The three positron based chemical entities, analogous to hydrogen atom, hydrogen molecule and hydride, are drawing increasing attention in the literature. While a lot is known about the binding energies, not much is known about the radii of the atoms and ions and bond lengths and structures of these exotic entities. This article brings for the first time exact values of these structural properties which are comparable with known data. It is hoped that the results will be of help for a better understanding and further development of the chemistry and physics of these exotic compounds and of their possible role as intermediates in chemical and interfacial processes

Raji Heyrovska

English

Nature Precedings

 1 
Atomic and molecular structures of positronium, dipositronium and 
positronium hydride   
 
Raji Heyrovska 
Institute of Biophysics, Academy of Sciences of the Czech Republic, 135 
Kralovopolska, 612 65 Brno, Czech Republic. Email: rheyrovs@hotmail.com 
 
Abstract 
The three positron based chemical entities, analogous to hydrogen atom, hydrogen 
molecule and hydride, are drawing increasing attention in the literature. While a lot is 
known about the binding energies, not much is known about the radii of the atoms and 
ions and bond lengths and structures of these exotic entities. This article brings for the 
first time exact values of these structural properties which are comparable with known 
data. It is hoped that the results will be of help for a better understanding and further 
development of the chemistry and physics of these exotic compounds and of their 
possible role as intermediates in chemical and interfacial processes. 
 
Subject terms: Chemical physics, Physical chemistry, Atomic physics, Nuclear physics, 
Structural chemistry  
 
 
 
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Positronium (Ps), dipositronium (Ps2) and positronium hydride (PsH), based on positron, 
an antimatter, have drawn considerable interest in recent literature. To cite a few, see.1-8 
Several years ago, the author had proposed9 that, since a neutron (n0) and a proton (p+) 
decay by emitting an electron (e-) and a positron (e+) respectively, an (e+ e-) pair, now 
known as positronium, could be the intermediate when neutrons and protons fuse 
together to form atomic nuclei. At distances (ae-e+) of the order of 10-15m, the (e-e+) pair 
annihilates releasing two gamma photons of energy, mec2 = 0.511 Mev each. This article 
brings for the first time the structures of the above exotic chemical entities based on 
exact atomic radii. It is hoped that the results will be of help for a better understanding 
and further development of the chemistry and physics of these short lived compounds 
and of their possible yet undiscovered important role as intermediates in chemical and 
interfacial processes, nucleosynthesis, and biophysics. 
    Positronium atom,1-8 is a positron (e+) bound to an electron (e-) at a distance of the 
order of 10-10 m as in a hydrogen atom, H in which a proton (p+) is bound to an electron 
(e-) at the distance of the Bohr radius, aH (~ 0.5 x 10-10 m). Dipositronium (Ps2) is a 
molecule consisting of a pair of positronium atoms,3,6 like H2 where a pair of (p+e-) 
combine to form the molecular hydrogen. Positronium hydride (PsH) is a diatomic 
molecule,4,6,7 which can be completely or partially covalent or ionic. Considered here are 
the ionization energy, Bohr radius and bond lengths in the above, based on exact atomic 
and ionic radii evaluated here for the first time.  The observed bond lengths of PsH are 
comparable with the corresponding sums of the ionic and atomic radii obtained here. 
    The ionization energy, Bohr radius and reduced mass relation for Ps can be 
considered similar1a,10 to that for H. The ground state energy, EH = eIH (where IH  is the 
ionization potential and e is the  elementary charge), Bohr radius, aH (which is the 
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distance between the electron and proton) and the reduced mass, µH,red = me-mp+/(me- + 
mp+) (where me- and mp+ are masses of the electron and proton respectively), are related 
as follows,11,12 
 
EH = (1/2)(e2/κaH) = (1/2)µH,red(αc)2 = eIH                                                   (1a-c) 
 
where, κ = 4piε0, the atomic unit of permittivity, ε0, the electric constant, α, the fine 
structure constant, h, the Planck constant and c, the speed of light in vacuum. On using 
the latest values,13 of the physical constants, it follows that 
 
µH,red = 0.910442 x 10-31 kg                                                                         (2) 
EH = (1/2)µH,red(αc)2 = 2.178686 x 10-18 J                                                   (3)   
IH = EH/e = 13.5983 V                                                                                 (4) 
aH = (1/2)(e/κIH) = 0.5295 x 10-10m = 0.05295 nm.                                    (5)    
  
    Bohr radius (aH) was shown to be the sum of two Golden sections aH,e- and aH,p+ 
pertaining to the electron and proton11,12: In equation (1c), IH = (Ip+ + Ie-)/2 where Ip = 
e/(κaH,p+) and Ie- = -e/(κaH,e-) are the components of the ionization potential, and aH,e- and 
aH,p+, are sections of aH pertaining to (p+) and (e-) respectively. Thus,  
 
IH = (1/2)(e/κaH) = (1/2)(e/κ)[(1/aH,p+) - (1/aH,e-)]                                        (6a,b)    
1/aH = (1/aH,p+) - (1/aH,e-); aH = aH,e- + aH,p+                                                  (7a,b)     
(aH,e-/aH,p+)2 - (aH,e-/aH,p+) = 1                                                                        (8) 
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    On combining equations (7a & b), one obtains equation (8), which is a Golden 
quadratic equation, whose positive root is aH,e-/aH,p+ = φ = (51/2 +1)/2 = 1.618034.., the 
Golden ratio, a mathematical constant.14 This shows Nature's unique construction of the 
atom, that aH is divided at the Golden point at the point of electrical neutrality Pel, into 
the two Golden sections  aH,e- and aH,p+ pertaining to the electron and proton. These 
Golden sections are given by, 
   
aH,e- = aH/φ = 0.3272 x 10-10m; aH,p+ = aH/φ2 = aH,e-/φ = 0.2022 x 10-10m        (9a,b)   
 
Figure 1a shows the hydrogen atom, with the two circles of radii aH,e- and aH,p+ 
pertaining to the electron and proton. 
    In the case of positronium, Ps, which consists of an electron and a positron, equations 
similar to those for H hold, but with the reduced mass,1a,10 µPs,red = me-/2,  
 
µPs,red = me+me-/(me+ + me-) = me-/2 = 4.554691 x 10-31kg.                         (10) 
EPs = (1/2)µPs,red(αc)2 = 1.089945 x 10-17 J                                                 (11)  
IPs = EPs/e = 6.803 V                                                                                   (12) 
aPs = (1/2)(e/κIPs) = 1.0583 x 10-10 m = 0.10583 nm                                   (13) 
 
The value of IPs is the same as that, 6.803 V in.1a,4,5 and aPs ~ 2aH. as reported in.2 
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    Bohr radius (aPs) of positronium is likewise the sum of the two Golden sections aPs,e- 
and aPs,e+ pertaining to the electron and positron. The Golden sections of aPs (= aPs,e+ + 
aPs,e-) give the radii of e- and e+ in Ps as in the case of H, 
 
aPs,e+ = aPs/φ2 = 0.4042 x 10-10 m; aPs,e- = aPs/φ = 0.6541 x 10-10 m              (14) 
 
Figure 2a shows the positronium with the two circles of radii aPs,e- and aPs,e+ pertaining to 
the electron and positron. 
    As in the case of the hydrogen molecule, H2, where a proton and an electron occupy 
the opposite corners11,12 (see Figure 1b) of a square, dipositronium, Ps2, can be 
considered as a square with an electron and positron on opposite corners as in Figure 2b. 
The length of the covalent bond in the H2 molecule, d(H-H), (see Figure 1c) is equal to 
the diagonal of the square with the Bohr radius, aB,H as a side11 and the covalent radius,15 
RH = d(H-H)/2, as given by, 
 
d(H-H) = 21/2aB,H = 0.07488 nm                                                                  (15) 
RH = 0.03744 nm                                                                                         (16) 
 
    The Golden sections of d(H-H) give the cationic and anionic radii11,12 of the ionic 
resonance forms15 of H2 (see Figure 1d) 
 
d(H-H) = d(H+) + d(H-) = 0.07488 nm                                                        (17) 
d(H+) = d(H-H)/φ2 = 21/2aH,e+ = 0.02860 nm                                                (18)  
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d(H-) = d(H-H)/φ = 21/2aH,e- = 0.04628 nm                                                   (19) 
 
    The covalent bond length, d(Ps-Ps) of Ps2 can similarly be obtained as the diagonal of 
a square with aB,Ps as the sides (see Figures 2b,c), 
 
d(Ps-Ps) = 21/2aB,Ps = 0.1497 nm                                                                 (20) 
RPs = d(Ps-Ps)/2 = 0.07485 nm                                                                   (21) 
 
where the two electrons and two positrons are at the opposite corners of the square. This 
bond length (see Figure 2c) is nearly 2d(H-H). The covalent radius, RPs (see Figure 2c) 
is comparable with half the diameter, 0.159 nm reported in.4  
    As in the case of hydrogen, the Golden sections of d(Ps-Ps) give the radii of the cation 
Ps+ and anion Ps- (see Figure 2d), 
 
d(Ps-Ps) = d(Ps+) + d(Ps-) = 0.1497 nm                                                       (22)                                                              
d(Ps+) = d(Ps-Ps)/φ2 = 21/2aPs,e+ = 0.0572 nm                                               (23) 
d(Ps-) = d(Ps-Ps)/φ = 21/2aPs,e- = 0.0925 nm                                                  (24) 
 
    The bond length of positronium hydride, PsH,4,6,7 consting of a positronium ion or 
atom and a hydrogen atom or ion depends on the many possible forms.7 In,4 PsH is 
considered as an atom and not molecule. However, since in PsH there are two distinct 
positive nuclei (a positron and a proton) sharing the two electrons, here it is considered 
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as a molecule. Four of the possible bond lengths obtained as the sums of the radii are 
given below (see Figures 3 a -d) as examples:  
 
d(Ps+H-) = d(Ps+) + d(H-) = 0.0572 + 0.0462 = 0.103 nm                          (25) 
d(PsH) = d(Ps) + d(H) = 0.075 + 0.037 = 0.112 nm.                                  (26) 
d(Ps-H+) = d(Ps-) + d(H+) = 0.0925 + 0.0286 = 0.121 nm.                         (27) 
d(Ps-H-) = d(Ps-) + d(H-) = 0.0925 + 0.0462 = 0.139 nm                           (28) 
 
    In the above, (Ps+H-), (Ps-H+) and (Ps-H-) are completely ionic bonds and PsH is a 
completely covalent bond. The bond length d(Ps-H+) = 0.121 nm in equation (27), is 
close to the electron-proton distance, 2.31aB,H = 0.122 nm in,6 and that of d(Ps-H-) of 
equation (28) corresponds to the positron-proton distance, 0.14 nm in.7 For a comparison 
of the above bond lengths with those of many hydrides, see.16-18 
 
Summary 
 
This work presents for the first time exact values of the radii, bond lengths, dimensions 
and structures of positronium, dipositronium and positronium hydride at the atomic level 
and it is hoped that this will be of fundamental help in the study of these exotic atoms 
and molecules consisting of antimatter and their possible role as important short lived 
intermediates in chemical, biochemical and nuclear processes 
 
 
 
 
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References  
1. a) http://en.wikipedia.org/wiki/Positronium and b) http://en.wikipedia.org/wiki/Di-
positronium and the literature therein. 
2. Deutsch, M. Evidence for the Formation of Positronium in Gasses. Phys. Rev. 
82:55 (1951).  
3. Cassidy, D.B.; Mills, A.P. Jr. The Production of Molecular Positronium. Nature, 449, 195–
197, (2007). 
4. Schrader, D. M. Compounds of positrons with koino-atoms and -molecules, in: 
Proceedings of the International School of Physics “Enrico Fermi”, 174, 337-398 
(2010), Physics with Many Positrons, edited by Dupasquier, A., Mills jr. A. P. and 
Brusa, R. S., (IOS, Amsterdam; SIF, Bologna) 
5. Cheng, X., Babikov, D. and Schrader, D. M., Binding-energy Predictions of Positrons 
and Atoms. Phys. Rev. A. A83, 032504 (2011) 
6. Usukura, J., Varga, K., and Suzuki, Y. Signature of the Existence of the Positronium 
Molecule. Phys. Rev. 58, 1918-1031 (1998); arXiv:physics/9804023v1 
7. Saito, S. L. Is positronium hydride atom or molecule? Nuclear Instruments and 
Methods in Physics Research B 171, 60-66 (2000)  
8. Many articles in: Proceedings of the 9th International Workshop on Positron and 
Positronium Chemistry, May 11-15, 2008, Wuhan University, China, 
http://aff.whu.edu.cn/whuppc9/files/PPC9%20proceedings.pdf 
9. Heyrovska, R. A Sub-atomic Redox Process that can Release Coulombic Energy and 
Cause Mass Defects of Elements and Nuclear Emissions. In: Proceedings II, J.H. 
Centennial Congr. on Polarography and 41st Meeting of the International Society of 
Electrochemistry, Prague, Extd. Abs. Fr-113 (1990) 
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20
11
.6
29
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 9 
10. http://en.wikipedia.org/wiki/Bohr_model 
11. Heyrovska, R. The Golden ratio, ionic and atomic radii and bond lengths.  
Mol. Phys. 103, 877 - 882 (2005) (Special Issue of in honor of Nicholas Handy) 
12. Heyrovska, R. The Golden ratio in the creations of Nature arises in the architecture 
of atoms and ions. Chapter 12 in: Innovations in Chemical Biology, Editor: Bilge Sener, 
Springer.com, (2009). 
13. http://physics.nist.gov/cuu/Constants/index.html 
14. Livio, M. The Golden ratio, the story of Phi, the World’s most astonishing number, 
Broadway Books, NY (2003).  
15. Pauling, L. The Nature of the Chemical Bond, Cornell Univ. Press, New York (1960). 
16. Heyrovska, R. The Golden Ratio, Atomic, Ionic and Molecular Capacities and 
Bonding Distances in Hydrides. In: International Joint meeting of ECS, USA and 
Japanese, Korean and Australian Societies, Honolulu, Hawaii 2004 - 2, Extd.  Abs. C2-
0551 (2004) http://www.electrochem.org/dl/ma/206/pdfs/0551.pdf 
17. Heyrovska. R. Golden sections of inter-atomic distances as exact ionic radii of 
atoms. http://precedings.nature.com/documents/2929/version/1  (2009) 
18. Heyrovska, R. and Narayan, S. Atomic Structures of Molecules Based on Additivity 
of Atomic and/or Ionic Radii. http://precedings.nature.com/documents/3292/version/1 
(2009). 
 
Acknowledgement 
 
The author thanks the Institute of Biophysics of the Academy of Sciences of the Czech 
Republic for the financial and moral support.  
 
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Competing financial interests 
There are no competing financial interests 
 
Legends for Figures. 
Figure 1. Hydrogen (H). a) H = e-p+; separated by Bohr radius (aH). The blue and pink 
circles have the radii, aH,e- and aH,p+ respectively and the Golden point (point of electrical 
neutrality, Pel) is marked by the arrow, b) a pair of (e-p+) forming H2, as a square of side 
aH, c) H2 molecule with bond length d(H-H) = 21/2aH and d) d(H-H) = d(H-) + d(H+). The 
blue and pink circles have the radii d(H-) and d(H+) respectively, and the Golden point is 
marked by the arrow. 
 
Figure 2. Positronium (Ps). a) Ps = e-e+; separated by Bohr radius (aPs ~ 2aH). The blue 
and pink circles have the radii, aPs,e- and aPs,e+ respectively and the Golden point (point of 
electrical neutrality, Pel) is marked by the arrow, b) a pair of (e-e+) forming Ps2, as a 
square of side aPs, c) Ps2 molecule with bond length d(Ps-Ps) = 21/2 aPs and d) d(Ps-Ps)  = 
d(Ps-) + d(Ps+). The blue and pink circles have the radii d(Ps-) and d(Ps+) respectively, 
and the Golden point is marked by the arrow. 
 
Figure 3. Positronium hydride. a) Ps+H-, b) PsH and c) Ps-H+, where the cation Ps+ and 
anion Ps- are from Fig. 2d and the cation, H+ and anion, H- are from Fig. 1d. The bond 
lengths are sums of the radii of the adjacent atoms or ions: a) d(Ps+H-) = 0.103 nm, b) 
d(PsH) = 0.112 nm. c) d(Ps-H+) = 0.121 nm and d) d(Ps-H-) = 0.139 nm. 
 
 
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Figures 1 - 3                    
 
 
a) Ps+H- 
b) PsH 
c) Ps-H+ 
d) Ps-H- 
c) Ps2 
d) Ps+Ps- 
b) (e-e+)2 
a) Ps = e-e+ a) H = e-p+ 
b) (e-p+)2 
c) H2 
d) H+H- 
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A Sub-atomic Redox Process that can Release Coulombic Energy and Cause Mass Defects of Elements and Nuclear Emissions. In:

a) http://en.wikipedia.org/wiki/Positronium and b) http://en.wikipedia.org/wiki/Dipositronium and the literature therein.

Atomic Structures of Molecules Based on Additivity of Atomic and/or Ionic Radii.

Compounds of positrons with koino-atoms and -molecules, in:

Evidence for the Formation of Positronium in Gasses.

Golden sections of inter-atomic distances as exact ionic radii of atoms. http://precedings.nature.com/documents/2929/version/1

Is positronium hydride atom or molecule? Nuclear Instruments and Methods in

Many articles in:

The Golden ratio in the creations of Nature arises in the architecture of atoms and ions.

The Golden Ratio, Atomic, Ionic and Molecular Capacities and Bonding Distances in Hydrides. In:

The Golden ratio, ionic and atomic radii and bond lengths.

The Golden ratio, the story of Phi, the World’s most astonishing number,

The Nature of the Chemical Bond,

The Production of Molecular Positronium.

Atomic and molecular structures of positronium, dipositronium and positronium hydride

http://precedings.nature.com/documents/6290/version/1

Atomic and molecular structures of positronium, dipositronium and positronium hydride

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