共1兲†D is far larger than the

others. The next leading term is T1共1兲†DT2共0兲. This is attributed to the larger values, compared to T1共0兲, of T2共0兲cluster ampli-tudes. The dipole polarizability calculations with relativistic coupled-cluster theory involve two sets of cluster ampli-tudes. These are the T共0兲and T共1兲cluster amplitudes. As men-tioned earlier, solving coupled-cluster equations is compute intensive.

One pattern discernible in the results is the better agree-ment between the T共1兲†D results and experimental data. The

deviations from the experimental data are large when we consider the total共CCSD兲 result. For Ne, the deviation from experimental data is 2%, where as it is 9% in the case of Xe atom. We attribute this to the approximation in Eq.共44兲 and

partly to the basis set. To confirm this, however, requires

detailed computations with higher order terms in the unitary transformation. This poses considerable computational chal-lenges and shall be addressed in future publications. We also implement the approximate triples excitation of the perturbed cluster amplitudes and contribution 共arising from the twen-tyfour diagrams兲 to␣ are listed in the table.

VI. CONCLUSION

We have done a systematic study of the electron correla-tion energy of neutral inert gas atoms using relativistic MBPT and coupled-cluster theory. Our MBPT results are based on larger basis sets consisting of higher symmetries than the previous works, and hence more reliable and accu-rate. Our study shows that in heavier atoms Kr and Xe, the inner core electrons in d shells dominates the electron corre-lation effects. This ought to be considered in high precision properties calculations. For example, the EDM calculations of Xe arising from nuclear Schiff moment. The dipole polar-izability calculated with the perturbed coupled-cluster show systematic deviation from the experimental data. However, the contribution from the leading term T1共1兲†D is in good

agreement with the experimental data. The deviations might decrease when higher order terms are incorporated in the properties calculations. From these results, it is evident that the basis set chosen is of good quality and appropriate for precision calculations.

ACKNOWLEDGMENTS

We wish to thank S. Chattopadhyay, S. Gautam, B. Sahoo, and S. A. Silotri for useful discussions. The results presented in the paper are based on computations using the HPC clus-ter of the Cenclus-ter for computational Maclus-terial Science, JN-CASR, Bangalore.

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TABLE VII. Dipole polarizability of the ground state of neutral rare-gas atoms共in a.u.兲.

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數據

FIG. 1. The diagrammatic representations of the one- and two- two-body wave operator. Lines with downward 共upward兲 arrows  repre-sent core 共virtual兲 single particle states.
FIG. 3. Coupled-cluster correlation energy diagrams. The dia- dia-gram 共e兲 is equal to zero when Dirac-Fock orbitals are used.
FIG. 6. Diagrams of approximate triples calculated perturba- perturba-tively: 共a兲 representation of approximate perturbed triples
table other than 关 35 兴 are from nonrelativistic calculations.
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