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Journal of Physics B Atomic Molecular and Optical Physics

Variational treatment of positronium molecules

M. M. H. El-Gogary, M. A. Abdel-Raouf, M Y H Hassan, S A Saleh

Published Nov 28, 1995• 10 citations

Abstract

A variational treatment for obtaining the binding energies, wavefunctions and internal distances of the positronium molecule (Ps 2 ) is presented. For this purpose, very modern Hylleraas-type trial wavefunctions, as well as different envelope functions ( upsilon m6 e - gamma upsilon where m 6 =0, 1, 2, 3 and upsilon is the distance between the two positrons), are employed. The resulting binding energies show excellent convergence when the number of components of the wavefunction considered is increased. Our results at m 6 =2 show that only 22 components of our wavefunction are sufficient for obtaining the binding energy omega P(s 2 )=-0.03 Ryd (=-0.41 eV) which is identical to the binding energy calculated by Ho via 400 components of his trial wavefunctions. The best convergence, however, has been achieved via 22 components of our wavefunctions at m 6 =1. In this case, omega P(s 2 )=-0.042 Ryd (=-0.573 eV). Comparison with the very recent value of omega P(s 2 )(=-0.435 eV) determined by Kinghorn and Poshusta (1993) and Kozlowski and Adamowicz (1993), using 300 components of their trial wavefunctions, supports the opinion that the exact binding energy of Ps 2 is less than -0.41 eV. Our average values for the internal distances of Ps 2 agree quite well with those determined by previous authors and emphasize the argument that the size of the molecule is decreased when the binding energy is lowered.

DOI: 10.1088/0953-4075/28/22/021

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New Journal of Physics•Feb 25, 2008•12 citations

A variational treatment for obtaining the binding energies, wavefunctions and internal distances of the positronium molecule (Ps 2 ) is presented. For this purpose, very modern Hylleraas-type trial wavefunctions, as well as different envelope functions ( upsilon m6 e - gamma upsilon where m 6 =0, 1, 2, 3 and upsilon is the distance between the two positrons), are employed. The resulting binding energies show excellent convergence when the number of components of the wavefunction considered is increased. Our results at m 6 =2 show that only 22 components of our wavefunction are sufficient for obtaining the binding energy omega P(s 2 )=-0.03 Ryd (=-0.41 eV) which is identical to the binding energy calculated by Ho via 400 components of his trial wavefunctions. The best convergence, however, has been achieved via 22 components of our wavefunctions at m 6 =1. In this case, omega P(s 2 )=-0.042 Ryd (=-0.573 eV). Comparison with the very recent value of omega P(s 2 )(=-0.435 eV) determined by Kinghorn and Poshusta (1993) and Kozlowski and Adamowicz (1993), using 300 components of their trial wavefunctions, supports the opinion that the exact binding energy of Ps 2 is less than -0.41 eV. Our average values for the internal distances of Ps 2 agree quite well with those determined by previous authors and emphasize the argument that the size of the molecule is decreased when the binding energy is lowered.

Positronium-hydrogen (Ps-H) scattering is of interest, as it is a fundamental four-body Coulomb problem. We have investigated low-energy Ps-H scattering below the Ps(n=2) excitation threshold using the Kohn variational method and variants of the method with a trial wavefunction that includes highly correlated Hylleraas-type short-range terms. We give an elegant formalism that combines all Kohn-type variational methods into a single form. Along with this, we have also developed a general formalism for Kohn-type matrix elements that allows us to evaluate arbitrary partial waves with a single codebase. Computational strategies we have developed and use in this work will also be discussed.With these methods, we have computed phase shifts for the first six partial waves for both the singlet and triplet states. The 1S and 1P phase shifts are highly accurate results and could potentially be viewed as benchmark results. Resonance positions and widths for the 1S-, 1P-, 1D-, and 1F-waves have been calculated.We present elastic integrated, elastic differential, and momentum transfer cross sections using all six partial waves and note interesting features of each. We use multiple effective range theories, including several that explicitly take into account the long-range van der Waals interaction, to investigate scattering lengths for the 1,3S and 1,3P partial waves and effective ranges for the 1,3S-wave.

The positronium molecule

({\mathrm{Ps}}_{2})

has not been experimentally observed yet because its tiny (4.5 eV) binding energy cannot be detected when the molecule annihilates by emitting two photons with energy of 0.51 MeV each. It is shown in this paper that the electric dipole transition between the recently found

L=1

excited state and the

L=0

ground state with its characteristic photon energy of 4.94 eV is a clear signature of the existence of the positronium molecule and the possibility of its experimental observation is realistic. The probability of this transition is about 17% of the total decay rate. Another Coulomb four-body system containing positron HPs (the positronium hydride or hydrogen positride), is also included for comparison.

A trial wavefunction, Psi 1 , which contains both linear and nonlinear parameters is constructed using an approximate target wavefunction, Psi 1 is related to a quantity, a v , which becomes the Kohn scattering length if the target wavefunction is exact, through an integral expression corresponding to a slight modification of the usual Kohn variational method. It is conjectured that if the values of a v display a local minimum when all the nonlinear parameters of Psi 1 are varied, then this local minimum value of a v is an upper bound on the exact scattering length. Calculations on the zero-energy positron-helium scattering system using Hylleraas-type approximations to the helium ground-state wavefunction are analysed using this criterion. The use of either a six-parameter or a ten-parameter Hylleraas-type approximation to the helium ground-state wavefunction can give good results in scattering calculations for the positron-helium system at zero energy.

In order to investigate the binding mechanism of weakly bound states of positronic alkali atoms, we calculate the energies and wavefunctions using the Gaussian expansion method (GEM) where a positronium (Ps)–alkali ion channel and a positron–alkali atom channel are explicitly introduced. The energies of the bound states are updated using a model potential that reproduces well the observed energy levels of alkali atoms. The binding mechanism of the positronic alkali atom is analyzed by the wavefunctions obtained. The structure of the positronic alkali atom has been regarded as a Ps cluster orbiting the alkali ion, which is described by the Ps–alkali ion channel. We point out that the fraction having the positron–alkali atom configuration is small but plays an indispensable role for the weakly bound system.