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Comment on the “Stability of the five-body bi-positronium ion Ps2e−” [Phys. Lett. A 372 (45) (2008) 6721–6726]
Physics Letters A 378 (2014) 529–530
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Comment
Comment on the “Stability of the five-body bi-positronium ion Ps2 e− ” [Phys. Lett. A 372 (45) (2008) 6721–6726] Kálmán Varga Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA
a r t i c l e
i n f o
Article history: Received 31 May 2013 Accepted 11 September 2013 Available online 27 December 2013 Communicated by A.R. Bishop
The stability of Ps2 e− (e+ , e+ , e− , e− , e− ) and Ps3 (e+ , e+ , e− , e− , e− , e− ) is predicted in Ref. [2]. On the other hand, in a recent paper it was shown that the Ps3 molecule is not stable [1]. The authors of Ref. [2] argue that by creating a wave function with a permutation symmetry corresponding to a Ps2 cluster plus an e− (a Ps2 cluster plus a Ps), the Ps− 2 (Ps3 ) is stable. The binding energies obtained in Ref. [2] resemble the binding energies of the Coulombic five and six boson systems presented in [4]. To check the claim of the authors of Ref. [1], and to resolve the controversy, we have recalculated these Coulombic five and six particle systems using the same correlated Gaussian basis functions [3] as used in Ref. [2]. To make the trial wave function completely general, the spins are not coupled and we did not use any symmetry. Two systems are investigated (the arrow in − − + + − the subscript denotes the spin state): Ps− 2 = (e↑ , e↓ , e↑ , e↓ , e↑ ) − + + − + − and Ps3 = (e− ↑ , e↓ , e↑ , e↓ , e↑ , e↓ ). In the case of Ps2 , one has two
− identical particles (e− ↑ and e↑ ), while in the case of Ps3 , one has − + − two pairs of identical particles (e− ↑ − e↑ and e↓ − e↓ ). Antisymmetrizing the wave function with respect to the exchange of identical particles (that is using a fermionic wave function), neither the five nor the six particle system has bound states according to our calculation. The energy of the five and six particle system converges to the energy of the Ps2 threshold (−0.516003 a.u.). This agrees with our earlier results [1,4], and also agrees with the result of Ref. [2] in the case what they call “atomic wave function”. On the other hand, symmetrizing the wave function with respect to the exchange of identical particles (that is using a bosonic wave function), both the five and the six particle systems are bound. The energies and square distances between particles are shown in Table 1. Our results are in good agreement with those of Ref. [2]. Both results are variational and the fact that our en-
DOI of original article: http://dx.doi.org/10.1016/j.physleta.2008.07.084. 0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.09.049
Table 1 Energy and expectation values (in atomic units) of Coulombic five and six particle systems. Bubin et al. [1]
Bosonic system
−0.55617 52.085
−0.55637 52.785
2 r++
36.207
36.691
2 r+−
33.436
33.895
−0.8132
−0.8214
− + + − (e− ↑ , e↓ , e↑ , e↓ , e↑ )
E 2 r−−
− + + − + (e− ↑ , e↓ , e↑ , e↓ , e↑ , e↓ )
E
ergy is somewhat lower means that our results are more accurate, which explains the slight differences in the square interparticle distances. Table 1 shows that the bound state results calculated for Ps2 e− and Ps3 in Ref. [2] correspond to bosonic wave functions. These bound states could exist in five or six body systems of bosons, e.g. a system containing π − and π + particles interacting with Coulomb forces. In summary, the Ps− 2 and Ps3 is not bound. In Ref. [2] the cluster picture violated the fermionic symmetry, and the calculated results correspond to bosonic wave functions. Acknowledgements This work was supported by NSF grants ECCS 1307368 and PHY 1314463. References [1] S. Bubin, O.V. Prezhdo, K. Varga, Instability of tripositronium, Phys. Rev. A 87 (2013) 054501, http://link.aps.org/doi/10.1103/PhysRevA.87.054501. [2] A.M. Frolov, D.M. Wardlaw, Stability of the five-body bi-positronium ion, Phys. Lett. A 372 (45) (2008) 6721–6726. [3] J. Mitroy, S. Bubin, W. Horiuchi, Y. Suzuki, L. Adamowicz, W. Cencek,
530
K. Varga / Physics Letters A 378 (2014) 529–530
K. Szalewicz, J. Komasa, D. Blume, K. Varga, Theory and application of explicitly correlated Gaussians, Rev. Mod. Phys. 85 (2013) 693–749, http://link.aps.org/ doi/10.1103/RevModPhys.85.693.
[4] K. Varga, Y. Suzuki, Stochastic variational method with a correlated Gaussian basis, Phys. Rev. A 53 (1996) 1907–1910, http://link.aps.org/doi/10.1103/ PhysRevA.53.1907.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Comment
Comment on the “Stability of the five-body bi-positronium ion Ps2 e− ” [Phys. Lett. A 372 (45) (2008) 6721–6726] Kálmán Varga Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA
a r t i c l e
i n f o
Article history: Received 31 May 2013 Accepted 11 September 2013 Available online 27 December 2013 Communicated by A.R. Bishop
The stability of Ps2 e− (e+ , e+ , e− , e− , e− ) and Ps3 (e+ , e+ , e− , e− , e− , e− ) is predicted in Ref. [2]. On the other hand, in a recent paper it was shown that the Ps3 molecule is not stable [1]. The authors of Ref. [2] argue that by creating a wave function with a permutation symmetry corresponding to a Ps2 cluster plus an e− (a Ps2 cluster plus a Ps), the Ps− 2 (Ps3 ) is stable. The binding energies obtained in Ref. [2] resemble the binding energies of the Coulombic five and six boson systems presented in [4]. To check the claim of the authors of Ref. [1], and to resolve the controversy, we have recalculated these Coulombic five and six particle systems using the same correlated Gaussian basis functions [3] as used in Ref. [2]. To make the trial wave function completely general, the spins are not coupled and we did not use any symmetry. Two systems are investigated (the arrow in − − + + − the subscript denotes the spin state): Ps− 2 = (e↑ , e↓ , e↑ , e↓ , e↑ ) − + + − + − and Ps3 = (e− ↑ , e↓ , e↑ , e↓ , e↑ , e↓ ). In the case of Ps2 , one has two
− identical particles (e− ↑ and e↑ ), while in the case of Ps3 , one has − + − two pairs of identical particles (e− ↑ − e↑ and e↓ − e↓ ). Antisymmetrizing the wave function with respect to the exchange of identical particles (that is using a fermionic wave function), neither the five nor the six particle system has bound states according to our calculation. The energy of the five and six particle system converges to the energy of the Ps2 threshold (−0.516003 a.u.). This agrees with our earlier results [1,4], and also agrees with the result of Ref. [2] in the case what they call “atomic wave function”. On the other hand, symmetrizing the wave function with respect to the exchange of identical particles (that is using a bosonic wave function), both the five and the six particle systems are bound. The energies and square distances between particles are shown in Table 1. Our results are in good agreement with those of Ref. [2]. Both results are variational and the fact that our en-
DOI of original article: http://dx.doi.org/10.1016/j.physleta.2008.07.084. 0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.09.049
Table 1 Energy and expectation values (in atomic units) of Coulombic five and six particle systems. Bubin et al. [1]
Bosonic system
−0.55617 52.085
−0.55637 52.785
2 r++
36.207
36.691
2 r+−
33.436
33.895
−0.8132
−0.8214
− + + − (e− ↑ , e↓ , e↑ , e↓ , e↑ )
E 2 r−−
− + + − + (e− ↑ , e↓ , e↑ , e↓ , e↑ , e↓ )
E
ergy is somewhat lower means that our results are more accurate, which explains the slight differences in the square interparticle distances. Table 1 shows that the bound state results calculated for Ps2 e− and Ps3 in Ref. [2] correspond to bosonic wave functions. These bound states could exist in five or six body systems of bosons, e.g. a system containing π − and π + particles interacting with Coulomb forces. In summary, the Ps− 2 and Ps3 is not bound. In Ref. [2] the cluster picture violated the fermionic symmetry, and the calculated results correspond to bosonic wave functions. Acknowledgements This work was supported by NSF grants ECCS 1307368 and PHY 1314463. References [1] S. Bubin, O.V. Prezhdo, K. Varga, Instability of tripositronium, Phys. Rev. A 87 (2013) 054501, http://link.aps.org/doi/10.1103/PhysRevA.87.054501. [2] A.M. Frolov, D.M. Wardlaw, Stability of the five-body bi-positronium ion, Phys. Lett. A 372 (45) (2008) 6721–6726. [3] J. Mitroy, S. Bubin, W. Horiuchi, Y. Suzuki, L. Adamowicz, W. Cencek,
530
K. Varga / Physics Letters A 378 (2014) 529–530
K. Szalewicz, J. Komasa, D. Blume, K. Varga, Theory and application of explicitly correlated Gaussians, Rev. Mod. Phys. 85 (2013) 693–749, http://link.aps.org/ doi/10.1103/RevModPhys.85.693.
[4] K. Varga, Y. Suzuki, Stochastic variational method with a correlated Gaussian basis, Phys. Rev. A 53 (1996) 1907–1910, http://link.aps.org/doi/10.1103/ PhysRevA.53.1907.