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Physical Review A

Bound-state properties of the positronium negative ion Ps−

Alexei M. Frolov

Published Oct 1, 1999• 72 citations
Publisher

Abstract

Various geometrical and energetical (bound-state) properties of the positronium negative ion $({\mathrm{Ps}}^{\ensuremath{-}}$ or ${e}^{\ensuremath{-}}{e}^{+}{e}^{\ensuremath{-}})$ are determined by using highly accurate variational wave functions. These wave functions have been constructed by applying the advanced two-stage strategy proposed by Frolov [Phys. Rev. A 57, 2436 (1998)]. The determined total energy of the ground state $E=\ensuremath{-}0.2620050702329757\mathrm{a}.\mathrm{u}.$ is the lowest and most accurate value obtained for this system to date (the corresponding binding energy equals $\ensuremath{-}0.326674721317821\mathrm{eV}).$ The computation of the second-order relativistic corrections $(\ensuremath{\simeq}{\ensuremath{\alpha}}^{2})$ to the total energy is discussed also. The general form of the Breit-Pauli Hamiltonian in the relative coordinates for the ${\mathrm{Ps}}^{\ensuremath{-}}$ ion is presented.

DOI
10.1103/physreva.60.2834

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Various geometrical and energetical (bound-state) properties of the positronium negative ion (Ps\ensuremath−({\mathrm{Ps}}^{\ensuremath{-}}(Ps\ensuremath− or e\ensuremath−e+e\ensuremath−){e}^{\ensuremath{-}}{e}^{+}{e}^{\ensuremath{-}})e\ensuremath−e+e\ensuremath−) are determined by using highly accurate variational wave functions. These wave functions have been constructed by applying the advanced two-stage strategy proposed by Frolov [Phys. Rev. A 57, 2436 (1998)]. The determined total energy of the ground state E=\ensuremath−0.2620050702329757a.u.E=\ensuremath{-}0.2620050702329757\mathrm{a}.\mathrm{u}.E=\ensuremath−0.2620050702329757a.u. is the lowest and most accurate value obtained for this system to date (the corresponding binding energy equals \ensuremath−0.326674721317821eV).\ensuremath{-}0.326674721317821\mathrm{eV}).\ensuremath−0.326674721317821eV). The computation of the second-order relativistic corrections (\ensuremath≃\ensuremathα2)(\ensuremath{\simeq}{\ensuremath{\alpha}}^{2})(\ensuremath≃\ensuremathα2) to the total energy is discussed also. The general form of the Breit-Pauli Hamiltonian in the relative coordinates for the Ps\ensuremath−{\mathrm{Ps}}^{\ensuremath{-}}Ps\ensuremath− ion is presented.
The leading-order relativistic corrections to the ground-state energy of the positronium molecule (Ps2)({\mathrm{Ps}}_{2})(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{\mathrm{Ps}}_{2}Ps2​.
In an effort to unravel the dependence of the dynamics of three-bound-particle systems on the relative masses of the particles, an adiabatic treatment in hyperspherical coordinates is used to study the ground state and some doubly excited states of Ps('-) (positronium negative ion) and other three-particle systems, including H('-) and M('-) (muonium negative ion). Two different methods for the solution of the adiabatic eigenvalue equation are presented; one uses a "prediagonalized" basis of hyperspherical harmonics, and the other uses an asymptotic basis which consists of hydrogenic functions for the positronium. Potential curves for Ps('-) and M('-) in the ('1)S('e), ('1)P('o), ('3)P('e), ('3)P('o) and ('1)D('e) states and for H('-) in the ('1)P('o) states are presented. A method for calculating the probability density as a function of the hyperspherical coordinates (R,(alpha),(theta)(,12)) is introduced and used for the P states of Ps('-). This helps to understand the different symmetries and to distinguish among them. A remarkable linear dependence of the binding energy of the ground state and of the doubly excited states on the reduced mass of the atom, discovered in this study, is presented and discussed.
The positronium ion Ps−−{\mathrm{Ps}}^{--}Ps−− is often qualitative described as a positronium Ps with a loosely bound second electron, since the value of the annihilation rate for Ps−−{\mathrm{Ps}}^{--}Ps−− is nearly equal to that of the spin-averaged neutral positronium. This similarity however has never been fully explained. To study its internal structure, we decompose the electron-positron distribution function of Ps−−{\mathrm{Ps}}^{--}Ps−− into the sum of two terms: one for the electron closest to the positron and the other for the farthest. We show that the inner electron behaves almost as if the outer electron were not present, forming a Ps substructure inside Ps−−{\mathrm{Ps}}^{--}Ps−− and that the effect of electron correlation is such that the outer electron tries to stay as close as possible to the positron but without interfering too much with the distribution of the inner electron. We show that this phenomenon is closely related to the problem of the critical stability of the system (Z+,e−−,e−−)({Z}^{+},\phantom{\rule{0.28em}{0ex}}{e}^{--},\phantom{\rule{0.28em}{0ex}}{e}^{--})(Z+,e−−,e−−) with varying ZZZ charge.
We reexamine the binding energies of three-particle systems containing two electrons and one positive particle of mass M, in an attempt to understand the approximate proportionality of the 1Se^{1}\mathrm{S}^{\mathrm{e}}1Se ground-state binding energies to the reduced masses, as pointed out by Botero and Greene [Phys. Rev. 56, 1366 (1986)]. Our investigation involves a careful evaluation of the contribution to the energy of the mass-polarization term ${H}_{\mathrm{mp}=\mathrm{\ensuremath{-}}(2/\mathrm{M}){\ensuremath{\nabla}}_{1}\mathrm{\ensuremath{\cdot}}{\ensuremath{\nabla}}_{2}}$ which is the only part of the Hamiltonian not scaling like the reduced mass. It does not appear that any fundamental principle is involved, since we find merely that the mass polarization decreases somewhat as the mass of the positive particle is reduced below the proton mass. In the case of the excited 3Pe^{3}\mathrm{P}^{\mathrm{e}}3Pe state, this reduction is not sufficient to allow binding when M approaches the electron mass. As a by-product, we compute accurately some properties of the recently observed negative muonium ion (${e}^{\mathrm{\ensuremath{-}}}$${\ensuremath{\mu}}^{+}$${e}^{\mathrm{\ensuremath{-}}}$).
Various energetic, geometric, and other properties for the ground and first vibrationally excited S states (L=0) of Coulomb three-body systems having unit charges are obtained by means of accurate exponential wave functions. The mass dependence of the properties for these systems has a relatively smooth form for the ground states. The results presented should be useful as a basis for further investigation of these three-body systems. The prethreshold (weakly bound) exotic nonsymmetric systems (d+{\mathit{d}}^{+}d+t+{\mathit{t}}^{+}t+${\mathit{p}}^{\mathrm{\ensuremath{-}}}$, ${\mathrm{\ensuremath{\mu}}}^{+}$${\mathrm{\ensuremath{\pi}}}^{+}$${\mathrm{\ensuremath{\mu}}}^{\mathrm{\ensuremath{-}}}$, and ${\mathrm{\ensuremath{\mu}}}^{+}$${\mathrm{\ensuremath{\pi}}}^{+}$${\mathrm{\ensuremath{\pi}}}^{\mathrm{\ensuremath{-}}}$) considered have sharp cluster structures, i.e., the lighter positive ion moves in the field of the neutral pair (e.g., d+{\mathit{d}}^{+}d+ in d+{\mathit{d}}^{+}d+t+{\mathit{t}}^{+}t+${\mathit{p}}^{\mathrm{\ensuremath{-}}}$ moves in the field of t+{\mathit{t}}^{+}t+${\mathit{p}}^{\mathrm{\ensuremath{-}}}$ ).
The stability and structure of the $({m}^{Z+},{e}^{\ensuremath{-}},{e}^{\ensuremath{-}},{e}^{+})$ system is studied as a function of the mass of the mZ+{m}^{Z+}mZ+ particle and for Z=1Z=1Z=1, 2, 3, and 10. The Z=1Z=1Z=1 system can be regarded as an analog of the APsA\mathrm{Ps}APs system (where AAA is a group I or IB atom of the periodic table) and was found to be stable for all values of m+{m}^{+}m+. This is supportive of the idea that all the group I and IB atoms can bind Ps. The $({m}^{2+},{e}^{\ensuremath{-}},{e}^{\ensuremath{-}},{e}^{+})$ system is stable for all ${m}^{2+}∕{m}_{e}\ensuremath{\leqslant}0.68$ and evolves into a configuration best described as $({m}^{2+},{\mathrm{Ps}}^{\ensuremath{-}})$ when ${m}^{2+}∕{m}_{e}\ensuremath{\rightarrow}0$. The $({m}^{3+},{e}^{\ensuremath{-}},{e}^{\ensuremath{-}},{e}^{+})$ system was stable for a mass range given by ${m}^{3+}∕{m}_{e}\ensuremath{\leqslant}0.066\phantom{\rule{0.2em}{0ex}}32$, which suggests that positrons could form Feshbach resonances in collisions with positive ions which are isolectronic with the group II and IIB columns of the periodic table. The $({m}^{3+},{e}^{\ensuremath{-}},{e}^{\ensuremath{-}},{e}^{+})$ system has the unusual property that it has a mass range where it becomes more compact while its binding energy simultaneously decreases. The $({m}^{10+},{e}^{\ensuremath{-}},{e}^{\ensuremath{-}},{e}^{+})$ system is also stable at m10+∕me=0.00254{m}^{10+}∕{m}_{e}=0.002\phantom{\rule{0.2em}{0ex}}54m10+∕me​=0.00254, which implies stability for all mass ratios less than 0.002 54. In total, the calculations suggest that the $({m}^{Z+},{e}^{\ensuremath{-}},{e}^{\ensuremath{-}},{e}^{+})$ system is stable whenever the ${m}^{Z+}+{\mathrm{Ps}}^{\ensuremath{-}}$ or $({m}^{Z+},{e}^{\ensuremath{-}})+\mathrm{Ps}$ breakups represent the lowest energy dissociation channel. As part of the analysis some improved estimates of the properties of the KPs ground state are reported.