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

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Relativistic Corrections to the Ground-State Energy of the Positronium Molecule View metadata, citation and similar papers at core.ac.uk brought to you by CORE 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, calcula- tions 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 calcula- tions 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.
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