Relativistic Corrections to the Ground-State Energy of the Positronium Molecule

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provided by Nazarbayev University Repository PHYSICAL REVIEW A 75, 062504 ͑2007͒

Relativistic corrections to the ground-state energy of the positronium molecule

Sergiy Bubin,1 Monika Stanke,1,2 Dariusz Ke¸dziera,3 and Ludwik Adamowicz1,4 1Department of Chemistry, University of Arizona, Tucson, Arizona 85721, USA 2Institute of Physics, Nicholas Copernicus University, ul. Grudziadzka 5, PL 87-100 Toruń, Poland 3Department of Chemistry, Nicholas Copernicus University, ul. Gagarina 7, PL 87-100 Toruń, Poland 4Department of Physics, University of Arizona, Tucson, Arizona 85721, USA ͑Received 2 February 2007; published 11 June 2007͒ ͑ ͒ The leading-order relativistic corrections to the ground-state energy of the positronium molecule Ps2 have been computed within the framework of perturbation theory. As the zero-order wave function we used a highly accurate nonrelativistic variational expansion in terms of 6000 explicitly correlated Gaussians that yielded the lowest variational upper bound for this system to date. We also report some expectation values representing the properties of Ps2.

DOI: 10.1103/PhysRevA.75.062504 PACS number͑s͒: 31.15.Pf, 31.30.Jv, 36.10.Dr

Positronium, Ps, is a system that consists of a positron and electron to Ps and form a stable Ps− anion. This was con- an electron. It is a lighter version of the hydrogen atom. ﬁrmed experimentally by Mills, who also measured the ra- However, due to the substantially smaller reduced mass, the diative decay rate for this system ͓2͔. The binding energy of Ps ground-state energy is only one-half of the hydrogen Ps−, which includes relativistic corrections, was determined ground-state energy and its interparticle distance is 2 times with high precision by Bhatia and Drachman ͓3͔ and, more − the value in H. Like the hydrogen atom, the energy, the wave recently, by Drake and Grigorescu ͓4͔.Ps can be subse- function, and the expectation values of most operators repre- quently combined with another positron forming a neutral senting physical properties of Ps can be determined exactly. system, which in analogy to the hydrogen molecule can be These include the expectation values of relativistic correc- called the positronium molecule or Ps2. The positronium ␣2 ␣ 2 ប molecule, though proven stable a long time ago by theoreti- tions of the order of , where =e / c is the ﬁne structure ͓ ͔ constant. Those relativistic corrections, obtained as mean cal calculations 5 , has not been directly observed in an experiment. Only recently an intense pulsed positron source values of the corresponding relativistic operators with the was developed using a buffer gas trap to accumulate large exact Ps ground-state wave function, number of positrons. This creates a dense plasma, which can 1 be spatially focused and used for the formation of positro- ␺ = e−r1/2 ͑1͒ ͱ ␲ nium molecules and positronium Bose-Einstein condensate 8 ͓6,7͔. This development may soon allow detection and ex- ͑ ͒ perimental characterization of Ps . This makes accurate cal- where r1 is the positron-electron distance are 2 culations of this system more relevant. It should be men- ͗ ˆ ͘ HMV = − 5/64, tioned that the Ps2 molecule is not a stable system with regard to electron-positron annihilation. However, calculations predict that its lifetime is almost 2 times as long as that ͗Hˆ ͘ = 1/8, D of the positronium atom ͑in the singlet state͓͒8–14͔. Apart from its experimental relevance, Ps is, and will ͗ ˆ ͘ 2 HOO = − 1/8, remain, an interesting model system for theoretical calculations and analysis. Calculations can answer the question 1/4 for singlet, whether Ps is a genuine molecule similar to H with a co- ͗Hˆ ͘ = ͭ ͮ 2 2 SS 1/12 for triplet, valent bond between the two positroniums, or it is a loosely bonded Ps cluster. Also, the calculations can very precisely 0 for singlet, determine the Ps2 dissociation energy and radiative lifetime. ͗Hˆ ͘ = ͭ ͮ ͑2͒ Unlike the H molecule, the Ps molecule is a weakly A 1/4 for triplet. 2 2 bound system and so far only the existence of two bound ͗ ˆ ͘ ͗ ˆ ͘ states has been predicted with theoretical calculations. The Here HMV represents the mass-velocity term, HD is the ͗ ˆ ͘ ͗ ˆ ͘ ground state of Ps2 has zero orbital angular momentum, and Darwin term, HOO is the orbit-orbit term, and HA is the both pairs of identical particles are in a singlet state. Re- ͑ annihilation channel term the expressions for all of them cently, another bound state with orbital angular momentum ͒ will be given below . The two possible spin couplings of Ps, L=1 and negative parity was predicted ͓15͔. The ground i.e., singlet and triplet, result in two different values for the state of the positronium molecule has been a subject of the- ͗ ˆ ͘ ͗ ˆ ͘ energy of the spin-spin interaction terms HSS and HA . oretical calculations for quite a long period of time ͑see, for With the above relativistic contributions and higher order example, ͓8–14,16–18͔ and references therein͒. This stable QED corrections, the energy of Ps is known with a very high ͑if we disregard the annihilation process͒ electron-positron precision. complex is interesting from many perspectives. It represents As was shown by Wheeler ͓1͔, it is possible to add an a fully nonadiabatic case of a hydrogenlike molecule and

1050-2947/2007/75͑6͒/062504͑9͒ 062504-1 ©2007 The American Physical Society BUBIN et al. PHYSICAL REVIEW A 75, 062504 ͑2007͒ plays an important role in the general theory of the quantum- and 4 are electrons, in which case Q1 =1, Q2 =1, Q3 =−1, and mechanical four-body problem. Much of the attention to- Q4 =−1. wards this system also comes from quantum electrodynamics ˆ ͓ ͔ Generally, Hrel consists of several contributions 27,28 , ͑QED͒, because this is a purely leptonic system and because ˆ ˆ ˆ ˆ ˆ ˆ ˆ ͑ ͒ the electron-positron annihilation in Ps2 may proceed with Hrel = HMV + HD + HOO + HSO + HSS + HA, 5 the emission of zero, one, two, three, or more photons. The ͑ ͒ ͑ ͒ studies of the ground state of Ps so far have been mainly that represent the mass-velocity MV , Darwin D , orbit- 2 orbit ͑OO͒, spin-orbit ͑SO͒, spin-spin ͑SS͒, and annihilation focused on obtaining a precise nonrelativistic binding energy ͑ ͒ ͑ and wave function, as well as some other properties such as channel A corrections. In the case of S-symmetry zero total angular momentum͒ of the zero-order ͑nonrelativistic͒ the electron-positron contact density, which determines the ͑ ͒ radiative lifetime. At the same time, the methodology of very wave function, the expressions for certain terms in 5 be- precise calculations on few-particle systems in recent years come somewhat simplified and the relativistic operators then has advanced to the level where accurate treatment of rela- take the following forms: N tivistic and QED corrections becomes necessary. Very accu- 1 1 ١4 , ͑6͒ rate calculations that include relativistic corrections have re- Hˆ =− ͚ MV 3 Ri cently achieved a high level of precision for small atoms 8 i=1 Mi ͑see, for example, Refs. ͓19–23͔͒ and, more recently, also for N N diatomic molecules ͓24,25͔. However, the number of such ␲ Q Q ˆ i j ␦͑ ͒ ͑ ͒ studies done for positronic systems remains very limited. To HD =− ͚ ͚ 2 Rij , 7 2 M our knowledge, the only high accuracy calculation of the i=1 j i i relativistic effects for a positronic system that contains more N N than three particles was the work of Yan and Ho ͓26͔, where 1 Q Q 1 1 ,ͪ ١͒ RЈ͑RЈ١ + ١ ١Јͩ Hˆ = ͚ ͚ i j leading relativistic corrections for positronium hydride were OO Ri Rj 2 ij ij Ri Rj 2 j=1 iϾj MiM j Rij R computed. ij In our previous work ͓14͔ we reported precise, the best to ͑8͒ date, nonrelativistic energies and some other properties of ˆ ͑ ͒ the positronium molecule. Having generated a very accurate HSO =0, 9 variational wave function, we now proceed to the evaluation of the relativistic corrections, the main goal of the present N N 8␲ Q Q ˆ i j Ј ␦͑ ͒ ͑ ͒ work. The calculations are performed within the framework HSS =− ͚ ͚ Si Sj Rij , 10 of the Breit-Pauli formalism, as has been done for small 3 i=1 jϾi MiM j atomic and molecular systems. As the calculations on the He, N N Li, and Be atoms show this formalism gives quite good re- Q Q 3 ˆ ␲ i j ͩ Ј ͪ␦͑ ͒ ͑ ͒ sults for light atomic systems. For such systems the relativ- H =−2 ͚ ͚ + S Sj Rij . 11 A M M 4 i istic effects are small, but not negligible, when such quanti- i=1 jϾi i j ties as the ionization potentials, electron affinities, and xx¯ pairs transition energies are calculated. ␦͑ ͒ In the above expressions, Rij is the three-dimensional In the Breit-Pauli formalism, a quantum system is de- Dirac delta function, the prime ͑ Ј͒ represents the vector or scribed by the Hamiltonian that is a sum of the nonrelativis- matrix transposition, and Si is the spin operator of the ith tic Hamiltonian and a small relativistic correction, particle. ˆ ˆ ␣2 ˆ ͑ ͒ The expressions for Hˆ , Hˆ , Hˆ , and Hˆ are valid for Htot = Hnonrel + Hrel, 3 MV OO SO SS any type of particles ͑if we disregard magnetic moment where ␣ is the fine structure constant. Due to the small mag- ͒ ˆ ˆ anomalies , while the expressions for HD and HA are given nitude of the relativistic correction it is possible to first solve with the assumption that all particles in the system are spin- the nonrelativistic problem and then account for the relativ- 1/2 fermions. It should also be emphasized that in the ex- istic effects using the first-order perturbation theory. This is ˆ the approach used in this work. pression for HA the summation is only over the particle- ͑ + −͒ The nonrelativistic Hamiltonian of an N-particle Coulomb antiparticle pairs such as e e . system in the laboratory frame has the following form ͑in In order to separate out the motion of the center of mass ͓ ͔ atomic units͒: we use a set of internal coordinates as described in 29,30 , N N N r =−R + R , 1 Q Q 1 1 2 ͒ ͑ i j ١2 ˆ Hnonrel =−͚ + ͚ ͚ . 4 M Ri R i=1 2 i i=1 jϾi ij r2 =−R1 + R3, Here R , M , Q are the position, mass, and charge of the ith i i i , ¯ is the gradient with respect to R ; and R =͉R ١ ;particle Ri i ij j −R ͉ are interparticle distances. In the case of the positro- i r =−R + R . ͑12͒ nium molecule all particle masses are equal to unity. We will n 1 N also be assuming that particles 1 and 2 are positrons, and 3 The coordinate transformation given by ͑12͒ sets particle

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n 1 in the origin of the new, internal coordinate frame. By q q 3 ˆ ␲ 0 i ͩ Ј ͪ␦͑ ͒ using this set of internal coordinates we reduce the N-particle HA =−2 ͚ + s0si ri problem to an n-pseudoparticle problem with n=N−1. i=1 m0mi 4 In the internal coordinates the nonrelativistic Hamiltonian xx¯ pairs ͑4͒ has the following form: n n q q 3 ␲ i j ͩ Ј ͪ␦͑ ͒ ͑ ͒ n n n n −2 ͚ ͚ + si sj rij . 18 1 1 1 q q i=1 Ͼ mimj 4 0 i j i ͪ ١ ١Ј ١2 ͩ ˆ Hnonrel =− ͚ + ͚ ͚ r + ͚ ␮ ri ri j xx¯ pairs 2 i=1 i i=1 ji m0 i=1 ri n In the above expressions s ϵS . q q i i+1 + ͚ i j , ͑13͒ The spin part of the positronium molecule ground-state iϽj rij wave function is ͉ ͉ ␹ 1 ͓␣͑ ͒␤͑ ͒ ␤͑ ͒␣͑ ͔͓͒␣͑ ͒␤͑ ͒ ␤͑ ͒␣͑ ͔͒ where rij= ri −rj . The masses and charges of the pseudopar- = 2 1 2 − 1 2 3 4 − 3 4 . ␮ ͑ ͒ ticles are defined as i =m0mi / m0 +mi and qi =Qi+1, respec- ͑19͒ tively. We also used the notation mi =Mi+1. An analysis of the ͑ ͒ ͑ ͒ nonrelativistic Hamiltonian 13 shows that for Ps2 it repre- By expanding and rearranging the expression 19 in terms ͑ ͒2 sents a motion of three pseudoparticles around the charge of of eigenfunctions of the Si +Sj operators, one will find that Ј the reference particle placed in the center of the coordinate the expectation values of Si Sj are equal to −3/4 in the case system. The Hamiltonian is isotropic ͑fully spherically sym- of a positron-positron or an electron-electron pair and zero in metric͒ and its eigenfunctions should form a functional basis the case of a positron-electron pair. for irreducible representations of the fully spherically sym- In order to solve the nonrelativistic problem with the metric group of rotations. In particular, the ground state Hamiltonian ͑13͒ we used the variational method and ex- should be described by a spherically symmetric function panded the wave function in terms of explicitly correlated with the S-symmetry. Due to isotropy of the Hamiltonian, it Gausssian functions that have the following form: commutes both with the operator representing the square of ͓ Ј͑ ͒ ͔ ͓ Ј͑ Ј ͒ ͔ ͑ ͒ the total angular momentum operator, and its z component. fk = exp − r Ak I3 r = exp − r LkLk I3 r . 20 Furthermore, since the Hamiltonian does not depend on spin, In the above expression, r isa3n-component position vector, it also commutes with the square of the total spin operator, as well as with its z component. Since the rotational motion of r1 the positrons is strongly coupled to the rotational motion of ͑ ͒ r = ΄r2 ΅, 21 the electrons, it is not possible to separate out the positron r3 rotation as it is possible for H2, where the rotation of the ϫ molecular frame formed by the protons can be approximately Ak is a symmetric, positive definite n n matrix of exponen- separated from the internal motion of the electrons. tial parameters that are unique for each basis function, I3 is It can be shown ͑see Refs. ͓31,32͔͒ that in the internal 3ϫ3 identity matrix, and denotes the Kronecker product. frame the terms included in Hˆ have the following form: For computational convenience, matrix Ak is represented in rel Ј the Cholesky-factored form as LkLk. In such a representation n 4 n 1 1 1 there are no constraints on the values of the elements of Lk, ͒ ͑ ͬ ١4 ͚ ͪ ١ ͚ͩ ͫ ˆ HMV =− r + r , 14 while elements of the original A matrix must obey certain 8 m3 i m3 i k 0 i=1 i=1 i constraints to maintain its positive definiteness. To account for the permutational and charge conjugation n n n ␲ 1 1 1 symmetry of the ground-state wave function of the positro- ˆ ͫ ͩ ͪ ␦͑ ͒ ͚ ␦͑ ͒ͬ HD =− ͚ 2 + 2 q0qi ri + ͚ 2 qiqj rij , nium molecule, we need to apply the following projection 2 m m m i=1 0 i i=1 j i i operator to the basis functions ͑20͒: ͑15͒ ͑ ˆ ˆ ͒͑ ˆ ͒͑ ˆ ͒ ͑ ͒ 1+P13P24 1+P12 1+P34 , 22 n n 1 q q 1 1 ˆ where Pij stands for the permutation of particles i and j. The ͪ ١͒ Ј͑ Ј١ ١ ١Ј ͩ j 0 ˆ HOO =− ͚ ͚ r + r r r r ri j 3 j j i j ˆ 2 i=1 j=1 m0mj rj rj result of applying the Pij operators to the basis functions is n n equivalent to certain transformations of matrices Lk that can 1 qiqj 1 1 be easily implemented in computer code. The symmetry of ,ͪ ١͒ rЈ͑rЈ١ + ١ ١Ј ͩ ͚ ͚ + ri rj 3 ij ij ri rj 2 i=1 jϾi mimj rij rij the Ps2 molecule has been discussed before by Kinghorn and Poshusta ͓16͔ and by Schrader ͓33,34͔. ͑ ͒ 16 In our calculations we used several basis sets of functions ͑20͒ obtained in our previous work on the positronium mol- n n n 8␲ q q 8␲ q q ecule ͓14͔. The largest basis set in that prior work consisted ˆ 0 i Ј ␦͑ ͒ i j Ј ␦͑ ͒ HSS =− ͚ s0si ri − ͚ ͚ si sj rij , of 5000 functions. In the present study, we followed the 3 m m 3 m m i=1 0 i i=1 jϾi i j method of generating and optimizing the basis functions de- ͑17͒ scribed in ͓14͔, and we increased the number of basis func-

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TABLE I. Expectation values of various operators for the ground state of Ps2 molecule computed with different basis sets in comparison with the same expectation values for two separated Ps atoms. Case ͑a͒ corresponds to the situation when both Ps atoms are in singlet state, while case ͑b͒ is when both are in triplet state. All numbers are in hartrees.

͗ ͗͗͗͗͗͗͘͘͘͘͘͘␣2 ͘ Basis size Hnonrel HMV HD HOO HSS HA Hnonrel+ Hrel 1000 −0.516003787849 −0.17182668 0.27398527 −0.25294619 7.86716ϫ10−3 0.41687828 −0.515989199222 2000 −0.516003789829 −0.17183022 0.27399394 −0.25294620 7.86569ϫ10−3 0.41689018 −0.515989200374 3000 −0.516003790168 −0.17183138 0.27399733 −0.25294620 7.86403ϫ10−3 0.41689402 −0.515989200478 4000 −0.516003790332 −0.17183701 0.27400780 −0.25294620 7.86403ϫ10−3 0.41690972 −0.515989199550 5000 −0.516003790416 −0.17183697 0.27400787 −0.25294620 7.86401ϫ10−3 0.41690981 −0.515989199624 6000 −0.516003790455 −0.17183704 0.27400797 −0.25294620 7.86397ϫ10−3 0.41690993 −0.515989199656

2Ps͑a͒ −0.5 −0.15625 0.25 −0.25 −0.5 0.0 −0.500034946201 2Ps͑b͒ −0.5 −0.15625 0.25 −0.25 0.16666667 0.5 −0.499999445298 tions to 6000. With that, the previous upper bound to the ticle distances. The values have been computed with our nonrelativistic ground-state energy of the positronium mol- nonrelativistic 6000-term wave function. Average interpar- ecule was further improved. With 6000 explicitly correlated ticle distances calculated for smaller basis sets were already Gaussians we obtained the nonrelativistic energy value of presented in ͓14͔. In Table II we used the following short- −0.516 003 790 455 hartree, which, we believe, is converged hand notations for the expectation values of the delta func- ͗␦ ͘ϵ͗␦͑ ͒͘ ͗␦ ͘ϵ͗␦͑ ͒␦͑ to about 1 part in 1010. tions: e+e− re+ −re− , e+e+e− re+ −re+ re+ ͒͘ ͗␦ ͘ϵ͗␦͑ ͒␦͑ ͒␦͑ ͒͘ Having obtained the accurate nonrelativistic wave func- −re− , e+e+e−e− re+ −re+ re+ −re− re− −re− . tion we computed expectation values of the operators that Closer examination of the results for Ps2 in Tables I and II ˆ reveals that the convergence of the expectation values for the constitute Hrel. The expressions for matrix elements of those ͑ ͒ ˆ operators containing singularities ͑such as Hˆ , Hˆ , Hˆ , operators with basis functions 20 , except for HOO, can be MV D SS ˆ ͒ found in ͓31͔. In fact, in ͓31͔ we presented the derivations and HA is signiﬁcantly worse than for those that do not have for a more general case of explicitly correlated Gaussians ͑ ˆ ˆ ͒ them Hnonrel, HOO, interparticle distances . This behavior containing premultipliers in the form of even powers of r1. does not come as a surprise since the Gaussian basis func- ˆ ͑ ͒ The formulas for matrix elements of HOO with functions 20 tions are known for not satisfying the interparticle cusp con- are given in the Appendix. In Table I we show the conver- ditions. gence of the expectation values of operators ͑14͒–͑18͒ with The interparticle distances allow insight into the structure increasing number of basis functions. As was mentioned of the Ps2 molecule. Although only two distinct average dis- above, the basis sets smaller than 6000 functions were taken tances, the positron-positron distance ͑equal to the electron- from our previous calculations ͓14͔. In Table I we also show electron distance͒ and the positron-electron distance, can be the total nonrelativistic and relativistic energies obtained determined, they show that the structure of Ps2 is very dif- with different basis sets, as well as the results for the Ps atom ferent from the H2 molecule. While in H2 the average dis- for comparison. tance between the protons is about 1.4 bohr, the average dis- In Table II we list expectation values of the interparticle tance between the positrons in Ps2 is over 6.0 bohr. This distances and the delta functions that depend on the interpar- indicates that this system is a loosely bound complex of two Ps atoms rather than a molecule with a covalent bond, such TABLE II. Nonrelativistic expectation values of the interparticle as H2. The much smaller binding energy in Ps2 than in H2 distances, their squares, and contact densities computed with the conﬁrms this conclusion. largest basis of 6000 Gaussians used in this work. All numbers are Using the total relativistic energy of the positronium mol- in atomic units, i.e., distances are in bohrs, two-particle delta func- ecule obtained with the largest basis set from Table I and the −3 tions are in bohrs , etc. energy of two noninteracting positronium atoms that include the relativistic corrections, one can compute that the binding Quantity Value ͑ ͒ energy of Ps2 is 0.015 954 25 hartree 0.434 137 3 eV . Even though higher order corrections for the ground state of the ͗re+e+͘ 6.033210260 positronium atom are known, for the sake of consistency we ͗r + −͘ 4.487154601 e e intentionally did not include them in the above result. ͗r2 ͘ 46.37487912 e+e+ To conclude, accurate calculations of the positronium ͗ 2 ͘ 29.11270462 re+e− molecule ground-state nonrelativistic energy and leading- ͗␦ ͘ ϫ −4 e+e+ 6.2579505 10 order relativistic corrections have been carried out using a −2 ͗␦e+e−͘ 2.2117759ϫ10 variational expansion in terms of explicitly correlated Gauss- −5 ͗␦e+e+e−͘ 9.10802ϫ10 ian basis functions. A new upper bound to the nonrelativistic −6 energy has been obtained. The next step in improving the ͗␦e+e+e−e−͘ 4.56149ϫ10 accuracy of the results would be the inclusion of the QED

062504-4 RELATIVISTIC CORRECTIONS TO THE GROUND-STATE… PHYSICAL REVIEW A 75, 062504 ͑2007͒

3 4 corrections proportional to ␣ and ␣ . It would also be de- ͑ ͒ ¯ ii For A=A I3 we have the following relations: sirable to implement a procedure of regularization ͑some- ͓ ͔͒ times called “Drachmanization” after Drachman 35 of the tr͑¯A͒ =3tr͑A͒, operators whose expectation values converge slowly. Such a procedure works very well in the case of Born-Oppenheimer calculations of atoms and molecules ͓36͔ and may also work ͉¯A͉ = ͉A͉3. well for fully nonadiabatic systems, such as the positronium ͑ ͒ molecule. iii We deﬁne matrix Jij as

This work has been supported by the National Science Eii, i = j, J = ͭ ͮ ij Foundation. The authors would like to thank University of Eii + Ejj − Eij − Eji i j, Arizona Center of Computing and Information Technology ϫ for using their supercomputer resources. where Eij is the n n matrix with 1 in the ijth position and 0 ͓͑ ͒␣ ␦ ␣␦ ͔ elsewhere. Eij ␤ = i j ␤ . It can be easily shown that

APPENDIX: EXPECTATION VALUE OF THE ORBIT- EijEij =0, ORBIT INTERACTION OPERATOR As we have done in our previous works ͓29–31͔, we will EijEji = Eii, keep expressions in matrix form when evaluating the integrals needed in the calculations of the expectation value of EijEjj = Eij, the orbit-orbit Hamiltonian, Hˆ . To simplify the notations OO when i j. we will deﬁne ͑iv͒ With the above-deﬁned quantities it can be shown Ј١ ١Ј١ ¯ that operators j, and r j can be expressed as follows: A = A I3, i ij , ١ rЈ͑¯E − ¯E ͒¯E = ١ ͒ rЈ͑¯E − ¯E = r − r ͒Ј١͑ = where A is a nϫn symmetric matrix of exponential param- rЈ١ ϫ ij j i j j ij jj ij jj jj eters, I3 isa3 3 identity matrix. With this our explicitly correlated Gaussian basis functions take the following form: ١ ¯١Ј ١Ј١ i j = Eij . ͑ ¯ ͒ fk = exp − rЈAkr . Finally, to remove the singularity of the orbit-orbit operator We will also introduce the following shorthand notation, 3 that appears due to the term 1/rij, we employ the following :commuation relation . ١ ϵ ١ ri i

Greek indices will be used to denote components and 1 ri A1͒͑ , −= ͬ , ١ͫ summation over repeated greek indices is assumed every- i 3 ri ri where below, unless otherwise noted. 1 1 r ij , ͑A2͒ = ͬ , ١ͫ−= ͬ , ١ͫ 1. Some auxiliary formulas j i 3 rij rij rij There are some simple relations used in deriving the ex- r r r pressions for the integrals, which we would like to list. where ij= i − j. ١ ͒ ͑ i By r we denote the gradient with respect to the vector of the pseudoparticle coordinates, r, and for f =exp͑−rЈ¯Ar͒ 2. Transformation of the orbit-orbit operator we have In the internal coordinate system the orbit-orbit operator has the following form: ␣͒ ¯Ј ͑ ␣١ ϵ ␣١ r f f =−2 r A f , n 1 q q 1 1 ͪ ١͒ Ј͑ Ј١ ١Ј١ ͩ i 0 ͒ ͑ ˆ HOO r =− ͚ i i + ri ri i i f =−2͑¯Ar͒␣f , 3␣١ 2 i=i m0mi ri ri n n ¯ ¯ 1 q0qi 1 1 ͪ ١͒ Ј͑ Ј١ ١Ј١ ͩ f␤١␣Ar͒␣ + ͑Ar͒͑␤f =−2f١␣١␤١ − ͚ ͚ i j + 3 ri ri i j 2 m m r r ¯ ␥ ¯ ¯ i=1 j i 0 i i i = ͓−2A␣ ␦␤␥ +4͑Ar͒␣͑Ar͒␤͔f n−1 n 1 q q 1 1 ͪ ١͒ Ј͑ Ј١ ١Ј١ ͩ i j ¯ ¯ ¯ = ͓4͑Ar͒␣͑Ar͒␤ −2A␣␤͔f , + ͚ ͚ i j + 3 rij rij i j . 2 i=1 ji mimj rij rij and Using the relations ͑A1͒ and ͑A2͒ we make the following ␣␤␦ ␣ ␤١ ␣ ␤١ 2 r12 =− 2 r2 =− . transformations:

062504-5 BUBIN et al. PHYSICAL REVIEW A 75, 062504 ͑2007͒

1 rЈ 1 ͒ ١͑ͬ ١ͪ ١ ͩͫ␣ ١ͪ ١ Јͩ i ١͒ Ј͑ Ј١ 3 ri ri i j = ri 3 i j = ri − i i j ␣, ri ri ri

1 rЈ 1 ͒ ١͑ͬ ١ͪ ١ͩͫ␣ ١ͪ Јͩ ij١ ١͒ Ј͑ Ј١ 3 rij rij i j = rij 3 i j = rij j i j ␣, rij rij rij which yields

n n n 1 q q 1 1 ␤ 1 q q 1 1 ␤ ͬ ͒ ١͑ ͒ ١͑ ͪ ١ͩ␣ ͒ ١͑␣͒ ١͑ ͫ 0 i ͬ ͒ ١͑ ͒ ١͑ ͪ ١ͩ␣ ͒ ١͑␣͒ ١͑ ͫ i 0 ͒ ͑ ˆ HOO r =− ͚ i i ␣ − ri i i ␤ i ␣ − ͚ ͚ i j ␣ − ri i i ␤ j ␣ 2 i=1 m0mi ri ri 2 i=1 ji m0mi ri ri n−1 n 1 q q 1 1 ␤ ͬ ͒ ١͑ ͒ ١͑ ͪ ١ͩ␣ ͒ ١͑␣͒ ١͑ ͫ i j + ͚ ͚ i j ␣ + rij j i ␤ j ␣ . 2 i=1 ji mimj rij rij

In the next step we rewrite the orbit-orbit operator using the Hence, in order to calculate the expectation value of above-described matrix notation, ˆ ͑ ͒ HOO r we need to evaluate the following integrals: n 1 q q 1 ¯ 1 ͒ ١ ¯١Ј͑ ͫ i 0 ͒ ͑ ˆ f ͘, ͑A4͉͒͒ ١ ١ЈB͑ ͉ HOO r =− ͚ Eii ͗f m m r k l 2 i=1 0 i i rg 1 ␤ ␤ ͒ ͑ ͬ ͒ ١ ¯͑ ͒ ١ ¯͑ ͪ ¯١Јͩ␣͒ ¯Ј ͑ − r E E E ␤ E ␣ A3a 1 ͒ ͑ ͘ ͉ ͒ ١ ¯͑ ͒ ١ ¯͑ ͪ ¯١Јͩ␣͒ ¯ii ii ii ii ͗ ͉͑ Ј ri fk r C D F ␤ G ␣ fl . A5 rg n n 1 q q 1 Next we will derive the formulas for the two integrals. ͒ ١ ¯١Ј͑ ͫ i 0 − ͚ ͚ Eij 2 i=1 ji m0mi ri ͦ ١ ¯١Ј 1 ͦ ␤ 1 3. Integral fk r „ B … fl‹ Š g ͒ ͑ ͬ ͒ ١ ¯͑ ͒ ١ ¯͑ ͪ ¯١Јͩ␣͒ ¯Ј ͑ − r E E E ␤ E ␣ A3b ١ ¯ ١ ii ii ii ij ri First we apply the operator ЈB to fl, ١␣١␤ ¯ ١␤ ͒ ¯͑␣١ ١ ¯١Ј n−1 n 1 q q 1 B fl = B ␣ ␤fl = B␣ ␤fl rЈ͑¯E͓ + ͒ ١ ١Ј¯E͑ ͫ ͚ ͚ i j + ij ij ␤ ␣ 2 m m r ¯ ͓ ͑ ¯ ͒␣͑¯ ͒ ͑¯ ͒ ͔ i=1 j i i j ij = B␣ 4 rЈAl Alr ␤ −2 Al ␤ fl ␤ ␤ ␤ ␣ ␣ 1 ͓͑ ¯ ͒␣¯ ͑¯ ͒ ¯ ͑¯ ͒ ͔ 4 rЈAl B␣ Alr ␤ −2B␣ Al ␤ fl= ͒ ͑ ͬ ͒ ١ ¯͑ ͒ ١ ¯͑ ͪ ¯١Јͩ ͔͒ ¯ − Ejj Eji Eii ␤ Ejj ␣ . A3c rij =4͑rЈ¯A ¯B¯A r͒f −6tr͑BA ͒f . To simplify the expression for the expectation value of the l l l l l ˆ ͑ ͒ With that we can now calculate the value of the integral, HOO r operator we use the following general integral for each of the three terms that appear in the expectation value, 1 1 1 .f ͘ =4͗f ͉ ͑rЈ¯A ¯B¯A r͉͒f ͘ −6tr͑BA ͒͗f ͉ ͉f ͉͘ ١ ١Ј¯B ͉ f͗ 1 1 ␤ k r l k r l l l l k r l ͘ g g g ͉ͬ ͒ ١ ¯͑ ͒ ١ ¯͑ ͪ ¯١Јͩ␣͒ ¯Ј ͑ ͒ ١ ¯١Ј͑ ͉ͫ ͗ fk B − r C D F ␤ G ␣ fl , rg rg ͑A6͒

͑ ͒ ¯ ¯ ¯ ¯ ¯ ¯ for term A3a : g = i, B = Eii, C = Eii, D = Eii, ¯ ␣ ¯ 1 ␤ ¯ ¯ flͦ␣ G١ ␤ ١ЈD F١ Integral fkͦ rЈC .4 Š „ … „ rg … „ … „ … ‹ ¯ ¯ ¯ ¯ F = Eii, G = Eii; We rewrite the integral in the following way: 1 ␤ ͘ ͉ ͒ ١ ¯͑ ͒ ١ ¯͑ ͪ ¯١Јͩ␣͒ ¯for term ͑A3b͒: g = i, ¯B = ¯E , C¯ = ¯E , D¯ = ¯E , ͗ ͉͑ Ј ij ii ii fk r C D F ␤ G ␣ fl rg ¯ ¯ ¯ ¯ F=Eii, G=Eij; 1 ͔ ͒ ١ ¯͑ ͒ ١ ¯͑␣͒ ¯١Ј¯ ͒␤͓ ͑ Ј͑ ␶ ͵ =− d D fk r C F ␤ G ␣fl , ͑ ͒ ¯ ¯ ¯ ͑¯ ¯ ͒ r for term A3c : g = ij, B = Eij, C = Eij − Ejj , g where the operator in the integral can be split into three ¯ ¯ ¯ ¯ ¯ ¯ D = Eji, F = Eii, G = Ejj. terms:

062504-6 RELATIVISTIC CORRECTIONS TO THE GROUND-STATE… PHYSICAL REVIEW A 75, 062504 ͑2007͒

͒ ͑ ͒ ¯ ¯ ¯ ͑ ͒ ١ ¯ ١͑ ͔ ͒ ١ ¯͑ ͒ ١ ¯͑␣͒ ¯ ͑ ͓␤͒ ¯ ١͑ ЈD fk rЈC F ␤ G ␣fl ЈM fl =4 rЈAlMAlr fl −6tr MAl fl. ͒ ١ ¯͑͒ ١ ¯ ¯ ͔͑ ␤͒ ¯ ١͓͑ = ЈD fk rЈCG F ␤fl term 1 Now, we can evaluate the expectation value of term 2 as ͒ ١ ¯͑ ͒ ١ ¯͔͑␣͒ ¯ ͑␤͒ ¯ ١͓͑ + fk ЈD rЈC F ␤ G ␣fl term 2 ͔ ͒ ١ ¯͑ ͒ ١ ¯͑␤͒ ¯ ١͓͑␣͒ ¯ ͑ + fk rЈC ЈD F ␤ G ␣fl term 3, 1 ͘ ͉ ͒ ١ ¯͑ ͒ ١ ¯͔͑␣͒ ¯ ͑␤͒ ¯ ١͓͑ ͉ ͗ fk ЈD rЈC F ␤ G ␣ fl and now we evaluate each term separately. rg 1 1 Ј¯ ¯ ¯ ¯¯¯ ͉͒ ͘ ͑ Ј Ј ͒͗ ͉ ͉ ͘ ͑ ͉ ͗ ١ ¯ ١ ¯ ¯١Ј¯ ␤ Ј a. Term 1: †„ D… fk‡„r CG …„F …␤fl =4 fk r AlGCDFAlr fl −6tr G C DFAl fk fl . rg rg First we make the following transformations: ͑A8͒ ␤͒ ¯ ¯ ͑ ␥١␤ ¯ ␤͒ ¯ ١͑ ЈD fk = D␥ fk =−2 rЈAkD fk, ¯ ␣ ¯ ␤ ¯ ¯ fl␣ G١ ␤ ١ЈD F١ c. Term 3: fk rЈC ‡ … „ … „ … „† … „ ͔ ͒ ¯ ¯ ¯¯͑ ͒ ¯¯͑͒ ¯¯ ¯ ͑ ͓ ͒ ١ ¯͑͒ ١ ¯ ¯ ͑ rЈCG F ␤fl = 4 rЈCGAlr FAlr ␤ −2 FAlGCr ␤ fl. Here we use the following relations: The ﬁrst of the above transformations can be easily derived and the second results from the following: ͔ ͒ ١ ¯͑ ͒ ١ ¯͑␤͒ ¯١Ј͓͑␣͒ ¯Ј ͑ ␳͓ ͑¯ ͒ ͑¯ ͒ ͑¯ ͒ ͔ r C D F ␤ G ␣fl ¯␣͒ ¯ ¯ ͑ ١ ␳١ ¯␣͒ ¯ ¯ ͑ rЈCG F␤ ␣ ␳fl = rЈCG F␤ 4 Alr ␣ Alr ␳ −2 Al ␣␳ fl ١ ١␳١␤ ͒ ¯¯͑␣͒ ¯ ¯Ј ͑ ͓ ͑ ¯ ¯ ͒␣͑¯ ͒ ¯ ␳͑¯ ͒ = r CG DF ␳ ␣ ␤fl; = 4 rЈCG Alr ␣F␤ Alr ␳ ¯ ␳͑¯ ͒ ͑¯ ¯ ͒␣͔ −2F␤ Al ␣␳ GCr fl which results from the following simple transformations: ͓ ͑ ¯ ¯¯ ͒͑¯¯ ͒ ͑¯¯ ¯ ¯ ͒ ͔ = 4 rЈCGAlr FAlr ␤ −2 FAlGCr ␤ fl. ͒ ١ ¯¯ ١͑͒ ١ ¯ ¯ ͑ ͒ ١ ¯͑␤͒ ¯ ١͑ ͒ ١ ¯͑␣͒ ¯ ͑ With that, the expectation value of term 1 can now be evalu- rЈC G ␣ ЈD F ␤fl = rЈCG ЈDF fl ated as ١ ١␳١␤ ͒ ¯¯͑␣͒ ¯ ¯ ͑ = rЈCG DF ␳ ␣ ␤fl, 1 ͒ ١ ¯͑͒ ١ ¯ ¯١Ј¯ ͒␤ ͔͑ Ј͓͑ ␶ ͵ d D fk r CG F ␤fl rg ␳ ␳ ␳ f =4͓͑¯A ͒ ͑¯A r͒␤ + ͑¯A r͒␣͑¯A ͒ ͔f␤١␣١ ١ 1 ͗ ͉ ͑ ¯ ¯¯ ͒͑ ¯ ¯¯¯ ͉͒ ͘ l l ␣ l l l ␤ l =−8 fk rЈCGAlr rЈAkDFAlr fl rg ͓ ͑ ¯ ͒␳͑¯ ͒ ͑¯ ͒ ͑ ¯ ͒␳͑¯ ͒ ͔ −4 2 rЈAl Alr ␣ Alr ␤ − rЈAl Al ␣␤ fl; 1 +4͗f ͉ ͑rЈ¯A D¯¯F¯A G¯ C¯ r͉͒f ͘. ͑A7͒ k r k l l g which results from the following transformations:

١ ¯ ١ ¯ ␣ ¯ ␤ ¯ ١ b. Term 2: fk†„ ЈD… „rЈC… ‡„F …␤„G …␣fl ͖ ͔ ͒ ¯͑ ͒ ¯͑ ͒ ¯͑ ١␳͕͓ ١ ١␳١ Here we make use of the following relations: ␣ ␤fl = 4 Alr ␣ Alr ␤ −2 Al ␣␤ fl ͖ ͔ ͒ ¯͑ ١␳͕͓ ͑¯ ͒ ␥ ͑¯ ͒ ␰ ¯ ␤ ¯ ␣ ¯ ␤¯ ␥␣ = 4 Al ␣ r␥ Al ␤ r␰ −2 Al ␣␤ fl , ١ЈD͒ ͑rЈC͒ = D␥ C͑ ͓͑¯ ͒ ␥␦ ␥͑¯ ͒ ͑¯ ͒ ͑¯ ͒ ␰␦ ␰͔ =4 Al ␣ ␳ Alr ␤ + Alr ␣ Al ␤ ␳ fl ͒ ١ ¯¯ ¯ ¯١Ј͑ ͒ ١ ¯͑ ͒ ١ ¯͔͑␣͒ ¯١Ј¯ ͒␤͑ Ј͓͑ D r C F ␤ G ␣fl = GCDF fl. ͓ ͑¯ ͒ ͑¯ ͒ ͑¯ ͒ ͔͑ ¯ ͒␳ −4 2 Alr ␣ Alr ␤ − Al ␣␤ rЈAl fl The ﬁrst of those relations is obtained as follows: ͓͑¯ ͒ ␳͑¯ ͒ ͑¯ ͒ ͑¯ ͒ ␳͔ =4 Al ␣ Alr ␤ + Alr ␣ Al ␤ fl ␤ ␣ ␤ ␣ ␤ ;␣␥ ¯r␳C¯ = D¯ ␦␥␳C¯ = D¯ C␥١ ¯١ЈD¯ ͒␤͑rЈC¯ ͒␣ = D͑ ␥ ␳ ␥ ␳ ␥ ͓ ͑ ¯ ͒␳͑¯ ͒ ͑¯ ͒ ͑ ¯ ͒␳͑¯ ͒ ͔ −4 2 rЈAl Alr ␣ Alr ␤ − rЈAl Al ␣␤ fl, and the second one results from the following transformations:

␤ ␥␣ ␤ ␥ ␣ ␤ f␣͒ ١ ¯G͑␤͒ ١ ١ЈD¯ ͒ ͑¯F͑ ͒ ¯rЈC͑ ͒ ١ ¯ ¯͑ ͒ ١ ¯͑ ¯ ͒ ١ ¯͑ ͒ ١ ¯͑͒ ¯ ¯͑ D␥ C F ␤ G ␣fl = D␥ F ␤ CG fl l 4͑rЈC¯ G¯¯A D¯¯F¯A r͒f +4͑rЈC¯ G¯¯A r͒tr͑D¯¯F¯A ͒f= ͒ ١ ¯͑␤ ¯␥͒ ¯ ¯ ١͑ = ЈGC D␥ F ␤fl l l l l l l ; −8͑rЈC¯ G¯¯A r͒͑rЈ¯A D¯¯F¯A r͒f +4͑rЈ¯A D¯¯F¯A G¯ C¯ r͒f ͒ ١ ¯¯ ¯ ¯ ١͑ = ЈGCDF fl. l l l l l l l From here, analogously to the expression obtained in ͑A4͒, we have which is obtained as follows:

062504-7 BUBIN et al. PHYSICAL REVIEW A 75, 062504 ͑2007͒

١ ١␳١␤ ͒ ¯¯͑␣͒ ¯ ¯ ͑ ͒ ١ ¯͑ ͒ ١ ¯͑␤͒ ¯ ١͑␣͒ ¯ ͑ rЈC ЈD F ␤ G ␣fl = rЈCG DF ␳ ␣ ␤fl ͑ ¯ ¯ ͒␣͑¯¯ ͒ ␤͓͑¯ ͒ ␳͑¯ ͒ ͑¯ ͒ ͑¯ ͒ ␳͔ ͑ ¯ ¯ ͒␣͑¯¯ ͒ ␤͓ ͑ ¯ ͒␳͑¯ ͒ ͑¯ ͒ =4 rЈCG DF ␳ Al ␣ Alr ␤ + Alr ␣ Al ␤ fl + rЈCG DF ␳ −8 rЈAl Alr ␣ Alr ␤ ͑ ¯ ͒␳͑¯ ͒ ͔ +4 rЈAl Al ␣␤ fl ͑ ¯ ¯ ͒␣͓͑¯ ͒ ␳͑¯¯ ͒ ␤͑¯ ͒ ͑¯ ͒ ͑¯¯ ͒ ␤͑¯ ͒ ␳͔ =4 rЈCG Al ␣ DF ␳ Alr ␤ + Alr ␣ DF ␳ Al ␤ fl ͑ ¯ ¯ ͒␣͑¯ ͒ ͑ ¯ ͒␳͑¯¯ ͒ ␤͑¯ ͒ ͑ ¯ ͒␳͑¯¯ ͒ ␤͑¯ ͒ ͑ ¯ ¯ ͒␣ −8 rЈCG Alr ␣ rЈAl DF ␳ Alr ␤fl +4 rЈAl DF ␳ Al ␣␤ rЈCG fl ͑ ¯ ¯¯ ¯¯¯ ͒ ͑ ¯ ¯¯ ͒ ͑¯¯¯ ͒ ͑ ¯ ¯¯ ͒͑ ¯ ¯¯¯ ͒ =4 rЈCGAlDFAlr fl +4 rЈCGAlr tr DFAl fl −8 rЈCGAlr rЈAlDFAlr fl ͑ ¯ ¯¯¯ ¯ ¯ ͒ +4 rЈAlDFAlGCr fl. The above three relations allow us to evaluate the expectation value of the operator in term 3 as 1 1 1 1 ͒ ¯¯ ¯ ͑ ͉ ͗ ͘ ͗ ͉ ͑ ¯ ¯¯ ¯¯¯ ͉͒ ͘ ͗ ͉ ͑ ¯ ¯¯¯ ¯ ¯ ͉͒ ͘ ͉ ͒ ١ ¯͑ ͒ ١ ¯͑␤͒ ¯ ١͑␣͒ ¯ ͑ ͉ ͗ fk rЈC ЈD F ␤ G ␣ fl =4 fk rЈCGAlDFAlr fl +4 fk rЈAlDFAlGCr fl −8 fk rЈCGAlr rg rg rg rg 1 ϫ͑ ¯ ¯¯¯ ͉͒ ͘ ͑ ͒͗ ͉ ͑ ¯ ¯¯ ͉͒ ͘ ͑ ͒ rЈAlDFAlr fl +12tr DFAl fk rЈCGAlr fl . A9 rg Now collecting the terms from ͑A7͒–͑A9͒ we determine the expectation value ͑A5͒ as 1 1 ␤ 1 1 1 ͒ ¯ ¯ ¯¯¯ ¯͘ ͗ ͉ ͑ Ј¯ ¯ ¯ ¯¯¯ ͉͒ ͘ ͗ ͉ ͑ Ј¯ ¯¯ ¯¯¯ ͉͒ ͘ ͗ ͉ ͑ Ј ͉ ͒ ١ ¯͑ ͒ ١ ¯͑ ͪ ¯١Јͩ␣͒ ¯Ј ͑ ͉ ͗ fk r C D F ␤ G ␣ fl =−4 fk r AlGCDFAlr fl −4 fk r CGAlDFAlr fl −4 fk r AlDFAlGCr rg rg rg rg rg 1 1 ϫ͉ ͘ ͗ ͉ ͑ ¯ ¯¯¯ ¯ ¯ ͉͒ ͘ ͑ ͒͗ ͉ ͉ ͘ ͑ ͒ fl −4 fk rЈAkDFAlGCr fl +6tr GЈCЈDFAl fk fl −12tr DFAl rg rg 1 1 1 ϫ͗ ͉ ͑ ¯ ¯¯ ͉͒ ͘ ͗ ͉ ͑ ¯ ¯¯ ͒͑ ¯ ¯¯¯ ͉͒ ͘ ͗ ͉ ͑ ¯ ¯¯ ͒ fk rЈCGAlr fl +8 fk rЈCGAlr rЈAlDFAlr fl +8 fk rЈCGAlr rg rg rg ϫ͑ ¯ ¯¯¯ ͉͒ ͘ ͑ ͒ rЈAkDFAlr fl . A10 Having evaluated the general integrals ͑A4͒ and ͑A5͒ we have all of the expressions needed to determine the expectation value ˆ ͑ ͒ of the HOO r operator.

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