1. Fishing for positronic compounds Dario Bressanini QMC in the Apuan Alps VII 2012 TTI Vallico Sotto Università dell’Insubria, Como, ITALY

2. Theory ahead of experiments e + 1930: Dirac theory e + 1933: Carl Anderson experiment (cosmic radiations) Ps (e + e - ) 1937 (1946): Ruark theory (coined the name) Ps (e + e - ) 1946: Wheeler theory (polyelectrons) Ps (e + e - ) 1951: Martin Deutsch experiment Two spin states Singlet (para-positronium) – 0.125 ns Triplet (ortho-positronium) – 142 ns Annihilation via  photons is inevitable… “resistance is futile” …but a lot can happen on the way

3. Time scales

4. e + and Ps spectroscopy Used in ► Polymer science ► Medical research ► Solid state, electronics ► hope to build a  -ray laser

5. Theory ahead of experiments Ps - (e + e - e - ) and Ps + (e + e + e - ) ► inconsistent nomenclature ► A - means Ae -, an added electron (as usual) ► A + means Ae +, an added positron (but not always ) ► 1946, 1947: Wheeler, Hylleraas theory ► 1981 Ps - seen in experiment by Mills HPs (p + e + e - e - ) ► 1947, Hylleraas & Ore theory ► Seen in experiment 1992 Schrader Many calculations on atomic bound states, very few experimental results (more with molecules)

6. Current status for atoms

7. Fishing

8. Computational techniques CI ► slow convergence ► frozen core ► many atoms ECG-SVM ► very accurate ► slow optimization ► 4 e - VMC-DMC ► Compact  ► VMC can be unbound ► statistical error Bressanini and Morosi: JCP 119, 7037 (2003) HPs

9. Prediction for atoms

10. Polyleptons

11. Pse+e+ Ps - Ps + Ps 2 - Ps 2 Ps 3 Ps 2 + Ps 3 - e-e- Ps 3 + Ps 4 10324 1 2 3 4 0 e-e- e+e+ Ps n (,+,-) e m + e n -

12. Ps 2 : e - e - e + e + 1946: Wheeler, unbound 1946: Ore, unbound 1947: Hylleraas & Ore: bound ► -0.504 a.u. 1947-1996: what is the energy? ► Energy did not converge with time 0.001 a.u. = 0.027 eV

13. The energy of Ps 2 1996-1997: Energy controversy resolved ► -0.51601(1) DMC (1997, Bressanini et al.) ► agrees with Frolov & Smith, hylleraas (1996) ► -0.516003788 Matyus & Reiher ECG 2012 0.001 a.u. = 0.027 eV

14. Ps 2 : e - e - e + e + 2007: finally seen in experiment ► Cassidy & Mills, Nature 449 195 (2007) ► 60 years after theoretical prediction Open the possibility to study BEC of Ps Not the end of the story…

15. Ps 2 : e - e - e + e + Symmetry of Ps 2 must include charge conjugation: e -  e + ► problems with early calculations 1993: isomorph to D 2h group (Kinghorn & Poshusta) ► 0 + (A 1 ) ground state ► 0 + (B 2 ), 0 + (E) excited states 1998: Varga, Usukura, Suzuki ECG ► 1 - (B 2 ) bound L=1 state E = -0.334408 a.u. 2012: L=1 state detected experimentally

16. Pse+e+ Ps - Ps + Ps 2 - Ps 2 Ps 3 Ps 2 + Ps 3 - e-e- Ps 3 + Ps 4 10324 1 2 3 4 0 e-e- e+e+ Higher systems? Ps n, Ps - n Ps 2 - ► Ps 2 + e -  Ps 2 - ► Ps + Ps -  Ps 2 - ► L=0 unbound (ECG) ► What about L>0 ?

17. Ps 2 - and beyond: general strategy Problems with Monte Carlo: ► No starting  (R) from HF/DFT Use a “Valence Bond-like”  (R) ►  (R)=A[  (Ps)  (Ps - )] or for other fragments VMC and DMC unbound. Dissociation ► Use a modified potential (preserving symmetry) ► V(R) = V(e -,e + ) + g (V(e -,e - ) + V(e +,e + )) ►  (R)=A[  (Ps)  (Ps - )  (interaction)] ► Consider the limit for g  1

18. Ps 2 - total energy E(L=1) < E(L=0)

19. Ps 2 - binding energy L=0 is unbound L=1 is probably unbound… …but with better nodes?

20. Ps 3 and Ps 3 - Preliminary results Ps 3 and Ps 3 - unbound so far Not explored yet all excited states

21. Pse+e+ Ps - Ps + Ps 2 - Ps 2 Ps 3 Ps 2 + Ps 3 - e-e- Ps 4 10324 1 2 3 4 0 e-e- e+e+ Higher systems? Ps n, Ps - n

22. Z=3 e + Li 1976: Hylleraas CI, e + Li unbound (Clary) 1996: DMC, e + Li unbound (Yoshida & Miyako) 1997: ECG, e + Li bound (Ryzhikh & Mitroy, Strasburger & Choinacki) Li + + Ps → e + Li BE=0.0025 a.u. (first neutral atom to bind e + ) 1999: DMC, e + Li bound (Mella, Morosi & Bressanini) e + Li Li +2 Li +3 LiLi + 10324 1 2 0 e-e- e+e+ Li - 5 6

23. Z=3 e + Li 1996: DMC, LiPs unbound, BE = -0.011(4) a.u. (Harju, Barbiellini & Nieminen) 1997: DMC, LiPs bound, BE = 0.028(5) a.u. (Yoshida & Miyako) 1998: DMC, LiPs bound, BE = 0.0096(8) (Bressanini, Mella & Morosi) 1998: ECG-SVM, LiPs bound BE = 0.01051 (Ryzhikh & Mitroy) Li + Ps → LiPs BE=0.01237 a.u. e + Li Li +2 Li +3 LiLi + 10324 1 2 0 e-e- e+e+ LiPs Li - 5 6

24. Z=3 e + Li Li + + Ps 2 → Li + Ps 2 SVM-FC BE = 0.009 a.u. DMC BE = 0.012 a.u. (preliminary) +Z=3, 4e - 2e + e + Li Li +2 Li +3 LiLi + 10324 1 2 0 e-e- e+e+ LiPs Li + Ps 2 Li - 5 Li - Ps LiPs 2 Li - Ps 2 6

25. Excited states Be Too small for my qmc

26. e + Z=1 HPs - HPs HPs 2 e + HPs H 10324 1 2 0 e-e- e+e+ H - Ps 2 5 H+H+ H-H- e+He+H H + Ps 3 HPs 3 3 H - Ps 3 +Z=1, 4e - 2e + H - + Ps 2 → H - Ps 2 Ps - + HPs → H - Ps 2 ECG-SVM (Varga not converged) BE = 0.004 a.u. DMC BE = 0.006 a.u. (preliminary) What about Z=2 HePs 2 ?

27. H - Ps 2  (H - )  (Ps 2 ) unbound  (Ps - )  (HPs) unbound  c    + (1-c)    bound

28. Fishing for positronic compounds Which atom? How many e - ? How many e + ? Which state?

29. e + Z=2 He ( 1s 2 1 S) does not bind e + He ( 1s2s 3 S) binds e + (very weakly) He - ( 1s2s2p 4 P o ) and He - ( 2p 3 4 S o ) do not bind e + e + He He + Ps He + 10324 1 2 0 e-e- e+e+ HePs 2 5 He +2 HeHe - HePsHe - Ps He - Ps 2 Excited state

30. LiPs 1s1s 2s2s 2p2p 1s+1s+ 2s+2s+ 2p+2p+ x UNBOUND Stable with respect to dissociation into Li( 2 P o ) + Ps( 2 P o ) E thr = -7.472656532 Preliminary DMC: BE = 0.47 mH, SVM = 0.21 mH, CI-FC = 0.02 mH

31. Molecules

32. H 2 e m + e n - H 2 Ps  H 2 + Ps  H + HPs H 2 Ps -  H 2 + Ps -  H - + HPs H 2 Pse+H2e+H2 H 2 Ps + H 2 Ps 2 - H2+H2+ H2-H2- 10324 1 2 0 e-e- e+e+ H 2 Ps - H 2 Ps 2 5 H2H2 Unbound

33. H 2 Ps -

34. Thank you Still a lot of work to do