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Physical Review A

Resonant states of positronium molecules

Y. K. Ho

Published Mar 1, 1989• 22 citations
Publisher

Abstract

The method of complex-coordinate rotation is applied to investigate autoionization states of positronium molecules. Using elaborate Hylleraas-type wave functions in which all six interparticle coordinates are used, resonance parameters for six S-wave resonant states are determined. It is believed that three of them belong to a Rydberg series in which the two positrons form a singlet spin state, and the other three resonances belong to the triplet spin state for the two positrons. All these two Rydberg series are converging on the ground state of the ${\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}$ threshold.

DOI
10.1103/physreva.39.2709

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Physical Review A•Jul 31, 2020•4 citations
The method of complex-coordinate rotation is applied to investigate autoionization states of positronium molecules. Using elaborate Hylleraas-type wave functions in which all six interparticle coordinates are used, resonance parameters for six S-wave resonant states are determined. It is believed that three of them belong to a Rydberg series in which the two positrons form a singlet spin state, and the other three resonances belong to the triplet spin state for the two positrons. All these two Rydberg series are converging on the ground state of the Ps\ensuremath−{\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}Ps\ensuremath− threshold.
The fully correlated calculations in Hylleraas coordinates for the nonrelativistic ground-state energy and the first three S-wave resonances of positronium hydride are reported. For the resonances, the positions and widths are determined using the method of complex-coordinate rotation. Our results significantly improve previous work. For example, the ground-state energy is about five parts in 109{10}^{9}109 in accuracy, the lowest variational upper bound reported to date. A comparison with other previous work is made.
The "molecule" positronium hydride (PsH), consisting of one atom each of positronium and hydrogen, has one well-known particle-stable state of total energy $E=\ensuremath{-}1.575$ Ry. In addition, for each LLL it must have an infinite Rydberg series of excited states whose principal configuration is (${e}^{+}{\mathrm{H}}^{\ensuremath{-}}$), converging to a limit at $E=\ensuremath{-}1.0555$ Ry, the energy of ${\mathrm{H}}^{\ensuremath{-}}$. All of these excited states are energetically able to dissociate into ground-state Ps + H, and the higher-lying states can also decay with emission of excited Ps and ground-state H. These Rydberg states should appear as resonances in Ps + H scattering. In this paper I compute the lowest SSS-wave resonance parameters, using an optical-potential formalism based on the (${e}^{+}{\mathrm{H}}^{\ensuremath{-}}$) configuration. The results are in rough agreement with those previously obtained using Kohn variational or complex-rotation techniques with highly correlated trial functions. The physical description of the PsH excited states is thus clarified.
A doubly excited autodissociating resonant state in positronium hydride has been discovered, which is situated in a Rydberg series converging to the ${\mathrm{H}}^{\ensuremath{-}}(2{s}^{2})+{e}^{+}$ threshold. Previous research has noted a scarcity of such highly excited resonant states, particularly those with a doubly excited nature. The exploration of these states was made possible by employing a projection operator method supported by the stochastic variational method. Accurate calculations of the resonant positions and widths were performed using the complex coordinate rotation method with explicitly correlated Gaussians as basis functions. Our analysis has revealed the intrinsic structural characteristics of these highly excited states. The probability density distribution demonstrates that most resonant states of positronium hydride tend to favor either a ${\mathrm{H}}^{\ensuremath{-}}+{e}^{+}$ or ${\mathrm{Ps}}^{\ensuremath{-}}+{\mathrm{H}}^{+}$ configuration. However, two doubly excited resonant states exhibit a unique characteristic involving both the ${\mathrm{H}}^{\ensuremath{-}}+{e}^{+}$ and ${\mathrm{Ps}}^{\ensuremath{-}}+{\mathrm{H}}^{+}$ configurations. By integrating the probability density distribution with a quantum defect formula, highly excited resonances can be accurately assigned to either the ${\mathrm{H}}^{\ensuremath{-}}(2{s}^{2})+{e}^{+}$ or ${\mathrm{Ps}}^{\ensuremath{-}}(1{s}^{2})+{\mathrm{H}}^{+}$ Rydberg series, thereby reducing discrepancies compared to earlier classifications.
The resonance states of the systems consisting of three identical bosons and of three fermions interacting with an attractive 1∕r1∕r1∕r potential are investigated using highly correlated exponential basis functions. The method of complex-coordinate rotation is used to extract energies and widths for the resonances lying below the n=2n=2n=2 excited state threshold of the respective two-body subsystem. For the system consisting of three identical bosons, we have obtained a total of 12 SSS-wave resonances. It is concluded that nine of them belong to one series of Rydberg states. For the system consisting of three fermions, we have obtained a total of 23 SSS-wave resonances in total spin three-halfs (the 4S^{4}S4S state) and 17 SSS-wave resonances total spin one-half (the 2S^{2}S2S states). The resonances for the 4S^{4}S4S states belong to three different series, while those for the 2S^{2}S2S states belong to two Rydberg series. The resonance energies and widths of these states for the three-body self-gravitating bosons and fermions are reported.
The projection method with explicitly correlated Gaussians is used to demonstrate the existence of two doubly excited states of the positronium molecule (Ps2{\mathrm{Ps}}_{2}Ps2​). One is below the Ps(n=2)+Ps(n=2)\mathrm{Ps}(n=2)\phantom{\rule{4pt}{0ex}}+\phantom{\rule{4pt}{0ex}}\mathrm{Ps}(n=2)Ps(n=2)+Ps(n=2) threshold, with A1{A}_{1}A1​ symmetry, and the other is below the Ps(n=2)+Ps(n=3)\mathrm{Ps}(n=2)+\mathrm{Ps}(n=3)Ps(n=2)+Ps(n=3) threshold, with EEE symmetry. These states exist as resonances in the Ps-Ps continuum. Moreover, the resonance positions and resonance widths of the two states are calculated using the complex rotation method with basis sets obtained via the orthogonalizing pseudoprojector method. The resonance positions obtained using the complex rotation method agree with the results of the orthogonalizing pseudoprojector method. We also investigate the various structural properties of these states as well as the decay probabilities of $2\ensuremath{\gamma}$ emission due to electron-positron annihilation.